Subgroups — C₃₁×#9791 #21466550

Order 7936.

474 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#495	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1568	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1527	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1469	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₁₆×C₃₁	C₄	1	1	yes	characteristic
7	C₂²×SD₁₆×C₃₁	C₄	1	2	yes	—
8	C₂²×D₈×C₃₁	C₄	1	1	yes	characteristic
9	C₂×C₃₁×#124	C₂²	1	2	yes	—
10	C₃₁×#495	—	2	1	no	—
11	C₃₁×#495	—	2	1	no	—
12	C₃₁×#495	—	2	1	no	—
13	C₃₁×#495	—	2	1	no	—
14	C₃₁×#432	C₂²	1	2	yes	—
15	C₃₁×#431	C₂²	1	1	yes	characteristic
16	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#335	—	2	1	no	—
19	C₃₁×#334	—	2	1	no	—
20	C₃₁×#329	—	2	4	no	—
21	C₃₁×#323	—	2	4	no	—
22	C₃₁×#322	—	2	1	no	—
23	C₃₁×#322	—	2	1	no	—
24	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
25	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
26	C₂×Q₁₆×C₃₁	—	2	1	no	—
27	C₂×Q₁₆×C₃₁	—	2	1	no	—
28	C₂×Q₁₆×C₃₁	—	4	2	no	—
29	C₂×SD₁₆×C₃₁	—	2	2	no	—
30	C₂×SD₁₆×C₃₁	—	2	2	no	—
31	C₂×SD₁₆×C₃₁	—	4	4	no	—
32	C₂×D₈×C₃₁	—	2	1	no	—
33	C₂×D₈×C₃₁	—	2	1	no	—
34	C₂×D₈×C₃₁	—	4	2	no	—
35	C₂×M₁₆×C₃₁	—	2	1	no	—
36	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₃₁×#124	—	2	2	no	—
39	C₃₁×#124	—	2	2	no	—
40	C₃₁×#124	—	2	2	no	—
41	C₃₁×#124	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
45	C₃₁×C₄⋊C₈	—	2	2	no	—
46	C₃₁×C₄⋊C₈	—	2	1	no	—
47	C₃₁×C₂²⋊C₈	—	2	2	no	—
48	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
49	C₃₁×#97	—	2	2	no	—
50	C₃₁×#97	D₄	1	2	yes	—
51	C₄×C₈×C₃₁	—	2	1	no	—
52	C₄×C₈×C₃₁	—	2	1	no	—
53	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
54	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
55	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
56	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
57	C₂⁴×C₃₁	—	2	2	no	—
58	C₂×Q₈×C₃₁	—	2	2	no	—
59	C₂×Q₈×C₃₁	—	2	2	no	—
60	C₂×Q₈×C₃₁	—	4	2	no	—
61	C₂×Q₈×C₃₁	—	4	2	no	—
62	C₂×D₄×C₃₁	—	2	2	no	—
63	C₂×D₄×C₃₁	—	2	2	no	—
64	C₂×D₄×C₃₁	—	4	2	no	—
65	C₂×D₄×C₃₁	—	4	2	no	—
66	C₂²×C₄×C₃₁	—	2	2	no	—
67	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
70	Q₁₆×C₃₁	—	4	2	no	—
71	Q₁₆×C₃₁	—	4	2	no	—
72	SD₁₆×C₃₁	—	4	4	no	—
73	SD₁₆×C₃₁	—	4	4	no	—
74	D₈×C₃₁	—	4	2	no	—
75	D₈×C₃₁	—	4	2	no	—
76	M₁₆×C₃₁	—	4	2	no	—
77	C₂×C₈×C₃₁	—	2	2	no	—
78	C₂×C₈×C₃₁	—	2	2	no	—
79	C₂×C₈×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	—	2	2	no	—
81	C₂×C₈×C₃₁	—	4	2	no	—
82	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
86	C₂²⋊C₄×C₃₁	—	4	2	no	—
87	C₂²⋊C₄×C₃₁	—	4	2	no	—
88	C₂²⋊C₄×C₃₁	—	4	2	no	—
89	C₂²⋊C₄×C₃₁	—	4	2	no	—
90	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₃₁	#41	1	1	yes	characteristic
94	#9791	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₃₁	—	4	2	no	—
96	C₂³×C₃₁	—	4	4	no	—
97	C₂³×C₃₁	—	8	2	no	—
98	C₂³×C₃₁	—	8	2	no	—
99	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
100	Q₈×C₃₁	—	4	2	no	—
101	Q₈×C₃₁	—	4	2	no	—
102	Q₈×C₃₁	—	8	2	no	—
103	D₄×C₃₁	—	4	2	no	—
104	D₄×C₃₁	—	4	2	no	—
105	D₄×C₃₁	—	8	2	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	1	no	—
108	C₂×C₄×C₃₁	—	2	1	no	—
109	C₂×C₄×C₃₁	—	4	1	no	—
110	C₂×C₄×C₃₁	—	4	2	no	—
111	C₂×C₄×C₃₁	—	8	2	no	—
112	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
115	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
116	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
117	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
118	C₈×C₃₁	—	2	1	no	—
119	C₈×C₃₁	—	2	1	no	—
120	C₈×C₃₁	—	4	2	no	—
121	C₈×C₃₁	—	4	2	no	—
122	C₈×C₃₁	—	8	2	no	—
123	C₈×C₃₁	#100	1	2	yes	—
124	C₈×C₃₁	#100	1	2	yes	—
125	C₂×#495	C₂×C₃₁	1	1	yes	characteristic
126	#1568	C₂×C₃₁	1	1	yes	characteristic, complemented
127	#1527	C₂×C₃₁	1	2	yes	complemented
128	#1469	C₂×C₃₁	1	2	yes	complemented
129	#1451	C₂×C₃₁	1	1	yes	characteristic, complemented
130	C₂²×C₃₁	—	2	1	no	—
131	C₂²×C₃₁	—	2	1	no	—
132	C₂²×C₃₁	—	8	2	no	—
133	C₂²×C₃₁	—	8	2	no	—
134	C₂²×C₃₁	—	8	2	no	—
135	C₂²×C₃₁	—	16	4	no	—
136	C₂²×C₃₁	#390	1	1	yes	characteristic
137	C₂²×C₃₁	#444	1	2	yes	—
138	C₄×C₃₁	—	2	1	no	—
139	C₄×C₃₁	—	2	1	no	—
140	C₄×C₃₁	—	2	1	no	—
141	C₄×C₃₁	—	2	1	no	—
142	C₄×C₃₁	—	2	1	no	—
143	C₄×C₃₁	—	8	2	no	—
144	C₄×C₃₁	C₂×#100	1	1	yes	characteristic
145	C₄×C₃₁	C₂×#100	1	1	yes	characteristic
146	C₂²×Q₁₆	C₄×C₃₁	1	1	yes	characteristic
147	C₂²×SD₁₆	C₄×C₃₁	1	2	yes	—
148	C₂²×D₈	C₄×C₃₁	1	1	yes	characteristic
149	C₂×#124	C₂²×C₃₁	1	2	yes	—
150	#495	—	2	1	no	—
151	#495	—	2	1	no	—
152	#495	—	2	1	no	—
153	#495	—	2	1	no	—
154	#432	C₂²×C₃₁	1	2	yes	—
155	#431	C₂²×C₃₁	1	1	yes	characteristic
156	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
157	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
158	#335	—	2	1	no	—
159	#334	—	2	1	no	—
160	#329	—	2	4	no	—
161	#323	—	2	4	no	—
162	#322	—	2	1	no	—
163	#322	—	2	1	no	—
164	C₂×C₃₁	—	2	2	no	—
165	C₂×C₃₁	—	16	2	no	—
166	C₂×C₃₁	#1965	1	1	yes	characteristic
167	C₂×C₃₁	#1975	1	1	yes	characteristic
168	C₂×C₃₁	#1976	1	1	yes	characteristic
169	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
170	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
171	C₂×Q₁₆	—	2	1	no	—
172	C₂×Q₁₆	—	2	1	no	—
173	C₂×Q₁₆	—	4	2	no	—
174	C₂×SD₁₆	—	2	2	no	—
175	C₂×SD₁₆	—	2	2	no	—
176	C₂×SD₁₆	—	4	4	no	—
177	C₂×D₈	—	2	1	no	—
178	C₂×D₈	—	2	1	no	—
179	C₂×D₈	—	4	2	no	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
182	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
183	#124	—	2	2	no	—
184	#124	—	2	2	no	—
185	#124	—	2	2	no	—
186	#124	—	2	2	no	—
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₂²⋊C₄	—	2	2	no	—
189	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
190	C₄⋊C₈	—	2	2	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₂²⋊C₈	—	2	2	no	—
193	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
194	#97	—	2	2	no	—
195	#97	D₄×C₃₁	1	2	yes	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
199	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
200	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
201	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
202	C₃₁	#9791	1	1	yes	characteristic, direct product factor, Sylow
203	C₂⁴	—	2	2	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂×Q₈	—	4	2	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
215	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
216	Q₁₆	—	4	2	no	—
217	Q₁₆	—	4	2	no	—
218	SD₁₆	—	4	4	no	—
219	SD₁₆	—	4	4	no	—
220	D₈	—	4	2	no	—
221	D₈	—	4	2	no	—
222	M₁₆	—	4	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
229	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
231	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
232	C₂²⋊C₄	—	4	2	no	—
233	C₂²⋊C₄	—	4	2	no	—
234	C₂²⋊C₄	—	4	2	no	—
235	C₂²⋊C₄	—	4	2	no	—
236	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
237	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
238	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
239	C₄²	#41×C₃₁	1	1	yes	characteristic
240	C₂³	—	4	2	no	—
241	C₂³	—	4	4	no	—
242	C₂³	—	8	2	no	—
243	C₂³	—	8	2	no	—
244	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
245	Q₈	—	4	2	no	—
246	Q₈	—	4	2	no	—
247	Q₈	—	8	2	no	—
248	D₄	—	4	2	no	—
249	D₄	—	4	2	no	—
250	D₄	—	8	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
258	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
259	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
260	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
261	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
262	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	4	2	no	—
266	C₈	—	4	2	no	—
267	C₈	—	8	2	no	—
268	C₈	C₃₁×#100	1	2	yes	—
269	C₈	C₃₁×#100	1	2	yes	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	2	no	—
274	C₂²	—	8	2	no	—
275	C₂²	—	16	4	no	—
276	C₂²	C₃₁×#390	1	1	yes	characteristic
277	C₂²	C₃₁×#444	1	2	yes	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	8	2	no	—
284	C₄	C₂×C₃₁×#100	1	1	yes	characteristic
285	C₄	C₂×C₃₁×#100	1	1	yes	characteristic
286	C₂	—	2	2	no	—
287	C₂	—	16	2	no	—
288	C₂	C₃₁×#1965	1	1	yes	characteristic
289	C₂	C₃₁×#1975	1	1	yes	characteristic
290	C₂	C₃₁×#1976	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	C₂²×D₈×C₃₁	C₄
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×#1568	C₂	semidirect
3. C₃₁×#1527	C₂	semidirect
4. C₃₁×#1469	C₂	semidirect
5. C₃₁×#1451	C₂	semidirect
94. #9791	C₃₁	direct (N × H)
126. #1568	C₂×C₃₁	semidirect
127. #1527	C₂×C₃₁	semidirect
128. #1469	C₂×C₃₁	semidirect
129. #1451	C₂×C₃₁	semidirect
202. C₃₁	#9791	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×#495	2
C₃₁×#1568	2
C₃₁×#1527	2
C₃₁×#1469	2
C₃₁×#1451	2
#9791	31

Subgroups — C31×#9791 #21466550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9791 #21466550