Group #30776466 — subgroups

Subgroups — #30776466

Order 17152.

158 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 91 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	C₁₆×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₆₇⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
3	Dic₆₇⋊C₃₂	C₂	1	2	yes	maximal
4	#22558342	C₂	1	2	yes	maximal
5	C₄×C₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₃₄	C₄	1	1	yes	characteristic
7	C₂×C₆₇⋊C₃₂	C₂²	1	2	yes	—
8	C₄⋊(C₆₇⋊C₁₆)	C₄	1	1	yes	characteristic
9	Dic₆₇⋊C₁₆	C₄	1	2	yes	—
10	C₁₆×Dic₆₇	C₂²	1	2	yes	—
11	C₂×C₃₂×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₃₂×C₆₇	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₄×Dic₁₃₄	C₈	1	1	yes	characteristic
15	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
16	C₄⋊(C₆₇⋊C₈)	C₈	1	1	yes	characteristic
17	Dic₆₇⋊C₈	C₈	1	2	yes	—
18	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
19	C₆₇⋊C₃₂	—	2	2	no	—
20	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
22	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₃₂×C₆₇	—	2	1	no	—
24	C₃₂×C₆₇	Q₈	1	2	yes	complemented
25	C₂×Dic₁₃₄	C₁₆	1	1	yes	characteristic
26	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
27	C₄⋊Dic₆₇	C₁₆	1	1	yes	characteristic
28	Dic₆₇⋊C₄	C₁₆	1	2	yes	—
29	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
30	C₆₇⋊C₁₆	—	2	2	no	—
31	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
33	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
34	C₁₆×C₆₇	—	2	1	no	—
35	C₁₆×C₆₇	C₂×Q₈	1	1	yes	characteristic
36	C₁₆×C₆₇	#41	1	1	yes	characteristic
37	C₂×C₄×C₆₇	C₂×C₁₆	1	1	yes	characteristic
38	C₂×C₄×C₆₇	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
40	C₂×Dic₆₇	C₂×C₁₆	1	2	yes	—
41	C₂×Dic₆₇	C₂×C₁₆	1	2	yes	—
42	Dic₁₃₄	—	2	2	no	—
43	C₈×C₆₇	—	2	1	no	—
44	C₈×C₆₇	C₄×Q₈	1	1	yes	characteristic
45	C₈×C₆₇	#131	1	1	yes	characteristic
46	C₆₇⋊C₈	—	2	2	no	—
47	C₂²×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
48	C₄×C₆₇	—	2	1	no	—
49	C₄×C₆₇	M₆₄	1	2	yes	—
50	C₄×C₆₇	C₈×Q₈	1	1	yes	characteristic
51	C₄×C₆₇	#504	1	1	yes	characteristic
52	Dic₆₇	—	2	2	no	—
53	Dic₆₇	—	2	2	no	—
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	—	67	1	no	—
59	C₄⋊C₃₂	—	67	2	no	—
60	C₄⋊C₃₂	—	67	1	no	—
61	#1397	—	67	2	no	—
62	C₄×C₃₂	D₆₇	1	1	yes	characteristic
63	C₆₇	#13660	1	1	yes	characteristic, complemented, Sylow
64	C₈×Q₈	—	67	1	no	—
65	C₂×C₃₂	—	67	2	no	—
66	C₂×C₃₂	C₂×D₆₇	1	1	yes	characteristic
67	C₂×C₃₂	C₂×D₆₇	1	1	yes	characteristic
68	C₄⋊C₁₆	—	67	2	no	—
69	C₄⋊C₁₆	—	67	1	no	—
70	C₄×C₁₆	—	67	2	no	—
71	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
72	C₄×Q₈	—	67	1	no	—
73	C₂×C₁₆	—	67	2	no	—
74	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
75	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
76	C₄⋊C₈	—	67	2	no	—
77	C₄⋊C₈	—	67	1	no	—
78	C₄×C₈	—	67	2	no	—
79	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
80	C₃₂	—	2	1	no	—
81	C₃₂	—	134	2	no	—
82	C₃₂	Dic₁₃₄	1	2	yes	complemented
83	C₂×Q₈	—	67	1	no	—
84	C₂×C₈	—	67	2	no	—
85	C₂×C₈	C₈×D₆₇	1	1	yes	characteristic
86	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
87	C₄⋊C₄	—	67	2	no	—
88	C₄⋊C₄	—	67	1	no	—
89	C₄²	—	67	2	no	—
90	C₄²	C₈×D₆₇	1	1	yes	characteristic
91	C₁₆	—	2	1	no	—
92	C₁₆	—	134	2	no	—
93	C₁₆	C₂×Dic₁₃₄	1	1	yes	characteristic
94	C₁₆	#11761628	1	1	yes	characteristic
95	Q₈	—	134	2	no	—
96	C₂×C₄	—	67	2	no	—
97	C₂×C₄	—	67	2	no	—
98	C₂×C₄	C₁₆×D₆₇	1	1	yes	characteristic
99	C₂×C₄	C₁₆×D₆₇	1	1	yes	characteristic
100	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	134	2	no	—
103	C₈	C₄×Dic₁₃₄	1	1	yes	characteristic
104	C₈	#14524049	1	1	yes	characteristic
105	C₂²	C₂×C₁₆×D₆₇	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	134	2	no	—
108	C₄	—	134	2	no	—
109	C₄	#16863229	1	2	yes	—
110	C₄	C₈×Dic₁₃₄	1	1	yes	characteristic
111	C₄	#16863812	1	1	yes	characteristic
112	C₂	C₁₆×Dic₁₃₄	1	1	yes	characteristic
113	C₂	C₂×#16863229	1	1	yes	characteristic
114	C₂	#22562502	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₃₄

C₂

yes

maximal, characteristic

C₄⋊(C₆₇⋊C₃₂)

C₂

yes

maximal, characteristic

Dic₆₇⋊C₃₂

C₂

yes

maximal

#22558342

C₂

yes

maximal

C₄×C₃₂×C₆₇

C₂

yes

maximal, characteristic

C₈×Dic₁₃₄

C₄

yes

characteristic

C₂×C₆₇⋊C₃₂

C₂²

yes

—

C₄⋊(C₆₇⋊C₁₆)

C₄

yes

characteristic

Dic₆₇⋊C₁₆

C₄

yes

—

C₁₆×Dic₆₇

C₂²

yes

—

C₂×C₃₂×C₆₇

C₂²

yes

characteristic

C₂×C₃₂×C₆₇

C₂²

yes

characteristic

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

C₄×Dic₁₃₄

C₈

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

—

C₄⋊(C₆₇⋊C₈)

C₈

yes

characteristic

Dic₆₇⋊C₈

C₈

yes

—

C₈×Dic₆₇

C₂×C₄

yes

—

C₆₇⋊C₃₂

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₇

C₂³

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₃₂×C₆₇

—

C₃₂×C₆₇

Q₈

yes

complemented

C₂×Dic₁₃₄

C₁₆

yes

characteristic

C₂×C₆₇⋊C₈

C₂×C₈

yes

—

C₄⋊Dic₆₇

C₁₆

yes

characteristic

Dic₆₇⋊C₄

C₁₆

yes

—

C₄×Dic₆₇

C₂×C₈

yes

—

C₆₇⋊C₁₆

—

C₂×C₈×C₆₇

C₂×C₈

yes

characteristic

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×C₈

yes

characteristic

C₁₆×C₆₇

—

C₁₆×C₆₇

C₂×Q₈

yes

characteristic

C₁₆×C₆₇

#41

yes

characteristic

C₂×C₄×C₆₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₈

yes

characteristic

C₂×Dic₆₇

C₂×C₁₆

yes

—

C₂×Dic₆₇

C₂×C₁₆

yes

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

C₄×Q₈

yes

characteristic

C₈×C₆₇

#131

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

C₂²×C₁₆

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

M₆₄

yes

—

C₄×C₆₇

C₈×Q₈

yes

characteristic

C₄×C₆₇

#504

yes

characteristic

Dic₆₇

—

Dic₆₇

—

#13660

—

maximal, Sylow

C₂×C₆₇

Q₈×C₁₆

yes

characteristic

C₂×C₆₇

C₂×M₆₄

yes

characteristic

C₂×C₆₇

#2258

yes

characteristic

Q₈×C₁₆

—

C₄⋊C₃₂

—

C₄⋊C₃₂

—

#1397

—

C₄×C₃₂

D₆₇

yes

characteristic

C₆₇

#13660

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₆₇

yes

characteristic

C₂×C₃₂

C₂×D₆₇

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₇

yes

characteristic

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₆₇

yes

characteristic

C₂×C₁₆

C₂²×D₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₆₇

yes

characteristic

C₃₂

—

C₃₂

—

134

—

C₃₂

Dic₁₃₄

yes

complemented

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

C₈×D₆₇

yes

characteristic

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₈×D₆₇

yes

characteristic

C₁₆

—

C₁₆

—

134

—

C₁₆

C₂×Dic₁₃₄

yes

characteristic

C₁₆

#11761628

yes

characteristic

Q₈

—

134

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₁₆×D₆₇

yes

characteristic

C₂×C₄

C₁₆×D₆₇

yes

characteristic

100

C₂×C₄

C₂×C₈×D₆₇

yes

characteristic

101

C₈

—

102

C₈

—

134

—

103

C₈

C₄×Dic₁₃₄

yes

characteristic

104

C₈

#14524049

yes

characteristic

105

C₂²

C₂×C₁₆×D₆₇

yes

characteristic

106

C₄

—

107

C₄

—

134

—

108

C₄

—

134

—

109

C₄

#16863229

yes

—

110

C₄

C₈×Dic₁₃₄

yes

characteristic

111

C₄

#16863812

yes

characteristic

112

C₂

C₁₆×Dic₁₃₄

yes

characteristic

113

C₂

C₂×#16863229

yes

characteristic

114

C₂

#22562502

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₆₇
commutator, socle, prime radical	C₂²×C₆₇	C₂²×C₁₆
Fitting	C₄×C₃₂×C₆₇	C₂
cosocle	C₂×C₁₆×C₆₇	C₂³
hypercenter	C₄×C₃₂	D₆₇
hypocenter	C₆₇	#13660
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₃₂

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₆₇

commutator, socle, prime radical

C₂²×C₆₇

C₂²×C₁₆

Fitting

C₄×C₃₂×C₆₇

C₂

cosocle

C₂×C₁₆×C₆₇

C₂³

hypercenter

C₄×C₃₂

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₁₆ ◁ C₄×C₃₂

N	H ≅ G/N	type
24. C₃₂×C₆₇	Q₈	semidirect
63. C₆₇	#13660	semidirect
82. C₃₂	Dic₁₃₄	semidirect

H ≅ G/N

type

24. C₃₂×C₆₇

Q₈

semidirect

63. C₆₇

#13660

semidirect

82. C₃₂

Dic₁₃₄

semidirect

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

subgroup	index
C₁₆×Dic₁₃₄	2
C₄⋊(C₆₇⋊C₃₂)	2
Dic₆₇⋊C₃₂	2
#22558342	2
C₄×C₃₂×C₆₇	2
#13660	67

subgroup

index

C₁₆×Dic₁₃₄

C₄⋊(C₆₇⋊C₃₂)

Dic₆₇⋊C₃₂

#22558342

C₄×C₃₂×C₆₇

#13660

Subgroups — #30776466

Conjugacy classes of subgroups

Complements

Maximal subgroups