Group #22568496 — subgroups

Subgroups — #22568496

Order 8576.

1068 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 237 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×#14524022	C₂	1	3	yes	maximal, complemented
2	#16863269	C₂	1	6	yes	maximal, complemented
3	#16863256	C₂	1	3	yes	maximal, complemented
4	#16863252	C₂	1	2	yes	maximal
5	C₂⁴×C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₃₄	C₂²	1	1	yes	characteristic, complemented
7	#14524022	—	2	12	no	—
8	C₂×C₄⋊Dic₆₇	C₂²	1	3	yes	complemented
9	C₂×Dic₆₇⋊C₄	C₂²	1	6	yes	complemented
10	C₂×C₂²⋊Dic₆₇	C₂²	1	6	yes	—
11	C₂×C₂²⋊Dic₆₇	C₂²	1	6	yes	—
12	C₆₇⋊#95	C₂²	1	6	yes	complemented
13	C₂⁵×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₆₇	—	2	12	no	—
15	C₂³×C₄×C₆₇	C₂²	1	3	yes	—
16	C₂³×C₄×C₆₇	C₂²	1	3	yes	—
17	C₂×Dic₁₃₄	—	2	6	no	—
18	C₂²×Dic₆₇	C₂³	1	6	yes	—
19	C₂²×Dic₆₇	C₂³	1	2	yes	—
20	C₄⋊Dic₆₇	—	2	6	no	—
21	Dic₆₇⋊C₄	—	2	12	no	—
22	C₂²⋊Dic₆₇	—	2	12	no	—
23	C₂²⋊Dic₆₇	—	2	12	no	—
24	C₂⁴×C₆₇	—	2	6	no	—
25	C₂⁴×C₆₇	—	2	6	no	—
26	C₂⁴×C₆₇	Q₈	1	4	yes	complemented
27	C₂⁴×C₆₇	C₂³	1	3	yes	—
28	C₂²×C₄×C₆₇	—	2	32	no	—
29	C₂²×C₄×C₆₇	—	2	6	no	—
30	C₂²×C₄×C₆₇	—	2	12	no	—
31	C₂²×C₄×C₆₇	—	2	12	no	—
32	C₂²×C₄×C₆₇	D₄	1	12	yes	—
33	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
34	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
35	C₂³×C₆₇	—	2	8	no	—
36	C₂³×C₆₇	—	2	24	no	—
37	C₂³×C₆₇	—	2	24	no	—
38	C₂³×C₆₇	—	2	6	no	—
39	C₂³×C₆₇	—	2	6	no	—
40	C₂³×C₆₇	C₂×D₄	1	6	yes	—
41	C₂³×C₆₇	C₂×Q₈	1	6	yes	—
42	C₂³×C₆₇	#41	1	6	yes	—
43	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
44	C₂×C₄×C₆₇	—	2	24	no	—
45	C₂×C₄×C₆₇	—	2	6	no	—
46	C₂×C₄×C₆₇	—	2	24	no	—
47	C₂×C₄×C₆₇	C₂×D₄	1	6	yes	—
48	C₂×C₄×C₆₇	C₂×D₄	1	6	yes	—
49	C₂×Dic₆₇	—	2	6	no	—
50	C₂×Dic₆₇	—	2	6	no	—
51	C₂×Dic₆₇	—	2	12	no	—
52	Dic₁₃₄	—	4	4	no	—
53	C₂²×C₆₇	—	2	32	no	—
54	C₂²×C₆₇	—	2	6	no	—
55	C₂²×C₆₇	—	2	12	no	—
56	C₂²×C₆₇	—	2	6	no	—
57	C₂²×C₆₇	—	2	12	no	—
58	C₂²×C₆₇	C₂²⋊Q₈	1	12	yes	complemented
59	C₂²×C₆₇	C₂²×D₄	1	3	yes	—
60	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
61	C₂²×C₆₇	C₂×#41	1	3	yes	—
62	C₄×C₆₇	—	2	6	no	—
63	C₄×C₆₇	#120	1	4	yes	—
64	Dic₆₇	—	4	2	no	—
65	Dic₆₇	—	4	6	no	—
66	C₂×C₆₇	—	2	12	no	—
67	C₂×C₆₇	C₂×#120	1	1	yes	characteristic
68	C₂×C₆₇	C₂×C₂²⋊Q₈	1	3	yes	—
69	C₂×C₆₇	#525	1	3	yes	—
70	#2848	—	67	1	no	maximal, Sylow
71	C₆₇	#2848	1	1	yes	characteristic, complemented, Sylow
72	C₂⁴×C₄	D₆₇	1	1	yes	characteristic
73	C₂×C₂²⋊Q₈	—	67	3	no	—
74	#385	—	67	6	no	—
75	#380	—	67	3	no	—
76	#379	—	67	2	no	—
77	C₂⁵	C₂×D₆₇	1	1	yes	characteristic
78	C₂²×Q₈	—	67	1	no	—
79	C₂³×C₄	—	2	12	no	—
80	C₂³×C₄	C₂×D₆₇	1	3	yes	—
81	C₂³×C₄	C₂×D₆₇	1	3	yes	—
82	C₂²⋊Q₈	—	134	12	no	—
83	C₂×C₄⋊C₄	—	67	6	no	—
84	C₂×C₄⋊C₄	—	67	3	no	—
85	C₂×C₂²⋊C₄	—	67	6	no	—
86	C₂×C₂²⋊C₄	—	67	6	no	—
87	#95	—	67	6	no	—
88	C₂⁴	—	2	6	no	—
89	C₂⁴	—	2	6	no	—
90	C₂⁴	Dic₁₃₄	1	4	yes	complemented
91	C₂⁴	C₂²×D₆₇	1	3	yes	—
92	C₂×Q₈	—	134	6	no	—
93	C₂²×C₄	—	2	32	no	—
94	C₂²×C₄	—	2	6	no	—
95	C₂²×C₄	—	2	12	no	—
96	C₂²×C₄	—	2	12	no	—
97	C₂²×C₄	—	67	6	no	—
98	C₂²×C₄	—	67	2	no	—
99	C₂²×C₄	C₂²⋊D₆₇	1	12	yes	—
100	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
101	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
102	C₄⋊C₄	—	134	12	no	—
103	C₄⋊C₄	—	134	6	no	—
104	C₂²⋊C₄	—	134	12	no	—
105	C₂²⋊C₄	—	134	12	no	—
106	C₂³	—	2	8	no	—
107	C₂³	—	2	24	no	—
108	C₂³	—	2	24	no	—
109	C₂³	—	2	6	no	—
110	C₂³	—	2	6	no	—
111	C₂³	C₂×C₂²⋊D₆₇	1	6	yes	—
112	C₂³	C₂×Dic₁₃₄	1	6	yes	—
113	C₂³	#11761628	1	6	yes	—
114	C₂³	C₂³×D₆₇	1	1	yes	characteristic
115	Q₈	—	268	4	no	—
116	C₂×C₄	—	2	24	no	—
117	C₂×C₄	—	2	6	no	—
118	C₂×C₄	—	2	24	no	—
119	C₂×C₄	—	134	6	no	—
120	C₂×C₄	—	134	6	no	—
121	C₂×C₄	—	134	12	no	—
122	C₂×C₄	C₂×C₂²⋊D₆₇	1	6	yes	—
123	C₂×C₄	C₂×C₂²⋊D₆₇	1	6	yes	—
124	C₂²	—	2	32	no	—
125	C₂²	—	2	6	no	—
126	C₂²	—	2	12	no	—
127	C₂²	—	2	6	no	—
128	C₂²	—	2	12	no	—
129	C₂²	#14524022	1	12	yes	complemented
130	C₂²	C₂²×C₂²⋊D₆₇	1	3	yes	—
131	C₂²	C₂²×Dic₁₃₄	1	1	yes	characteristic
132	C₂²	C₂×#11761628	1	3	yes	—
133	C₄	—	2	6	no	—
134	C₄	—	268	2	no	—
135	C₄	—	268	6	no	—
136	C₄	#14524012	1	4	yes	—
137	C₂	—	2	12	no	—
138	C₂	C₂×#14524012	1	1	yes	characteristic
139	C₂	C₂×#14524022	1	3	yes	—
140	C₂	#16863907	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524022

C₂

yes

maximal, complemented

#16863269

C₂

yes

maximal, complemented

#16863256

C₂

yes

maximal, complemented

#16863252

C₂

yes

maximal

C₂⁴×C₄×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

characteristic, complemented

#14524022

—

C₂×C₄⋊Dic₆₇

C₂²

yes

complemented

C₂×Dic₆₇⋊C₄

C₂²

yes

complemented

C₂×C₂²⋊Dic₆₇

C₂²

yes

—

C₂×C₂²⋊Dic₆₇

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

complemented

C₂⁵×C₆₇

C₂²

yes

characteristic

C₂³×C₄×C₆₇

—

C₂³×C₄×C₆₇

C₂²

yes

—

C₂³×C₄×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₂⁴×C₆₇

—

C₂⁴×C₆₇

—

C₂⁴×C₆₇

Q₈

yes

complemented

C₂⁴×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

—

C₂³×C₆₇

C₂×Q₈

yes

—

C₂³×C₆₇

#41

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²⋊Q₈

yes

complemented

C₂²×C₆₇

C₂²×D₄

yes

—

C₂²×C₆₇

C₂²×Q₈

yes

characteristic

C₂²×C₆₇

C₂×#41

yes

—

C₄×C₆₇

—

C₄×C₆₇

#120

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂×#120

yes

characteristic

C₂×C₆₇

C₂×C₂²⋊Q₈

yes

—

C₂×C₆₇

#525

yes

—

#2848

—

maximal, Sylow

C₆₇

#2848

yes

characteristic, complemented, Sylow

C₂⁴×C₄

D₆₇

yes

characteristic

C₂×C₂²⋊Q₈

—

#385

—

#380

—

#379

—

C₂⁵

C₂×D₆₇

yes

characteristic

C₂²×Q₈

—

C₂³×C₄

—

C₂³×C₄

C₂×D₆₇

yes

—

C₂³×C₄

C₂×D₆₇

yes

—

C₂²⋊Q₈

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

#95

—

C₂⁴

—

C₂⁴

—

C₂⁴

Dic₁₃₄

yes

complemented

C₂⁴

C₂²×D₆₇

yes

—

C₂×Q₈

—

134

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₆₇

yes

—

100

C₂²×C₄

C₂²×D₆₇

yes

—

101

C₂²×C₄

C₂²×D₆₇

yes

characteristic

102

C₄⋊C₄

—

134

—

103

C₄⋊C₄

—

134

—

104

C₂²⋊C₄

—

134

—

105

C₂²⋊C₄

—

134

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

C₂×C₂²⋊D₆₇

yes

—

112

C₂³

C₂×Dic₁₃₄

yes

—

113

C₂³

#11761628

yes

—

114

C₂³

C₂³×D₆₇

yes

characteristic

115

Q₈

—

268

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

134

—

120

C₂×C₄

—

134

—

121

C₂×C₄

—

134

—

122

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

123

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#14524022

yes

complemented

130

C₂²

C₂²×C₂²⋊D₆₇

yes

—

131

C₂²

C₂²×Dic₁₃₄

yes

characteristic

132

C₂²

C₂×#11761628

yes

—

133

C₄

—

134

C₄

—

268

—

135

C₄

—

268

—

136

C₄

#14524012

yes

—

137

C₂

—

138

C₂

C₂×#14524012

yes

characteristic

139

C₂

C₂×#14524022

yes

—

140

C₂

#16863907

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	C₂⁴×C₄×C₆₇	C₂
hypercenter	C₂⁴×C₄	D₆₇
hypocenter	C₆₇	#2848
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₇

commutator, socle

C₂³×C₆₇

C₂⁴

Fitting

C₂⁴×C₄×C₆₇

C₂

hypercenter

C₂⁴×C₄

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂³ ◁ C₂⁴×C₄

N	H ≅ G/N	type
1. C₂×#14524022	C₂	semidirect
2. #16863269	C₂	semidirect
3. #16863256	C₂	semidirect
6. C₂²×Dic₁₃₄	C₂²	semidirect
8. C₂×C₄⋊Dic₆₇	C₂²	semidirect
9. C₂×Dic₆₇⋊C₄	C₂²	semidirect
12. C₆₇⋊#95	C₂²	semidirect
26. C₂⁴×C₆₇	Q₈	semidirect
58. C₂²×C₆₇	C₂²⋊Q₈	semidirect
71. C₆₇	#2848	semidirect
90. C₂⁴	Dic₁₃₄	semidirect
129. C₂²	#14524022	semidirect

H ≅ G/N

type

1. C₂×#14524022

C₂

semidirect

2. #16863269

C₂

semidirect

3. #16863256

C₂

semidirect

6. C₂²×Dic₁₃₄

C₂²

semidirect

8. C₂×C₄⋊Dic₆₇

C₂²

semidirect

9. C₂×Dic₆₇⋊C₄

C₂²

semidirect

12. C₆₇⋊#95

C₂²

semidirect

26. C₂⁴×C₆₇

Q₈

semidirect

58. C₂²×C₆₇

C₂²⋊Q₈

semidirect

71. C₆₇

#2848

semidirect

90. C₂⁴

Dic₁₃₄

semidirect

129. C₂²

#14524022

semidirect

Subgroups — #22568496

Conjugacy classes of subgroups

Complements