Subgroups — C₃₁×#9695 #21466454

Order 7936.

882 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2867	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#331	C₂	1	4	yes	maximal, complemented
5	C₂×C₃₁×#127	—	2	1	no	—
6	C₂×C₃₁×#127	—	2	1	no	—
7	C₂×C₃₁×#127	—	2	2	no	—
8	C₂×C₃₁×#127	C₄	1	4	yes	—
9	C₂×C₃₁×#127	C₂²	1	2	yes	—
10	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₃₁	—	2	1	no	—
14	C₄³×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#336	—	2	1	no	—
16	C₃₁×#331	—	2	8	no	—
17	C₂²×D₄×C₃₁	—	2	4	no	—
18	C₂²×D₄×C₃₁	—	2	4	no	—
19	C₂²×D₄×C₃₁	—	2	2	no	—
20	C₂²×D₄×C₃₁	—	2	2	no	—
21	C₂²×D₄×C₃₁	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₃₁	—	2	2	no	—
23	C₂²×C₈×C₃₁	—	2	2	no	—
24	C₃₁×#127	—	2	4	no	—
25	C₃₁×#127	—	2	4	no	—
26	C₃₁×#127	—	4	4	no	—
27	C₃₁×#127	—	4	8	no	—
28	C₃₁×#127	—	4	2	no	—
29	C₃₁×#127	—	4	4	no	—
30	C₃₁×#127	—	4	4	no	—
31	C₃₁×#127	—	4	2	no	—
32	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₃₁	D₄	1	2	yes	—
36	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
37	C₃₁×C₄⋊C₈	—	2	4	no	—
38	C₃₁×C₄⋊C₈	—	2	2	no	—
39	C₃₁×C₄⋊C₈	—	4	4	no	—
40	C₃₁×C₄⋊C₈	D₄	1	4	yes	complemented
41	C₄×C₈×C₃₁	—	4	2	no	—
42	C₂⁴×C₃₁	—	2	4	no	—
43	C₂×D₄×C₃₁	—	2	8	no	—
44	C₂×D₄×C₃₁	—	4	8	no	—
45	C₂×D₄×C₃₁	—	4	8	no	—
46	C₂×D₄×C₃₁	—	4	8	no	—
47	C₂×D₄×C₃₁	—	4	8	no	—
48	C₂×D₄×C₃₁	—	4	8	no	—
49	C₂×D₄×C₃₁	—	4	8	no	—
50	C₂×D₄×C₃₁	—	4	8	no	—
51	C₂×D₄×C₃₁	—	4	4	no	—
52	C₂×D₄×C₃₁	—	4	2	no	—
53	C₂×D₄×C₃₁	—	4	2	no	—
54	C₂×D₄×C₃₁	—	4	2	no	—
55	C₂×D₄×C₃₁	—	4	4	no	—
56	C₂×D₄×C₃₁	—	4	2	no	—
57	C₂×D₄×C₃₁	—	4	4	no	—
58	C₂×D₄×C₃₁	—	4	4	no	—
59	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₃₁	—	2	4	no	—
65	C₂×C₈×C₃₁	—	4	2	no	—
66	C₂×C₈×C₃₁	—	4	2	no	—
67	C₂×C₈×C₃₁	—	4	2	no	—
68	C₂×C₈×C₃₁	—	4	4	no	—
69	C₄²×C₃₁	—	2	4	no	—
70	C₄²×C₃₁	—	2	2	no	—
71	C₄²×C₃₁	D₈	1	2	yes	—
72	C₄²×C₃₁	D₈	1	4	yes	—
73	C₄²×C₃₁	SD₁₆	1	4	yes	—
74	C₄²×C₃₁	SD₁₆	1	2	yes	—
75	C₄²×C₃₁	C₂×D₄	1	2	yes	—
76	C₄²×C₃₁	#41	1	2	yes	—
77	#9695	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₃₁	—	4	4	no	—
79	C₂³×C₃₁	—	4	8	no	—
80	C₂³×C₃₁	—	4	4	no	—
81	C₂³×C₃₁	—	4	8	no	—
82	C₂³×C₃₁	—	4	4	no	—
83	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	D₄×C₃₁	—	8	8	no	—
85	D₄×C₃₁	—	8	16	no	—
86	D₄×C₃₁	—	8	16	no	—
87	D₄×C₃₁	—	8	4	no	—
88	D₄×C₃₁	—	8	4	no	—
89	D₄×C₃₁	—	8	4	no	—
90	D₄×C₃₁	—	8	2	no	—
91	D₄×C₃₁	—	8	2	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	#100	1	4	yes	—
96	C₂×C₄×C₃₁	#102	1	4	yes	—
97	C₂×C₄×C₃₁	#102	1	4	yes	—
98	C₂×C₄×C₃₁	#102	1	4	yes	—
99	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₃₁	#121	1	2	yes	—
101	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
102	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
103	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
105	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
107	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
108	C₈×C₃₁	—	4	4	no	—
109	C₈×C₃₁	—	8	2	no	—
110	#2867	C₂×C₃₁	1	1	yes	characteristic
111	#1842	C₂×C₃₁	1	1	yes	characteristic, complemented
112	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
113	C₂×#331	C₂×C₃₁	1	4	yes	complemented
114	C₂²×C₃₁	—	8	8	no	—
115	C₂²×C₃₁	—	8	8	no	—
116	C₂²×C₃₁	—	8	4	no	—
117	C₂²×C₃₁	—	8	4	no	—
118	C₂²×C₃₁	—	8	4	no	—
119	C₂²×C₃₁	#390	1	1	yes	characteristic
120	C₂²×C₃₁	C₂×#100	1	2	yes	—
121	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
122	C₂²×C₃₁	C₂×#102	1	2	yes	—
123	C₂²×C₃₁	#418	1	1	yes	characteristic
124	C₄×C₃₁	—	2	4	no	—
125	C₄×C₃₁	—	2	2	no	—
126	C₄×C₃₁	#331	1	8	yes	complemented
127	C₄×C₃₁	#459	1	2	yes	—
128	C₄×C₃₁	#460	1	2	yes	—
129	C₄×C₃₁	#486	1	2	yes	—
130	C₄×C₃₁	#492	1	1	yes	characteristic
131	C₄×C₃₁	#493	1	1	yes	characteristic
132	C₂×#127	—	2	1	no	—
133	C₂×#127	—	2	1	no	—
134	C₂×#127	—	2	2	no	—
135	C₂×#127	C₄×C₃₁	1	4	yes	—
136	C₂×#127	C₂²×C₃₁	1	2	yes	—
137	C₂×C₄⋊C₈	—	2	2	no	—
138	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
139	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
140	C₂×C₄×C₈	—	2	1	no	—
141	C₄³	C₂²×C₃₁	1	1	yes	characteristic
142	#336	—	2	1	no	—
143	#331	—	2	8	no	—
144	C₂×C₃₁	—	16	4	no	—
145	C₂×C₃₁	C₂×#331	1	2	yes	—
146	C₂×C₃₁	#1545	1	2	yes	—
147	C₂×C₃₁	#1957	1	1	yes	characteristic
148	C₂×C₃₁	#1965	1	1	yes	characteristic
149	C₂×C₃₁	#1982	1	1	yes	characteristic
150	C₂²×D₄	—	2	4	no	—
151	C₂²×D₄	—	2	4	no	—
152	C₂²×D₄	—	2	2	no	—
153	C₂²×D₄	—	2	2	no	—
154	C₂²×D₄	C₂×C₄×C₃₁	1	4	yes	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	2	2	no	—
157	#127	—	2	4	no	—
158	#127	—	2	4	no	—
159	#127	—	4	8	no	—
160	#127	—	4	4	no	—
161	#127	—	4	2	no	—
162	#127	—	4	4	no	—
163	#127	—	4	4	no	—
164	#127	—	4	2	no	—
165	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
166	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
167	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
168	C₂×C₄²	D₄×C₃₁	1	2	yes	—
169	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
170	C₄⋊C₈	—	2	4	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	4	4	no	—
173	C₄⋊C₈	D₄×C₃₁	1	4	yes	complemented
174	C₄×C₈	—	4	2	no	—
175	C₃₁	#9695	1	1	yes	characteristic, direct product factor, Sylow
176	C₂⁴	—	2	4	no	—
177	C₂×D₄	—	2	8	no	—
178	C₂×D₄	—	4	8	no	—
179	C₂×D₄	—	4	8	no	—
180	C₂×D₄	—	4	8	no	—
181	C₂×D₄	—	4	8	no	—
182	C₂×D₄	—	4	8	no	—
183	C₂×D₄	—	4	8	no	—
184	C₂×D₄	—	4	8	no	—
185	C₂×D₄	—	4	4	no	—
186	C₂×D₄	—	4	2	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂×D₄	—	4	4	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂×D₄	—	4	4	no	—
192	C₂×D₄	—	4	4	no	—
193	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
194	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
195	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	4	4	no	—
203	C₄²	—	2	4	no	—
204	C₄²	—	2	2	no	—
205	C₄²	D₈×C₃₁	1	2	yes	—
206	C₄²	D₈×C₃₁	1	4	yes	—
207	C₄²	SD₁₆×C₃₁	1	4	yes	—
208	C₄²	SD₁₆×C₃₁	1	2	yes	—
209	C₄²	C₂×D₄×C₃₁	1	2	yes	—
210	C₄²	#41×C₃₁	1	2	yes	—
211	C₂³	—	4	8	no	—
212	C₂³	—	4	8	no	—
213	C₂³	—	4	4	no	—
214	C₂³	—	4	4	no	—
215	C₂³	—	4	4	no	—
216	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
217	D₄	—	8	8	no	—
218	D₄	—	8	16	no	—
219	D₄	—	8	4	no	—
220	D₄	—	8	4	no	—
221	D₄	—	8	4	no	—
222	D₄	—	8	2	no	—
223	D₄	—	8	2	no	—
224	D₄	—	8	16	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	C₃₁×#100	1	4	yes	—
229	C₂×C₄	C₃₁×#102	1	4	yes	—
230	C₂×C₄	C₃₁×#102	1	4	yes	—
231	C₂×C₄	C₃₁×#102	1	4	yes	—
232	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
233	C₂×C₄	C₃₁×#121	1	2	yes	—
234	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
235	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
236	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
237	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
238	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
239	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
240	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
241	C₈	—	4	4	no	—
242	C₈	—	8	2	no	—
243	C₂²	—	8	8	no	—
244	C₂²	—	8	4	no	—
245	C₂²	—	8	4	no	—
246	C₂²	—	8	4	no	—
247	C₂²	—	8	8	no	—
248	C₂²	C₃₁×#390	1	1	yes	characteristic
249	C₂²	C₂×C₃₁×#100	1	2	yes	—
250	C₂²	C₂×C₃₁×#102	1	2	yes	—
251	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
252	C₂²	C₃₁×#418	1	1	yes	characteristic
253	C₄	—	2	4	no	—
254	C₄	—	2	2	no	—
255	C₄	C₃₁×#331	1	8	yes	complemented
256	C₄	C₃₁×#459	1	2	yes	—
257	C₄	C₃₁×#460	1	2	yes	—
258	C₄	C₃₁×#486	1	2	yes	—
259	C₄	C₃₁×#492	1	1	yes	characteristic
260	C₄	C₃₁×#493	1	1	yes	characteristic
261	C₂	—	16	4	no	—
262	C₂	C₂×C₃₁×#331	1	2	yes	—
263	C₂	C₃₁×#1545	1	2	yes	—
264	C₂	C₃₁×#1957	1	1	yes	characteristic
265	C₂	C₃₁×#1965	1	1	yes	characteristic
266	C₂	C₃₁×#1982	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₃₁×#2867	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1842	C₂	semidirect
3. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
4. C₂×C₃₁×#331	C₂	semidirect
40. C₃₁×C₄⋊C₈	D₄	semidirect
77. #9695	C₃₁	direct (N × H)
111. #1842	C₂×C₃₁	semidirect
112. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
113. C₂×#331	C₂×C₃₁	semidirect
126. C₄×C₃₁	#331	semidirect
173. C₄⋊C₈	D₄×C₃₁	semidirect
175. C₃₁	#9695	direct (N × H)
255. C₄	C₃₁×#331	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2867	2
C₃₁×#1842	2
C₄×C₃₁×C₄⋊C₈	2
C₂×C₃₁×#331	2
#9695	31

Subgroups — C31×#9695 #21466454

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9695 #21466454