Group #22573353 — subgroups

Subgroups — #22573353

Order 8576.

360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 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	#16864201	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863837	C₂	1	3	yes	maximal, complemented
3	#16863831	C₂	1	3	yes	maximal, complemented
4	#16863826	C₂	1	3	yes	maximal, complemented
5	C₂×#14523986	C₂	1	3	yes	maximal, complemented
6	#16863813	C₂	1	1	yes	maximal, characteristic, complemented
7	#578×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14524089	—	2	1	no	—
9	#14524071	—	2	3	no	—
10	#14524065	C₂²	1	3	yes	—
11	C₂×D₅₃₆	C₂²	1	3	yes	—
12	#14523989	C₂²	1	3	yes	complemented
13	#14523987	C₂²	1	1	yes	characteristic, complemented
14	#14523986	C₂²	1	3	yes	complemented
15	#14523986	C₂²	1	6	yes	complemented
16	#14523984	C₂²	1	3	yes	complemented
17	#14523983	C₂²	1	3	yes	complemented
18	C₂×C₆₇⋊C₁₆	C₂²	1	3	yes	complemented
19	2₋¹⁺⁴×C₆₇	—	2	1	no	—
20	#137×C₆₇	—	2	3	no	—
21	#135×C₆₇	C₂²	1	3	yes	complemented
22	C₂×Q₁₆×C₆₇	C₂²	1	3	yes	complemented
23	#131×C₆₇	C₂²	1	1	yes	characteristic
24	#11761629	—	4	1	no	—
25	#11761628	—	2	3	no	—
26	C₂×D₂₆₈	—	2	3	no	—
27	D₄×D₆₇	—	4	3	no	—
28	Dic₂₆₈	C₂³	1	1	yes	characteristic
29	#11761617	—	2	3	no	—
30	D₅₃₆	—	2	3	no	—
31	D₅₃₆	C₂³	1	3	yes	—
32	C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
33	C₆₇⋊C₁₆	C₂³	1	3	yes	—
34	#41×C₆₇	—	2	3	no	—
35	#41×C₆₇	—	4	3	no	—
36	#41×C₆₇	D₄	1	1	yes	characteristic
37	C₂×Q₈×C₆₇	—	2	3	no	—
38	C₂×Q₈×C₆₇	—	4	1	no	—
39	Q₁₆×C₆₇	—	2	3	no	—
40	Q₁₆×C₆₇	C₂³	1	3	yes	—
41	SD₁₆×C₆₇	—	2	3	no	—
42	D₈×C₆₇	C₂³	1	1	yes	characteristic
43	M₁₆×C₆₇	D₄	1	3	yes	—
44	C₂×C₈×C₆₇	C₂³	1	3	yes	—
45	C₂²×D₆₇	—	4	3	no	—
46	Q₈×C₆₇	—	2	3	no	—
47	Q₈×C₆₇	—	4	3	no	—
48	Q₈×C₆₇	D₈	1	1	yes	characteristic
49	D₄×C₆₇	—	2	1	no	—
50	D₄×C₆₇	—	4	3	no	—
51	D₄×C₆₇	D₈	1	3	yes	—
52	C₂×C₄×C₆₇	—	4	3	no	—
53	C₂×C₄×C₆₇	—	4	3	no	—
54	C₂×C₄×C₆₇	C₂×D₄	1	3	yes	—
55	C₂²⋊D₆₇	—	4	3	no	—
56	D₂₆₈	—	2	3	no	—
57	D₂₆₈	—	4	3	no	—
58	C₄×D₆₇	—	4	3	no	—
59	Dic₁₃₄	—	2	1	no	—
60	C₈×C₆₇	C₂×D₄	1	3	yes	—
61	C₈×C₆₇	C₂⁴	1	1	yes	characteristic
62	C₂²×C₆₇	—	4	1	no	—
63	C₂²×C₆₇	C₂×D₈	1	3	yes	—
64	C₂×D₆₇	—	4	3	no	—
65	C₂×D₆₇	—	16	3	no	—
66	C₄×C₆₇	—	4	3	no	—
67	C₄×C₆₇	C₂×D₈	1	3	yes	—
68	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
69	Dic₆₇	—	4	1	no	—
70	C₂×C₆₇	—	2	3	no	—
71	C₂×C₆₇	—	8	1	no	—
72	C₂×C₆₇	C₂²×D₈	1	1	yes	characteristic
73	D₆₇	—	8	3	no	—
74	#3416	—	67	1	no	maximal, Sylow
75	C₆₇	#3416	1	1	yes	characteristic, complemented, Sylow
76	#578	D₆₇	1	1	yes	characteristic, complemented
77	#576	—	67	1	no	—
78	#510	—	67	3	no	—
79	#509	—	67	3	no	—
80	#508	—	67	3	no	—
81	C₂×SD₃₂	—	67	3	no	—
82	#504	—	67	1	no	—
83	2₋¹⁺⁴	—	2	1	no	—
84	H(5,𝔽₂)	—	134	1	no	—
85	#137	—	2	3	no	—
86	AGL(1,ℤ/8)	—	134	3	no	—
87	#135	—	67	3	no	—
88	#135	C₂×D₆₇	1	3	yes	complemented
89	C₂×Q₁₆	C₂×D₆₇	1	3	yes	complemented
90	C₂×D₈	—	67	3	no	—
91	#131	C₂×D₆₇	1	1	yes	characteristic
92	Q₃₂	—	67	3	no	—
93	SD₃₂	—	67	3	no	—
94	SD₃₂	—	67	6	no	—
95	SD₃₂	—	67	1	no	—
96	D₁₆	—	67	3	no	—
97	M₃₂	—	67	3	no	—
98	C₂×C₁₆	—	67	3	no	—
99	#41	—	2	3	no	—
100	#41	—	4	3	no	—
101	#41	—	268	1	no	—
102	#41	—	134	3	no	—
103	#41	C₂²⋊D₆₇	1	1	yes	characteristic
104	C₂×Q₈	—	2	3	no	—
105	C₂×Q₈	—	4	1	no	—
106	C₂×D₄	—	268	3	no	—
107	C₂×D₄	—	134	3	no	—
108	Q₁₆	—	2	3	no	—
109	Q₁₆	—	67	1	no	—
110	Q₁₆	C₂²×D₆₇	1	3	yes	—
111	SD₁₆	—	2	3	no	—
112	SD₁₆	—	134	3	no	—
113	D₈	—	67	3	no	—
114	D₈	—	134	3	no	—
115	D₈	C₂²×D₆₇	1	1	yes	characteristic
116	M₁₆	C₂²⋊D₆₇	1	3	yes	—
117	C₂×C₈	C₂²×D₆₇	1	3	yes	—
118	C₁₆	—	67	1	no	—
119	C₁₆	—	67	3	no	—
120	C₂³	—	268	3	no	—
121	Q₈	—	2	3	no	—
122	Q₈	—	4	3	no	—
123	Q₈	—	134	1	no	—
124	Q₈	#11761613	1	1	yes	characteristic
125	D₄	—	2	1	no	—
126	D₄	—	4	3	no	—
127	D₄	—	268	3	no	—
128	D₄	—	268	3	no	—
129	D₄	—	134	3	no	—
130	D₄	#11761613	1	3	yes	—
131	C₂×C₄	—	4	3	no	—
132	C₂×C₄	—	4	3	no	—
133	C₂×C₄	—	268	3	no	—
134	C₂×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
135	C₈	C₂×C₂²⋊D₆₇	1	3	yes	—
136	C₈	C₂³×D₆₇	1	1	yes	characteristic
137	C₂²	—	4	1	no	—
138	C₂²	—	268	3	no	—
139	C₂²	—	1072	3	no	—
140	C₂²	C₂×#11761613	1	3	yes	—
141	C₄	—	4	3	no	—
142	C₄	—	268	1	no	—
143	C₄	C₂×#11761613	1	3	yes	—
144	C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
145	C₂	—	2	3	no	—
146	C₂	—	8	1	no	—
147	C₂	—	536	3	no	—
148	C₂	C₂²×#11761613	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16864201

C₂

yes

maximal, characteristic, complemented

#16863837

C₂

yes

maximal, complemented

#16863831

C₂

yes

maximal, complemented

#16863826

C₂

yes

maximal, complemented

C₂×#14523986

C₂

yes

maximal, complemented

#16863813

C₂

yes

maximal, characteristic, complemented

#578×C₆₇

C₂

yes

maximal, characteristic, complemented

#14524089

—

#14524071

—

#14524065

C₂²

yes

—

C₂×D₅₃₆

C₂²

yes

—

#14523989

C₂²

yes

complemented

#14523987

C₂²

yes

characteristic, complemented

#14523986

C₂²

yes

complemented

#14523986

C₂²

yes

complemented

#14523984

C₂²

yes

complemented

#14523983

C₂²

yes

complemented

C₂×C₆₇⋊C₁₆

C₂²

yes

complemented

2₋¹⁺⁴×C₆₇

—

#137×C₆₇

—

#135×C₆₇

C₂²

yes

complemented

C₂×Q₁₆×C₆₇

C₂²

yes

complemented

#131×C₆₇

C₂²

yes

characteristic

#11761629

—

#11761628

—

C₂×D₂₆₈

—

D₄×D₆₇

—

Dic₂₆₈

C₂³

yes

characteristic

#11761617

—

D₅₃₆

—

D₅₃₆

C₂³

yes

—

C₆₇⋊C₁₆

C₂³

yes

characteristic

C₆₇⋊C₁₆

C₂³

yes

—

#41×C₆₇

—

#41×C₆₇

—

#41×C₆₇

D₄

yes

characteristic

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

C₂³

yes

—

SD₁₆×C₆₇

—

D₈×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

C₂³

yes

—

C₂²×D₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

D₈

yes

characteristic

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

D₈

yes

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂²⋊D₆₇

—

D₂₆₈

—

D₂₆₈

—

C₄×D₆₇

—

Dic₁₃₄

—

C₈×C₆₇

C₂×D₄

yes

—

C₈×C₆₇

C₂⁴

yes

characteristic

C₂²×C₆₇

—

C₂²×C₆₇

C₂×D₈

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×D₈

yes

—

C₄×C₆₇

C₂²×D₄

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂²×D₈

yes

characteristic

D₆₇

—

#3416

—

maximal, Sylow

C₆₇

#3416

yes

characteristic, complemented, Sylow

#578

D₆₇

yes

characteristic, complemented

—

—

—

—

C₂×SD₃₂

—

#504

—

2₋¹⁺⁴

—

H(5,𝔽₂)

—

134

—

#137

—

AGL(1,ℤ/8)

—

134

—

#135

—

#135

C₂×D₆₇

yes

complemented

C₂×Q₁₆

C₂×D₆₇

yes

complemented

C₂×D₈

—

#131

C₂×D₆₇

yes

characteristic

Q₃₂

—

SD₃₂

—

SD₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

C₂×C₁₆

—

#41

—

100

#41

—

101

#41

—

268

—

102

#41

—

134

—

103

#41

C₂²⋊D₆₇

yes

characteristic

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×D₄

—

268

—

107

C₂×D₄

—

134

—

108

Q₁₆

—

109

Q₁₆

—

110

Q₁₆

C₂²×D₆₇

yes

—

111

SD₁₆

—

112

SD₁₆

—

134

—

113

D₈

—

114

D₈

—

134

—

115

D₈

C₂²×D₆₇

yes

characteristic

116

M₁₆

C₂²⋊D₆₇

yes

—

117

C₂×C₈

C₂²×D₆₇

yes

—

118

C₁₆

—

119

C₁₆

—

120

C₂³

—

268

—

121

Q₈

—

122

Q₈

—

123

Q₈

—

134

—

124

Q₈

#11761613

yes

characteristic

125

D₄

—

126

D₄

—

127

D₄

—

268

—

128

D₄

—

268

—

129

D₄

—

134

—

130

D₄

#11761613

yes

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

268

—

134

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

135

C₈

C₂×C₂²⋊D₆₇

yes

—

136

C₈

C₂³×D₆₇

yes

characteristic

137

C₂²

—

138

C₂²

—

268

—

139

C₂²

—

1072

—

140

C₂²

C₂×#11761613

yes

—

141

C₄

—

142

C₄

—

268

—

143

C₄

C₂×#11761613

yes

—

144

C₄

C₂²×C₂²⋊D₆₇

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

536

—

148

C₂

C₂²×#11761613

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×#11761613
commutator	C₈×C₆₇	C₂⁴
Frattini	C₈	C₂³×D₆₇
Fitting	#578×C₆₇	C₂
socle	C₂×C₆₇	C₂²×D₈
hypercenter	#578	D₆₇
hypocenter	C₆₇	#3416
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #578

subgroup

G/H

center

C₂

C₂²×#11761613

commutator

C₈×C₆₇

C₂⁴

Frattini

C₈

C₂³×D₆₇

Fitting

#578×C₆₇

C₂

socle

C₂×C₆₇

C₂²×D₈

D₆₇

C₆₇

G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #578

N	H ≅ G/N	type
1. #16864201	C₂	semidirect
2. #16863837	C₂	semidirect
3. #16863831	C₂	semidirect
4. #16863826	C₂	semidirect
5. C₂×#14523986	C₂	semidirect
6. #16863813	C₂	semidirect
7. #578×C₆₇	C₂	semidirect
12. #14523989	C₂²	semidirect
13. #14523987	C₂²	semidirect
14. #14523986	C₂²	semidirect
15. #14523986	C₂²	semidirect
16. #14523984	C₂²	semidirect
17. #14523983	C₂²	semidirect
18. C₂×C₆₇⋊C₁₆	C₂²	semidirect
21. #135×C₆₇	C₂²	semidirect
22. C₂×Q₁₆×C₆₇	C₂²	semidirect
75. C₆₇	#3416	semidirect
76. #578	D₆₇	semidirect
88. #135	C₂×D₆₇	semidirect
89. C₂×Q₁₆	C₂×D₆₇	semidirect

H ≅ G/N

type

1. #16864201

C₂

semidirect

2. #16863837

C₂

semidirect

3. #16863831

C₂

semidirect

4. #16863826

C₂

semidirect

5. C₂×#14523986

C₂

semidirect

6. #16863813

C₂

semidirect

7. #578×C₆₇

C₂

semidirect

12. #14523989

C₂²

semidirect

13. #14523987

C₂²

semidirect

14. #14523986

C₂²

semidirect

15. #14523986

C₂²

semidirect

16. #14523984

C₂²

semidirect

17. #14523983

C₂²

semidirect

18. C₂×C₆₇⋊C₁₆

C₂²

semidirect

21. #135×C₆₇

C₂²

semidirect

22. C₂×Q₁₆×C₆₇

C₂²

semidirect

75. C₆₇

#3416

semidirect

76. #578

D₆₇

semidirect

88. #135

C₂×D₆₇

semidirect

89. C₂×Q₁₆

C₂×D₆₇

semidirect

Subgroups — #22573353

Conjugacy classes of subgroups

Complements