Group #22570732 — subgroups

Subgroups — #22570732

Order 8576.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#16863949	C₂	1	2	yes	maximal
2	#16863804	C₂	1	4	yes	maximal, complemented
3	#16863803	C₂	1	4	yes	maximal, complemented
4	#16863434	C₂	1	1	yes	maximal, characteristic
5	#16863432	C₂	1	2	yes	maximal
6	C₂×#14523953	C₂	1	1	yes	maximal, characteristic
7	#427×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	#14524042	C₂²	1	4	yes	—
10	#14524034	C₂²	1	4	yes	—
11	#14524020	—	2	4	no	—
12	C₄×Dic₁₃₄	—	2	4	no	—
13	#14524002	C₂²	1	2	yes	—
14	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
15	#14523977	C₂²	1	2	yes	—
16	#14523976	C₂²	1	4	yes	—
17	#14523975	C₂²	1	2	yes	—
18	#14523974	C₂²	1	4	yes	—
19	#14523953	C₂²	1	2	yes	—
20	#14523953	C₂²	1	2	yes	—
21	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	#117×C₆₇	C₂²	1	2	yes	—
23	#107×C₆₇	C₂²	1	2	yes	—
24	#106×C₆₇	C₂²	1	2	yes	—
25	C₂×Dic₁₃₄	—	2	4	no	—
26	C₂²×Dic₆₇	D₄	1	2	yes	—
27	C₂×C₆₇⋊C₈	Q₈	1	4	yes	complemented
28	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
29	C₄⋊Dic₆₇	—	2	4	no	—
30	C₄⋊Dic₆₇	C₂³	1	4	yes	—
31	Dic₆₇⋊C₄	—	2	4	no	—
32	Dic₆₇⋊C₄	—	2	4	no	—
33	C₂²⋊Dic₆₇	—	2	2	no	—
34	C₄×Dic₆₇	—	2	2	no	—
35	C₄×Dic₆₇	D₄	1	2	yes	—
36	C₄×Dic₆₇	C₂³	1	2	yes	—
37	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₆₇	Q₈	1	4	yes	complemented
39	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
42	C₂²⋊C₄×C₆₇	—	2	2	no	—
43	C₄²×C₆₇	—	2	2	no	—
44	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₆₇	—	2	4	no	—
46	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₆₇	—	2	4	no	—
51	C₂×Dic₆₇	—	4	2	no	—
52	C₂×Dic₆₇	C₂×D₄	1	4	yes	—
53	Dic₁₃₄	—	4	8	no	—
54	C₈×C₆₇	C₂×Q₈	1	2	yes	—
55	C₈×C₆₇	C₂×Q₈	1	2	yes	—
56	C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
57	C₂²×C₆₇	—	2	2	no	—
58	C₂²×C₆₇	C₄⋊Q₈	1	2	yes	—
59	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
60	C₄×C₆₇	—	4	4	no	—
61	C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
62	C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
63	C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
64	Dic₆₇	—	2	4	no	—
65	Dic₆₇	—	4	4	no	—
66	C₂×C₆₇	—	2	2	no	—
67	C₂×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₆₇	#575	1	2	yes	—
69	#3162	—	67	1	no	maximal, Sylow
70	C₆₇	#3162	1	1	yes	characteristic, complemented, Sylow
71	#533	—	67	2	no	—
72	#501	—	67	4	no	—
73	#501	—	67	4	no	—
74	#427	—	67	1	no	—
75	#427	—	67	2	no	—
76	#427	D₆₇	1	1	yes	characteristic
77	C₂×#97	—	67	1	no	—
78	C₂²×C₈	—	67	1	no	—
79	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
80	C₄⋊Q₈	—	67	4	no	—
81	#125	—	67	4	no	—
82	C₂²⋊Q₈	—	134	4	no	—
83	C₄×Q₈	—	134	4	no	—
84	#117	—	67	2	no	—
85	#117	C₂×D₆₇	1	2	yes	—
86	C₂×C₄²	—	67	1	no	—
87	#107	—	67	2	no	—
88	#107	—	67	4	no	—
89	#107	C₂×D₆₇	1	2	yes	—
90	#106	—	67	2	no	—
91	#106	—	67	4	no	—
92	#106	C₂×D₆₇	1	2	yes	—
93	#97	—	67	2	no	—
94	#97	—	67	2	no	—
95	C₂×Q₈	—	134	4	no	—
96	C₂²×C₄	—	67	2	no	—
97	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
98	C₂×C₈	—	67	4	no	—
99	C₂×C₈	—	67	2	no	—
100	C₂×C₈	Dic₁₃₄	1	4	yes	complemented
101	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
103	C₄⋊C₄	—	67	4	no	—
104	C₄⋊C₄	—	134	4	no	—
105	C₄⋊C₄	—	134	4	no	—
106	C₄⋊C₄	—	134	4	no	—
107	C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₂²⋊C₄	—	134	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	67	2	no	—
112	C₄²	—	67	2	no	—
113	C₄²	—	134	2	no	—
114	C₂³	D₄×D₆₇	1	1	yes	characteristic
115	Q₈	—	268	8	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	268	2	no	—
118	C₂×C₄	—	67	4	no	—
119	C₂×C₄	—	134	4	no	—
120	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
121	C₂×C₄	Q₈×D₆₇	1	2	yes	—
122	C₂×C₄	C₂×Dic₁₃₄	1	2	yes	—
123	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
124	C₈	—	67	4	no	—
125	C₈	C₂×Dic₁₃₄	1	2	yes	—
126	C₈	C₂×Dic₁₃₄	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	#14524042	1	2	yes	—
129	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
130	C₄	—	4	4	no	—
131	C₄	—	268	4	no	—
132	C₄	—	134	4	no	—
133	C₄	#14524042	1	2	yes	—
134	C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
135	C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
136	C₂	—	2	2	no	—
137	C₂	C₂×#14524042	1	1	yes	characteristic
138	C₂	#16864193	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863949

C₂

yes

maximal

#16863804

C₂

yes

maximal, complemented

#16863803

C₂

yes

maximal, complemented

#16863434

C₂

yes

maximal, characteristic

#16863432

C₂

yes

maximal

C₂×#14523953

C₂

yes

maximal, characteristic

#427×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

#14524042

C₂²

yes

—

#14524034

C₂²

yes

—

#14524020

—

C₄×Dic₁₃₄

—

#14524002

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

#14523977

C₂²

yes

—

#14523976

C₂²

yes

—

#14523975

C₂²

yes

—

#14523974

C₂²

yes

—

#14523953

C₂²

yes

—

#14523953

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

#117×C₆₇

C₂²

yes

—

#107×C₆₇

C₂²

yes

—

#106×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

D₄

yes

—

C₂×C₆₇⋊C₈

Q₈

yes

complemented

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

D₄

yes

—

C₄×Dic₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

Q₈

yes

complemented

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

C₂³

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×D₄

yes

—

Dic₁₃₄

—

C₈×C₆₇

C₂×Q₈

yes

—

C₈×C₆₇

C₂×Q₈

yes

—

C₆₇⋊C₈

C₂×Q₈

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄⋊Q₈

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₄⋊Q₈

yes

—

C₄×C₆₇

C₂²×Q₈

yes

characteristic

C₄×C₆₇

C₂²×Q₈

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₆₇

#575

yes

—

#3162

—

maximal, Sylow

C₆₇

#3162

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₆₇

yes

characteristic

C₂×#97

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

134

—

C₄×Q₈

—

134

—

#117

—

#117

C₂×D₆₇

yes

—

C₂×C₄²

—

#107

—

#107

—

#107

C₂×D₆₇

yes

—

#106

—

#106

—

#106

C₂×D₆₇

yes

—

#97

—

#97

—

C₂×Q₈

—

134

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

Dic₁₃₄

yes

complemented

101

C₂×C₈

C₂²×D₆₇

yes

characteristic

102

C₂×C₈

C₂²×D₆₇

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

134

—

105

C₄⋊C₄

—

134

—

106

C₄⋊C₄

—

134

—

107

C₄⋊C₄

C₂²×D₆₇

yes

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

134

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

134

—

114

C₂³

D₄×D₆₇

yes

characteristic

115

Q₈

—

268

—

116

C₂×C₄

—

117

C₂×C₄

—

268

—

118

C₂×C₄

—

119

C₂×C₄

—

134

—

120

C₂×C₄

D₄×D₆₇

yes

characteristic

121

C₂×C₄

Q₈×D₆₇

yes

—

122

C₂×C₄

C₂×Dic₁₃₄

yes

—

123

C₂×C₄

C₂³×D₆₇

yes

characteristic

124

C₈

—

125

C₈

C₂×Dic₁₃₄

yes

—

126

C₈

C₂×Dic₁₃₄

yes

—

127

C₂²

—

128

C₂²

#14524042

yes

—

129

C₂²

C₂×D₄×D₆₇

yes

characteristic

130

C₄

—

131

C₄

—

268

—

132

C₄

—

134

—

133

C₄

#14524042

yes

—

134

C₄

C₂×Q₈×D₆₇

yes

characteristic

135

C₄

C₂²×Dic₁₃₄

yes

characteristic

136

C₂

—

137

C₂

C₂×#14524042

yes

characteristic

138

C₂

#16864193

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#427×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#427	D₆₇
hypocenter	C₆₇	#3162
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

#427×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #16863804	C₂	semidirect
3. #16863803	C₂	semidirect
27. C₂×C₆₇⋊C₈	Q₈	semidirect
38. C₂×C₈×C₆₇	Q₈	semidirect
70. C₆₇	#3162	semidirect
100. C₂×C₈	Dic₁₃₄	semidirect

H ≅ G/N

type

2. #16863804

C₂

semidirect

3. #16863803

C₂

semidirect

27. C₂×C₆₇⋊C₈

Q₈

semidirect

38. C₂×C₈×C₆₇

Q₈

semidirect

70. C₆₇

#3162

semidirect

100. C₂×C₈

Dic₁₃₄

semidirect

Subgroups — #22570732

Conjugacy classes of subgroups

Complements