Group #22560935 — subgroups

Subgroups — #22560935

Order 8576.

248 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 109 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×#14523975	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	#16863292	C₂	1	2	yes	maximal
4	#16863158	C₂	1	2	yes	maximal
5	C₂×#106×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
8	C₂×Dic₆₇⋊C₄	—	2	2	no	—
9	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
10	#14523975	C₄	1	4	yes	complemented
11	C₈×Dic₆₇	—	2	2	no	—
12	C₆₇⋊#95	—	2	2	no	—
13	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
15	#106×C₆₇	—	2	2	no	—
16	C₂²×Dic₆₇	—	2	2	no	—
17	C₂²×Dic₆₇	D₄	1	2	yes	—
18	C₂×C₆₇⋊C₈	—	2	1	no	—
19	C₂×C₆₇⋊C₈	—	2	1	no	—
20	C₂×C₆₇⋊C₈	Q₈	1	2	yes	—
21	C₄⋊Dic₆₇	—	2	2	no	—
22	C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
23	Dic₆₇⋊C₄	—	4	4	no	—
24	C₄×Dic₆₇	—	2	1	no	—
25	C₄×Dic₆₇	—	2	1	no	—
26	C₂²×C₄×C₆₇	—	2	2	no	—
27	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₆₇	D₄	1	2	yes	—
30	C₂×C₈×C₆₇	Q₈	1	2	yes	—
31	C₄⋊C₄×C₆₇	—	2	2	no	—
32	C₄⋊C₄×C₆₇	—	2	2	no	—
33	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₆₇	—	4	2	no	—
35	C₂×C₄×C₆₇	—	4	2	no	—
36	C₂×C₄×C₆₇	—	4	2	no	—
37	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
42	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
43	C₂×Dic₆₇	—	2	2	no	—
44	C₂×Dic₆₇	—	2	2	no	—
45	C₂×Dic₆₇	—	2	4	no	—
46	C₂×Dic₆₇	—	4	2	no	—
47	C₂×Dic₆₇	—	4	2	no	—
48	C₂×Dic₆₇	SD₁₆	1	4	yes	—
49	C₈×C₆₇	C₄⋊C₄	1	4	yes	complemented
50	C₆₇⋊C₈	—	2	2	no	—
51	C₂²×C₆₇	C₄×D₄	1	1	yes	characteristic
52	C₂²×C₆₇	#121	1	1	yes	characteristic
53	C₂²×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
54	C₂²×C₆₇	C₂×SD₁₆	1	2	yes	—
55	C₂²×C₆₇	#135	1	2	yes	—
56	C₄×C₆₇	—	8	2	no	—
57	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
59	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
60	C₄×C₆₇	#125	1	1	yes	characteristic
61	Dic₆₇	—	2	4	no	—
62	Dic₆₇	—	4	4	no	—
63	C₂×C₆₇	#389	1	1	yes	characteristic
64	C₂×C₆₇	C₄×SD₁₆	1	2	yes	—
65	C₂×C₆₇	#465	1	2	yes	—
66	C₂×C₆₇	#498	1	1	yes	characteristic
67	C₂×C₆₇	#499	1	1	yes	characteristic
68	#1941	—	67	1	no	maximal, Sylow
69	C₆₇	#1941	1	1	yes	characteristic, complemented, Sylow
70	C₂×#106	—	67	1	no	—
71	C₂×#106	D₆₇	1	1	yes	characteristic
72	C₂×C₄×C₈	—	67	1	no	—
73	#389	—	67	2	no	—
74	#340	—	67	2	no	—
75	C₂²×C₈	—	67	1	no	—
76	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	67	2	no	—
78	C₂×C₄⋊C₄	—	134	2	no	—
79	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
80	C₂×C₄²	—	67	1	no	—
81	#106	—	2	2	no	—
82	#106	—	67	4	no	—
83	C₄×C₈	—	134	2	no	—
84	#95	—	134	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	67	2	no	—
87	C₂²×C₄	—	134	2	no	—
88	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
89	C₂×C₈	—	67	2	no	—
90	C₂×C₈	—	134	1	no	—
91	C₂×C₈	—	134	1	no	—
92	C₂×C₈	Dic₁₃₄	1	2	yes	—
93	C₂×C₈	C₄×D₆₇	1	2	yes	—
94	C₂×C₈	C₂²⋊D₆₇	1	2	yes	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	268	4	no	—
98	C₄⋊C₄	—	67	4	no	—
99	C₄⋊C₄	—	134	2	no	—
100	C₄²	—	134	1	no	—
101	C₄²	—	134	1	no	—
102	C₂³	D₄×D₆₇	1	1	yes	characteristic
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	268	2	no	—
107	C₂×C₄	—	268	2	no	—
108	C₂×C₄	—	67	4	no	—
109	C₂×C₄	—	134	4	no	—
110	C₂×C₄	—	134	2	no	—
111	C₂×C₄	—	134	2	no	—
112	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
113	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
114	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
115	C₂×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
116	C₂×C₄	#11761627	1	1	yes	characteristic
117	C₂×C₄	#11761629	1	1	yes	characteristic
118	C₈	—	134	2	no	—
119	C₈	Dic₆₇⋊C₄	1	4	yes	complemented
120	C₂²	#14524005	1	1	yes	characteristic
121	C₂²	#14524014	1	1	yes	characteristic
122	C₂²	#14524042	1	1	yes	characteristic
123	C₂²	SD₁₆×D₆₇	1	2	yes	—
124	C₂²	#14524064	1	2	yes	—
125	C₄	—	8	2	no	—
126	C₄	—	268	4	no	—
127	C₄	—	134	4	no	—
128	C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
129	C₄	#14524008	1	1	yes	characteristic
130	C₄	#14524019	1	1	yes	characteristic
131	C₄	#14524034	1	1	yes	characteristic
132	C₂	#16863292	1	1	yes	characteristic
133	C₂	#16863483	1	2	yes	—
134	C₂	#16863622	1	2	yes	—
135	C₂	#16863793	1	1	yes	characteristic
136	C₂	#16863796	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523975

C₂

yes

maximal, characteristic

C₂×C₈×Dic₆₇

C₂

yes

maximal, characteristic

#16863292

C₂

yes

maximal

#16863158

C₂

yes

maximal

C₂×#106×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×Dic₆₇⋊C₄

—

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

#14523975

C₄

yes

complemented

C₈×Dic₆₇

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

#106×C₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

D₄

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

Q₈

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

Q₈

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

SD₁₆

yes

—

C₈×C₆₇

C₄⋊C₄

yes

complemented

C₆₇⋊C₈

—

C₂²×C₆₇

C₄×D₄

yes

characteristic

C₂²×C₆₇

#121

yes

characteristic

C₂²×C₆₇

C₄⋊Q₈

yes

characteristic

C₂²×C₆₇

C₂×SD₁₆

yes

—

C₂²×C₆₇

#135

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₇

C₄×Q₈

yes

characteristic

C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₄×C₆₇

#125

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#389

yes

characteristic

C₂×C₆₇

C₄×SD₁₆

yes

—

C₂×C₆₇

#465

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×#106

—

C₂×#106

D₆₇

yes

characteristic

C₂×C₄×C₈

—

#389

—

#340

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄²

—

#106

—

#106

—

C₄×C₈

—

134

—

#95

—

134

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

134

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₂×C₈

Dic₁₃₄

yes

—

C₂×C₈

C₄×D₆₇

yes

—

C₂×C₈

C₂²⋊D₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

C₄⋊C₄

—

134

—

100

C₄²

—

134

—

101

C₄²

—

134

—

102

C₂³

D₄×D₆₇

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

268

—

107

C₂×C₄

—

268

—

108

C₂×C₄

—

109

C₂×C₄

—

134

—

110

C₂×C₄

—

134

—

111

C₂×C₄

—

134

—

112

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

113

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

114

C₂×C₄

Q₈×D₆₇

yes

characteristic

115

C₂×C₄

C₂×Dic₁₃₄

yes

characteristic

116

C₂×C₄

#11761627

yes

characteristic

117

C₂×C₄

#11761629

yes

characteristic

118

C₈

—

134

—

119

C₈

Dic₆₇⋊C₄

yes

complemented

120

C₂²

#14524005

yes

characteristic

121

C₂²

#14524014

yes

characteristic

122

C₂²

#14524042

yes

characteristic

123

C₂²

SD₁₆×D₆₇

yes

—

124

C₂²

#14524064

yes

—

125

C₄

—

126

C₄

—

268

—

127

C₄

—

134

—

128

C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

129

C₄

#14524008

yes

characteristic

130

C₄

#14524019

yes

characteristic

131

C₄

#14524034

yes

characteristic

132

C₂

#16863292

yes

characteristic

133

C₂

#16863483

yes

—

134

C₂

#16863622

yes

—

135

C₂

#16863793

yes

characteristic

136

C₂

#16863796

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×#106×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×#106	D₆₇
hypocenter	C₆₇	#1941
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×#106×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
10. #14523975	C₄	semidirect
49. C₈×C₆₇	C₄⋊C₄	semidirect
69. C₆₇	#1941	semidirect
119. C₈	Dic₆₇⋊C₄	semidirect

H ≅ G/N

type

10. #14523975

C₄

semidirect

49. C₈×C₆₇

C₄⋊C₄

semidirect

69. C₆₇

#1941

semidirect

119. C₈

Dic₆₇⋊C₄

semidirect

Subgroups — #22560935

Conjugacy classes of subgroups

Complements