Subgroups — C₃₁×#9762 #21466521

Order 7936.

462 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1726	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1568	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1530	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1505	C₂	1	2	yes	maximal, complemented
6	C₃₁×#532	C₂²	1	2	yes	—
7	C₃₁×#500	C₄	1	1	yes	characteristic, complemented
8	C₃₁×#493	C₄	1	1	yes	characteristic, complemented
9	C₃₁×#486	C₄	1	2	yes	complemented
10	C₃₁×#466	—	2	1	no	—
11	C₃₁×#466	—	2	1	no	—
12	C₄×D₈×C₃₁	—	2	1	no	—
13	C₄×D₈×C₃₁	—	2	1	no	—
14	C₃₁×#432	C₂²	1	2	yes	—
15	C₂×C₃₁×#97	—	2	1	no	—
16	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₃₁	—	2	1	no	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#376	C₂²	1	1	yes	characteristic
20	C₃₁×#335	—	2	1	no	—
21	C₃₁×#334	—	2	1	no	—
22	C₃₁×#329	—	2	4	no	—
23	C₃₁×#327	—	2	4	no	—
24	C₂×#41×C₃₁	—	2	2	no	—
25	C₂×D₈×C₃₁	—	2	1	no	—
26	C₂×D₈×C₃₁	—	2	1	no	—
27	C₂²×C₈×C₃₁	—	2	1	no	—
28	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
30	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
31	C₃₁×#127	C₂×C₄	1	2	yes	—
32	C₃₁×#124	—	2	2	no	—
33	C₃₁×#121	—	2	2	no	—
34	C₃₁×#121	—	2	2	no	—
35	C₄×D₄×C₃₁	—	2	2	no	—
36	C₄×D₄×C₃₁	—	2	2	no	—
37	C₂×C₄²×C₃₁	—	2	1	no	—
38	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
39	C₃₁×#107	—	2	1	no	—
40	C₃₁×#107	—	2	1	no	—
41	C₃₁×C₄⋊C₈	—	2	2	no	—
42	C₃₁×#102	—	2	2	no	—
43	C₃₁×#102	—	2	2	no	—
44	C₃₁×C₂²⋊C₈	—	2	2	no	—
45	C₃₁×#97	—	2	4	no	—
46	C₃₁×#97	—	2	2	no	—
47	C₃₁×#97	—	2	1	no	—
48	C₃₁×#97	D₄	1	2	yes	—
49	C₄×C₈×C₃₁	—	2	4	no	—
50	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
53	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
54	C₃₁×#95	—	2	1	no	—
55	C₃₁×#95	D₄	1	2	yes	—
56	#41×C₃₁	—	8	4	no	—
57	C₂×Q₈×C₃₁	—	2	2	no	—
58	C₂×D₄×C₃₁	—	2	2	no	—
59	C₂×D₄×C₃₁	—	2	2	no	—
60	C₂×D₄×C₃₁	—	2	2	no	—
61	C₂×D₄×C₃₁	—	4	2	no	—
62	C₂²×C₄×C₃₁	—	2	2	no	—
63	C₂²×C₄×C₃₁	—	4	2	no	—
64	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
67	D₈×C₃₁	—	4	2	no	—
68	D₈×C₃₁	—	4	2	no	—
69	C₂×C₈×C₃₁	—	2	2	no	—
70	C₂×C₈×C₃₁	—	2	2	no	—
71	C₂×C₈×C₃₁	—	2	2	no	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	—	4	2	no	—
75	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₄⋊C₄×C₃₁	—	2	2	no	—
80	C₄⋊C₄×C₃₁	—	2	2	no	—
81	C₂²⋊C₄×C₃₁	—	4	2	no	—
82	C₂²⋊C₄×C₃₁	—	4	2	no	—
83	C₄²×C₃₁	—	2	2	no	—
84	C₄²×C₃₁	—	2	2	no	—
85	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₃₁	#41	1	1	yes	characteristic
89	#9762	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	—	4	2	no	—
91	C₂³×C₃₁	#121	1	1	yes	characteristic
92	Q₈×C₃₁	—	8	2	no	—
93	D₄×C₃₁	—	4	2	no	—
94	D₄×C₃₁	—	4	2	no	—
95	D₄×C₃₁	—	8	2	no	—
96	D₄×C₃₁	—	8	2	no	—
97	D₄×C₃₁	—	8	4	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	1	no	—
100	C₂×C₄×C₃₁	—	2	1	no	—
101	C₂×C₄×C₃₁	—	2	4	no	—
102	C₂×C₄×C₃₁	—	4	1	no	—
103	C₂×C₄×C₃₁	—	4	2	no	—
104	C₂×C₄×C₃₁	—	4	2	no	—
105	C₂×C₄×C₃₁	—	4	2	no	—
106	C₂×C₄×C₃₁	—	8	4	no	—
107	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
111	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
112	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
113	C₈×C₃₁	—	2	4	no	—
114	C₈×C₃₁	—	2	1	no	—
115	C₈×C₃₁	—	2	1	no	—
116	C₈×C₃₁	—	8	2	no	—
117	C₈×C₃₁	#104	1	2	yes	—
118	C₈×C₃₁	#104	1	2	yes	—
119	#3169	C₂×C₃₁	1	1	yes	characteristic
120	#1726	C₂×C₃₁	1	1	yes	characteristic, complemented
121	#1568	C₂×C₃₁	1	1	yes	characteristic, complemented
122	#1530	C₂×C₃₁	1	2	yes	complemented
123	#1505	C₂×C₃₁	1	2	yes	complemented
124	C₂²×C₃₁	—	2	1	no	—
125	C₂²×C₃₁	—	2	1	no	—
126	C₂²×C₃₁	—	8	2	no	—
127	C₂²×C₃₁	—	8	2	no	—
128	C₂²×C₃₁	—	8	2	no	—
129	C₂²×C₃₁	#390	1	1	yes	characteristic
130	C₂²×C₃₁	#472	1	2	yes	—
131	C₄×C₃₁	—	2	1	no	—
132	C₄×C₃₁	—	2	1	no	—
133	C₄×C₃₁	—	2	1	no	—
134	C₄×C₃₁	—	2	1	no	—
135	C₄×C₃₁	—	2	1	no	—
136	C₄×C₃₁	—	4	4	no	—
137	C₄×C₃₁	—	8	2	no	—
138	C₄×C₃₁	C₂×#104	1	1	yes	characteristic
139	C₄×C₃₁	C₂×#104	1	1	yes	characteristic
140	#532	C₂²×C₃₁	1	2	yes	—
141	#500	C₄×C₃₁	1	1	yes	characteristic, complemented
142	#493	C₄×C₃₁	1	1	yes	characteristic, complemented
143	#486	C₄×C₃₁	1	2	yes	complemented
144	#466	—	2	1	no	—
145	#466	—	2	1	no	—
146	C₄×D₈	—	2	1	no	—
147	C₄×D₈	—	2	1	no	—
148	#432	C₂²×C₃₁	1	2	yes	—
149	C₂×#97	—	2	1	no	—
150	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
151	C₂×C₄×C₈	—	2	1	no	—
152	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
153	#376	C₂²×C₃₁	1	1	yes	characteristic
154	#335	—	2	1	no	—
155	#334	—	2	1	no	—
156	#329	—	2	4	no	—
157	#327	—	2	4	no	—
158	C₂×C₃₁	—	2	2	no	—
159	C₂×C₃₁	—	16	2	no	—
160	C₂×C₃₁	#1962	1	1	yes	characteristic
161	C₂×C₃₁	#1976	1	1	yes	characteristic
162	C₂×C₃₁	#1978	1	1	yes	characteristic
163	C₂×#41	—	2	2	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂×D₈	—	2	1	no	—
166	C₂²×C₈	—	2	1	no	—
167	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
168	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
169	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
170	#127	C₂×C₄×C₃₁	1	2	yes	—
171	#124	—	2	2	no	—
172	#121	—	2	2	no	—
173	#121	—	2	2	no	—
174	C₄×D₄	—	2	2	no	—
175	C₄×D₄	—	2	2	no	—
176	C₂×C₄²	—	2	1	no	—
177	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
178	#107	—	2	1	no	—
179	#107	—	2	1	no	—
180	C₄⋊C₈	—	2	2	no	—
181	#102	—	2	2	no	—
182	#102	—	2	2	no	—
183	C₂²⋊C₈	—	2	2	no	—
184	#97	—	2	4	no	—
185	#97	—	2	2	no	—
186	#97	—	2	1	no	—
187	#97	D₄×C₃₁	1	2	yes	—
188	C₄×C₈	—	2	4	no	—
189	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
190	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
191	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
192	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
193	#95	—	2	1	no	—
194	#95	D₄×C₃₁	1	2	yes	—
195	C₃₁	#9762	1	1	yes	characteristic, direct product factor, Sylow
196	#41	—	8	4	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
206	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
207	D₈	—	4	2	no	—
208	D₈	—	4	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
216	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
217	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₂²⋊C₄	—	4	2	no	—
222	C₂²⋊C₄	—	4	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	2	no	—
225	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
226	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
227	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
228	C₄²	#41×C₃₁	1	1	yes	characteristic
229	C₂³	—	4	2	no	—
230	C₂³	C₃₁×#121	1	1	yes	characteristic
231	Q₈	—	8	2	no	—
232	D₄	—	4	2	no	—
233	D₄	—	4	2	no	—
234	D₄	—	8	2	no	—
235	D₄	—	8	2	no	—
236	D₄	—	8	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
247	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
248	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
249	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
250	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
251	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
252	C₈	—	2	4	no	—
253	C₈	—	2	1	no	—
254	C₈	—	2	1	no	—
255	C₈	—	8	2	no	—
256	C₈	C₃₁×#104	1	2	yes	—
257	C₈	C₃₁×#104	1	2	yes	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	8	2	no	—
262	C₂²	—	8	2	no	—
263	C₂²	C₃₁×#390	1	1	yes	characteristic
264	C₂²	C₃₁×#472	1	2	yes	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	4	4	no	—
271	C₄	—	8	2	no	—
272	C₄	C₂×C₃₁×#104	1	1	yes	characteristic
273	C₄	C₂×C₃₁×#104	1	1	yes	characteristic
274	C₂	—	2	2	no	—
275	C₂	—	16	2	no	—
276	C₂	C₃₁×#1962	1	1	yes	characteristic
277	C₂	C₃₁×#1976	1	1	yes	characteristic
278	C₂	C₃₁×#1978	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#390
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₃₁×#3169	C₂
LCS	G ▷ C₄² ▷ 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×#1726	C₂	semidirect
3. C₃₁×#1568	C₂	semidirect
4. C₃₁×#1530	C₂	semidirect
5. C₃₁×#1505	C₂	semidirect
7. C₃₁×#500	C₄	semidirect
8. C₃₁×#493	C₄	semidirect
9. C₃₁×#486	C₄	semidirect
89. #9762	C₃₁	direct (N × H)
120. #1726	C₂×C₃₁	semidirect
121. #1568	C₂×C₃₁	semidirect
122. #1530	C₂×C₃₁	semidirect
123. #1505	C₂×C₃₁	semidirect
141. #500	C₄×C₃₁	semidirect
142. #493	C₄×C₃₁	semidirect
143. #486	C₄×C₃₁	semidirect
195. C₃₁	#9762	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×#3169	2
C₃₁×#1726	2
C₃₁×#1568	2
C₃₁×#1530	2
C₃₁×#1505	2
#9762	31

Subgroups — C31×#9762 #21466521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9762 #21466521