Subgroups — C₃₁×#18615 #21475374

Order 7936.

930 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3506	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3237	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3182	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3101	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1650	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1642	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1638	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1631	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1553	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1490	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1485	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#559	—	2	2	no	—
13	C₃₁×#556	C₂²	1	2	yes	—
14	C₃₁×#555	—	2	1	no	—
15	C₃₁×#553	—	2	2	no	—
16	C₃₁×#550	C₂²	1	2	yes	—
17	C₃₁×#547	—	2	1	no	—
18	C₃₁×#538	—	2	2	no	—
19	C₃₁×#535	—	2	1	no	—
20	C₃₁×#533	C₂²	1	1	yes	characteristic
21	C₃₁×#533	C₂²	1	1	yes	characteristic
22	C₃₁×#532	C₂²	1	1	yes	characteristic
23	C₃₁×#525	—	2	2	no	—
24	C₃₁×#520	—	2	1	no	—
25	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#501	—	2	1	no	—
27	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
28	C₃₁×#492	C₂²	1	1	yes	characteristic, complemented
29	C₃₁×#491	—	2	1	no	—
30	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
31	C₃₁×#486	C₂²	1	1	yes	characteristic, complemented
32	C₃₁×#485	—	2	2	no	—
33	C₃₁×#483	—	2	1	no	—
34	C₃₁×#477	C₂²	1	2	yes	complemented
35	C₃₁×#476	C₂²	1	2	yes	—
36	C₃₁×#475	C₂²	1	2	yes	complemented
37	C₃₁×#468	—	2	2	no	—
38	C₃₁×#461	C₂²	1	2	yes	—
39	C₃₁×#450	—	2	1	no	—
40	C₃₁×#440	—	2	2	no	—
41	C₄×SD₁₆×C₃₁	—	2	2	no	—
42	C₃₁×#432	C₂²	1	1	yes	characteristic
43	C₃₁×#431	C₂²	1	1	yes	characteristic
44	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
45	C₃₁×#332	C₂²	1	2	yes	complemented
46	C₃₁×#331	C₂²	1	2	yes	complemented
47	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	complemented
48	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	complemented
49	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
50	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
51	C₂×#41×C₃₁	—	2	2	no	—
52	C₂×#41×C₃₁	—	2	1	no	—
53	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
54	C₂³×C₄×C₃₁	—	2	1	no	—
55	C₂×SD₁₆×C₃₁	—	2	2	no	—
56	C₂×SD₁₆×C₃₁	—	2	2	no	—
57	C₂×M₁₆×C₃₁	—	2	1	no	—
58	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
59	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
60	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
61	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
62	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
63	C₃₁×#127	—	2	2	no	—
64	C₃₁×#127	C₂³	1	1	yes	characteristic
65	C₃₁×#126	—	2	4	no	—
66	C₃₁×#125	—	2	1	no	—
67	C₃₁×#125	—	2	1	no	—
68	C₃₁×#124	—	2	4	no	—
69	C₃₁×#124	—	2	2	no	—
70	C₃₁×#124	—	2	1	no	—
71	C₃₁×#123	—	2	4	no	—
72	C₃₁×#123	—	2	4	no	—
73	C₃₁×C₂²⋊Q₈	—	2	2	no	—
74	C₃₁×C₂²⋊Q₈	—	2	1	no	—
75	C₃₁×C₂²⋊Q₈	—	2	1	no	—
76	C₃₁×C₂²⋊Q₈	—	2	1	no	—
77	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
78	C₃₁×#121	—	2	2	no	—
79	C₃₁×#121	—	2	2	no	—
80	C₃₁×#121	—	2	4	no	—
81	C₃₁×#121	—	2	1	no	—
82	C₃₁×#121	—	2	2	no	—
83	C₃₁×#121	D₄	1	2	yes	—
84	C₃₁×#120	—	2	4	no	—
85	C₄×Q₈×C₃₁	—	2	2	no	—
86	C₄×Q₈×C₃₁	—	2	1	no	—
87	C₄×Q₈×C₃₁	—	2	1	no	—
88	C₄×Q₈×C₃₁	—	2	1	no	—
89	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
90	C₄×D₄×C₃₁	—	2	4	no	—
91	C₄×D₄×C₃₁	—	2	4	no	—
92	C₄×D₄×C₃₁	—	2	2	no	—
93	C₄×D₄×C₃₁	—	2	1	no	—
94	C₄×D₄×C₃₁	—	2	1	no	—
95	C₄×D₄×C₃₁	—	2	1	no	—
96	C₄×D₄×C₃₁	D₄	1	2	yes	—
97	C₄×D₄×C₃₁	D₄	1	2	yes	—
98	C₄×D₄×C₃₁	C₂³	1	2	yes	—
99	C₃₁×#117	—	2	2	no	—
100	C₃₁×#117	—	2	1	no	—
101	C₃₁×#117	—	2	2	no	—
102	C₃₁×#117	—	2	1	no	—
103	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
105	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
106	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
107	C₃₁×#107	—	2	2	no	—
108	C₃₁×#107	—	2	2	no	—
109	C₃₁×#106	—	2	2	no	—
110	C₃₁×#106	—	2	2	no	—
111	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
112	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
113	C₃₁×#103	—	2	2	no	—
114	C₃₁×#103	—	2	2	no	—
115	C₃₁×#103	—	2	2	no	—
116	C₃₁×#103	—	2	2	no	—
117	C₃₁×#102	—	2	2	no	—
118	C₃₁×#102	—	2	2	no	—
119	C₃₁×C₂²⋊C₈	—	2	1	no	—
120	C₃₁×C₂²⋊C₈	—	2	1	no	—
121	C₃₁×#97	—	2	1	no	—
122	C₃₁×#97	—	2	1	no	—
123	C₄×C₈×C₃₁	—	2	1	no	—
124	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
125	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
126	C₂⁴×C₃₁	—	2	1	no	—
127	#41×C₃₁	—	4	4	no	—
128	#41×C₃₁	—	8	4	no	—
129	C₂×Q₈×C₃₁	—	2	1	no	—
130	C₂×Q₈×C₃₁	—	2	1	no	—
131	C₂×Q₈×C₃₁	—	2	1	no	—
132	C₂×Q₈×C₃₁	—	2	1	no	—
133	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
134	C₂×D₄×C₃₁	—	2	2	no	—
135	C₂×D₄×C₃₁	—	2	4	no	—
136	C₂×D₄×C₃₁	—	2	4	no	—
137	C₂×D₄×C₃₁	—	2	1	no	—
138	C₂×D₄×C₃₁	—	2	1	no	—
139	C₂×D₄×C₃₁	—	2	1	no	—
140	C₂×D₄×C₃₁	—	4	2	no	—
141	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
142	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₃₁	—	2	4	no	—
144	C₂²×C₄×C₃₁	—	2	1	no	—
145	C₂²×C₄×C₃₁	—	2	2	no	—
146	C₂²×C₄×C₃₁	—	2	2	no	—
147	C₂²×C₄×C₃₁	—	2	2	no	—
148	C₂²×C₄×C₃₁	—	8	2	no	—
149	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
152	SD₁₆×C₃₁	—	4	4	no	—
153	SD₁₆×C₃₁	—	4	4	no	—
154	M₁₆×C₃₁	—	4	2	no	—
155	C₂×C₈×C₃₁	—	2	1	no	—
156	C₂×C₈×C₃₁	—	2	2	no	—
157	C₂×C₈×C₃₁	—	2	1	no	—
158	C₂×C₈×C₃₁	—	2	2	no	—
159	C₄⋊C₄×C₃₁	—	2	4	no	—
160	C₄⋊C₄×C₃₁	—	2	1	no	—
161	C₄⋊C₄×C₃₁	—	2	1	no	—
162	C₄⋊C₄×C₃₁	—	2	1	no	—
163	C₄⋊C₄×C₃₁	—	2	2	no	—
164	C₄⋊C₄×C₃₁	—	2	1	no	—
165	C₄⋊C₄×C₃₁	—	2	1	no	—
166	C₄⋊C₄×C₃₁	—	2	2	no	—
167	C₄⋊C₄×C₃₁	—	2	1	no	—
168	C₄⋊C₄×C₃₁	—	2	1	no	—
169	C₄⋊C₄×C₃₁	—	2	1	no	—
170	C₄⋊C₄×C₃₁	—	2	1	no	—
171	C₄⋊C₄×C₃₁	—	4	1	no	—
172	C₄⋊C₄×C₃₁	—	4	1	no	—
173	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
174	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
175	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
176	C₂²⋊C₄×C₃₁	—	2	4	no	—
177	C₂²⋊C₄×C₃₁	—	2	4	no	—
178	C₂²⋊C₄×C₃₁	—	2	4	no	—
179	C₂²⋊C₄×C₃₁	—	2	2	no	—
180	C₂²⋊C₄×C₃₁	—	2	2	no	—
181	C₂²⋊C₄×C₃₁	—	2	2	no	—
182	C₂²⋊C₄×C₃₁	—	2	2	no	—
183	C₂²⋊C₄×C₃₁	—	2	2	no	—
184	C₂²⋊C₄×C₃₁	—	4	1	no	—
185	C₄²×C₃₁	—	2	4	no	—
186	C₄²×C₃₁	—	2	2	no	—
187	C₄²×C₃₁	—	4	1	no	—
188	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
189	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
190	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
191	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
192	#18615	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂³×C₃₁	—	2	2	no	—
194	C₂³×C₃₁	—	2	2	no	—
195	C₂³×C₃₁	—	4	2	no	—
196	C₂³×C₃₁	—	8	1	no	—
197	C₂³×C₃₁	—	8	1	no	—
198	C₂³×C₃₁	#120	1	1	yes	characteristic
199	Q₈×C₃₁	—	2	4	no	—
200	Q₈×C₃₁	—	4	2	no	—
201	Q₈×C₃₁	—	4	2	no	—
202	Q₈×C₃₁	—	4	2	no	—
203	Q₈×C₃₁	—	8	1	no	—
204	D₄×C₃₁	—	2	2	no	—
205	D₄×C₃₁	—	2	2	no	—
206	D₄×C₃₁	—	4	2	no	—
207	D₄×C₃₁	—	4	2	no	—
208	D₄×C₃₁	—	4	4	no	—
209	D₄×C₃₁	—	4	2	no	—
210	D₄×C₃₁	—	4	2	no	—
211	D₄×C₃₁	—	8	4	no	—
212	D₄×C₃₁	—	8	4	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	2	no	—
219	C₂×C₄×C₃₁	—	4	1	no	—
220	C₂×C₄×C₃₁	—	4	4	no	—
221	C₂×C₄×C₃₁	—	4	4	no	—
222	C₂×C₄×C₃₁	—	4	1	no	—
223	C₂×C₄×C₃₁	—	4	1	no	—
224	C₂×C₄×C₃₁	—	8	2	no	—
225	C₂×C₄×C₃₁	—	8	1	no	—
226	C₂×C₄×C₃₁	—	8	4	no	—
227	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
228	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
229	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
230	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
232	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
233	C₈×C₃₁	—	2	2	no	—
234	C₈×C₃₁	—	4	1	no	—
235	C₈×C₃₁	—	4	2	no	—
236	C₈×C₃₁	—	4	2	no	—
237	#3506	C₂×C₃₁	1	1	yes	characteristic
238	#3237	C₂×C₃₁	1	1	yes	characteristic, complemented
239	#3182	C₂×C₃₁	1	1	yes	characteristic, complemented
240	#3101	C₂×C₃₁	1	1	yes	characteristic, complemented
241	#1650	C₂×C₃₁	1	2	yes	complemented
242	#1642	C₂×C₃₁	1	2	yes	complemented
243	#1638	C₂×C₃₁	1	2	yes	complemented
244	#1631	C₂×C₃₁	1	2	yes	complemented
245	#1553	C₂×C₃₁	1	1	yes	characteristic, complemented
246	#1490	C₂×C₃₁	1	1	yes	characteristic, complemented
247	#1485	C₂×C₃₁	1	1	yes	characteristic, complemented
248	C₂²×C₃₁	—	2	1	no	—
249	C₂²×C₃₁	—	2	1	no	—
250	C₂²×C₃₁	—	2	1	no	—
251	C₂²×C₃₁	—	4	2	no	—
252	C₂²×C₃₁	—	4	2	no	—
253	C₂²×C₃₁	—	4	2	no	—
254	C₂²×C₃₁	—	4	2	no	—
255	C₂²×C₃₁	—	4	2	no	—
256	C₂²×C₃₁	—	4	2	no	—
257	C₂²×C₃₁	—	8	4	no	—
258	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
259	C₄×C₃₁	—	2	1	no	—
260	C₄×C₃₁	—	2	1	no	—
261	C₄×C₃₁	—	2	1	no	—
262	C₄×C₃₁	—	2	1	no	—
263	C₄×C₃₁	—	4	2	no	—
264	C₄×C₃₁	—	4	2	no	—
265	C₄×C₃₁	—	4	2	no	—
266	C₄×C₃₁	—	4	2	no	—
267	C₄×C₃₁	—	8	1	no	—
268	C₄×C₃₁	—	8	1	no	—
269	C₄×C₃₁	#575	1	2	yes	—
270	C₄×C₃₁	#577	1	2	yes	—
271	#559	—	2	2	no	—
272	#556	C₂²×C₃₁	1	2	yes	—
273	#555	—	2	1	no	—
274	#553	—	2	2	no	—
275	#550	C₂²×C₃₁	1	2	yes	—
276	#547	—	2	1	no	—
277	#538	—	2	2	no	—
278	#535	—	2	1	no	—
279	#533	C₂²×C₃₁	1	1	yes	characteristic
280	#533	C₂²×C₃₁	1	1	yes	characteristic
281	#532	C₂²×C₃₁	1	1	yes	characteristic
282	#525	—	2	2	no	—
283	#520	—	2	1	no	—
284	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
285	#501	—	2	1	no	—
286	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
287	#492	C₂²×C₃₁	1	1	yes	characteristic, complemented
288	#491	—	2	1	no	—
289	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
290	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
291	#485	—	2	2	no	—
292	#483	—	2	1	no	—
293	#477	C₂²×C₃₁	1	2	yes	complemented
294	#476	C₂²×C₃₁	1	2	yes	—
295	#475	C₂²×C₃₁	1	2	yes	complemented
296	#468	—	2	2	no	—
297	#461	C₂²×C₃₁	1	2	yes	—
298	#450	—	2	1	no	—
299	#440	—	2	2	no	—
300	C₄×SD₁₆	—	2	2	no	—
301	#432	C₂²×C₃₁	1	1	yes	characteristic
302	#431	C₂²×C₃₁	1	1	yes	characteristic
303	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
304	#332	C₂²×C₃₁	1	2	yes	complemented
305	#331	C₂²×C₃₁	1	2	yes	complemented
306	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	complemented
307	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	complemented
308	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
309	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
310	C₂×C₃₁	—	4	1	no	—
311	C₂×C₃₁	—	8	2	no	—
312	C₂×C₃₁	—	8	2	no	—
313	C₂×C₃₁	#3005	1	1	yes	characteristic
314	C₂×C₃₁	#3009	1	1	yes	characteristic
315	C₂×C₃₁	#3018	1	1	yes	characteristic
316	C₂×#41	—	2	2	no	—
317	C₂×#41	—	2	1	no	—
318	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
319	C₂³×C₄	—	2	1	no	—
320	C₂×SD₁₆	—	2	2	no	—
321	C₂×SD₁₆	—	2	2	no	—
322	C₂×M₁₆	—	2	1	no	—
323	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
324	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
325	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
326	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
327	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
328	#127	—	2	2	no	—
329	#127	C₂³×C₃₁	1	1	yes	characteristic
330	#126	—	2	4	no	—
331	#125	—	2	1	no	—
332	#125	—	2	1	no	—
333	#124	—	2	4	no	—
334	#124	—	2	2	no	—
335	#124	—	2	1	no	—
336	#123	—	2	4	no	—
337	#123	—	2	4	no	—
338	C₂²⋊Q₈	—	2	2	no	—
339	C₂²⋊Q₈	—	2	1	no	—
340	C₂²⋊Q₈	—	2	1	no	—
341	C₂²⋊Q₈	—	2	1	no	—
342	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
343	#121	—	2	2	no	—
344	#121	—	2	2	no	—
345	#121	—	2	4	no	—
346	#121	—	2	1	no	—
347	#121	—	2	2	no	—
348	#121	D₄×C₃₁	1	2	yes	—
349	#120	—	2	4	no	—
350	C₄×Q₈	—	2	2	no	—
351	C₄×Q₈	—	2	1	no	—
352	C₄×Q₈	—	2	1	no	—
353	C₄×Q₈	—	2	1	no	—
354	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
355	C₄×D₄	—	2	4	no	—
356	C₄×D₄	—	2	4	no	—
357	C₄×D₄	—	2	2	no	—
358	C₄×D₄	—	2	1	no	—
359	C₄×D₄	—	2	1	no	—
360	C₄×D₄	—	2	1	no	—
361	C₄×D₄	D₄×C₃₁	1	2	yes	—
362	C₄×D₄	D₄×C₃₁	1	2	yes	—
363	C₄×D₄	C₂³×C₃₁	1	2	yes	—
364	#117	—	2	2	no	—
365	#117	—	2	1	no	—
366	#117	—	2	2	no	—
367	#117	—	2	1	no	—
368	C₂×C₄⋊C₄	—	2	2	no	—
369	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
370	C₂×C₂²⋊C₄	—	2	1	no	—
371	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
372	#107	—	2	2	no	—
373	#107	—	2	2	no	—
374	#106	—	2	2	no	—
375	#106	—	2	2	no	—
376	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
377	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
378	#103	—	2	2	no	—
379	#103	—	2	2	no	—
380	#103	—	2	2	no	—
381	#103	—	2	2	no	—
382	#102	—	2	2	no	—
383	#102	—	2	2	no	—
384	C₂²⋊C₈	—	2	1	no	—
385	C₂²⋊C₈	—	2	1	no	—
386	#97	—	2	1	no	—
387	#97	—	2	1	no	—
388	C₄×C₈	—	2	1	no	—
389	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
390	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
391	C₃₁	#18615	1	1	yes	characteristic, direct product factor, Sylow
392	C₂⁴	—	2	1	no	—
393	#41	—	4	4	no	—
394	#41	—	8	4	no	—
395	C₂×Q₈	—	2	1	no	—
396	C₂×Q₈	—	2	1	no	—
397	C₂×Q₈	—	2	1	no	—
398	C₂×Q₈	—	2	1	no	—
399	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	2	4	no	—
403	C₂×D₄	—	2	1	no	—
404	C₂×D₄	—	2	1	no	—
405	C₂×D₄	—	2	1	no	—
406	C₂×D₄	—	4	2	no	—
407	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
408	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
409	C₂²×C₄	—	2	4	no	—
410	C₂²×C₄	—	2	1	no	—
411	C₂²×C₄	—	2	2	no	—
412	C₂²×C₄	—	2	2	no	—
413	C₂²×C₄	—	2	2	no	—
414	C₂²×C₄	—	8	2	no	—
415	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
416	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
417	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
418	SD₁₆	—	4	4	no	—
419	SD₁₆	—	4	4	no	—
420	M₁₆	—	4	2	no	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	2	1	no	—
424	C₂×C₈	—	2	2	no	—
425	C₄⋊C₄	—	2	4	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	2	2	no	—
430	C₄⋊C₄	—	2	1	no	—
431	C₄⋊C₄	—	2	1	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	1	no	—
434	C₄⋊C₄	—	2	1	no	—
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	2	1	no	—
437	C₄⋊C₄	—	4	1	no	—
438	C₄⋊C₄	—	4	1	no	—
439	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
440	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
441	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
442	C₂²⋊C₄	—	2	4	no	—
443	C₂²⋊C₄	—	2	4	no	—
444	C₂²⋊C₄	—	2	4	no	—
445	C₂²⋊C₄	—	2	2	no	—
446	C₂²⋊C₄	—	2	2	no	—
447	C₂²⋊C₄	—	2	2	no	—
448	C₂²⋊C₄	—	2	2	no	—
449	C₂²⋊C₄	—	2	2	no	—
450	C₂²⋊C₄	—	4	1	no	—
451	C₄²	—	2	4	no	—
452	C₄²	—	2	2	no	—
453	C₄²	—	4	1	no	—
454	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
455	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
456	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
457	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
458	C₂³	—	2	2	no	—
459	C₂³	—	2	2	no	—
460	C₂³	—	4	2	no	—
461	C₂³	—	8	1	no	—
462	C₂³	—	8	1	no	—
463	C₂³	C₃₁×#120	1	1	yes	characteristic
464	Q₈	—	2	4	no	—
465	Q₈	—	4	2	no	—
466	Q₈	—	4	2	no	—
467	Q₈	—	4	2	no	—
468	Q₈	—	8	1	no	—
469	D₄	—	2	2	no	—
470	D₄	—	2	2	no	—
471	D₄	—	4	2	no	—
472	D₄	—	4	2	no	—
473	D₄	—	4	4	no	—
474	D₄	—	4	2	no	—
475	D₄	—	4	2	no	—
476	D₄	—	8	4	no	—
477	D₄	—	8	4	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	4	1	no	—
485	C₂×C₄	—	4	4	no	—
486	C₂×C₄	—	4	4	no	—
487	C₂×C₄	—	4	1	no	—
488	C₂×C₄	—	4	1	no	—
489	C₂×C₄	—	8	2	no	—
490	C₂×C₄	—	8	1	no	—
491	C₂×C₄	—	8	4	no	—
492	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
493	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
494	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
495	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
496	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
497	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
498	C₈	—	2	2	no	—
499	C₈	—	4	1	no	—
500	C₈	—	4	2	no	—
501	C₈	—	4	2	no	—
502	C₂²	—	2	1	no	—
503	C₂²	—	2	1	no	—
504	C₂²	—	2	1	no	—
505	C₂²	—	4	2	no	—
506	C₂²	—	4	2	no	—
507	C₂²	—	4	2	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	4	2	no	—
510	C₂²	—	4	2	no	—
511	C₂²	—	8	4	no	—
512	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
513	C₄	—	2	1	no	—
514	C₄	—	2	1	no	—
515	C₄	—	2	1	no	—
516	C₄	—	2	1	no	—
517	C₄	—	4	2	no	—
518	C₄	—	4	2	no	—
519	C₄	—	4	2	no	—
520	C₄	—	4	2	no	—
521	C₄	—	8	1	no	—
522	C₄	—	8	1	no	—
523	C₄	C₃₁×#575	1	2	yes	—
524	C₄	C₃₁×#577	1	2	yes	—
525	C₂	—	4	1	no	—
526	C₂	—	8	2	no	—
527	C₂	—	8	2	no	—
528	C₂	C₃₁×#3005	1	1	yes	characteristic
529	C₂	C₃₁×#3009	1	1	yes	characteristic
530	C₂	C₃₁×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3506	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3237	C₂	semidirect
3. C₃₁×#3182	C₂	semidirect
4. C₃₁×#3101	C₂	semidirect
5. C₃₁×#1650	C₂	semidirect
6. C₃₁×#1642	C₂	semidirect
7. C₃₁×#1638	C₂	semidirect
8. C₃₁×#1631	C₂	semidirect
9. C₃₁×#1553	C₂	semidirect
10. C₃₁×#1490	C₂	semidirect
11. C₃₁×#1485	C₂	semidirect
27. C₃₁×#500	C₂²	semidirect
28. C₃₁×#492	C₂²	semidirect
30. C₃₁×#487	C₂²	semidirect
31. C₃₁×#486	C₂²	semidirect
34. C₃₁×#477	C₂²	semidirect
36. C₃₁×#475	C₂²	semidirect
45. C₃₁×#332	C₂²	semidirect
46. C₃₁×#331	C₂²	semidirect
47. C₃₁×Q₈⋊C₈	C₂²	semidirect
48. C₃₁×Q₈⋊C₈	C₂²	semidirect
192. #18615	C₃₁	direct (N × H)
238. #3237	C₂×C₃₁	semidirect
239. #3182	C₂×C₃₁	semidirect
240. #3101	C₂×C₃₁	semidirect
241. #1650	C₂×C₃₁	semidirect
242. #1642	C₂×C₃₁	semidirect
243. #1638	C₂×C₃₁	semidirect
244. #1631	C₂×C₃₁	semidirect
245. #1553	C₂×C₃₁	semidirect
246. #1490	C₂×C₃₁	semidirect
247. #1485	C₂×C₃₁	semidirect
286. #500	C₂²×C₃₁	semidirect
287. #492	C₂²×C₃₁	semidirect
289. #487	C₂²×C₃₁	semidirect
290. #486	C₂²×C₃₁	semidirect
293. #477	C₂²×C₃₁	semidirect
295. #475	C₂²×C₃₁	semidirect
304. #332	C₂²×C₃₁	semidirect
305. #331	C₂²×C₃₁	semidirect
306. Q₈⋊C₈	C₂²×C₃₁	semidirect
307. Q₈⋊C₈	C₂²×C₃₁	semidirect
391. C₃₁	#18615	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3506	2
C₃₁×#3237	2
C₃₁×#3182	2
C₃₁×#3101	2
C₃₁×#1650	2
C₃₁×#1642	2
C₃₁×#1638	2
C₃₁×#1631	2
C₃₁×#1553	2
C₃₁×#1490	2
C₃₁×#1485	2
#18615	31

Subgroups — C31×#18615 #21475374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18615 #21475374