Subgroups — C₃₁×#33682 #21490441

Order 7936.

580 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3176	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2948	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2234	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2228	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2206	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2184	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2177	C₂	1	2	yes	maximal, complemented
8	C₄×C₃₁×D₁₆	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2164	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#540	—	2	1	no	—
12	C₃₁×#518	—	2	1	no	—
13	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
14	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
15	C₃₁×#499	C₂²	1	1	yes	characteristic, complemented
16	C₃₁×#493	C₂²	1	1	yes	characteristic, complemented
17	C₃₁×#490	—	2	1	no	—
18	C₃₁×#469	—	2	1	no	—
19	C₃₁×#466	C₂²	1	2	yes	—
20	C₃₁×#442	—	2	1	no	—
21	C₃₁×#442	—	2	1	no	—
22	C₃₁×#442	—	2	1	no	—
23	C₄×D₈×C₃₁	C₂²	1	2	yes	—
24	C₄×D₈×C₃₁	C₂²	1	2	yes	—
25	C₄×D₈×C₃₁	C₂²	1	2	yes	complemented
26	C₃₁×#427	C₂²	1	1	yes	characteristic
27	C₃₁×#418	—	2	1	no	—
28	C₃₁×#405	C₂²	1	1	yes	characteristic
29	C₃₁×#367	C₂²	1	2	yes	complemented
30	C₃₁×#366	C₂²	1	2	yes	complemented
31	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	complemented
32	C₃₁×#357	C₂²	1	2	yes	—
33	C₃₁×#357	C₂²	1	2	yes	—
34	C₃₁×#357	C₂²	1	2	yes	complemented
35	C₃₁×#357	C₂²	1	2	yes	complemented
36	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
37	C₃₁×#346	C₂²	1	1	yes	characteristic, complemented
38	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
39	C₂²×D₄×C₃₁	—	2	1	no	—
40	C₂×D₈×C₃₁	—	2	4	no	—
41	C₂×D₈×C₃₁	—	2	1	no	—
42	C₂×D₈×C₃₁	—	2	1	no	—
43	C₂×D₈×C₃₁	C₂³	1	2	yes	—
44	C₂×D₈×C₃₁	C₂³	1	2	yes	—
45	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×#127	—	2	1	no	—
48	C₃₁×#125	—	2	1	no	—
49	C₃₁×#123	—	4	2	no	—
50	C₃₁×#121	—	2	2	no	—
51	C₃₁×#121	—	4	2	no	—
52	C₄×D₄×C₃₁	—	2	2	no	—
53	C₄×D₄×C₃₁	—	2	2	no	—
54	C₄×D₄×C₃₁	—	2	2	no	—
55	C₄×D₄×C₃₁	—	4	2	no	—
56	C₃₁×#117	—	2	1	no	—
57	C₃₁×#117	D₄	1	1	yes	characteristic
58	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
59	C₃₁×D₁₆	—	2	4	no	—
60	C₃₁×D₁₆	—	2	4	no	—
61	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
62	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
63	C₃₁×#107	C₂³	1	2	yes	—
64	C₃₁×#107	C₂³	1	2	yes	—
65	C₃₁×#106	—	2	1	no	—
66	C₃₁×#106	—	2	1	no	—
67	C₃₁×#102	—	2	2	no	—
68	C₃₁×#102	—	2	2	no	—
69	C₃₁×#102	—	2	2	no	—
70	C₃₁×#102	—	2	2	no	—
71	C₃₁×#97	D₄	1	1	yes	characteristic
72	C₃₁×#97	D₄	1	1	yes	characteristic
73	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₂⁴×C₃₁	—	4	1	no	—
76	C₂×D₄×C₃₁	—	2	4	no	—
77	C₂×D₄×C₃₁	—	2	2	no	—
78	C₂×D₄×C₃₁	—	2	2	no	—
79	C₂×D₄×C₃₁	—	4	2	no	—
80	C₂×D₄×C₃₁	—	4	2	no	—
81	C₂×D₄×C₃₁	—	4	1	no	—
82	C₂×D₄×C₃₁	—	4	2	no	—
83	C₂²×C₄×C₃₁	—	4	2	no	—
84	C₂²×C₄×C₃₁	—	4	2	no	—
85	C₂²×C₄×C₃₁	—	4	2	no	—
86	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
87	D₈×C₃₁	—	2	4	no	—
88	D₈×C₃₁	—	2	2	no	—
89	D₈×C₃₁	—	2	2	no	—
90	D₈×C₃₁	—	2	2	no	—
91	D₈×C₃₁	—	4	2	no	—
92	D₈×C₃₁	#41	1	4	yes	—
93	M₁₆×C₃₁	—	4	1	no	—
94	C₂×C₈×C₃₁	—	2	2	no	—
95	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₃₁	—	2	2	no	—
100	C₄⋊C₄×C₃₁	—	2	2	no	—
101	C₄⋊C₄×C₃₁	—	4	2	no	—
102	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
103	C₂²⋊C₄×C₃₁	—	4	1	no	—
104	C₂²⋊C₄×C₃₁	—	4	2	no	—
105	C₂²⋊C₄×C₃₁	—	4	2	no	—
106	C₂²⋊C₄×C₃₁	—	4	2	no	—
107	C₂²⋊C₄×C₃₁	—	4	2	no	—
108	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
109	C₄²×C₃₁	—	4	1	no	—
110	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₃₁	—	2	2	no	—
113	C₁₆×C₃₁	—	2	2	no	—
114	#33682	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₃₁	—	4	4	no	—
116	C₂³×C₃₁	—	4	2	no	—
117	C₂³×C₃₁	—	4	2	no	—
118	C₂³×C₃₁	—	16	2	no	—
119	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
120	D₄×C₃₁	—	2	4	no	—
121	D₄×C₃₁	—	4	4	no	—
122	D₄×C₃₁	—	4	2	no	—
123	D₄×C₃₁	—	4	2	no	—
124	D₄×C₃₁	—	4	2	no	—
125	D₄×C₃₁	—	16	2	no	—
126	D₄×C₃₁	—	16	2	no	—
127	C₂×C₄×C₃₁	—	2	1	no	—
128	C₂×C₄×C₃₁	—	2	1	no	—
129	C₂×C₄×C₃₁	—	4	2	no	—
130	C₂×C₄×C₃₁	—	4	2	no	—
131	C₂×C₄×C₃₁	—	16	2	no	—
132	C₂×C₄×C₃₁	—	16	2	no	—
133	C₂×C₄×C₃₁	—	16	2	no	—
134	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
138	C₈×C₃₁	—	2	1	no	—
139	C₈×C₃₁	—	2	1	no	—
140	C₈×C₃₁	—	2	1	no	—
141	C₈×C₃₁	C₂×#41	1	2	yes	—
142	#3176	C₂×C₃₁	1	1	yes	characteristic, complemented
143	#2948	C₂×C₃₁	1	1	yes	characteristic, complemented
144	#2234	C₂×C₃₁	1	2	yes	complemented
145	#2228	C₂×C₃₁	1	2	yes	complemented
146	#2206	C₂×C₃₁	1	2	yes	complemented
147	#2184	C₂×C₃₁	1	2	yes	complemented
148	#2177	C₂×C₃₁	1	2	yes	complemented
149	C₄×D₁₆	C₂×C₃₁	1	2	yes	complemented
150	#2164	C₂×C₃₁	1	1	yes	characteristic, complemented
151	C₂²×C₃₁	—	2	1	no	—
152	C₂²×C₃₁	—	2	2	no	—
153	C₂²×C₃₁	—	4	4	no	—
154	C₂²×C₃₁	—	8	2	no	—
155	C₂²×C₃₁	—	8	2	no	—
156	C₂²×C₃₁	—	8	2	no	—
157	C₂²×C₃₁	—	8	2	no	—
158	C₂²×C₃₁	—	8	2	no	—
159	C₂²×C₃₁	—	16	4	no	—
160	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
161	C₄×C₃₁	—	2	1	no	—
162	C₄×C₃₁	—	2	2	no	—
163	C₄×C₃₁	—	2	2	no	—
164	C₄×C₃₁	—	8	2	no	—
165	C₄×C₃₁	—	8	2	no	—
166	C₄×C₃₁	#525	1	1	yes	characteristic
167	C₄×C₃₁	#575	1	1	yes	characteristic
168	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
169	#540	—	2	1	no	—
170	#518	—	2	1	no	—
171	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
172	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
173	#499	C₂²×C₃₁	1	1	yes	characteristic, complemented
174	#493	C₂²×C₃₁	1	1	yes	characteristic, complemented
175	#490	—	2	1	no	—
176	#469	—	2	1	no	—
177	#466	C₂²×C₃₁	1	2	yes	—
178	#442	—	2	1	no	—
179	#442	—	2	1	no	—
180	#442	—	2	1	no	—
181	C₄×D₈	C₂²×C₃₁	1	2	yes	—
182	C₄×D₈	C₂²×C₃₁	1	2	yes	—
183	C₄×D₈	C₂²×C₃₁	1	2	yes	complemented
184	#427	C₂²×C₃₁	1	1	yes	characteristic
185	#418	—	2	1	no	—
186	#405	C₂²×C₃₁	1	1	yes	characteristic
187	#367	C₂²×C₃₁	1	2	yes	complemented
188	#366	C₂²×C₃₁	1	2	yes	complemented
189	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	complemented
190	#357	C₂²×C₃₁	1	2	yes	—
191	#357	C₂²×C₃₁	1	2	yes	—
192	#357	C₂²×C₃₁	1	2	yes	complemented
193	#357	C₂²×C₃₁	1	2	yes	complemented
194	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
195	#346	C₂²×C₃₁	1	1	yes	characteristic, complemented
196	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
197	C₂×C₃₁	—	4	1	no	—
198	C₂×C₃₁	—	8	4	no	—
199	C₂×C₃₁	—	16	2	no	—
200	C₂×C₃₁	#3100	1	1	yes	characteristic
201	C₂×C₃₁	#3415	1	2	yes	—
202	C₂²×D₄	—	2	1	no	—
203	C₂×D₈	—	2	4	no	—
204	C₂×D₈	—	2	1	no	—
205	C₂×D₈	—	2	1	no	—
206	C₂×D₈	C₂³×C₃₁	1	2	yes	—
207	C₂×D₈	C₂³×C₃₁	1	2	yes	—
208	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
209	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
210	#127	—	2	1	no	—
211	#125	—	2	1	no	—
212	#123	—	4	2	no	—
213	#121	—	2	2	no	—
214	#121	—	4	2	no	—
215	C₄×D₄	—	2	2	no	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	—	2	2	no	—
218	C₄×D₄	—	4	2	no	—
219	#117	—	2	1	no	—
220	#117	D₄×C₃₁	1	1	yes	characteristic
221	C₂×C₂²⋊C₄	—	4	2	no	—
222	D₁₆	—	2	4	no	—
223	D₁₆	—	2	4	no	—
224	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
225	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
226	#107	C₂³×C₃₁	1	2	yes	—
227	#107	C₂³×C₃₁	1	2	yes	—
228	#106	—	2	1	no	—
229	#106	—	2	1	no	—
230	#102	—	2	2	no	—
231	#102	—	2	2	no	—
232	#102	—	2	2	no	—
233	#102	—	2	2	no	—
234	#97	D₄×C₃₁	1	1	yes	characteristic
235	#97	D₄×C₃₁	1	1	yes	characteristic
236	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
237	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
238	C₃₁	#33682	1	1	yes	characteristic, direct product factor, Sylow
239	C₂⁴	—	4	1	no	—
240	C₂×D₄	—	2	4	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂×D₄	—	4	2	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	—	4	2	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
251	D₈	—	2	4	no	—
252	D₈	—	2	2	no	—
253	D₈	—	2	2	no	—
254	D₈	—	2	2	no	—
255	D₈	—	4	2	no	—
256	D₈	#41×C₃₁	1	4	yes	—
257	M₁₆	—	4	1	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
262	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
267	C₂²⋊C₄	—	4	1	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
273	C₄²	—	4	1	no	—
274	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
275	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
276	C₁₆	—	2	2	no	—
277	C₁₆	—	2	2	no	—
278	C₂³	—	4	4	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	4	2	no	—
281	C₂³	—	16	2	no	—
282	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
283	D₄	—	2	4	no	—
284	D₄	—	4	4	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	16	2	no	—
289	D₄	—	16	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	16	2	no	—
295	C₂×C₄	—	16	2	no	—
296	C₂×C₄	—	16	2	no	—
297	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
298	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
299	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
300	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	C₂×#41×C₃₁	1	2	yes	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	4	4	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	8	2	no	—
310	C₂²	—	8	2	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	16	4	no	—
314	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	C₃₁×#525	1	1	yes	characteristic
321	C₄	C₃₁×#575	1	1	yes	characteristic
322	C₂	—	4	1	no	—
323	C₂	—	8	4	no	—
324	C₂	—	16	2	no	—
325	C₂	C₃₁×#3100	1	1	yes	characteristic
326	C₂	C₃₁×#3415	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#117 ◁ C₃₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3176	C₂	semidirect
2. C₃₁×#2948	C₂	semidirect
3. C₃₁×#2234	C₂	semidirect
4. C₃₁×#2228	C₂	semidirect
5. C₃₁×#2206	C₂	semidirect
6. C₃₁×#2184	C₂	semidirect
7. C₃₁×#2177	C₂	semidirect
8. C₄×C₃₁×D₁₆	C₂	semidirect
9. C₃₁×#2164	C₂	semidirect
15. C₃₁×#499	C₂²	semidirect
16. C₃₁×#493	C₂²	semidirect
25. C₄×D₈×C₃₁	C₂²	semidirect
29. C₃₁×#367	C₂²	semidirect
30. C₃₁×#366	C₂²	semidirect
31. C₃₁×C₄⋊C₁₆	C₂²	semidirect
34. C₃₁×#357	C₂²	semidirect
35. C₃₁×#357	C₂²	semidirect
37. C₃₁×#346	C₂²	semidirect
38. C₄×C₁₆×C₃₁	C₂²	semidirect
114. #33682	C₃₁	direct (N × H)
142. #3176	C₂×C₃₁	semidirect
143. #2948	C₂×C₃₁	semidirect
144. #2234	C₂×C₃₁	semidirect
145. #2228	C₂×C₃₁	semidirect
146. #2206	C₂×C₃₁	semidirect
147. #2184	C₂×C₃₁	semidirect
148. #2177	C₂×C₃₁	semidirect
149. C₄×D₁₆	C₂×C₃₁	semidirect
150. #2164	C₂×C₃₁	semidirect
173. #499	C₂²×C₃₁	semidirect
174. #493	C₂²×C₃₁	semidirect
183. C₄×D₈	C₂²×C₃₁	semidirect
187. #367	C₂²×C₃₁	semidirect
188. #366	C₂²×C₃₁	semidirect
189. C₄⋊C₁₆	C₂²×C₃₁	semidirect
192. #357	C₂²×C₃₁	semidirect
193. #357	C₂²×C₃₁	semidirect
195. #346	C₂²×C₃₁	semidirect
196. C₄×C₁₆	C₂²×C₃₁	semidirect
238. C₃₁	#33682	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×#3176	2
C₃₁×#2948	2
C₃₁×#2234	2
C₃₁×#2228	2
C₃₁×#2206	2
C₃₁×#2184	2
C₃₁×#2177	2
C₄×C₃₁×D₁₆	2
C₃₁×#2164	2
#33682	31

Subgroups — C31×#33682 #21490441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33682 #21490441