Group #22561197 — subgroups

Subgroups — #22561197

Order 8576.

260 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 97 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×#14524043	C₂	1	1	yes	maximal, characteristic
2	C₄×#11761612	C₂	1	1	yes	maximal, characteristic
3	#16863164	C₂	1	4	yes	maximal
4	#423×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₃₄	C₄	1	2	yes	—
6	C₂×#11761612	C₂²	1	2	yes	—
7	#14524043	—	2	1	no	—
8	#14524043	—	2	1	no	—
9	#14524043	—	2	1	no	—
10	#14524043	—	2	1	no	—
11	C₂×C₄⋊Dic₆₇	C₄	1	2	yes	—
12	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
13	#14523954	—	2	1	no	—
14	C₄×C₆₇⋊C₈	—	2	1	no	—
15	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₆₇	—	2	1	no	—
18	C₄⋊C₈×C₆₇	—	2	1	no	—
19	C₂×Dic₁₃₄	—	2	2	no	—
20	C₂×Dic₁₃₄	—	2	2	no	—
21	C₂×Dic₁₃₄	—	4	4	no	—
22	C₂²×Dic₆₇	C₂×C₄	1	4	yes	—
23	#11761612	—	2	2	no	—
24	#11761612	Q₈	1	4	yes	complemented
25	C₂×C₆₇⋊C₈	—	2	2	no	—
26	C₄⋊Dic₆₇	—	2	2	no	—
27	C₄⋊Dic₆₇	—	2	2	no	—
28	C₄⋊Dic₆₇	—	2	2	no	—
29	C₄⋊Dic₆₇	—	2	2	no	—
30	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	M₁₆×C₆₇	—	2	4	no	—
34	C₂×C₈×C₆₇	—	2	2	no	—
35	C₄²×C₆₇	D₄	1	1	yes	characteristic
36	C₄²×C₆₇	D₄	1	1	yes	characteristic
37	C₄²×C₆₇	D₄	1	1	yes	characteristic
38	C₄²×C₆₇	D₄	1	1	yes	characteristic
39	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₆₇	—	2	2	no	—
41	C₂×C₄×C₆₇	—	2	2	no	—
42	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₆₇	#41	1	2	yes	—
50	C₂×Dic₆₇	—	2	4	no	—
51	C₂×Dic₆₇	—	4	4	no	—
52	Dic₁₃₄	—	4	4	no	—
53	Dic₁₃₄	—	4	4	no	—
54	C₈×C₆₇	—	4	2	no	—
55	C₆₇⋊C₈	—	2	4	no	—
56	C₂²×C₆₇	—	2	1	no	—
57	C₂²×C₆₇	—	2	1	no	—
58	C₂²×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₆₇	C₄×Q₈	1	2	yes	—
60	C₄×C₆₇	—	2	1	no	—
61	C₄×C₆₇	—	2	1	no	—
62	C₄×C₆₇	#100	1	2	yes	—
63	C₄×C₆₇	#101	1	2	yes	—
64	C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₆₇	#124	1	1	yes	characteristic
66	C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
67	Dic₆₇	—	4	4	no	—
68	C₂×C₆₇	—	2	2	no	—
69	C₂×C₆₇	#391	1	1	yes	characteristic
70	C₂×C₆₇	C₂×#100	1	1	yes	characteristic
71	C₂×C₆₇	C₂×#101	1	1	yes	characteristic
72	#1997	—	67	1	no	maximal, Sylow
73	C₆₇	#1997	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄⋊Q₈	—	67	1	no	—
75	#423	D₆₇	1	1	yes	characteristic
76	C₄×M₁₆	—	67	1	no	—
77	#343	—	67	4	no	—
78	C₂²×Q₈	—	67	2	no	—
79	C₂×M₁₆	—	67	2	no	—
80	C₂×M₁₆	C₂×D₆₇	1	2	yes	—
81	C₄⋊Q₈	—	134	1	no	—
82	C₄⋊Q₈	—	134	1	no	—
83	C₄⋊Q₈	—	134	1	no	—
84	C₄⋊Q₈	—	134	1	no	—
85	C₂×C₄⋊C₄	—	67	2	no	—
86	C₂×C₄⋊C₄	—	67	2	no	—
87	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
88	C₄⋊C₈	—	2	1	no	—
89	C₄⋊C₈	—	2	1	no	—
90	#97	—	134	1	no	—
91	C₄×C₈	—	134	1	no	—
92	C₂×Q₈	—	268	4	no	—
93	C₂×Q₈	—	134	2	no	—
94	C₂×Q₈	—	134	2	no	—
95	C₂²×C₄	—	67	4	no	—
96	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
97	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
99	M₁₆	—	2	4	no	—
100	M₁₆	—	67	4	no	—
101	M₁₆	—	134	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	134	2	no	—
104	C₄⋊C₄	—	134	2	no	—
105	C₄⋊C₄	—	134	2	no	—
106	C₄⋊C₄	—	134	2	no	—
107	C₄⋊C₄	—	134	2	no	—
108	C₄²	D₂₆₈	1	1	yes	characteristic
109	C₄²	D₂₆₈	1	1	yes	characteristic
110	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
111	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
112	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
113	Q₈	—	268	4	no	—
114	Q₈	—	268	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	268	4	no	—
118	C₂×C₄	—	134	4	no	—
119	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
120	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
121	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
122	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
124	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
125	C₂×C₄	Q₈×D₆₇	1	2	yes	—
126	C₂×C₄	#11761627	1	2	yes	—
127	C₈	—	4	2	no	—
128	C₈	—	134	4	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
132	C₂²	#14524008	1	2	yes	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	268	4	no	—
136	C₄	#14523959	1	2	yes	—
137	C₄	#14523961	1	2	yes	—
138	C₄	#14524023	1	2	yes	—
139	C₄	#14524030	1	1	yes	characteristic
140	C₄	#14524041	1	1	yes	characteristic
141	C₂	—	2	2	no	—
142	C₂	#16863301	1	1	yes	characteristic
143	C₂	C₂×#14523959	1	1	yes	characteristic
144	C₂	C₂×#14523961	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524043

C₂

yes

maximal, characteristic

C₄×#11761612

C₂

yes

maximal, characteristic

#16863164

C₂

yes

maximal

#423×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₄

yes

—

C₂×#11761612

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Dic₆₇

C₄

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

#14523954

—

C₄×C₆₇⋊C₈

—

C₂×M₁₆×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

C₂×C₄

yes

—

#11761612

—

#11761612

Q₈

yes

complemented

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₂³×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₇

C₄×Q₈

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#100

yes

—

C₄×C₆₇

#101

yes

—

C₄×C₆₇

C₂²⋊Q₈

yes

—

C₄×C₆₇

#124

yes

characteristic

C₄×C₆₇

C₄⋊Q₈

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#423

D₆₇

yes

characteristic

C₄×M₁₆

—

#343

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

—

C₄⋊Q₈

—

134

—

C₄⋊Q₈

—

134

—

C₄⋊Q₈

—

134

—

C₄⋊Q₈

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

134

—

C₄×C₈

—

134

—

C₂×Q₈

—

268

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂²×C₄

—

C₂²×C₄

C₄×D₆₇

yes

characteristic

C₂²×C₄

C₄×D₆₇

yes

characteristic

C₂²×C₄

C₂²×D₆₇

yes

characteristic

M₁₆

—

100

M₁₆

—

101

M₁₆

—

134

—

102

C₂×C₈

—

103

C₂×C₈

—

134

—

104

C₄⋊C₄

—

134

—

105

C₄⋊C₄

—

134

—

106

C₄⋊C₄

—

134

—

107

C₄⋊C₄

—

134

—

108

C₄²

D₂₆₈

yes

characteristic

109

C₄²

D₂₆₈

yes

characteristic

110

C₄²

C₂²⋊D₆₇

yes

characteristic

111

C₄²

C₂²⋊D₆₇

yes

characteristic

112

C₂³

C₂×C₄×D₆₇

yes

characteristic

113

Q₈

—

268

—

114

Q₈

—

268

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

268

—

118

C₂×C₄

—

134

—

119

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

120

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

121

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

122

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

124

C₂×C₄

C₂×D₂₆₈

yes

characteristic

125

C₂×C₄

Q₈×D₆₇

yes

—

126

C₂×C₄

#11761627

yes

—

127

C₈

—

128

C₈

—

134

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

132

C₂²

#14524008

yes

—

133

C₄

—

134

C₄

—

135

C₄

—

268

—

136

C₄

#14523959

yes

—

137

C₄

#14523961

yes

—

138

C₄

#14524023

yes

—

139

C₄

#14524030

yes

characteristic

140

C₄

#14524041

yes

characteristic

141

C₂

—

142

C₂

#16863301

yes

characteristic

143

C₂

C₂×#14523959

yes

characteristic

144

C₂

C₂×#14523961

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₆₇)⋊C₄
commutator	C₂³×C₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#423×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₄
hypercenter	#423	D₆₇
hypocenter	C₆₇	#1997
LCS	G ▷ C₂³×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×(C₂×D₆₇)⋊C₄

commutator

C₂³×C₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#423×C₆₇

C₂

socle

C₂²×C₆₇

C₂×C₂²⋊C₄

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
24. #11761612	Q₈	semidirect
73. C₆₇	#1997	semidirect

H ≅ G/N

type

24. #11761612

Q₈

semidirect

73. C₆₇

#1997

semidirect

Subgroups — #22561197

Conjugacy classes of subgroups

Complements