Group #30764868 — subgroups

Subgroups — #30764868

Order 17152.

260 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 163 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	#22562008	C₂	1	2	yes	maximal
2	#22558127	C₂	1	2	yes	maximal, complemented
3	#22558126	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1450	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₆₇⋊C₁₆)	C₄	1	4	yes	—
7	C₂²⋊(C₆₇⋊C₁₆)	C₄	1	4	yes	—
8	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
9	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
10	#431×C₆₇	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	#322×C₆₇	C₄	1	2	yes	—
13	C₈²×C₆₇	C₂²	1	1	yes	characteristic
14	C₈²×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
16	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
17	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
19	C₄⋊C₈×C₆₇	C₈	1	4	yes	—
20	C₂²⋊C₈×C₆₇	C₈	1	4	yes	—
21	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
26	C₆₇⋊C₁₆	—	2	4	no	—
27	C₆₇⋊C₁₆	—	2	4	no	—
28	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
29	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
30	C₂×C₈×C₆₇	—	2	4	no	—
31	C₂×C₈×C₆₇	C₄²	1	2	yes	—
32	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
34	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
35	C₄²×C₆₇	C₄²	1	1	yes	characteristic
36	C₄²×C₆₇	C₂×C₈	1	2	yes	—
37	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₂³×C₆₇	C₄×C₈	1	1	yes	characteristic
39	C₂×C₄×C₆₇	—	2	2	no	—
40	C₂×C₄×C₆₇	—	2	4	no	—
41	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
42	C₂×C₄×C₆₇	C₄×C₈	1	1	yes	characteristic
43	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
44	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
45	C₈×C₆₇	—	2	2	no	—
46	C₈×C₆₇	—	2	4	no	—
47	C₈×C₆₇	—	2	4	no	—
48	C₈×C₆₇	#131	1	4	yes	—
49	C₂²×C₆₇	—	2	2	no	—
50	C₂²×C₆₇	—	2	1	no	—
51	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
52	C₄×C₆₇	—	2	2	no	—
53	C₄×C₆₇	—	2	1	no	—
54	C₄×C₆₇	#405	1	2	yes	—
55	C₄×C₆₇	#504	1	4	yes	—
56	#11412	—	67	1	no	maximal, Sylow
57	C₂×C₆₇	—	4	1	no	—
58	C₂×C₆₇	#1450	1	1	yes	characteristic
59	C₂×C₆₇	#2108	1	2	yes	—
60	#2162	—	67	2	no	—
61	#1450	D₆₇	1	1	yes	characteristic
62	#1312	—	67	2	no	—
63	#1311	—	67	1	no	—
64	C₈×C₁₆	—	67	1	no	—
65	C₆₇	#11412	1	1	yes	characteristic, complemented, Sylow
66	#431	Dic₆₇	1	2	yes	—
67	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
68	C₄⋊C₁₆	—	67	4	no	—
69	C₂²⋊C₁₆	—	67	4	no	—
70	C₄×C₁₆	—	67	2	no	—
71	C₄×C₁₆	—	67	2	no	—
72	#322	Dic₆₇	1	2	yes	—
73	C₈²	C₂×D₆₇	1	1	yes	characteristic
74	C₈²	C₂×D₆₇	1	1	yes	characteristic
75	C₂²×C₈	C₄×D₆₇	1	2	yes	—
76	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
77	C₂×C₁₆	—	67	4	no	—
78	C₂×C₁₆	—	67	4	no	—
79	C₄⋊C₈	C₆₇⋊C₈	1	4	yes	—
80	C₂²⋊C₈	C₆₇⋊C₈	1	4	yes	—
81	C₄×C₈	C₄×D₆₇	1	2	yes	—
82	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
83	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
84	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
85	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
86	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
87	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
90	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
91	C₂×C₈	C₂×C₆₇⋊C₈	1	4	yes	—
92	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
93	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
94	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
95	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
96	C₁₆	—	134	4	no	—
97	C₁₆	—	134	4	no	—
98	C₂³	C₄×C₆₇⋊C₈	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	C₄×C₆₇⋊C₈	1	2	yes	—
102	C₂×C₄	C₄×C₆₇⋊C₈	1	1	yes	characteristic
103	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
104	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
105	C₈	—	2	4	no	—
106	C₈	—	2	4	no	—
107	C₈	—	2	2	no	—
108	C₈	#14524049	1	4	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	—	2	1	no	—
111	C₂²	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	2	1	no	—
114	C₄	#16863351	1	2	yes	—
115	C₄	#16863813	1	4	yes	—
116	C₂	—	4	1	no	—
117	C₂	#22558492	1	1	yes	characteristic
118	C₂	#22561806	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22562008

C₂

yes

maximal

#22558127

C₂

yes

maximal, complemented

#22558126

C₂

yes

maximal, characteristic, complemented

C₈×C₆₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₆₇×#1450

C₂

yes

maximal, characteristic

C₄⋊(C₆₇⋊C₁₆)

C₄

yes

—

C₂²⋊(C₆₇⋊C₁₆)

C₄

yes

—

C₄×C₆₇⋊C₁₆

C₂²

yes

—

C₄×C₆₇⋊C₁₆

C₂²

yes

—

#431×C₆₇

C₄

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#322×C₆₇

C₄

yes

—

C₈²×C₆₇

C₂²

yes

characteristic

C₈²×C₆₇

C₂²

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

—

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₇

C₈

yes

—

C₂²⋊C₈×C₆₇

C₈

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₄²

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₈

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄²

yes

—

C₂×C₈×C₆₇

C₂×C₈

yes

—

C₂×C₈×C₆₇

C₂×C₈

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

—

C₄²×C₆₇

C₄²

yes

characteristic

C₄²×C₆₇

C₂×C₈

yes

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₂³×C₆₇

C₄×C₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×C₈

yes

—

C₂×C₄×C₆₇

C₄×C₈

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₈

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#131

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄×C₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#405

yes

—

C₄×C₆₇

#504

yes

—

#11412

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1450

yes

characteristic

C₂×C₆₇

#2108

yes

—

#2162

—

#1450

D₆₇

yes

characteristic

#1312

—

#1311

—

C₈×C₁₆

—

C₆₇

#11412

yes

characteristic, complemented, Sylow

#431

Dic₆₇

yes

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#322

Dic₆₇

yes

—

C₈²

C₂×D₆₇

yes

characteristic

C₈²

C₂×D₆₇

yes

characteristic

C₂²×C₈

C₄×D₆₇

yes

—

C₂×C₄²

C₂×Dic₆₇

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

C₆₇⋊C₈

yes

—

C₂²⋊C₈

C₆₇⋊C₈

yes

—

C₄×C₈

C₄×D₆₇

yes

—

C₄×C₈

C₂×Dic₆₇

yes

characteristic

C₄×C₈

C₂×Dic₆₇

yes

—

C₄×C₈

C₂×Dic₆₇

yes

—

C₄×C₈

C₂²×D₆₇

yes

characteristic

C₂²×C₄

C₄×Dic₆₇

yes

characteristic

C₂²×C₄

C₂×C₆₇⋊C₈

yes

—

C₂×C₈

—

C₂×C₈

C₄×Dic₆₇

yes

—

C₂×C₈

C₂×C₆₇⋊C₈

yes

—

C₂×C₈

C₂×C₆₇⋊C₈

yes

—

C₂×C₈

C₂×C₄×D₆₇

yes

—

C₄²

C₄×Dic₆₇

yes

characteristic

C₄²

C₂×C₆₇⋊C₈

yes

—

C₄²

C₂²×Dic₆₇

yes

characteristic

C₁₆

—

134

—

C₁₆

—

134

—

C₂³

C₄×C₆₇⋊C₈

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₄×C₆₇⋊C₈

yes

—

102

C₂×C₄

C₄×C₆₇⋊C₈

yes

characteristic

103

C₂×C₄

C₂×C₄×Dic₆₇

yes

characteristic

104

C₂×C₄

C₂²×C₆₇⋊C₈

yes

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

#14524049

yes

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂×C₄×C₆₇⋊C₈

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

#16863351

yes

—

115

C₄

#16863813

yes

—

116

C₂

—

117

C₂

#22558492

yes

characteristic

118

C₂

#22561806

yes

—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₆₇
commutator, socle	C₂²×C₆₇	C₂×C₄×C₈
Fitting	C₆₇×#1450	C₂
prime radical	C₂³×C₆₇	C₄×C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#1450	D₆₇
hypocenter	C₆₇	#11412
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄×C₈ ◁ #1450

subgroup

G/H

center, Frattini

C₄×C₈

C₂²×D₆₇

commutator, socle

C₂²×C₆₇

C₂×C₄×C₈

Fitting

C₆₇×#1450

C₂

prime radical

C₂³×C₆₇

C₄×C₈

cosocle

C₄×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₄×C₈ ◁ #1450

N	H ≅ G/N	type
2. #22558127	C₂	semidirect
3. #22558126	C₂	semidirect
4. C₈×C₆₇⋊C₁₆	C₂	semidirect
65. C₆₇	#11412	semidirect

H ≅ G/N

type

2. #22558127

C₂

semidirect

3. #22558126

C₂

semidirect

4. C₈×C₆₇⋊C₁₆

C₂

semidirect

65. C₆₇

#11412

semidirect

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

subgroup	index
#22562008	2
#22558127	2
#22558126	2
C₈×C₆₇⋊C₁₆	2
C₆₇×#1450	2
#11412	67

subgroup

index

#22562008

#22558127

#22558126

C₈×C₆₇⋊C₁₆

C₆₇×#1450

#11412

Subgroups — #30764868

Conjugacy classes of subgroups

Complements

Maximal subgroups