Group #22569340 — subgroups

Subgroups — #22569340

Order 8576.

394 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 179 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	C₄×#11761629	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863475	C₂	1	3	yes	maximal, complemented
4	#16863473	C₂	1	3	yes	maximal, complemented
5	#16863447	C₂	1	3	yes	maximal, complemented
6	#16863418	C₂	1	3	yes	maximal, complemented
7	C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11761629	C₄	1	1	yes	characteristic
9	C₂×#11761611	C₂²	1	6	yes	—
10	Q₈×Dic₆₇	C₄	1	1	yes	characteristic
11	C₄×D₂₆₈	C₂²	1	3	yes	—
12	#14524005	C₄	1	3	yes	—
13	#14524001	C₄	1	3	yes	—
14	C₄²×D₆₇	C₂²	1	3	yes	—
15	C₄⋊(C₆₇⋊C₈)	C₂²	1	3	yes	—
16	Dic₆₇⋊C₈	C₂²	1	3	yes	—
17	Dic₆₇⋊C₈	C₂²	1	1	yes	characteristic
18	(C₂×D₆₇)⋊C₈	C₂²	1	3	yes	—
19	(C₂×D₆₇)⋊C₈	C₂²	1	3	yes	—
20	C₄×C₆₇⋊C₈	C₂²	1	3	yes	—
21	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₆₇	C₂²	1	3	yes	—
23	C₄×C₈×C₆₇	C₂²	1	3	yes	—
24	#11761629	—	2	4	no	—
25	C₂×D₂₆₈	C₂×C₄	1	3	yes	—
26	C₂×C₄×D₆₇	C₂×C₄	1	3	yes	—
27	C₂×C₄×D₆₇	C₂×C₄	1	3	yes	—
28	C₂×C₄×D₆₇	C₂³	1	3	yes	—
29	#11761611	—	4	6	no	—
30	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
31	C₂×C₆₇⋊C₈	C₂³	1	3	yes	—
32	C₄⋊Dic₆₇	C₂×C₄	1	3	yes	—
33	(C₂×D₆₇)⋊C₄	C₂×C₄	1	6	yes	—
34	C₄×Dic₆₇	C₂×C₄	1	3	yes	—
35	C₄×Dic₆₇	C₂×C₄	1	3	yes	—
36	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
37	C₂×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×C₆₇	C₂³	1	3	yes	—
39	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
41	C₄²×C₆₇	C₂³	1	3	yes	—
42	C₂²×D₆₇	C₂²×C₄	1	3	yes	—
43	Q₈×C₆₇	M₁₆	1	4	yes	complemented
44	C₂×C₄×C₆₇	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₆₇	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₆₇	C₂²×C₄	1	3	yes	—
49	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
50	D₂₆₈	—	2	6	no	—
51	C₄×D₆₇	—	2	6	no	—
52	C₄×D₆₇	—	2	3	no	—
53	C₄×D₆₇	—	2	3	no	—
54	C₄×D₆₇	—	2	6	no	—
55	C₈×C₆₇	—	2	3	no	—
56	C₈×C₆₇	—	2	1	no	—
57	C₆₇⋊C₈	—	2	3	no	—
58	C₆₇⋊C₈	—	2	1	no	—
59	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
60	C₂×D₆₇	—	2	3	no	—
61	C₂×D₆₇	—	2	3	no	—
62	C₂×D₆₇	—	2	3	no	—
63	C₄×C₆₇	—	2	3	no	—
64	C₄×C₆₇	C₂×M₁₆	1	6	yes	—
65	C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
66	C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
67	Dic₆₇	—	2	3	no	—
68	Dic₆₇	—	2	1	no	—
69	Dic₆₇	#131	1	2	yes	—
70	C₂×C₆₇	#520	1	1	yes	characteristic
71	C₂×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
72	C₂×C₆₇	C₂×#131	1	1	yes	characteristic
73	D₆₇	—	4	3	no	—
74	#2991	—	67	1	no	maximal, Sylow
75	C₆₇	#2991	1	1	yes	characteristic, complemented, Sylow
76	C₄×#41	—	67	1	no	—
77	C₈×Q₈	—	67	1	no	—
78	C₈×Q₈	D₆₇	1	1	yes	characteristic, complemented
79	#436	—	67	3	no	—
80	#436	—	67	3	no	—
81	#431	—	67	3	no	—
82	#423	—	67	3	no	—
83	C₂×#41	—	67	1	no	—
84	C₂×M₁₆	—	67	6	no	—
85	C₄×Q₈	—	67	1	no	—
86	C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
87	C₄×D₄	—	67	3	no	—
88	C₄×D₄	—	67	3	no	—
89	#117	—	67	3	no	—
90	C₂×C₄²	—	67	3	no	—
91	C₄⋊C₈	—	67	3	no	—
92	C₄⋊C₈	—	67	1	no	—
93	C₄⋊C₈	—	67	3	no	—
94	C₄⋊C₈	C₂×D₆₇	1	3	yes	—
95	C₂²⋊C₈	—	67	3	no	—
96	C₂²⋊C₈	—	67	3	no	—
97	C₄×C₈	—	67	3	no	—
98	C₄×C₈	C₂×D₆₇	1	3	yes	—
99	#41	—	134	4	no	—
100	C₂×Q₈	C₄×D₆₇	1	1	yes	characteristic
101	C₂×D₄	—	67	3	no	—
102	C₂²×C₄	—	67	3	no	—
103	C₂²×C₄	—	67	3	no	—
104	C₂²×C₄	—	67	3	no	—
105	M₁₆	—	268	6	no	—
106	C₂×C₈	—	67	1	no	—
107	C₂×C₈	—	67	3	no	—
108	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
109	C₂×C₈	C₂²×D₆₇	1	3	yes	—
110	C₄⋊C₄	—	67	3	no	—
111	C₄⋊C₄	C₄×D₆₇	1	3	yes	—
112	C₂²⋊C₄	—	67	6	no	—
113	C₄²	—	67	3	no	—
114	C₄²	—	67	3	no	—
115	C₄²	—	67	1	no	—
116	C₄²	C₂²×D₆₇	1	3	yes	—
117	C₂³	—	67	3	no	—
118	Q₈	#11761611	1	4	yes	complemented
119	D₄	—	134	6	no	—
120	C₂×C₄	—	67	1	no	—
121	C₂×C₄	—	67	1	no	—
122	C₂×C₄	—	67	3	no	—
123	C₂×C₄	—	134	6	no	—
124	C₂×C₄	—	134	6	no	—
125	C₂×C₄	—	134	3	no	—
126	C₂×C₄	—	134	3	no	—
127	C₂×C₄	C₂×C₄×D₆₇	1	3	yes	—
128	C₂×C₄	C₂×C₄×D₆₇	1	3	yes	—
129	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
130	C₈	—	2	3	no	—
131	C₈	—	2	1	no	—
132	C₈	—	134	3	no	—
133	C₈	—	134	1	no	—
134	C₂²	—	134	3	no	—
135	C₂²	—	134	3	no	—
136	C₂²	—	134	3	no	—
137	C₂²	C₂²×C₄×D₆₇	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	67	2	no	—
140	C₄	—	134	3	no	—
141	C₄	—	134	1	no	—
142	C₄	C₂×#11761611	1	6	yes	—
143	C₄	#14524089	1	1	yes	characteristic
144	C₄	#14524091	1	1	yes	characteristic
145	C₂	—	268	3	no	—
146	C₂	#16863878	1	1	yes	characteristic
147	C₂	C₂²×#11761611	1	1	yes	characteristic
148	C₂	#131×D₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#11761629

C₂

yes

maximal, characteristic

Q₈×C₆₇⋊C₈

C₂

yes

maximal, characteristic, complemented

#16863475

C₂

yes

maximal, complemented

#16863473

C₂

yes

maximal, complemented

#16863447

C₂

yes

maximal, complemented

#16863418

C₂

yes

maximal, complemented

C₈×Q₈×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761629

C₄

yes

characteristic

C₂×#11761611

C₂²

yes

—

Q₈×Dic₆₇

C₄

yes

characteristic

C₄×D₂₆₈

C₂²

yes

—

#14524005

C₄

yes

—

#14524001

C₄

yes

—

C₄²×D₆₇

C₂²

yes

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

Dic₆₇⋊C₈

C₂²

yes

—

Dic₆₇⋊C₈

C₂²

yes

characteristic

(C₂×D₆₇)⋊C₈

C₂²

yes

—

(C₂×D₆₇)⋊C₈

C₂²

yes

—

C₄×C₆₇⋊C₈

C₂²

yes

—

C₄×Q₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

—

#11761629

—

C₂×D₂₆₈

C₂×C₄

yes

—

C₂×C₄×D₆₇

C₂×C₄

yes

—

C₂×C₄×D₆₇

C₂×C₄

yes

—

C₂×C₄×D₆₇

C₂³

yes

—

#11761611

—

C₂×C₆₇⋊C₈

C₂³

yes

characteristic

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

(C₂×D₆₇)⋊C₄

C₂×C₄

yes

—

C₄×Dic₆₇

C₂×C₄

yes

—

C₄×Dic₆₇

C₂×C₄

yes

—

C₄×Dic₆₇

C₂³

yes

characteristic

C₂×Q₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

C₂³

yes

—

C₂²×D₆₇

C₂²×C₄

yes

—

Q₈×C₆₇

M₁₆

yes

complemented

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

C₂²×C₄

yes

characteristic

C₂×Dic₆₇

C₂²×C₄

yes

—

C₂×Dic₆₇

C₂²×C₄

yes

characteristic

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

C₄×D₆₇

—

C₄×D₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₂³×C₄

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×M₁₆

yes

—

C₄×C₆₇

H(5,𝔽₂)

yes

characteristic

C₄×C₆₇

2₋¹⁺⁴

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

#131

yes

—

C₂×C₆₇

#520

yes

characteristic

C₂×C₆₇

C₂²×M₁₆

yes

characteristic

C₂×C₆₇

C₂×#131

yes

characteristic

D₆₇

—

#2991

—

maximal, Sylow

C₆₇

#2991

yes

characteristic, complemented, Sylow

C₄×#41

—

C₈×Q₈

—

C₈×Q₈

D₆₇

yes

characteristic, complemented

—

—

—

—

C₂×#41

—

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

C₂×D₆₇

yes

characteristic

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₇

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₆₇

yes

—

#41

—

134

—

100

C₂×Q₈

C₄×D₆₇

yes

characteristic

101

C₂×D₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

M₁₆

—

268

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²×D₆₇

yes

characteristic

109

C₂×C₈

C₂²×D₆₇

yes

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₄×D₆₇

yes

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₄²

C₂²×D₆₇

yes

—

117

C₂³

—

118

Q₈

#11761611

yes

complemented

119

D₄

—

134

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

134

—

124

C₂×C₄

—

134

—

125

C₂×C₄

—

134

—

126

C₂×C₄

—

134

—

127

C₂×C₄

C₂×C₄×D₆₇

yes

—

128

C₂×C₄

C₂×C₄×D₆₇

yes

—

129

C₂×C₄

C₂³×D₆₇

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

134

—

133

C₈

—

134

—

134

C₂²

—

134

—

135

C₂²

—

134

—

136

C₂²

—

134

—

137

C₂²

C₂²×C₄×D₆₇

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

134

—

141

C₄

—

134

—

142

C₄

C₂×#11761611

yes

—

143

C₄

#14524089

yes

characteristic

144

C₄

#14524091

yes

characteristic

145

C₂

—

268

—

146

C₂

#16863878

yes

characteristic

147

C₂

C₂²×#11761611

yes

characteristic

148

C₂

#131×D₆₇

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₄
Fitting	C₈×Q₈×C₆₇	C₂
hypercenter	C₈×Q₈	D₆₇
hypocenter	C₆₇	#2991
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₆₇

commutator, socle

C₂²×C₆₇

C₂³×C₄

Fitting

C₈×Q₈×C₆₇

C₂

hypercenter

C₈×Q₈

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₄ ◁ C₈×Q₈

N	H ≅ G/N	type
2. Q₈×C₆₇⋊C₈	C₂	semidirect
3. #16863475	C₂	semidirect
4. #16863473	C₂	semidirect
5. #16863447	C₂	semidirect
6. #16863418	C₂	semidirect
7. C₈×Q₈×C₆₇	C₂	semidirect
43. Q₈×C₆₇	M₁₆	semidirect
75. C₆₇	#2991	semidirect
78. C₈×Q₈	D₆₇	semidirect
118. Q₈	#11761611	semidirect

H ≅ G/N

type

2. Q₈×C₆₇⋊C₈

C₂

semidirect

3. #16863475

C₂

semidirect

4. #16863473

C₂

semidirect

5. #16863447

C₂

semidirect

6. #16863418

C₂

semidirect

7. C₈×Q₈×C₆₇

C₂

semidirect

43. Q₈×C₆₇

M₁₆

semidirect

75. C₆₇

#2991

semidirect

78. C₈×Q₈

D₆₇

semidirect

118. Q₈

#11761611

semidirect

Subgroups — #22569340

Conjugacy classes of subgroups

Complements