Group #30776318 — subgroups

Subgroups — #30776318

Order 17152.

206 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 123 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	C₂²×C₆₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#22561805	C₂	1	1	yes	maximal, characteristic
3	Dic₆₇⋊C₃₂	C₂	1	4	yes	maximal, complemented
4	C₂×M₆₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	C₄	1	1	yes	characteristic
6	#16863351	C₄	1	1	yes	characteristic
7	C₂×C₆₇⋊C₃₂	—	2	2	no	—
8	C₂×C₆₇⋊C₃₂	C₂²	1	2	yes	—
9	#16863171	C₄	1	2	yes	—
10	C₁₆×Dic₆₇	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	M₆₄×C₆₇	—	2	2	no	—
13	C₂×C₃₂×C₆₇	C₂²	1	2	yes	—
14	C₂×#11761612	C₈	1	1	yes	characteristic
15	#14524002	C₈	1	1	yes	characteristic
16	#14523983	—	2	2	no	—
17	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
18	#14523953	C₈	1	2	yes	—
19	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
20	C₆₇⋊C₃₂	—	2	2	no	—
21	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
24	C₂×C₁₆×C₆₇	Q₈	1	2	yes	—
25	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
26	C₃₂×C₆₇	—	2	2	no	—
27	#11761612	—	2	2	no	—
28	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
29	C₄⋊Dic₆₇	C₁₆	1	2	yes	—
30	C₂²⋊Dic₆₇	C₁₆	1	2	yes	—
31	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
32	C₆₇⋊C₁₆	—	2	2	no	—
33	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
37	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₁₆×C₆₇	C₄⋊C₄	1	2	yes	—
39	C₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₆₇	C₂×Q₈	1	1	yes	characteristic
41	C₂³×C₆₇	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
43	C₂×C₄×C₆₇	C₂×C₁₆	1	1	yes	characteristic
44	C₂×C₄×C₆₇	M₃₂	1	2	yes	—
45	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
46	C₂×Dic₆₇	C₂×C₁₆	1	4	yes	—
47	C₈×C₆₇	C₄⋊C₈	1	2	yes	—
48	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₈×C₆₇	C₂×M₁₆	1	1	yes	characteristic
50	C₆₇⋊C₈	—	2	2	no	—
51	C₂²×C₆₇	—	2	2	no	—
52	C₂²×C₆₇	C₄⋊C₁₆	1	2	yes	—
53	C₂²×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
54	C₄×C₆₇	C₄⋊C₁₆	1	2	yes	—
55	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
56	C₄×C₆₇	C₂×M₃₂	1	1	yes	characteristic
57	Dic₆₇	—	2	4	no	—
58	#13628	—	67	1	no	maximal, Sylow
59	C₂×C₆₇	—	2	2	no	—
60	C₂×C₆₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
61	C₂×C₆₇	#2258	1	2	yes	—
62	C₂×M₆₄	D₆₇	1	1	yes	characteristic
63	C₂²×C₃₂	—	67	1	no	—
64	#2108	—	67	1	no	—
65	C₄⋊C₃₂	—	67	4	no	—
66	C₆₇	#13628	1	1	yes	characteristic, complemented, Sylow
67	C₂×M₃₂	—	67	1	no	—
68	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
69	#405	—	67	1	no	—
70	M₆₄	—	2	2	no	—
71	C₂×C₃₂	—	67	2	no	—
72	C₂×C₃₂	—	134	2	no	—
73	C₂×C₃₂	C₂×D₆₇	1	2	yes	—
74	#346	—	67	2	no	—
75	C₄×C₁₆	—	67	2	no	—
76	C₂×M₁₆	—	67	1	no	—
77	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
78	#117	—	67	1	no	—
79	M₃₂	—	134	2	no	—
80	C₂×C₁₆	—	67	2	no	—
81	C₂×C₁₆	Dic₁₃₄	1	2	yes	—
82	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
83	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
84	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
85	#97	—	67	2	no	—
86	C₄×C₈	—	67	2	no	—
87	C₃₂	—	2	2	no	—
88	C₃₂	—	134	2	no	—
89	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
90	M₁₆	—	134	2	no	—
91	C₂×C₈	—	67	2	no	—
92	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
93	C₂×C₈	C₈×D₆₇	1	1	yes	characteristic
94	C₂×C₈	#11761611	1	2	yes	—
95	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
96	C₄⋊C₄	—	67	2	no	—
97	C₂²⋊C₄	—	67	2	no	—
98	C₄²	—	67	2	no	—
99	C₁₆	—	134	2	no	—
100	C₁₆	Dic₆₇⋊C₄	1	2	yes	—
101	C₁₆	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
102	C₁₆	C₂×Dic₁₃₄	1	1	yes	characteristic
103	C₂³	C₁₆×D₆₇	1	1	yes	characteristic
104	C₂×C₄	—	67	4	no	—
105	C₂×C₄	Dic₆₇⋊C₈	1	2	yes	—
106	C₂×C₄	C₁₆×D₆₇	1	1	yes	characteristic
107	C₂×C₄	#14523982	1	2	yes	—
108	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
109	C₈	—	134	2	no	—
110	C₈	Dic₆₇⋊C₈	1	2	yes	—
111	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
112	C₈	C₂×#11761611	1	1	yes	characteristic
113	C₂²	—	2	2	no	—
114	C₂²	Dic₆₇⋊C₁₆	1	2	yes	—
115	C₂²	C₂×C₁₆×D₆₇	1	1	yes	characteristic
116	C₄	—	134	4	no	—
117	C₄	Dic₆₇⋊C₁₆	1	2	yes	—
118	C₄	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
119	C₄	C₂×#14523982	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×Dic₆₇⋊C₁₆	1	1	yes	characteristic
122	C₂	#22562504	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₆₇⋊C₃₂

C₂

yes

maximal, characteristic

#22561805

C₂

yes

maximal, characteristic

Dic₆₇⋊C₃₂

C₂

yes

maximal, complemented

C₂×M₆₄×C₆₇

C₂

yes

maximal, characteristic

C₂×#14523983

C₄

yes

characteristic

#16863351

C₄

yes

characteristic

C₂×C₆₇⋊C₃₂

—

C₂×C₆₇⋊C₃₂

C₂²

yes

—

#16863171

C₄

yes

—

C₁₆×Dic₆₇

C₂²

yes

—

C₂²×C₁₆×C₆₇

C₂²

yes

characteristic

M₆₄×C₆₇

—

C₂×C₃₂×C₆₇

C₂²

yes

—

C₂×#11761612

C₈

yes

characteristic

#14524002

C₈

yes

characteristic

#14523983

—

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

—

#14523953

C₈

yes

—

C₈×Dic₆₇

C₂×C₄

yes

—

C₆₇⋊C₃₂

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₇

D₄

yes

—

C₂×C₁₆×C₆₇

Q₈

yes

—

C₂×C₁₆×C₆₇

C₂³

yes

characteristic

C₃₂×C₆₇

—

#11761612

—

C₂×C₆₇⋊C₈

C₂×C₈

yes

—

C₄⋊Dic₆₇

C₁₆

yes

—

C₂²⋊Dic₆₇

C₁₆

yes

—

C₄×Dic₆₇

C₂×C₈

yes

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₂×C₈

yes

characteristic

C₂×C₈×C₆₇

M₁₆

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₁₆×C₆₇

C₄⋊C₄

yes

—

C₁₆×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

C₂×Q₈

yes

characteristic

C₂³×C₆₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₈

yes

—

C₂×C₄×C₆₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₆₇

M₃₂

yes

—

C₂×C₄×C₆₇

C₂²×C₈

yes

characteristic

C₂×Dic₆₇

C₂×C₁₆

yes

—

C₈×C₆₇

C₄⋊C₈

yes

—

C₈×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₈×C₆₇

C₂×M₁₆

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄⋊C₁₆

yes

—

C₂²×C₆₇

C₂²×C₁₆

yes

characteristic

C₄×C₆₇

C₄⋊C₁₆

yes

—

C₄×C₆₇

C₂×C₄⋊C₈

yes

characteristic

C₄×C₆₇

C₂×M₃₂

yes

characteristic

Dic₆₇

—

#13628

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₆₇

#2258

yes

—

C₂×M₆₄

D₆₇

yes

characteristic

C₂²×C₃₂

—

#2108

—

C₄⋊C₃₂

—

C₆₇

#13628

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₆₇

yes

characteristic

#405

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

134

—

C₂×C₃₂

C₂×D₆₇

yes

—

#346

—

C₄×C₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₄×D₆₇

yes

characteristic

#117

—

M₃₂

—

134

—

C₂×C₁₆

—

C₂×C₁₆

Dic₁₃₄

yes

—

C₂×C₁₆

C₄×D₆₇

yes

characteristic

C₂×C₁₆

C₂²⋊D₆₇

yes

—

C₂×C₁₆

C₂²×D₆₇

yes

characteristic

#97

—

C₄×C₈

—

C₃₂

—

C₃₂

—

134

—

C₂²×C₄

C₈×D₆₇

yes

characteristic

M₁₆

—

134

—

C₂×C₈

—

C₂×C₈

Dic₆₇⋊C₄

yes

—

C₂×C₈

C₈×D₆₇

yes

characteristic

C₂×C₈

#11761611

yes

—

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

134

—

100

C₁₆

Dic₆₇⋊C₄

yes

—

101

C₁₆

C₂×C₂²⋊D₆₇

yes

characteristic

102

C₁₆

C₂×Dic₁₃₄

yes

characteristic

103

C₂³

C₁₆×D₆₇

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

Dic₆₇⋊C₈

yes

—

106

C₂×C₄

C₁₆×D₆₇

yes

characteristic

107

C₂×C₄

#14523982

yes

—

108

C₂×C₄

C₂×C₈×D₆₇

yes

characteristic

109

C₈

—

134

—

110

C₈

Dic₆₇⋊C₈

yes

—

111

C₈

C₂×Dic₆₇⋊C₄

yes

characteristic

112

C₈

C₂×#11761611

yes

characteristic

113

C₂²

—

114

C₂²

Dic₆₇⋊C₁₆

yes

—

115

C₂²

C₂×C₁₆×D₆₇

yes

characteristic

116

C₄

—

134

—

117

C₄

Dic₆₇⋊C₁₆

yes

—

118

C₄

C₂×Dic₆₇⋊C₈

yes

characteristic

119

C₄

C₂×#14523982

yes

characteristic

120

C₂

—

121

C₂

C₂×Dic₆₇⋊C₁₆

yes

characteristic

122

C₂

#22562504

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₆₇
commutator, socle	C₂²×C₆₇	C₂²×C₁₆
Fitting	C₂×M₆₄×C₆₇	C₂
prime radical	C₂³×C₆₇	C₂×C₁₆
cosocle	C₂×C₁₆×C₆₇	C₂³
hypercenter	C₂×M₆₄	D₆₇
hypocenter	C₆₇	#13628
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₆₇

commutator, socle

C₂²×C₆₇

C₂²×C₁₆

Fitting

C₂×M₆₄×C₆₇

C₂

prime radical

C₂³×C₆₇

C₂×C₁₆

cosocle

C₂×C₁₆×C₆₇

C₂³

hypercenter

C₂×M₆₄

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

N	H ≅ G/N	type
3. Dic₆₇⋊C₃₂	C₂	semidirect
66. C₆₇	#13628	semidirect

H ≅ G/N

type

3. Dic₆₇⋊C₃₂

C₂

semidirect

66. C₆₇

#13628

semidirect

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

subgroup	index
C₂²×C₆₇⋊C₃₂	2
#22561805	2
Dic₆₇⋊C₃₂	2
C₂×M₆₄×C₆₇	2
#13628	67

subgroup

index

C₂²×C₆₇⋊C₃₂

#22561805

Dic₆₇⋊C₃₂

C₂×M₆₄×C₆₇

#13628

Subgroups — #30776318

Conjugacy classes of subgroups

Complements

Maximal subgroups