Group #30764809 — subgroups

Subgroups — #30764809

Order 17152.

354 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 199 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×#14523982	C₂	1	2	yes	maximal
2	C₈²×D₆₇	C₂	1	1	yes	maximal, characteristic
3	#22558126	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₆×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523982	C₄	1	4	yes	—
7	C₄×C₈×D₆₇	C₄	1	2	yes	—
8	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
9	#16863171	C₄	1	4	yes	—
10	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
11	C₈×C₆₇⋊C₈	C₄	1	2	yes	—
12	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
14	C₈²×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₆₇	C₈	1	4	yes	—
16	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
17	C₄²×D₆₇	C₂×C₄	1	1	yes	characteristic
18	#14523982	C₈	1	16	yes	complemented
19	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
20	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₈×Dic₆₇	C₈	1	4	yes	—
23	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
25	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	C₂×C₄×D₆₇	C₄²	1	1	yes	characteristic
28	C₂×C₄×D₆₇	C₂×C₈	1	2	yes	—
29	C₂×C₆₇⋊C₈	C₄²	1	2	yes	—
30	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
31	C₈×D₆₇	—	2	8	no	—
32	C₈×D₆₇	C₂×C₈	1	8	yes	—
33	C₄×Dic₆₇	C₄²	1	1	yes	characteristic
34	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
35	C₆₇⋊C₁₆	C₂×C₈	1	8	yes	complemented
36	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
37	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
38	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
39	C₁₆×C₆₇	C₂×C₈	1	8	yes	complemented
40	C₂²×D₆₇	C₄×C₈	1	1	yes	characteristic
41	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
42	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
43	C₂×Dic₆₇	C₄×C₈	1	2	yes	—
44	C₂×Dic₆₇	C₄×C₈	1	1	yes	characteristic
45	C₄×D₆₇	—	2	4	no	—
46	C₄×D₆₇	C₄×C₈	1	4	yes	—
47	C₈×C₆₇	—	2	4	no	—
48	C₈×C₆₇	C₂²×C₈	1	4	yes	—
49	C₆₇⋊C₈	C₄×C₈	1	4	yes	—
50	C₆₇⋊C₈	M₃₂	1	8	yes	complemented
51	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
52	C₂×D₆₇	—	2	2	no	—
53	C₂×D₆₇	C₈²	1	2	yes	—
54	C₄×C₆₇	C₂×C₄×C₈	1	2	yes	—
55	C₄×C₆₇	C₂×M₃₂	1	4	yes	—
56	Dic₆₇	—	2	2	no	—
57	Dic₆₇	C₈²	1	2	yes	—
58	C₈×M₃₂	—	67	1	no	maximal, Sylow
59	C₂×C₆₇	C₂×C₈²	1	1	yes	characteristic
60	C₂×C₆₇	C₄×M₃₂	1	2	yes	—
61	D₆₇	—	2	2	no	—
62	C₄×M₃₂	—	67	2	no	—
63	C₂×C₈²	—	67	1	no	—
64	#1311	—	67	1	no	—
65	#1311	D₆₇	1	1	yes	characteristic, complemented
66	C₈×C₁₆	—	67	2	no	—
67	C₆₇	C₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
68	C₂×M₃₂	—	67	4	no	—
69	C₂×C₄×C₈	—	67	2	no	—
70	C₂×C₄×C₈	—	67	1	no	—
71	#346	—	67	4	no	—
72	C₄×C₁₆	—	67	2	no	—
73	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
74	C₈²	—	67	2	no	—
75	C₈²	—	67	1	no	—
76	C₈²	C₂×D₆₇	1	1	yes	characteristic
77	C₂²×C₈	—	67	2	no	—
78	C₂²×C₈	—	67	4	no	—
79	C₂×C₄²	—	67	1	no	—
80	M₃₂	—	67	16	no	—
81	C₂×C₁₆	—	67	4	no	—
82	C₂×C₁₆	C₄×D₆₇	1	4	yes	—
83	C₄×C₈	—	67	4	no	—
84	C₄×C₈	—	67	2	no	—
85	C₄×C₈	—	67	1	no	—
86	C₄×C₈	—	67	2	no	—
87	C₄×C₈	C₄×D₆₇	1	2	yes	—
88	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
89	C₂²×C₄	—	67	1	no	—
90	C₂²×C₄	—	67	2	no	—
91	C₂×C₈	—	67	2	no	—
92	C₂×C₈	—	67	4	no	—
93	C₂×C₈	—	67	8	no	—
94	C₂×C₈	—	134	8	no	—
95	C₂×C₈	C₈×D₆₇	1	4	yes	—
96	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
97	C₄²	—	67	1	no	—
98	C₄²	—	67	2	no	—
99	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
100	C₁₆	—	67	8	no	—
101	C₁₆	C₈×D₆₇	1	8	yes	complemented
102	C₂³	—	67	1	no	—
103	C₂×C₄	—	67	1	no	—
104	C₂×C₄	—	67	4	no	—
105	C₂×C₄	—	67	2	no	—
106	C₂×C₄	—	134	4	no	—
107	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
108	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₈	—	67	8	no	—
111	C₈	—	67	4	no	—
112	C₈	C₂×C₈×D₆₇	1	4	yes	—
113	C₂²	—	67	2	no	—
114	C₂²	—	134	2	no	—
115	C₂²	C₄×C₈×D₆₇	1	1	yes	characteristic
116	C₄	—	67	2	no	—
117	C₄	—	134	2	no	—
118	C₄	C₄×C₈×D₆₇	1	2	yes	—
119	C₄	M₃₂×D₆₇	1	4	yes	—
120	C₂	—	134	2	no	—
121	C₂	C₈²×D₆₇	1	1	yes	characteristic
122	C₂	#22561802	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14523982

C₂

yes

maximal

C₈²×D₆₇

C₂

yes

maximal, characteristic

#22558126

C₂

yes

maximal, characteristic, complemented

C₁₆×C₆₇⋊C₈

C₂

yes

maximal, complemented

C₆₇×#1311

C₂

yes

maximal, characteristic, complemented

C₂×#14523982

C₄

yes

—

C₄×C₈×D₆₇

C₄

yes

—

C₄×C₈×D₆₇

C₂²

yes

characteristic

#16863171

C₄

yes

—

C₄×C₆₇⋊C₁₆

C₂²

yes

—

C₈×C₆₇⋊C₈

C₄

yes

—

C₈×C₆₇⋊C₈

C₂²

yes

characteristic

C₄×C₁₆×C₆₇

C₂²

yes

—

C₈²×C₆₇

C₂²

yes

characteristic

C₂×C₈×D₆₇

C₈

yes

—

C₂×C₈×D₆₇

C₂×C₄

yes

—

C₄²×D₆₇

C₂×C₄

yes

characteristic

#14523982

C₈

yes

complemented

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₄×C₆₇⋊C₈

C₂×C₄

yes

—

C₈×Dic₆₇

C₈

yes

—

C₈×Dic₆₇

C₂×C₄

yes

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄×D₆₇

C₄²

yes

characteristic

C₂×C₄×D₆₇

C₂×C₈

yes

—

C₂×C₆₇⋊C₈

C₄²

yes

—

C₂×C₆₇⋊C₈

C₂×C₈

yes

—

C₈×D₆₇

—

C₈×D₆₇

C₂×C₈

yes

—

C₄×Dic₆₇

C₄²

yes

characteristic

C₄×Dic₆₇

C₂×C₈

yes

—

C₆₇⋊C₁₆

C₂×C₈

yes

complemented

C₂×C₈×C₆₇

C₂×C₈

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₁₆×C₆₇

C₂×C₈

yes

complemented

C₂²×D₆₇

C₄×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₈

yes

—

C₂×Dic₆₇

C₄×C₈

yes

—

C₂×Dic₆₇

C₄×C₈

yes

characteristic

C₄×D₆₇

—

C₄×D₆₇

C₄×C₈

yes

—

C₈×C₆₇

—

C₈×C₆₇

C₂²×C₈

yes

—

C₆₇⋊C₈

C₄×C₈

yes

—

C₆₇⋊C₈

M₃₂

yes

complemented

C₂²×C₆₇

C₂×C₄×C₈

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

C₈²

yes

—

C₄×C₆₇

C₂×C₄×C₈

yes

—

C₄×C₆₇

C₂×M₃₂

yes

—

Dic₆₇

—

Dic₆₇

C₈²

yes

—

C₈×M₃₂

—

maximal, Sylow

C₂×C₆₇

C₂×C₈²

yes

characteristic

C₂×C₆₇

C₄×M₃₂

yes

—

D₆₇

—

C₄×M₃₂

—

C₂×C₈²

—

#1311

—

#1311

D₆₇

yes

characteristic, complemented

C₈×C₁₆

—

C₆₇

C₈×M₃₂

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₇

yes

—

C₈²

—

C₈²

—

C₈²

C₂×D₆₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₆₇

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₆₇

yes

—

C₄×C₈

C₂²×D₆₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

C₈×D₆₇

yes

—

C₂×C₈

C₂×C₄×D₆₇

yes

—

C₄²

—

C₄²

—

C₄²

C₂×C₄×D₆₇

yes

characteristic

100

C₁₆

—

101

C₁₆

C₈×D₆₇

yes

complemented

102

C₂³

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

134

—

107

C₂×C₄

C₄²×D₆₇

yes

characteristic

108

C₂×C₄

C₂×C₈×D₆₇

yes

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

C₂×C₈×D₆₇

yes

—

113

C₂²

—

114

C₂²

—

134

—

115

C₂²

C₄×C₈×D₆₇

yes

characteristic

116

C₄

—

117

C₄

—

134

—

118

C₄

C₄×C₈×D₆₇

yes

—

119

C₄

M₃₂×D₆₇

yes

—

120

C₂

—

134

—

121

C₂

C₈²×D₆₇

yes

characteristic

122

C₂

#22561802

yes

—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₆₇
commutator	C₂×C₆₇	C₂×C₈²
Fitting	C₆₇×#1311	C₂
socle	C₂²×C₆₇	C₂×C₄×C₈
prime radical	C₂²×D₆₇	C₄×C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#1311	D₆₇
hypocenter	C₆₇	C₈×M₃₂
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄×C₈ ◁ #1311

subgroup

G/H

center, Frattini

C₄×C₈

C₂²×D₆₇

commutator

C₂×C₆₇

C₂×C₈²

Fitting

C₆₇×#1311

C₂

socle

C₂²×C₆₇

C₂×C₄×C₈

prime radical

C₂²×D₆₇

C₄×C₈

cosocle

C₄×C₈×C₆₇

C₂³

hypercenter

#1311

D₆₇

hypocenter

C₆₇

C₈×M₃₂

LCS

G ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄×C₈ ◁ #1311

N	H ≅ G/N	type
3. #22558126	C₂	semidirect
4. C₁₆×C₆₇⋊C₈	C₂	semidirect
5. C₆₇×#1311	C₂	semidirect
18. #14523982	C₈	semidirect
35. C₆₇⋊C₁₆	C₂×C₈	semidirect
39. C₁₆×C₆₇	C₂×C₈	semidirect
50. C₆₇⋊C₈	M₃₂	semidirect
65. #1311	D₆₇	semidirect
67. C₆₇	C₈×M₃₂	semidirect
101. C₁₆	C₈×D₆₇	semidirect

H ≅ G/N

type

3. #22558126

C₂

semidirect

4. C₁₆×C₆₇⋊C₈

C₂

semidirect

5. C₆₇×#1311

C₂

semidirect

18. #14523982

C₈

semidirect

35. C₆₇⋊C₁₆

C₂×C₈

semidirect

39. C₁₆×C₆₇

C₂×C₈

semidirect

50. C₆₇⋊C₈

M₃₂

semidirect

65. #1311

D₆₇

semidirect

67. C₆₇

C₈×M₃₂

semidirect

101. C₁₆

C₈×D₆₇

semidirect

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

subgroup	index
C₄×#14523982	2
C₈²×D₆₇	2
#22558126	2
C₁₆×C₆₇⋊C₈	2
C₆₇×#1311	2
C₈×M₃₂	67

subgroup

index

C₄×#14523982

C₈²×D₆₇

#22558126

C₁₆×C₆₇⋊C₈

C₆₇×#1311

C₈×M₃₂

Subgroups — #30764809

Conjugacy classes of subgroups

Complements

Maximal subgroups