Subgroups — C₃₁×#20654 #21477413

Order 7936.

1482 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#421	C₂	1	2	yes	maximal
4	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1806	C₂	1	8	yes	maximal, complemented
6	C₃₁×#1733	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₄×#41×C₃₁	—	2	1	no	—
10	C₄×#41×C₃₁	—	2	1	no	—
11	C₄×#41×C₃₁	—	2	2	no	—
12	C₄×#41×C₃₁	C₄	1	2	yes	complemented
13	C₄×#41×C₃₁	C₄	1	2	yes	complemented
14	C₄×#41×C₃₁	C₂²	1	4	yes	—
15	C₂×C₄×Q₈×C₃₁	C₄	1	2	yes	complemented
16	C₂×C₄×D₄×C₃₁	—	2	2	no	—
17	C₂×C₄×D₄×C₃₁	C₄	1	2	yes	complemented
18	C₂×C₃₁×#117	—	2	1	no	—
19	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#117	C₂²	1	2	yes	—
21	C₂²×C₄²×C₃₁	—	2	1	no	—
22	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#423	—	2	1	no	—
24	C₃₁×#423	—	2	1	no	—
25	C₃₁×#423	C₂²	1	4	yes	complemented
26	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
27	C₃₁×#421	—	2	8	no	—
28	C₂×C₃₁×#104	C₂²	1	8	yes	—
29	C₃₁×#384	—	2	1	no	—
30	C₃₁×#382	—	2	1	no	—
31	C₃₁×#380	—	2	2	no	—
32	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
33	C₃₁×#339	C₂²	1	8	yes	complemented
34	C₂×#41×C₃₁	—	2	1	no	—
35	C₂×#41×C₃₁	—	2	1	no	—
36	C₂×#41×C₃₁	—	2	2	no	—
37	C₂×#41×C₃₁	—	2	2	no	—
38	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
39	C₂×#41×C₃₁	D₄	1	4	yes	—
40	C₂×#41×C₃₁	Q₈	1	4	yes	—
41	C₂×#41×C₃₁	C₂³	1	2	yes	—
42	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×D₄×C₃₁	—	2	1	no	—
44	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₃₁	—	2	1	no	—
46	C₂³×C₄×C₃₁	—	2	2	no	—
47	C₂³×C₄×C₃₁	D₄	1	2	yes	—
48	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₃₁	—	2	2	no	—
50	C₂×M₁₆×C₃₁	—	2	4	no	—
51	C₂×M₁₆×C₃₁	—	2	4	no	—
52	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
53	C₂²×C₈×C₃₁	—	2	2	no	—
54	C₄×Q₈×C₃₁	—	2	2	no	—
55	C₄×Q₈×C₃₁	—	2	2	no	—
56	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
57	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
58	C₄×D₄×C₃₁	—	2	4	no	—
59	C₄×D₄×C₃₁	—	2	4	no	—
60	C₄×D₄×C₃₁	—	2	4	no	—
61	C₄×D₄×C₃₁	—	2	4	no	—
62	C₄×D₄×C₃₁	—	2	2	no	—
63	C₄×D₄×C₃₁	—	2	2	no	—
64	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
65	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
66	C₃₁×#117	—	2	1	no	—
67	C₃₁×#117	—	2	1	no	—
68	C₃₁×#117	—	2	4	no	—
69	C₃₁×#117	—	2	4	no	—
70	C₃₁×#117	—	2	2	no	—
71	C₃₁×#117	—	2	2	no	—
72	C₃₁×#117	—	2	4	no	—
73	C₃₁×#117	C₂×C₄	1	2	yes	—
74	C₃₁×#117	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
78	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
79	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
80	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
81	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
82	C₂×C₄²×C₃₁	—	2	2	no	—
83	C₂×C₄²×C₃₁	—	2	1	no	—
84	C₂×C₄²×C₃₁	—	2	2	no	—
85	C₂×C₄²×C₃₁	—	2	2	no	—
86	C₂×C₄²×C₃₁	—	2	4	no	—
87	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
88	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
89	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
91	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
92	C₂×C₄²×C₃₁	D₄	1	2	yes	—
93	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
94	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
95	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
96	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
97	C₃₁×C₄⋊C₈	—	2	4	no	—
98	C₃₁×C₄⋊C₈	—	2	4	no	—
99	C₃₁×#104	—	2	16	no	—
100	C₃₁×#95	—	2	4	no	—
101	C₂⁴×C₃₁	—	2	1	no	—
102	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
103	#41×C₃₁	—	2	4	no	—
104	#41×C₃₁	—	2	4	no	—
105	#41×C₃₁	—	2	4	no	—
106	#41×C₃₁	—	2	4	no	—
107	#41×C₃₁	—	2	4	no	—
108	#41×C₃₁	—	2	2	no	—
109	#41×C₃₁	—	2	2	no	—
110	#41×C₃₁	C₄⋊C₄	1	8	yes	complemented
111	#41×C₃₁	C₂×D₄	1	4	yes	—
112	#41×C₃₁	C₂×Q₈	1	4	yes	—
113	C₂×Q₈×C₃₁	—	2	1	no	—
114	C₂×Q₈×C₃₁	—	2	2	no	—
115	C₂×Q₈×C₃₁	C₄⋊C₄	1	4	yes	complemented
116	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
117	C₂×D₄×C₃₁	—	2	2	no	—
118	C₂×D₄×C₃₁	—	2	4	no	—
119	C₂×D₄×C₃₁	—	2	4	no	—
120	C₂×D₄×C₃₁	—	2	2	no	—
121	C₂×D₄×C₃₁	—	2	1	no	—
122	C₂×D₄×C₃₁	—	2	2	no	—
123	C₂×D₄×C₃₁	C₄⋊C₄	1	4	yes	complemented
124	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	—	2	2	no	—
127	C₂²×C₄×C₃₁	—	2	4	no	—
128	C₂²×C₄×C₃₁	—	2	4	no	—
129	C₂²×C₄×C₃₁	—	2	2	no	—
130	C₂²×C₄×C₃₁	—	2	2	no	—
131	C₂²×C₄×C₃₁	—	2	4	no	—
132	C₂²×C₄×C₃₁	—	2	4	no	—
133	C₂²×C₄×C₃₁	—	2	4	no	—
134	C₂²×C₄×C₃₁	—	4	4	no	—
135	C₂²×C₄×C₃₁	—	4	8	no	—
136	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
137	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
138	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
139	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
140	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
144	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
148	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
149	C₂²×C₄×C₃₁	#41	1	2	yes	—
150	C₂²×C₄×C₃₁	#41	1	2	yes	—
151	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
152	M₁₆×C₃₁	—	2	4	no	—
153	M₁₆×C₃₁	—	2	4	no	—
154	M₁₆×C₃₁	—	4	4	no	—
155	C₂×C₈×C₃₁	—	2	4	no	—
156	C₂×C₈×C₃₁	—	4	4	no	—
157	C₄⋊C₄×C₃₁	—	2	4	no	—
158	C₄⋊C₄×C₃₁	—	2	4	no	—
159	C₄⋊C₄×C₃₁	—	2	2	no	—
160	C₄⋊C₄×C₃₁	—	4	8	no	—
161	C₄⋊C₄×C₃₁	—	4	4	no	—
162	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
163	C₂²⋊C₄×C₃₁	—	2	2	no	—
164	C₂²⋊C₄×C₃₁	—	2	2	no	—
165	C₂²⋊C₄×C₃₁	—	2	4	no	—
166	C₂²⋊C₄×C₃₁	—	2	4	no	—
167	C₂²⋊C₄×C₃₁	—	4	4	no	—
168	C₄²×C₃₁	—	2	2	no	—
169	C₄²×C₃₁	—	2	4	no	—
170	C₄²×C₃₁	—	2	4	no	—
171	C₄²×C₃₁	—	2	2	no	—
172	C₄²×C₃₁	—	2	4	no	—
173	C₄²×C₃₁	—	2	4	no	—
174	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
175	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
176	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
177	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
178	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
179	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
180	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
181	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
182	C₄²×C₃₁	C₂×D₄	1	2	yes	—
183	C₄²×C₃₁	C₂×D₄	1	2	yes	—
184	#20654	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
185	C₂³×C₃₁	—	2	2	no	—
186	C₂³×C₃₁	—	2	4	no	—
187	C₂³×C₃₁	—	4	2	no	—
188	C₂³×C₃₁	—	4	4	no	—
189	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
190	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
191	C₂³×C₃₁	#121	1	2	yes	—
192	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
193	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
194	Q₈×C₃₁	—	2	2	no	—
195	Q₈×C₃₁	—	2	4	no	—
196	Q₈×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
197	D₄×C₃₁	—	2	4	no	—
198	D₄×C₃₁	—	2	4	no	—
199	D₄×C₃₁	—	2	8	no	—
200	D₄×C₃₁	—	2	2	no	—
201	D₄×C₃₁	—	2	4	no	—
202	D₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	2	4	no	—
205	C₂×C₄×C₃₁	—	2	4	no	—
206	C₂×C₄×C₃₁	—	2	4	no	—
207	C₂×C₄×C₃₁	—	2	8	no	—
208	C₂×C₄×C₃₁	—	2	4	no	—
209	C₂×C₄×C₃₁	—	2	4	no	—
210	C₂×C₄×C₃₁	—	2	4	no	—
211	C₂×C₄×C₃₁	—	2	4	no	—
212	C₂×C₄×C₃₁	—	4	2	no	—
213	C₂×C₄×C₃₁	—	4	2	no	—
214	C₂×C₄×C₃₁	—	4	2	no	—
215	C₂×C₄×C₃₁	—	4	8	no	—
216	C₂×C₄×C₃₁	—	4	4	no	—
217	C₂×C₄×C₃₁	—	4	8	no	—
218	C₂×C₄×C₃₁	—	4	8	no	—
219	C₂×C₄×C₃₁	—	4	4	no	—
220	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
221	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
222	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
223	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
224	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
225	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
226	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
227	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
228	C₂×C₄×C₃₁	#117	1	2	yes	—
229	C₂×C₄×C₃₁	#117	1	2	yes	—
230	C₂×C₄×C₃₁	#121	1	2	yes	—
231	C₂×C₄×C₃₁	#121	1	2	yes	—
232	C₂×C₄×C₃₁	#121	1	2	yes	—
233	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
234	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
235	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
236	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
239	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
240	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
241	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
242	C₈×C₃₁	—	4	4	no	—
243	C₂×C₄×#41	C₂×C₃₁	1	1	yes	characteristic
244	C₂×#423	C₂×C₃₁	1	1	yes	characteristic, complemented
245	C₂×#421	C₂×C₃₁	1	2	yes	—
246	#2285	C₂×C₃₁	1	1	yes	characteristic, complemented
247	#1806	C₂×C₃₁	1	8	yes	complemented
248	#1733	C₂×C₃₁	1	2	yes	complemented
249	C₂²×C₃₁	—	2	2	no	—
250	C₂²×C₃₁	—	2	2	no	—
251	C₂²×C₃₁	—	2	2	no	—
252	C₂²×C₃₁	—	2	4	no	—
253	C₂²×C₃₁	—	4	2	no	—
254	C₂²×C₃₁	—	4	2	no	—
255	C₂²×C₃₁	—	4	8	no	—
256	C₂²×C₃₁	—	4	4	no	—
257	C₂²×C₃₁	#380	1	2	yes	—
258	C₂²×C₃₁	#380	1	2	yes	—
259	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
260	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
261	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
262	C₄×C₃₁	—	2	2	no	—
263	C₄×C₃₁	—	2	2	no	—
264	C₄×C₃₁	—	2	2	no	—
265	C₄×C₃₁	—	2	4	no	—
266	C₄×C₃₁	—	4	4	no	—
267	C₄×C₃₁	—	8	4	no	—
268	C₄×C₃₁	#380	1	2	yes	—
269	C₄×C₃₁	#380	1	2	yes	—
270	C₄×C₃₁	#421	1	4	yes	complemented
271	C₄×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
272	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
273	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
274	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
275	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
276	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
277	C₄×#41	—	2	1	no	—
278	C₄×#41	—	2	1	no	—
279	C₄×#41	—	2	2	no	—
280	C₄×#41	C₄×C₃₁	1	2	yes	complemented
281	C₄×#41	C₄×C₃₁	1	2	yes	complemented
282	C₄×#41	C₂²×C₃₁	1	4	yes	—
283	C₂×C₄×Q₈	C₄×C₃₁	1	2	yes	complemented
284	C₂×C₄×D₄	—	2	2	no	—
285	C₂×C₄×D₄	C₄×C₃₁	1	2	yes	complemented
286	C₂×#117	—	2	1	no	—
287	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
288	C₂×#117	C₂²×C₃₁	1	2	yes	—
289	C₂²×C₄²	—	2	1	no	—
290	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
291	#423	—	2	1	no	—
292	#423	—	2	1	no	—
293	#423	C₂²×C₃₁	1	4	yes	complemented
294	C₂×C₄⋊C₈	—	2	2	no	—
295	#421	—	2	8	no	—
296	C₂×#104	C₂²×C₃₁	1	8	yes	—
297	#384	—	2	1	no	—
298	#382	—	2	1	no	—
299	#380	—	2	2	no	—
300	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
301	#339	C₂²×C₃₁	1	8	yes	complemented
302	C₂×C₃₁	—	2	1	no	—
303	C₂×C₃₁	—	2	2	no	—
304	C₂×C₃₁	—	2	1	no	—
305	C₂×C₃₁	—	2	2	no	—
306	C₂×C₃₁	—	4	4	no	—
307	C₂×C₃₁	C₂×#380	1	1	yes	characteristic
308	C₂×C₃₁	C₂×#421	1	2	yes	—
309	C₂×#41	—	2	1	no	—
310	C₂×#41	—	2	1	no	—
311	C₂×#41	—	2	2	no	—
312	C₂×#41	—	2	2	no	—
313	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
314	C₂×#41	D₄×C₃₁	1	4	yes	—
315	C₂×#41	Q₈×C₃₁	1	4	yes	—
316	C₂×#41	C₂³×C₃₁	1	2	yes	—
317	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
318	C₂²×D₄	—	2	1	no	—
319	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
320	C₂³×C₄	—	2	1	no	—
321	C₂³×C₄	—	2	2	no	—
322	C₂³×C₄	D₄×C₃₁	1	2	yes	—
323	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
324	C₂×M₁₆	—	2	2	no	—
325	C₂×M₁₆	—	2	4	no	—
326	C₂×M₁₆	—	2	4	no	—
327	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
328	C₂²×C₈	—	2	2	no	—
329	C₄×Q₈	—	2	2	no	—
330	C₄×Q₈	—	2	2	no	—
331	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
332	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
333	C₄×D₄	—	2	4	no	—
334	C₄×D₄	—	2	4	no	—
335	C₄×D₄	—	2	4	no	—
336	C₄×D₄	—	2	4	no	—
337	C₄×D₄	—	2	2	no	—
338	C₄×D₄	—	2	2	no	—
339	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
340	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
341	#117	—	2	1	no	—
342	#117	—	2	1	no	—
343	#117	—	2	4	no	—
344	#117	—	2	4	no	—
345	#117	—	2	2	no	—
346	#117	—	2	2	no	—
347	#117	—	2	4	no	—
348	#117	C₂×C₄×C₃₁	1	2	yes	—
349	#117	C₂³×C₃₁	1	2	yes	—
350	C₂×C₄⋊C₄	—	2	2	no	—
351	C₂×C₄⋊C₄	—	2	2	no	—
352	C₂×C₄⋊C₄	—	2	4	no	—
353	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
354	C₂×C₂²⋊C₄	—	2	1	no	—
355	C₂×C₂²⋊C₄	—	2	2	no	—
356	C₂×C₂²⋊C₄	—	2	2	no	—
357	C₂×C₄²	—	2	2	no	—
358	C₂×C₄²	—	2	1	no	—
359	C₂×C₄²	—	2	2	no	—
360	C₂×C₄²	—	2	2	no	—
361	C₂×C₄²	—	2	4	no	—
362	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
363	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
364	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
365	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
366	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
367	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
368	C₂×C₄²	D₄×C₃₁	1	2	yes	—
369	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
370	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
371	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
372	C₄⋊C₈	—	2	4	no	—
373	C₄⋊C₈	—	2	4	no	—
374	#104	—	2	16	no	—
375	#95	—	2	4	no	—
376	C₃₁	#20654	1	1	yes	characteristic, direct product factor, Sylow
377	C₂⁴	—	2	1	no	—
378	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
379	#41	—	2	4	no	—
380	#41	—	2	4	no	—
381	#41	—	2	4	no	—
382	#41	—	2	4	no	—
383	#41	—	2	4	no	—
384	#41	—	2	2	no	—
385	#41	—	2	2	no	—
386	#41	C₄⋊C₄×C₃₁	1	8	yes	complemented
387	#41	C₂×D₄×C₃₁	1	4	yes	—
388	#41	C₂×Q₈×C₃₁	1	4	yes	—
389	C₂×Q₈	—	2	1	no	—
390	C₂×Q₈	—	2	2	no	—
391	C₂×Q₈	C₄⋊C₄×C₃₁	1	4	yes	complemented
392	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
393	C₂×D₄	—	2	2	no	—
394	C₂×D₄	—	2	4	no	—
395	C₂×D₄	—	2	4	no	—
396	C₂×D₄	—	2	2	no	—
397	C₂×D₄	—	2	1	no	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	C₄⋊C₄×C₃₁	1	4	yes	complemented
400	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
401	C₂²×C₄	—	2	1	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	4	no	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	2	2	no	—
407	C₂²×C₄	—	2	4	no	—
408	C₂²×C₄	—	2	4	no	—
409	C₂²×C₄	—	2	4	no	—
410	C₂²×C₄	—	4	4	no	—
411	C₂²×C₄	—	4	8	no	—
412	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
413	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
414	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
415	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
416	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
417	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
418	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
419	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
420	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
421	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
422	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
423	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
424	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
425	C₂²×C₄	#41×C₃₁	1	2	yes	—
426	C₂²×C₄	#41×C₃₁	1	2	yes	—
427	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
428	M₁₆	—	2	4	no	—
429	M₁₆	—	2	4	no	—
430	M₁₆	—	4	4	no	—
431	C₂×C₈	—	2	4	no	—
432	C₂×C₈	—	4	4	no	—
433	C₄⋊C₄	—	2	4	no	—
434	C₄⋊C₄	—	2	4	no	—
435	C₄⋊C₄	—	2	2	no	—
436	C₄⋊C₄	—	4	8	no	—
437	C₄⋊C₄	—	4	4	no	—
438	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
439	C₂²⋊C₄	—	2	2	no	—
440	C₂²⋊C₄	—	2	2	no	—
441	C₂²⋊C₄	—	2	4	no	—
442	C₂²⋊C₄	—	2	4	no	—
443	C₂²⋊C₄	—	4	4	no	—
444	C₄²	—	2	2	no	—
445	C₄²	—	2	4	no	—
446	C₄²	—	2	4	no	—
447	C₄²	—	2	2	no	—
448	C₄²	—	2	4	no	—
449	C₄²	—	2	4	no	—
450	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
451	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
452	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
453	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
454	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
455	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
456	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
457	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
458	C₄²	C₂×D₄×C₃₁	1	2	yes	—
459	C₄²	C₂×D₄×C₃₁	1	2	yes	—
460	C₂³	—	2	2	no	—
461	C₂³	—	2	4	no	—
462	C₂³	—	4	2	no	—
463	C₂³	—	4	4	no	—
464	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
465	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
466	C₂³	C₃₁×#121	1	2	yes	—
467	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
468	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
469	Q₈	—	2	2	no	—
470	Q₈	—	2	4	no	—
471	Q₈	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
472	D₄	—	2	4	no	—
473	D₄	—	2	4	no	—
474	D₄	—	2	8	no	—
475	D₄	—	2	2	no	—
476	D₄	—	2	4	no	—
477	D₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	8	no	—
486	C₂×C₄	—	2	4	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	8	no	—
491	C₂×C₄	—	4	4	no	—
492	C₂×C₄	—	4	8	no	—
493	C₂×C₄	—	4	8	no	—
494	C₂×C₄	—	4	4	no	—
495	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
496	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
497	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
498	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
499	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
500	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
501	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
502	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
503	C₂×C₄	C₃₁×#117	1	2	yes	—
504	C₂×C₄	C₃₁×#117	1	2	yes	—
505	C₂×C₄	C₃₁×#121	1	2	yes	—
506	C₂×C₄	C₃₁×#121	1	2	yes	—
507	C₂×C₄	C₃₁×#121	1	2	yes	—
508	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
509	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
510	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
511	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
512	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
513	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
514	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
515	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
516	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
517	C₈	—	4	4	no	—
518	C₂²	—	2	2	no	—
519	C₂²	—	2	2	no	—
520	C₂²	—	2	2	no	—
521	C₂²	—	2	4	no	—
522	C₂²	—	4	2	no	—
523	C₂²	—	4	2	no	—
524	C₂²	—	4	8	no	—
525	C₂²	—	4	4	no	—
526	C₂²	C₃₁×#380	1	2	yes	—
527	C₂²	C₃₁×#380	1	2	yes	—
528	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
529	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
530	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
531	C₄	—	2	2	no	—
532	C₄	—	2	2	no	—
533	C₄	—	2	2	no	—
534	C₄	—	2	4	no	—
535	C₄	—	4	4	no	—
536	C₄	—	8	4	no	—
537	C₄	C₃₁×#380	1	2	yes	—
538	C₄	C₃₁×#380	1	2	yes	—
539	C₄	C₃₁×#421	1	4	yes	complemented
540	C₄	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
541	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
542	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
543	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
544	C₂	—	2	1	no	—
545	C₂	—	2	2	no	—
546	C₂	—	2	1	no	—
547	C₂	—	2	2	no	—
548	C₂	—	4	4	no	—
549	C₂	C₂×C₃₁×#380	1	1	yes	characteristic
550	C₂	C₂×C₃₁×#421	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#423	C₂	semidirect
4. C₃₁×#2285	C₂	semidirect
5. C₃₁×#1806	C₂	semidirect
6. C₃₁×#1733	C₂	semidirect
12. C₄×#41×C₃₁	C₄	semidirect
13. C₄×#41×C₃₁	C₄	semidirect
15. C₂×C₄×Q₈×C₃₁	C₄	semidirect
17. C₂×C₄×D₄×C₃₁	C₄	semidirect
25. C₃₁×#423	C₂²	semidirect
32. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
33. C₃₁×#339	C₂²	semidirect
110. #41×C₃₁	C₄⋊C₄	semidirect
115. C₂×Q₈×C₃₁	C₄⋊C₄	semidirect
123. C₂×D₄×C₃₁	C₄⋊C₄	semidirect
184. #20654	C₃₁	direct (N × H)
244. C₂×#423	C₂×C₃₁	semidirect
246. #2285	C₂×C₃₁	semidirect
247. #1806	C₂×C₃₁	semidirect
248. #1733	C₂×C₃₁	semidirect
270. C₄×C₃₁	#421	semidirect
280. C₄×#41	C₄×C₃₁	semidirect
281. C₄×#41	C₄×C₃₁	semidirect
283. C₂×C₄×Q₈	C₄×C₃₁	semidirect
285. C₂×C₄×D₄	C₄×C₃₁	semidirect
293. #423	C₂²×C₃₁	semidirect
300. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
301. #339	C₂²×C₃₁	semidirect
376. C₃₁	#20654	direct (N × H)
386. #41	C₄⋊C₄×C₃₁	semidirect
391. C₂×Q₈	C₄⋊C₄×C₃₁	semidirect
399. C₂×D₄	C₄⋊C₄×C₃₁	semidirect
539. C₄	C₃₁×#421	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₃₁	2
C₂×C₃₁×#423	2
C₂×C₃₁×#421	2
C₃₁×#2285	2
C₃₁×#1806	2
C₃₁×#1733	2
#20654	31

Subgroups — C31×#20654 #21477413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20654 #21477413