Group #22562142 — subgroups

Subgroups — #22562142

Order 8576.

216 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₂²×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523989	C₂	1	2	yes	maximal, complemented
3	#16863199	C₂	1	2	yes	maximal, complemented
4	C₂²⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
5	#467×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₃₄	—	2	1	no	—
7	C₂×Dic₂₆₈	—	2	4	no	—
8	C₂×Dic₂₆₈	—	2	1	no	—
9	C₂×Dic₂₆₈	C₂²	1	2	yes	—
10	#14523989	—	2	4	no	—
11	C₂×C₆₇⋊C₁₆	C₂²	1	2	yes	—
12	C₂×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²⋊Q₈×C₆₇	—	2	1	no	—
15	#107×C₆₇	C₂²	1	1	yes	characteristic
16	#103×C₆₇	—	2	1	no	—
17	C₂×Dic₁₃₄	—	2	4	no	—
18	C₂×Dic₁₃₄	—	2	2	no	—
19	C₂×Dic₁₃₄	—	4	2	no	—
20	C₂×Dic₁₃₄	—	4	1	no	—
21	C₂²×Dic₆₇	—	4	1	no	—
22	Dic₂₆₈	—	2	4	no	—
23	Dic₂₆₈	—	2	2	no	—
24	Dic₂₆₈	D₄	1	4	yes	—
25	C₆₇⋊C₁₆	—	2	2	no	—
26	C₂×Q₈×C₆₇	—	2	1	no	—
27	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
28	Q₁₆×C₆₇	—	2	2	no	—
29	C₂×C₈×C₆₇	—	2	2	no	—
30	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₆₇	—	2	1	no	—
33	C₄⋊C₄×C₆₇	—	4	1	no	—
34	C₂²⋊C₄×C₆₇	—	4	1	no	—
35	C₂³×C₆₇	D₈	1	1	yes	characteristic
36	Q₈×C₆₇	—	4	1	no	—
37	Q₈×C₆₇	—	4	1	no	—
38	C₂×C₄×C₆₇	—	2	1	no	—
39	C₂×C₄×C₆₇	—	2	1	no	—
40	C₂×C₄×C₆₇	—	4	1	no	—
41	C₂×C₄×C₆₇	—	4	1	no	—
42	C₂×C₄×C₆₇	D₈	1	1	yes	characteristic
43	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₆₇	—	4	4	no	—
45	C₂×Dic₆₇	—	4	2	no	—
46	Dic₁₃₄	—	2	4	no	—
47	Dic₁₃₄	—	4	4	no	—
48	Dic₁₃₄	—	4	2	no	—
49	C₈×C₆₇	—	2	1	no	—
50	C₈×C₆₇	C₂×D₄	1	2	yes	—
51	C₂²×C₆₇	—	2	1	no	—
52	C₂²×C₆₇	—	2	1	no	—
53	C₂²×C₆₇	Q₃₂	1	2	yes	complemented
54	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
55	C₄×C₆₇	—	2	1	no	—
56	C₄×C₆₇	—	8	1	no	—
57	C₄×C₆₇	—	8	1	no	—
58	C₄×C₆₇	#120	1	1	yes	characteristic
59	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
60	Dic₆₇	—	4	4	no	—
61	C₂×C₆₇	—	2	2	no	—
62	C₂×C₆₇	#447	1	1	yes	characteristic
63	C₂×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
64	C₂×C₆₇	#510	1	1	yes	characteristic
65	#2188	—	67	1	no	maximal, Sylow
66	C₆₇	#2188	1	1	yes	characteristic, complemented, Sylow
67	C₂²×Q₁₆	—	67	1	no	—
68	C₂×Q₃₂	—	67	2	no	—
69	#467	D₆₇	1	1	yes	characteristic
70	#358	—	67	2	no	—
71	C₂²⋊C₁₆	—	67	1	no	—
72	C₂²×Q₈	—	134	1	no	—
73	C₂×Q₁₆	—	67	2	no	—
74	C₂×Q₁₆	—	134	4	no	—
75	C₂×Q₁₆	—	134	1	no	—
76	C₂×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
77	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
78	C₂²⋊Q₈	—	2	1	no	—
79	Q₃₂	—	134	4	no	—
80	C₂×C₁₆	—	67	2	no	—
81	#107	C₂×D₆₇	1	1	yes	characteristic
82	#103	—	2	1	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	268	2	no	—
85	C₂×Q₈	—	268	1	no	—
86	C₂×Q₈	—	134	4	no	—
87	C₂×Q₈	—	134	2	no	—
88	C₂²×C₄	—	268	1	no	—
89	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
90	Q₁₆	—	2	2	no	—
91	Q₁₆	—	67	4	no	—
92	Q₁₆	—	134	4	no	—
93	Q₁₆	—	134	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
96	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	4	1	no	—
99	C₂²⋊C₄	—	4	1	no	—
100	C₁₆	—	134	2	no	—
101	C₂³	#11761613	1	1	yes	characteristic
102	Q₈	—	4	1	no	—
103	Q₈	—	4	1	no	—
104	Q₈	—	268	4	no	—
105	Q₈	—	268	2	no	—
106	Q₈	—	134	4	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	268	4	no	—
112	C₂×C₄	—	268	2	no	—
113	C₂×C₄	#11761613	1	1	yes	characteristic
114	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
115	C₈	—	2	1	no	—
116	C₈	D₄×D₆₇	1	2	yes	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	#14523989	1	2	yes	complemented
120	C₂²	C₂×#11761613	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	8	1	no	—
123	C₄	—	8	1	no	—
124	C₄	—	268	4	no	—
125	C₄	#14524011	1	1	yes	characteristic
126	C₄	#14524068	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	#16863529	1	1	yes	characteristic
129	C₂	C₂×#14523989	1	1	yes	characteristic
130	C₂	#16863836	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₆₈

C₂

yes

maximal, characteristic

C₂×#14523989

C₂

yes

maximal, complemented

#16863199

C₂

yes

maximal, complemented

C₂²⋊(C₆₇⋊C₁₆)

C₂

yes

maximal, characteristic

#467×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

C₂²

yes

—

#14523989

—

C₂×C₆₇⋊C₁₆

C₂²

yes

—

C₂×Q₁₆×C₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂²⋊Q₈×C₆₇

—

#107×C₆₇

C₂²

yes

characteristic

#103×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

D₄

yes

—

C₆₇⋊C₁₆

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₂³×C₆₇

D₈

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

C₂×D₄

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

Q₃₂

yes

complemented

C₂²×C₆₇

C₂×D₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#447

yes

characteristic

C₂×C₆₇

C₂×Q₃₂

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×Q₃₂

—

#467

D₆₇

yes

characteristic

#358

—

C₂²⋊C₁₆

—

C₂²×Q₈

—

134

—

C₂×Q₁₆

—

C₂×Q₁₆

—

134

—

C₂×Q₁₆

—

134

—

C₂×Q₁₆

C₂×D₆₇

yes

characteristic

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₂²⋊Q₈

—

Q₃₂

—

134

—

C₂×C₁₆

—

#107

C₂×D₆₇

yes

characteristic

#103

—

C₂×Q₈

—

C₂×Q₈

—

268

—

C₂×Q₈

—

268

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂²×C₄

—

268

—

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

134

—

Q₁₆

—

134

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

100

C₁₆

—

134

—

101

C₂³

#11761613

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

268

—

105

Q₈

—

268

—

106

Q₈

—

134

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

268

—

112

C₂×C₄

—

268

—

113

C₂×C₄

#11761613

yes

characteristic

114

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

115

C₈

—

116

C₈

D₄×D₆₇

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

#14523989

yes

complemented

120

C₂²

C₂×#11761613

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

268

—

125

C₄

#14524011

yes

characteristic

126

C₄

#14524068

yes

characteristic

127

C₂

—

128

C₂

#16863529

yes

characteristic

129

C₂

C₂×#14523989

yes

characteristic

130

C₂

#16863836

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#467×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	#467	D₆₇
hypocenter	C₆₇	#2188
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #467

subgroup

G/H

center

C₂²

C₂×#11761613

commutator

C₂×C₈×C₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

#467×C₆₇

C₂

socle

C₂²×C₆₇

C₂×D₈

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #467

N	H ≅ G/N	type
2. C₂×#14523989	C₂	semidirect
3. #16863199	C₂	semidirect
53. C₂²×C₆₇	Q₃₂	semidirect
66. C₆₇	#2188	semidirect
119. C₂²	#14523989	semidirect

H ≅ G/N

type

2. C₂×#14523989

C₂

semidirect

3. #16863199

C₂

semidirect

53. C₂²×C₆₇

Q₃₂

semidirect

66. C₆₇

#2188

semidirect

119. C₂²

#14523989

semidirect

Subgroups — #22562142

Conjugacy classes of subgroups

Complements