Subgroups — C₃₁×#61413 #21518172

Order 7936.

1886 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3510	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3401	C₂	1	8	yes	maximal, complemented
3	C₃₁×#3396	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3393	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3156	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3155	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃₁×#495	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3043	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3042	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3040	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2960	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#2950	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#2948	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₃₁×#135	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₃₁×#137	C₂²	1	2	yes	—
16	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
17	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
18	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
20	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#559	C₂²	1	8	yes	complemented
22	C₃₁×#556	C₂²	1	8	yes	complemented
23	C₃₁×#551	—	2	8	no	—
24	C₃₁×#542	—	2	4	no	—
25	C₃₁×#536	C₂²	1	2	yes	—
26	C₃₁×#535	C₂²	1	2	yes	—
27	C₃₁×#533	C₂²	1	1	yes	characteristic
28	C₃₁×#529	—	2	4	no	—
29	C₂×C₃₁×#124	C₂²	1	2	yes	—
30	C₃₁×#519	C₂²	1	2	yes	—
31	C₃₁×#518	C₂²	1	2	yes	—
32	C₄×#41×C₃₁	C₂²	1	2	yes	—
33	C₃₁×#497	C₂²	1	4	yes	complemented
34	C₃₁×#496	C₂²	1	4	yes	—
35	C₃₁×#495	C₂²	1	2	yes	—
36	C₃₁×#495	C₂²	1	4	yes	—
37	C₃₁×#495	C₂²	1	4	yes	—
38	C₃₁×#495	C₂²	1	2	yes	complemented
39	C₃₁×#494	C₂²	1	2	yes	complemented
40	C₃₁×#493	C₂²	1	2	yes	—
41	C₃₁×#464	C₂²	1	8	yes	complemented
42	C₃₁×#464	C₂²	1	8	yes	complemented
43	C₃₁×#463	C₂²	1	8	yes	complemented
44	C₃₁×#463	C₂²	1	8	yes	complemented
45	C₃₁×#461	C₂²	1	8	yes	complemented
46	C₃₁×#460	C₂²	1	8	yes	complemented
47	C₃₁×#446	C₂²	1	4	yes	complemented
48	C₈×Q₈×C₃₁	C₂²	1	4	yes	complemented
49	C₃₁×#442	C₂²	1	4	yes	—
50	C₃₁×#441	C₂²	1	4	yes	complemented
51	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	complemented
52	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	complemented
53	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
54	C₄×D₈×C₃₁	C₂²	1	4	yes	—
55	C₄×D₈×C₃₁	C₂²	1	2	yes	complemented
56	C₃₁×#427	C₂²	1	1	yes	characteristic
57	C₃₁×#424	C₂²	1	1	yes	characteristic
58	C₃₁×#424	C₂²	1	1	yes	characteristic
59	C₃₁×#423	C₂²	1	1	yes	characteristic
60	C₃₁×#419	C₂²	1	2	yes	—
61	C₃₁×#418	C₂²	1	2	yes	—
62	C₃₁×#416	C₂²	1	2	yes	—
63	C₃₁×#405	C₂²	1	1	yes	characteristic
64	C₃₁×#405	C₂²	1	1	yes	characteristic
65	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
66	C₂×#41×C₃₁	C₂³	1	2	yes	—
67	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
68	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
69	C₃₁×#137	—	4	4	no	—
70	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
71	C₃₁×#135	—	2	4	no	—
72	C₂×Q₁₆×C₃₁	—	2	4	no	—
73	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
75	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
76	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
77	C₂×SD₁₆×C₃₁	—	4	4	no	—
78	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
79	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
80	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
81	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
82	C₂×D₈×C₃₁	—	2	4	no	—
83	C₂×D₈×C₃₁	C₂³	1	2	yes	—
84	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×D₈×C₃₁	C₂³	1	2	yes	—
86	C₂×D₈×C₃₁	C₂³	1	4	yes	—
87	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
88	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
89	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
90	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
91	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
92	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
93	C₃₁×#127	C₂³	1	4	yes	—
94	C₃₁×#126	—	2	8	no	—
95	C₃₁×#126	—	2	8	no	—
96	C₃₁×#125	D₄	1	2	yes	—
97	C₃₁×#124	—	2	8	no	—
98	C₃₁×#124	—	2	2	no	—
99	C₃₁×#124	—	2	2	no	—
100	C₃₁×#124	C₂³	1	4	yes	—
101	C₃₁×#124	C₂³	1	8	yes	—
102	C₃₁×#124	C₂³	1	4	yes	—
103	C₃₁×#123	—	2	8	no	—
104	C₃₁×#123	—	2	8	no	—
105	C₃₁×#123	—	2	4	no	—
106	C₃₁×C₂²⋊Q₈	—	2	8	no	—
107	C₃₁×C₂²⋊Q₈	—	2	4	no	—
108	C₃₁×C₂²⋊Q₈	D₄	1	4	yes	—
109	C₃₁×#121	—	2	4	no	—
110	C₃₁×#120	—	2	4	no	—
111	C₄×Q₈×C₃₁	—	2	4	no	—
112	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
113	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
114	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
115	C₄×D₄×C₃₁	—	2	4	no	—
116	C₄×D₄×C₃₁	—	2	4	no	—
117	C₄×D₄×C₃₁	—	2	8	no	—
118	C₄×D₄×C₃₁	C₂³	1	8	yes	—
119	C₄×D₄×C₃₁	C₂³	1	4	yes	—
120	C₃₁×#117	—	2	8	no	—
121	C₃₁×#117	—	2	4	no	—
122	C₃₁×#117	—	2	2	no	—
123	C₃₁×#117	C₂³	1	2	yes	—
124	C₃₁×#117	C₂³	1	1	yes	characteristic
125	C₃₁×#117	C₂³	1	1	yes	characteristic
126	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
127	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
128	C₂×C₄²×C₃₁	—	2	2	no	—
129	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
130	C₃₁×#107	C₂³	1	2	yes	—
131	C₃₁×#106	C₂³	1	2	yes	—
132	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
133	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
134	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
135	C₃₁×#103	C₂³	1	4	yes	—
136	C₃₁×#103	C₂³	1	8	yes	—
137	C₃₁×#102	C₂³	1	4	yes	—
138	C₃₁×#102	C₂³	1	8	yes	—
139	C₃₁×#97	C₂³	1	2	yes	—
140	C₃₁×#97	C₂³	1	2	yes	—
141	C₄×C₈×C₃₁	C₂³	1	2	yes	—
142	C₄×C₈×C₃₁	C₂³	1	2	yes	—
143	C₄×C₈×C₃₁	C₂³	1	2	yes	—
144	C₄×C₈×C₃₁	C₂³	1	2	yes	—
145	C₂⁴×C₃₁	—	2	2	no	—
146	#41×C₃₁	—	2	4	no	—
147	#41×C₃₁	—	2	4	no	—
148	C₂×Q₈×C₃₁	—	2	2	no	—
149	C₂×Q₈×C₃₁	—	2	4	no	—
150	C₂×Q₈×C₃₁	—	2	4	no	—
151	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
152	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
153	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
154	C₂×D₄×C₃₁	—	2	2	no	—
155	C₂×D₄×C₃₁	—	2	2	no	—
156	C₂×D₄×C₃₁	—	2	8	no	—
157	C₂×D₄×C₃₁	—	2	4	no	—
158	C₂×D₄×C₃₁	—	2	4	no	—
159	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
160	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₃₁	—	2	2	no	—
162	C₂²×C₄×C₃₁	—	2	2	no	—
163	C₂²×C₄×C₃₁	—	2	4	no	—
164	C₂²×C₄×C₃₁	—	2	8	no	—
165	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
167	Q₁₆×C₃₁	—	2	4	no	—
168	Q₁₆×C₃₁	—	2	4	no	—
169	Q₁₆×C₃₁	—	2	2	no	—
170	Q₁₆×C₃₁	—	4	4	no	—
171	SD₁₆×C₃₁	—	2	4	no	—
172	SD₁₆×C₃₁	—	4	8	no	—
173	SD₁₆×C₃₁	—	4	4	no	—
174	SD₁₆×C₃₁	—	4	4	no	—
175	D₈×C₃₁	—	2	4	no	—
176	D₈×C₃₁	—	2	4	no	—
177	D₈×C₃₁	—	2	2	no	—
178	D₈×C₃₁	—	4	4	no	—
179	M₁₆×C₃₁	—	2	2	no	—
180	M₁₆×C₃₁	—	2	2	no	—
181	C₂×C₈×C₃₁	—	2	2	no	—
182	C₂×C₈×C₃₁	#41	1	4	yes	—
183	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
184	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
185	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
186	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
187	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
188	C₄⋊C₄×C₃₁	—	2	4	no	—
189	C₄⋊C₄×C₃₁	—	2	8	no	—
190	C₄⋊C₄×C₃₁	—	2	8	no	—
191	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
192	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
193	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
194	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
195	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
196	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
197	C₂²⋊C₄×C₃₁	—	2	2	no	—
198	C₂²⋊C₄×C₃₁	—	2	8	no	—
199	C₂²⋊C₄×C₃₁	—	2	8	no	—
200	C₂²⋊C₄×C₃₁	—	2	8	no	—
201	C₂²⋊C₄×C₃₁	—	2	4	no	—
202	C₂²⋊C₄×C₃₁	—	2	8	no	—
203	C₂²⋊C₄×C₃₁	—	2	8	no	—
204	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
205	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
206	C₄²×C₃₁	—	2	4	no	—
207	C₄²×C₃₁	—	2	2	no	—
208	C₄²×C₃₁	—	2	8	no	—
209	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
210	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
211	C₄²×C₃₁	C₂⁴	1	2	yes	—
212	C₄²×C₃₁	C₂⁴	1	2	yes	—
213	C₄²×C₃₁	C₂⁴	1	2	yes	—
214	#61413	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
215	C₂³×C₃₁	—	2	4	no	—
216	C₂³×C₃₁	—	2	2	no	—
217	C₂³×C₃₁	—	4	4	no	—
218	C₂³×C₃₁	—	8	4	no	—
219	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
220	Q₈×C₃₁	—	2	2	no	—
221	Q₈×C₃₁	—	2	4	no	—
222	Q₈×C₃₁	—	2	4	no	—
223	Q₈×C₃₁	—	2	2	no	—
224	Q₈×C₃₁	—	4	4	no	—
225	Q₈×C₃₁	—	4	4	no	—
226	D₄×C₃₁	—	2	4	no	—
227	D₄×C₃₁	—	2	2	no	—
228	D₄×C₃₁	—	2	2	no	—
229	D₄×C₃₁	—	2	4	no	—
230	D₄×C₃₁	—	4	4	no	—
231	D₄×C₃₁	—	4	4	no	—
232	D₄×C₃₁	—	8	8	no	—
233	C₂×C₄×C₃₁	—	2	4	no	—
234	C₂×C₄×C₃₁	—	2	4	no	—
235	C₂×C₄×C₃₁	—	2	2	no	—
236	C₂×C₄×C₃₁	—	4	2	no	—
237	C₂×C₄×C₃₁	—	4	4	no	—
238	C₂×C₄×C₃₁	—	4	4	no	—
239	C₂×C₄×C₃₁	—	8	4	no	—
240	C₂×C₄×C₃₁	—	8	8	no	—
241	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
242	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
243	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
244	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
245	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
247	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
248	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
249	C₈×C₃₁	—	2	2	no	—
250	C₈×C₃₁	—	2	2	no	—
251	C₈×C₃₁	—	2	2	no	—
252	C₈×C₃₁	C₂×#41	1	2	yes	—
253	C₈×C₃₁	C₂×#41	1	2	yes	—
254	#3510	C₂×C₃₁	1	2	yes	complemented
255	#3401	C₂×C₃₁	1	8	yes	complemented
256	#3396	C₂×C₃₁	1	4	yes	complemented
257	#3393	C₂×C₃₁	1	4	yes	complemented
258	#3156	C₂×C₃₁	1	1	yes	characteristic, complemented
259	#3155	C₂×C₃₁	1	1	yes	characteristic
260	C₂×#495	C₂×C₃₁	1	1	yes	characteristic, complemented
261	#3043	C₂×C₃₁	1	2	yes	complemented
262	#3042	C₂×C₃₁	1	2	yes	complemented
263	#3040	C₂×C₃₁	1	2	yes	complemented
264	#2960	C₂×C₃₁	1	1	yes	characteristic, complemented
265	#2950	C₂×C₃₁	1	1	yes	characteristic, complemented
266	#2948	C₂×C₃₁	1	1	yes	characteristic, complemented
267	C₄×#135	C₂×C₃₁	1	1	yes	characteristic, complemented
268	C₂²×C₃₁	—	2	2	no	—
269	C₂²×C₃₁	—	4	4	no	—
270	C₂²×C₃₁	—	4	8	no	—
271	C₂²×C₃₁	—	4	2	no	—
272	C₂²×C₃₁	—	4	4	no	—
273	C₂²×C₃₁	—	16	4	no	—
274	C₂²×C₃₁	#550	1	2	yes	—
275	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
276	C₄×C₃₁	—	2	2	no	—
277	C₄×C₃₁	—	2	2	no	—
278	C₄×C₃₁	—	2	4	no	—
279	C₄×C₃₁	—	2	4	no	—
280	C₄×C₃₁	—	4	4	no	—
281	C₄×C₃₁	—	4	2	no	—
282	C₄×C₃₁	—	4	4	no	—
283	C₄×C₃₁	#550	1	2	yes	—
284	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
285	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
286	C₂×#137	C₂²×C₃₁	1	2	yes	—
287	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
288	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
289	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
290	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
291	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
292	#559	C₂²×C₃₁	1	8	yes	complemented
293	#556	C₂²×C₃₁	1	8	yes	complemented
294	#551	—	2	8	no	—
295	#542	—	2	4	no	—
296	#536	C₂²×C₃₁	1	2	yes	—
297	#535	C₂²×C₃₁	1	2	yes	—
298	#533	C₂²×C₃₁	1	1	yes	characteristic
299	#529	—	2	4	no	—
300	C₂×#124	C₂²×C₃₁	1	2	yes	—
301	#519	C₂²×C₃₁	1	2	yes	—
302	#518	C₂²×C₃₁	1	2	yes	—
303	C₄×#41	C₂²×C₃₁	1	2	yes	—
304	#497	C₂²×C₃₁	1	4	yes	complemented
305	#496	C₂²×C₃₁	1	4	yes	—
306	#495	C₂²×C₃₁	1	2	yes	—
307	#495	C₂²×C₃₁	1	4	yes	—
308	#495	C₂²×C₃₁	1	4	yes	—
309	#495	C₂²×C₃₁	1	2	yes	complemented
310	#494	C₂²×C₃₁	1	2	yes	complemented
311	#493	C₂²×C₃₁	1	2	yes	—
312	#464	C₂²×C₃₁	1	8	yes	complemented
313	#464	C₂²×C₃₁	1	8	yes	complemented
314	#463	C₂²×C₃₁	1	8	yes	complemented
315	#463	C₂²×C₃₁	1	8	yes	complemented
316	#461	C₂²×C₃₁	1	8	yes	complemented
317	#460	C₂²×C₃₁	1	8	yes	complemented
318	#446	C₂²×C₃₁	1	4	yes	complemented
319	C₈×Q₈	C₂²×C₃₁	1	4	yes	complemented
320	#442	C₂²×C₃₁	1	4	yes	—
321	#441	C₂²×C₃₁	1	4	yes	complemented
322	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	complemented
323	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	complemented
324	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
325	C₄×D₈	C₂²×C₃₁	1	4	yes	—
326	C₄×D₈	C₂²×C₃₁	1	2	yes	complemented
327	#427	C₂²×C₃₁	1	1	yes	characteristic
328	#424	C₂²×C₃₁	1	1	yes	characteristic
329	#424	C₂²×C₃₁	1	1	yes	characteristic
330	#423	C₂²×C₃₁	1	1	yes	characteristic
331	#419	C₂²×C₃₁	1	2	yes	—
332	#418	C₂²×C₃₁	1	2	yes	—
333	#416	C₂²×C₃₁	1	2	yes	—
334	#405	C₂²×C₃₁	1	1	yes	characteristic
335	#405	C₂²×C₃₁	1	1	yes	characteristic
336	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
337	C₂×C₃₁	—	2	2	no	—
338	C₂×C₃₁	—	8	2	no	—
339	C₂×C₃₁	—	8	4	no	—
340	C₂×C₃₁	C₂×#550	1	1	yes	characteristic
341	C₂×C₃₁	#3584	1	2	yes	—
342	C₂×#41	C₂³×C₃₁	1	2	yes	—
343	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
344	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
345	#137	—	4	4	no	—
346	AGL(1,ℤ/8)	—	4	4	no	—
347	#135	—	2	4	no	—
348	C₂×Q₁₆	—	2	4	no	—
349	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
350	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
351	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
352	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
353	C₂×SD₁₆	—	4	4	no	—
354	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
355	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
356	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
357	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
358	C₂×D₈	—	2	4	no	—
359	C₂×D₈	C₂³×C₃₁	1	2	yes	—
360	C₂×D₈	C₂³×C₃₁	1	2	yes	—
361	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
362	C₂×D₈	C₂³×C₃₁	1	4	yes	—
363	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
364	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
365	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
366	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
367	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
368	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
369	#127	C₂³×C₃₁	1	4	yes	—
370	#126	—	2	8	no	—
371	#126	—	2	8	no	—
372	#125	D₄×C₃₁	1	2	yes	—
373	#124	—	2	8	no	—
374	#124	—	2	2	no	—
375	#124	—	2	2	no	—
376	#124	C₂³×C₃₁	1	8	yes	—
377	#124	C₂³×C₃₁	1	4	yes	—
378	#124	C₂³×C₃₁	1	4	yes	—
379	#123	—	2	8	no	—
380	#123	—	2	8	no	—
381	#123	—	2	4	no	—
382	C₂²⋊Q₈	—	2	8	no	—
383	C₂²⋊Q₈	—	2	4	no	—
384	C₂²⋊Q₈	D₄×C₃₁	1	4	yes	—
385	#121	—	2	4	no	—
386	#120	—	2	4	no	—
387	C₄×Q₈	—	2	4	no	—
388	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
389	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
390	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
391	C₄×D₄	—	2	4	no	—
392	C₄×D₄	—	2	4	no	—
393	C₄×D₄	—	2	8	no	—
394	C₄×D₄	C₂³×C₃₁	1	4	yes	—
395	C₄×D₄	C₂³×C₃₁	1	8	yes	—
396	#117	—	2	8	no	—
397	#117	—	2	4	no	—
398	#117	—	2	2	no	—
399	#117	C₂³×C₃₁	1	2	yes	—
400	#117	C₂³×C₃₁	1	1	yes	characteristic
401	#117	C₂³×C₃₁	1	1	yes	characteristic
402	C₂×C₂²⋊C₄	—	2	4	no	—
403	C₂×C₂²⋊C₄	—	2	4	no	—
404	C₂×C₄²	—	2	2	no	—
405	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
406	#107	C₂³×C₃₁	1	2	yes	—
407	#106	C₂³×C₃₁	1	2	yes	—
408	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
409	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
410	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
411	#103	C₂³×C₃₁	1	4	yes	—
412	#103	C₂³×C₃₁	1	8	yes	—
413	#102	C₂³×C₃₁	1	4	yes	—
414	#102	C₂³×C₃₁	1	8	yes	—
415	#97	C₂³×C₃₁	1	2	yes	—
416	#97	C₂³×C₃₁	1	2	yes	—
417	C₄×C₈	C₂³×C₃₁	1	2	yes	—
418	C₄×C₈	C₂³×C₃₁	1	2	yes	—
419	C₄×C₈	C₂³×C₃₁	1	2	yes	—
420	C₄×C₈	C₂³×C₃₁	1	2	yes	—
421	C₃₁	#61413	1	1	yes	characteristic, direct product factor, Sylow
422	C₂⁴	—	2	2	no	—
423	#41	—	2	4	no	—
424	#41	—	2	4	no	—
425	C₂×Q₈	—	2	2	no	—
426	C₂×Q₈	—	2	4	no	—
427	C₂×Q₈	—	2	4	no	—
428	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
429	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
430	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
431	C₂×D₄	—	2	2	no	—
432	C₂×D₄	—	2	2	no	—
433	C₂×D₄	—	2	8	no	—
434	C₂×D₄	—	2	4	no	—
435	C₂×D₄	—	2	4	no	—
436	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
437	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	2	4	no	—
441	C₂²×C₄	—	2	8	no	—
442	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
443	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
444	Q₁₆	—	2	4	no	—
445	Q₁₆	—	2	4	no	—
446	Q₁₆	—	2	2	no	—
447	Q₁₆	—	4	4	no	—
448	SD₁₆	—	2	4	no	—
449	SD₁₆	—	4	8	no	—
450	SD₁₆	—	4	4	no	—
451	SD₁₆	—	4	4	no	—
452	D₈	—	2	4	no	—
453	D₈	—	2	4	no	—
454	D₈	—	2	2	no	—
455	D₈	—	4	4	no	—
456	M₁₆	—	2	2	no	—
457	M₁₆	—	2	2	no	—
458	C₂×C₈	—	2	2	no	—
459	C₂×C₈	#41×C₃₁	1	4	yes	—
460	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
461	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
462	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
463	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
464	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
465	C₄⋊C₄	—	2	4	no	—
466	C₄⋊C₄	—	2	8	no	—
467	C₄⋊C₄	—	2	8	no	—
468	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
469	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
470	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
471	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
472	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
473	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
474	C₂²⋊C₄	—	2	2	no	—
475	C₂²⋊C₄	—	2	8	no	—
476	C₂²⋊C₄	—	2	8	no	—
477	C₂²⋊C₄	—	2	8	no	—
478	C₂²⋊C₄	—	2	4	no	—
479	C₂²⋊C₄	—	2	8	no	—
480	C₂²⋊C₄	—	2	8	no	—
481	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
482	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
483	C₄²	—	2	4	no	—
484	C₄²	—	2	2	no	—
485	C₄²	—	2	8	no	—
486	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
487	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
488	C₄²	C₂⁴×C₃₁	1	2	yes	—
489	C₄²	C₂⁴×C₃₁	1	2	yes	—
490	C₄²	C₂⁴×C₃₁	1	2	yes	—
491	C₂³	—	2	4	no	—
492	C₂³	—	2	2	no	—
493	C₂³	—	4	4	no	—
494	C₂³	—	8	4	no	—
495	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
496	Q₈	—	2	2	no	—
497	Q₈	—	2	4	no	—
498	Q₈	—	2	4	no	—
499	Q₈	—	2	2	no	—
500	Q₈	—	4	4	no	—
501	Q₈	—	4	4	no	—
502	D₄	—	2	4	no	—
503	D₄	—	2	2	no	—
504	D₄	—	2	2	no	—
505	D₄	—	2	4	no	—
506	D₄	—	4	4	no	—
507	D₄	—	4	4	no	—
508	D₄	—	8	8	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	4	2	no	—
513	C₂×C₄	—	4	4	no	—
514	C₂×C₄	—	4	4	no	—
515	C₂×C₄	—	8	4	no	—
516	C₂×C₄	—	8	8	no	—
517	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
518	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
519	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
520	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
521	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
522	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
523	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
524	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
525	C₈	—	2	2	no	—
526	C₈	—	2	2	no	—
527	C₈	—	2	2	no	—
528	C₈	C₂×#41×C₃₁	1	2	yes	—
529	C₈	C₂×#41×C₃₁	1	2	yes	—
530	C₂²	—	2	2	no	—
531	C₂²	—	4	2	no	—
532	C₂²	—	4	4	no	—
533	C₂²	—	4	4	no	—
534	C₂²	—	4	8	no	—
535	C₂²	—	16	4	no	—
536	C₂²	C₃₁×#550	1	2	yes	—
537	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
538	C₄	—	2	2	no	—
539	C₄	—	2	2	no	—
540	C₄	—	2	4	no	—
541	C₄	—	2	4	no	—
542	C₄	—	4	2	no	—
543	C₄	—	4	4	no	—
544	C₄	—	4	4	no	—
545	C₄	C₃₁×#550	1	2	yes	—
546	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
547	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
548	C₂	—	2	2	no	—
549	C₂	—	8	4	no	—
550	C₂	—	8	2	no	—
551	C₂	C₂×C₃₁×#550	1	1	yes	characteristic
552	C₂	C₃₁×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#3155	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3510	C₂	semidirect
2. C₃₁×#3401	C₂	semidirect
3. C₃₁×#3396	C₂	semidirect
4. C₃₁×#3393	C₂	semidirect
5. C₃₁×#3156	C₂	semidirect
7. C₂×C₃₁×#495	C₂	semidirect
8. C₃₁×#3043	C₂	semidirect
9. C₃₁×#3042	C₂	semidirect
10. C₃₁×#3040	C₂	semidirect
11. C₃₁×#2960	C₂	semidirect
12. C₃₁×#2950	C₂	semidirect
13. C₃₁×#2948	C₂	semidirect
14. C₄×C₃₁×#135	C₂	semidirect
21. C₃₁×#559	C₂²	semidirect
22. C₃₁×#556	C₂²	semidirect
33. C₃₁×#497	C₂²	semidirect
38. C₃₁×#495	C₂²	semidirect
39. C₃₁×#494	C₂²	semidirect
41. C₃₁×#464	C₂²	semidirect
42. C₃₁×#464	C₂²	semidirect
43. C₃₁×#463	C₂²	semidirect
44. C₃₁×#463	C₂²	semidirect
45. C₃₁×#461	C₂²	semidirect
46. C₃₁×#460	C₂²	semidirect
47. C₃₁×#446	C₂²	semidirect
48. C₈×Q₈×C₃₁	C₂²	semidirect
50. C₃₁×#441	C₂²	semidirect
51. C₄×Q₁₆×C₃₁	C₂²	semidirect
52. C₄×Q₁₆×C₃₁	C₂²	semidirect
53. C₄×SD₁₆×C₃₁	C₂²	semidirect
55. C₄×D₈×C₃₁	C₂²	semidirect
214. #61413	C₃₁	direct (N × H)
254. #3510	C₂×C₃₁	semidirect
255. #3401	C₂×C₃₁	semidirect
256. #3396	C₂×C₃₁	semidirect
257. #3393	C₂×C₃₁	semidirect
258. #3156	C₂×C₃₁	semidirect
260. C₂×#495	C₂×C₃₁	semidirect
261. #3043	C₂×C₃₁	semidirect
262. #3042	C₂×C₃₁	semidirect
263. #3040	C₂×C₃₁	semidirect
264. #2960	C₂×C₃₁	semidirect
265. #2950	C₂×C₃₁	semidirect
266. #2948	C₂×C₃₁	semidirect
267. C₄×#135	C₂×C₃₁	semidirect
292. #559	C₂²×C₃₁	semidirect
293. #556	C₂²×C₃₁	semidirect
304. #497	C₂²×C₃₁	semidirect
309. #495	C₂²×C₃₁	semidirect
310. #494	C₂²×C₃₁	semidirect
312. #464	C₂²×C₃₁	semidirect
313. #464	C₂²×C₃₁	semidirect
314. #463	C₂²×C₃₁	semidirect
315. #463	C₂²×C₃₁	semidirect
316. #461	C₂²×C₃₁	semidirect
317. #460	C₂²×C₃₁	semidirect
318. #446	C₂²×C₃₁	semidirect
319. C₈×Q₈	C₂²×C₃₁	semidirect
321. #441	C₂²×C₃₁	semidirect
322. C₄×Q₁₆	C₂²×C₃₁	semidirect
323. C₄×Q₁₆	C₂²×C₃₁	semidirect
324. C₄×SD₁₆	C₂²×C₃₁	semidirect
326. C₄×D₈	C₂²×C₃₁	semidirect
421. C₃₁	#61413	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×#3510	2
C₃₁×#3401	2
C₃₁×#3396	2
C₃₁×#3393	2
C₃₁×#3156	2
C₃₁×#3155	2
C₂×C₃₁×#495	2
C₃₁×#3043	2
C₃₁×#3042	2
C₃₁×#3040	2
C₃₁×#2960	2
C₃₁×#2950	2
C₃₁×#2948	2
C₄×C₃₁×#135	2
#61413	31

Subgroups — C31×#61413 #21518172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61413 #21518172