Group #22573069 — subgroups

Subgroups — Q₈×#11761616 #22573069

Order 8576.

370 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 149 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	1	yes	maximal, characteristic, complemented
2	#16863792	C₂	1	3	yes	maximal, complemented
3	#16863686	C₂	1	3	yes	maximal
4	#16863678	C₂	1	3	yes	maximal, complemented
5	Q₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#11761616	C₂	1	3	yes	maximal
7	D₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#11761616	C₂²	1	1	yes	characteristic
9	#14524043	C₂²	1	3	yes	—
10	#14524042	—	2	3	no	—
11	Q₈×Dic₆₇	—	2	1	no	—
12	C₄×Dic₁₃₄	C₂²	1	3	yes	—
13	#14523974	C₂²	1	3	yes	—
14	#14523974	C₂²	1	6	yes	—
15	C₄⋊(C₆₇⋊C₈)	C₂²	1	3	yes	—
16	#14523966	C₂²	1	3	yes	—
17	#14523964	C₂²	1	3	yes	—
18	C₄×C₆₇⋊C₈	C₂²	1	3	yes	—
19	C₂²×Q₈×C₆₇	—	2	1	no	—
20	C₄⋊Q₈×C₆₇	C₂²	1	3	yes	—
21	C₂²⋊Q₈×C₆₇	—	2	3	no	—
22	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₆₇	C₂²	1	3	yes	—
24	C₂×Dic₁₃₄	—	2	3	no	—
25	C₂×Dic₁₃₄	C₂³	1	1	yes	characteristic
26	#11761616	Q₈	1	4	yes	direct product factor
27	C₂×C₆₇⋊C₈	C₂³	1	3	yes	—
28	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₆₇	—	2	3	no	—
30	C₄⋊Dic₆₇	C₂³	1	3	yes	—
31	Dic₆₇⋊C₄	—	2	3	no	—
32	C₄×Dic₆₇	—	2	3	no	—
33	C₂×Q₈×C₆₇	—	2	3	no	—
34	C₂×Q₈×C₆₇	—	4	4	no	—
35	C₂×Q₈×C₆₇	D₄	1	1	yes	characteristic
36	C₂×D₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	—	2	3	no	—
38	C₄⋊C₄×C₆₇	—	2	3	no	—
39	C₄⋊C₄×C₆₇	D₄	1	3	yes	—
40	C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
41	C₂²⋊C₄×C₆₇	—	2	3	no	—
42	C₄²×C₆₇	C₂³	1	3	yes	—
43	C₂³×C₆₇	—	2	1	no	—
44	Q₈×C₆₇	—	4	6	no	—
45	Q₈×C₆₇	SD₁₆	1	4	yes	complemented
46	D₄×C₆₇	—	2	1	no	—
47	D₄×C₆₇	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₆₇	—	2	3	no	—
49	C₂×C₄×C₆₇	—	4	6	no	—
50	C₂×C₄×C₆₇	C₂×D₄	1	3	yes	—
51	C₂×C₄×C₆₇	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₆₇	—	2	1	no	—
54	C₂×Dic₆₇	—	2	3	no	—
55	Dic₁₃₄	—	2	1	no	—
56	Dic₁₃₄	—	8	3	no	—
57	Dic₁₃₄	C₂×Q₈	1	2	yes	complemented
58	C₆₇⋊C₈	—	2	3	no	—
59	C₆₇⋊C₈	C₂×Q₈	1	2	yes	complemented
60	C₂²×C₆₇	—	2	2	no	—
61	C₂²×C₆₇	—	4	1	no	—
62	C₂²×C₆₇	—	4	1	no	—
63	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₆₇	—	2	3	no	—
65	C₄×C₆₇	—	4	3	no	—
66	C₄×C₆₇	C₂×SD₁₆	1	6	yes	—
67	C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
68	C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
69	Dic₆₇	—	2	2	no	—
70	Dic₆₇	—	4	3	no	—
71	C₂×C₆₇	—	4	2	no	—
72	C₂×C₆₇	D₄×Q₈	1	1	yes	characteristic
73	C₂×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
74	C₂×C₆₇	#577	1	1	yes	characteristic
75	Q₈×SD₁₆	—	67	1	no	maximal, Sylow
76	C₆₇	Q₈×SD₁₆	1	1	yes	characteristic, complemented, Sylow
77	Q₈²	—	67	1	no	—
78	D₄×Q₈	D₆₇	1	1	yes	characteristic
79	#498	—	67	3	no	—
80	#476	—	67	3	no	—
81	#475	—	67	3	no	—
82	C₈×Q₈	—	67	1	no	—
83	C₄×SD₁₆	—	67	3	no	—
84	C₂²×Q₈	—	2	1	no	—
85	C₂×SD₁₆	—	67	1	no	—
86	C₄⋊Q₈	—	67	3	no	—
87	C₄⋊Q₈	—	134	3	no	—
88	C₄⋊Q₈	C₂×D₆₇	1	3	yes	—
89	C₂²⋊Q₈	—	2	3	no	—
90	C₄×Q₈	—	67	3	no	—
91	C₄×Q₈	—	134	1	no	—
92	C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
93	C₄×D₄	C₂×D₆₇	1	3	yes	—
94	#106	—	67	3	no	—
95	#106	—	67	6	no	—
96	C₄⋊C₈	—	67	3	no	—
97	#103	—	67	3	no	—
98	#102	—	67	3	no	—
99	C₄×C₈	—	67	3	no	—
100	C₂×Q₈	—	2	3	no	—
101	C₂×Q₈	—	4	4	no	—
102	C₂×Q₈	—	67	1	no	—
103	C₂×Q₈	—	134	3	no	—
104	C₂×Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
105	C₂×D₄	C₂²×D₆₇	1	1	yes	characteristic
106	C₂²×C₄	—	2	3	no	—
107	SD₁₆	—	67	4	no	—
108	C₂×C₈	—	67	1	no	—
109	C₂×C₈	—	67	3	no	—
110	C₄⋊C₄	—	2	3	no	—
111	C₄⋊C₄	—	67	3	no	—
112	C₄⋊C₄	—	134	3	no	—
113	C₄⋊C₄	—	134	3	no	—
114	C₄⋊C₄	C₂²⋊D₆₇	1	3	yes	—
115	C₄⋊C₄	C₂²×D₆₇	1	3	yes	—
116	C₂²⋊C₄	—	2	3	no	—
117	C₄²	—	134	3	no	—
118	C₄²	C₂²×D₆₇	1	3	yes	—
119	C₂³	—	2	1	no	—
120	Q₈	—	4	6	no	—
121	Q₈	—	536	3	no	—
122	Q₈	—	67	2	no	—
123	Q₈	—	134	1	no	—
124	Q₈	#11761616	1	4	yes	direct product factor
125	D₄	—	2	1	no	—
126	D₄	Q₈×D₆₇	1	2	yes	—
127	C₂×C₄	—	2	3	no	—
128	C₂×C₄	—	4	6	no	—
129	C₂×C₄	—	134	3	no	—
130	C₂×C₄	—	134	1	no	—
131	C₂×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
132	C₂×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
133	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
134	C₈	—	67	2	no	—
135	C₈	—	134	3	no	—
136	C₂²	—	2	2	no	—
137	C₂²	—	4	1	no	—
138	C₂²	—	4	1	no	—
139	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
140	C₄	—	2	3	no	—
141	C₄	—	4	3	no	—
142	C₄	—	268	3	no	—
143	C₄	—	134	2	no	—
144	C₄	C₂×#11761616	1	6	yes	—
145	C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
146	C₄	#14524091	1	1	yes	characteristic
147	C₂	—	4	2	no	—
148	C₂	Q₈×C₂²⋊D₆₇	1	1	yes	characteristic
149	C₂	C₂²×#11761616	1	1	yes	characteristic
150	C₂	#16864204	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₃₄

C₂

yes

maximal, characteristic, complemented

#16863792

C₂

yes

maximal, complemented

#16863686

C₂

yes

maximal

#16863678

C₂

yes

maximal, complemented

Q₈×C₆₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#11761616

C₂

yes

maximal

D₄×Q₈×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761616

C₂²

yes

characteristic

#14524043

C₂²

yes

—

#14524042

—

Q₈×Dic₆₇

—

C₄×Dic₁₃₄

C₂²

yes

—

#14523974

C₂²

yes

—

#14523974

C₂²

yes

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

#14523966

C₂²

yes

—

#14523964

C₂²

yes

—

C₄×C₆₇⋊C₈

C₂²

yes

—

C₂²×Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

C₂²

yes

—

C₂²⋊Q₈×C₆₇

—

C₄×Q₈×C₆₇

C₂²

yes

characteristic

C₄×D₄×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂³

yes

characteristic

#11761616

Q₈

yes

direct product factor

C₂×C₆₇⋊C₈

C₂³

yes

—

C₂×C₆₇⋊C₈

C₂³

yes

characteristic

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

D₄

yes

characteristic

C₂×D₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

D₄

yes

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

SD₁₆

yes

complemented

D₄×C₆₇

—

D₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

C₂×Q₈

yes

complemented

C₆₇⋊C₈

—

C₆₇⋊C₈

C₂×Q₈

yes

complemented

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×SD₁₆

yes

—

C₄×C₆₇

C₂²×Q₈

yes

characteristic

C₄×C₆₇

2₋¹⁺⁴

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

D₄×Q₈

yes

characteristic

C₂×C₆₇

C₂²×SD₁₆

yes

characteristic

C₂×C₆₇

#577

yes

characteristic

Q₈×SD₁₆

—

maximal, Sylow

C₆₇

Q₈×SD₁₆

yes

characteristic, complemented, Sylow

Q₈²

—

D₄×Q₈

D₆₇

yes

—

—

—

C₈×Q₈

—

C₄×SD₁₆

—

C₂²×Q₈

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

C₄⋊Q₈

C₂×D₆₇

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

134

—

C₄×Q₈

C₂×D₆₇

yes

characteristic

C₄×D₄

C₂×D₆₇

yes

—

#106

—

#106

—

C₄⋊C₈

—

#103

—

#102

—

C₄×C₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

134

—

104

C₂×Q₈

C₂²⋊D₆₇

yes

characteristic

105

C₂×D₄

C₂²×D₆₇

yes

characteristic

106

C₂²×C₄

—

107

SD₁₆

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

134

—

113

C₄⋊C₄

—

134

—

114

C₄⋊C₄

C₂²⋊D₆₇

yes

—

115

C₄⋊C₄

C₂²×D₆₇

yes

—

116

C₂²⋊C₄

—

117

C₄²

—

134

—

118

C₄²

C₂²×D₆₇

yes

—

119

C₂³

—

120

Q₈

—

121

Q₈

—

536

—

122

Q₈

—

123

Q₈

—

134

—

124

Q₈

#11761616

yes

direct product factor

125

D₄

—

126

D₄

Q₈×D₆₇

yes

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

134

—

130

C₂×C₄

—

134

—

131

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

132

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

133

C₂×C₄

C₂³×D₆₇

yes

characteristic

134

C₈

—

135

C₈

—

134

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

C₂²×C₂²⋊D₆₇

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

268

—

143

C₄

—

134

—

144

C₄

C₂×#11761616

yes

—

145

C₄

C₂×Q₈×D₆₇

yes

characteristic

146

C₄

#14524091

yes

characteristic

147

C₂

—

148

C₂

Q₈×C₂²⋊D₆₇

yes

characteristic

149

C₂

C₂²×#11761616

yes

characteristic

150

C₂

#16864204

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	D₄×Q₈×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	D₄×Q₈	D₆₇
hypocenter	C₆₇	Q₈×SD₁₆
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

D₄×Q₈×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

hypercenter

D₄×Q₈

D₆₇

hypocenter

C₆₇

Q₈×SD₁₆

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

N	H ≅ G/N	type
1. Q₈×Dic₁₃₄	C₂	semidirect
2. #16863792	C₂	semidirect
4. #16863678	C₂	semidirect
5. Q₈×C₆₇⋊C₈	C₂	semidirect
26. #11761616	Q₈	direct (N × H)
45. Q₈×C₆₇	SD₁₆	semidirect
57. Dic₁₃₄	C₂×Q₈	semidirect
59. C₆₇⋊C₈	C₂×Q₈	semidirect
76. C₆₇	Q₈×SD₁₆	semidirect
124. Q₈	#11761616	direct (N × H)

H ≅ G/N

type

1. Q₈×Dic₁₃₄

C₂

semidirect

2. #16863792

C₂

semidirect

4. #16863678

C₂

semidirect

5. Q₈×C₆₇⋊C₈

C₂

semidirect

26. #11761616

Q₈

direct (N × H)

45. Q₈×C₆₇

SD₁₆

semidirect

57. Dic₁₃₄

C₂×Q₈

semidirect

59. C₆₇⋊C₈

C₂×Q₈

semidirect

76. C₆₇

Q₈×SD₁₆

semidirect

124. Q₈

#11761616

direct (N × H)

Subgroups — Q₈×#11761616 #22573069

Conjugacy classes of subgroups

Complements

Subgroups — Q8×#11761616 #22573069

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×#11761616 #22573069