Subgroups — C₃₁×#9796 #21466555

Order 7936.

494 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1966	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1965	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1546	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1545	C₂	1	1	yes	maximal, characteristic
7	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
10	C₃₁×#486	—	2	1	no	—
11	C₃₁×#486	—	2	1	no	—
12	C₃₁×#486	—	2	2	no	—
13	C₃₁×#431	C₂²	1	1	yes	characteristic
14	C₃₁×#423	C₂²	1	1	yes	characteristic
15	C₃₁×#423	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#102	C₄	1	4	yes	—
17	C₂×C₃₁×#101	—	2	2	no	—
18	C₂×C₃₁×#100	—	2	2	no	—
19	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#335	—	2	2	no	—
22	C₃₁×#332	—	2	2	no	—
23	C₃₁×#331	—	2	2	no	—
24	C₃₁×#322	—	2	1	no	—
25	C₈²×C₃₁	—	2	1	no	—
26	C₂²×Q₈×C₃₁	—	2	1	no	—
27	C₂²×D₄×C₃₁	—	2	1	no	—
28	C₂²×D₄×C₃₁	—	2	1	no	—
29	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₃₁	—	2	1	no	—
31	C₂×M₁₆×C₃₁	—	2	1	no	—
32	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
33	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
34	C₃₁×C₄⋊Q₈	—	2	1	no	—
35	C₃₁×C₄⋊Q₈	—	2	1	no	—
36	C₃₁×C₄⋊Q₈	—	2	1	no	—
37	C₃₁×C₄⋊Q₈	—	2	1	no	—
38	C₃₁×#127	—	2	1	no	—
39	C₃₁×#127	—	2	1	no	—
40	C₃₁×#127	—	2	1	no	—
41	C₃₁×#127	—	2	1	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
43	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
45	C₃₁×C₄⋊C₈	—	2	1	no	—
46	C₃₁×C₄⋊C₈	—	2	1	no	—
47	C₃₁×C₄⋊C₈	—	2	1	no	—
48	C₃₁×C₄⋊C₈	—	2	1	no	—
49	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
50	C₃₁×#102	—	2	4	no	—
51	C₃₁×#102	—	2	4	no	—
52	C₃₁×#101	—	4	4	no	—
53	C₃₁×#100	—	4	4	no	—
54	C₃₁×C₂²⋊C₈	—	2	1	no	—
55	C₃₁×#97	—	2	1	no	—
56	C₄×C₈×C₃₁	—	2	1	no	—
57	C₄×C₈×C₃₁	—	2	1	no	—
58	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
59	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
60	C₄×C₈×C₃₁	D₄	1	2	yes	—
61	C₂⁴×C₃₁	—	2	2	no	—
62	C₂×Q₈×C₃₁	—	4	1	no	—
63	C₂×Q₈×C₃₁	—	4	1	no	—
64	C₂×Q₈×C₃₁	—	4	2	no	—
65	C₂×Q₈×C₃₁	—	4	2	no	—
66	C₂×D₄×C₃₁	—	2	2	no	—
67	C₂×D₄×C₃₁	—	2	2	no	—
68	C₂×D₄×C₃₁	—	4	4	no	—
69	C₂×D₄×C₃₁	—	4	1	no	—
70	C₂×D₄×C₃₁	—	4	1	no	—
71	C₂×D₄×C₃₁	—	4	1	no	—
72	C₂×D₄×C₃₁	—	4	1	no	—
73	C₂×D₄×C₃₁	—	4	2	no	—
74	C₂×D₄×C₃₁	—	4	2	no	—
75	C₂×D₄×C₃₁	—	8	2	no	—
76	C₂²×C₄×C₃₁	—	2	2	no	—
77	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
80	M₁₆×C₃₁	—	2	2	no	—
81	M₁₆×C₃₁	—	4	2	no	—
82	M₁₆×C₃₁	—	4	2	no	—
83	M₁₆×C₃₁	D₈	1	4	yes	—
84	C₂×C₈×C₃₁	—	2	2	no	—
85	C₂×C₈×C₃₁	—	2	2	no	—
86	C₂×C₈×C₃₁	—	4	2	no	—
87	C₂×C₈×C₃₁	—	4	1	no	—
88	C₂×C₈×C₃₁	—	4	1	no	—
89	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
90	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
91	C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	2	2	no	—
93	C₄⋊C₄×C₃₁	—	4	1	no	—
94	C₄⋊C₄×C₃₁	—	4	1	no	—
95	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₃₁	#41	1	1	yes	characteristic
99	#9796	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₃₁	—	4	2	no	—
101	C₂³×C₃₁	—	4	4	no	—
102	C₂³×C₃₁	—	8	2	no	—
103	C₂³×C₃₁	—	8	2	no	—
104	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
105	Q₈×C₃₁	—	8	2	no	—
106	Q₈×C₃₁	—	8	1	no	—
107	Q₈×C₃₁	—	8	1	no	—
108	D₄×C₃₁	—	4	4	no	—
109	D₄×C₃₁	—	8	2	no	—
110	D₄×C₃₁	—	8	2	no	—
111	D₄×C₃₁	—	8	2	no	—
112	D₄×C₃₁	—	8	2	no	—
113	D₄×C₃₁	—	8	1	no	—
114	D₄×C₃₁	—	8	1	no	—
115	C₂×C₄×C₃₁	—	2	2	no	—
116	C₂×C₄×C₃₁	—	2	2	no	—
117	C₂×C₄×C₃₁	—	4	2	no	—
118	C₂×C₄×C₃₁	—	4	4	no	—
119	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
122	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
123	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
124	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₃₁	#135	1	2	yes	—
127	C₈×C₃₁	—	2	2	no	—
128	C₈×C₃₁	—	2	2	no	—
129	C₈×C₃₁	—	2	4	no	—
130	C₈×C₃₁	—	4	2	no	—
131	C₈×C₃₁	—	8	1	no	—
132	C₈×C₃₁	—	8	1	no	—
133	C₂×#486	C₂×C₃₁	1	1	yes	characteristic
134	#1966	C₂×C₃₁	1	1	yes	characteristic, complemented
135	#1965	C₂×C₃₁	1	1	yes	characteristic
136	#1567	C₂×C₃₁	1	1	yes	characteristic, complemented
137	#1546	C₂×C₃₁	1	1	yes	characteristic, complemented
138	#1545	C₂×C₃₁	1	1	yes	characteristic
139	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
140	C₂²×C₃₁	—	2	1	no	—
141	C₂²×C₃₁	—	2	1	no	—
142	C₂²×C₃₁	—	8	2	no	—
143	C₂²×C₃₁	—	8	2	no	—
144	C₂²×C₃₁	—	8	2	no	—
145	C₂²×C₃₁	—	16	4	no	—
146	C₂²×C₃₁	#390	1	1	yes	characteristic
147	C₂²×C₃₁	C₄×D₈	1	2	yes	—
148	C₄×C₃₁	—	2	1	no	—
149	C₄×C₃₁	—	2	1	no	—
150	C₄×C₃₁	—	8	2	no	—
151	C₄×C₃₁	#413	1	2	yes	—
152	C₄×C₃₁	#466	1	2	yes	—
153	C₄×C₃₁	#471	1	2	yes	—
154	C₄×C₃₁	#493	1	1	yes	characteristic
155	C₄×C₃₁	#495	1	1	yes	characteristic
156	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
157	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
158	#486	—	2	1	no	—
159	#486	—	2	1	no	—
160	#486	—	2	2	no	—
161	#431	C₂²×C₃₁	1	1	yes	characteristic
162	#423	C₂²×C₃₁	1	1	yes	characteristic
163	#423	C₂²×C₃₁	1	1	yes	characteristic
164	C₂×#102	C₄×C₃₁	1	4	yes	—
165	C₂×#101	—	2	2	no	—
166	C₂×#100	—	2	2	no	—
167	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
168	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
169	#335	—	2	2	no	—
170	#332	—	2	2	no	—
171	#331	—	2	2	no	—
172	#322	—	2	1	no	—
173	C₈²	—	2	1	no	—
174	C₂×C₃₁	—	2	2	no	—
175	C₂×C₃₁	—	16	2	no	—
176	C₂×C₃₁	#1967	1	1	yes	characteristic
177	C₂×C₃₁	#1970	1	1	yes	characteristic
178	C₂×C₃₁	#1977	1	1	yes	characteristic
179	C₂²×Q₈	—	2	1	no	—
180	C₂²×D₄	—	2	1	no	—
181	C₂²×D₄	—	2	1	no	—
182	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	2	1	no	—
185	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
186	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
187	C₄⋊Q₈	—	2	1	no	—
188	C₄⋊Q₈	—	2	1	no	—
189	C₄⋊Q₈	—	2	1	no	—
190	C₄⋊Q₈	—	2	1	no	—
191	#127	—	2	1	no	—
192	#127	—	2	1	no	—
193	#127	—	2	1	no	—
194	#127	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
197	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	—	2	1	no	—
201	C₄⋊C₈	—	2	1	no	—
202	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
203	#102	—	2	4	no	—
204	#102	—	2	4	no	—
205	#101	—	4	4	no	—
206	#100	—	4	4	no	—
207	C₂²⋊C₈	—	2	1	no	—
208	#97	—	2	1	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
212	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
213	C₄×C₈	D₄×C₃₁	1	2	yes	—
214	C₃₁	#9796	1	1	yes	characteristic, direct product factor, Sylow
215	C₂⁴	—	2	2	no	—
216	C₂×Q₈	—	4	1	no	—
217	C₂×Q₈	—	4	1	no	—
218	C₂×Q₈	—	4	2	no	—
219	C₂×Q₈	—	4	2	no	—
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	4	4	no	—
223	C₂×D₄	—	4	1	no	—
224	C₂×D₄	—	4	1	no	—
225	C₂×D₄	—	4	1	no	—
226	C₂×D₄	—	4	1	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	8	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
233	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
234	M₁₆	—	2	2	no	—
235	M₁₆	—	4	2	no	—
236	M₁₆	—	4	2	no	—
237	M₁₆	D₈×C₃₁	1	4	yes	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
244	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	4	1	no	—
248	C₄⋊C₄	—	4	1	no	—
249	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
250	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
252	C₄²	#41×C₃₁	1	1	yes	characteristic
253	C₂³	—	4	2	no	—
254	C₂³	—	4	4	no	—
255	C₂³	—	8	2	no	—
256	C₂³	—	8	2	no	—
257	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
258	Q₈	—	8	2	no	—
259	Q₈	—	8	1	no	—
260	Q₈	—	8	1	no	—
261	D₄	—	4	4	no	—
262	D₄	—	8	2	no	—
263	D₄	—	8	2	no	—
264	D₄	—	8	2	no	—
265	D₄	—	8	2	no	—
266	D₄	—	8	1	no	—
267	D₄	—	8	1	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
273	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
274	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
275	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
276	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
277	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
278	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
279	C₂×C₄	C₃₁×#135	1	2	yes	—
280	C₈	—	2	2	no	—
281	C₈	—	2	2	no	—
282	C₈	—	2	4	no	—
283	C₈	—	4	2	no	—
284	C₈	—	8	1	no	—
285	C₈	—	8	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	8	2	no	—
289	C₂²	—	8	2	no	—
290	C₂²	—	8	2	no	—
291	C₂²	—	16	4	no	—
292	C₂²	C₃₁×#390	1	1	yes	characteristic
293	C₂²	C₄×D₈×C₃₁	1	2	yes	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	8	2	no	—
297	C₄	C₃₁×#413	1	2	yes	—
298	C₄	C₃₁×#466	1	2	yes	—
299	C₄	C₃₁×#471	1	2	yes	—
300	C₄	C₃₁×#493	1	1	yes	characteristic
301	C₄	C₃₁×#495	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	16	2	no	—
304	C₂	C₃₁×#1967	1	1	yes	characteristic
305	C₂	C₃₁×#1970	1	1	yes	characteristic
306	C₂	C₃₁×#1977	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#390
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1966	C₂	semidirect
4. C₃₁×#1567	C₂	semidirect
5. C₃₁×#1546	C₂	semidirect
7. C₈×M₁₆×C₃₁	C₂	semidirect
99. #9796	C₃₁	direct (N × H)
134. #1966	C₂×C₃₁	semidirect
136. #1567	C₂×C₃₁	semidirect
137. #1546	C₂×C₃₁	semidirect
139. C₈×M₁₆	C₂×C₃₁	semidirect
214. C₃₁	#9796	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#486	2
C₃₁×#1966	2
C₃₁×#1965	2
C₃₁×#1567	2
C₃₁×#1546	2
C₃₁×#1545	2
C₈×M₁₆×C₃₁	2
#9796	31

Subgroups — C31×#9796 #21466555

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9796 #21466555