Group #30776324 — subgroups

Subgroups — #30776324

Order 17152.

198 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 143 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	C₂×#16863230	C₂	1	2	yes	maximal
2	#22558407	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2108	C₂	1	1	yes	maximal, characteristic
4	#16863230	—	2	2	no	—
5	#16863230	C₂²	1	4	yes	—
6	C₂×C₆₇⋊C₃₂	—	2	2	no	—
7	C₂×M₃₂×C₆₇	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
9	#405×C₆₇	C₄	1	1	yes	characteristic
10	#346×C₆₇	C₄	1	2	yes	—
11	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
12	C₆₇⋊C₃₂	—	2	4	no	—
13	C₂×M₁₆×C₆₇	C₈	1	1	yes	characteristic
14	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
15	#117×C₆₇	C₈	1	1	yes	characteristic
16	M₃₂×C₆₇	—	2	2	no	—
17	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
18	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
20	C₂×C₁₆×C₆₇	Q₈	1	2	yes	—
21	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
22	#97×C₆₇	C₈	1	2	yes	—
23	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
25	M₁₆×C₆₇	C₁₆	1	4	yes	—
26	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
28	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
29	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
30	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
31	C₄⋊C₄×C₆₇	—	2	1	no	—
32	C₂²⋊C₄×C₆₇	—	2	1	no	—
33	C₄²×C₆₇	C₂×C₈	1	2	yes	—
34	C₁₆×C₆₇	—	2	2	no	—
35	C₁₆×C₆₇	C₄⋊C₄	1	2	yes	—
36	C₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
37	C₁₆×C₆₇	C₂×Q₈	1	1	yes	characteristic
38	C₂³×C₆₇	M₃₂	1	1	yes	characteristic
39	C₂×C₄×C₆₇	—	2	2	no	—
40	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
41	C₂×C₄×C₆₇	C₂×C₁₆	1	2	yes	—
42	C₂×C₄×C₆₇	M₃₂	1	1	yes	characteristic
43	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
44	C₈×C₆₇	C₄⋊C₈	1	2	yes	—
45	C₈×C₆₇	C₂×C₁₆	1	4	yes	—
46	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₈×C₆₇	C₂×M₁₆	1	1	yes	characteristic
48	C₂²×C₆₇	—	4	1	no	—
49	C₂²×C₆₇	C₄⋊C₁₆	1	2	yes	—
50	C₂²×C₆₇	C₂×M₃₂	1	1	yes	characteristic
51	C₄×C₆₇	—	4	2	no	—
52	C₄×C₆₇	C₄⋊C₁₆	1	2	yes	—
53	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
54	C₄×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
55	#13630	—	67	1	no	maximal, Sylow
56	C₂×C₆₇	—	2	1	no	—
57	C₂×C₆₇	—	2	2	no	—
58	C₂×C₆₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
59	C₂×M₆₄	—	67	2	no	—
60	#2108	D₆₇	1	1	yes	characteristic
61	#1422	—	67	4	no	—
62	C₆₇	#13630	1	1	yes	characteristic, complemented, Sylow
63	C₂×M₃₂	Dic₆₇	1	1	yes	characteristic
64	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
65	#405	Dic₆₇	1	1	yes	characteristic
66	M₆₄	—	67	4	no	—
67	M₆₄	—	134	2	no	—
68	C₂×C₃₂	—	134	2	no	—
69	#346	Dic₆₇	1	2	yes	—
70	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
71	C₂×M₁₆	C₆₇⋊C₈	1	1	yes	characteristic
72	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
73	#117	C₆₇⋊C₈	1	1	yes	characteristic
74	M₃₂	—	2	2	no	—
75	C₂×C₁₆	Dic₁₃₄	1	2	yes	—
76	C₂×C₁₆	D₂₆₈	1	2	yes	—
77	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
78	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
79	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
80	#97	C₆₇⋊C₈	1	2	yes	—
81	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
82	C₃₂	—	134	4	no	—
83	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
84	M₁₆	C₆₇⋊C₁₆	1	4	yes	—
85	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
86	C₂×C₈	C₂×C₆₇⋊C₈	1	1	yes	characteristic
87	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
88	C₂×C₈	#11761612	1	2	yes	—
89	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₂²⋊C₄	—	2	1	no	—
92	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
93	C₁₆	—	2	2	no	—
94	C₁₆	C₄⋊Dic₆₇	1	2	yes	—
95	C₁₆	C₂×D₂₆₈	1	1	yes	characteristic
96	C₁₆	C₂×Dic₁₃₄	1	1	yes	characteristic
97	C₂³	#14523983	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
100	C₂×C₄	C₂×C₆₇⋊C₁₆	1	2	yes	—
101	C₂×C₄	#14523983	1	1	yes	characteristic
102	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
103	C₈	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
104	C₈	C₂×C₆₇⋊C₁₆	1	4	yes	—
105	C₈	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
106	C₈	C₂×#11761612	1	1	yes	characteristic
107	C₂²	—	4	1	no	—
108	C₂²	C₄⋊(C₆₇⋊C₁₆)	1	2	yes	—
109	C₂²	C₂×#14523983	1	1	yes	characteristic
110	C₄	—	4	2	no	—
111	C₄	C₄⋊(C₆₇⋊C₁₆)	1	2	yes	—
112	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
113	C₄	C₂²×C₆₇⋊C₁₆	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	2	2	no	—
116	C₂	C₂×C₄⋊(C₆₇⋊C₁₆)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863230

C₂

yes

maximal

#22558407

C₂

yes

maximal, complemented

C₆₇×#2108

C₂

yes

maximal, characteristic

#16863230

—

#16863230

C₂²

yes

—

C₂×C₆₇⋊C₃₂

—

C₂×M₃₂×C₆₇

C₄

yes

characteristic

C₂²×C₁₆×C₆₇

C₂²

yes

characteristic

#405×C₆₇

C₄

yes

characteristic

#346×C₆₇

C₄

yes

—

C₄×C₁₆×C₆₇

C₂²

yes

—

C₆₇⋊C₃₂

—

C₂×M₁₆×C₆₇

C₈

yes

characteristic

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

#117×C₆₇

C₈

yes

characteristic

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₇

D₄

yes

—

C₂×C₁₆×C₆₇

Q₈

yes

—

C₂×C₁₆×C₆₇

C₂³

yes

characteristic

#97×C₆₇

C₈

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂×C₈

yes

characteristic

M₁₆×C₆₇

C₁₆

yes

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₂×C₈

yes

characteristic

C₂×C₈×C₆₇

C₂×C₈

yes

—

C₂×C₈×C₆₇

M₁₆

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×C₈

yes

—

C₁₆×C₆₇

—

C₁₆×C₆₇

C₄⋊C₄

yes

—

C₁₆×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

C₂×Q₈

yes

characteristic

C₂³×C₆₇

M₃₂

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄⋊C₈

yes

—

C₂×C₄×C₆₇

C₂×C₁₆

yes

—

C₂×C₄×C₆₇

M₃₂

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₈

yes

characteristic

C₈×C₆₇

C₄⋊C₈

yes

—

C₈×C₆₇

C₂×C₁₆

yes

—

C₈×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₈×C₆₇

C₂×M₁₆

yes

characteristic

C₂²×C₆₇

—

C₂²×C₆₇

C₄⋊C₁₆

yes

—

C₂²×C₆₇

C₂×M₃₂

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₄⋊C₁₆

yes

—

C₄×C₆₇

C₂×C₄⋊C₈

yes

characteristic

C₄×C₆₇

C₂²×C₁₆

yes

characteristic

#13630

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂×C₄⋊C₁₆

yes

characteristic

C₂×M₆₄

—

#2108

D₆₇

yes

characteristic

#1422

—

C₆₇

#13630

yes

characteristic, complemented, Sylow

C₂×M₃₂

Dic₆₇

yes

characteristic

C₂²×C₁₆

C₂×D₆₇

yes

characteristic

#405

Dic₆₇

yes

characteristic

M₆₄

—

M₆₄

—

134

—

C₂×C₃₂

—

134

—

#346

Dic₆₇

yes

—

C₄×C₁₆

C₂×D₆₇

yes

—

C₂×M₁₆

C₆₇⋊C₈

yes

characteristic

C₂²×C₈

C₂×Dic₆₇

yes

characteristic

#117

C₆₇⋊C₈

yes

characteristic

M₃₂

—

C₂×C₁₆

Dic₁₃₄

yes

—

C₂×C₁₆

D₂₆₈

yes

—

C₂×C₁₆

C₂×Dic₆₇

yes

—

C₂×C₁₆

C₂×Dic₆₇

yes

characteristic

C₂×C₁₆

C₂²×D₆₇

yes

characteristic

#97

C₆₇⋊C₈

yes

—

C₄×C₈

C₂×Dic₆₇

yes

—

C₃₂

—

134

—

C₂²×C₄

C₂×C₆₇⋊C₈

yes

characteristic

M₁₆

C₆₇⋊C₁₆

yes

—

C₂×C₈

C₄⋊Dic₆₇

yes

—

C₂×C₈

C₂×C₆₇⋊C₈

yes

characteristic

C₂×C₈

C₂×C₆₇⋊C₈

yes

—

C₂×C₈

#11761612

yes

—

C₂×C₈

C₂²×Dic₆₇

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×C₆₇⋊C₈

yes

—

C₁₆

—

C₁₆

C₄⋊Dic₆₇

yes

—

C₁₆

C₂×D₂₆₈

yes

characteristic

C₁₆

C₂×Dic₁₃₄

yes

characteristic

C₂³

#14523983

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄⋊(C₆₇⋊C₈)

yes

—

100

C₂×C₄

C₂×C₆₇⋊C₁₆

yes

—

101

C₂×C₄

#14523983

yes

characteristic

102

C₂×C₄

C₂²×C₆₇⋊C₈

yes

characteristic

103

C₈

C₄⋊(C₆₇⋊C₈)

yes

—

104

C₈

C₂×C₆₇⋊C₁₆

yes

—

105

C₈

C₂×C₄⋊Dic₆₇

yes

characteristic

106

C₈

C₂×#11761612

yes

characteristic

107

C₂²

—

108

C₂²

C₄⋊(C₆₇⋊C₁₆)

yes

—

109

C₂²

C₂×#14523983

yes

characteristic

110

C₄

—

111

C₄

C₄⋊(C₆₇⋊C₁₆)

yes

—

112

C₄

C₂×C₄⋊(C₆₇⋊C₈)

yes

characteristic

113

C₄

C₂²×C₆₇⋊C₁₆

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

C₂×C₄⋊(C₆₇⋊C₁₆)

yes

characteristic

	subgroup	G/H
center	C₁₆	C₂×D₂₆₈
commutator	C₄×C₆₇	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₆₇
Fitting	C₆₇×#2108	C₂
socle	C₂×C₆₇	C₂×C₄⋊C₁₆
prime radical	C₂³×C₆₇	M₃₂
cosocle	C₂×C₁₆×C₆₇	C₂³
hypercenter	#2108	D₆₇
hypocenter	C₆₇	#13630
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ #2108

subgroup

G/H

center

C₁₆

C₂×D₂₆₈

commutator

C₄×C₆₇

C₂²×C₁₆

Frattini

C₂×C₁₆

C₂²×D₆₇

Fitting

C₆₇×#2108

C₂

socle

C₂×C₆₇

C₂×C₄⋊C₁₆

prime radical

C₂³×C₆₇

M₃₂

cosocle

C₂×C₁₆×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ #2108

N	H ≅ G/N	type
2. #22558407	C₂	semidirect
62. C₆₇	#13630	semidirect

H ≅ G/N

type

2. #22558407

C₂

semidirect

62. C₆₇

#13630

semidirect

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

subgroup	index
C₂×#16863230	2
#22558407	2
C₆₇×#2108	2
#13630	67

subgroup

index

C₂×#16863230

#22558407

C₆₇×#2108

#13630

Subgroups — #30776324

Conjugacy classes of subgroups

Complements

Maximal subgroups