Subgroups — C₂²×C₃₁×#357 #21490133

Order 7936.

1470 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃₁×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#357	C₂	1	12	yes	maximal, direct product factor
5	C₂³×D₄×C₃₁	—	2	1	no	—
6	C₂²×D₈×C₃₁	—	2	3	no	—
7	C₂²×D₈×C₃₁	—	2	4	no	—
8	C₂²×D₈×C₃₁	C₄	1	8	yes	complemented
9	C₂²×D₈×C₃₁	C₂²	1	6	yes	—
10	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
12	C₂²×C₁₆×C₃₁	—	2	4	no	—
13	C₂²×C₁₆×C₃₁	C₂²	1	6	yes	complemented
14	C₂×C₃₁×#107	—	2	3	no	—
15	C₂×C₃₁×#107	C₂²	1	6	yes	complemented
16	C₃₁×#357	C₂²	1	16	yes	direct product factor
17	C₂⁵×C₃₁	—	4	1	no	—
18	C₂²×D₄×C₃₁	—	2	8	no	—
19	C₂²×D₄×C₃₁	—	2	6	no	—
20	C₂²×D₄×C₃₁	—	4	3	no	—
21	C₂²×D₄×C₃₁	—	4	4	no	—
22	C₂³×C₄×C₃₁	—	4	1	no	—
23	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
24	C₂×D₈×C₃₁	—	2	24	no	—
25	C₂×D₈×C₃₁	—	2	6	no	—
26	C₂×D₈×C₃₁	—	2	12	no	—
27	C₂×D₈×C₃₁	C₂×C₄	1	24	yes	complemented
28	C₂×D₈×C₃₁	C₂³	1	4	yes	—
29	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₃₁	D₄	1	3	yes	—
31	C₂²×C₈×C₃₁	D₄	1	4	yes	—
32	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
33	C₂×C₄⋊C₄×C₃₁	—	2	6	no	—
34	C₂×C₄⋊C₄×C₃₁	—	4	3	no	—
35	C₂×C₁₆×C₃₁	—	2	12	no	—
36	C₂×C₁₆×C₃₁	C₂³	1	4	yes	complemented
37	C₃₁×#107	—	2	6	no	—
38	C₃₁×#107	C₂³	1	4	yes	complemented
39	C₂⁴×C₃₁	—	4	8	no	—
40	C₂⁴×C₃₁	—	4	6	no	—
41	C₂⁴×C₃₁	—	8	4	no	—
42	C₂⁴×C₃₁	—	8	4	no	—
43	C₂⁴×C₃₁	D₈	1	1	yes	characteristic
44	C₂×D₄×C₃₁	—	2	24	no	—
45	C₂×D₄×C₃₁	—	2	4	no	—
46	C₂×D₄×C₃₁	—	4	24	no	—
47	C₂×D₄×C₃₁	—	4	6	no	—
48	C₂×D₄×C₃₁	—	4	12	no	—
49	C₂²×C₄×C₃₁	—	4	6	no	—
50	C₂²×C₄×C₃₁	—	8	4	no	—
51	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
52	C₂²×C₄×C₃₁	D₈	1	3	yes	—
53	C₂²×C₄×C₃₁	SD₁₆	1	4	yes	—
54	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
55	D₈×C₃₁	—	2	24	no	—
56	D₈×C₃₁	—	2	4	no	—
57	D₈×C₃₁	C₂²×C₄	1	8	yes	complemented
58	C₂×C₈×C₃₁	C₂²⋊C₄	1	12	yes	—
59	C₂×C₈×C₃₁	C₂²×C₄	1	6	yes	—
60	C₂×C₈×C₃₁	C₂×D₄	1	3	yes	—
61	C₂×C₈×C₃₁	C₂×D₄	1	6	yes	—
62	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
63	C₄⋊C₄×C₃₁	—	2	4	no	—
64	C₄⋊C₄×C₃₁	—	4	6	no	—
65	C₁₆×C₃₁	—	2	4	no	—
66	C₂²×#357	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₃₁	—	4	4	no	—
68	C₂³×C₃₁	—	4	24	no	—
69	C₂³×C₃₁	—	8	32	no	—
70	C₂³×C₃₁	—	8	12	no	—
71	C₂³×C₃₁	—	8	12	no	—
72	C₂³×C₃₁	#102	1	4	yes	—
73	C₂³×C₃₁	D₁₆	1	4	yes	—
74	C₂³×C₃₁	SD₃₂	1	4	yes	—
75	C₂³×C₃₁	C₂×D₈	1	3	yes	—
76	D₄×C₃₁	—	2	8	no	—
77	D₄×C₃₁	—	4	4	no	—
78	D₄×C₃₁	—	4	24	no	—
79	C₂×C₄×C₃₁	—	4	4	no	—
80	C₂×C₄×C₃₁	—	8	12	no	—
81	C₂×C₄×C₃₁	#102	1	12	yes	—
82	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
83	C₂×C₄×C₃₁	C₂×D₈	1	3	yes	—
84	C₂×C₄×C₃₁	C₂×SD₁₆	1	6	yes	—
85	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
86	C₈×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
87	C₈×C₃₁	C₂³×C₄	1	1	yes	characteristic
88	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
89	C₂³×D₈	C₂×C₃₁	1	1	yes	characteristic
90	C₂³×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
91	C₂²×#107	C₂×C₃₁	1	1	yes	characteristic, complemented
92	C₂×#357	C₂×C₃₁	1	12	yes	direct product factor
93	C₂²×C₃₁	—	4	8	no	—
94	C₂²×C₃₁	—	8	4	no	—
95	C₂²×C₃₁	—	8	4	no	—
96	C₂²×C₃₁	—	8	48	no	—
97	C₂²×C₃₁	#357	1	16	yes	direct product factor
98	C₂²×C₃₁	C₂×#102	1	6	yes	—
99	C₂²×C₃₁	C₂×D₁₆	1	6	yes	—
100	C₂²×C₃₁	C₂×SD₃₂	1	6	yes	—
101	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
102	C₄×C₃₁	—	8	4	no	—
103	C₄×C₃₁	C₂×#102	1	6	yes	—
104	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
105	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
106	C₂³×D₄	—	2	1	no	—
107	C₂²×D₈	—	2	3	no	—
108	C₂²×D₈	—	2	4	no	—
109	C₂²×D₈	C₄×C₃₁	1	8	yes	complemented
110	C₂²×D₈	C₂²×C₃₁	1	6	yes	—
111	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
112	C₂²×C₄⋊C₄	—	2	1	no	—
113	C₂²×C₁₆	—	2	4	no	—
114	C₂²×C₁₆	C₂²×C₃₁	1	6	yes	complemented
115	C₂×#107	—	2	3	no	—
116	C₂×#107	C₂²×C₃₁	1	6	yes	complemented
117	#357	C₂²×C₃₁	1	16	yes	direct product factor
118	C₂×C₃₁	—	8	8	no	—
119	C₂×C₃₁	C₂×#357	1	12	yes	direct product factor
120	C₂×C₃₁	C₂²×#102	1	1	yes	characteristic
121	C₂×C₃₁	C₂²×D₁₆	1	1	yes	characteristic
122	C₂×C₃₁	C₂²×SD₃₂	1	1	yes	characteristic
123	C₂⁵	—	4	1	no	—
124	C₂²×D₄	—	2	8	no	—
125	C₂²×D₄	—	2	6	no	—
126	C₂²×D₄	—	4	3	no	—
127	C₂²×D₄	—	4	4	no	—
128	C₂³×C₄	—	4	1	no	—
129	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
130	C₂×D₈	—	2	24	no	—
131	C₂×D₈	—	2	6	no	—
132	C₂×D₈	—	2	12	no	—
133	C₂×D₈	C₂×C₄×C₃₁	1	24	yes	complemented
134	C₂×D₈	C₂³×C₃₁	1	4	yes	—
135	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
136	C₂²×C₈	D₄×C₃₁	1	3	yes	—
137	C₂²×C₈	D₄×C₃₁	1	4	yes	—
138	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
139	C₂×C₄⋊C₄	—	2	6	no	—
140	C₂×C₄⋊C₄	—	4	3	no	—
141	C₂×C₁₆	—	2	12	no	—
142	C₂×C₁₆	C₂³×C₃₁	1	4	yes	complemented
143	#107	—	2	6	no	—
144	#107	C₂³×C₃₁	1	4	yes	complemented
145	C₃₁	C₂²×#357	1	1	yes	characteristic, direct product factor, Sylow
146	C₂⁴	—	4	8	no	—
147	C₂⁴	—	4	6	no	—
148	C₂⁴	—	8	4	no	—
149	C₂⁴	—	8	4	no	—
150	C₂⁴	D₈×C₃₁	1	1	yes	characteristic
151	C₂×D₄	—	2	24	no	—
152	C₂×D₄	—	2	4	no	—
153	C₂×D₄	—	4	24	no	—
154	C₂×D₄	—	4	6	no	—
155	C₂×D₄	—	4	12	no	—
156	C₂²×C₄	—	4	6	no	—
157	C₂²×C₄	—	8	4	no	—
158	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
159	C₂²×C₄	D₈×C₃₁	1	3	yes	—
160	C₂²×C₄	SD₁₆×C₃₁	1	4	yes	—
161	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
162	D₈	—	2	24	no	—
163	D₈	—	2	4	no	—
164	D₈	C₂²×C₄×C₃₁	1	8	yes	complemented
165	C₂×C₈	C₂²⋊C₄×C₃₁	1	12	yes	—
166	C₂×C₈	C₂²×C₄×C₃₁	1	6	yes	—
167	C₂×C₈	C₂×D₄×C₃₁	1	3	yes	—
168	C₂×C₈	C₂×D₄×C₃₁	1	6	yes	—
169	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	4	6	no	—
172	C₁₆	—	2	4	no	—
173	C₂³	—	4	24	no	—
174	C₂³	—	4	4	no	—
175	C₂³	—	8	12	no	—
176	C₂³	—	8	12	no	—
177	C₂³	—	8	32	no	—
178	C₂³	C₃₁×#102	1	4	yes	—
179	C₂³	C₃₁×D₁₆	1	4	yes	—
180	C₂³	C₃₁×SD₃₂	1	4	yes	—
181	C₂³	C₂×D₈×C₃₁	1	3	yes	—
182	D₄	—	2	8	no	—
183	D₄	—	4	24	no	—
184	D₄	—	4	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	8	12	no	—
187	C₂×C₄	C₃₁×#102	1	12	yes	—
188	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
189	C₂×C₄	C₂×D₈×C₃₁	1	3	yes	—
190	C₂×C₄	C₂×SD₁₆×C₃₁	1	6	yes	—
191	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
192	C₈	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
193	C₈	C₂³×C₄×C₃₁	1	1	yes	characteristic
194	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
195	C₂²	—	4	8	no	—
196	C₂²	—	8	4	no	—
197	C₂²	—	8	4	no	—
198	C₂²	—	8	48	no	—
199	C₂²	C₃₁×#357	1	16	yes	direct product factor
200	C₂²	C₂×C₃₁×#102	1	6	yes	—
201	C₂²	C₂×C₃₁×D₁₆	1	6	yes	—
202	C₂²	C₂×C₃₁×SD₃₂	1	6	yes	—
203	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
204	C₄	—	8	4	no	—
205	C₄	C₂×C₃₁×#102	1	6	yes	—
206	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
207	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
208	C₂	—	8	8	no	—
209	C₂	C₂×C₃₁×#357	1	12	yes	direct product factor
210	C₂	C₂²×C₃₁×#102	1	1	yes	characteristic
211	C₂	C₂²×C₃₁×D₁₆	1	1	yes	characteristic
212	C₂	C₂²×C₃₁×SD₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃₁	D₈
commutator	C₈	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂³×D₈×C₃₁	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₁₆×C₃₁	C₂	semidirect
3. C₂²×C₃₁×#107	C₂	semidirect
4. C₂×C₃₁×#357	C₂	direct (N × H)
8. C₂²×D₈×C₃₁	C₄	semidirect
13. C₂²×C₁₆×C₃₁	C₂²	semidirect
15. C₂×C₃₁×#107	C₂²	semidirect
16. C₃₁×#357	C₂²	direct (N × H)
27. C₂×D₈×C₃₁	C₂×C₄	semidirect
36. C₂×C₁₆×C₃₁	C₂³	semidirect
38. C₃₁×#107	C₂³	semidirect
57. D₈×C₃₁	C₂²×C₄	semidirect
66. C₂²×#357	C₃₁	direct (N × H)
90. C₂³×C₁₆	C₂×C₃₁	semidirect
91. C₂²×#107	C₂×C₃₁	semidirect
92. C₂×#357	C₂×C₃₁	direct (N × H)
97. C₂²×C₃₁	#357	direct (N × H)
109. C₂²×D₈	C₄×C₃₁	semidirect
114. C₂²×C₁₆	C₂²×C₃₁	semidirect
116. C₂×#107	C₂²×C₃₁	semidirect
117. #357	C₂²×C₃₁	direct (N × H)
119. C₂×C₃₁	C₂×#357	direct (N × H)
133. C₂×D₈	C₂×C₄×C₃₁	semidirect
142. C₂×C₁₆	C₂³×C₃₁	semidirect
144. #107	C₂³×C₃₁	semidirect
145. C₃₁	C₂²×#357	direct (N × H)
164. D₈	C₂²×C₄×C₃₁	semidirect
199. C₂²	C₃₁×#357	direct (N × H)
209. C₂	C₂×C₃₁×#357	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₈×C₃₁	2
C₂³×C₁₆×C₃₁	2
C₂²×C₃₁×#107	2
C₂×C₃₁×#357	2
C₂²×#357	31

Subgroups — C22×C31×#357 #21490133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₁×#357 #21490133