Group #22565238 — subgroups

Subgroups — #22565238

Order 8576.

556 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 197 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×#14524042	C₂	1	4	yes	maximal
2	#16863316	C₂	1	4	yes	maximal
3	#16863292	C₂	1	4	yes	maximal
4	C₄×C₄⋊Dic₆₇	C₂	1	2	yes	maximal
5	#384×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₃₄	C₂²	1	4	yes	—
7	#14524042	—	2	16	no	—
8	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
10	C₂×Dic₆₇⋊C₄	C₂²	1	8	yes	—
11	C₂×C₄×Dic₆₇	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	4	yes	—
15	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₃₄	—	2	16	no	—
18	C₂×Dic₁₃₄	—	4	4	no	—
19	C₂²×Dic₆₇	C₂³	1	4	yes	—
20	C₂²×Dic₆₇	C₂³	1	4	yes	—
21	C₄⋊Dic₆₇	D₄	1	8	yes	—
22	C₄⋊Dic₆₇	Q₈	1	8	yes	complemented
23	Dic₆₇⋊C₄	—	2	16	no	—
24	C₄×Dic₆₇	—	2	8	no	—
25	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
26	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
28	C₄⋊C₄×C₆₇	—	2	8	no	—
29	C₄⋊C₄×C₆₇	—	4	2	no	—
30	C₄²×C₆₇	Q₈	1	4	yes	complemented
31	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
32	C₂×C₄×C₆₇	—	2	1	no	—
33	C₂×C₄×C₆₇	—	2	1	no	—
34	C₂×C₄×C₆₇	—	2	8	no	—
35	C₂×C₄×C₆₇	—	2	4	no	—
36	C₂×C₄×C₆₇	C₂×D₄	1	4	yes	—
37	C₂×C₄×C₆₇	C₂×Q₈	1	4	yes	—
38	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₆₇	C₂×Q₈	1	4	yes	—
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	C₂×D₄	1	8	yes	—
45	C₂×Dic₆₇	C₂×Q₈	1	8	yes	—
46	Dic₁₃₄	—	4	16	no	—
47	C₂²×C₆₇	C₂²×D₄	1	2	yes	—
48	C₂²×C₆₇	C₂²×Q₈	1	2	yes	—
49	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
50	C₂²×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
51	C₂²×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
52	C₄×C₆₇	—	2	2	no	—
53	C₄×C₆₇	—	4	4	no	—
54	C₄×C₆₇	C₄⋊Q₈	1	8	yes	complemented
55	Dic₆₇	—	2	8	no	—
56	Dic₆₇	—	2	8	no	—
57	C₂×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
58	C₂×C₆₇	#544	1	1	yes	characteristic
59	C₂×C₆₇	D₄²	1	1	yes	characteristic
60	C₂×C₆₇	D₄×Q₈	1	2	yes	—
61	C₂×C₆₇	Q₈²	1	1	yes	characteristic
62	#2551	—	67	1	no	maximal, Sylow
63	C₆₇	#2551	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊Q₈	—	67	4	no	—
65	#395	—	67	4	no	—
66	#389	—	67	4	no	—
67	#384	D₆₇	1	1	yes	characteristic
68	C₄×C₄⋊C₄	—	67	2	no	—
69	C₂²×Q₈	—	67	4	no	—
70	C₄⋊Q₈	—	134	16	no	—
71	C₂×C₄⋊C₄	—	67	2	no	—
72	C₂×C₄⋊C₄	—	67	2	no	—
73	C₂×C₄⋊C₄	—	67	8	no	—
74	C₂×C₄⋊C₄	C₂×D₆₇	1	4	yes	—
75	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
76	C₂×C₄²	—	67	4	no	—
77	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
78	#95	—	67	4	no	—
79	#95	—	67	4	no	—
80	C₂×Q₈	—	268	4	no	—
81	C₂×Q₈	—	134	16	no	—
82	C₂²×C₄	—	67	4	no	—
83	C₂²×C₄	—	67	4	no	—
84	C₂²×C₄	C₂²×D₆₇	1	4	yes	—
85	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
86	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
87	C₄⋊C₄	—	2	8	no	—
88	C₄⋊C₄	—	4	2	no	—
89	C₄⋊C₄	—	67	8	no	—
90	C₄⋊C₄	—	67	8	no	—
91	C₄⋊C₄	—	134	16	no	—
92	C₄²	—	134	8	no	—
93	C₄²	Dic₁₃₄	1	4	yes	complemented
94	C₂³	C₂³×D₆₇	1	1	yes	characteristic
95	Q₈	—	268	16	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	8	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	67	8	no	—
101	C₂×C₄	—	67	8	no	—
102	C₂×C₄	—	134	4	no	—
103	C₂×C₄	—	134	4	no	—
104	C₂×C₄	—	134	4	no	—
105	C₂×C₄	—	134	4	no	—
106	C₂×C₄	D₄×D₆₇	1	4	yes	—
107	C₂×C₄	Q₈×D₆₇	1	4	yes	—
108	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
109	C₂×C₄	C₂×Dic₁₃₄	1	4	yes	—
110	C₂²	C₂×D₄×D₆₇	1	2	yes	—
111	C₂²	C₂×Q₈×D₆₇	1	2	yes	—
112	C₂²	C₂²×Dic₁₃₄	1	1	yes	characteristic
113	C₂²	#14524088	1	1	yes	characteristic
114	C₂²	#14524091	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	4	4	no	—
117	C₄	—	134	8	no	—
118	C₄	—	134	8	no	—
119	C₄	#14524042	1	8	yes	complemented
120	C₂	C₂×#14524042	1	2	yes	—
121	C₂	#16864027	1	1	yes	characteristic
122	C₂	#16864031	1	1	yes	characteristic
123	C₂	#16864061	1	2	yes	—
124	C₂	#16864126	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524042

C₂

yes

maximal

#16863316

C₂

yes

maximal

#16863292

C₂

yes

maximal

C₄×C₄⋊Dic₆₇

C₂

yes

maximal

#384×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

—

#14524042

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×Dic₆₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

D₄

yes

—

C₄⋊Dic₆₇

Q₈

yes

complemented

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

Q₈

yes

complemented

C₂³×C₆₇

C₂⁴

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×D₄

yes

—

C₂×Dic₆₇

C₂×Q₈

yes

—

Dic₁₃₄

—

C₂²×C₆₇

C₂²×D₄

yes

—

C₂²×C₆₇

C₂²×Q₈

yes

—

C₂²×C₆₇

C₂²×Q₈

yes

characteristic

C₂²×C₆₇

H(5,𝔽₂)

yes

characteristic

C₂²×C₆₇

2₋¹⁺⁴

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₄⋊Q₈

yes

complemented

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

C₂×C₄⋊Q₈

yes

—

C₂×C₆₇

#544

yes

characteristic

C₂×C₆₇

D₄²

yes

characteristic

C₂×C₆₇

D₄×Q₈

yes

—

C₂×C₆₇

Q₈²

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#395

—

#389

—

#384

D₆₇

yes

characteristic

C₄×C₄⋊C₄

—

C₂²×Q₈

—

C₄⋊Q₈

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

#95

—

#95

—

C₂×Q₈

—

268

—

C₂×Q₈

—

134

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂²×C₄

C₂²×D₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

134

—

C₄²

—

134

—

C₄²

Dic₁₃₄

yes

complemented

C₂³

C₂³×D₆₇

yes

characteristic

Q₈

—

268

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

134

—

103

C₂×C₄

—

134

—

104

C₂×C₄

—

134

—

105

C₂×C₄

—

134

—

106

C₂×C₄

D₄×D₆₇

yes

—

107

C₂×C₄

Q₈×D₆₇

yes

—

108

C₂×C₄

C₂×Dic₁₃₄

yes

—

109

C₂×C₄

C₂×Dic₁₃₄

yes

—

110

C₂²

C₂×D₄×D₆₇

yes

—

111

C₂²

C₂×Q₈×D₆₇

yes

—

112

C₂²

C₂²×Dic₁₃₄

yes

characteristic

113

C₂²

#14524088

yes

characteristic

114

C₂²

#14524091

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

134

—

118

C₄

—

134

—

119

C₄

#14524042

yes

complemented

120

C₂

C₂×#14524042

yes

—

121

C₂

#16864027

yes

characteristic

122

C₂

#16864031

yes

characteristic

123

C₂

#16864061

yes

—

124

C₂

#16864126

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	#384×C₆₇	C₂
hypercenter	#384	D₆₇
hypocenter	C₆₇	#2551
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₇

commutator, socle

C₂³×C₆₇

C₂⁴

Fitting

#384×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
22. C₄⋊Dic₆₇	Q₈	semidirect
30. C₄²×C₆₇	Q₈	semidirect
54. C₄×C₆₇	C₄⋊Q₈	semidirect
63. C₆₇	#2551	semidirect
93. C₄²	Dic₁₃₄	semidirect
119. C₄	#14524042	semidirect

H ≅ G/N

type

22. C₄⋊Dic₆₇

Q₈

semidirect

30. C₄²×C₆₇

Q₈

semidirect

54. C₄×C₆₇

C₄⋊Q₈

semidirect

63. C₆₇

#2551

semidirect

93. C₄²

Dic₁₃₄

semidirect

119. C₄

#14524042

semidirect

Subgroups — #22565238

Conjugacy classes of subgroups

Complements