Group #22558893 — subgroups

Subgroups — #22558893

Order 8576.

240 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 93 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	#16863420	C₂	1	1	yes	maximal, characteristic
3	#16863143	C₂	1	4	yes	maximal, complemented
4	#431×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₃₄	—	2	1	no	—
6	C₂×#11761612	—	2	1	no	—
7	#14524043	—	2	1	no	—
8	#14524043	—	2	1	no	—
9	#14524043	C₄	1	2	yes	—
10	#14524043	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₆₇	—	2	1	no	—
12	C₂×C₄⋊Dic₆₇	C₄	1	2	yes	—
13	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
14	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
17	C₂²⋊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	—	2	2	no	—
23	#11761612	—	4	2	no	—
24	C₂×C₆₇⋊C₈	—	2	2	no	—
25	C₄⋊Dic₆₇	—	2	2	no	—
26	C₄⋊Dic₆₇	—	2	2	no	—
27	C₄⋊Dic₆₇	—	2	2	no	—
28	C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
32	C₂×C₈×C₆₇	—	2	2	no	—
33	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₆₇	D₄	1	1	yes	characteristic
35	C₄²×C₆₇	D₄	1	1	yes	characteristic
36	C₄²×C₆₇	C₂³	1	1	yes	characteristic
37	C₂³×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₆₇	—	2	2	no	—
39	C₂×C₄×C₆₇	—	2	2	no	—
40	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₆₇	SD₁₆	1	2	yes	—
44	C₂×C₄×C₆₇	Q₁₆	1	2	yes	—
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×Dic₆₇	—	2	4	no	—
49	C₂×Dic₆₇	—	4	4	no	—
50	Dic₁₃₄	—	4	4	no	—
51	Dic₁₃₄	—	4	4	no	—
52	C₈×C₆₇	—	4	2	no	—
53	C₆₇⋊C₈	—	4	2	no	—
54	C₂²×C₆₇	—	2	1	no	—
55	C₂²×C₆₇	—	2	1	no	—
56	C₂²×C₆₇	#103	1	2	yes	—
57	C₂²×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₆₇	—	2	1	no	—
59	C₄×C₆₇	—	2	1	no	—
60	C₄×C₆₇	#101	1	2	yes	—
61	C₄×C₆₇	#103	1	2	yes	—
62	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
64	C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
65	Dic₆₇	—	4	4	no	—
66	C₂×C₆₇	—	2	2	no	—
67	C₂×C₆₇	C₂×#101	1	1	yes	characteristic
68	C₂×C₆₇	C₂×#103	1	1	yes	characteristic
69	C₂×C₆₇	#418	1	1	yes	characteristic
70	#1549	—	67	1	no	maximal, Sylow
71	C₆₇	#1549	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₄⋊Q₈	—	67	1	no	—
73	#431	D₆₇	1	1	yes	characteristic
74	#423	—	67	1	no	—
75	#332	—	67	4	no	—
76	C₂²×Q₈	—	134	1	no	—
77	C₂×M₁₆	—	134	1	no	—
78	C₄⋊Q₈	—	67	2	no	—
79	C₄⋊Q₈	—	67	2	no	—
80	C₄⋊Q₈	—	134	1	no	—
81	C₄⋊Q₈	—	134	1	no	—
82	C₂×C₄⋊C₄	—	67	2	no	—
83	C₂×C₄⋊C₄	—	134	1	no	—
84	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
85	C₄⋊C₈	—	67	2	no	—
86	C₄⋊C₈	—	134	1	no	—
87	C₄⋊C₈	C₂×D₆₇	1	2	yes	—
88	C₂²⋊C₈	—	2	1	no	—
89	C₄×C₈	—	2	1	no	—
90	C₂×Q₈	—	268	4	no	—
91	C₂×Q₈	—	134	2	no	—
92	C₂×Q₈	—	134	2	no	—
93	C₂²×C₄	—	134	2	no	—
94	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
95	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
97	M₁₆	—	268	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	134	2	no	—
100	C₄⋊C₄	—	67	4	no	—
101	C₄⋊C₄	—	134	2	no	—
102	C₄⋊C₄	—	134	2	no	—
103	C₄⋊C₄	—	134	2	no	—
104	C₄²	C₄×D₆₇	1	1	yes	characteristic
105	C₄²	D₂₆₈	1	1	yes	characteristic
106	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
107	C₄²	C₂²×D₆₇	1	1	yes	characteristic
108	C₂³	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
109	Q₈	—	268	4	no	—
110	Q₈	—	268	4	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	268	4	no	—
114	C₂×C₄	—	134	4	no	—
115	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
116	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
117	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
118	C₂×C₄	#11761617	1	2	yes	—
119	C₂×C₄	Dic₂₆₈	1	2	yes	—
120	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
122	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
123	C₈	—	4	2	no	—
124	C₈	—	268	2	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	#14523968	1	2	yes	—
128	C₂²	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	268	4	no	—
132	C₄	#14523961	1	2	yes	—
133	C₄	#14523968	1	2	yes	—
134	C₄	C₂×#11761617	1	1	yes	characteristic
135	C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
136	C₄	#14524068	1	2	yes	—
137	C₂	—	2	2	no	—
138	C₂	C₂×#14523961	1	1	yes	characteristic
139	C₂	C₂×#14523968	1	1	yes	characteristic
140	C₂	#16863400	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524043

C₂

yes

maximal, characteristic

#16863420

C₂

yes

maximal, characteristic

#16863143

C₂

yes

maximal, complemented

#431×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

—

—

—

—

C₄

yes

—

#14524043

C₂²

yes

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₄

yes

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

C₂³

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

SD₁₆

yes

—

C₂×C₄×C₆₇

Q₁₆

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#103

yes

—

C₂²×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#101

yes

—

C₄×C₆₇

#103

yes

—

C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₄×C₆₇

C₂×Q₁₆

yes

characteristic

C₄×C₆₇

AGL(1,ℤ/8)

yes

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#431

D₆₇

yes

characteristic

#423

—

#332

—

C₂²×Q₈

—

134

—

C₂×M₁₆

—

134

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

C₄⋊Q₈

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₄⋊C₈

C₂×D₆₇

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×Q₈

—

268

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

C₄×D₆₇

yes

characteristic

C₂²×C₄

D₂₆₈

yes

characteristic

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

M₁₆

—

268

—

C₂×C₈

—

C₂×C₈

—

134

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

134

—

102

C₄⋊C₄

—

134

—

103

C₄⋊C₄

—

134

—

104

C₄²

C₄×D₆₇

yes

characteristic

105

C₄²

D₂₆₈

yes

characteristic

106

C₄²

C₂²⋊D₆₇

yes

characteristic

107

C₄²

C₂²×D₆₇

yes

characteristic

108

C₂³

(C₂×D₆₇)⋊C₄

yes

characteristic

109

Q₈

—

268

—

110

Q₈

—

268

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

268

—

114

C₂×C₄

—

134

—

115

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

116

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

117

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

118

C₂×C₄

#11761617

yes

—

119

C₂×C₄

Dic₂₆₈

yes

—

120

C₂×C₄

C₂×C₄×D₆₇

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

122

C₂×C₄

C₂×D₂₆₈

yes

characteristic

123

C₈

—

124

C₈

—

268

—

125

C₂²

—

126

C₂²

—

127

C₂²

#14523968

yes

—

128

C₂²

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

268

—

132

C₄

#14523961

yes

—

133

C₄

#14523968

yes

—

134

C₄

C₂×#11761617

yes

characteristic

135

C₄

C₂×Dic₂₆₈

yes

characteristic

136

C₄

#14524068

yes

—

137

C₂

—

138

C₂

C₂×#14523961

yes

characteristic

139

C₂

C₂×#14523968

yes

characteristic

140

C₂

#16863400

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₆₇)⋊C₄
commutator	C₂×C₄×C₆₇	C₂²×C₄
Frattini	C₄²	C₂²×D₆₇
Fitting	#431×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₄
hypercenter	#431	D₆₇
hypocenter	C₆₇	#1549
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂²

C₂×(C₂×D₆₇)⋊C₄

commutator

C₂×C₄×C₆₇

C₂²×C₄

Frattini

C₄²

C₂²×D₆₇

Fitting

#431×C₆₇

C₂

socle

C₂²×C₆₇

C₂×C₂²⋊C₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #16863143	C₂	semidirect
71. C₆₇	#1549	semidirect

H ≅ G/N

type

3. #16863143

C₂

semidirect

71. C₆₇

#1549

semidirect

Subgroups — #22558893

Conjugacy classes of subgroups

Complements