Group #30764867 — subgroups

Subgroups — #30764867

Order 17152.

260 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 131 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	#22562006	C₂	1	2	yes	maximal
2	#22558491	C₂	1	1	yes	maximal, characteristic
3	#22558128	C₂	1	2	yes	maximal, complemented
4	#22558126	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863447	C₄	1	2	yes	—
7	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
8	Dic₆₇⋊C₁₆	C₄	1	4	yes	—
9	(C₂×D₆₇)⋊C₁₆	C₄	1	4	yes	—
10	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
11	#16863118	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	C₂×C₄	1	2	yes	—
16	C₄²×D₆₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
18	Dic₆₇⋊C₈	C₈	1	4	yes	—
19	(C₂×D₆₇)⋊C₈	C₈	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	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
24	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
26	C₂×C₄×D₆₇	C₄²	1	1	yes	characteristic
27	C₂×C₄×D₆₇	C₂×C₈	1	2	yes	—
28	C₂×C₆₇⋊C₈	C₄²	1	2	yes	—
29	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
30	C₈×D₆₇	—	2	4	no	—
31	C₄×Dic₆₇	C₄²	1	1	yes	characteristic
32	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
33	C₆₇⋊C₁₆	—	2	4	no	—
34	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
35	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
36	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
37	C₁₆×C₆₇	—	2	4	no	—
38	C₂²×D₆₇	C₄×C₈	1	1	yes	characteristic
39	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
40	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
41	C₂×Dic₆₇	C₄×C₈	1	2	yes	—
42	C₂×Dic₆₇	C₄×C₈	1	1	yes	characteristic
43	C₄×D₆₇	—	2	2	no	—
44	C₄×D₆₇	—	2	4	no	—
45	C₈×C₆₇	—	2	4	no	—
46	C₈×C₆₇	#131	1	4	yes	—
47	C₆₇⋊C₈	—	2	2	no	—
48	C₆₇⋊C₈	—	2	4	no	—
49	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
50	C₂×D₆₇	—	2	2	no	—
51	C₂×D₆₇	—	2	1	no	—
52	C₄×C₆₇	#405	1	2	yes	—
53	C₄×C₆₇	#504	1	4	yes	—
54	Dic₆₇	—	2	2	no	—
55	Dic₆₇	—	2	1	no	—
56	#11412	—	67	1	no	maximal, Sylow
57	C₂×C₆₇	#1450	1	1	yes	characteristic
58	C₂×C₆₇	#2108	1	2	yes	—
59	D₆₇	—	4	1	no	—
60	#2162	—	67	2	no	—
61	#1450	—	67	1	no	—
62	#1312	—	67	2	no	—
63	#1311	—	67	1	no	—
64	C₈×C₁₆	D₆₇	1	1	yes	characteristic, complemented
65	C₆₇	#11412	1	1	yes	characteristic, complemented, Sylow
66	#431	—	67	2	no	—
67	C₂×C₄×C₈	—	67	1	no	—
68	C₄⋊C₁₆	—	67	4	no	—
69	C₂²⋊C₁₆	—	67	4	no	—
70	C₄×C₁₆	—	67	2	no	—
71	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
72	#322	—	67	2	no	—
73	C₈²	—	67	1	no	—
74	C₈²	C₂×D₆₇	1	1	yes	characteristic
75	C₂²×C₈	—	67	2	no	—
76	C₂×C₄²	—	67	1	no	—
77	C₂×C₁₆	—	67	4	no	—
78	C₂×C₁₆	C₄×D₆₇	1	4	yes	—
79	C₄⋊C₈	—	67	4	no	—
80	C₂²⋊C₈	—	67	4	no	—
81	C₄×C₈	—	67	2	no	—
82	C₄×C₈	—	67	2	no	—
83	C₄×C₈	—	67	1	no	—
84	C₄×C₈	C₄×D₆₇	1	2	yes	—
85	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
86	C₂²×C₄	—	67	1	no	—
87	C₂²×C₄	—	67	2	no	—
88	C₂×C₈	—	67	2	no	—
89	C₂×C₈	—	67	4	no	—
90	C₂×C₈	—	134	4	no	—
91	C₂×C₈	C₈×D₆₇	1	4	yes	—
92	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
93	C₄²	—	67	1	no	—
94	C₄²	—	67	2	no	—
95	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
96	C₁₆	—	2	4	no	—
97	C₁₆	—	134	4	no	—
98	C₂³	—	67	1	no	—
99	C₂×C₄	—	67	2	no	—
100	C₂×C₄	—	67	1	no	—
101	C₂×C₄	—	134	4	no	—
102	C₂×C₄	—	134	2	no	—
103	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
104	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
105	C₈	—	2	4	no	—
106	C₈	—	134	4	no	—
107	C₈	—	134	2	no	—
108	C₈	#14524049	1	4	yes	—
109	C₂²	—	134	2	no	—
110	C₂²	—	134	1	no	—
111	C₂²	C₄×C₈×D₆₇	1	1	yes	characteristic
112	C₄	—	134	2	no	—
113	C₄	—	134	1	no	—
114	C₄	#16863350	1	2	yes	—
115	C₄	#16863812	1	4	yes	—
116	C₂	—	268	1	no	—
117	C₂	#22558491	1	1	yes	characteristic
118	C₂	#22561804	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22562006

C₂

yes

maximal

#22558491

C₂

yes

maximal, characteristic

#22558128

C₂

yes

maximal, complemented

#22558126

C₂

yes

maximal, characteristic, complemented

C₈×C₁₆×C₆₇

C₂

yes

maximal, characteristic, complemented

#16863447

C₄

yes

—

C₄×C₈×D₆₇

C₂²

yes

characteristic

Dic₆₇⋊C₁₆

C₄

yes

—

(C₂×D₆₇)⋊C₁₆

C₄

yes

—

C₄×C₆₇⋊C₁₆

C₂²

yes

—

#16863118

C₄

yes

—

C₈×C₆₇⋊C₈

C₂²

yes

characteristic

C₄×C₁₆×C₆₇

C₂²

yes

—

C₈²×C₆₇

C₂²

yes

characteristic

C₂×C₈×D₆₇

C₂×C₄

yes

—

C₄²×D₆₇

C₂×C₄

yes

characteristic

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

—

Dic₆₇⋊C₈

C₈

yes

—

(C₂×D₆₇)⋊C₈

C₈

yes

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

characteristic

C₄×C₆₇⋊C₈

C₂×C₄

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

—

C₄×Dic₆₇

C₄²

yes

characteristic

C₄×Dic₆₇

C₂×C₈

yes

—

C₆₇⋊C₁₆

—

C₂×C₈×C₆₇

C₂×C₈

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₁₆×C₆₇

—

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

—

C₈×C₆₇

—

C₈×C₆₇

#131

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₂×C₄×C₈

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

#405

yes

—

C₄×C₆₇

#504

yes

—

Dic₆₇

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

D₆₇

—

—

—

—

—

C₈×C₁₆

D₆₇

yes

characteristic, complemented

C₆₇

#11412

yes

characteristic, complemented, Sylow

#431

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₇

yes

—

#322

—

C₈²

—

C₈²

C₂×D₆₇

yes

characteristic

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₆₇

yes

—

C₄⋊C₈

—

C₂²⋊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

—

134

—

C₂×C₈

C₈×D₆₇

yes

—

C₂×C₈

C₂×C₄×D₆₇

yes

—

C₄²

—

C₄²

—

C₄²

C₂×C₄×D₆₇

yes

characteristic

C₁₆

—

C₁₆

—

134

—

C₂³

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

134

—

102

C₂×C₄

—

134

—

103

C₂×C₄

C₄²×D₆₇

yes

characteristic

104

C₂×C₄

C₂×C₈×D₆₇

yes

—

105

C₈

—

106

C₈

—

134

—

107

C₈

—

134

—

108

C₈

#14524049

yes

—

109

C₂²

—

134

—

110

C₂²

—

134

—

111

C₂²

C₄×C₈×D₆₇

yes

characteristic

112

C₄

—

134

—

113

C₄

—

134

—

114

C₄

#16863350

yes

—

115

C₄

#16863812

yes

—

116

C₂

—

268

—

117

C₂

#22558491

yes

characteristic

118

C₂

#22561804

yes

—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₆₇
commutator, socle	C₂²×C₆₇	C₂×C₄×C₈
Fitting	C₈×C₁₆×C₆₇	C₂
prime radical	C₂²×D₆₇	C₄×C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₈×C₁₆	D₆₇
hypocenter	C₆₇	#11412
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄×C₈ ◁ C₈×C₁₆

subgroup

G/H

center, Frattini

C₄×C₈

C₂²×D₆₇

commutator, socle

C₂²×C₆₇

C₂×C₄×C₈

Fitting

C₈×C₁₆×C₆₇

C₂

prime radical

C₂²×D₆₇

C₄×C₈

cosocle

C₄×C₈×C₆₇

C₂³

hypercenter

C₈×C₁₆

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₄×C₈ ◁ C₈×C₁₆

N	H ≅ G/N	type
3. #22558128	C₂	semidirect
4. #22558126	C₂	semidirect
5. C₈×C₁₆×C₆₇	C₂	semidirect
64. C₈×C₁₆	D₆₇	semidirect
65. C₆₇	#11412	semidirect

H ≅ G/N

type

3. #22558128

C₂

semidirect

4. #22558126

C₂

semidirect

5. C₈×C₁₆×C₆₇

C₂

semidirect

64. C₈×C₁₆

D₆₇

semidirect

65. C₆₇

#11412

semidirect

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

subgroup	index
#22562006	2
#22558491	2
#22558128	2
#22558126	2
C₈×C₁₆×C₆₇	2
#11412	67

subgroup

index

C₈×C₁₆×C₆₇

Subgroups — #30764867

Conjugacy classes of subgroups

Complements

Maximal subgroups