Group #22563045 — subgroups

Subgroups — Q₈×C₂²⋊Dic₆₇ #22563045

Order 8576.

820 conjugacy classes of subgroups; only the 333 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 333 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×Q₈×Dic₆₇	C₂	1	2	yes	maximal, complemented
2	#16863303	C₂	1	6	yes	maximal, complemented
3	#16863256	C₂	1	3	yes	maximal
4	C₄×C₂²⋊Dic₆₇	C₂	1	3	yes	maximal
5	C₂³×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₆₇	—	2	8	no	—
7	C₂×C₄⋊Dic₆₇	C₂²	1	6	yes	—
8	C₂×C₂²⋊Dic₆₇	C₂²	1	3	yes	—
9	C₂×C₂²⋊Dic₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₆₇	C₂²	1	6	yes	—
11	C₆₇⋊#95	C₂²	1	6	yes	—
12	C₆₇⋊#95	C₂²	1	6	yes	—
13	C₂²×Q₈×C₆₇	—	2	8	no	—
14	C₂²×Q₈×C₆₇	C₄	1	8	yes	complemented
15	C₂²×Q₈×C₆₇	C₂²	1	3	yes	—
16	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₆₇	C₂²	1	3	yes	—
18	C₂²×Dic₆₇	C₂³	1	6	yes	—
19	C₂²×Dic₆₇	C₂³	1	2	yes	—
20	C₄⋊Dic₆₇	—	2	12	no	—
21	C₂²⋊Dic₆₇	—	2	6	no	—
22	C₂²⋊Dic₆₇	Q₈	1	4	yes	direct product factor
23	C₄×Dic₆₇	—	2	12	no	—
24	C₂⁴×C₆₇	C₂³	1	1	yes	characteristic
25	C₂×Q₈×C₆₇	—	2	32	no	—
26	C₂×Q₈×C₆₇	—	2	12	no	—
27	C₂×Q₈×C₆₇	C₂×C₄	1	12	yes	—
28	C₂×Q₈×C₆₇	C₂×C₄	1	4	yes	—
29	C₂×Q₈×C₆₇	D₄	1	8	yes	—
30	C₂²×C₄×C₆₇	—	2	12	no	—
31	C₂²×C₄×C₆₇	C₂×C₄	1	12	yes	—
32	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
33	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
34	C₂³×C₆₇	—	2	2	no	—
35	C₂³×C₆₇	—	2	2	no	—
36	C₂³×C₆₇	C₂²×C₄	1	2	yes	—
37	C₂³×C₆₇	C₂×Q₈	1	2	yes	—
38	C₂³×C₆₇	#41	1	2	yes	—
39	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
40	Q₈×C₆₇	—	2	24	no	—
41	Q₈×C₆₇	C₂²⋊C₄	1	16	yes	complemented
42	C₂×C₄×C₆₇	—	2	6	no	—
43	C₂×C₄×C₆₇	—	2	24	no	—
44	C₂×C₄×C₆₇	C₂²×C₄	1	6	yes	—
45	C₂×C₄×C₆₇	C₂²×C₄	1	6	yes	—
46	C₂×C₄×C₆₇	C₂×D₄	1	12	yes	—
47	C₂×Dic₆₇	—	2	2	no	—
48	C₂×Dic₆₇	—	2	2	no	—
49	C₂×Dic₆₇	—	2	6	no	—
50	C₂×Dic₆₇	—	2	6	no	—
51	C₂×Dic₆₇	—	2	6	no	—
52	C₂×Dic₆₇	C₂×Q₈	1	4	yes	complemented
53	C₂²×C₆₇	—	2	2	no	—
54	C₂²×C₆₇	—	2	2	no	—
55	C₂²×C₆₇	—	2	4	no	—
56	C₂²×C₆₇	—	2	4	no	—
57	C₂²×C₆₇	C₄×Q₈	1	4	yes	complemented
58	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
59	C₂²×C₆₇	C₂²×D₄	1	2	yes	—
60	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
61	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
62	C₂²×C₆₇	2₋¹⁺⁴	1	2	yes	—
63	C₄×C₆₇	—	2	6	no	—
64	C₄×C₆₇	C₂×C₂²⋊C₄	1	12	yes	—
65	Dic₆₇	—	2	4	no	—
66	Dic₆₇	—	4	6	no	—
67	C₂×C₆₇	—	2	4	no	—
68	C₂×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
70	C₂×C₆₇	#519	1	1	yes	characteristic
71	C₂×C₆₇	#548	1	2	yes	—
72	C₂×C₆₇	D₄×Q₈	1	2	yes	—
73	Q₈×C₂²⋊C₄	—	67	1	no	maximal, Sylow
74	C₆₇	Q₈×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
75	C₂³×Q₈	D₆₇	1	1	yes	characteristic
76	C₂×C₄×Q₈	—	67	2	no	—
77	#391	—	67	6	no	—
78	#380	—	67	3	no	—
79	C₄×C₂²⋊C₄	—	67	3	no	—
80	C₂²×Q₈	—	2	8	no	—
81	C₂²×Q₈	Dic₆₇	1	8	yes	complemented
82	C₂²×Q₈	C₂×D₆₇	1	3	yes	—
83	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
84	C₂³×C₄	C₂×D₆₇	1	3	yes	—
85	C₄×Q₈	—	134	8	no	—
86	C₂×C₄⋊C₄	—	67	6	no	—
87	C₂×C₂²⋊C₄	—	67	1	no	—
88	C₂×C₂²⋊C₄	—	67	3	no	—
89	C₂×C₄²	—	67	6	no	—
90	#95	—	67	6	no	—
91	#95	—	67	6	no	—
92	C₂⁴	C₂²×D₆₇	1	1	yes	characteristic
93	C₂×Q₈	—	2	32	no	—
94	C₂×Q₈	—	2	12	no	—
95	C₂×Q₈	C₂×Dic₆₇	1	12	yes	—
96	C₂×Q₈	C₂×Dic₆₇	1	4	yes	—
97	C₂×Q₈	C₂²⋊D₆₇	1	8	yes	—
98	C₂²×C₄	—	2	12	no	—
99	C₂²×C₄	—	67	2	no	—
100	C₂²×C₄	—	67	6	no	—
101	C₂²×C₄	C₂×Dic₆₇	1	12	yes	—
102	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
103	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
104	C₄⋊C₄	—	134	12	no	—
105	C₂²⋊C₄	—	67	4	no	—
106	C₂²⋊C₄	—	134	6	no	—
107	C₄²	—	134	12	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	C₂²×Dic₆₇	1	2	yes	—
111	C₂³	Q₈×D₆₇	1	2	yes	—
112	C₂³	#11761629	1	2	yes	—
113	C₂³	C₂³×D₆₇	1	1	yes	characteristic
114	Q₈	—	2	24	no	—
115	Q₈	C₂²⋊Dic₆₇	1	16	yes	direct product factor
116	C₂×C₄	—	2	6	no	—
117	C₂×C₄	—	2	24	no	—
118	C₂×C₄	—	67	4	no	—
119	C₂×C₄	—	134	6	no	—
120	C₂×C₄	—	134	6	no	—
121	C₂×C₄	—	134	6	no	—
122	C₂×C₄	—	134	2	no	—
123	C₂×C₄	—	134	2	no	—
124	C₂×C₄	C₂²×Dic₆₇	1	6	yes	—
125	C₂×C₄	C₂²×Dic₆₇	1	6	yes	—
126	C₂×C₄	C₂×C₂²⋊D₆₇	1	12	yes	—
127	C₂²	—	2	4	no	—
128	C₂²	—	2	4	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	2	2	no	—
131	C₂²	Q₈×Dic₆₇	1	4	yes	complemented
132	C₂²	C₂³×Dic₆₇	1	1	yes	characteristic
133	C₂²	C₂²×C₂²⋊D₆₇	1	2	yes	—
134	C₂²	C₂×Q₈×D₆₇	1	1	yes	characteristic
135	C₂²	C₂×#11761629	1	1	yes	characteristic
136	C₂²	#14524091	1	2	yes	—
137	C₄	—	2	6	no	—
138	C₄	—	268	6	no	—
139	C₄	—	134	4	no	—
140	C₄	C₂×C₂²⋊Dic₆₇	1	12	yes	—
141	C₂	—	2	4	no	—
142	C₂	C₂²×C₂²⋊Dic₆₇	1	1	yes	characteristic
143	C₂	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
144	C₂	#16863874	1	1	yes	characteristic
145	C₂	#16864057	1	2	yes	—
146	C₂	Q₈×C₂²⋊D₆₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₆₇

C₂

yes

maximal, complemented

#16863303

C₂

yes

maximal, complemented

#16863256

C₂

yes

maximal

C₄×C₂²⋊Dic₆₇

C₂

yes

maximal

C₂³×Q₈×C₆₇

C₂

yes

maximal, characteristic

Q₈×Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₂²⋊Dic₆₇

C₂²

yes

—

C₂×C₂²⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄×Dic₆₇

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₆₇

—

C₂²×Q₈×C₆₇

C₄

yes

complemented

C₂²×Q₈×C₆₇

C₂²

yes

—

C₂²×Q₈×C₆₇

C₂²

yes

characteristic

C₂³×C₄×C₆₇

C₂²

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

Q₈

yes

direct product factor

C₄×Dic₆₇

—

C₂⁴×C₆₇

C₂³

yes

characteristic

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂×C₄

yes

—

C₂×Q₈×C₆₇

C₂×C₄

yes

—

C₂×Q₈×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂²×C₄

yes

—

C₂³×C₆₇

C₂×Q₈

yes

—

C₂³×C₆₇

#41

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

C₂²⋊C₄

yes

complemented

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×Q₈

yes

complemented

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×Q₈

yes

complemented

C₂²×C₆₇

C₂³×C₄

yes

characteristic

C₂²×C₆₇

C₂²×D₄

yes

—

C₂²×C₆₇

C₂²×Q₈

yes

characteristic

C₂²×C₆₇

C₂×#41

yes

characteristic

C₂²×C₆₇

2₋¹⁺⁴

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₂²⋊C₄

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂²×C₂²⋊C₄

yes

characteristic

C₂×C₆₇

C₂×C₄×Q₈

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

D₄×Q₈

yes

—

Q₈×C₂²⋊C₄

—

maximal, Sylow

C₆₇

Q₈×C₂²⋊C₄

yes

characteristic, complemented, Sylow

C₂³×Q₈

D₆₇

yes

characteristic

C₂×C₄×Q₈

—

#391

—

#380

—

C₄×C₂²⋊C₄

—

C₂²×Q₈

—

C₂²×Q₈

Dic₆₇

yes

complemented

C₂²×Q₈

C₂×D₆₇

yes

—

C₂²×Q₈

C₂×D₆₇

yes

characteristic

C₂³×C₄

C₂×D₆₇

yes

—

C₄×Q₈

—

134

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

#95

—

C₂⁴

C₂²×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₆₇

yes

—

C₂×Q₈

C₂×Dic₆₇

yes

—

C₂×Q₈

C₂²⋊D₆₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂×Dic₆₇

yes

—

102

C₂²×C₄

C₂²×D₆₇

yes

—

103

C₂²×C₄

C₂²×D₆₇

yes

—

104

C₄⋊C₄

—

134

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

—

134

—

107

C₄²

—

134

—

108

C₂³

—

109

C₂³

—

110

C₂³

C₂²×Dic₆₇

yes

—

111

C₂³

Q₈×D₆₇

yes

—

112

C₂³

#11761629

yes

—

113

C₂³

C₂³×D₆₇

yes

characteristic

114

Q₈

—

115

Q₈

C₂²⋊Dic₆₇

yes

direct product factor

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

134

—

120

C₂×C₄

—

134

—

121

C₂×C₄

—

134

—

122

C₂×C₄

—

134

—

123

C₂×C₄

—

134

—

124

C₂×C₄

C₂²×Dic₆₇

yes

—

125

C₂×C₄

C₂²×Dic₆₇

yes

—

126

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

Q₈×Dic₆₇

yes

complemented

132

C₂²

C₂³×Dic₆₇

yes

characteristic

133

C₂²

C₂²×C₂²⋊D₆₇

yes

—

134

C₂²

C₂×Q₈×D₆₇

yes

characteristic

135

C₂²

C₂×#11761629

yes

characteristic

136

C₂²

#14524091

yes

—

137

C₄

—

138

C₄

—

268

—

139

C₄

—

134

—

140

C₄

C₂×C₂²⋊Dic₆₇

yes

—

141

C₂

—

142

C₂

C₂²×C₂²⋊Dic₆₇

yes

characteristic

143

C₂

C₂×Q₈×Dic₆₇

yes

characteristic

144

C₂

#16863874

yes

characteristic

145

C₂

#16864057

yes

—

146

C₂

Q₈×C₂²⋊D₆₇

yes

—

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

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₇

commutator

C₂²×C₆₇

C₂³×C₄

Fitting

C₂³×Q₈×C₆₇

C₂

socle

C₂³×C₆₇

C₂⁴

hypercenter

C₂³×Q₈

D₆₇

hypocenter

C₆₇

Q₈×C₂²⋊C₄

LCS

G ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂³×Q₈

N	H ≅ G/N	type
1. C₂×Q₈×Dic₆₇	C₂	semidirect
2. #16863303	C₂	semidirect
14. C₂²×Q₈×C₆₇	C₄	semidirect
22. C₂²⋊Dic₆₇	Q₈	direct (N × H)
41. Q₈×C₆₇	C₂²⋊C₄	semidirect
52. C₂×Dic₆₇	C₂×Q₈	semidirect
57. C₂²×C₆₇	C₄×Q₈	semidirect
74. C₆₇	Q₈×C₂²⋊C₄	semidirect
81. C₂²×Q₈	Dic₆₇	semidirect
115. Q₈	C₂²⋊Dic₆₇	direct (N × H)
131. C₂²	Q₈×Dic₆₇	semidirect

H ≅ G/N

type

1. C₂×Q₈×Dic₆₇

C₂

semidirect

2. #16863303

C₂

semidirect

14. C₂²×Q₈×C₆₇

C₄

semidirect

22. C₂²⋊Dic₆₇

Q₈

direct (N × H)

41. Q₈×C₆₇

C₂²⋊C₄

semidirect

52. C₂×Dic₆₇

C₂×Q₈

semidirect

57. C₂²×C₆₇

C₄×Q₈

semidirect

74. C₆₇

Q₈×C₂²⋊C₄

semidirect

81. C₂²×Q₈

Dic₆₇

semidirect

115. Q₈

C₂²⋊Dic₆₇

direct (N × H)

131. C₂²

Q₈×Dic₆₇

semidirect

Subgroups — Q₈×C₂²⋊Dic₆₇ #22563045

Conjugacy classes of subgroups

Complements

Subgroups — Q8×C22⋊Dic67 #22563045

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×C₂²⋊Dic₆₇ #22563045