Group #22574019 — subgroups

Subgroups — #22574019

Order 8576.

964 conjugacy classes of subgroups; only the 553 normal subgroups are shown.

normal subgroups only

This group has 964 conjugacy classes of subgroups — too many to draw. Only its 553 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	Q₈×Dic₁₃₄	C₂	1	4	yes	maximal, complemented
2	#16864125	C₂	1	4	yes	maximal, complemented
3	#16864122	C₂	1	8	yes	maximal, complemented
4	#16864027	C₂	1	6	yes	maximal
5	#16863953	C₂	1	6	yes	maximal
6	#16863874	C₂	1	2	yes	maximal
7	C₂×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#14524043	C₂²	1	6	yes	—
9	#14524043	C₂²	1	6	yes	—
10	#14524042	C₂²	1	24	yes	—
11	#14524035	C₂²	1	12	yes	—
12	#14524034	C₂²	1	24	yes	—
13	#14524022	C₂²	1	12	yes	—
14	#14524022	C₂²	1	4	yes	—
15	Q₈×Dic₆₇	C₂²	1	8	yes	—
16	Q₈×Dic₆₇	C₂²	1	8	yes	—
17	C₄×Dic₁₃₄	C₂²	1	12	yes	—
18	C₄×Dic₁₃₄	C₂²	1	12	yes	—
19	#14524002	C₂²	1	12	yes	—
20	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₆₇	C₂²	1	4	yes	—
22	C₄×Q₈×C₆₇	C₂²	1	4	yes	—
23	C₂×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
24	C₂×C₄²×C₆₇	C₂²	1	3	yes	—
25	C₂×Dic₁₃₄	C₂³	1	4	yes	—
26	C₂×Dic₁₃₄	C₂³	1	12	yes	—
27	C₄⋊Dic₆₇	C₂³	1	12	yes	—
28	C₄⋊Dic₆₇	C₂³	1	12	yes	—
29	C₄⋊Dic₆₇	C₂³	1	4	yes	—
30	C₄⋊Dic₆₇	C₂³	1	12	yes	—
31	Dic₆₇⋊C₄	C₂³	1	8	yes	—
32	Dic₆₇⋊C₄	C₂³	1	24	yes	—
33	C₂²⋊Dic₆₇	C₂³	1	8	yes	—
34	C₄×Dic₆₇	C₂³	1	24	yes	—
35	C₂×Q₈×C₆₇	Q₈	1	8	yes	complemented
36	C₂×Q₈×C₆₇	C₂³	1	4	yes	—
37	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
39	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
40	C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
41	C₄²×C₆₇	C₂³	1	6	yes	—
42	C₄²×C₆₇	C₂³	1	6	yes	—
43	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
44	Q₈×C₆₇	C₂×Q₈	1	16	yes	—
45	C₂×C₄×C₆₇	—	2	2	no	—
46	C₂×C₄×C₆₇	—	4	3	no	—
47	C₂×C₄×C₆₇	C₂×Q₈	1	12	yes	—
48	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₂⁴	1	6	yes	—
50	C₂×C₄×C₆₇	C₂⁴	1	6	yes	—
51	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₆₇	C₂⁴	1	8	yes	—
53	C₂×Dic₆₇	C₂⁴	1	8	yes	—
54	Dic₁₃₄	—	2	8	no	—
55	Dic₁₃₄	—	4	12	no	—
56	C₂²×C₆₇	—	2	1	no	—
57	C₂²×C₆₇	—	2	1	no	—
58	C₂²×C₆₇	C₂²×Q₈	1	2	yes	—
59	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
60	C₄×C₆₇	—	2	1	no	—
61	C₄×C₆₇	—	2	6	no	—
62	C₄×C₆₇	C₂²×Q₈	1	12	yes	—
63	C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
64	Dic₆₇	—	2	8	no	—
65	Dic₆₇	—	2	8	no	—
66	C₂×C₆₇	—	2	2	no	—
67	C₂×C₆₇	C₂³×Q₈	1	1	yes	characteristic
68	C₂×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
69	C₂×C₆₇	#585	1	1	yes	characteristic
70	#3502	—	67	1	no	maximal, Sylow
71	C₆₇	#3502	1	1	yes	characteristic, complemented, Sylow
72	Q₈²	—	67	4	no	—
73	#557	—	67	8	no	—
74	#557	—	67	4	no	—
75	#544	—	67	6	no	—
76	#533	—	67	6	no	—
77	#519	—	67	2	no	—
78	C₂×C₄×Q₈	D₆₇	1	1	yes	characteristic
79	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
80	C₄⋊Q₈	—	67	24	no	—
81	C₄⋊Q₈	—	67	6	no	—
82	C₄⋊Q₈	—	67	6	no	—
83	#125	—	67	24	no	—
84	#125	—	67	12	no	—
85	C₂²⋊Q₈	—	67	4	no	—
86	C₂²⋊Q₈	—	67	12	no	—
87	C₄×Q₈	—	67	8	no	—
88	C₄×Q₈	—	67	8	no	—
89	C₄×Q₈	—	67	12	no	—
90	C₄×Q₈	—	67	12	no	—
91	C₄×Q₈	C₂×D₆₇	1	4	yes	—
92	C₄×Q₈	C₂×D₆₇	1	4	yes	—
93	#117	—	67	12	no	—
94	C₂×C₄⋊C₄	C₂×D₆₇	1	3	yes	—
95	C₂×C₄²	C₂×D₆₇	1	3	yes	—
96	C₂×Q₈	—	67	12	no	—
97	C₂×Q₈	—	67	4	no	—
98	C₂×Q₈	Dic₁₃₄	1	8	yes	complemented
99	C₂×Q₈	C₂²×D₆₇	1	4	yes	—
100	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
101	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
102	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
103	C₄⋊C₄	—	67	12	no	—
104	C₄⋊C₄	—	67	12	no	—
105	C₄⋊C₄	—	67	8	no	—
106	C₄⋊C₄	—	67	4	no	—
107	C₄⋊C₄	—	67	12	no	—
108	C₄⋊C₄	—	67	24	no	—
109	C₄⋊C₄	C₂²×D₆₇	1	12	yes	—
110	C₂²⋊C₄	—	67	8	no	—
111	C₄²	—	67	24	no	—
112	C₄²	C₂²×D₆₇	1	6	yes	—
113	C₄²	C₂²×D₆₇	1	6	yes	—
114	C₂³	C₂³×D₆₇	1	1	yes	characteristic
115	Q₈	—	268	12	no	—
116	Q₈	—	134	8	no	—
117	Q₈	C₂×Dic₁₃₄	1	16	yes	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	3	no	—
120	C₂×C₄	—	67	8	no	—
121	C₂×C₄	—	67	8	no	—
122	C₂×C₄	C₂×Dic₁₃₄	1	12	yes	—
123	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
124	C₂×C₄	C₂³×D₆₇	1	6	yes	—
125	C₂×C₄	C₂³×D₆₇	1	6	yes	—
126	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	C₂²×Dic₁₃₄	1	2	yes	—
130	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	6	no	—
133	C₄	—	134	8	no	—
134	C₄	—	134	8	no	—
135	C₄	C₂²×Dic₁₃₄	1	12	yes	—
136	C₄	#14524091	1	2	yes	—
137	C₂	—	2	2	no	—
138	C₂	C₂³×Dic₁₃₄	1	1	yes	characteristic
139	C₂	C₂×#14524091	1	1	yes	characteristic
140	C₂	#16864232	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₃₄

C₂

yes

maximal, complemented

#16864125

C₂

yes

maximal, complemented

#16864122

C₂

yes

maximal, complemented

#16864027

C₂

yes

maximal

#16863953

C₂

yes

maximal

#16863874

C₂

yes

maximal

C₂×C₄×Q₈×C₆₇

C₂

yes

maximal, characteristic

#14524043

C₂²

yes

—

#14524043

C₂²

yes

—

#14524042

C₂²

yes

—

#14524035

C₂²

yes

—

#14524034

C₂²

yes

—

#14524022

C₂²

yes

—

#14524022

C₂²

yes

—

Q₈×Dic₆₇

C₂²

yes

—

Q₈×Dic₆₇

C₂²

yes

—

C₄×Dic₁₃₄

C₂²

yes

—

C₄×Dic₁₃₄

C₂²

yes

—

#14524002

C₂²

yes

—

C₂²×Q₈×C₆₇

C₂²

yes

characteristic

C₄×Q₈×C₆₇

C₂²

yes

—

C₄×Q₈×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

C₂³

yes

—

C₂×Dic₁₃₄

C₂³

yes

—

C₄⋊Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

C₂³

yes

—

Dic₆₇⋊C₄

C₂³

yes

—

C₂²⋊Dic₆₇

C₂³

yes

—

C₄×Dic₆₇

C₂³

yes

—

C₂×Q₈×C₆₇

Q₈

yes

complemented

C₂×Q₈×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₄²×C₆₇

C₂³

yes

—

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

Q₈×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×C₄×C₆₇

C₂⁴

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

C₂⁴

yes

—

C₂×Dic₆₇

C₂⁴

yes

—

Dic₁₃₄

—

Dic₁₃₄

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×Q₈

yes

—

C₂²×C₆₇

C₂⁵

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²×Q₈

yes

—

C₄×C₆₇

2₋¹⁺⁴

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂³×Q₈

yes

characteristic

C₂×C₆₇

C₂×2₋¹⁺⁴

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

Q₈²

—

—

—

—

—

—

C₂×C₄×Q₈

D₆₇

yes

characteristic

C₂²×Q₈

C₂×D₆₇

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₆₇

yes

—

C₄×Q₈

C₂×D₆₇

yes

—

#117

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄²

C₂×D₆₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₁₃₄

yes

complemented

C₂×Q₈

C₂²×D₆₇

yes

—

100

C₂²×C₄

C₂²×D₆₇

yes

characteristic

101

C₂²×C₄

C₂²×D₆₇

yes

—

102

C₂²×C₄

C₂²×D₆₇

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

C₂²×D₆₇

yes

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₄²

C₂²×D₆₇

yes

—

113

C₄²

C₂²×D₆₇

yes

—

114

C₂³

C₂³×D₆₇

yes

characteristic

115

Q₈

—

268

—

116

Q₈

—

134

—

117

Q₈

C₂×Dic₁₃₄

yes

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂×Dic₁₃₄

yes

—

123

C₂×C₄

C₂³×D₆₇

yes

characteristic

124

C₂×C₄

C₂³×D₆₇

yes

—

125

C₂×C₄

C₂³×D₆₇

yes

—

126

C₂×C₄

C₂³×D₆₇

yes

characteristic

127

C₂²

—

128

C₂²

—

129

C₂²

C₂²×Dic₁₃₄

yes

—

130

C₂²

C₂⁴×D₆₇

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

—

134

C₄

—

134

—

135

C₄

C₂²×Dic₁₃₄

yes

—

136

C₄

#14524091

yes

—

137

C₂

—

138

C₂

C₂³×Dic₁₃₄

yes

characteristic

139

C₂

C₂×#14524091

yes

characteristic

140

C₂

#16864232

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₆₇
commutator, socle	C₂²×C₆₇	C₂⁵
Fitting	C₂×C₄×Q₈×C₆₇	C₂
hypercenter	C₂×C₄×Q₈	D₆₇
hypocenter	C₆₇	#3502
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₆₇

commutator, socle

C₂²×C₆₇

C₂⁵

Fitting

C₂×C₄×Q₈×C₆₇

C₂

hypercenter

C₂×C₄×Q₈

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
1. Q₈×Dic₁₃₄	C₂	semidirect
2. #16864125	C₂	semidirect
3. #16864122	C₂	semidirect
35. C₂×Q₈×C₆₇	Q₈	semidirect
71. C₆₇	#3502	semidirect
98. C₂×Q₈	Dic₁₃₄	semidirect

H ≅ G/N

type

1. Q₈×Dic₁₃₄

C₂

semidirect

2. #16864125

C₂

semidirect

3. #16864122

C₂

semidirect

35. C₂×Q₈×C₆₇

Q₈

semidirect

71. C₆₇

#3502

semidirect

98. C₂×Q₈

Dic₁₃₄

semidirect

Subgroups — #22574019

Conjugacy classes of subgroups

Complements