Subgroups — C₃₁×#9642 #21466401

Order 7936.

698 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#127	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×D₄⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₂×C₃₁×#127	C₄	1	1	yes	characteristic
6	C₂×C₄×D₄×C₃₁	—	2	2	no	—
7	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
8	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
10	C₃₁×#390	—	2	2	no	—
11	C₃₁×#390	C₄	1	2	yes	—
12	C₃₁×#384	C₄	1	1	yes	characteristic
13	C₄³×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#336	—	2	2	no	—
15	C₃₁×#336	—	2	2	no	—
16	C₃₁×D₄⋊C₈	—	2	8	no	—
17	C₂²×D₄×C₃₁	—	2	2	no	—
18	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₃₁	—	2	2	no	—
20	C₂²×C₈×C₃₁	—	2	2	no	—
21	C₂²×C₈×C₃₁	—	2	2	no	—
22	C₃₁×#127	—	2	2	no	—
23	C₃₁×#127	C₈	1	4	yes	complemented
24	C₄×D₄×C₃₁	—	2	4	no	—
25	C₄×D₄×C₃₁	—	2	4	no	—
26	C₄×D₄×C₃₁	—	4	8	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
30	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
31	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₃₁	D₄	1	2	yes	—
34	C₂×C₄²×C₃₁	D₄	1	2	yes	—
35	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
36	C₃₁×C₄⋊C₈	—	2	4	no	—
37	C₄×C₈×C₃₁	—	2	2	no	—
38	C₄×C₈×C₃₁	D₄	1	4	yes	complemented
39	C₂⁴×C₃₁	—	2	2	no	—
40	C₂×D₄×C₃₁	—	2	8	no	—
41	C₂×D₄×C₃₁	—	2	4	no	—
42	C₂×D₄×C₃₁	—	2	2	no	—
43	C₂×D₄×C₃₁	—	2	4	no	—
44	C₂×D₄×C₃₁	—	4	4	no	—
45	C₂×D₄×C₃₁	—	4	4	no	—
46	C₂×D₄×C₃₁	C₂×C₈	1	4	yes	—
47	C₂²×C₄×C₃₁	—	2	2	no	—
48	C₂²×C₄×C₃₁	—	4	2	no	—
49	C₂²×C₄×C₃₁	—	4	2	no	—
50	C₂²×C₄×C₃₁	—	4	8	no	—
51	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₃₁	—	2	4	no	—
57	C₂×C₈×C₃₁	—	4	4	no	—
58	C₂×C₈×C₃₁	—	4	2	no	—
59	C₂×C₈×C₃₁	—	4	4	no	—
60	C₄⋊C₄×C₃₁	—	2	4	no	—
61	C₄⋊C₄×C₃₁	—	4	4	no	—
62	C₂²⋊C₄×C₃₁	—	4	8	no	—
63	C₂²⋊C₄×C₃₁	—	4	4	no	—
64	C₄²×C₃₁	—	2	2	no	—
65	C₄²×C₃₁	—	2	4	no	—
66	C₄²×C₃₁	—	4	2	no	—
67	C₄²×C₃₁	C₂×C₈	1	2	yes	—
68	C₄²×C₃₁	M₁₆	1	2	yes	—
69	C₄²×C₃₁	C₂×D₄	1	2	yes	—
70	C₄²×C₃₁	#41	1	2	yes	—
71	#9642	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂³×C₃₁	—	2	4	no	—
73	C₂³×C₃₁	—	4	2	no	—
74	C₂³×C₃₁	—	4	2	no	—
75	C₂³×C₃₁	—	4	4	no	—
76	C₂³×C₃₁	—	4	4	no	—
77	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	D₄×C₃₁	—	2	8	no	—
79	D₄×C₃₁	—	4	16	no	—
80	D₄×C₃₁	—	4	4	no	—
81	C₂×C₄×C₃₁	—	2	2	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	2	2	no	—
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	4	2	no	—
88	C₂×C₄×C₃₁	—	4	4	no	—
89	C₂×C₄×C₃₁	—	8	4	no	—
90	C₂×C₄×C₃₁	—	8	4	no	—
91	C₂×C₄×C₃₁	C₂²⋊C₈	1	4	yes	—
92	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₃₁	#121	1	2	yes	—
94	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
95	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
98	C₂×C₄×C₃₁	#131	1	2	yes	—
99	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
100	C₈×C₃₁	—	4	4	no	—
101	C₈×C₃₁	—	8	2	no	—
102	C₄×#127	C₂×C₃₁	1	1	yes	characteristic
103	#1831	C₂×C₃₁	1	1	yes	characteristic, complemented
104	#1744	C₂×C₃₁	1	1	yes	characteristic, complemented
105	C₂×D₄⋊C₈	C₂×C₃₁	1	4	yes	complemented
106	C₂²×C₃₁	—	4	4	no	—
107	C₂²×C₃₁	—	4	8	no	—
108	C₂²×C₃₁	—	8	2	no	—
109	C₂²×C₃₁	—	8	2	no	—
110	C₂²×C₃₁	—	8	4	no	—
111	C₂²×C₃₁	#390	1	1	yes	characteristic
112	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
113	C₂²×C₃₁	#408	1	1	yes	characteristic
114	C₂²×C₃₁	#418	1	2	yes	—
115	C₂²×C₃₁	#421	1	2	yes	—
116	C₄×C₃₁	—	2	2	no	—
117	C₄×C₃₁	—	2	4	no	—
118	C₄×C₃₁	—	4	2	no	—
119	C₄×C₃₁	—	8	2	no	—
120	C₄×C₃₁	C₈×D₄	1	2	yes	—
121	C₄×C₃₁	#436	1	2	yes	—
122	C₄×C₃₁	#490	1	2	yes	—
123	C₄×C₃₁	#496	1	2	yes	—
124	C₂×#127	C₄×C₃₁	1	1	yes	characteristic
125	C₂×C₄×D₄	—	2	2	no	—
126	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
127	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
128	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
129	#390	—	2	2	no	—
130	#390	C₄×C₃₁	1	2	yes	—
131	#384	C₄×C₃₁	1	1	yes	characteristic
132	C₄³	C₂²×C₃₁	1	1	yes	characteristic
133	#336	—	2	2	no	—
134	#336	—	2	2	no	—
135	D₄⋊C₈	—	2	8	no	—
136	C₂×C₃₁	—	8	4	no	—
137	C₂×C₃₁	#1478	1	2	yes	—
138	C₂×C₃₁	#1496	1	2	yes	—
139	C₂×C₃₁	#1954	1	1	yes	characteristic
140	C₂×C₃₁	#1961	1	1	yes	characteristic
141	C₂×C₃₁	#1963	1	1	yes	characteristic
142	C₂²×D₄	—	2	2	no	—
143	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
144	C₂³×C₄	—	2	2	no	—
145	C₂²×C₈	—	2	2	no	—
146	C₂²×C₈	—	2	2	no	—
147	#127	—	2	2	no	—
148	#127	C₈×C₃₁	1	4	yes	complemented
149	C₄×D₄	—	2	4	no	—
150	C₄×D₄	—	2	4	no	—
151	C₄×D₄	—	4	8	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
154	C₂×C₂²⋊C₄	—	2	4	no	—
155	C₂×C₂²⋊C₄	—	2	2	no	—
156	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
157	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
158	C₂×C₄²	D₄×C₃₁	1	2	yes	—
159	C₂×C₄²	D₄×C₃₁	1	2	yes	—
160	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
161	C₄⋊C₈	—	2	4	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	D₄×C₃₁	1	4	yes	complemented
164	C₃₁	#9642	1	1	yes	characteristic, direct product factor, Sylow
165	C₂⁴	—	2	2	no	—
166	C₂×D₄	—	2	8	no	—
167	C₂×D₄	—	2	4	no	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	2	4	no	—
170	C₂×D₄	—	4	4	no	—
171	C₂×D₄	—	4	4	no	—
172	C₂×D₄	C₂×C₈×C₃₁	1	4	yes	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	—	4	8	no	—
177	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
178	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
179	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	4	4	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	4	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₂²⋊C₄	—	4	8	no	—
189	C₂²⋊C₄	—	4	4	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	2	4	no	—
192	C₄²	—	4	2	no	—
193	C₄²	C₂×C₈×C₃₁	1	2	yes	—
194	C₄²	M₁₆×C₃₁	1	2	yes	—
195	C₄²	C₂×D₄×C₃₁	1	2	yes	—
196	C₄²	#41×C₃₁	1	2	yes	—
197	C₂³	—	2	4	no	—
198	C₂³	—	4	2	no	—
199	C₂³	—	4	4	no	—
200	C₂³	—	4	4	no	—
201	C₂³	—	4	2	no	—
202	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
203	D₄	—	2	8	no	—
204	D₄	—	4	16	no	—
205	D₄	—	4	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	8	4	no	—
215	C₂×C₄	—	8	4	no	—
216	C₂×C₄	C₃₁×C₂²⋊C₈	1	4	yes	—
217	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
218	C₂×C₄	C₃₁×#121	1	2	yes	—
219	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
220	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
221	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
222	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
223	C₂×C₄	C₃₁×#131	1	2	yes	—
224	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
225	C₈	—	4	4	no	—
226	C₈	—	8	2	no	—
227	C₂²	—	4	4	no	—
228	C₂²	—	4	8	no	—
229	C₂²	—	8	2	no	—
230	C₂²	—	8	2	no	—
231	C₂²	—	8	4	no	—
232	C₂²	C₃₁×#390	1	1	yes	characteristic
233	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
234	C₂²	C₃₁×#408	1	1	yes	characteristic
235	C₂²	C₃₁×#418	1	2	yes	—
236	C₂²	C₃₁×#421	1	2	yes	—
237	C₄	—	2	2	no	—
238	C₄	—	2	4	no	—
239	C₄	—	4	2	no	—
240	C₄	—	8	2	no	—
241	C₄	C₈×D₄×C₃₁	1	2	yes	—
242	C₄	C₃₁×#436	1	2	yes	—
243	C₄	C₃₁×#490	1	2	yes	—
244	C₄	C₃₁×#496	1	2	yes	—
245	C₂	—	8	4	no	—
246	C₂	C₃₁×#1478	1	2	yes	—
247	C₂	C₃₁×#1496	1	2	yes	—
248	C₂	C₃₁×#1954	1	1	yes	characteristic
249	C₂	C₃₁×#1961	1	1	yes	characteristic
250	C₂	C₃₁×#1963	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂³×C₃₁	C₂×C₂²⋊C₄
prime radical	C₂×C₃₁×#127	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1831	C₂	semidirect
3. C₃₁×#1744	C₂	semidirect
4. C₂×C₃₁×D₄⋊C₈	C₂	semidirect
23. C₃₁×#127	C₈	semidirect
38. C₄×C₈×C₃₁	D₄	semidirect
71. #9642	C₃₁	direct (N × H)
103. #1831	C₂×C₃₁	semidirect
104. #1744	C₂×C₃₁	semidirect
105. C₂×D₄⋊C₈	C₂×C₃₁	semidirect
148. #127	C₈×C₃₁	semidirect
163. C₄×C₈	D₄×C₃₁	semidirect
164. C₃₁	#9642	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#127	2
C₃₁×#1831	2
C₃₁×#1744	2
C₂×C₃₁×D₄⋊C₈	2
#9642	31

Subgroups — C31×#9642 #21466401

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9642 #21466401