Subgroups — C₃₁×#60884 #21517643

Order 7936.

1622 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3509	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3504	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3242	C₂	1	8	yes	maximal, complemented
4	C₃₁×#3237	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3236	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3166	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#3162	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3139	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#491	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3097	C₂	1	4	yes	maximal
11	C₃₁×#3082	C₂	1	4	yes	maximal, complemented
12	C₃₁×#2923	C₂	1	1	yes	maximal, characteristic
13	C₃₁×#557	—	2	2	no	—
14	C₃₁×#557	—	2	2	no	—
15	C₃₁×#556	—	2	2	no	—
16	C₃₁×#554	—	2	4	no	—
17	C₃₁×#554	—	2	2	no	—
18	C₃₁×#552	—	2	4	no	—
19	C₃₁×#544	C₂²	1	2	yes	—
20	C₃₁×#544	C₂²	1	4	yes	—
21	C₃₁×#541	C₂²	1	4	yes	—
22	C₃₁×#536	C₂²	1	2	yes	—
23	C₃₁×#533	C₂²	1	4	yes	—
24	C₃₁×#533	C₂²	1	1	yes	characteristic
25	C₃₁×#533	C₂²	1	1	yes	characteristic
26	C₃₁×#533	C₂²	1	2	yes	—
27	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#125	C₂²	1	2	yes	—
30	C₃₁×#520	C₂²	1	2	yes	—
31	C₃₁×#519	C₂²	1	2	yes	—
32	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#501	C₂²	1	2	yes	—
34	C₃₁×#501	C₂²	1	2	yes	—
35	C₃₁×#501	C₂²	1	4	yes	—
36	C₃₁×#501	C₂²	1	4	yes	—
37	C₃₁×#499	C₂²	1	4	yes	—
38	C₃₁×#491	C₂²	1	4	yes	—
39	C₃₁×#491	C₂²	1	4	yes	—
40	C₃₁×#491	C₂²	1	2	yes	—
41	C₃₁×#491	C₂²	1	2	yes	—
42	C₃₁×#487	C₂²	1	4	yes	—
43	C₃₁×#485	C₂²	1	8	yes	—
44	C₃₁×#485	C₂²	1	8	yes	—
45	C₃₁×#484	C₂²	1	8	yes	—
46	C₃₁×#483	C₂²	1	8	yes	—
47	C₃₁×#479	C₂²	1	8	yes	—
48	C₃₁×#479	C₂²	1	8	yes	—
49	C₃₁×#477	C₂²	1	8	yes	—
50	C₃₁×#475	C₂²	1	8	yes	—
51	C₃₁×#433	C₂²	1	4	yes	—
52	C₃₁×#432	C₂²	1	4	yes	—
53	C₃₁×#428	C₂²	1	4	yes	—
54	C₃₁×#427	C₂²	1	4	yes	—
55	C₃₁×#424	C₂²	1	2	yes	—
56	C₃₁×#419	C₂²	1	4	yes	—
57	C₂×C₃₁×#103	C₂²	1	4	yes	—
58	C₃₁×#408	C₂²	1	2	yes	—
59	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
60	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
61	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
63	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
64	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
65	C₃₁×C₄⋊Q₈	—	2	2	no	—
66	C₃₁×C₄⋊Q₈	—	2	2	no	—
67	C₃₁×C₄⋊Q₈	—	2	4	no	—
68	C₃₁×C₄⋊Q₈	—	2	2	no	—
69	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
70	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
71	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
72	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
73	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
74	C₃₁×#126	—	2	8	no	—
75	C₃₁×#125	—	2	8	no	—
76	C₃₁×#125	—	2	2	no	—
77	C₃₁×#125	—	2	4	no	—
78	C₃₁×#125	—	2	4	no	—
79	C₃₁×#125	—	2	2	no	—
80	C₃₁×#125	—	2	4	no	—
81	C₃₁×#125	C₂³	1	4	yes	—
82	C₃₁×#125	C₂³	1	2	yes	—
83	C₃₁×#125	C₂³	1	2	yes	—
84	C₃₁×#125	C₂³	1	1	yes	characteristic
85	C₃₁×#125	C₂³	1	1	yes	characteristic
86	C₃₁×#125	C₂³	1	2	yes	—
87	C₃₁×#124	—	2	2	no	—
88	C₃₁×#123	—	2	4	no	—
89	C₃₁×C₂²⋊Q₈	—	2	2	no	—
90	C₃₁×C₂²⋊Q₈	—	2	4	no	—
91	C₃₁×C₂²⋊Q₈	—	2	2	no	—
92	C₃₁×C₂²⋊Q₈	—	2	4	no	—
93	C₃₁×C₂²⋊Q₈	—	2	4	no	—
94	C₃₁×C₂²⋊Q₈	—	2	4	no	—
95	C₃₁×C₂²⋊Q₈	—	2	4	no	—
96	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
97	C₄×Q₈×C₃₁	—	2	2	no	—
98	C₄×Q₈×C₃₁	—	2	4	no	—
99	C₄×Q₈×C₃₁	—	2	4	no	—
100	C₄×Q₈×C₃₁	—	2	4	no	—
101	C₄×Q₈×C₃₁	—	2	2	no	—
102	C₄×Q₈×C₃₁	—	2	2	no	—
103	C₄×Q₈×C₃₁	D₄	1	2	yes	—
104	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
105	C₄×D₄×C₃₁	—	2	4	no	—
106	C₄×D₄×C₃₁	—	2	2	no	—
107	C₄×D₄×C₃₁	D₄	1	4	yes	—
108	C₄×D₄×C₃₁	D₄	1	2	yes	—
109	C₃₁×#117	—	2	4	no	—
110	C₃₁×#117	—	2	4	no	—
111	C₃₁×#117	C₂³	1	2	yes	—
112	C₃₁×#117	C₂³	1	2	yes	—
113	C₃₁×#117	C₂³	1	4	yes	—
114	C₃₁×#117	C₂³	1	1	yes	characteristic
115	C₃₁×#117	C₂³	1	2	yes	—
116	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
117	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
118	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
120	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
121	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
123	C₃₁×#107	C₂³	1	4	yes	—
124	C₃₁×#107	C₂³	1	4	yes	—
125	C₃₁×#107	C₂³	1	8	yes	—
126	C₃₁×#106	C₂³	1	4	yes	—
127	C₃₁×#106	C₂³	1	4	yes	—
128	C₃₁×#106	C₂³	1	8	yes	—
129	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
130	C₃₁×#103	C₂³	1	8	yes	—
131	C₃₁×#103	C₂³	1	8	yes	—
132	C₃₁×#103	C₂³	1	8	yes	—
133	C₃₁×#103	C₂³	1	8	yes	—
134	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
135	C₃₁×#97	C₂³	1	2	yes	—
136	C₃₁×#97	C₂³	1	2	yes	—
137	C₃₁×#97	C₂³	1	2	yes	—
138	C₄×C₈×C₃₁	C₂³	1	2	yes	—
139	#41×C₃₁	—	2	4	no	—
140	C₂×Q₈×C₃₁	—	2	2	no	—
141	C₂×Q₈×C₃₁	—	2	2	no	—
142	C₂×Q₈×C₃₁	—	2	4	no	—
143	C₂×Q₈×C₃₁	—	2	2	no	—
144	C₂×Q₈×C₃₁	—	2	8	no	—
145	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
147	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
148	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	—	2	2	no	—
150	C₂²×C₄×C₃₁	—	2	4	no	—
151	C₂²×C₄×C₃₁	—	2	2	no	—
152	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
154	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
156	M₁₆×C₃₁	—	2	4	no	—
157	C₂×C₈×C₃₁	—	2	4	no	—
158	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
159	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
160	C₄⋊C₄×C₃₁	—	2	4	no	—
161	C₄⋊C₄×C₃₁	—	2	2	no	—
162	C₄⋊C₄×C₃₁	—	2	4	no	—
163	C₄⋊C₄×C₃₁	—	2	4	no	—
164	C₄⋊C₄×C₃₁	—	2	4	no	—
165	C₄⋊C₄×C₃₁	—	2	4	no	—
166	C₄⋊C₄×C₃₁	—	2	4	no	—
167	C₄⋊C₄×C₃₁	—	2	4	no	—
168	C₄⋊C₄×C₃₁	—	2	2	no	—
169	C₄⋊C₄×C₃₁	—	2	2	no	—
170	C₄⋊C₄×C₃₁	—	2	8	no	—
171	C₄⋊C₄×C₃₁	—	2	2	no	—
172	C₄⋊C₄×C₃₁	—	2	4	no	—
173	C₄⋊C₄×C₃₁	—	2	2	no	—
174	C₄⋊C₄×C₃₁	—	2	4	no	—
175	C₄⋊C₄×C₃₁	—	2	8	no	—
176	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
177	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
178	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
179	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
180	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
181	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
182	C₂²⋊C₄×C₃₁	—	2	2	no	—
183	C₂²⋊C₄×C₃₁	—	2	8	no	—
184	C₂²⋊C₄×C₃₁	—	2	4	no	—
185	C₂²⋊C₄×C₃₁	—	2	2	no	—
186	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
187	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
188	C₄²×C₃₁	—	2	8	no	—
189	C₄²×C₃₁	—	2	4	no	—
190	C₄²×C₃₁	—	2	2	no	—
191	C₄²×C₃₁	—	2	2	no	—
192	C₄²×C₃₁	C₂×D₄	1	4	yes	—
193	C₄²×C₃₁	C₂×D₄	1	2	yes	—
194	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
195	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
196	C₄²×C₃₁	C₂⁴	1	2	yes	—
197	#60884	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
199	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
200	Q₈×C₃₁	—	2	2	no	—
201	Q₈×C₃₁	—	2	8	no	—
202	Q₈×C₃₁	—	4	4	no	—
203	Q₈×C₃₁	—	4	4	no	—
204	Q₈×C₃₁	—	8	4	no	—
205	Q₈×C₃₁	—	8	2	no	—
206	D₄×C₃₁	—	2	4	no	—
207	D₄×C₃₁	—	2	2	no	—
208	C₂×C₄×C₃₁	—	2	4	no	—
209	C₂×C₄×C₃₁	—	2	4	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	2	4	no	—
212	C₂×C₄×C₃₁	—	2	4	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	4	2	no	—
215	C₂×C₄×C₃₁	—	4	4	no	—
216	C₂×C₄×C₃₁	—	4	4	no	—
217	C₂×C₄×C₃₁	—	8	2	no	—
218	C₂×C₄×C₃₁	—	8	4	no	—
219	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
220	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
221	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
222	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
223	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
225	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
226	C₈×C₃₁	—	2	4	no	—
227	C₈×C₃₁	—	2	4	no	—
228	#3509	C₂×C₃₁	1	1	yes	characteristic
229	#3504	C₂×C₃₁	1	1	yes	characteristic
230	#3242	C₂×C₃₁	1	8	yes	complemented
231	#3237	C₂×C₃₁	1	4	yes	complemented
232	#3236	C₂×C₃₁	1	4	yes	complemented
233	#3166	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#3162	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#3139	C₂×C₃₁	1	1	yes	characteristic, complemented
236	C₂×#491	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#3097	C₂×C₃₁	1	4	yes	—
238	#3082	C₂×C₃₁	1	4	yes	complemented
239	#2923	C₂×C₃₁	1	1	yes	characteristic
240	C₂²×C₃₁	—	2	2	no	—
241	C₂²×C₃₁	—	2	4	no	—
242	C₂²×C₃₁	—	2	2	no	—
243	C₂²×C₃₁	#536	1	2	yes	—
244	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
245	C₄×C₃₁	—	2	2	no	—
246	C₄×C₃₁	—	2	4	no	—
247	C₄×C₃₁	—	2	2	no	—
248	C₄×C₃₁	—	2	2	no	—
249	C₄×C₃₁	—	4	4	no	—
250	C₄×C₃₁	—	4	2	no	—
251	C₄×C₃₁	—	4	4	no	—
252	C₄×C₃₁	—	4	2	no	—
253	C₄×C₃₁	—	4	4	no	—
254	C₄×C₃₁	#536	1	2	yes	—
255	C₄×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
256	#557	—	2	2	no	—
257	#557	—	2	2	no	—
258	#556	—	2	2	no	—
259	#554	—	2	4	no	—
260	#554	—	2	2	no	—
261	#552	—	2	4	no	—
262	#544	C₂²×C₃₁	1	2	yes	—
263	#544	C₂²×C₃₁	1	4	yes	—
264	#541	C₂²×C₃₁	1	4	yes	—
265	#536	C₂²×C₃₁	1	2	yes	—
266	#533	C₂²×C₃₁	1	4	yes	—
267	#533	C₂²×C₃₁	1	1	yes	characteristic
268	#533	C₂²×C₃₁	1	1	yes	characteristic
269	#533	C₂²×C₃₁	1	2	yes	—
270	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×#125	C₂²×C₃₁	1	2	yes	—
273	#520	C₂²×C₃₁	1	2	yes	—
274	#519	C₂²×C₃₁	1	2	yes	—
275	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
276	#501	C₂²×C₃₁	1	2	yes	—
277	#501	C₂²×C₃₁	1	2	yes	—
278	#501	C₂²×C₃₁	1	4	yes	—
279	#501	C₂²×C₃₁	1	4	yes	—
280	#499	C₂²×C₃₁	1	4	yes	—
281	#491	C₂²×C₃₁	1	4	yes	—
282	#491	C₂²×C₃₁	1	4	yes	—
283	#491	C₂²×C₃₁	1	2	yes	—
284	#491	C₂²×C₃₁	1	2	yes	—
285	#487	C₂²×C₃₁	1	4	yes	—
286	#485	C₂²×C₃₁	1	8	yes	—
287	#485	C₂²×C₃₁	1	8	yes	—
288	#484	C₂²×C₃₁	1	8	yes	—
289	#483	C₂²×C₃₁	1	8	yes	—
290	#479	C₂²×C₃₁	1	8	yes	—
291	#479	C₂²×C₃₁	1	8	yes	—
292	#477	C₂²×C₃₁	1	8	yes	—
293	#475	C₂²×C₃₁	1	8	yes	—
294	#433	C₂²×C₃₁	1	4	yes	—
295	#432	C₂²×C₃₁	1	4	yes	—
296	#428	C₂²×C₃₁	1	4	yes	—
297	#427	C₂²×C₃₁	1	4	yes	—
298	#424	C₂²×C₃₁	1	2	yes	—
299	#419	C₂²×C₃₁	1	4	yes	—
300	C₂×#103	C₂²×C₃₁	1	4	yes	—
301	#408	C₂²×C₃₁	1	2	yes	—
302	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
303	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
304	C₂×C₃₁	—	2	2	no	—
305	C₂×C₃₁	—	4	2	no	—
306	C₂×C₃₁	C₂×#536	1	1	yes	characteristic
307	C₂×C₃₁	#3586	1	2	yes	—
308	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
309	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
310	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
311	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
312	C₄⋊Q₈	—	2	2	no	—
313	C₄⋊Q₈	—	2	2	no	—
314	C₄⋊Q₈	—	2	4	no	—
315	C₄⋊Q₈	—	2	2	no	—
316	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
317	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
318	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
319	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
320	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
321	#126	—	2	8	no	—
322	#125	—	2	8	no	—
323	#125	—	2	2	no	—
324	#125	—	2	4	no	—
325	#125	—	2	4	no	—
326	#125	—	2	2	no	—
327	#125	—	2	4	no	—
328	#125	C₂³×C₃₁	1	2	yes	—
329	#125	C₂³×C₃₁	1	1	yes	characteristic
330	#125	C₂³×C₃₁	1	4	yes	—
331	#125	C₂³×C₃₁	1	2	yes	—
332	#125	C₂³×C₃₁	1	1	yes	characteristic
333	#125	C₂³×C₃₁	1	2	yes	—
334	#124	—	2	2	no	—
335	#123	—	2	4	no	—
336	C₂²⋊Q₈	—	2	2	no	—
337	C₂²⋊Q₈	—	2	4	no	—
338	C₂²⋊Q₈	—	2	2	no	—
339	C₂²⋊Q₈	—	2	4	no	—
340	C₂²⋊Q₈	—	2	4	no	—
341	C₂²⋊Q₈	—	2	4	no	—
342	C₂²⋊Q₈	—	2	4	no	—
343	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
344	C₄×Q₈	—	2	2	no	—
345	C₄×Q₈	—	2	4	no	—
346	C₄×Q₈	—	2	4	no	—
347	C₄×Q₈	—	2	4	no	—
348	C₄×Q₈	—	2	2	no	—
349	C₄×Q₈	—	2	2	no	—
350	C₄×Q₈	D₄×C₃₁	1	2	yes	—
351	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
352	C₄×D₄	—	2	4	no	—
353	C₄×D₄	—	2	2	no	—
354	C₄×D₄	D₄×C₃₁	1	4	yes	—
355	C₄×D₄	D₄×C₃₁	1	2	yes	—
356	#117	—	2	4	no	—
357	#117	—	2	4	no	—
358	#117	C₂³×C₃₁	1	2	yes	—
359	#117	C₂³×C₃₁	1	2	yes	—
360	#117	C₂³×C₃₁	1	4	yes	—
361	#117	C₂³×C₃₁	1	1	yes	characteristic
362	#117	C₂³×C₃₁	1	2	yes	—
363	C₂×C₄⋊C₄	—	2	2	no	—
364	C₂×C₄⋊C₄	—	2	4	no	—
365	C₂×C₄⋊C₄	—	2	2	no	—
366	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
367	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
368	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
369	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
370	#107	C₂³×C₃₁	1	4	yes	—
371	#107	C₂³×C₃₁	1	4	yes	—
372	#107	C₂³×C₃₁	1	8	yes	—
373	#106	C₂³×C₃₁	1	4	yes	—
374	#106	C₂³×C₃₁	1	4	yes	—
375	#106	C₂³×C₃₁	1	8	yes	—
376	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
377	#103	C₂³×C₃₁	1	8	yes	—
378	#103	C₂³×C₃₁	1	8	yes	—
379	#103	C₂³×C₃₁	1	8	yes	—
380	#103	C₂³×C₃₁	1	8	yes	—
381	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
382	#97	C₂³×C₃₁	1	2	yes	—
383	#97	C₂³×C₃₁	1	2	yes	—
384	#97	C₂³×C₃₁	1	2	yes	—
385	C₄×C₈	C₂³×C₃₁	1	2	yes	—
386	C₃₁	#60884	1	1	yes	characteristic, direct product factor, Sylow
387	#41	—	2	4	no	—
388	C₂×Q₈	—	2	2	no	—
389	C₂×Q₈	—	2	2	no	—
390	C₂×Q₈	—	2	4	no	—
391	C₂×Q₈	—	2	2	no	—
392	C₂×Q₈	—	2	8	no	—
393	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
394	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
395	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
396	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	4	no	—
399	C₂²×C₄	—	2	2	no	—
400	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
401	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
402	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
403	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
404	M₁₆	—	2	4	no	—
405	C₂×C₈	—	2	4	no	—
406	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
407	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
408	C₄⋊C₄	—	2	4	no	—
409	C₄⋊C₄	—	2	2	no	—
410	C₄⋊C₄	—	2	4	no	—
411	C₄⋊C₄	—	2	4	no	—
412	C₄⋊C₄	—	2	4	no	—
413	C₄⋊C₄	—	2	4	no	—
414	C₄⋊C₄	—	2	4	no	—
415	C₄⋊C₄	—	2	4	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	8	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	—	2	4	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	2	4	no	—
423	C₄⋊C₄	—	2	8	no	—
424	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
425	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
426	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
427	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
428	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
429	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
430	C₂²⋊C₄	—	2	2	no	—
431	C₂²⋊C₄	—	2	8	no	—
432	C₂²⋊C₄	—	2	4	no	—
433	C₂²⋊C₄	—	2	2	no	—
434	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
435	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
436	C₄²	—	2	8	no	—
437	C₄²	—	2	4	no	—
438	C₄²	—	2	2	no	—
439	C₄²	—	2	2	no	—
440	C₄²	C₂×D₄×C₃₁	1	4	yes	—
441	C₄²	C₂×D₄×C₃₁	1	2	yes	—
442	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
443	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
444	C₄²	C₂⁴×C₃₁	1	2	yes	—
445	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
446	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
447	Q₈	—	2	2	no	—
448	Q₈	—	2	8	no	—
449	Q₈	—	4	4	no	—
450	Q₈	—	4	4	no	—
451	Q₈	—	8	4	no	—
452	Q₈	—	8	2	no	—
453	D₄	—	2	4	no	—
454	D₄	—	2	2	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	—	4	4	no	—
463	C₂×C₄	—	4	4	no	—
464	C₂×C₄	—	8	2	no	—
465	C₂×C₄	—	8	4	no	—
466	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
467	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
468	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
469	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
470	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
471	C₂×C₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
472	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
473	C₈	—	2	4	no	—
474	C₈	—	2	4	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	2	2	no	—
477	C₂²	—	2	4	no	—
478	C₂²	C₃₁×#536	1	2	yes	—
479	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
480	C₄	—	2	2	no	—
481	C₄	—	2	2	no	—
482	C₄	—	2	4	no	—
483	C₄	—	2	2	no	—
484	C₄	—	4	4	no	—
485	C₄	—	4	4	no	—
486	C₄	—	4	2	no	—
487	C₄	—	4	4	no	—
488	C₄	—	4	2	no	—
489	C₄	C₃₁×#536	1	2	yes	—
490	C₄	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
491	C₂	—	2	2	no	—
492	C₂	—	4	2	no	—
493	C₂	C₂×C₃₁×#536	1	1	yes	characteristic
494	C₂	C₃₁×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂×#41×C₃₁	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#3242	C₂	semidirect
4. C₃₁×#3237	C₂	semidirect
5. C₃₁×#3236	C₂	semidirect
6. C₃₁×#3166	C₂	semidirect
7. C₃₁×#3162	C₂	semidirect
8. C₃₁×#3139	C₂	semidirect
9. C₂×C₃₁×#491	C₂	semidirect
11. C₃₁×#3082	C₂	semidirect
197. #60884	C₃₁	direct (N × H)
230. #3242	C₂×C₃₁	semidirect
231. #3237	C₂×C₃₁	semidirect
232. #3236	C₂×C₃₁	semidirect
233. #3166	C₂×C₃₁	semidirect
234. #3162	C₂×C₃₁	semidirect
235. #3139	C₂×C₃₁	semidirect
236. C₂×#491	C₂×C₃₁	semidirect
238. #3082	C₂×C₃₁	semidirect
386. C₃₁	#60884	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×#3509	2
C₃₁×#3504	2
C₃₁×#3242	2
C₃₁×#3237	2
C₃₁×#3236	2
C₃₁×#3166	2
C₃₁×#3162	2
C₃₁×#3139	2
C₂×C₃₁×#491	2
C₃₁×#3097	2
C₃₁×#3082	2
C₃₁×#2923	2
#60884	31

Subgroups — C31×#60884 #21517643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60884 #21517643