Subgroups — C₃₁×#17454 #21474213

Order 7936.

1138 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×2₋¹⁺⁴	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×#1491	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×#1487	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1479	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×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×2₋¹⁺⁴	C₄	1	1	yes	characteristic
13	C₃₁×#520	—	2	2	no	—
14	C₃₁×#520	—	2	1	no	—
15	C₃₁×#520	C₄	1	2	yes	—
16	C₃₁×#520	C₄	1	1	yes	characteristic
17	C₃₁×#520	C₄	1	1	yes	characteristic
18	C₃₁×#519	—	2	1	no	—
19	C₃₁×#519	C₄	1	2	yes	—
20	C₃₁×#519	C₄	1	1	yes	characteristic
21	C₄×#41×C₃₁	—	2	2	no	—
22	C₄×#41×C₃₁	—	2	1	no	—
23	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
24	C₄×#41×C₃₁	C₂²	1	2	yes	—
25	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
27	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×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	—
36	C₃₁×#431	C₂²	1	1	yes	characteristic
37	C₃₁×#431	C₂²	1	1	yes	characteristic
38	C₃₁×#431	C₂²	1	2	yes	—
39	C₃₁×#423	C₂²	1	2	yes	—
40	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
42	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
43	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
45	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
46	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
47	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
48	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
49	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
50	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
51	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
52	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
53	C₃₁×2₋¹⁺⁴	—	2	4	no	—
54	C₃₁×2₋¹⁺⁴	—	2	4	no	—
55	C₂×#41×C₃₁	—	2	1	no	—
56	C₂×#41×C₃₁	—	2	2	no	—
57	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
58	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₂²×Q₈×C₃₁	—	2	1	no	—
61	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
62	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
63	C₂×M₁₆×C₃₁	—	2	2	no	—
64	C₂×M₁₆×C₃₁	—	2	2	no	—
65	C₂²×C₈×C₃₁	—	2	1	no	—
66	C₂²×C₈×C₃₁	—	2	1	no	—
67	C₄×Q₈×C₃₁	—	2	1	no	—
68	C₄×Q₈×C₃₁	—	2	2	no	—
69	C₄×Q₈×C₃₁	—	2	1	no	—
70	C₄×Q₈×C₃₁	—	2	2	no	—
71	C₄×Q₈×C₃₁	—	2	1	no	—
72	C₄×Q₈×C₃₁	—	2	1	no	—
73	C₄×Q₈×C₃₁	—	2	2	no	—
74	C₄×Q₈×C₃₁	—	2	1	no	—
75	C₄×Q₈×C₃₁	—	2	2	no	—
76	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
77	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
78	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
79	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
80	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
81	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
82	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
83	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
84	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
85	C₄×D₄×C₃₁	—	2	2	no	—
86	C₄×D₄×C₃₁	—	2	1	no	—
87	C₄×D₄×C₃₁	—	2	1	no	—
88	C₄×D₄×C₃₁	—	2	2	no	—
89	C₄×D₄×C₃₁	—	2	1	no	—
90	C₄×D₄×C₃₁	—	2	2	no	—
91	C₄×D₄×C₃₁	—	2	2	no	—
92	C₄×D₄×C₃₁	—	2	2	no	—
93	C₄×D₄×C₃₁	—	2	2	no	—
94	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
95	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
96	C₄×D₄×C₃₁	D₄	1	2	yes	—
97	C₄×D₄×C₃₁	D₄	1	2	yes	—
98	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
99	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
100	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
101	C₃₁×#117	—	2	2	no	—
102	C₃₁×#117	—	2	1	no	—
103	C₃₁×#117	—	2	2	no	—
104	C₃₁×#117	—	2	2	no	—
105	C₃₁×#117	—	2	2	no	—
106	C₃₁×#117	—	2	1	no	—
107	C₃₁×#117	—	2	2	no	—
108	C₃₁×#117	C₂×C₄	1	2	yes	—
109	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
110	C₃₁×#117	C₂×C₄	1	2	yes	—
111	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
113	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
114	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
116	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
117	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
118	C₂×C₄²×C₃₁	—	2	1	no	—
119	C₂×C₄²×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	#41×C₃₁	—	2	16	no	—
143	#41×C₃₁	—	2	8	no	—
144	#41×C₃₁	—	2	4	no	—
145	#41×C₃₁	—	2	4	no	—
146	#41×C₃₁	—	2	2	no	—
147	#41×C₃₁	—	2	2	no	—
148	#41×C₃₁	—	2	2	no	—
149	#41×C₃₁	M₁₆	1	4	yes	complemented
150	C₂×Q₈×C₃₁	—	2	8	no	—
151	C₂×Q₈×C₃₁	—	2	4	no	—
152	C₂×Q₈×C₃₁	—	2	4	no	—
153	C₂×Q₈×C₃₁	—	2	2	no	—
154	C₂×Q₈×C₃₁	—	2	1	no	—
155	C₂×Q₈×C₃₁	—	2	2	no	—
156	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
157	C₂×Q₈×C₃₁	M₁₆	1	4	yes	complemented
158	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
159	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
160	C₂×D₄×C₃₁	—	2	2	no	—
161	C₂×D₄×C₃₁	—	2	1	no	—
162	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
163	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
164	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
165	C₂²×C₄×C₃₁	—	2	2	no	—
166	C₂²×C₄×C₃₁	—	2	1	no	—
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	1	no	—
173	C₂²×C₄×C₃₁	—	2	2	no	—
174	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
175	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
176	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
178	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
180	M₁₆×C₃₁	—	4	4	no	—
181	M₁₆×C₃₁	—	8	2	no	—
182	C₂×C₈×C₃₁	—	2	1	no	—
183	C₂×C₈×C₃₁	—	2	1	no	—
184	C₂×C₈×C₃₁	—	2	2	no	—
185	C₂×C₈×C₃₁	—	2	1	no	—
186	C₂×C₈×C₃₁	—	2	1	no	—
187	C₂×C₈×C₃₁	—	2	2	no	—
188	C₂×C₈×C₃₁	—	4	2	no	—
189	C₂×C₈×C₃₁	—	8	1	no	—
190	C₄⋊C₄×C₃₁	—	2	2	no	—
191	C₄⋊C₄×C₃₁	—	2	2	no	—
192	C₄⋊C₄×C₃₁	—	2	2	no	—
193	C₄⋊C₄×C₃₁	—	2	2	no	—
194	C₄⋊C₄×C₃₁	—	2	1	no	—
195	C₄⋊C₄×C₃₁	—	2	1	no	—
196	C₄⋊C₄×C₃₁	—	2	2	no	—
197	C₄⋊C₄×C₃₁	—	2	2	no	—
198	C₄⋊C₄×C₃₁	—	2	1	no	—
199	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
200	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
201	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
202	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
203	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
204	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
205	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
206	C₂²⋊C₄×C₃₁	—	2	2	no	—
207	C₂²⋊C₄×C₃₁	—	2	1	no	—
208	C₂²⋊C₄×C₃₁	—	2	1	no	—
209	C₂²⋊C₄×C₃₁	—	2	2	no	—
210	C₂²⋊C₄×C₃₁	—	2	1	no	—
211	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
212	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
213	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
214	C₄²×C₃₁	—	2	1	no	—
215	C₄²×C₃₁	—	2	2	no	—
216	C₄²×C₃₁	—	2	2	no	—
217	C₄²×C₃₁	—	2	2	no	—
218	C₄²×C₃₁	—	2	2	no	—
219	C₄²×C₃₁	—	2	2	no	—
220	C₄²×C₃₁	—	2	1	no	—
221	C₄²×C₃₁	—	2	2	no	—
222	C₄²×C₃₁	—	2	1	no	—
223	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
224	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
225	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
226	C₄²×C₃₁	C₂×D₄	1	2	yes	—
227	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
228	C₄²×C₃₁	C₂×D₄	1	2	yes	—
229	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
230	#17454	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
231	C₂³×C₃₁	—	2	1	no	—
232	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
233	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
234	Q₈×C₃₁	—	2	8	no	—
235	Q₈×C₃₁	—	2	1	no	—
236	Q₈×C₃₁	—	2	2	no	—
237	Q₈×C₃₁	—	2	2	no	—
238	Q₈×C₃₁	—	2	4	no	—
239	Q₈×C₃₁	C₂×M₁₆	1	2	yes	—
240	Q₈×C₃₁	C₂×M₁₆	1	4	yes	—
241	D₄×C₃₁	—	2	8	no	—
242	D₄×C₃₁	—	2	4	no	—
243	D₄×C₃₁	—	2	4	no	—
244	D₄×C₃₁	—	2	1	no	—
245	D₄×C₃₁	—	2	2	no	—
246	D₄×C₃₁	C₂×M₁₆	1	2	yes	—
247	C₂×C₄×C₃₁	—	2	8	no	—
248	C₂×C₄×C₃₁	—	2	8	no	—
249	C₂×C₄×C₃₁	—	2	2	no	—
250	C₂×C₄×C₃₁	—	2	1	no	—
251	C₂×C₄×C₃₁	—	2	4	no	—
252	C₂×C₄×C₃₁	—	2	2	no	—
253	C₂×C₄×C₃₁	—	2	2	no	—
254	C₂×C₄×C₃₁	—	2	1	no	—
255	C₂×C₄×C₃₁	—	2	2	no	—
256	C₂×C₄×C₃₁	—	2	1	no	—
257	C₂×C₄×C₃₁	—	2	2	no	—
258	C₂×C₄×C₃₁	—	2	4	no	—
259	C₂×C₄×C₃₁	—	2	1	no	—
260	C₂×C₄×C₃₁	—	2	2	no	—
261	C₂×C₄×C₃₁	—	2	1	no	—
262	C₂×C₄×C₃₁	—	4	1	no	—
263	C₂×C₄×C₃₁	—	4	2	no	—
264	C₂×C₄×C₃₁	—	4	1	no	—
265	C₂×C₄×C₃₁	—	4	1	no	—
266	C₂×C₄×C₃₁	—	8	1	no	—
267	C₂×C₄×C₃₁	—	8	2	no	—
268	C₂×C₄×C₃₁	—	8	1	no	—
269	C₂×C₄×C₃₁	—	8	2	no	—
270	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
271	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
272	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
273	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
274	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
275	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
277	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
278	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
279	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
280	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
281	C₈×C₃₁	—	2	2	no	—
282	C₈×C₃₁	—	4	1	no	—
283	C₈×C₃₁	—	4	2	no	—
284	C₈×C₃₁	—	4	2	no	—
285	C₈×C₃₁	—	4	1	no	—
286	C₈×C₃₁	—	4	1	no	—
287	C₄×2₋¹⁺⁴	C₂×C₃₁	1	1	yes	characteristic
288	#2989	C₂×C₃₁	1	1	yes	characteristic, complemented
289	#2965	C₂×C₃₁	1	1	yes	characteristic, complemented
290	#1491	C₂×C₃₁	1	2	yes	complemented
291	#1489	C₂×C₃₁	1	2	yes	complemented
292	#1488	C₂×C₃₁	1	2	yes	complemented
293	#1487	C₂×C₃₁	1	2	yes	complemented
294	#1479	C₂×C₃₁	1	1	yes	characteristic, complemented
295	#1477	C₂×C₃₁	1	1	yes	characteristic, complemented
296	#1476	C₂×C₃₁	1	1	yes	characteristic, complemented
297	C₂×Q₈⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
298	C₂²×C₃₁	—	2	1	no	—
299	C₂²×C₃₁	—	2	1	no	—
300	C₂²×C₃₁	—	2	2	no	—
301	C₂²×C₃₁	—	2	2	no	—
302	C₂²×C₃₁	—	2	2	no	—
303	C₂²×C₃₁	—	2	2	no	—
304	C₂²×C₃₁	—	4	1	no	—
305	C₂²×C₃₁	—	4	1	no	—
306	C₂²×C₃₁	#407	1	2	yes	—
307	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
308	C₄×C₃₁	—	2	4	no	—
309	C₄×C₃₁	—	2	2	no	—
310	C₄×C₃₁	—	2	2	no	—
311	C₄×C₃₁	—	2	1	no	—
312	C₄×C₃₁	—	2	2	no	—
313	C₄×C₃₁	—	2	1	no	—
314	C₄×C₃₁	—	2	1	no	—
315	C₄×C₃₁	—	2	2	no	—
316	C₄×C₃₁	—	4	1	no	—
317	C₄×C₃₁	—	4	2	no	—
318	C₄×C₃₁	—	4	1	no	—
319	C₄×C₃₁	#407	1	2	yes	—
320	C₄×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
321	C₄×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
322	C₄×C₃₁	#577	1	1	yes	characteristic
323	C₄×C₃₁	#578	1	1	yes	characteristic
324	C₂×2₋¹⁺⁴	C₄×C₃₁	1	1	yes	characteristic
325	#520	—	2	2	no	—
326	#520	—	2	1	no	—
327	#520	C₄×C₃₁	1	2	yes	—
328	#520	C₄×C₃₁	1	1	yes	characteristic
329	#520	C₄×C₃₁	1	1	yes	characteristic
330	#519	—	2	1	no	—
331	#519	C₄×C₃₁	1	2	yes	—
332	#519	C₄×C₃₁	1	1	yes	characteristic
333	C₄×#41	—	2	2	no	—
334	C₄×#41	—	2	1	no	—
335	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
336	C₄×#41	C₂²×C₃₁	1	2	yes	—
337	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
338	C₂×C₄×Q₈	—	2	1	no	—
339	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
340	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
341	#446	—	2	1	no	—
342	#446	—	2	1	no	—
343	#436	—	2	1	no	—
344	#436	—	2	1	no	—
345	#435	—	2	2	no	—
346	#435	—	2	2	no	—
347	#432	C₂²×C₃₁	1	2	yes	—
348	#431	C₂²×C₃₁	1	1	yes	characteristic
349	#431	C₂²×C₃₁	1	1	yes	characteristic
350	#431	C₂²×C₃₁	1	2	yes	—
351	#423	C₂²×C₃₁	1	2	yes	—
352	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
353	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
354	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
355	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
356	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
357	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
358	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
359	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
360	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
361	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
362	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
363	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
364	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
365	C₂×C₃₁	—	2	2	no	—
366	C₂×C₃₁	—	4	2	no	—
367	C₂×C₃₁	—	4	2	no	—
368	C₂×C₃₁	C₂×#407	1	1	yes	characteristic
369	C₂×C₃₁	#2898	1	1	yes	characteristic
370	C₂×C₃₁	#2901	1	1	yes	characteristic
371	2₋¹⁺⁴	—	2	4	no	—
372	2₋¹⁺⁴	—	2	4	no	—
373	C₂×#41	—	2	1	no	—
374	C₂×#41	—	2	2	no	—
375	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
376	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
377	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
378	C₂²×Q₈	—	2	1	no	—
379	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
380	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
381	C₂×M₁₆	—	2	2	no	—
382	C₂×M₁₆	—	2	2	no	—
383	C₂²×C₈	—	2	1	no	—
384	C₂²×C₈	—	2	1	no	—
385	C₄×Q₈	—	2	1	no	—
386	C₄×Q₈	—	2	2	no	—
387	C₄×Q₈	—	2	1	no	—
388	C₄×Q₈	—	2	2	no	—
389	C₄×Q₈	—	2	1	no	—
390	C₄×Q₈	—	2	1	no	—
391	C₄×Q₈	—	2	2	no	—
392	C₄×Q₈	—	2	1	no	—
393	C₄×Q₈	—	2	2	no	—
394	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
395	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
396	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
397	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
398	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
399	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
400	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
401	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
402	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
403	C₄×D₄	—	2	2	no	—
404	C₄×D₄	—	2	1	no	—
405	C₄×D₄	—	2	1	no	—
406	C₄×D₄	—	2	2	no	—
407	C₄×D₄	—	2	1	no	—
408	C₄×D₄	—	2	2	no	—
409	C₄×D₄	—	2	2	no	—
410	C₄×D₄	—	2	2	no	—
411	C₄×D₄	—	2	2	no	—
412	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
413	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
414	C₄×D₄	D₄×C₃₁	1	2	yes	—
415	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
416	C₄×D₄	D₄×C₃₁	1	2	yes	—
417	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
418	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
419	#117	—	2	2	no	—
420	#117	—	2	1	no	—
421	#117	—	2	2	no	—
422	#117	—	2	2	no	—
423	#117	—	2	2	no	—
424	#117	—	2	1	no	—
425	#117	—	2	2	no	—
426	#117	C₂×C₄×C₃₁	1	2	yes	—
427	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
428	#117	C₂×C₄×C₃₁	1	2	yes	—
429	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
430	C₂×C₄⋊C₄	—	2	1	no	—
431	C₂×C₄⋊C₄	—	2	2	no	—
432	C₂×C₄⋊C₄	—	2	2	no	—
433	C₂×C₄⋊C₄	—	2	1	no	—
434	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
435	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
436	C₂×C₄²	—	2	1	no	—
437	C₂×C₄²	—	2	2	no	—
438	C₂×C₄²	—	2	1	no	—
439	C₂×C₄²	—	2	2	no	—
440	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
441	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
442	C₄⋊C₈	—	2	1	no	—
443	C₄⋊C₈	—	2	2	no	—
444	C₄⋊C₈	—	2	2	no	—
445	C₄⋊C₈	—	2	1	no	—
446	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
447	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
448	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
449	C₂²⋊C₈	—	2	1	no	—
450	C₂²⋊C₈	—	2	1	no	—
451	C₂²⋊C₈	—	2	2	no	—
452	C₂²⋊C₈	—	2	2	no	—
453	#97	—	2	2	no	—
454	#97	—	2	2	no	—
455	C₄×C₈	—	2	1	no	—
456	C₄×C₈	—	2	1	no	—
457	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
458	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
459	C₄×C₈	C₂³×C₃₁	1	2	yes	—
460	C₃₁	#17454	1	1	yes	characteristic, direct product factor, Sylow
461	#41	—	2	16	no	—
462	#41	—	2	8	no	—
463	#41	—	2	4	no	—
464	#41	—	2	4	no	—
465	#41	—	2	2	no	—
466	#41	—	2	2	no	—
467	#41	—	2	2	no	—
468	#41	M₁₆×C₃₁	1	4	yes	complemented
469	C₂×Q₈	—	2	8	no	—
470	C₂×Q₈	—	2	4	no	—
471	C₂×Q₈	—	2	4	no	—
472	C₂×Q₈	—	2	2	no	—
473	C₂×Q₈	—	2	1	no	—
474	C₂×Q₈	—	2	2	no	—
475	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
476	C₂×Q₈	M₁₆×C₃₁	1	4	yes	complemented
477	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
478	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
479	C₂×D₄	—	2	2	no	—
480	C₂×D₄	—	2	1	no	—
481	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
482	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
483	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
484	C₂²×C₄	—	2	2	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	—	2	2	no	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	2	no	—
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	2	2	no	—
493	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
494	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
495	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
496	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
497	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
498	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
499	M₁₆	—	4	4	no	—
500	M₁₆	—	8	2	no	—
501	C₂×C₈	—	2	1	no	—
502	C₂×C₈	—	2	1	no	—
503	C₂×C₈	—	2	1	no	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	—	2	2	no	—
506	C₂×C₈	—	2	2	no	—
507	C₂×C₈	—	4	2	no	—
508	C₂×C₈	—	8	1	no	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	2	2	no	—
511	C₄⋊C₄	—	2	2	no	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	—	2	1	no	—
514	C₄⋊C₄	—	2	1	no	—
515	C₄⋊C₄	—	2	2	no	—
516	C₄⋊C₄	—	2	2	no	—
517	C₄⋊C₄	—	2	1	no	—
518	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
519	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
520	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
521	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
522	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
523	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
524	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
525	C₂²⋊C₄	—	2	2	no	—
526	C₂²⋊C₄	—	2	1	no	—
527	C₂²⋊C₄	—	2	1	no	—
528	C₂²⋊C₄	—	2	2	no	—
529	C₂²⋊C₄	—	2	1	no	—
530	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
531	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
532	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
533	C₄²	—	2	1	no	—
534	C₄²	—	2	2	no	—
535	C₄²	—	2	2	no	—
536	C₄²	—	2	2	no	—
537	C₄²	—	2	2	no	—
538	C₄²	—	2	2	no	—
539	C₄²	—	2	1	no	—
540	C₄²	—	2	2	no	—
541	C₄²	—	2	1	no	—
542	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
543	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
544	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
545	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
546	C₄²	C₂×D₄×C₃₁	1	2	yes	—
547	C₄²	C₂×D₄×C₃₁	1	2	yes	—
548	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
549	C₂³	—	2	1	no	—
550	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
551	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
552	Q₈	—	2	8	no	—
553	Q₈	—	2	2	no	—
554	Q₈	—	2	4	no	—
555	Q₈	—	2	1	no	—
556	Q₈	—	2	2	no	—
557	Q₈	C₂×M₁₆×C₃₁	1	2	yes	—
558	Q₈	C₂×M₁₆×C₃₁	1	4	yes	—
559	D₄	—	2	8	no	—
560	D₄	—	2	4	no	—
561	D₄	—	2	4	no	—
562	D₄	—	2	2	no	—
563	D₄	—	2	1	no	—
564	D₄	C₂×M₁₆×C₃₁	1	2	yes	—
565	C₂×C₄	—	2	8	no	—
566	C₂×C₄	—	2	8	no	—
567	C₂×C₄	—	2	4	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	4	1	no	—
581	C₂×C₄	—	4	2	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	8	1	no	—
585	C₂×C₄	—	8	2	no	—
586	C₂×C₄	—	8	1	no	—
587	C₂×C₄	—	8	2	no	—
588	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
589	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
590	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
591	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
592	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
593	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
594	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
595	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
596	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
597	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
598	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
599	C₈	—	2	2	no	—
600	C₈	—	4	1	no	—
601	C₈	—	4	2	no	—
602	C₈	—	4	2	no	—
603	C₈	—	4	1	no	—
604	C₈	—	4	1	no	—
605	C₂²	—	2	1	no	—
606	C₂²	—	2	1	no	—
607	C₂²	—	2	2	no	—
608	C₂²	—	2	2	no	—
609	C₂²	—	2	2	no	—
610	C₂²	—	2	2	no	—
611	C₂²	—	4	1	no	—
612	C₂²	—	4	1	no	—
613	C₂²	C₃₁×#407	1	2	yes	—
614	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
615	C₄	—	2	4	no	—
616	C₄	—	2	2	no	—
617	C₄	—	2	2	no	—
618	C₄	—	2	1	no	—
619	C₄	—	2	2	no	—
620	C₄	—	2	1	no	—
621	C₄	—	2	1	no	—
622	C₄	—	2	2	no	—
623	C₄	—	4	1	no	—
624	C₄	—	4	2	no	—
625	C₄	—	4	1	no	—
626	C₄	C₃₁×#407	1	2	yes	—
627	C₄	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
628	C₄	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
629	C₄	C₃₁×#577	1	1	yes	characteristic
630	C₄	C₃₁×#578	1	1	yes	characteristic
631	C₂	—	2	2	no	—
632	C₂	—	4	2	no	—
633	C₂	—	4	2	no	—
634	C₂	C₂×C₃₁×#407	1	1	yes	characteristic
635	C₂	C₃₁×#2898	1	1	yes	characteristic
636	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×C₃₁×2₋¹⁺⁴	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×#1491	C₂	semidirect
5. C₃₁×#1489	C₂	semidirect
6. C₃₁×#1488	C₂	semidirect
7. C₃₁×#1487	C₂	semidirect
8. C₃₁×#1479	C₂	semidirect
9. C₃₁×#1477	C₂	semidirect
10. C₃₁×#1476	C₂	semidirect
11. C₂×C₃₁×Q₈⋊C₈	C₂	semidirect
149. #41×C₃₁	M₁₆	semidirect
157. C₂×Q₈×C₃₁	M₁₆	semidirect
230. #17454	C₃₁	direct (N × H)
288. #2989	C₂×C₃₁	semidirect
289. #2965	C₂×C₃₁	semidirect
290. #1491	C₂×C₃₁	semidirect
291. #1489	C₂×C₃₁	semidirect
292. #1488	C₂×C₃₁	semidirect
293. #1487	C₂×C₃₁	semidirect
294. #1479	C₂×C₃₁	semidirect
295. #1477	C₂×C₃₁	semidirect
296. #1476	C₂×C₃₁	semidirect
297. C₂×Q₈⋊C₈	C₂×C₃₁	semidirect
460. C₃₁	#17454	direct (N × H)
468. #41	M₁₆×C₃₁	semidirect
476. C₂×Q₈	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×C₃₁×2₋¹⁺⁴	2
C₃₁×#2989	2
C₃₁×#2965	2
C₃₁×#1491	2
C₃₁×#1489	2
C₃₁×#1488	2
C₃₁×#1487	2
C₃₁×#1479	2
C₃₁×#1477	2
C₃₁×#1476	2
C₂×C₃₁×Q₈⋊C₈	2
#17454	31

Subgroups — C31×#17454 #21474213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17454 #21474213