Subgroups — C₃₁×#9734 #21466493

Order 7936.

654 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1982	C₂	1	2	yes	maximal, complemented
3	C₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#331	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₃₁	—	2	1	no	—
7	C₂²×D₈×C₃₁	—	2	1	no	—
8	C₂×C₃₁×#127	C₂²	1	2	yes	—
9	C₃₁×#493	—	2	2	no	—
10	C₃₁×#493	C₄	1	4	yes	complemented
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#102	—	2	4	no	—
14	C₂×C₄×C₈×C₃₁	—	2	1	no	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#336	—	2	1	no	—
18	C₃₁×#335	—	2	2	no	—
19	C₃₁×#331	—	2	4	no	—
20	C₂²×D₄×C₃₁	—	2	2	no	—
21	C₂²×D₄×C₃₁	—	2	2	no	—
22	C₂²×D₄×C₃₁	—	2	2	no	—
23	C₂×D₈×C₃₁	—	2	4	no	—
24	C₂×D₈×C₃₁	—	2	4	no	—
25	C₂×D₈×C₃₁	—	4	2	no	—
26	C₂×D₈×C₃₁	—	4	2	no	—
27	C₂×D₈×C₃₁	—	4	2	no	—
28	C₂×D₈×C₃₁	—	4	2	no	—
29	C₂²×C₈×C₃₁	—	2	1	no	—
30	C₂²×C₈×C₃₁	—	2	2	no	—
31	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
33	C₃₁×#127	—	2	2	no	—
34	C₃₁×#127	—	2	2	no	—
35	C₃₁×#127	—	2	2	no	—
36	C₃₁×#127	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
38	C₂×C₄²×C₃₁	—	2	1	no	—
39	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
40	C₃₁×C₄⋊C₈	—	2	2	no	—
41	C₃₁×C₄⋊C₈	—	2	2	no	—
42	C₃₁×C₄⋊C₈	—	2	1	no	—
43	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
44	C₃₁×#102	—	4	8	no	—
45	C₄×C₈×C₃₁	—	4	2	no	—
46	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₃₁	D₄	1	2	yes	—
48	C₃₁×#95	—	2	1	no	—
49	C₂⁴×C₃₁	—	4	2	no	—
50	C₂×D₄×C₃₁	—	2	4	no	—
51	C₂×D₄×C₃₁	—	2	4	no	—
52	C₂×D₄×C₃₁	—	2	4	no	—
53	C₂×D₄×C₃₁	—	4	2	no	—
54	C₂×D₄×C₃₁	—	4	2	no	—
55	C₂×D₄×C₃₁	—	4	2	no	—
56	C₂×D₄×C₃₁	—	4	4	no	—
57	C₂×D₄×C₃₁	—	4	2	no	—
58	C₂×D₄×C₃₁	—	4	2	no	—
59	C₂×D₄×C₃₁	—	4	4	no	—
60	C₂×D₄×C₃₁	—	4	2	no	—
61	C₂×D₄×C₃₁	—	4	2	no	—
62	C₂×D₄×C₃₁	—	8	4	no	—
63	C₂²×C₄×C₃₁	—	2	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	8	no	—
68	D₈×C₃₁	—	4	8	no	—
69	C₂×C₈×C₃₁	—	2	1	no	—
70	C₂×C₈×C₃₁	—	2	1	no	—
71	C₂×C₈×C₃₁	—	2	2	no	—
72	C₂×C₈×C₃₁	—	4	1	no	—
73	C₂×C₈×C₃₁	—	4	1	no	—
74	C₂×C₈×C₃₁	—	4	2	no	—
75	C₂×C₈×C₃₁	—	4	4	no	—
76	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₃₁	D₈	1	2	yes	—
79	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
80	C₄⋊C₄×C₃₁	—	2	2	no	—
81	C₄⋊C₄×C₃₁	D₈	1	4	yes	complemented
82	C₄²×C₃₁	—	4	1	no	—
83	C₄²×C₃₁	—	4	1	no	—
84	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₃₁	#41	1	1	yes	characteristic
88	#9734	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₃₁	—	4	4	no	—
90	C₂³×C₃₁	—	8	2	no	—
91	C₂³×C₃₁	—	8	2	no	—
92	C₂³×C₃₁	—	8	4	no	—
93	C₂³×C₃₁	—	8	4	no	—
94	C₂³×C₃₁	#121	1	1	yes	characteristic
95	D₄×C₃₁	—	4	4	no	—
96	D₄×C₃₁	—	4	4	no	—
97	D₄×C₃₁	—	4	8	no	—
98	D₄×C₃₁	—	8	4	no	—
99	D₄×C₃₁	—	8	4	no	—
100	D₄×C₃₁	—	8	2	no	—
101	D₄×C₃₁	—	8	2	no	—
102	D₄×C₃₁	—	8	2	no	—
103	D₄×C₃₁	—	8	2	no	—
104	C₂×C₄×C₃₁	—	2	2	no	—
105	C₂×C₄×C₃₁	—	2	4	no	—
106	C₂×C₄×C₃₁	—	4	2	no	—
107	C₂×C₄×C₃₁	—	4	2	no	—
108	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
112	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
113	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₃₁	#135	1	2	yes	—
118	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₈×C₃₁	—	2	2	no	—
121	C₈×C₃₁	—	4	2	no	—
122	C₈×C₃₁	—	8	2	no	—
123	C₈×C₃₁	#102	1	4	yes	complemented
124	C₂×#493	C₂×C₃₁	1	1	yes	characteristic
125	#1982	C₂×C₃₁	1	2	yes	complemented
126	C₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
127	C₂×#335	C₂×C₃₁	1	1	yes	characteristic, complemented
128	C₂×#331	C₂×C₃₁	1	2	yes	complemented
129	C₂²×C₃₁	—	8	4	no	—
130	C₂²×C₃₁	—	8	4	no	—
131	C₂²×C₃₁	—	8	4	no	—
132	C₂²×C₃₁	—	16	4	no	—
133	C₂²×C₃₁	—	16	4	no	—
134	C₂²×C₃₁	#390	1	1	yes	characteristic
135	C₂²×C₃₁	#459	1	1	yes	characteristic
136	C₂²×C₃₁	#460	1	1	yes	characteristic
137	C₂²×C₃₁	#466	1	2	yes	—
138	C₂²×C₃₁	#472	1	2	yes	—
139	C₄×C₃₁	—	2	1	no	—
140	C₄×C₃₁	—	2	1	no	—
141	C₄×C₃₁	—	4	4	no	—
142	C₄×C₃₁	C₂×#102	1	1	yes	characteristic
143	C₄×C₃₁	#418	1	1	yes	characteristic
144	C₄×C₃₁	C₄×D₈	1	2	yes	—
145	C₄×C₃₁	#459	1	1	yes	characteristic
146	C₄×C₃₁	#460	1	1	yes	characteristic
147	C₄×C₃₁	#493	1	1	yes	characteristic
148	C₄×C₃₁	#495	1	1	yes	characteristic
149	C₂²×D₈	—	2	1	no	—
150	C₂²×D₈	—	2	1	no	—
151	C₂×#127	C₂²×C₃₁	1	2	yes	—
152	#493	—	2	2	no	—
153	#493	C₄×C₃₁	1	4	yes	complemented
154	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
155	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
156	C₂×#102	—	2	4	no	—
157	C₂×C₄×C₈	—	2	1	no	—
158	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
159	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
160	#336	—	2	1	no	—
161	#335	—	2	2	no	—
162	#331	—	2	4	no	—
163	C₂×C₃₁	—	16	4	no	—
164	C₂×C₃₁	#1667	1	2	yes	—
165	C₂×C₃₁	#1668	1	2	yes	—
166	C₂×C₃₁	#1970	1	1	yes	characteristic
167	C₂×C₃₁	#1978	1	1	yes	characteristic
168	C₂×C₃₁	#1982	1	1	yes	characteristic
169	C₂²×D₄	—	2	2	no	—
170	C₂²×D₄	—	2	2	no	—
171	C₂²×D₄	—	2	2	no	—
172	C₂×D₈	—	2	4	no	—
173	C₂×D₈	—	2	4	no	—
174	C₂×D₈	—	4	2	no	—
175	C₂×D₈	—	4	2	no	—
176	C₂×D₈	—	4	2	no	—
177	C₂×D₈	—	4	2	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
181	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
182	#127	—	2	2	no	—
183	#127	—	2	2	no	—
184	#127	—	2	2	no	—
185	#127	C₂×C₄×C₃₁	1	4	yes	—
186	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
187	C₂×C₄²	—	2	1	no	—
188	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
189	C₄⋊C₈	—	2	2	no	—
190	C₄⋊C₈	—	2	2	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
193	#102	—	4	8	no	—
194	C₄×C₈	—	4	2	no	—
195	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
196	C₄×C₈	D₄×C₃₁	1	2	yes	—
197	#95	—	2	1	no	—
198	C₃₁	#9734	1	1	yes	characteristic, direct product factor, Sylow
199	C₂⁴	—	4	2	no	—
200	C₂×D₄	—	2	4	no	—
201	C₂×D₄	—	2	4	no	—
202	C₂×D₄	—	2	4	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂×D₄	—	4	4	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂×D₄	—	4	4	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	8	4	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
216	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
217	D₈	—	4	8	no	—
218	D₈	—	4	8	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	4	no	—
226	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
227	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
228	C₂×C₈	D₈×C₃₁	1	2	yes	—
229	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	D₈×C₃₁	1	4	yes	complemented
232	C₄²	—	4	1	no	—
233	C₄²	—	4	1	no	—
234	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
235	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
236	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
237	C₄²	#41×C₃₁	1	1	yes	characteristic
238	C₂³	—	4	4	no	—
239	C₂³	—	8	2	no	—
240	C₂³	—	8	2	no	—
241	C₂³	—	8	4	no	—
242	C₂³	—	8	4	no	—
243	C₂³	C₃₁×#121	1	1	yes	characteristic
244	D₄	—	4	4	no	—
245	D₄	—	4	4	no	—
246	D₄	—	4	8	no	—
247	D₄	—	8	4	no	—
248	D₄	—	8	4	no	—
249	D₄	—	8	2	no	—
250	D₄	—	8	2	no	—
251	D₄	—	8	2	no	—
252	D₄	—	8	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	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	C₄×D₄×C₃₁	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	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
264	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
265	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
266	C₂×C₄	C₃₁×#135	1	2	yes	—
267	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
268	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	4	2	no	—
271	C₈	—	8	2	no	—
272	C₈	C₃₁×#102	1	4	yes	complemented
273	C₂²	—	8	4	no	—
274	C₂²	—	8	4	no	—
275	C₂²	—	8	4	no	—
276	C₂²	—	16	4	no	—
277	C₂²	—	16	4	no	—
278	C₂²	C₃₁×#390	1	1	yes	characteristic
279	C₂²	C₃₁×#459	1	1	yes	characteristic
280	C₂²	C₃₁×#460	1	1	yes	characteristic
281	C₂²	C₃₁×#466	1	2	yes	—
282	C₂²	C₃₁×#472	1	2	yes	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	4	4	no	—
286	C₄	C₂×C₃₁×#102	1	1	yes	characteristic
287	C₄	C₃₁×#418	1	1	yes	characteristic
288	C₄	C₄×D₈×C₃₁	1	2	yes	—
289	C₄	C₃₁×#459	1	1	yes	characteristic
290	C₄	C₃₁×#460	1	1	yes	characteristic
291	C₄	C₃₁×#493	1	1	yes	characteristic
292	C₄	C₃₁×#495	1	1	yes	characteristic
293	C₂	—	16	4	no	—
294	C₂	C₃₁×#1667	1	2	yes	—
295	C₂	C₃₁×#1668	1	2	yes	—
296	C₂	C₃₁×#1970	1	1	yes	characteristic
297	C₂	C₃₁×#1978	1	1	yes	characteristic
298	C₂	C₃₁×#1982	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#121
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂×C₃₁×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1982	C₂	semidirect
3. C₈×C₄⋊C₄×C₃₁	C₂	semidirect
4. C₂×C₃₁×#335	C₂	semidirect
5. C₂×C₃₁×#331	C₂	semidirect
10. C₃₁×#493	C₄	semidirect
81. C₄⋊C₄×C₃₁	D₈	semidirect
88. #9734	C₃₁	direct (N × H)
123. C₈×C₃₁	#102	semidirect
125. #1982	C₂×C₃₁	semidirect
126. C₈×C₄⋊C₄	C₂×C₃₁	semidirect
127. C₂×#335	C₂×C₃₁	semidirect
128. C₂×#331	C₂×C₃₁	semidirect
153. #493	C₄×C₃₁	semidirect
198. C₃₁	#9734	direct (N × H)
231. C₄⋊C₄	D₈×C₃₁	semidirect
272. C₈	C₃₁×#102	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#493	2
C₃₁×#1982	2
C₈×C₄⋊C₄×C₃₁	2
C₂×C₃₁×#335	2
C₂×C₃₁×#331	2
#9734	31

Subgroups — C31×#9734 #21466493

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9734 #21466493