Group #22568670 — subgroups

Subgroups — #22568670

Order 8576.

800 conjugacy classes of subgroups; only the 289 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 289 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	#16863871	C₂	1	2	yes	maximal, complemented
2	#16863370	C₂	1	4	yes	maximal, complemented
3	C₂×#14523958	C₂	1	8	yes	maximal, complemented
4	C₂×H(5,𝔽₂)×C₆₇	C₂	1	1	yes	maximal, characteristic
5	D₄×Dic₆₇	—	2	8	no	—
6	#14524002	C₂²	1	4	yes	complemented
7	C₂×C₂²⋊Dic₆₇	C₂²	1	8	yes	complemented
8	#14523958	C₂²	1	16	yes	complemented
9	H(5,𝔽₂)×C₆₇	—	2	8	no	—
10	C₂×#41×C₆₇	C₄	1	4	yes	complemented
11	C₂×#41×C₆₇	C₂²	1	2	yes	—
12	C₂²×D₄×C₆₇	C₄	1	4	yes	complemented
13	C₂²×D₄×C₆₇	C₂²	1	4	yes	—
14	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₆₇	—	2	8	no	—
16	C₄⋊Dic₆₇	—	2	4	no	—
17	C₂²⋊Dic₆₇	—	2	8	no	—
18	C₂²⋊Dic₆₇	C₂³	1	8	yes	—
19	C₄×Dic₆₇	—	2	4	no	—
20	C₂⁴×C₆₇	C₂×C₄	1	4	yes	complemented
21	C₂⁴×C₆₇	C₂³	1	2	yes	—
22	#41×C₆₇	—	2	16	no	—
23	#41×C₆₇	—	2	8	no	—
24	C₂×Q₈×C₆₇	C₂×C₄	1	2	yes	complemented
25	C₂×D₄×C₆₇	—	2	16	no	—
26	C₂×D₄×C₆₇	—	2	16	no	—
27	C₂×D₄×C₆₇	C₂×C₄	1	4	yes	—
28	C₂×D₄×C₆₇	C₂×C₄	1	2	yes	complemented
29	C₂×D₄×C₆₇	C₂×C₄	1	8	yes	complemented
30	C₂×D₄×C₆₇	D₄	1	8	yes	—
31	C₂×D₄×C₆₇	C₂³	1	4	yes	—
32	C₂²×C₄×C₆₇	C₂×C₄	1	4	yes	complemented
33	C₂²×C₄×C₆₇	C₂×C₄	1	4	yes	complemented
34	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
35	C₂³×C₆₇	—	2	8	no	—
36	C₂³×C₆₇	—	8	2	no	—
37	C₂³×C₆₇	—	8	4	no	—
38	C₂³×C₆₇	C₂²×C₄	1	4	yes	—
39	C₂³×C₆₇	C₂²×C₄	1	4	yes	—
40	C₂³×C₆₇	C₂×D₄	1	8	yes	—
41	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
42	Q₈×C₆₇	—	2	4	no	—
43	D₄×C₆₇	—	2	16	no	—
44	D₄×C₆₇	—	2	8	no	—
45	D₄×C₆₇	—	2	4	no	—
46	D₄×C₆₇	C₂²⋊C₄	1	16	yes	—
47	C₂×C₄×C₆₇	—	2	8	no	—
48	C₂×C₄×C₆₇	—	2	8	no	—
49	C₂×C₄×C₆₇	C₂²×C₄	1	4	yes	—
50	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₆₇	C₂×D₄	1	4	yes	—
52	C₂×Dic₆₇	—	2	8	no	—
53	C₂×Dic₆₇	—	8	8	no	—
54	C₂²×C₆₇	—	2	4	no	—
55	C₂²×C₆₇	—	4	8	no	—
56	C₂²×C₆₇	—	4	4	no	—
57	C₂²×C₆₇	—	4	1	no	—
58	C₂²×C₆₇	—	4	4	no	—
59	C₂²×C₆₇	—	8	8	no	—
60	C₂²×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
61	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
62	C₂²×C₆₇	C₂²×D₄	1	2	yes	—
63	C₄×C₆₇	—	2	4	no	—
64	C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
65	Dic₆₇	—	4	8	no	—
66	C₂×C₆₇	—	2	2	no	—
67	C₂×C₆₇	—	2	1	no	—
68	C₂×C₆₇	—	2	8	no	—
69	C₂×C₆₇	—	4	4	no	—
70	C₂×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
71	#2883	—	67	1	no	maximal, Sylow
72	C₆₇	#2883	1	1	yes	characteristic, complemented, Sylow
73	C₂×H(5,𝔽₂)	D₆₇	1	1	yes	characteristic
74	#518	—	67	2	no	—
75	#410	—	67	4	no	—
76	C₂×#99	—	67	8	no	—
77	H(5,𝔽₂)	—	2	8	no	—
78	C₂×#41	Dic₆₇	1	4	yes	complemented
79	C₂×#41	C₂×D₆₇	1	2	yes	—
80	C₂²×D₄	Dic₆₇	1	4	yes	complemented
81	C₂²×D₄	C₂×D₆₇	1	4	yes	—
82	C₂²×D₄	C₂×D₆₇	1	1	yes	characteristic
83	C₄×D₄	—	134	8	no	—
84	#117	—	67	4	no	—
85	C₂×C₂²⋊C₄	—	67	8	no	—
86	#99	—	67	16	no	—
87	C₂⁴	C₂×Dic₆₇	1	4	yes	complemented
88	C₂⁴	C₂²×D₆₇	1	2	yes	—
89	#41	—	2	16	no	—
90	#41	—	2	8	no	—
91	C₂×Q₈	C₂×Dic₆₇	1	2	yes	complemented
92	C₂×D₄	—	2	16	no	—
93	C₂×D₄	—	2	16	no	—
94	C₂×D₄	C₂×Dic₆₇	1	4	yes	—
95	C₂×D₄	C₂×Dic₆₇	1	2	yes	complemented
96	C₂×D₄	C₂×Dic₆₇	1	8	yes	complemented
97	C₂×D₄	C₂²⋊D₆₇	1	8	yes	—
98	C₂×D₄	C₂²×D₆₇	1	4	yes	—
99	C₂²×C₄	—	134	8	no	—
100	C₂²×C₄	C₂×Dic₆₇	1	4	yes	complemented
101	C₂²×C₄	C₂×Dic₆₇	1	4	yes	complemented
102	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
103	C₄⋊C₄	—	134	4	no	—
104	C₂²⋊C₄	—	67	8	no	—
105	C₂²⋊C₄	—	134	8	no	—
106	C₄²	—	134	4	no	—
107	C₂³	—	2	8	no	—
108	C₂³	—	8	2	no	—
109	C₂³	—	8	4	no	—
110	C₂³	C₂²×Dic₆₇	1	4	yes	—
111	C₂³	C₂²×Dic₆₇	1	4	yes	—
112	C₂³	C₂×C₂²⋊D₆₇	1	8	yes	—
113	C₂³	C₂³×D₆₇	1	1	yes	characteristic
114	Q₈	—	2	4	no	—
115	D₄	—	2	16	no	—
116	D₄	—	2	8	no	—
117	D₄	—	2	4	no	—
118	D₄	C₂²⋊Dic₆₇	1	16	yes	—
119	C₂×C₄	—	2	8	no	—
120	C₂×C₄	—	2	8	no	—
121	C₂×C₄	—	536	8	no	—
122	C₂×C₄	—	134	8	no	—
123	C₂×C₄	C₂²×Dic₆₇	1	4	yes	—
124	C₂×C₄	C₂²×Dic₆₇	1	2	yes	—
125	C₂×C₄	C₂×C₂²⋊D₆₇	1	4	yes	—
126	C₂²	—	2	4	no	—
127	C₂²	—	4	8	no	—
128	C₂²	—	4	4	no	—
129	C₂²	—	4	1	no	—
130	C₂²	—	4	4	no	—
131	C₂²	—	8	8	no	—
132	C₂²	C₂×C₂²⋊Dic₆₇	1	8	yes	—
133	C₂²	C₂³×Dic₆₇	1	1	yes	characteristic
134	C₂²	C₂²×C₂²⋊D₆₇	1	2	yes	—
135	C₄	—	2	4	no	—
136	C₄	—	268	8	no	—
137	C₄	C₂×C₂²⋊Dic₆₇	1	4	yes	—
138	C₂	—	2	2	no	—
139	C₂	—	2	1	no	—
140	C₂	—	2	8	no	—
141	C₂	—	4	4	no	—
142	C₂	C₂²×C₂²⋊Dic₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863871

C₂

yes

maximal, complemented

#16863370

C₂

yes

maximal, complemented

C₂×#14523958

C₂

yes

maximal, complemented

C₂×H(5,𝔽₂)×C₆₇

C₂

yes

maximal, characteristic

D₄×Dic₆₇

—

#14524002

C₂²

yes

complemented

C₂×C₂²⋊Dic₆₇

C₂²

yes

complemented

#14523958

C₂²

yes

complemented

H(5,𝔽₂)×C₆₇

—

C₂×#41×C₆₇

C₄

yes

complemented

C₂×#41×C₆₇

C₂²

yes

—

C₂²×D₄×C₆₇

C₄

yes

complemented

C₂²×D₄×C₆₇

C₂²

yes

—

C₂²×D₄×C₆₇

C₂²

yes

characteristic

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

C₂³

yes

—

C₄×Dic₆₇

—

C₂⁴×C₆₇

C₂×C₄

yes

complemented

C₂⁴×C₆₇

C₂³

yes

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

C₂×C₄

yes

complemented

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

C₂×C₄

yes

—

C₂×D₄×C₆₇

C₂×C₄

yes

complemented

C₂×D₄×C₆₇

C₂×C₄

yes

complemented

C₂×D₄×C₆₇

D₄

yes

—

C₂×D₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

complemented

C₂²×C₄×C₆₇

C₂×C₄

yes

complemented

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂²×C₄

yes

—

C₂³×C₆₇

C₂²×C₄

yes

—

C₂³×C₆₇

C₂×D₄

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

C₂²⋊C₄

yes

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₂²⋊C₄

yes

—

C₂²×C₆₇

C₂³×C₄

yes

characteristic

C₂²×C₆₇

C₂²×D₄

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₂²⋊C₄

yes

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂²×C₂²⋊C₄

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×H(5,𝔽₂)

D₆₇

yes

characteristic

#518

—

#410

—

C₂×#99

—

H(5,𝔽₂)

—

C₂×#41

Dic₆₇

yes

complemented

C₂×#41

C₂×D₆₇

yes

—

C₂²×D₄

Dic₆₇

yes

complemented

C₂²×D₄

C₂×D₆₇

yes

—

C₂²×D₄

C₂×D₆₇

yes

characteristic

C₄×D₄

—

134

—

#117

—

C₂×C₂²⋊C₄

—

#99

—

C₂⁴

C₂×Dic₆₇

yes

complemented

C₂⁴

C₂²×D₆₇

yes

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₆₇

yes

complemented

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₆₇

yes

—

C₂×D₄

C₂×Dic₆₇

yes

complemented

C₂×D₄

C₂×Dic₆₇

yes

complemented

C₂×D₄

C₂²⋊D₆₇

yes

—

C₂×D₄

C₂²×D₆₇

yes

—

C₂²×C₄

—

134

—

100

C₂²×C₄

C₂×Dic₆₇

yes

complemented

101

C₂²×C₄

C₂×Dic₆₇

yes

complemented

102

C₂²×C₄

C₂²×D₆₇

yes

characteristic

103

C₄⋊C₄

—

134

—

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

134

—

106

C₄²

—

134

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

C₂²×Dic₆₇

yes

—

111

C₂³

C₂²×Dic₆₇

yes

—

112

C₂³

C₂×C₂²⋊D₆₇

yes

—

113

C₂³

C₂³×D₆₇

yes

characteristic

114

Q₈

—

115

D₄

—

116

D₄

—

117

D₄

—

118

D₄

C₂²⋊Dic₆₇

yes

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

536

—

122

C₂×C₄

—

134

—

123

C₂×C₄

C₂²×Dic₆₇

yes

—

124

C₂×C₄

C₂²×Dic₆₇

yes

—

125

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×C₂²⋊Dic₆₇

yes

—

133

C₂²

C₂³×Dic₆₇

yes

characteristic

134

C₂²

C₂²×C₂²⋊D₆₇

yes

—

135

C₄

—

136

C₄

—

268

—

137

C₄

C₂×C₂²⋊Dic₆₇

yes

—

138

C₂

—

139

C₂

—

140

C₂

—

141

C₂

—

142

C₂

C₂²×C₂²⋊Dic₆₇

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊Dic₆₇
commutator	C₂²×C₆₇	C₂³×C₄
Frattini	C₂³	C₂³×D₆₇
Fitting	C₂×H(5,𝔽₂)×C₆₇	C₂
socle	C₂×C₆₇	C₂²×C₂²⋊C₄
hypercenter	C₂×H(5,𝔽₂)	D₆₇
hypocenter	C₆₇	#2883
LCS	G ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂²×D₄ ◁ C₂×H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊Dic₆₇

commutator

C₂²×C₆₇

C₂³×C₄

Frattini

C₂³

C₂³×D₆₇

Fitting

C₂×H(5,𝔽₂)×C₆₇

C₂

socle

C₂×C₆₇

C₂²×C₂²⋊C₄

hypercenter

C₂×H(5,𝔽₂)

D₆₇

hypocenter

C₆₇

#2883

LCS

G ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ C₂²×D₄ ◁ C₂×H(5,𝔽₂)

N	H ≅ G/N	type
1. #16863871	C₂	semidirect
2. #16863370	C₂	semidirect
3. C₂×#14523958	C₂	semidirect
6. #14524002	C₂²	semidirect
7. C₂×C₂²⋊Dic₆₇	C₂²	semidirect
8. #14523958	C₂²	semidirect
10. C₂×#41×C₆₇	C₄	semidirect
12. C₂²×D₄×C₆₇	C₄	semidirect
20. C₂⁴×C₆₇	C₂×C₄	semidirect
24. C₂×Q₈×C₆₇	C₂×C₄	semidirect
28. C₂×D₄×C₆₇	C₂×C₄	semidirect
29. C₂×D₄×C₆₇	C₂×C₄	semidirect
32. C₂²×C₄×C₆₇	C₂×C₄	semidirect
33. C₂²×C₄×C₆₇	C₂×C₄	semidirect
72. C₆₇	#2883	semidirect
78. C₂×#41	Dic₆₇	semidirect
80. C₂²×D₄	Dic₆₇	semidirect
87. C₂⁴	C₂×Dic₆₇	semidirect
91. C₂×Q₈	C₂×Dic₆₇	semidirect
95. C₂×D₄	C₂×Dic₆₇	semidirect
96. C₂×D₄	C₂×Dic₆₇	semidirect
100. C₂²×C₄	C₂×Dic₆₇	semidirect
101. C₂²×C₄	C₂×Dic₆₇	semidirect

H ≅ G/N

type

1. #16863871

C₂

semidirect

2. #16863370

C₂

semidirect

3. C₂×#14523958

C₂

semidirect

6. #14524002

C₂²

semidirect

7. C₂×C₂²⋊Dic₆₇

C₂²

semidirect

8. #14523958

C₂²

semidirect

10. C₂×#41×C₆₇

C₄

semidirect

12. C₂²×D₄×C₆₇

C₄

semidirect

20. C₂⁴×C₆₇

C₂×C₄

semidirect

24. C₂×Q₈×C₆₇

C₂×C₄

semidirect

28. C₂×D₄×C₆₇

C₂×C₄

semidirect

29. C₂×D₄×C₆₇

C₂×C₄

semidirect

32. C₂²×C₄×C₆₇

C₂×C₄

semidirect

33. C₂²×C₄×C₆₇

C₂×C₄

semidirect

72. C₆₇

#2883

semidirect

78. C₂×#41

Dic₆₇

semidirect

80. C₂²×D₄

Dic₆₇

semidirect

87. C₂⁴

C₂×Dic₆₇

semidirect

91. C₂×Q₈

C₂×Dic₆₇

semidirect

95. C₂×D₄

C₂×Dic₆₇

semidirect

96. C₂×D₄

C₂×Dic₆₇

semidirect

100. C₂²×C₄

C₂×Dic₆₇

semidirect

101. C₂²×C₄

C₂×Dic₆₇

semidirect

Subgroups — #22568670

Conjugacy classes of subgroups

Complements