Group #30776467 — subgroups

Subgroups — #30776467

Order 17152.

158 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₆₇⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₆₇⋊C₃₂)	C₂	1	2	yes	maximal
3	#22558343	C₂	1	2	yes	maximal
4	C₄×C₆₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic
5	Q₈×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₆₇⋊C₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₆₇⋊C₃₂	C₂²	1	2	yes	—
8	C₂×C₆₇⋊C₃₂	C₂²	1	1	yes	characteristic
9	C₈×Q₈×C₆₇	C₄	1	1	yes	characteristic
10	C₄⋊C₁₆×C₆₇	C₄	1	2	yes	—
11	C₄⋊C₁₆×C₆₇	C₄	1	1	yes	characteristic
12	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
14	C₆₇⋊C₃₂	—	2	1	no	—
15	C₆₇⋊C₃₂	—	2	2	no	—
16	C₆₇⋊C₃₂	Q₈	1	2	yes	complemented
17	C₄×Q₈×C₆₇	C₈	1	1	yes	characteristic
18	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
21	C₄⋊C₈×C₆₇	C₈	1	2	yes	—
22	C₄⋊C₈×C₆₇	C₈	1	1	yes	characteristic
23	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
25	C₂×Q₈×C₆₇	C₁₆	1	1	yes	characteristic
26	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
29	C₄⋊C₄×C₆₇	C₁₆	1	2	yes	—
30	C₄⋊C₄×C₆₇	C₁₆	1	1	yes	characteristic
31	C₄²×C₆₇	C₂×C₈	1	2	yes	—
32	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
33	C₁₆×C₆₇	—	2	1	no	—
34	C₁₆×C₆₇	—	2	2	no	—
35	C₁₆×C₆₇	C₂×Q₈	1	1	yes	characteristic
36	C₁₆×C₆₇	#41	1	1	yes	characteristic
37	Q₈×C₆₇	—	2	2	no	—
38	C₂×C₄×C₆₇	C₂×C₁₆	1	2	yes	—
39	C₂×C₄×C₆₇	C₂×C₁₆	1	1	yes	characteristic
40	C₂×C₄×C₆₇	C₂×C₁₆	1	2	yes	—
41	C₂×C₄×C₆₇	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
43	C₈×C₆₇	—	2	1	no	—
44	C₈×C₆₇	—	2	2	no	—
45	C₈×C₆₇	C₄×Q₈	1	1	yes	characteristic
46	C₈×C₆₇	#131	1	1	yes	characteristic
47	C₂²×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
48	C₄×C₆₇	—	2	2	no	—
49	C₄×C₆₇	—	2	2	no	—
50	C₄×C₆₇	—	2	1	no	—
51	C₄×C₆₇	M₆₄	1	2	yes	—
52	C₄×C₆₇	C₈×Q₈	1	1	yes	characteristic
53	C₄×C₆₇	#504	1	1	yes	characteristic
54	#13660	—	67	1	no	maximal, Sylow
55	C₂×C₆₇	Q₈×C₁₆	1	1	yes	characteristic
56	C₂×C₆₇	C₂×M₆₄	1	1	yes	characteristic
57	C₂×C₆₇	#2258	1	1	yes	characteristic
58	Q₈×C₁₆	D₆₇	1	1	yes	characteristic
59	C₄⋊C₃₂	—	67	1	no	—
60	C₄⋊C₃₂	—	67	2	no	—
61	#1397	—	67	2	no	—
62	C₄×C₃₂	—	67	1	no	—
63	C₆₇	#13660	1	1	yes	characteristic, complemented, Sylow
64	C₈×Q₈	Dic₆₇	1	1	yes	characteristic
65	C₂×C₃₂	—	67	1	no	—
66	C₂×C₃₂	—	67	1	no	—
67	C₂×C₃₂	—	67	2	no	—
68	C₄⋊C₁₆	Dic₆₇	1	1	yes	characteristic
69	C₄⋊C₁₆	Dic₆₇	1	2	yes	—
70	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
71	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
72	C₄×Q₈	C₆₇⋊C₈	1	1	yes	characteristic
73	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
74	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
75	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
76	C₄⋊C₈	C₆₇⋊C₈	1	2	yes	—
77	C₄⋊C₈	C₆₇⋊C₈	1	1	yes	characteristic
78	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
79	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
80	C₃₂	—	67	2	no	—
81	C₃₂	—	134	1	no	—
82	C₃₂	—	134	2	no	—
83	C₂×Q₈	C₆₇⋊C₁₆	1	1	yes	characteristic
84	C₂×C₈	C₂×C₆₇⋊C₈	1	1	yes	characteristic
85	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
86	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
87	C₄⋊C₄	C₆₇⋊C₁₆	1	2	yes	—
88	C₄⋊C₄	C₆₇⋊C₁₆	1	1	yes	characteristic
89	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
90	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
91	C₁₆	—	2	1	no	—
92	C₁₆	—	2	2	no	—
93	C₁₆	Q₈×D₆₇	1	1	yes	characteristic
94	C₁₆	#11761629	1	1	yes	characteristic
95	Q₈	—	2	2	no	—
96	C₂×C₄	C₂×C₆₇⋊C₁₆	1	2	yes	—
97	C₂×C₄	C₂×C₆₇⋊C₁₆	1	1	yes	characteristic
98	C₂×C₄	C₂×C₆₇⋊C₁₆	1	2	yes	—
99	C₂×C₄	C₂×C₆₇⋊C₁₆	1	1	yes	characteristic
100	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	2	2	no	—
103	C₈	Q₈×Dic₆₇	1	1	yes	characteristic
104	C₈	#14524050	1	1	yes	characteristic
105	C₂²	C₂²×C₆₇⋊C₁₆	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	2	2	no	—
108	C₄	—	2	1	no	—
109	C₄	#16863230	1	2	yes	—
110	C₄	Q₈×C₆₇⋊C₈	1	1	yes	characteristic
111	C₄	#16863813	1	1	yes	characteristic
112	C₂	Q₈×C₆₇⋊C₁₆	1	1	yes	characteristic
113	C₂	C₂×#16863230	1	1	yes	characteristic
114	C₂	#22562503	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊(C₆₇⋊C₃₂)

C₂

yes

maximal, characteristic

C₄⋊(C₆₇⋊C₃₂)

C₂

yes

maximal

#22558343

C₂

yes

maximal

C₄×C₆₇⋊C₃₂

C₂

yes

maximal, characteristic

Q₈×C₁₆×C₆₇

C₂

yes

maximal, characteristic

C₂×C₆₇⋊C₃₂

C₂²

yes

characteristic

C₂×C₆₇⋊C₃₂

C₂²

yes

—

C₂×C₆₇⋊C₃₂

C₂²

yes

characteristic

C₈×Q₈×C₆₇

C₄

yes

characteristic

C₄⋊C₁₆×C₆₇

C₄

yes

—

C₄⋊C₁₆×C₆₇

C₄

yes

characteristic

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

C₄×C₁₆×C₆₇

C₂²

yes

—

C₆₇⋊C₃₂

—

C₆₇⋊C₃₂

—

C₆₇⋊C₃₂

Q₈

yes

complemented

C₄×Q₈×C₆₇

C₈

yes

characteristic

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₇

C₂×C₄

yes

—

C₂×C₁₆×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

C₈

yes

—

C₄⋊C₈×C₆₇

C₈

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₂×Q₈×C₆₇

C₁₆

yes

characteristic

C₂×C₈×C₆₇

C₂×C₈

yes

characteristic

C₂×C₈×C₆₇

C₂×C₈

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₆₇

C₁₆

yes

—

C₄⋊C₄×C₆₇

C₁₆

yes

characteristic

C₄²×C₆₇

C₂×C₈

yes

—

C₄²×C₆₇

C₂×C₈

yes

characteristic

C₁₆×C₆₇

—

C₁₆×C₆₇

—

C₁₆×C₆₇

C₂×Q₈

yes

characteristic

C₁₆×C₆₇

#41

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

C₂×C₁₆

yes

—

C₂×C₄×C₆₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×C₁₆

yes

—

C₂×C₄×C₆₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₈

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₄×Q₈

yes

characteristic

C₈×C₆₇

#131

yes

characteristic

C₂²×C₆₇

C₂²×C₁₆

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

M₆₄

yes

—

C₄×C₆₇

C₈×Q₈

yes

C₄×C₆₇

yes

—

C₂×C₆₇

Q₈×C₁₆

yes

characteristic

C₂×C₆₇

C₂×M₆₄

yes

characteristic

C₂×C₆₇

#2258

yes

characteristic

Q₈×C₁₆

D₆₇

yes

characteristic

C₄⋊C₃₂

—

C₄⋊C₃₂

—

#1397

—

C₄×C₃₂

—

C₆₇

#13660

yes

characteristic, complemented, Sylow

C₈×Q₈

Dic₆₇

yes

characteristic

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₄⋊C₁₆

Dic₆₇

yes

characteristic

C₄⋊C₁₆

Dic₆₇

yes

—

C₄×C₁₆

C₂×D₆₇

yes

characteristic

C₄×C₁₆

C₂×D₆₇

yes

—

C₄×Q₈

C₆₇⋊C₈

yes

characteristic

C₂×C₁₆

C₂×Dic₆₇

yes

characteristic

C₂×C₁₆

C₂×Dic₆₇

yes

—

C₂×C₁₆

C₂²×D₆₇

yes

characteristic

C₄⋊C₈

C₆₇⋊C₈

yes

—

C₄⋊C₈

C₆₇⋊C₈

yes

characteristic

C₄×C₈

C₂×Dic₆₇

yes

characteristic

C₄×C₈

C₂×Dic₆₇

yes

—

C₃₂

—

C₃₂

—

134

—

C₃₂

—

134

—

C₂×Q₈

C₆₇⋊C₁₆

yes

characteristic

C₂×C₈

C₂×C₆₇⋊C₈

yes

characteristic

C₂×C₈

C₂×C₆₇⋊C₈

yes

—

C₂×C₈

C₂²×Dic₆₇

yes

characteristic

C₄⋊C₄

C₆₇⋊C₁₆

yes

—

C₄⋊C₄

C₆₇⋊C₁₆

yes

characteristic

C₄²

C₂×C₆₇⋊C₈

yes

—

C₄²

C₂×C₆₇⋊C₈

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

Q₈×D₆₇

yes

characteristic

C₁₆

#11761629

yes

characteristic

Q₈

—

C₂×C₄

C₂×C₆₇⋊C₁₆

yes

—

C₂×C₄

C₂×C₆₇⋊C₁₆

yes

characteristic

C₂×C₄

C₂×C₆₇⋊C₁₆

yes

—

C₂×C₄

C₂×C₆₇⋊C₁₆

yes

characteristic

100

C₂×C₄

C₂²×C₆₇⋊C₈

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

Q₈×Dic₆₇

yes

characteristic

104

C₈

#14524050

yes

characteristic

105

C₂²

C₂²×C₆₇⋊C₁₆

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

#16863230

yes

—

110

C₄

Q₈×C₆₇⋊C₈

yes

characteristic

111

C₄

#16863813

yes

characteristic

112

C₂

Q₈×C₆₇⋊C₁₆

yes

characteristic

113

C₂

C₂×#16863230

yes

characteristic

114

C₂

#22562503

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₆₇
commutator, socle, prime radical	C₂²×C₆₇	C₂²×C₁₆
Fitting	Q₈×C₁₆×C₆₇	C₂
cosocle	C₂×C₁₆×C₆₇	C₂³
hypercenter	Q₈×C₁₆	D₆₇
hypocenter	C₆₇	#13660
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₁₆ ◁ Q₈×C₁₆

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₆₇

commutator, socle, prime radical

C₂²×C₆₇

C₂²×C₁₆

Fitting

Q₈×C₁₆×C₆₇

C₂

cosocle

C₂×C₁₆×C₆₇

C₂³

hypercenter

Q₈×C₁₆

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₁₆ ◁ Q₈×C₁₆

N	H ≅ G/N	type
16. C₆₇⋊C₃₂	Q₈	semidirect
63. C₆₇	#13660	semidirect

H ≅ G/N

type

16. C₆₇⋊C₃₂

Q₈

semidirect

63. C₆₇

#13660

semidirect

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

subgroup	index
C₄⋊(C₆₇⋊C₃₂)	2
#22558343	2
C₄×C₆₇⋊C₃₂	2
Q₈×C₁₆×C₆₇	2
#13660	67

subgroup

index

C₄⋊(C₆₇⋊C₃₂)

#22558343

C₄×C₆₇⋊C₃₂

Q₈×C₁₆×C₆₇

#13660

Subgroups — #30776467

Conjugacy classes of subgroups

Complements

Maximal subgroups