Subgroups — C₃₁×#33656 #21490415

Order 7936.

556 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3054	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#430	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2239	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2238	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2155	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2148	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#349	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₃₁	—	2	1	no	—
10	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#525	—	2	1	no	—
12	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
13	C₃₁×#470	C₂²	1	2	yes	—
14	C₃₁×#469	C₂²	1	2	yes	—
15	C₃₁×#468	—	2	2	no	—
16	C₃₁×#430	—	2	2	no	—
17	C₃₁×#430	C₂²	1	4	yes	—
18	C₂×C₃₁×#108	—	2	1	no	—
19	C₂×C₃₁×#108	C₂²	1	2	yes	—
20	C₃₁×#427	C₂²	1	1	yes	characteristic
21	C₃₁×#416	—	2	1	no	—
22	C₃₁×#365	C₂²	1	8	yes	complemented
23	C₃₁×#362	C₂²	1	4	yes	complemented
24	C₃₁×#361	C₂²	1	4	yes	complemented
25	C₃₁×#349	C₂²	1	4	yes	—
26	C₂×#41×C₃₁	—	2	1	no	—
27	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
28	C₃₁×#135	—	2	2	no	—
29	C₃₁×#135	—	2	2	no	—
30	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
31	C₂×SD₁₆×C₃₁	—	2	1	no	—
32	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₃₁	—	2	2	no	—
34	C₂×M₁₆×C₃₁	—	2	2	no	—
35	C₂×M₁₆×C₃₁	—	2	4	no	—
36	C₂×M₁₆×C₃₁	—	2	2	no	—
37	C₂×M₁₆×C₃₁	—	4	2	no	—
38	C₂×M₁₆×C₃₁	—	4	1	no	—
39	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
40	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
41	C₂²×C₈×C₃₁	—	4	1	no	—
42	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₃₁×#123	—	4	1	no	—
45	C₃₁×C₂²⋊Q₈	—	2	2	no	—
46	C₃₁×#121	—	2	2	no	—
47	C₃₁×#120	—	4	1	no	—
48	C₄×D₄×C₃₁	—	4	2	no	—
49	C₃₁×#117	—	2	1	no	—
50	C₃₁×M₃₂	—	2	2	no	—
51	C₃₁×M₃₂	C₂³	1	4	yes	—
52	C₂×C₁₆×C₃₁	—	2	2	no	—
53	C₃₁×#108	—	2	4	no	—
54	C₃₁×#108	—	2	2	no	—
55	C₃₁×#108	C₂³	1	4	yes	—
56	C₃₁×#107	—	2	1	no	—
57	C₃₁×#106	C₂³	1	2	yes	—
58	C₃₁×#103	—	2	2	no	—
59	C₃₁×#102	—	2	2	no	—
60	C₂⁴×C₃₁	SD₁₆	1	1	yes	characteristic
61	#41×C₃₁	—	4	1	no	—
62	#41×C₃₁	—	4	1	no	—
63	#41×C₃₁	—	4	1	no	—
64	#41×C₃₁	—	4	1	no	—
65	C₂×Q₈×C₃₁	—	2	1	no	—
66	C₂×D₄×C₃₁	—	2	1	no	—
67	C₂×D₄×C₃₁	—	4	2	no	—
68	C₂×D₄×C₃₁	—	4	1	no	—
69	C₂²×C₄×C₃₁	—	2	1	no	—
70	C₂²×C₄×C₃₁	—	2	1	no	—
71	C₂²×C₄×C₃₁	—	2	1	no	—
72	C₂²×C₄×C₃₁	—	2	1	no	—
73	C₂²×C₄×C₃₁	—	4	1	no	—
74	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
75	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
78	Q₁₆×C₃₁	—	2	2	no	—
79	SD₁₆×C₃₁	—	4	2	no	—
80	D₈×C₃₁	—	2	2	no	—
81	M₁₆×C₃₁	—	2	4	no	—
82	M₁₆×C₃₁	—	2	4	no	—
83	M₁₆×C₃₁	—	2	4	no	—
84	M₁₆×C₃₁	—	4	4	no	—
85	M₁₆×C₃₁	—	4	2	no	—
86	C₂×C₈×C₃₁	—	2	1	no	—
87	C₂×C₈×C₃₁	—	2	1	no	—
88	C₂×C₈×C₃₁	—	4	4	no	—
89	C₂×C₈×C₃₁	—	4	2	no	—
90	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₃₁	#41	1	2	yes	—
93	C₂×C₈×C₃₁	#41	1	2	yes	—
94	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₃₁	—	2	2	no	—
96	C₄⋊C₄×C₃₁	—	4	2	no	—
97	C₂²⋊C₄×C₃₁	—	4	1	no	—
98	C₂²⋊C₄×C₃₁	—	4	2	no	—
99	C₂²⋊C₄×C₃₁	—	4	2	no	—
100	C₄²×C₃₁	—	4	1	no	—
101	C₁₆×C₃₁	—	2	4	no	—
102	#33656	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₃₁	—	2	1	no	—
104	C₂³×C₃₁	—	2	1	no	—
105	C₂³×C₃₁	—	4	1	no	—
106	C₂³×C₃₁	—	8	1	no	—
107	C₂³×C₃₁	C₂×SD₁₆	1	2	yes	—
108	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
109	Q₈×C₃₁	—	4	1	no	—
110	Q₈×C₃₁	—	4	1	no	—
111	D₄×C₃₁	—	4	1	no	—
112	D₄×C₃₁	—	4	1	no	—
113	D₄×C₃₁	—	8	4	no	—
114	D₄×C₃₁	—	8	1	no	—
115	D₄×C₃₁	—	8	1	no	—
116	C₂×C₄×C₃₁	—	2	1	no	—
117	C₂×C₄×C₃₁	—	2	1	no	—
118	C₂×C₄×C₃₁	—	2	2	no	—
119	C₂×C₄×C₃₁	—	2	2	no	—
120	C₂×C₄×C₃₁	—	2	2	no	—
121	C₂×C₄×C₃₁	—	2	2	no	—
122	C₂×C₄×C₃₁	—	4	2	no	—
123	C₂×C₄×C₃₁	—	4	1	no	—
124	C₂×C₄×C₃₁	—	8	1	no	—
125	C₂×C₄×C₃₁	—	8	1	no	—
126	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
127	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
128	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
129	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
133	C₈×C₃₁	—	2	2	no	—
134	C₈×C₃₁	—	4	4	no	—
135	C₈×C₃₁	#123	1	2	yes	—
136	C₈×C₃₁	C₂×#41	1	2	yes	—
137	#3054	C₂×C₃₁	1	1	yes	characteristic
138	C₂×#430	C₂×C₃₁	1	1	yes	characteristic, complemented
139	#2239	C₂×C₃₁	1	4	yes	complemented
140	#2238	C₂×C₃₁	1	4	yes	complemented
141	#2155	C₂×C₃₁	1	2	yes	complemented
142	#2148	C₂×C₃₁	1	2	yes	complemented
143	C₂×#349	C₂×C₃₁	1	1	yes	characteristic, complemented
144	C₂²×C₃₁	—	2	2	no	—
145	C₂²×C₃₁	—	4	1	no	—
146	C₂²×C₃₁	—	4	2	no	—
147	C₂²×C₃₁	—	4	2	no	—
148	C₂²×C₃₁	—	8	1	no	—
149	C₂²×C₃₁	—	8	2	no	—
150	C₂²×C₃₁	—	8	1	no	—
151	C₂²×C₃₁	#481	1	1	yes	characteristic
152	C₂²×C₃₁	#481	1	1	yes	characteristic
153	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
154	C₄×C₃₁	—	2	2	no	—
155	C₄×C₃₁	—	8	2	no	—
156	C₄×C₃₁	—	8	1	no	—
157	C₄×C₃₁	#481	1	1	yes	characteristic
158	C₄×C₃₁	#481	1	1	yes	characteristic
159	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
160	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
162	C₂²×M₁₆	—	2	1	no	—
163	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
164	#525	—	2	1	no	—
165	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
166	#470	C₂²×C₃₁	1	2	yes	—
167	#469	C₂²×C₃₁	1	2	yes	—
168	#468	—	2	2	no	—
169	#430	—	2	2	no	—
170	#430	C₂²×C₃₁	1	4	yes	—
171	C₂×#108	—	2	1	no	—
172	C₂×#108	C₂²×C₃₁	1	2	yes	—
173	#427	C₂²×C₃₁	1	1	yes	characteristic
174	#416	—	2	1	no	—
175	#365	C₂²×C₃₁	1	8	yes	complemented
176	#362	C₂²×C₃₁	1	4	yes	complemented
177	#361	C₂²×C₃₁	1	4	yes	complemented
178	#349	C₂²×C₃₁	1	4	yes	—
179	C₂×C₃₁	—	2	1	no	—
180	C₂×C₃₁	—	2	1	no	—
181	C₂×C₃₁	—	2	1	no	—
182	C₂×C₃₁	—	4	2	no	—
183	C₂×C₃₁	—	16	1	no	—
184	C₂×C₃₁	C₂×#481	1	1	yes	characteristic
185	C₂×#41	—	2	1	no	—
186	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
187	#135	—	2	2	no	—
188	#135	—	2	2	no	—
189	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
190	C₂×SD₁₆	—	2	1	no	—
191	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	—	2	4	no	—
196	C₂×M₁₆	—	4	2	no	—
197	C₂×M₁₆	—	4	1	no	—
198	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
199	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
200	C₂²×C₈	—	4	1	no	—
201	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
203	#123	—	4	1	no	—
204	C₂²⋊Q₈	—	2	2	no	—
205	#121	—	2	2	no	—
206	#120	—	4	1	no	—
207	C₄×D₄	—	4	2	no	—
208	#117	—	2	1	no	—
209	M₃₂	—	2	2	no	—
210	M₃₂	C₂³×C₃₁	1	4	yes	—
211	C₂×C₁₆	—	2	2	no	—
212	#108	—	2	4	no	—
213	#108	—	2	2	no	—
214	#108	C₂³×C₃₁	1	4	yes	—
215	#107	—	2	1	no	—
216	#106	C₂³×C₃₁	1	2	yes	—
217	#103	—	2	2	no	—
218	#102	—	2	2	no	—
219	C₃₁	#33656	1	1	yes	characteristic, direct product factor, Sylow
220	C₂⁴	SD₁₆×C₃₁	1	1	yes	characteristic
221	#41	—	4	1	no	—
222	#41	—	4	1	no	—
223	#41	—	4	1	no	—
224	#41	—	4	1	no	—
225	C₂×Q₈	—	2	1	no	—
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
235	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
237	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
238	Q₁₆	—	2	2	no	—
239	SD₁₆	—	4	2	no	—
240	D₈	—	2	2	no	—
241	M₁₆	—	2	4	no	—
242	M₁₆	—	2	4	no	—
243	M₁₆	—	2	4	no	—
244	M₁₆	—	4	4	no	—
245	M₁₆	—	4	2	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	4	4	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
251	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
252	C₂×C₈	#41×C₃₁	1	2	yes	—
253	C₂×C₈	#41×C₃₁	1	2	yes	—
254	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	4	1	no	—
258	C₂²⋊C₄	—	4	2	no	—
259	C₂²⋊C₄	—	4	2	no	—
260	C₄²	—	4	1	no	—
261	C₁₆	—	2	4	no	—
262	C₂³	—	2	1	no	—
263	C₂³	—	2	1	no	—
264	C₂³	—	4	1	no	—
265	C₂³	—	8	1	no	—
266	C₂³	C₂×SD₁₆×C₃₁	1	2	yes	—
267	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
268	Q₈	—	4	1	no	—
269	Q₈	—	4	1	no	—
270	D₄	—	4	1	no	—
271	D₄	—	4	1	no	—
272	D₄	—	8	4	no	—
273	D₄	—	8	1	no	—
274	D₄	—	8	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	8	1	no	—
284	C₂×C₄	—	8	1	no	—
285	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
286	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
287	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
288	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
289	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
290	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
291	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
292	C₈	—	2	2	no	—
293	C₈	—	4	4	no	—
294	C₈	C₃₁×#123	1	2	yes	—
295	C₈	C₂×#41×C₃₁	1	2	yes	—
296	C₂²	—	2	2	no	—
297	C₂²	—	4	1	no	—
298	C₂²	—	4	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	8	1	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	8	1	no	—
303	C₂²	C₃₁×#481	1	1	yes	characteristic
304	C₂²	C₃₁×#481	1	1	yes	characteristic
305	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	1	no	—
309	C₄	C₃₁×#481	1	1	yes	characteristic
310	C₄	C₃₁×#481	1	1	yes	characteristic
311	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
312	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
313	C₂	—	2	1	no	—
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	4	2	no	—
317	C₂	—	16	1	no	—
318	C₂	C₂×C₃₁×#481	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	#481
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂×C₃₁	C₂×#481
prime radical	C₃₁×#3054	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×M₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#430	C₂	semidirect
3. C₃₁×#2239	C₂	semidirect
4. C₃₁×#2238	C₂	semidirect
5. C₃₁×#2155	C₂	semidirect
6. C₃₁×#2148	C₂	semidirect
7. C₂×C₃₁×#349	C₂	semidirect
22. C₃₁×#365	C₂²	semidirect
23. C₃₁×#362	C₂²	semidirect
24. C₃₁×#361	C₂²	semidirect
102. #33656	C₃₁	direct (N × H)
138. C₂×#430	C₂×C₃₁	semidirect
139. #2239	C₂×C₃₁	semidirect
140. #2238	C₂×C₃₁	semidirect
141. #2155	C₂×C₃₁	semidirect
142. #2148	C₂×C₃₁	semidirect
143. C₂×#349	C₂×C₃₁	semidirect
175. #365	C₂²×C₃₁	semidirect
176. #362	C₂²×C₃₁	semidirect
177. #361	C₂²×C₃₁	semidirect
219. C₃₁	#33656	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×#3054	2
C₂×C₃₁×#430	2
C₃₁×#2239	2
C₃₁×#2238	2
C₃₁×#2155	2
C₃₁×#2148	2
C₂×C₃₁×#349	2
#33656	31

Subgroups — C31×#33656 #21490415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33656 #21490415