Subgroups — C₃₁×#60619 #21517378

Order 7936.

2206 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3458	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#535	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3140	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3138	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3136	C₂	1	16	yes	maximal, complemented
6	C₃₁×#2896	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#418	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2873	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
10	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
12	C₃₁×#544	—	2	2	no	—
13	C₃₁×#544	C₂²	1	8	yes	complemented
14	C₃₁×#543	—	2	2	no	—
15	C₃₁×#535	—	2	4	no	—
16	C₃₁×#535	C₂²	1	8	yes	—
17	C₃₁×#533	—	2	4	no	—
18	C₃₁×#533	C₂²	1	4	yes	complemented
19	C₂×C₃₁×#127	C₂²	1	2	yes	—
20	C₂×C₃₁×#124	C₂²	1	2	yes	—
21	C₂×C₃₁×#121	—	2	2	no	—
22	C₂×C₃₁×#120	—	2	2	no	—
23	C₂×C₃₁×#117	C₂²	1	2	yes	—
24	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
25	C₃₁×#490	C₂²	1	16	yes	complemented
26	C₃₁×#489	C₂²	1	16	yes	complemented
27	C₃₁×#488	C₂²	1	16	yes	complemented
28	C₃₁×#486	C₂²	1	16	yes	complemented
29	C₃₁×#419	C₂²	1	4	yes	complemented
30	C₃₁×#418	C₂²	1	8	yes	—
31	C₃₁×#418	C₂²	1	8	yes	complemented
32	C₃₁×#418	C₂²	1	4	yes	complemented
33	C₃₁×#417	C₂²	1	8	yes	complemented
34	C₃₁×#416	C₂²	1	8	yes	complemented
35	C₂×C₃₁×#102	C₂²	1	8	yes	complemented
36	C₃₁×#405	C₂²	1	8	yes	complemented
37	C₄×M₁₆×C₃₁	C₂²	1	4	yes	complemented
38	C₂⁵×C₃₁	—	2	1	no	—
39	C₂×#41×C₃₁	—	2	2	no	—
40	C₂×#41×C₃₁	—	2	8	no	—
41	C₂×#41×C₃₁	C₂³	1	4	yes	—
42	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₃₁	—	2	1	no	—
44	C₂²×D₄×C₃₁	—	2	1	no	—
45	C₂²×D₄×C₃₁	—	2	2	no	—
46	C₂²×D₄×C₃₁	—	2	4	no	—
47	C₂²×D₄×C₃₁	—	2	8	no	—
48	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
49	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
50	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₃₁	—	2	1	no	—
52	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₃₁	—	2	8	no	—
54	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
55	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
56	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
57	C₃₁×C₄⋊Q₈	—	2	8	no	—
58	C₃₁×C₄⋊Q₈	C₂³	1	8	yes	complemented
59	C₃₁×#127	—	2	4	no	—
60	C₃₁×#127	C₂³	1	8	yes	—
61	C₃₁×#125	—	2	8	no	—
62	C₃₁×#125	C₂³	1	8	yes	complemented
63	C₃₁×#124	—	2	4	no	—
64	C₃₁×#124	C₂³	1	8	yes	—
65	C₃₁×C₂²⋊Q₈	—	2	8	no	—
66	C₃₁×C₂²⋊Q₈	—	2	16	no	—
67	C₃₁×#121	—	2	16	no	—
68	C₃₁×#120	—	2	16	no	—
69	C₄×Q₈×C₃₁	—	2	8	no	—
70	C₃₁×#117	—	2	2	no	—
71	C₃₁×#117	—	2	2	no	—
72	C₃₁×#117	D₄	1	4	yes	—
73	C₃₁×#117	C₂³	1	4	yes	—
74	C₃₁×#117	C₂³	1	4	yes	—
75	C₃₁×#117	C₂³	1	4	yes	—
76	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
77	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
78	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
80	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
81	C₂×C₄²×C₃₁	—	2	2	no	—
82	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
83	C₃₁×#103	C₂³	1	16	yes	complemented
84	C₃₁×#102	C₂³	1	32	yes	—
85	C₃₁×#102	C₂³	1	16	yes	complemented
86	C₃₁×#97	C₂³	1	8	yes	complemented
87	C₄×C₈×C₃₁	C₂³	1	8	yes	complemented
88	C₂⁴×C₃₁	—	2	2	no	—
89	C₂⁴×C₃₁	—	2	1	no	—
90	C₂⁴×C₃₁	—	2	4	no	—
91	C₂⁴×C₃₁	—	4	4	no	—
92	C₂⁴×C₃₁	—	8	4	no	—
93	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
94	#41×C₃₁	—	2	8	no	—
95	#41×C₃₁	—	2	8	no	—
96	#41×C₃₁	—	4	8	no	—
97	C₂×Q₈×C₃₁	—	2	1	no	—
98	C₂×Q₈×C₃₁	—	2	8	no	—
99	C₂×Q₈×C₃₁	—	2	4	no	—
100	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
101	C₂×D₄×C₃₁	—	2	4	no	—
102	C₂×D₄×C₃₁	—	2	4	no	—
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	—	2	1	no	—
105	C₂×D₄×C₃₁	—	2	16	no	—
106	C₂×D₄×C₃₁	—	2	8	no	—
107	C₂×D₄×C₃₁	—	2	8	no	—
108	C₂×D₄×C₃₁	—	2	8	no	—
109	C₂×D₄×C₃₁	—	2	4	no	—
110	C₂×D₄×C₃₁	—	4	4	no	—
111	C₂×D₄×C₃₁	—	4	4	no	—
112	C₂×D₄×C₃₁	—	4	8	no	—
113	C₂×D₄×C₃₁	—	8	8	no	—
114	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
115	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	2	4	no	—
118	C₂²×C₄×C₃₁	—	2	4	no	—
119	C₂²×C₄×C₃₁	—	4	4	no	—
120	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
121	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	#41	1	8	yes	—
124	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
125	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
126	M₁₆×C₃₁	—	2	8	no	—
127	M₁₆×C₃₁	—	2	8	no	—
128	C₂×C₈×C₃₁	—	2	8	no	—
129	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
130	C₄⋊C₄×C₃₁	—	2	4	no	—
131	C₄⋊C₄×C₃₁	—	2	8	no	—
132	C₄⋊C₄×C₃₁	—	2	16	no	—
133	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
134	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
135	C₂²⋊C₄×C₃₁	—	2	4	no	—
136	C₂²⋊C₄×C₃₁	—	2	4	no	—
137	C₂²⋊C₄×C₃₁	—	2	16	no	—
138	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
139	C₄²×C₃₁	—	2	4	no	—
140	C₄²×C₃₁	—	2	4	no	—
141	C₄²×C₃₁	C₂×D₄	1	4	yes	—
142	C₄²×C₃₁	C₂⁴	1	4	yes	—
143	#60619	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂³×C₃₁	—	2	4	no	—
145	C₂³×C₃₁	—	2	2	no	—
146	C₂³×C₃₁	—	4	2	no	—
147	C₂³×C₃₁	—	4	4	no	—
148	C₂³×C₃₁	—	4	8	no	—
149	C₂³×C₃₁	—	4	2	no	—
150	C₂³×C₃₁	—	4	4	no	—
151	C₂³×C₃₁	—	4	8	no	—
152	C₂³×C₃₁	—	8	8	no	—
153	C₂³×C₃₁	—	8	16	no	—
154	C₂³×C₃₁	#124	1	4	yes	—
155	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
156	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
157	Q₈×C₃₁	—	2	4	no	—
158	Q₈×C₃₁	—	4	2	no	—
159	Q₈×C₃₁	—	8	8	no	—
160	D₄×C₃₁	—	2	8	no	—
161	D₄×C₃₁	—	2	4	no	—
162	D₄×C₃₁	—	4	8	no	—
163	D₄×C₃₁	—	4	8	no	—
164	D₄×C₃₁	—	4	4	no	—
165	D₄×C₃₁	—	4	2	no	—
166	D₄×C₃₁	—	8	16	no	—
167	C₂×C₄×C₃₁	—	2	8	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	4	4	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	4	8	no	—
172	C₂×C₄×C₃₁	—	8	4	no	—
173	C₂×C₄×C₃₁	#124	1	4	yes	—
174	C₂×C₄×C₃₁	#124	1	8	yes	—
175	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
176	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
178	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
179	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
180	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
181	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
182	C₈×C₃₁	—	2	8	no	—
183	#3458	C₂×C₃₁	1	1	yes	characteristic, complemented
184	C₂×#535	C₂×C₃₁	1	1	yes	characteristic
185	#3140	C₂×C₃₁	1	4	yes	complemented
186	#3138	C₂×C₃₁	1	4	yes	complemented
187	#3136	C₂×C₃₁	1	16	yes	complemented
188	#2896	C₂×C₃₁	1	2	yes	complemented
189	C₂×#418	C₂×C₃₁	1	2	yes	complemented
190	#2873	C₂×C₃₁	1	1	yes	characteristic, complemented
191	C₂²×C₃₁	—	2	2	no	—
192	C₂²×C₃₁	—	2	4	no	—
193	C₂²×C₃₁	—	4	4	no	—
194	C₂²×C₃₁	—	4	4	no	—
195	C₂²×C₃₁	—	4	8	no	—
196	C₂²×C₃₁	—	4	2	no	—
197	C₂²×C₃₁	—	4	4	no	—
198	C₂²×C₃₁	—	8	4	no	—
199	C₂²×C₃₁	—	8	8	no	—
200	C₂²×C₃₁	—	8	16	no	—
201	C₂²×C₃₁	C₂×#124	1	2	yes	—
202	C₂²×C₃₁	C₂×#124	1	4	yes	—
203	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
204	C₄×C₃₁	—	2	8	no	—
205	C₄×C₃₁	—	4	8	no	—
206	C₄×C₃₁	—	4	2	no	—
207	C₄×C₃₁	C₂×#124	1	4	yes	—
208	C₄×C₃₁	C₂×#124	1	2	yes	—
209	C₄×C₃₁	C₂²×#41	1	2	yes	—
210	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
211	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
212	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
213	#544	—	2	2	no	—
214	#544	C₂²×C₃₁	1	8	yes	complemented
215	#543	—	2	2	no	—
216	#535	—	2	4	no	—
217	#535	C₂²×C₃₁	1	8	yes	—
218	#533	—	2	4	no	—
219	#533	C₂²×C₃₁	1	4	yes	complemented
220	C₂×#127	C₂²×C₃₁	1	2	yes	—
221	C₂×#124	C₂²×C₃₁	1	2	yes	—
222	C₂×#121	—	2	2	no	—
223	C₂×#120	—	2	2	no	—
224	C₂×#117	C₂²×C₃₁	1	2	yes	—
225	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
226	#490	C₂²×C₃₁	1	16	yes	complemented
227	#489	C₂²×C₃₁	1	16	yes	complemented
228	#488	C₂²×C₃₁	1	16	yes	complemented
229	#486	C₂²×C₃₁	1	16	yes	complemented
230	#419	C₂²×C₃₁	1	4	yes	complemented
231	#418	C₂²×C₃₁	1	8	yes	—
232	#418	C₂²×C₃₁	1	8	yes	complemented
233	#418	C₂²×C₃₁	1	4	yes	complemented
234	#417	C₂²×C₃₁	1	8	yes	complemented
235	#416	C₂²×C₃₁	1	8	yes	complemented
236	C₂×#102	C₂²×C₃₁	1	8	yes	complemented
237	#405	C₂²×C₃₁	1	8	yes	complemented
238	C₄×M₁₆	C₂²×C₃₁	1	4	yes	complemented
239	C₂×C₃₁	—	2	2	no	—
240	C₂×C₃₁	—	2	4	no	—
241	C₂×C₃₁	—	8	2	no	—
242	C₂×C₃₁	—	8	4	no	—
243	C₂×C₃₁	C₂²×#124	1	1	yes	characteristic
244	C₂×C₃₁	#3585	1	2	yes	—
245	C₂⁵	—	2	1	no	—
246	C₂×#41	—	2	2	no	—
247	C₂×#41	—	2	8	no	—
248	C₂×#41	C₂³×C₃₁	1	4	yes	—
249	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
250	C₂²×D₄	—	2	1	no	—
251	C₂²×D₄	—	2	1	no	—
252	C₂²×D₄	—	2	2	no	—
253	C₂²×D₄	—	2	4	no	—
254	C₂²×D₄	—	2	8	no	—
255	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
256	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
257	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
258	C₂³×C₄	—	2	1	no	—
259	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
260	C₂×M₁₆	—	2	8	no	—
261	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
262	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
263	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
264	C₄⋊Q₈	—	2	8	no	—
265	C₄⋊Q₈	C₂³×C₃₁	1	8	yes	complemented
266	#127	—	2	4	no	—
267	#127	C₂³×C₃₁	1	8	yes	—
268	#125	—	2	8	no	—
269	#125	C₂³×C₃₁	1	8	yes	complemented
270	#124	—	2	4	no	—
271	#124	C₂³×C₃₁	1	8	yes	—
272	C₂²⋊Q₈	—	2	8	no	—
273	C₂²⋊Q₈	—	2	16	no	—
274	#121	—	2	16	no	—
275	#120	—	2	16	no	—
276	C₄×Q₈	—	2	8	no	—
277	#117	—	2	2	no	—
278	#117	—	2	2	no	—
279	#117	D₄×C₃₁	1	4	yes	—
280	#117	C₂³×C₃₁	1	4	yes	—
281	#117	C₂³×C₃₁	1	4	yes	—
282	#117	C₂³×C₃₁	1	4	yes	—
283	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
284	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
285	C₂×C₂²⋊C₄	—	2	2	no	—
286	C₂×C₂²⋊C₄	—	2	4	no	—
287	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
288	C₂×C₄²	—	2	2	no	—
289	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
290	#103	C₂³×C₃₁	1	16	yes	complemented
291	#102	C₂³×C₃₁	1	32	yes	—
292	#102	C₂³×C₃₁	1	16	yes	complemented
293	#97	C₂³×C₃₁	1	8	yes	complemented
294	C₄×C₈	C₂³×C₃₁	1	8	yes	complemented
295	C₃₁	#60619	1	1	yes	characteristic, direct product factor, Sylow
296	C₂⁴	—	2	2	no	—
297	C₂⁴	—	2	1	no	—
298	C₂⁴	—	2	4	no	—
299	C₂⁴	—	4	4	no	—
300	C₂⁴	—	8	4	no	—
301	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
302	#41	—	2	8	no	—
303	#41	—	2	8	no	—
304	#41	—	4	8	no	—
305	C₂×Q₈	—	2	1	no	—
306	C₂×Q₈	—	2	8	no	—
307	C₂×Q₈	—	2	4	no	—
308	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
309	C₂×D₄	—	2	4	no	—
310	C₂×D₄	—	2	4	no	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	—	2	1	no	—
313	C₂×D₄	—	2	16	no	—
314	C₂×D₄	—	2	8	no	—
315	C₂×D₄	—	2	8	no	—
316	C₂×D₄	—	2	8	no	—
317	C₂×D₄	—	2	4	no	—
318	C₂×D₄	—	4	4	no	—
319	C₂×D₄	—	4	4	no	—
320	C₂×D₄	—	4	8	no	—
321	C₂×D₄	—	8	8	no	—
322	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
323	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	4	no	—
326	C₂²×C₄	—	2	4	no	—
327	C₂²×C₄	—	4	4	no	—
328	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
329	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
330	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
331	C₂²×C₄	#41×C₃₁	1	8	yes	—
332	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
333	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
334	M₁₆	—	2	8	no	—
335	M₁₆	—	2	8	no	—
336	C₂×C₈	—	2	8	no	—
337	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
338	C₄⋊C₄	—	2	4	no	—
339	C₄⋊C₄	—	2	8	no	—
340	C₄⋊C₄	—	2	16	no	—
341	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
342	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
343	C₂²⋊C₄	—	2	4	no	—
344	C₂²⋊C₄	—	2	4	no	—
345	C₂²⋊C₄	—	2	16	no	—
346	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
347	C₄²	—	2	4	no	—
348	C₄²	—	2	4	no	—
349	C₄²	C₂×D₄×C₃₁	1	4	yes	—
350	C₄²	C₂⁴×C₃₁	1	4	yes	—
351	C₂³	—	2	4	no	—
352	C₂³	—	2	2	no	—
353	C₂³	—	4	2	no	—
354	C₂³	—	4	4	no	—
355	C₂³	—	4	8	no	—
356	C₂³	—	4	2	no	—
357	C₂³	—	4	4	no	—
358	C₂³	—	4	8	no	—
359	C₂³	—	8	8	no	—
360	C₂³	—	8	16	no	—
361	C₂³	C₃₁×#124	1	4	yes	—
362	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
363	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
364	Q₈	—	2	4	no	—
365	Q₈	—	4	2	no	—
366	Q₈	—	8	8	no	—
367	D₄	—	2	8	no	—
368	D₄	—	2	4	no	—
369	D₄	—	4	8	no	—
370	D₄	—	4	8	no	—
371	D₄	—	4	4	no	—
372	D₄	—	4	2	no	—
373	D₄	—	8	16	no	—
374	C₂×C₄	—	2	8	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	4	4	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	8	no	—
379	C₂×C₄	—	8	4	no	—
380	C₂×C₄	C₃₁×#124	1	4	yes	—
381	C₂×C₄	C₃₁×#124	1	8	yes	—
382	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
383	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
384	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
385	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
386	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
387	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
388	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
389	C₈	—	2	8	no	—
390	C₂²	—	2	2	no	—
391	C₂²	—	2	4	no	—
392	C₂²	—	4	4	no	—
393	C₂²	—	4	2	no	—
394	C₂²	—	4	4	no	—
395	C₂²	—	4	4	no	—
396	C₂²	—	4	8	no	—
397	C₂²	—	8	8	no	—
398	C₂²	—	8	4	no	—
399	C₂²	—	8	16	no	—
400	C₂²	C₂×C₃₁×#124	1	2	yes	—
401	C₂²	C₂×C₃₁×#124	1	4	yes	—
402	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
403	C₄	—	2	8	no	—
404	C₄	—	4	2	no	—
405	C₄	—	4	8	no	—
406	C₄	C₂×C₃₁×#124	1	4	yes	—
407	C₄	C₂×C₃₁×#124	1	2	yes	—
408	C₄	C₂²×#41×C₃₁	1	2	yes	—
409	C₂	—	2	2	no	—
410	C₂	—	2	4	no	—
411	C₂	—	8	4	no	—
412	C₂	—	8	2	no	—
413	C₂	C₂²×C₃₁×#124	1	1	yes	characteristic
414	C₂	C₃₁×#3585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#535	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3458	C₂	semidirect
3. C₃₁×#3140	C₂	semidirect
4. C₃₁×#3138	C₂	semidirect
5. C₃₁×#3136	C₂	semidirect
6. C₃₁×#2896	C₂	semidirect
7. C₂×C₃₁×#418	C₂	semidirect
8. C₃₁×#2873	C₂	semidirect
13. C₃₁×#544	C₂²	semidirect
18. C₃₁×#533	C₂²	semidirect
25. C₃₁×#490	C₂²	semidirect
26. C₃₁×#489	C₂²	semidirect
27. C₃₁×#488	C₂²	semidirect
28. C₃₁×#486	C₂²	semidirect
29. C₃₁×#419	C₂²	semidirect
31. C₃₁×#418	C₂²	semidirect
32. C₃₁×#418	C₂²	semidirect
33. C₃₁×#417	C₂²	semidirect
34. C₃₁×#416	C₂²	semidirect
35. C₂×C₃₁×#102	C₂²	semidirect
36. C₃₁×#405	C₂²	semidirect
37. C₄×M₁₆×C₃₁	C₂²	semidirect
58. C₃₁×C₄⋊Q₈	C₂³	semidirect
62. C₃₁×#125	C₂³	semidirect
83. C₃₁×#103	C₂³	semidirect
85. C₃₁×#102	C₂³	semidirect
86. C₃₁×#97	C₂³	semidirect
87. C₄×C₈×C₃₁	C₂³	semidirect
143. #60619	C₃₁	direct (N × H)
183. #3458	C₂×C₃₁	semidirect
185. #3140	C₂×C₃₁	semidirect
186. #3138	C₂×C₃₁	semidirect
187. #3136	C₂×C₃₁	semidirect
188. #2896	C₂×C₃₁	semidirect
189. C₂×#418	C₂×C₃₁	semidirect
190. #2873	C₂×C₃₁	semidirect
214. #544	C₂²×C₃₁	semidirect
219. #533	C₂²×C₃₁	semidirect
226. #490	C₂²×C₃₁	semidirect
227. #489	C₂²×C₃₁	semidirect
228. #488	C₂²×C₃₁	semidirect
229. #486	C₂²×C₃₁	semidirect
230. #419	C₂²×C₃₁	semidirect
232. #418	C₂²×C₃₁	semidirect
233. #418	C₂²×C₃₁	semidirect
234. #417	C₂²×C₃₁	semidirect
235. #416	C₂²×C₃₁	semidirect
236. C₂×#102	C₂²×C₃₁	semidirect
237. #405	C₂²×C₃₁	semidirect
238. C₄×M₁₆	C₂²×C₃₁	semidirect
265. C₄⋊Q₈	C₂³×C₃₁	semidirect
269. #125	C₂³×C₃₁	semidirect
290. #103	C₂³×C₃₁	semidirect
292. #102	C₂³×C₃₁	semidirect
293. #97	C₂³×C₃₁	semidirect
294. C₄×C₈	C₂³×C₃₁	semidirect
295. C₃₁	#60619	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×#3458	2
C₂×C₃₁×#535	2
C₃₁×#3140	2
C₃₁×#3138	2
C₃₁×#3136	2
C₃₁×#2896	2
C₂×C₃₁×#418	2
C₃₁×#2873	2
#60619	31

Subgroups — C31×#60619 #21517378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60619 #21517378