Group #22570954 — subgroups

Subgroups — #22570954

Order 8576.

358 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 133 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	#16863953	C₂	1	2	yes	maximal
2	#16863794	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863776	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863747	C₂	1	2	yes	maximal, complemented
5	#16863654	C₂	1	4	yes	maximal
6	#16863504	C₂	1	4	yes	maximal, complemented
7	C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₆₈	C₂²	1	4	yes	—
9	#14524043	C₂²	1	2	yes	—
10	#14524043	C₂²	1	2	yes	—
11	#14524035	—	2	2	no	—
12	#14524022	—	2	2	no	—
13	#14524022	C₂²	1	4	yes	—
14	C₄×Dic₁₃₄	—	2	2	no	—
15	C₄×Dic₁₃₄	C₂²	1	4	yes	—
16	#14524002	—	2	2	no	—
17	#14523975	C₂²	1	4	yes	—
18	#14523968	C₂²	1	8	yes	—
19	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	#97×C₆₇	C₂²	1	2	yes	—
22	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₃₄	—	2	2	no	—
25	C₂×Dic₁₃₄	C₂³	1	4	yes	—
26	Dic₂₆₈	—	2	8	no	—
27	C₄⋊Dic₆₇	—	2	2	no	—
28	C₄⋊Dic₆₇	—	2	2	no	—
29	C₄⋊Dic₆₇	—	2	2	no	—
30	C₄⋊Dic₆₇	C₂³	1	4	yes	—
31	Dic₆₇⋊C₄	—	2	4	no	—
32	Dic₆₇⋊C₄	—	2	4	no	—
33	C₂²⋊Dic₆₇	—	2	4	no	—
34	C₄×Dic₆₇	—	2	4	no	—
35	C₂²×C₄×C₆₇	D₄	1	2	yes	—
36	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	M₁₆×C₆₇	—	2	4	no	—
38	C₂×C₈×C₆₇	C₂³	1	2	yes	—
39	C₂×C₈×C₆₇	C₂³	1	2	yes	—
40	C₄²×C₆₇	D₄	1	2	yes	—
41	C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₆₇	—	2	4	no	—
45	C₂×C₄×C₆₇	—	2	2	no	—
46	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₆₇	—	2	4	no	—
51	C₂×Dic₆₇	—	2	4	no	—
52	Dic₁₃₄	—	2	8	no	—
53	Dic₁₃₄	—	4	4	no	—
54	C₈×C₆₇	—	2	2	no	—
55	C₈×C₆₇	#41	1	4	yes	—
56	C₂²×C₆₇	—	2	1	no	—
57	C₂²×C₆₇	—	2	2	no	—
58	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₆₇	—	2	1	no	—
60	C₄×C₆₇	—	2	2	no	—
61	C₄×C₆₇	#137	1	4	yes	—
62	C₄×C₆₇	C₂×#41	1	2	yes	—
63	Dic₆₇	—	4	4	no	—
64	Dic₆₇	—	4	4	no	—
65	C₂×C₆₇	—	4	1	no	—
66	C₂×C₆₇	#532	1	1	yes	characteristic
67	C₂×C₆₇	C₂×#137	1	2	yes	—
68	#3185	—	67	1	no	maximal, Sylow
69	C₆₇	#3185	1	1	yes	characteristic, complemented, Sylow
70	#533	—	67	2	no	—
71	#498	—	67	1	no	—
72	#494	—	67	1	no	—
73	#487	—	67	2	no	—
74	#470	—	67	4	no	—
75	#441	—	67	4	no	—
76	C₄×M₁₆	D₆₇	1	1	yes	characteristic
77	C₂×Q₁₆	—	67	4	no	—
78	C₂×M₁₆	C₂×D₆₇	1	2	yes	—
79	C₄⋊Q₈	—	67	2	no	—
80	C₄⋊Q₈	—	67	2	no	—
81	#125	—	134	2	no	—
82	C₂²⋊Q₈	—	67	4	no	—
83	C₂²⋊Q₈	—	134	2	no	—
84	C₄×Q₈	—	67	4	no	—
85	C₄×Q₈	—	134	2	no	—
86	#117	—	134	2	no	—
87	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
88	#106	—	67	4	no	—
89	#103	—	67	8	no	—
90	#97	C₂×D₆₇	1	2	yes	—
91	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
92	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
93	C₂×Q₈	—	67	4	no	—
94	C₂×Q₈	—	134	2	no	—
95	C₂²×C₄	D₂₆₈	1	2	yes	—
96	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
97	Q₁₆	—	134	8	no	—
98	M₁₆	—	2	4	no	—
99	C₂×C₈	C₂²×D₆₇	1	2	yes	—
100	C₂×C₈	C₂²×D₆₇	1	2	yes	—
101	C₄⋊C₄	—	67	4	no	—
102	C₄⋊C₄	—	134	2	no	—
103	C₄⋊C₄	—	134	2	no	—
104	C₄⋊C₄	—	134	4	no	—
105	C₄⋊C₄	—	134	4	no	—
106	C₄⋊C₄	—	134	2	no	—
107	C₂²⋊C₄	—	134	4	no	—
108	C₄²	—	134	4	no	—
109	C₄²	D₂₆₈	1	2	yes	—
110	C₄²	C₂²×D₆₇	1	1	yes	characteristic
111	C₄²	C₂²×D₆₇	1	1	yes	characteristic
112	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
113	Q₈	—	268	4	no	—
114	Q₈	—	134	8	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	134	4	no	—
118	C₂×C₄	—	134	4	no	—
119	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
120	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
121	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
122	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	#11761628	1	4	yes	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	1	no	—
127	C₂²	C₂²×D₂₆₈	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	2	1	no	—
130	C₄	—	268	4	no	—
131	C₄	—	268	4	no	—
132	C₄	#14524076	1	4	yes	—
133	C₄	C₂×#11761628	1	2	yes	—
134	C₂	—	4	1	no	—
135	C₂	#16863946	1	1	yes	characteristic
136	C₂	C₂×#14524076	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863953

C₂

yes

maximal

#16863794

C₂

yes

maximal, characteristic, complemented

#16863776

C₂

yes

maximal, characteristic, complemented

#16863747

C₂

yes

maximal, complemented

#16863654

C₂

yes

maximal

#16863504

C₂

yes

maximal, complemented

C₄×M₁₆×C₆₇

C₂

yes

maximal, characteristic

C₂×Dic₂₆₈

C₂²

yes

—

#14524043

C₂²

yes

—

#14524043

C₂²

yes

—

#14524035

—

#14524022

—

#14524022

C₂²

yes

—

C₄×Dic₁₃₄

—

C₄×Dic₁₃₄

C₂²

yes

—

#14524002

—

#14523975

C₂²

yes

—

#14523968

C₂²

yes

—

C₂×M₁₆×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

#97×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂³

yes

—

Dic₂₆₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

—

C₂×C₈×C₆₇

C₂³

yes

—

C₄²×C₆₇

D₄

yes

—

C₄²×C₆₇

C₂³

yes

characteristic

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

#41

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#137

yes

—

C₄×C₆₇

C₂×#41

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#532

yes

characteristic

C₂×C₆₇

C₂×#137

yes

—

#3185

—

maximal, Sylow

C₆₇

#3185

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

C₄×M₁₆

D₆₇

yes

characteristic

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

134

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

134

—

C₄×Q₈

—

C₄×Q₈

—

134

—

#117

—

134

—

C₂×C₄²

C₂×D₆₇

yes

—

—

C₂×D₆₇

yes

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂²×C₄

D₂₆₈

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

Q₁₆

—

134

—

M₁₆

—

C₂×C₈

C₂²×D₆₇

yes

—

100

C₂×C₈

C₂²×D₆₇

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

134

—

103

C₄⋊C₄

—

134

—

104

C₄⋊C₄

—

134

—

105

C₄⋊C₄

—

134

—

106

C₄⋊C₄

—

134

—

107

C₂²⋊C₄

—

134

—

108

C₄²

—

134

—

109

C₄²

D₂₆₈

yes

—

110

C₄²

C₂²×D₆₇

yes

characteristic

111

C₄²

C₂²×D₆₇

yes

characteristic

112

C₂³

C₂×D₂₆₈

yes

characteristic

113

Q₈

—

268

—

114

Q₈

—

134

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

134

—

118

C₂×C₄

—

134

—

119

C₂×C₄

C₂×D₂₆₈

yes

—

120

C₂×C₄

C₂×D₂₆₈

yes

characteristic

121

C₂×C₄

C₂×D₂₆₈

yes

—

122

C₂×C₄

C₂³×D₆₇

yes

characteristic

123

C₈

—

124

C₈

#11761628

yes

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂²×D₂₆₈

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

268

—

131

C₄

—

268

—

132

C₄

#14524076

yes

—

133

C₄

C₂×#11761628

yes

—

134

C₂

—

135

C₂

#16863946

yes

characteristic

136

C₂

C₂×#14524076

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₄×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	C₄×M₁₆	D₆₇
hypocenter	C₆₇	#3185
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂²×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₄×M₁₆×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

hypercenter

C₄×M₁₆

D₆₇

hypocenter

C₆₇

#3185

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #16863794	C₂	semidirect
3. #16863776	C₂	semidirect
4. #16863747	C₂	semidirect
6. #16863504	C₂	semidirect
69. C₆₇	#3185	semidirect

H ≅ G/N

type

2. #16863794

C₂

semidirect

3. #16863776

C₂

semidirect

4. #16863747

C₂

semidirect

6. #16863504

C₂

semidirect

69. C₆₇

#3185

semidirect

Subgroups — #22570954

Conjugacy classes of subgroups

Complements