Group #30770833 — subgroups

Subgroups — #30770833

Order 17152.

390 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 227 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×#16863172	C₂	1	2	yes	maximal
2	#22558287	C₂	1	4	yes	maximal
3	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₆₇⋊C₁₆	C₂²	1	4	yes	—
5	#16863172	C₄	1	8	yes	complemented
6	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₄³×C₆₇	C₄	1	1	yes	characteristic
11	C₂×C₆₇⋊C₁₆	—	2	8	no	—
12	C₂×C₆₇⋊C₁₆	C₂×C₄	1	8	yes	—
13	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
14	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₆₇	C₈	1	4	yes	—
17	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₆₇	—	2	4	no	—
20	C₄×C₈×C₆₇	—	2	4	no	—
21	C₄×C₈×C₆₇	C₈	1	4	yes	—
22	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
23	C₆₇⋊C₁₆	—	2	8	no	—
24	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
25	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
26	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₈×C₆₇	—	2	2	no	—
28	C₂×C₈×C₆₇	—	2	2	no	—
29	C₂×C₈×C₆₇	C₄²	1	2	yes	—
30	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
31	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
32	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
33	C₂×C₈×C₆₇	#41	1	4	yes	—
34	C₄²×C₆₇	—	2	8	no	—
35	C₄²×C₆₇	—	2	2	no	—
36	C₄²×C₆₇	—	2	2	no	—
37	C₄²×C₆₇	C₄²	1	2	yes	—
38	C₄²×C₆₇	C₂×C₈	1	2	yes	—
39	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
40	C₂×C₄×C₆₇	—	2	4	no	—
41	C₂×C₄×C₆₇	—	2	4	no	—
42	C₂×C₄×C₆₇	—	2	4	no	—
43	C₂×C₄×C₆₇	C₄×C₈	1	4	yes	—
44	C₂×C₄×C₆₇	M₃₂	1	8	yes	—
45	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
46	C₂×C₄×C₆₇	#117	1	2	yes	—
47	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
49	C₈×C₆₇	—	2	4	no	—
50	C₈×C₆₇	#97	1	4	yes	—
51	C₈×C₆₇	#117	1	4	yes	—
52	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
53	C₂²×C₆₇	#431	1	2	yes	—
54	C₂²×C₆₇	C₂×M₃₂	1	4	yes	—
55	C₄×C₆₇	—	2	4	no	—
56	C₄×C₆₇	—	2	8	no	—
57	C₄×C₆₇	#376	1	1	yes	characteristic
58	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
59	C₄×C₆₇	#431	1	2	yes	—
60	#12591	—	67	1	no	maximal, Sylow
61	C₂×C₆₇	#1744	1	1	yes	characteristic
62	C₂×C₆₇	C₄×M₃₂	1	2	yes	—
63	C₂×C₆₇	#2163	1	4	yes	—
64	C₂×#346	—	67	2	no	—
65	C₄²×C₈	D₆₇	1	1	yes	characteristic
66	#1374	—	67	4	no	—
67	C₆₇	#12591	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₁₆	—	67	4	no	—
69	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
70	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
72	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
73	C₄³	Dic₆₇	1	1	yes	characteristic
74	#346	—	67	8	no	—
75	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
76	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
77	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
78	C₂×C₄²	C₆₇⋊C₈	1	4	yes	—
79	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
80	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
81	C₂×C₁₆	—	67	8	no	—
82	C₂×C₁₆	—	134	8	no	—
83	C₄×C₈	—	2	4	no	—
84	C₄×C₈	—	2	4	no	—
85	C₄×C₈	C₆₇⋊C₈	1	4	yes	—
86	C₄×C₈	C₄×D₆₇	1	4	yes	—
87	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
88	C₂²×C₄	C₂×C₆₇⋊C₈	1	4	yes	—
89	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
93	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
94	C₂×C₈	#11761612	1	4	yes	—
95	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
96	C₂×C₈	#11761628	1	4	yes	—
97	C₄²	—	2	8	no	—
98	C₄²	—	2	2	no	—
99	C₄²	—	2	2	no	—
100	C₄²	C₄×Dic₆₇	1	2	yes	—
101	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
102	C₁₆	—	134	8	no	—
103	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	C₄×C₆₇⋊C₈	1	4	yes	—
108	C₂×C₄	#14523983	1	8	yes	—
109	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
110	C₂×C₄	#14524002	1	2	yes	—
111	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
112	C₂×C₄	C₂×#11761612	1	2	yes	—
113	C₈	—	2	4	no	—
114	C₈	#14523954	1	4	yes	—
115	C₈	#14524000	1	4	yes	—
116	C₂²	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
117	C₂²	#16863449	1	2	yes	—
118	C₂²	C₂×#14523983	1	4	yes	—
119	C₄	—	2	4	no	—
120	C₄	—	2	8	no	—
121	C₄	#16863242	1	1	yes	characteristic
122	C₄	C₂×#14523954	1	1	yes	characteristic
123	C₄	#16863449	1	2	yes	—
124	C₂	#22559955	1	1	yes	characteristic
125	C₂	C₄×#14523983	1	2	yes	—
126	C₂	#22562012	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863172

C₂

yes

maximal

#22558287

C₂

yes

maximal

C₄²×C₈×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₁₆

C₂²

yes

—

#16863172

C₄

yes

complemented

C₂×C₄×C₈×C₆₇

C₄

yes

—

C₂×C₄×C₈×C₆₇

C₄

yes

characteristic

C₂×C₄×C₈×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄³×C₆₇

C₄

yes

characteristic

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

C₈

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

C₈

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂×C₈

yes

—

C₂²×C₄×C₆₇

C₂×C₈

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄²

yes

—

C₂×C₈×C₆₇

C₂×C₈

yes

—

C₂×C₈×C₆₇

M₁₆

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₄²

yes

—

C₄²×C₆₇

C₂×C₈

yes

—

C₂³×C₆₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×C₈

yes

—

C₂×C₄×C₆₇

M₃₂

yes

—

C₂×C₄×C₆₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₈×C₆₇

—

C₈×C₆₇

#97

yes

—

C₈×C₆₇

#117

yes

—

C₂²×C₆₇

C₂×C₄×C₈

yes

characteristic

C₂²×C₆₇

#431

yes

—

C₂²×C₆₇

C₂×M₃₂

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#376

yes

characteristic

C₄×C₆₇

C₂×#97

yes

characteristic

C₄×C₆₇

#431

yes

—

—

C₂×C₆₇

yes

C₂×C₆₇

C₄×M₃₂

yes

—

C₂×C₆₇

#2163

yes

—

C₂×#346

—

C₄²×C₈

D₆₇

yes

characteristic

#1374

—

C₆₇

#12591

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×C₄×C₈

Dic₆₇

yes

—

C₂×C₄×C₈

Dic₆₇

yes

characteristic

C₂×C₄×C₈

C₂×D₆₇

yes

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

C₄³

Dic₆₇

yes

characteristic

#346

—

C₂²×C₈

C₂×Dic₆₇

yes

—

C₂²×C₈

C₂×Dic₆₇

yes

characteristic

C₂²×C₈

C₂²×D₆₇

yes

characteristic

C₂×C₄²

C₆₇⋊C₈

yes

—

C₂×C₄²

C₂×Dic₆₇

yes

—

C₂×C₄²

C₂×Dic₆₇

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

134

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₆₇⋊C₈

yes

—

C₄×C₈

C₄×D₆₇

yes

—

C₂²×C₄

C₂×C₆₇⋊C₈

yes

—

C₂²×C₄

C₂×C₆₇⋊C₈

yes

—

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×Dic₆₇

yes

—

C₂×C₈

C₂×C₆₇⋊C₈

yes

—

C₂×C₈

#11761612

yes

—

C₂×C₈

C₂×C₄×D₆₇

yes

—

C₂×C₈

#11761628

yes

—

C₄²

—

C₄²

—

C₄²

—

100

C₄²

C₄×Dic₆₇

yes

—

101

C₄²

C₂×C₆₇⋊C₈

yes

—

102

C₁₆

—

134

—

103

C₂³

C₂²×C₆₇⋊C₈

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₄×C₆₇⋊C₈

yes

—

108

C₂×C₄

#14523983

yes

—

109

C₂×C₄

C₂×C₄×Dic₆₇

yes

characteristic

110

C₂×C₄

#14524002

yes

—

111

C₂×C₄

C₂²×C₆₇⋊C₈

yes

characteristic

112

C₂×C₄

C₂×#11761612

yes

—

113

C₈

—

114

C₈

#14523954

yes

—

115

C₈

#14524000

yes

—

116

C₂²

C₂×C₄×C₆₇⋊C₈

yes

characteristic

117

C₂²

#16863449

yes

—

118

C₂²

C₂×#14523983

yes

—

119

C₄

—

120

C₄

—

121

C₄

#16863242

yes

characteristic

122

C₄

C₂×#14523954

yes

characteristic

123

C₄

#16863449

yes

—

124

C₂

#22559955

yes

characteristic

125

C₂

C₄×#14523983

yes

—

126

C₂

#22562012

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂×C₄×C₈
Fitting	C₄²×C₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₄²×C₈	D₆₇
hypocenter	C₆₇	#12591
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₆₇

commutator

C₂²×C₆₇

C₂×C₄×C₈

Fitting

C₄²×C₈×C₆₇

C₂

socle, prime radical

C₂³×C₆₇

C₂²×C₈

cosocle

C₂²×C₈×C₆₇

C₂³

hypercenter

C₄²×C₈

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

N	H ≅ G/N	type
5. #16863172	C₄	semidirect
67. C₆₇	#12591	semidirect

H ≅ G/N

type

5. #16863172

C₄

semidirect

67. C₆₇

#12591

semidirect

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

subgroup	index
C₂×#16863172	2
#22558287	2
C₄²×C₈×C₆₇	2
#12591	67

subgroup

index

C₂×#16863172

#22558287

C₄²×C₈×C₆₇

#12591

Subgroups — #30770833

Conjugacy classes of subgroups

Complements

Maximal subgroups