Group #22570589 — subgroups

Subgroups — #22570589

Order 8576.

380 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 141 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	#16863953	C₂	1	1	yes	maximal, characteristic
2	#16863764	C₂	1	4	yes	maximal, complemented
3	#16863746	C₂	1	4	yes	maximal, complemented
4	#16863405	C₂	1	4	yes	maximal
5	C₄×#11761612	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#11761612	C₂²	1	2	yes	—
8	#14524043	C₂²	1	2	yes	—
9	#14524035	C₂²	1	2	yes	—
10	#14524022	—	2	2	no	—
11	C₄×Dic₁₃₄	—	2	2	no	—
12	#14524002	C₂²	1	2	yes	—
13	#14523967	C₂²	1	16	yes	—
14	#14523954	C₂²	1	2	yes	—
15	C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
16	C₂²×Q₈×C₆₇	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₆₇	—	2	2	no	—
18	C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
21	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×Dic₁₃₄	—	2	2	no	—
23	#11761612	—	2	4	no	—
24	C₂×C₆₇⋊C₈	C₂³	1	4	yes	—
25	C₄⋊Dic₆₇	—	2	2	no	—
26	C₄⋊Dic₆₇	C₂³	1	4	yes	—
27	Dic₆₇⋊C₄	—	2	4	no	—
28	C₂²⋊Dic₆₇	—	2	2	no	—
29	C₄×Dic₆₇	—	2	2	no	—
30	C₂×Q₈×C₆₇	—	2	2	no	—
31	C₂×Q₈×C₆₇	—	2	8	no	—
32	C₂×Q₈×C₆₇	C₂³	1	4	yes	—
33	C₂²×C₄×C₆₇	—	2	2	no	—
34	C₂²×C₄×C₆₇	D₄	1	2	yes	—
35	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₆₇	—	2	4	no	—
37	C₄⋊C₄×C₆₇	—	2	4	no	—
38	C₄²×C₆₇	D₄	1	2	yes	—
39	C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₆₇	—	2	8	no	—
43	Q₈×C₆₇	—	4	4	no	—
44	C₂×C₄×C₆₇	—	2	4	no	—
45	C₂×C₄×C₆₇	—	2	4	no	—
46	C₂×C₄×C₆₇	—	4	4	no	—
47	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₆₇	#41	1	4	yes	—
51	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₆₇	—	2	4	no	—
53	Dic₁₃₄	—	4	4	no	—
54	C₆₇⋊C₈	—	2	4	no	—
55	C₂²×C₆₇	—	2	2	no	—
56	C₂²×C₆₇	#124	1	2	yes	—
57	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₆₇	—	2	2	no	—
59	C₄×C₆₇	—	4	4	no	—
60	C₄×C₆₇	#124	1	2	yes	—
61	C₄×C₆₇	#137	1	4	yes	—
62	C₄×C₆₇	C₂×#41	1	2	yes	—
63	Dic₆₇	—	4	4	no	—
64	C₂×C₆₇	—	2	2	no	—
65	C₂×C₆₇	C₂×#124	1	1	yes	characteristic
66	C₂×C₆₇	C₂×#137	1	2	yes	—
67	#3139	—	67	1	no	maximal, Sylow
68	C₆₇	#3139	1	1	yes	characteristic, complemented, Sylow
69	#533	—	67	1	no	—
70	C₂×C₄⋊Q₈	D₆₇	1	1	yes	characteristic
71	#491	—	67	4	no	—
72	#487	—	67	4	no	—
73	#419	—	67	4	no	—
74	C₄×M₁₆	—	67	1	no	—
75	C₂²×Q₈	C₂×D₆₇	1	2	yes	—
76	C₂×M₁₆	—	67	2	no	—
77	C₄⋊Q₈	—	2	2	no	—
78	C₄⋊Q₈	—	67	2	no	—
79	C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
80	C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
81	#125	—	67	2	no	—
82	C₂²⋊Q₈	—	134	2	no	—
83	C₄×Q₈	—	134	2	no	—
84	#117	—	67	2	no	—
85	C₂×C₄⋊C₄	—	2	2	no	—
86	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
87	#103	—	67	16	no	—
88	#97	—	67	2	no	—
89	C₄×C₈	—	67	2	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	2	8	no	—
92	C₂×Q₈	—	134	2	no	—
93	C₂×Q₈	C₂²×D₆₇	1	4	yes	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	C₂²⋊D₆₇	1	2	yes	—
96	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
97	M₁₆	—	134	4	no	—
98	C₂×C₈	—	67	4	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	67	4	no	—
102	C₄⋊C₄	—	134	4	no	—
103	C₄⋊C₄	—	134	2	no	—
104	C₂²⋊C₄	—	134	2	no	—
105	C₄²	—	134	2	no	—
106	C₄²	C₂²⋊D₆₇	1	2	yes	—
107	C₄²	C₂²×D₆₇	1	1	yes	characteristic
108	C₄²	C₂²×D₆₇	1	1	yes	characteristic
109	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	268	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	—	134	4	no	—
117	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
120	C₂×C₄	#11761627	1	4	yes	—
121	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
122	C₈	—	134	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#14524030	1	2	yes	—
125	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	4	4	no	—
128	C₄	—	268	4	no	—
129	C₄	#14524030	1	2	yes	—
130	C₄	#14524074	1	4	yes	—
131	C₄	C₂×#11761627	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	C₂×#14524030	1	1	yes	characteristic
134	C₂	C₂×#14524074	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863953

C₂

yes

maximal, characteristic

#16863764

C₂

yes

maximal, complemented

#16863746

C₂

yes

maximal, complemented

#16863405

C₂

yes

maximal

C₄×#11761612

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₂²

yes

—

#14524043

C₂²

yes

—

#14524035

C₂²

yes

—

#14524022

—

C₄×Dic₁₃₄

—

#14524002

C₂²

yes

—

#14523967

C₂²

yes

—

#14523954

C₂²

yes

—

C₄×C₆₇⋊C₈

C₂²

yes

—

C₂²×Q₈×C₆₇

C₂²

yes

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

C₂²

yes

—

C₄⋊Q₈×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

#11761612

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

D₄

yes

—

C₄²×C₆₇

C₂³

yes

characteristic

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

Dic₁₃₄

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#124

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#124

yes

—

C₄×C₆₇

#137

yes

—

C₄×C₆₇

C₂×#41

yes

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂×#124

yes

characteristic

C₂×C₆₇

C₂×#137

yes

—

#3139

—

maximal, Sylow

C₆₇

#3139

yes

characteristic, complemented, Sylow

#533

—

C₂×C₄⋊Q₈

D₆₇

yes

—

—

—

C₄×M₁₆

—

C₂²×Q₈

C₂×D₆₇

yes

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₇

yes

—

C₄⋊Q₈

C₂×D₆₇

yes

—

#125

—

C₂²⋊Q₈

—

134

—

C₄×Q₈

—

134

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

#103

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×Q₈

C₂²×D₆₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

M₁₆

—

134

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

134

—

103

C₄⋊C₄

—

134

—

104

C₂²⋊C₄

—

134

—

105

C₄²

—

134

—

106

C₄²

C₂²⋊D₆₇

yes

—

107

C₄²

C₂²×D₆₇

yes

characteristic

108

C₄²

C₂²×D₆₇

yes

characteristic

109

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

268

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

134

—

117

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

118

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

120

C₂×C₄

#11761627

yes

—

121

C₂×C₄

C₂³×D₆₇

yes

characteristic

122

C₈

—

134

—

123

C₂²

—

124

C₂²

#14524030

yes

—

125

C₂²

C₂²×C₂²⋊D₆₇

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

268

—

129

C₄

#14524030

yes

—

130

C₄

#14524074

yes

—

131

C₄

C₂×#11761627

yes

—

132

C₂

—

133

C₂

C₂×#14524030

yes

characteristic

134

C₂

C₂×#14524074

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂×C₄⋊Q₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	C₂×C₄⋊Q₈	D₆₇
hypocenter	C₆₇	#3139
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₂×C₄⋊Q₈×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

hypercenter

C₂×C₄⋊Q₈

D₆₇

hypocenter

C₆₇

#3139

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #16863764	C₂	semidirect
3. #16863746	C₂	semidirect
68. C₆₇	#3139	semidirect

H ≅ G/N

type

2. #16863764

C₂

semidirect

3. #16863746

C₂

semidirect

68. C₆₇

#3139

semidirect

Subgroups — #22570589

Conjugacy classes of subgroups

Complements