Subgroups — C₂×C₃₁×#2158 #21490166

Order 7936.

694 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#427	C₂	1	2	yes	maximal
3	C₃₁×#2158	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₁×#367	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#366	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₃₁×#117	—	2	2	no	—
8	C₂²×C₁₆×C₃₁	C₄	1	8	yes	complemented
9	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₃₁	C₂²	1	4	yes	—
12	C₃₁×#427	—	2	4	no	—
13	C₃₁×#427	C₂²	1	8	yes	—
14	C₂×C₃₁×#107	C₂²	1	4	yes	—
15	C₂×C₃₁×#106	—	2	2	no	—
16	C₃₁×#367	C₂²	1	8	yes	complemented
17	C₃₁×#366	C₂²	1	8	yes	complemented
18	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₃₁	D₄	1	2	yes	—
22	C₂²×C₈×C₃₁	Q₈	1	4	yes	—
23	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
25	C₃₁×#117	—	2	8	no	—
26	C₃₁×#117	—	4	4	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
28	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
29	C₂×C₄²×C₃₁	—	4	2	no	—
30	C₂×C₁₆×C₃₁	C₂×C₄	1	4	yes	—
31	C₂×C₁₆×C₃₁	C₂×C₄	1	4	yes	—
32	C₂×C₁₆×C₃₁	C₂×C₄	1	16	yes	complemented
33	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
34	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
35	C₃₁×#107	—	2	4	no	—
36	C₃₁×#107	C₂³	1	8	yes	—
37	C₃₁×#106	—	2	8	no	—
38	C₂⁴×C₃₁	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₃₁	—	4	4	no	—
40	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₃₁	D₈	1	2	yes	—
42	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	Q₁₆	1	4	yes	—
44	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₃₁	C₄⋊C₄	1	4	yes	—
47	C₂×C₈×C₃₁	C₄⋊C₄	1	8	yes	—
48	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
50	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
54	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
55	C₄⋊C₄×C₃₁	—	2	8	no	—
56	C₄⋊C₄×C₃₁	—	4	4	no	—
57	C₂²⋊C₄×C₃₁	—	4	8	no	—
58	C₄²×C₃₁	—	4	8	no	—
59	C₁₆×C₃₁	C₂²×C₄	1	4	yes	—
60	C₁₆×C₃₁	C₂²×C₄	1	4	yes	—
61	C₂×#2158	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂³×C₃₁	—	2	4	no	—
63	C₂³×C₃₁	#107	1	4	yes	—
64	C₂³×C₃₁	C₂×D₈	1	2	yes	—
65	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₃₁	—	4	8	no	—
67	C₂×C₄×C₃₁	—	8	8	no	—
68	C₂×C₄×C₃₁	#107	1	8	yes	—
69	C₂×C₄×C₃₁	#107	1	4	yes	—
70	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
71	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
73	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₃₁	C₂×Q₁₆	1	4	yes	—
75	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
76	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
77	C₈×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
78	C₈×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
79	C₈×C₃₁	C₂³×C₄	1	1	yes	characteristic
80	C₈×C₃₁	C₂²×Q₈	1	1	yes	characteristic
81	C₂³×C₁₆	C₂×C₃₁	1	1	yes	characteristic
82	C₂×#427	C₂×C₃₁	1	2	yes	—
83	#2158	C₂×C₃₁	1	8	yes	direct product factor
84	C₂×#367	C₂×C₃₁	1	2	yes	complemented
85	C₂×#366	C₂×C₃₁	1	2	yes	complemented
86	C₂²×C₃₁	—	2	4	no	—
87	C₂²×C₃₁	—	2	8	no	—
88	C₂²×C₃₁	C₂×#107	1	4	yes	—
89	C₂²×C₃₁	C₂×#107	1	2	yes	—
90	C₂²×C₃₁	#508	1	4	yes	—
91	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
92	C₄×C₃₁	—	8	8	no	—
93	C₄×C₃₁	C₂×#107	1	2	yes	—
94	C₄×C₃₁	C₂×#107	1	4	yes	—
95	C₄×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
96	C₄×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
97	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
98	C₂×#117	—	2	2	no	—
99	C₂²×C₁₆	C₄×C₃₁	1	8	yes	complemented
100	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
101	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
102	C₂²×C₁₆	C₂²×C₃₁	1	4	yes	—
103	#427	—	2	4	no	—
104	#427	C₂²×C₃₁	1	8	yes	—
105	C₂×#107	C₂²×C₃₁	1	4	yes	—
106	C₂×#106	—	2	2	no	—
107	#367	C₂²×C₃₁	1	8	yes	complemented
108	#366	C₂²×C₃₁	1	8	yes	complemented
109	C₂×C₃₁	—	2	4	no	—
110	C₂×C₃₁	#2158	1	4	yes	direct product factor
111	C₂×C₃₁	C₂²×#107	1	1	yes	characteristic
112	C₂×C₃₁	C₂×#508	1	2	yes	—
113	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
114	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
115	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
116	C₂²×C₈	D₄×C₃₁	1	2	yes	—
117	C₂²×C₈	Q₈×C₃₁	1	4	yes	—
118	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
119	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
120	#117	—	2	8	no	—
121	#117	—	4	4	no	—
122	C₂×C₄⋊C₄	—	2	4	no	—
123	C₂×C₂²⋊C₄	—	4	2	no	—
124	C₂×C₄²	—	4	2	no	—
125	C₂×C₁₆	C₂×C₄×C₃₁	1	4	yes	—
126	C₂×C₁₆	C₂×C₄×C₃₁	1	4	yes	—
127	C₂×C₁₆	C₂×C₄×C₃₁	1	16	yes	complemented
128	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
129	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
130	#107	—	2	4	no	—
131	#107	C₂³×C₃₁	1	8	yes	—
132	#106	—	2	8	no	—
133	C₃₁	C₂×#2158	1	1	yes	characteristic, direct product factor, Sylow
134	C₂⁴	D₈×C₃₁	1	1	yes	characteristic
135	C₂²×C₄	—	4	4	no	—
136	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
137	C₂²×C₄	D₈×C₃₁	1	2	yes	—
138	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
139	C₂²×C₄	Q₁₆×C₃₁	1	4	yes	—
140	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
142	C₂×C₈	C₄⋊C₄×C₃₁	1	4	yes	—
143	C₂×C₈	C₄⋊C₄×C₃₁	1	8	yes	—
144	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
145	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
146	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
147	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
148	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
149	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
150	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
151	C₄⋊C₄	—	2	8	no	—
152	C₄⋊C₄	—	4	4	no	—
153	C₂²⋊C₄	—	4	8	no	—
154	C₄²	—	4	8	no	—
155	C₁₆	C₂²×C₄×C₃₁	1	4	yes	—
156	C₁₆	C₂²×C₄×C₃₁	1	4	yes	—
157	C₂³	—	2	4	no	—
158	C₂³	C₃₁×#107	1	4	yes	—
159	C₂³	C₂×D₈×C₃₁	1	2	yes	—
160	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
161	C₂×C₄	—	4	8	no	—
162	C₂×C₄	—	8	8	no	—
163	C₂×C₄	C₃₁×#107	1	8	yes	—
164	C₂×C₄	C₃₁×#107	1	4	yes	—
165	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
166	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
167	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
168	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
169	C₂×C₄	C₂×Q₁₆×C₃₁	1	4	yes	—
170	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
171	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
172	C₈	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
173	C₈	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
174	C₈	C₂³×C₄×C₃₁	1	1	yes	characteristic
175	C₈	C₂²×Q₈×C₃₁	1	1	yes	characteristic
176	C₂²	—	2	4	no	—
177	C₂²	—	2	8	no	—
178	C₂²	C₂×C₃₁×#107	1	4	yes	—
179	C₂²	C₂×C₃₁×#107	1	2	yes	—
180	C₂²	C₃₁×#508	1	4	yes	—
181	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
182	C₄	—	8	8	no	—
183	C₄	C₂×C₃₁×#107	1	4	yes	—
184	C₄	C₂×C₃₁×#107	1	2	yes	—
185	C₄	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
186	C₄	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
187	C₂	—	2	4	no	—
188	C₂	C₃₁×#2158	1	4	yes	direct product factor
189	C₂	C₂²×C₃₁×#107	1	1	yes	characteristic
190	C₂	C₂×C₃₁×#508	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	D₈
commutator	C₈	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂×D₈
prime radical	C₂⁴×C₃₁	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2158	C₂	direct (N × H)
4. C₂×C₃₁×#367	C₂	semidirect
5. C₂×C₃₁×#366	C₂	semidirect
8. C₂²×C₁₆×C₃₁	C₄	semidirect
16. C₃₁×#367	C₂²	semidirect
17. C₃₁×#366	C₂²	semidirect
32. C₂×C₁₆×C₃₁	C₂×C₄	semidirect
61. C₂×#2158	C₃₁	direct (N × H)
83. #2158	C₂×C₃₁	direct (N × H)
84. C₂×#367	C₂×C₃₁	semidirect
85. C₂×#366	C₂×C₃₁	semidirect
99. C₂²×C₁₆	C₄×C₃₁	semidirect
107. #367	C₂²×C₃₁	semidirect
108. #366	C₂²×C₃₁	semidirect
110. C₂×C₃₁	#2158	direct (N × H)
127. C₂×C₁₆	C₂×C₄×C₃₁	semidirect
133. C₃₁	C₂×#2158	direct (N × H)
188. C₂	C₃₁×#2158	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×C₁₆×C₃₁	2
C₂×C₃₁×#427	2
C₃₁×#2158	2
C₂×C₃₁×#367	2
C₂×C₃₁×#366	2
C₂×#2158	31

Subgroups — C2×C31×#2158 #21490166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2158 #21490166