Group #22560005 — subgroups

Subgroups — C₄×#14523962 #22560005

Order 8576.

296 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 129 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×#14523962	C₂	1	2	yes	maximal
2	C₄×#11761612	C₂	1	2	yes	maximal
3	#16863165	C₂	1	2	yes	maximal
4	C₂×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#11761612	C₂²	1	4	yes	—
6	#14523962	C₄	1	8	yes	direct product factor
7	#14523954	—	2	2	no	—
8	C₄×C₆₇⋊C₈	—	2	2	no	—
9	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₆₇	—	2	2	no	—
11	C₄×Q₈×C₆₇	—	2	2	no	—
12	C₂×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
13	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₆₇	C₄	1	2	yes	—
15	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	#11761612	—	2	4	no	—
17	#11761612	C₂×C₄	1	8	yes	—
18	C₂×C₆₇⋊C₈	—	2	4	no	—
19	C₂×Q₈×C₆₇	—	2	2	no	—
20	C₂×Q₈×C₆₇	—	2	2	no	—
21	C₂×Q₈×C₆₇	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₆₇	—	2	4	no	—
28	C₄⋊C₄×C₆₇	—	2	2	no	—
29	C₄²×C₆₇	—	2	2	no	—
30	C₄²×C₆₇	—	2	2	no	—
31	C₄²×C₆₇	D₄	1	2	yes	—
32	C₄²×C₆₇	D₄	1	2	yes	—
33	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
34	Q₈×C₆₇	—	2	8	no	—
35	C₂×C₄×C₆₇	—	2	2	no	—
36	C₂×C₄×C₆₇	—	2	2	no	—
37	C₂×C₄×C₆₇	—	2	2	no	—
38	C₂×C₄×C₆₇	—	2	2	no	—
39	C₂×C₄×C₆₇	—	4	1	no	—
40	C₂×C₄×C₆₇	—	4	2	no	—
41	C₂×C₄×C₆₇	C₄²	1	4	yes	—
42	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₆₇	#41	1	2	yes	—
48	C₆₇⋊C₈	—	2	8	no	—
49	C₂²×C₆₇	—	2	1	no	—
50	C₂²×C₆₇	—	2	1	no	—
51	C₂²×C₆₇	C₂×C₄²	1	1	yes	characteristic
52	C₂²×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₆₇	#117	1	1	yes	characteristic
54	C₄×C₆₇	—	2	4	no	—
55	C₄×C₆₇	—	2	1	no	—
56	C₄×C₆₇	—	2	2	no	—
57	C₄×C₆₇	—	4	2	no	—
58	C₄×C₆₇	#101	1	2	yes	complemented
59	C₄×C₆₇	C₄×D₄	1	4	yes	—
60	C₂×C₆₇	—	2	1	no	—
61	C₂×C₆₇	—	2	1	no	—
62	C₂×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₆₇	C₂×#101	1	1	yes	characteristic
64	C₂×C₆₇	#413	1	1	yes	characteristic
65	C₄×#101	—	67	1	no	maximal, Sylow
66	C₆₇	C₄×#101	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₄×Q₈	D₆₇	1	1	yes	characteristic
68	C₂×#101	—	67	2	no	—
69	C₄×M₁₆	—	67	2	no	—
70	#343	—	67	2	no	—
71	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
72	C₂×M₁₆	—	67	4	no	—
73	C₄×Q₈	—	2	2	no	—
74	C₄×Q₈	—	2	2	no	—
75	C₂×C₄⋊C₄	Dic₆₇	1	2	yes	—
76	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
77	C₂×C₄²	Dic₆₇	1	2	yes	—
78	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
79	#101	—	67	8	no	—
80	#97	—	134	2	no	—
81	C₄×C₈	—	134	2	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	C₄×D₆₇	1	4	yes	—
85	C₂²×C₄	C₂×Dic₆₇	1	2	yes	—
86	C₂²×C₄	C₂×Dic₆₇	1	1	yes	characteristic
87	C₂²×C₄	C₂×Dic₆₇	1	2	yes	—
88	C₂²×C₄	C₂×Dic₆₇	1	1	yes	characteristic
89	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
90	M₁₆	—	67	8	no	—
91	M₁₆	—	134	4	no	—
92	C₂×C₈	—	134	4	no	—
93	C₄⋊C₄	—	2	4	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄²	—	2	2	no	—
96	C₄²	—	2	2	no	—
97	C₄²	C₂²⋊D₆₇	1	2	yes	—
98	C₄²	C₂²⋊D₆₇	1	2	yes	—
99	C₂³	C₂²×Dic₆₇	1	1	yes	characteristic
100	Q₈	—	2	8	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	C₄×Dic₆₇	1	4	yes	—
108	C₂×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
109	C₂×C₄	C₂²⋊Dic₆₇	1	2	yes	—
110	C₂×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
111	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
113	C₂×C₄	#11761628	1	2	yes	—
114	C₈	—	134	8	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	C₂×C₄×Dic₆₇	1	1	yes	characteristic
118	C₂²	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
119	C₂²	#14524002	1	1	yes	characteristic
120	C₄	—	2	4	no	—
121	C₄	—	2	1	no	—
122	C₄	—	2	2	no	—
123	C₄	—	4	2	no	—
124	C₄	#14523962	1	2	yes	direct product factor
125	C₄	C₄×C₂²⋊D₆₇	1	4	yes	—
126	C₂	—	2	1	no	—
127	C₂	—	2	1	no	—
128	C₂	C₄×C₂²⋊Dic₆₇	1	1	yes	characteristic
129	C₂	C₂×#14523962	1	1	yes	characteristic
130	C₂	#16863380	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523962

C₂

yes

maximal

C₄×#11761612

C₂

yes

maximal

#16863165

C₂

yes

maximal

C₂×C₄×Q₈×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₂²

yes

—

#14523962

C₄

yes

direct product factor

#14523954

—

C₄×C₆₇⋊C₈

—

C₂²×Q₈×C₆₇

C₂²

yes

characteristic

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₄

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×C₄²×C₆₇

C₄

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

#11761612

—

#11761612

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

D₄

yes

—

C₄²×C₆₇

D₄

yes

—

C₂³×C₆₇

C₂²×C₄

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄²

yes

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄²

yes

characteristic

C₂²×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₇

#117

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#101

yes

complemented

C₄×C₆₇

C₄×D₄

yes

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₄×C₂²⋊C₄

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

D₆₇

yes

characteristic

C₂×#101

—

C₄×M₁₆

—

#343

—

C₂²×Q₈

C₂×D₆₇

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₆₇

yes

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

C₂×C₄²

Dic₆₇

yes

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

#101

—

#97

—

134

—

C₄×C₈

—

134

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₄×D₆₇

yes

—

C₂²×C₄

C₂×Dic₆₇

yes

—

C₂²×C₄

C₂×Dic₆₇

yes

characteristic

C₂²×C₄

C₂×Dic₆₇

yes

—

C₂²×C₄

C₂×Dic₆₇

yes

characteristic

C₂²×C₄

C₂²×D₆₇

yes

characteristic

M₁₆

—

M₁₆

—

134

—

C₂×C₈

—

134

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₆₇

yes

—

C₄²

C₂²⋊D₆₇

yes

—

C₂³

C₂²×Dic₆₇

yes

characteristic

100

Q₈

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₄×Dic₆₇

yes

—

108

C₂×C₄

C₂²⋊Dic₆₇

yes

characteristic

109

C₂×C₄

C₂²⋊Dic₆₇

yes

—

110

C₂×C₄

C₂²⋊Dic₆₇

yes

characteristic

111

C₂×C₄

C₂×C₄×D₆₇

yes

—

112

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

113

C₂×C₄

#11761628

yes

—

114

C₈

—

134

—

115

C₂²

—

116

C₂²

—

117

C₂²

C₂×C₄×Dic₆₇

yes

characteristic

118

C₂²

C₂×C₂²⋊Dic₆₇

yes

characteristic

119

C₂²

#14524002

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

#14523962

yes

direct product factor

125

C₄

C₄×C₂²⋊D₆₇

yes

—

126

C₂

—

127

C₂

—

128

C₂

C₄×C₂²⋊Dic₆₇

yes

characteristic

129

C₂

C₂×#14523962

yes

characteristic

130

C₂

#16863380

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₆₇
commutator	C₂²×C₆₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×C₄×Q₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₄
hypercenter	C₂×C₄×Q₈	D₆₇
hypocenter	C₆₇	C₄×#101
LCS	G ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₆₇

commutator

C₂²×C₆₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×C₄×Q₈×C₆₇

C₂

socle

C₂²×C₆₇

C₂×C₂²⋊C₄

hypercenter

C₂×C₄×Q₈

D₆₇

hypocenter

C₆₇

C₄×#101

LCS

G ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
6. #14523962	C₄	direct (N × H)
58. C₄×C₆₇	#101	semidirect
66. C₆₇	C₄×#101	semidirect
124. C₄	#14523962	direct (N × H)

H ≅ G/N

type

6. #14523962

C₄

direct (N × H)

58. C₄×C₆₇

#101

semidirect

66. C₆₇

C₄×#101

semidirect

124. C₄

#14523962

direct (N × H)

Subgroups — C₄×#14523962 #22560005

Conjugacy classes of subgroups

Complements

Subgroups — C4×#14523962 #22560005

Conjugacy classes of subgroups

Complements

Subgroups — C₄×#14523962 #22560005