Group #22562439 — subgroups

Subgroups — #22562439

Order 8576.

138 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 47 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	#16863797	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863779	C₂	1	1	yes	maximal, characteristic
3	#16863201	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863199	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863197	C₂	1	1	yes	maximal, characteristic
6	C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₆₈	C₂²	1	1	yes	characteristic
9	C₂×#11761616	—	2	1	no	—
10	#14524034	—	2	1	no	—
11	#14524030	—	2	1	no	—
12	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	#14523977	C₂²	1	1	yes	characteristic
14	#14523974	—	2	1	no	—
15	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
16	C₂×D₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	#107×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₃₄	—	2	1	no	—
20	Dic₂₆₈	—	2	1	no	—
21	Dic₂₆₈	—	2	1	no	—
22	#11761616	—	4	2	no	—
23	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
24	C₄⋊Dic₆₇	—	2	1	no	—
25	C₄⋊Dic₆₇	—	4	1	no	—
26	Dic₆₇⋊C₄	—	4	1	no	—
27	C₂²⋊Dic₆₇	—	4	1	no	—
28	C₂²⋊Dic₆₇	—	4	1	no	—
29	C₄×Dic₆₇	D₄	1	1	yes	characteristic
30	C₆₇⋊C₁₆	—	2	1	no	—
31	C₂×D₄×C₆₇	—	2	1	no	—
32	D₈×C₆₇	—	2	1	no	—
33	D₈×C₆₇	—	2	1	no	—
34	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₆₇	—	2	1	no	—
36	C₁₆×C₆₇	—	2	1	no	—
37	C₂³×C₆₇	—	4	1	no	—
38	D₄×C₆₇	—	4	1	no	—
39	D₄×C₆₇	—	4	1	no	—
40	C₂×C₄×C₆₇	—	4	1	no	—
41	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₆₇	—	4	1	no	—
43	C₂×Dic₆₇	—	4	1	no	—
44	C₂×Dic₆₇	D₈	1	1	yes	characteristic
45	C₂×Dic₆₇	D₈	1	1	yes	characteristic
46	Dic₁₃₄	—	4	1	no	—
47	Dic₁₃₄	—	4	1	no	—
48	C₈×C₆₇	#41	1	1	yes	characteristic
49	C₈×C₆₇	#41	1	1	yes	characteristic
50	C₆₇⋊C₈	SD₁₆	1	2	yes	—
51	C₂²×C₆₇	—	8	1	no	—
52	C₂²×C₆₇	—	8	1	no	—
53	C₂²×C₆₇	—	8	1	no	—
54	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
55	C₄×C₆₇	—	8	1	no	—
56	C₄×C₆₇	#124	1	1	yes	characteristic
57	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
58	Dic₆₇	—	2	1	no	—
59	Dic₆₇	—	2	1	no	—
60	Dic₆₇	—	8	1	no	—
61	Dic₆₇	—	8	1	no	—
62	C₂×C₆₇	—	16	1	no	—
63	C₂×C₆₇	#486	1	1	yes	characteristic
64	C₂×C₆₇	#508	1	1	yes	characteristic
65	C₂×C₆₇	#508	1	1	yes	characteristic
66	#2242	—	67	1	no	maximal, Sylow
67	C₆₇	#2242	1	1	yes	characteristic, complemented, Sylow
68	#499	—	67	1	no	—
69	#495	—	67	1	no	—
70	#358	—	67	1	no	—
71	#358	—	67	1	no	—
72	#357	—	67	1	no	—
73	#357	D₆₇	1	1	yes	characteristic
74	C₄×C₁₆	—	67	1	no	—
75	C₂×Q₁₆	—	67	1	no	—
76	C₂×SD₁₆	—	134	1	no	—
77	C₂×D₈	C₂×D₆₇	1	1	yes	characteristic
78	#125	—	134	1	no	—
79	#124	—	134	1	no	—
80	C₂×C₁₆	—	67	1	no	—
81	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
82	#107	—	67	1	no	—
83	#107	C₂×D₆₇	1	1	yes	characteristic
84	#106	—	134	1	no	—
85	C₄×C₈	—	67	1	no	—
86	C₂×Q₈	—	134	1	no	—
87	C₂×D₄	—	2	1	no	—
88	Q₁₆	—	134	1	no	—
89	Q₁₆	—	134	1	no	—
90	SD₁₆	—	268	2	no	—
91	D₈	—	2	1	no	—
92	D₈	—	2	1	no	—
93	C₂×C₈	—	67	1	no	—
94	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	268	1	no	—
97	C₄⋊C₄	—	268	1	no	—
98	C₄⋊C₄	—	134	1	no	—
99	C₂²⋊C₄	—	268	1	no	—
100	C₂²⋊C₄	—	268	1	no	—
101	C₄²	—	67	1	no	—
102	C₁₆	—	2	1	no	—
103	C₁₆	—	134	1	no	—
104	C₂³	—	4	1	no	—
105	Q₈	—	268	1	no	—
106	Q₈	—	268	1	no	—
107	D₄	—	4	1	no	—
108	D₄	—	4	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	268	1	no	—
111	C₂×C₄	—	268	1	no	—
112	C₂×C₄	—	67	1	no	—
113	C₂×C₄	—	67	1	no	—
114	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
115	C₈	—	67	2	no	—
116	C₈	#11761627	1	1	yes	characteristic
117	C₈	#11761628	1	1	yes	characteristic
118	C₂²	—	8	1	no	—
119	C₂²	—	8	1	no	—
120	C₂²	—	8	1	no	—
121	C₂²	D₈×D₆₇	1	1	yes	characteristic
122	C₄	—	8	1	no	—
123	C₄	—	536	1	no	—
124	C₄	—	536	1	no	—
125	C₄	—	134	1	no	—
126	C₄	—	134	1	no	—
127	C₄	#14524029	1	1	yes	characteristic
128	C₄	SD₁₆×D₆₇	1	1	yes	characteristic
129	C₂	—	16	1	no	—
130	C₂	#16863741	1	1	yes	characteristic
131	C₂	#16863825	1	1	yes	characteristic
132	C₂	#16863827	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863797

C₂

yes

maximal, characteristic, complemented

#16863779

C₂

yes

maximal, characteristic

#16863201

C₂

yes

maximal, characteristic, complemented

#16863199

C₂

yes

maximal, characteristic, complemented

#16863197

C₂

yes

maximal, characteristic

C₁₆×Dic₆₇

C₂

yes

maximal, characteristic, complemented

#357×C₆₇

C₂

yes

maximal, characteristic

C₂×Dic₂₆₈

C₂²

yes

—

—

—

C₂×C₆₇⋊C₁₆

C₂²

yes

characteristic

#14523977

C₂²

yes

characteristic

#14523974

—

C₈×Dic₆₇

C₂²

yes

characteristic

C₂×D₈×C₆₇

C₂²

yes

characteristic

C₂×C₁₆×C₆₇

C₂²

yes

characteristic

#107×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

Dic₂₆₈

—

Dic₂₆₈

—

#11761616

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

D₄

yes

characteristic

C₆₇⋊C₁₆

—

C₂×D₄×C₆₇

—

D₈×C₆₇

—

D₈×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₁₆×C₆₇

—

C₂³×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

D₈

yes

characteristic

C₂×Dic₆₇

D₈

yes

characteristic

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

yes

C₈×C₆₇

yes

C₆₇⋊C₈

SD₁₆

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×D₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

#124

yes

characteristic

C₄×C₆₇

C₂×SD₁₆

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₆₇

yes

characteristic

C₄×C₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

134

—

C₂×D₈

C₂×D₆₇

yes

characteristic

#125

—

134

—

#124

—

134

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₇

yes

characteristic

#107

—

#107

C₂×D₆₇

yes

characteristic

#106

—

134

—

C₄×C₈

—

C₂×Q₈

—

134

—

C₂×D₄

—

Q₁₆

—

134

—

Q₁₆

—

134

—

SD₁₆

—

268

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

134

—

C₂²⋊C₄

—

268

—

100

C₂²⋊C₄

—

268

—

101

C₄²

—

102

C₁₆

—

103

C₁₆

—

134

—

104

C₂³

—

105

Q₈

—

268

—

106

Q₈

—

268

—

107

D₄

—

108

D₄

—

109

C₂×C₄

—

110

C₂×C₄

—

268

—

111

C₂×C₄

—

268

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

D₄×D₆₇

yes

characteristic

115

C₈

—

116

C₈

#11761627

yes

characteristic

117

C₈

#11761628

yes

characteristic

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

D₈×D₆₇

yes

characteristic

122

C₄

—

123

C₄

—

536

—

124

C₄

—

536

—

125

C₄

—

134

—

126

C₄

—

134

—

127

C₄

#14524029

yes

characteristic

128

C₄

SD₁₆×D₆₇

yes

characteristic

129

C₂

—

130

C₂

#16863741

yes

characteristic

131

C₂

#16863825

yes

characteristic

132

C₂

#16863827

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₆₇
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#357×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	#357	D₆₇
hypocenter	C₆₇	#2242
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₆₇

commutator

C₂×C₈×C₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

#357×C₆₇

C₂

socle

C₂²×C₆₇

C₂×D₈

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
1. #16863797	C₂	semidirect
3. #16863201	C₂	semidirect
4. #16863199	C₂	semidirect
6. C₁₆×Dic₆₇	C₂	semidirect
67. C₆₇	#2242	semidirect

H ≅ G/N

type

1. #16863797

C₂

semidirect

3. #16863201

C₂

semidirect

4. #16863199

C₂

semidirect

6. C₁₆×Dic₆₇

C₂

semidirect

67. C₆₇

#2242

semidirect

Subgroups — #22562439

Conjugacy classes of subgroups

Complements