Subgroups — C₃₁×#17453 #21474212

Order 7936.

1222 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2989	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2965	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1490	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1489	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1488	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1486	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1478	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1477	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1476	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×H(5,𝔽₂)	C₄	1	1	yes	characteristic
13	C₃₁×#520	—	2	1	no	—
14	C₃₁×#520	C₄	1	2	yes	—
15	C₃₁×#520	C₄	1	1	yes	characteristic
16	C₃₁×#520	C₄	1	1	yes	characteristic
17	C₃₁×#518	—	2	2	no	—
18	C₃₁×#518	—	2	1	no	—
19	C₃₁×#518	C₄	1	2	yes	—
20	C₃₁×#518	C₄	1	1	yes	characteristic
21	C₄×#41×C₃₁	—	2	1	no	—
22	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
23	C₄×#41×C₃₁	C₂²	1	2	yes	—
24	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₃₁	—	2	2	no	—
26	C₂×C₄×D₄×C₃₁	—	2	1	no	—
27	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#446	—	2	1	no	—
30	C₃₁×#446	—	2	1	no	—
31	C₃₁×#436	—	2	1	no	—
32	C₃₁×#436	—	2	1	no	—
33	C₃₁×#435	—	2	2	no	—
34	C₃₁×#435	—	2	2	no	—
35	C₃₁×#432	C₂²	1	2	yes	complemented
36	C₃₁×#431	C₂²	1	1	yes	characteristic, complemented
37	C₃₁×#431	C₂²	1	1	yes	characteristic, complemented
38	C₃₁×#431	C₂²	1	2	yes	complemented
39	C₃₁×#423	C₂²	1	2	yes	complemented
40	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
41	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
42	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
43	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
44	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
45	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	complemented
46	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
47	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
48	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
49	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
50	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
51	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
52	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
53	C₃₁×H(5,𝔽₂)	—	2	4	no	—
54	C₃₁×H(5,𝔽₂)	—	2	4	no	—
55	C₂×#41×C₃₁	—	2	1	no	—
56	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
57	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂²×D₄×C₃₁	—	2	2	no	—
60	C₂²×D₄×C₃₁	—	2	1	no	—
61	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
62	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
63	C₂³×C₄×C₃₁	—	2	1	no	—
64	C₂³×C₄×C₃₁	—	2	2	no	—
65	C₂×M₁₆×C₃₁	—	2	2	no	—
66	C₂×M₁₆×C₃₁	—	2	2	no	—
67	C₂²×C₈×C₃₁	—	2	1	no	—
68	C₂²×C₈×C₃₁	—	2	1	no	—
69	C₄×Q₈×C₃₁	—	2	1	no	—
70	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
71	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
72	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
73	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
74	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₄×D₄×C₃₁	—	2	2	no	—
76	C₄×D₄×C₃₁	—	2	1	no	—
77	C₄×D₄×C₃₁	—	2	1	no	—
78	C₄×D₄×C₃₁	—	2	2	no	—
79	C₄×D₄×C₃₁	—	2	1	no	—
80	C₄×D₄×C₃₁	—	2	2	no	—
81	C₄×D₄×C₃₁	—	2	2	no	—
82	C₄×D₄×C₃₁	—	2	1	no	—
83	C₄×D₄×C₃₁	—	2	2	no	—
84	C₄×D₄×C₃₁	—	2	1	no	—
85	C₄×D₄×C₃₁	—	2	2	no	—
86	C₄×D₄×C₃₁	—	2	2	no	—
87	C₄×D₄×C₃₁	—	2	2	no	—
88	C₄×D₄×C₃₁	—	2	1	no	—
89	C₄×D₄×C₃₁	—	2	2	no	—
90	C₄×D₄×C₃₁	—	2	1	no	—
91	C₄×D₄×C₃₁	—	2	2	no	—
92	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
93	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
94	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
95	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
96	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
97	C₄×D₄×C₃₁	D₄	1	2	yes	—
98	C₄×D₄×C₃₁	D₄	1	2	yes	—
99	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
100	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
101	C₄×D₄×C₃₁	C₂³	1	2	yes	—
102	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
103	C₃₁×#117	—	2	2	no	—
104	C₃₁×#117	—	2	1	no	—
105	C₃₁×#117	—	2	1	no	—
106	C₃₁×#117	—	2	2	no	—
107	C₃₁×#117	C₂×C₄	1	2	yes	—
108	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
109	C₃₁×#117	C₂×C₄	1	2	yes	—
110	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
112	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
113	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
114	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
115	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
116	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
117	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
118	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
119	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
120	C₂×C₄²×C₃₁	—	2	1	no	—
121	C₂×C₄²×C₃₁	—	2	2	no	—
122	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
124	C₃₁×C₄⋊C₈	—	2	1	no	—
125	C₃₁×C₄⋊C₈	—	2	2	no	—
126	C₃₁×C₄⋊C₈	—	2	2	no	—
127	C₃₁×C₄⋊C₈	—	2	1	no	—
128	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
129	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
130	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
131	C₃₁×C₂²⋊C₈	—	2	1	no	—
132	C₃₁×C₂²⋊C₈	—	2	1	no	—
133	C₃₁×C₂²⋊C₈	—	2	2	no	—
134	C₃₁×C₂²⋊C₈	—	2	2	no	—
135	C₃₁×#97	—	2	2	no	—
136	C₃₁×#97	—	2	2	no	—
137	C₄×C₈×C₃₁	—	2	1	no	—
138	C₄×C₈×C₃₁	—	2	1	no	—
139	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
140	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
141	C₄×C₈×C₃₁	C₂³	1	2	yes	—
142	C₂⁴×C₃₁	—	2	2	no	—
143	C₂⁴×C₃₁	—	2	1	no	—
144	#41×C₃₁	—	2	8	no	—
145	#41×C₃₁	—	2	4	no	—
146	#41×C₃₁	—	2	4	no	—
147	#41×C₃₁	—	2	2	no	—
148	#41×C₃₁	—	2	2	no	—
149	#41×C₃₁	—	2	2	no	—
150	#41×C₃₁	M₁₆	1	4	yes	complemented
151	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
152	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
153	C₂×D₄×C₃₁	—	2	16	no	—
154	C₂×D₄×C₃₁	—	2	8	no	—
155	C₂×D₄×C₃₁	—	2	2	no	—
156	C₂×D₄×C₃₁	—	2	1	no	—
157	C₂×D₄×C₃₁	—	2	4	no	—
158	C₂×D₄×C₃₁	—	2	4	no	—
159	C₂×D₄×C₃₁	—	2	2	no	—
160	C₂×D₄×C₃₁	—	2	1	no	—
161	C₂×D₄×C₃₁	—	2	2	no	—
162	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
163	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
164	C₂×D₄×C₃₁	M₁₆	1	4	yes	complemented
165	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
166	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
167	C₂²×C₄×C₃₁	—	2	2	no	—
168	C₂²×C₄×C₃₁	—	2	1	no	—
169	C₂²×C₄×C₃₁	—	2	2	no	—
170	C₂²×C₄×C₃₁	—	2	1	no	—
171	C₂²×C₄×C₃₁	—	2	1	no	—
172	C₂²×C₄×C₃₁	—	2	2	no	—
173	C₂²×C₄×C₃₁	—	2	1	no	—
174	C₂²×C₄×C₃₁	—	2	2	no	—
175	C₂²×C₄×C₃₁	—	2	2	no	—
176	C₂²×C₄×C₃₁	—	2	2	no	—
177	C₂²×C₄×C₃₁	—	2	2	no	—
178	C₂²×C₄×C₃₁	—	8	1	no	—
179	C₂²×C₄×C₃₁	—	8	2	no	—
180	C₂²×C₄×C₃₁	—	8	1	no	—
181	C₂²×C₄×C₃₁	—	8	2	no	—
182	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
183	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
184	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
185	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
186	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
188	M₁₆×C₃₁	—	4	4	no	—
189	M₁₆×C₃₁	—	8	2	no	—
190	C₂×C₈×C₃₁	—	2	1	no	—
191	C₂×C₈×C₃₁	—	2	1	no	—
192	C₂×C₈×C₃₁	—	2	2	no	—
193	C₂×C₈×C₃₁	—	2	1	no	—
194	C₂×C₈×C₃₁	—	2	1	no	—
195	C₂×C₈×C₃₁	—	2	2	no	—
196	C₂×C₈×C₃₁	—	4	2	no	—
197	C₂×C₈×C₃₁	—	8	1	no	—
198	C₄⋊C₄×C₃₁	—	2	1	no	—
199	C₄⋊C₄×C₃₁	—	2	2	no	—
200	C₄⋊C₄×C₃₁	—	2	2	no	—
201	C₄⋊C₄×C₃₁	—	2	1	no	—
202	C₄⋊C₄×C₃₁	—	2	1	no	—
203	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
204	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
205	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
206	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
207	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
208	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
209	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
210	C₂²⋊C₄×C₃₁	—	2	2	no	—
211	C₂²⋊C₄×C₃₁	—	2	1	no	—
212	C₂²⋊C₄×C₃₁	—	2	1	no	—
213	C₂²⋊C₄×C₃₁	—	2	2	no	—
214	C₂²⋊C₄×C₃₁	—	2	2	no	—
215	C₂²⋊C₄×C₃₁	—	2	2	no	—
216	C₂²⋊C₄×C₃₁	—	2	2	no	—
217	C₂²⋊C₄×C₃₁	—	2	2	no	—
218	C₂²⋊C₄×C₃₁	—	2	1	no	—
219	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
220	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
221	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
222	C₄²×C₃₁	—	2	1	no	—
223	C₄²×C₃₁	—	2	2	no	—
224	C₄²×C₃₁	—	2	2	no	—
225	C₄²×C₃₁	—	2	1	no	—
226	C₄²×C₃₁	—	2	1	no	—
227	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
228	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
229	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
230	C₄²×C₃₁	C₂×D₄	1	2	yes	—
231	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
232	C₄²×C₃₁	C₂×D₄	1	2	yes	—
233	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
234	#17453	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
235	C₂³×C₃₁	—	2	8	no	—
236	C₂³×C₃₁	—	2	2	no	—
237	C₂³×C₃₁	—	2	1	no	—
238	C₂³×C₃₁	—	2	4	no	—
239	C₂³×C₃₁	—	8	1	no	—
240	C₂³×C₃₁	—	8	2	no	—
241	C₂³×C₃₁	—	8	1	no	—
242	C₂³×C₃₁	—	8	2	no	—
243	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
244	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
245	Q₈×C₃₁	—	2	1	no	—
246	Q₈×C₃₁	—	2	2	no	—
247	Q₈×C₃₁	C₂×M₁₆	1	2	yes	—
248	D₄×C₃₁	—	2	8	no	—
249	D₄×C₃₁	—	2	8	no	—
250	D₄×C₃₁	—	2	4	no	—
251	D₄×C₃₁	—	2	4	no	—
252	D₄×C₃₁	—	2	2	no	—
253	D₄×C₃₁	—	2	1	no	—
254	D₄×C₃₁	—	2	2	no	—
255	D₄×C₃₁	—	2	4	no	—
256	D₄×C₃₁	C₂×M₁₆	1	2	yes	—
257	D₄×C₃₁	C₂×M₁₆	1	4	yes	—
258	C₂×C₄×C₃₁	—	2	8	no	—
259	C₂×C₄×C₃₁	—	2	1	no	—
260	C₂×C₄×C₃₁	—	2	2	no	—
261	C₂×C₄×C₃₁	—	2	2	no	—
262	C₂×C₄×C₃₁	—	2	1	no	—
263	C₂×C₄×C₃₁	—	2	2	no	—
264	C₂×C₄×C₃₁	—	2	1	no	—
265	C₂×C₄×C₃₁	—	2	2	no	—
266	C₂×C₄×C₃₁	—	2	4	no	—
267	C₂×C₄×C₃₁	—	2	1	no	—
268	C₂×C₄×C₃₁	—	2	2	no	—
269	C₂×C₄×C₃₁	—	2	1	no	—
270	C₂×C₄×C₃₁	—	4	1	no	—
271	C₂×C₄×C₃₁	—	4	2	no	—
272	C₂×C₄×C₃₁	—	4	2	no	—
273	C₂×C₄×C₃₁	—	4	1	no	—
274	C₂×C₄×C₃₁	—	4	2	no	—
275	C₂×C₄×C₃₁	—	4	1	no	—
276	C₂×C₄×C₃₁	—	8	1	no	—
277	C₂×C₄×C₃₁	—	8	2	no	—
278	C₂×C₄×C₃₁	—	8	2	no	—
279	C₂×C₄×C₃₁	—	8	2	no	—
280	C₂×C₄×C₃₁	—	8	2	no	—
281	C₂×C₄×C₃₁	—	8	1	no	—
282	C₂×C₄×C₃₁	—	8	2	no	—
283	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
284	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
285	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
286	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
287	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
288	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
289	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
290	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
291	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
292	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
293	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
294	C₈×C₃₁	—	2	2	no	—
295	C₈×C₃₁	—	4	1	no	—
296	C₈×C₃₁	—	4	2	no	—
297	C₈×C₃₁	—	4	2	no	—
298	C₈×C₃₁	—	4	1	no	—
299	C₈×C₃₁	—	4	1	no	—
300	C₄×H(5,𝔽₂)	C₂×C₃₁	1	1	yes	characteristic
301	#2989	C₂×C₃₁	1	1	yes	characteristic, complemented
302	#2965	C₂×C₃₁	1	1	yes	characteristic, complemented
303	#1490	C₂×C₃₁	1	2	yes	complemented
304	#1489	C₂×C₃₁	1	2	yes	complemented
305	#1488	C₂×C₃₁	1	2	yes	complemented
306	#1486	C₂×C₃₁	1	2	yes	complemented
307	#1478	C₂×C₃₁	1	1	yes	characteristic, complemented
308	#1477	C₂×C₃₁	1	1	yes	characteristic, complemented
309	#1476	C₂×C₃₁	1	1	yes	characteristic, complemented
310	C₂×D₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
311	C₂²×C₃₁	—	2	1	no	—
312	C₂²×C₃₁	—	2	1	no	—
313	C₂²×C₃₁	—	2	2	no	—
314	C₂²×C₃₁	—	2	4	no	—
315	C₂²×C₃₁	—	2	2	no	—
316	C₂²×C₃₁	—	2	2	no	—
317	C₂²×C₃₁	—	2	2	no	—
318	C₂²×C₃₁	—	4	2	no	—
319	C₂²×C₃₁	—	4	1	no	—
320	C₂²×C₃₁	—	4	2	no	—
321	C₂²×C₃₁	—	4	1	no	—
322	C₂²×C₃₁	—	8	4	no	—
323	C₂²×C₃₁	—	8	8	no	—
324	C₂²×C₃₁	#407	1	2	yes	—
325	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
326	C₄×C₃₁	—	2	2	no	—
327	C₄×C₃₁	—	2	2	no	—
328	C₄×C₃₁	—	2	1	no	—
329	C₄×C₃₁	—	2	2	no	—
330	C₄×C₃₁	—	2	1	no	—
331	C₄×C₃₁	—	2	1	no	—
332	C₄×C₃₁	—	2	2	no	—
333	C₄×C₃₁	—	4	2	no	—
334	C₄×C₃₁	—	4	1	no	—
335	C₄×C₃₁	—	4	1	no	—
336	C₄×C₃₁	#407	1	2	yes	—
337	C₄×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
338	C₄×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
339	C₄×C₃₁	#576	1	1	yes	characteristic
340	C₄×C₃₁	#577	1	1	yes	characteristic
341	C₂×H(5,𝔽₂)	C₄×C₃₁	1	1	yes	characteristic
342	#520	—	2	1	no	—
343	#520	C₄×C₃₁	1	2	yes	—
344	#520	C₄×C₃₁	1	1	yes	characteristic
345	#520	C₄×C₃₁	1	1	yes	characteristic
346	#518	—	2	2	no	—
347	#518	—	2	1	no	—
348	#518	C₄×C₃₁	1	2	yes	—
349	#518	C₄×C₃₁	1	1	yes	characteristic
350	C₄×#41	—	2	1	no	—
351	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
352	C₄×#41	C₂²×C₃₁	1	2	yes	—
353	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
354	C₂×C₄×D₄	—	2	2	no	—
355	C₂×C₄×D₄	—	2	1	no	—
356	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
357	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
358	#446	—	2	1	no	—
359	#446	—	2	1	no	—
360	#436	—	2	1	no	—
361	#436	—	2	1	no	—
362	#435	—	2	2	no	—
363	#435	—	2	2	no	—
364	#432	C₂²×C₃₁	1	2	yes	complemented
365	#431	C₂²×C₃₁	1	1	yes	characteristic, complemented
366	#431	C₂²×C₃₁	1	1	yes	characteristic, complemented
367	#431	C₂²×C₃₁	1	2	yes	complemented
368	#423	C₂²×C₃₁	1	2	yes	complemented
369	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
370	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
371	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
372	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
373	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
374	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	complemented
375	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
376	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
377	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
378	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
379	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
380	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
381	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
382	C₂×C₃₁	—	2	2	no	—
383	C₂×C₃₁	—	4	2	no	—
384	C₂×C₃₁	—	4	4	no	—
385	C₂×C₃₁	—	4	2	no	—
386	C₂×C₃₁	C₂×#407	1	1	yes	characteristic
387	C₂×C₃₁	#2897	1	1	yes	characteristic
388	C₂×C₃₁	#2901	1	1	yes	characteristic
389	H(5,𝔽₂)	—	2	4	no	—
390	H(5,𝔽₂)	—	2	4	no	—
391	C₂×#41	—	2	1	no	—
392	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
393	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
394	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
395	C₂²×D₄	—	2	2	no	—
396	C₂²×D₄	—	2	1	no	—
397	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
398	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
399	C₂³×C₄	—	2	1	no	—
400	C₂³×C₄	—	2	2	no	—
401	C₂×M₁₆	—	2	2	no	—
402	C₂×M₁₆	—	2	2	no	—
403	C₂²×C₈	—	2	1	no	—
404	C₂²×C₈	—	2	1	no	—
405	C₄×Q₈	—	2	1	no	—
406	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
407	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
408	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
409	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
410	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
411	C₄×D₄	—	2	2	no	—
412	C₄×D₄	—	2	1	no	—
413	C₄×D₄	—	2	1	no	—
414	C₄×D₄	—	2	2	no	—
415	C₄×D₄	—	2	1	no	—
416	C₄×D₄	—	2	2	no	—
417	C₄×D₄	—	2	2	no	—
418	C₄×D₄	—	2	1	no	—
419	C₄×D₄	—	2	2	no	—
420	C₄×D₄	—	2	1	no	—
421	C₄×D₄	—	2	2	no	—
422	C₄×D₄	—	2	2	no	—
423	C₄×D₄	—	2	2	no	—
424	C₄×D₄	—	2	1	no	—
425	C₄×D₄	—	2	2	no	—
426	C₄×D₄	—	2	1	no	—
427	C₄×D₄	—	2	2	no	—
428	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
429	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
430	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
431	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
432	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
433	C₄×D₄	D₄×C₃₁	1	2	yes	—
434	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
435	C₄×D₄	D₄×C₃₁	1	2	yes	—
436	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
437	C₄×D₄	C₂³×C₃₁	1	2	yes	—
438	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
439	#117	—	2	2	no	—
440	#117	—	2	1	no	—
441	#117	—	2	1	no	—
442	#117	—	2	2	no	—
443	#117	C₂×C₄×C₃₁	1	2	yes	—
444	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
445	#117	C₂×C₄×C₃₁	1	2	yes	—
446	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
447	C₂×C₄⋊C₄	—	2	2	no	—
448	C₂×C₄⋊C₄	—	2	1	no	—
449	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
450	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
451	C₂×C₂²⋊C₄	—	2	1	no	—
452	C₂×C₂²⋊C₄	—	2	2	no	—
453	C₂×C₂²⋊C₄	—	2	2	no	—
454	C₂×C₂²⋊C₄	—	2	2	no	—
455	C₂×C₂²⋊C₄	—	2	2	no	—
456	C₂×C₄²	—	2	1	no	—
457	C₂×C₄²	—	2	2	no	—
458	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
459	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
460	C₄⋊C₈	—	2	1	no	—
461	C₄⋊C₈	—	2	2	no	—
462	C₄⋊C₈	—	2	2	no	—
463	C₄⋊C₈	—	2	1	no	—
464	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
465	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
466	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
467	C₂²⋊C₈	—	2	1	no	—
468	C₂²⋊C₈	—	2	1	no	—
469	C₂²⋊C₈	—	2	2	no	—
470	C₂²⋊C₈	—	2	2	no	—
471	#97	—	2	2	no	—
472	#97	—	2	2	no	—
473	C₄×C₈	—	2	1	no	—
474	C₄×C₈	—	2	1	no	—
475	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
476	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
477	C₄×C₈	C₂³×C₃₁	1	2	yes	—
478	C₃₁	#17453	1	1	yes	characteristic, direct product factor, Sylow
479	C₂⁴	—	2	2	no	—
480	C₂⁴	—	2	1	no	—
481	#41	—	2	8	no	—
482	#41	—	2	4	no	—
483	#41	—	2	4	no	—
484	#41	—	2	2	no	—
485	#41	—	2	2	no	—
486	#41	—	2	2	no	—
487	#41	M₁₆×C₃₁	1	4	yes	complemented
488	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
489	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
490	C₂×D₄	—	2	16	no	—
491	C₂×D₄	—	2	8	no	—
492	C₂×D₄	—	2	2	no	—
493	C₂×D₄	—	2	1	no	—
494	C₂×D₄	—	2	4	no	—
495	C₂×D₄	—	2	4	no	—
496	C₂×D₄	—	2	2	no	—
497	C₂×D₄	—	2	1	no	—
498	C₂×D₄	—	2	2	no	—
499	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
500	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
501	C₂×D₄	M₁₆×C₃₁	1	4	yes	complemented
502	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
503	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
504	C₂²×C₄	—	2	2	no	—
505	C₂²×C₄	—	2	1	no	—
506	C₂²×C₄	—	2	2	no	—
507	C₂²×C₄	—	2	1	no	—
508	C₂²×C₄	—	2	1	no	—
509	C₂²×C₄	—	2	2	no	—
510	C₂²×C₄	—	2	1	no	—
511	C₂²×C₄	—	2	2	no	—
512	C₂²×C₄	—	2	2	no	—
513	C₂²×C₄	—	2	2	no	—
514	C₂²×C₄	—	2	2	no	—
515	C₂²×C₄	—	8	1	no	—
516	C₂²×C₄	—	8	2	no	—
517	C₂²×C₄	—	8	1	no	—
518	C₂²×C₄	—	8	2	no	—
519	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
520	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
521	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
522	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
523	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
524	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
525	M₁₆	—	4	4	no	—
526	M₁₆	—	8	2	no	—
527	C₂×C₈	—	2	1	no	—
528	C₂×C₈	—	2	1	no	—
529	C₂×C₈	—	2	1	no	—
530	C₂×C₈	—	2	1	no	—
531	C₂×C₈	—	2	2	no	—
532	C₂×C₈	—	2	2	no	—
533	C₂×C₈	—	4	2	no	—
534	C₂×C₈	—	8	1	no	—
535	C₄⋊C₄	—	2	1	no	—
536	C₄⋊C₄	—	2	2	no	—
537	C₄⋊C₄	—	2	2	no	—
538	C₄⋊C₄	—	2	1	no	—
539	C₄⋊C₄	—	2	1	no	—
540	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
541	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
542	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
543	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
544	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
545	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
546	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
547	C₂²⋊C₄	—	2	2	no	—
548	C₂²⋊C₄	—	2	1	no	—
549	C₂²⋊C₄	—	2	1	no	—
550	C₂²⋊C₄	—	2	2	no	—
551	C₂²⋊C₄	—	2	2	no	—
552	C₂²⋊C₄	—	2	2	no	—
553	C₂²⋊C₄	—	2	2	no	—
554	C₂²⋊C₄	—	2	2	no	—
555	C₂²⋊C₄	—	2	1	no	—
556	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
557	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
558	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
559	C₄²	—	2	1	no	—
560	C₄²	—	2	2	no	—
561	C₄²	—	2	2	no	—
562	C₄²	—	2	1	no	—
563	C₄²	—	2	1	no	—
564	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
565	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
566	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
567	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
568	C₄²	C₂×D₄×C₃₁	1	2	yes	—
569	C₄²	C₂×D₄×C₃₁	1	2	yes	—
570	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
571	C₂³	—	2	8	no	—
572	C₂³	—	2	2	no	—
573	C₂³	—	2	1	no	—
574	C₂³	—	2	4	no	—
575	C₂³	—	8	1	no	—
576	C₂³	—	8	2	no	—
577	C₂³	—	8	1	no	—
578	C₂³	—	8	2	no	—
579	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
580	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
581	Q₈	—	2	2	no	—
582	Q₈	—	2	1	no	—
583	Q₈	C₂×M₁₆×C₃₁	1	2	yes	—
584	D₄	—	2	8	no	—
585	D₄	—	2	8	no	—
586	D₄	—	2	4	no	—
587	D₄	—	2	4	no	—
588	D₄	—	2	2	no	—
589	D₄	—	2	4	no	—
590	D₄	—	2	2	no	—
591	D₄	—	2	1	no	—
592	D₄	C₂×M₁₆×C₃₁	1	2	yes	—
593	D₄	C₂×M₁₆×C₃₁	1	4	yes	—
594	C₂×C₄	—	2	8	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	4	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	1	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	1	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	2	no	—
608	C₂×C₄	—	4	2	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	2	no	—
611	C₂×C₄	—	4	1	no	—
612	C₂×C₄	—	8	1	no	—
613	C₂×C₄	—	8	2	no	—
614	C₂×C₄	—	8	2	no	—
615	C₂×C₄	—	8	2	no	—
616	C₂×C₄	—	8	2	no	—
617	C₂×C₄	—	8	1	no	—
618	C₂×C₄	—	8	2	no	—
619	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
620	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
621	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
622	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
623	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
624	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
625	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
626	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
627	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
628	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
629	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
630	C₈	—	2	2	no	—
631	C₈	—	4	1	no	—
632	C₈	—	4	2	no	—
633	C₈	—	4	2	no	—
634	C₈	—	4	1	no	—
635	C₈	—	4	1	no	—
636	C₂²	—	2	1	no	—
637	C₂²	—	2	1	no	—
638	C₂²	—	2	2	no	—
639	C₂²	—	2	4	no	—
640	C₂²	—	2	2	no	—
641	C₂²	—	2	2	no	—
642	C₂²	—	2	2	no	—
643	C₂²	—	4	2	no	—
644	C₂²	—	4	1	no	—
645	C₂²	—	4	2	no	—
646	C₂²	—	4	1	no	—
647	C₂²	—	8	4	no	—
648	C₂²	—	8	8	no	—
649	C₂²	C₃₁×#407	1	2	yes	—
650	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
651	C₄	—	2	2	no	—
652	C₄	—	2	2	no	—
653	C₄	—	2	1	no	—
654	C₄	—	2	2	no	—
655	C₄	—	2	1	no	—
656	C₄	—	2	1	no	—
657	C₄	—	2	2	no	—
658	C₄	—	4	2	no	—
659	C₄	—	4	1	no	—
660	C₄	—	4	1	no	—
661	C₄	C₃₁×#407	1	2	yes	—
662	C₄	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
663	C₄	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
664	C₄	C₃₁×#576	1	1	yes	characteristic
665	C₄	C₃₁×#577	1	1	yes	characteristic
666	C₂	—	2	2	no	—
667	C₂	—	4	2	no	—
668	C₂	—	4	4	no	—
669	C₂	—	4	2	no	—
670	C₂	C₂×C₃₁×#407	1	1	yes	characteristic
671	C₂	C₃₁×#2897	1	1	yes	characteristic
672	C₂	C₃₁×#2901	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
prime radical	C₂×C₃₁×H(5,𝔽₂)	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₃₁×#2989	C₂	semidirect
3. C₃₁×#2965	C₂	semidirect
4. C₃₁×#1490	C₂	semidirect
5. C₃₁×#1489	C₂	semidirect
6. C₃₁×#1488	C₂	semidirect
7. C₃₁×#1486	C₂	semidirect
8. C₃₁×#1478	C₂	semidirect
9. C₃₁×#1477	C₂	semidirect
10. C₃₁×#1476	C₂	semidirect
11. C₂×C₃₁×D₄⋊C₈	C₂	semidirect
35. C₃₁×#432	C₂²	semidirect
36. C₃₁×#431	C₂²	semidirect
37. C₃₁×#431	C₂²	semidirect
38. C₃₁×#431	C₂²	semidirect
39. C₃₁×#423	C₂²	semidirect
40. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
41. C₄×M₁₆×C₃₁	C₂²	semidirect
42. C₂×C₄×C₈×C₃₁	C₂²	semidirect
43. C₃₁×Q₈⋊C₈	C₂²	semidirect
44. C₃₁×Q₈⋊C₈	C₂²	semidirect
45. C₃₁×Q₈⋊C₈	C₂²	semidirect
46. C₃₁×D₄⋊C₈	C₂²	semidirect
47. C₃₁×D₄⋊C₈	C₂²	semidirect
48. C₃₁×D₄⋊C₈	C₂²	semidirect
49. C₃₁×D₄⋊C₈	C₂²	semidirect
50. C₃₁×D₄⋊C₈	C₂²	semidirect
51. C₃₁×D₄⋊C₈	C₂²	semidirect
52. C₃₁×D₄⋊C₈	C₂²	semidirect
150. #41×C₃₁	M₁₆	semidirect
164. C₂×D₄×C₃₁	M₁₆	semidirect
234. #17453	C₃₁	direct (N × H)
301. #2989	C₂×C₃₁	semidirect
302. #2965	C₂×C₃₁	semidirect
303. #1490	C₂×C₃₁	semidirect
304. #1489	C₂×C₃₁	semidirect
305. #1488	C₂×C₃₁	semidirect
306. #1486	C₂×C₃₁	semidirect
307. #1478	C₂×C₃₁	semidirect
308. #1477	C₂×C₃₁	semidirect
309. #1476	C₂×C₃₁	semidirect
310. C₂×D₄⋊C₈	C₂×C₃₁	semidirect
364. #432	C₂²×C₃₁	semidirect
365. #431	C₂²×C₃₁	semidirect
366. #431	C₂²×C₃₁	semidirect
367. #431	C₂²×C₃₁	semidirect
368. #423	C₂²×C₃₁	semidirect
369. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
370. C₄×M₁₆	C₂²×C₃₁	semidirect
371. C₂×C₄×C₈	C₂²×C₃₁	semidirect
372. Q₈⋊C₈	C₂²×C₃₁	semidirect
373. Q₈⋊C₈	C₂²×C₃₁	semidirect
374. Q₈⋊C₈	C₂²×C₃₁	semidirect
375. D₄⋊C₈	C₂²×C₃₁	semidirect
376. D₄⋊C₈	C₂²×C₃₁	semidirect
377. D₄⋊C₈	C₂²×C₃₁	semidirect
378. D₄⋊C₈	C₂²×C₃₁	semidirect
379. D₄⋊C₈	C₂²×C₃₁	semidirect
380. D₄⋊C₈	C₂²×C₃₁	semidirect
381. D₄⋊C₈	C₂²×C₃₁	semidirect
478. C₃₁	#17453	direct (N × H)
487. #41	M₁₆×C₃₁	semidirect
501. C₂×D₄	M₁₆×C₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₃₁×H(5,𝔽₂)	2
C₃₁×#2989	2
C₃₁×#2965	2
C₃₁×#1490	2
C₃₁×#1489	2
C₃₁×#1488	2
C₃₁×#1486	2
C₃₁×#1478	2
C₃₁×#1477	2
C₃₁×#1476	2
C₂×C₃₁×D₄⋊C₈	2
#17453	31

Subgroups — C31×#17453 #21474212

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17453 #21474212