Group #22568484 — subgroups

Subgroups — #22568484

Order 8576.

340 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 133 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	#16863330	C₂	1	4	yes	maximal
2	#16863329	C₂	1	4	yes	maximal
3	#16863328	C₂	1	4	yes	maximal
4	#16863262	C₂	1	2	yes	maximal
5	#382×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₆₇	C₂²	1	4	yes	—
9	C₂×Dic₆₇⋊C₄	C₂²	1	4	yes	—
10	C₂×Dic₆₇⋊C₄	C₂²	1	4	yes	—
11	C₆₇⋊#95	C₂²	1	4	yes	—
12	C₆₇⋊#95	C₂²	1	4	yes	—
13	C₆₇⋊#95	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	#95×C₆₇	C₂²	1	4	yes	—
17	C₂²×Dic₆₇	C₂³	1	4	yes	—
18	C₂²×Dic₆₇	C₂³	1	4	yes	—
19	C₄⋊Dic₆₇	—	4	2	no	—
20	C₄⋊Dic₆₇	—	4	4	no	—
21	C₄⋊Dic₆₇	—	4	2	no	—
22	Dic₆₇⋊C₄	—	4	4	no	—
23	Dic₆₇⋊C₄	—	4	4	no	—
24	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
25	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
27	C₄⋊C₄×C₆₇	—	4	2	no	—
28	C₄²×C₆₇	Q₈	1	4	yes	—
29	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
30	C₂×C₄×C₆₇	—	2	1	no	—
31	C₂×C₄×C₆₇	—	2	2	no	—
32	C₂×C₄×C₆₇	—	2	2	no	—
33	C₂×C₄×C₆₇	—	2	1	no	—
34	C₂×C₄×C₆₇	—	2	4	no	—
35	C₂×C₄×C₆₇	—	2	4	no	—
36	C₂×C₄×C₆₇	—	2	4	no	—
37	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₆₇	C₂×Q₈	1	4	yes	—
39	C₂×Dic₆₇	—	2	4	no	—
40	C₂×Dic₆₇	—	2	4	no	—
41	C₂×Dic₆₇	—	2	4	no	—
42	C₂×Dic₆₇	—	2	4	no	—
43	C₂×Dic₆₇	—	2	4	no	—
44	C₂×Dic₆₇	—	2	4	no	—
45	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
46	C₂²×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
47	C₂²×C₆₇	H(5,𝔽₂)	1	2	yes	—
48	C₂²×C₆₇	2₋¹⁺⁴	1	2	yes	—
49	C₂²×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
50	C₄×C₆₇	—	2	4	no	—
51	C₄×C₆₇	—	2	2	no	—
52	C₄×C₆₇	—	4	4	no	—
53	Dic₆₇	—	4	4	no	—
54	Dic₆₇	—	4	4	no	—
55	C₂×C₆₇	#544	1	2	yes	—
56	C₂×C₆₇	#544	1	1	yes	characteristic
57	C₂×C₆₇	#562	1	2	yes	—
58	C₂×C₆₇	#562	1	1	yes	characteristic
59	C₂×C₆₇	#564	1	1	yes	characteristic
60	#2845	—	67	1	no	maximal, Sylow
61	C₆₇	#2845	1	1	yes	characteristic, complemented, Sylow
62	#398	—	67	4	no	—
63	#398	—	67	4	no	—
64	#398	—	67	4	no	—
65	#382	—	67	2	no	—
66	#382	D₆₇	1	1	yes	characteristic
67	C₂×C₄⋊C₄	—	67	4	no	—
68	C₂×C₄⋊C₄	—	67	4	no	—
69	C₂×C₄⋊C₄	—	67	2	no	—
70	C₂×C₄⋊C₄	—	67	2	no	—
71	C₂×C₄⋊C₄	—	67	4	no	—
72	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
73	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
74	#95	—	67	4	no	—
75	#95	—	67	4	no	—
76	#95	—	67	4	no	—
77	#95	C₂×D₆₇	1	4	yes	—
78	C₂²×C₄	—	67	4	no	—
79	C₂²×C₄	—	67	4	no	—
80	C₂²×C₄	C₂²×D₆₇	1	4	yes	—
81	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
82	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
83	C₄⋊C₄	—	4	2	no	—
84	C₄⋊C₄	—	268	4	no	—
85	C₄⋊C₄	—	268	4	no	—
86	C₄⋊C₄	—	268	2	no	—
87	C₄⋊C₄	—	268	2	no	—
88	C₄⋊C₄	—	268	4	no	—
89	C₄²	Dic₁₃₄	1	4	yes	—
90	C₂³	C₂³×D₆₇	1	1	yes	characteristic
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	134	4	no	—
99	C₂×C₄	—	134	4	no	—
100	C₂×C₄	—	134	4	no	—
101	C₂×C₄	—	134	4	no	—
102	C₂×C₄	—	134	4	no	—
103	C₂×C₄	—	134	4	no	—
104	C₂×C₄	C₂×Dic₁₃₄	1	4	yes	—
105	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
106	C₂²	C₂²×Dic₁₃₄	1	1	yes	characteristic
107	C₂²	#14524088	1	1	yes	characteristic
108	C₂²	#14524089	1	2	yes	—
109	C₂²	#14524090	1	2	yes	—
110	C₂²	#14524091	1	1	yes	characteristic
111	C₄	—	2	4	no	—
112	C₄	—	2	2	no	—
113	C₄	—	4	4	no	—
114	C₄	—	268	4	no	—
115	C₄	—	268	4	no	—
116	C₂	#16864024	1	2	yes	—
117	C₂	#16864027	1	1	yes	characteristic
118	C₂	#16864140	1	2	yes	—
119	C₂	#16864144	1	1	yes	characteristic
120	C₂	C₆₇⋊#564	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863330

C₂

yes

maximal

#16863329

C₂

yes

maximal

#16863328

C₂

yes

maximal

#16863262

C₂

yes

maximal

#382×C₆₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×Dic₆₇⋊C₄

C₂²

yes

—

C₂×Dic₆₇⋊C₄

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

#95×C₆₇

C₂²

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

Q₈

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂²×C₆₇

C₂²×Q₈

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₂²×C₆₇

2₋¹⁺⁴

yes

—

C₂²×C₆₇

2₋¹⁺⁴

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#544

yes

—

C₂×C₆₇

#544

yes

characteristic

C₂×C₆₇

#562

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

—

D₆₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄²

C₂×D₆₇

yes

—

—

—

C₂×D₆₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂²×C₄

C₂²×D₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

268

—

C₄²

Dic₁₃₄

yes

—

C₂³

C₂³×D₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

100

C₂×C₄

—

134

—

101

C₂×C₄

—

134

—

102

C₂×C₄

—

134

—

103

C₂×C₄

—

134

—

104

C₂×C₄

C₂×Dic₁₃₄

yes

—

105

C₂×C₄

C₂×Dic₁₃₄

yes

—

106

C₂²

C₂²×Dic₁₃₄

yes

characteristic

107

C₂²

#14524088

yes

characteristic

108

C₂²

#14524089

yes

—

109

C₂²

#14524090

yes

—

110

C₂²

#14524091

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

268

—

115

C₄

—

268

—

116

C₂

#16864024

yes

—

117

C₂

#16864027

yes

characteristic

118

C₂

#16864140

yes

—

119

C₂

#16864144

yes

characteristic

120

C₂

C₆₇⋊#564

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	#382×C₆₇	C₂
hypercenter	#382	D₆₇
hypocenter	C₆₇	#2845
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₇

commutator, socle

C₂³×C₆₇

C₂⁴

Fitting

#382×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
61. C₆₇	#2845	semidirect

H ≅ G/N

type

61. C₆₇

#2845

semidirect

Subgroups — #22568484

Conjugacy classes of subgroups

Complements