Group #16864004 — subgroups

Subgroups — #16864004

Order 4288.

192 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 95 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	#14524041	C₂	1	1	yes	maximal, characteristic, complemented
2	#14524038	C₂	1	2	yes	maximal
3	#14524037	C₂	1	2	yes	maximal
4	#14524035	C₂	1	2	yes	maximal, complemented
5	#14524034	C₂	1	1	yes	maximal, characteristic, complemented
6	#14524033	C₂	1	2	yes	maximal, complemented
7	#14524023	C₂	1	1	yes	maximal, characteristic
8	#14524019	C₂	1	1	yes	maximal, characteristic
9	C₄×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic, complemented
10	#14524000	C₂	1	1	yes	maximal, characteristic
11	C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₁₃₄	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₆₇	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₆₇	C₂²	1	2	yes	—
16	C₄⋊Dic₆₇	C₂²	1	2	yes	—
17	Dic₆₇⋊C₄	C₂²	1	1	yes	characteristic
18	Dic₆₇⋊C₄	C₂²	1	1	yes	characteristic
19	Dic₆₇⋊C₄	C₂²	1	2	yes	—
20	Dic₆₇⋊C₄	C₂²	1	2	yes	—
21	Dic₆₇⋊C₄	C₂²	1	2	yes	—
22	Dic₆₇⋊C₄	C₂²	1	2	yes	—
23	Dic₆₇⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₆₇)⋊C₄	C₂²	1	1	yes	characteristic
25	(C₂×D₆₇)⋊C₄	C₂²	1	1	yes	characteristic
26	(C₂×D₆₇)⋊C₄	C₂²	1	2	yes	—
27	(C₂×D₆₇)⋊C₄	C₂²	1	2	yes	—
28	C₄×Dic₆₇	C₂²	1	1	yes	characteristic
29	C₄×Dic₆₇	C₂²	1	2	yes	—
30	C₂×Q₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
33	C₄²×C₆₇	C₂²	1	1	yes	characteristic
34	C₄²×C₆₇	C₂²	1	2	yes	—
35	C₂²×D₆₇	C₂³	1	1	yes	characteristic
36	Q₈×C₆₇	—	2	2	no	—
37	C₂×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₆₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₆₇	C₂³	1	2	yes	—
40	C₂×C₄×C₆₇	C₂³	1	2	yes	—
41	C₂×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	C₂×Dic₆₇	C₂³	1	1	yes	characteristic
43	C₂×Dic₆₇	C₂³	1	1	yes	characteristic
44	C₂×Dic₆₇	C₂³	1	2	yes	—
45	C₂×Dic₆₇	C₂³	1	2	yes	—
46	C₂×Dic₆₇	C₂³	1	1	yes	characteristic
47	C₄×D₆₇	—	2	2	no	—
48	Dic₁₃₄	—	2	2	no	—
49	C₂²×C₆₇	C₂⁴	1	1	yes	characteristic
50	C₂×D₆₇	—	2	1	no	—
51	C₂×D₆₇	—	2	1	no	—
52	C₂×D₆₇	—	2	1	no	—
53	C₄×C₆₇	—	2	2	no	—
54	C₄×C₆₇	—	2	1	no	—
55	C₄×C₆₇	—	2	2	no	—
56	C₄×C₆₇	—	2	1	no	—
57	C₄×C₆₇	#41	1	2	yes	—
58	Dic₆₇	—	2	2	no	—
59	Dic₆₇	—	2	1	no	—
60	Dic₆₇	—	2	1	no	—
61	Dic₆₇	—	2	1	no	—
62	Dic₆₇	—	2	2	no	—
63	C₂×C₆₇	C₂×#41	1	1	yes	characteristic
64	C₂×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
65	C₂×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
66	D₆₇	—	4	1	no	—
67	C₆₇	#541	1	1	yes	characteristic, complemented, Sylow
68	#541	—	67	1	no	maximal, Sylow
69	C₄⋊Q₈	—	67	1	no	—
70	#126	—	67	2	no	—
71	#126	—	67	2	no	—
72	#125	—	67	2	no	—
73	#125	—	67	1	no	—
74	#125	—	67	2	no	—
75	C₂²⋊Q₈	—	67	1	no	—
76	C₂²⋊Q₈	—	67	1	no	—
77	C₄×Q₈	—	67	1	no	—
78	C₄×Q₈	D₆₇	1	1	yes	characteristic, complemented
79	#117	—	67	1	no	—
80	C₂×Q₈	—	67	1	no	—
81	C₂×Q₈	C₂×D₆₇	1	1	yes	characteristic
82	C₂²×C₄	—	67	1	no	—
83	C₄⋊C₄	—	67	1	no	—
84	C₄⋊C₄	—	67	1	no	—
85	C₄⋊C₄	—	67	1	no	—
86	C₄⋊C₄	—	67	2	no	—
87	C₄⋊C₄	—	67	2	no	—
88	C₄⋊C₄	—	67	2	no	—
89	C₄⋊C₄	—	67	2	no	—
90	C₄⋊C₄	—	67	2	no	—
91	C₄⋊C₄	—	67	2	no	—
92	C₄⋊C₄	—	67	2	no	—
93	C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
94	C₄⋊C₄	C₂×D₆₇	1	2	yes	—
95	C₂²⋊C₄	—	67	1	no	—
96	C₂²⋊C₄	—	67	1	no	—
97	C₂²⋊C₄	—	67	2	no	—
98	C₂²⋊C₄	—	67	2	no	—
99	C₄²	—	67	1	no	—
100	C₄²	—	67	2	no	—
101	C₄²	C₂×D₆₇	1	1	yes	characteristic
102	C₄²	C₂×D₆₇	1	2	yes	—
103	C₂³	—	67	1	no	—
104	Q₈	—	2	2	no	—
105	Q₈	—	134	2	no	—
106	C₂×C₄	—	67	1	no	—
107	C₂×C₄	—	67	1	no	—
108	C₂×C₄	—	67	2	no	—
109	C₂×C₄	—	67	2	no	—
110	C₂×C₄	—	67	1	no	—
111	C₂×C₄	—	134	2	no	—
112	C₂×C₄	C₂²×D₆₇	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₆₇	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₆₇	1	2	yes	—
115	C₂×C₄	C₂²×D₆₇	1	2	yes	—
116	C₂×C₄	C₂²×D₆₇	1	1	yes	characteristic
117	C₂²	—	134	1	no	—
118	C₂²	—	134	1	no	—
119	C₂²	—	134	1	no	—
120	C₂²	C₂³×D₆₇	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	134	1	no	—
126	C₄	—	134	2	no	—
127	C₄	—	134	1	no	—
128	C₄	—	134	1	no	—
129	C₄	—	134	2	no	—
130	C₄	#11761628	1	2	yes	—
131	C₂	—	268	1	no	—
132	C₂	C₂×#11761628	1	1	yes	characteristic
133	C₂	#14524090	1	1	yes	characteristic
134	C₂	#14524091	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14524041

C₂

yes

maximal, characteristic, complemented

#14524038

C₂

yes

maximal

#14524037

C₂

yes

maximal

#14524035

C₂

yes

maximal, complemented

#14524034

C₂

yes

maximal, characteristic, complemented

#14524033

C₂

yes

maximal, complemented

#14524023

C₂

yes

maximal, characteristic

#14524019

C₂

yes

maximal, characteristic

C₄×Dic₁₃₄

C₂

yes

maximal, characteristic, complemented

#14524000

C₂

yes

maximal, characteristic

C₄×Q₈×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₃₄

C₂²

yes

characteristic

C₂×C₄×D₆₇

C₂²

yes

characteristic

C₄⋊Dic₆₇

C₂²

yes

characteristic

C₄⋊Dic₆₇

C₂²

yes

—

C₄⋊Dic₆₇

C₂²

yes

—

Dic₆₇⋊C₄

C₂²

yes

characteristic

Dic₆₇⋊C₄

C₂²

yes

characteristic

Dic₆₇⋊C₄

C₂²

yes

—

Dic₆₇⋊C₄

C₂²

yes

—

Dic₆₇⋊C₄

C₂²

yes

—

Dic₆₇⋊C₄

C₂²

yes

—

Dic₆₇⋊C₄

C₂²

yes

—

(C₂×D₆₇)⋊C₄

C₂²

yes

characteristic

(C₂×D₆₇)⋊C₄

C₂²

yes

characteristic

(C₂×D₆₇)⋊C₄

C₂²

yes

—

(C₂×D₆₇)⋊C₄

C₂²

yes

—

C₄×Dic₆₇

C₂²

yes

characteristic

C₄×Dic₆₇

C₂²

yes

—

C₂×Q₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₄⋊C₄×C₆₇

C₂²

yes

—

C₄²×C₆₇

C₂²

yes

characteristic

C₄²×C₆₇

C₂²

yes

—

C₂²×D₆₇

C₂³

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₄×C₆₇

C₂³

yes

—

C₂×C₄×C₆₇

C₂³

yes

—

C₂×C₄×C₆₇

C₂³

yes

characteristic

C₂×Dic₆₇

C₂³

yes

characteristic

C₂×Dic₆₇

C₂³

yes

characteristic

C₂×Dic₆₇

C₂³

yes

—

C₂×Dic₆₇

C₂³

yes

—

C₂×Dic₆₇

C₂³

yes

characteristic

C₄×D₆₇

—

Dic₁₃₄

—

C₂²×C₆₇

C₂⁴

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#41

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

C₂×#41

yes

characteristic

C₂×C₆₇

2₋¹⁺⁴

yes

characteristic

C₂×C₆₇

2₋¹⁺⁴

yes

characteristic

D₆₇

—

C₆₇

#541

yes

characteristic, complemented, Sylow

#541

—

maximal, Sylow

C₄⋊Q₈

—

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

D₆₇

yes

characteristic, complemented

#117

—

C₂×Q₈

—

C₂×Q₈

C₂×D₆₇

yes

characteristic

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₆₇

yes

characteristic

C₄⋊C₄

C₂×D₆₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

100

C₄²

—

101

C₄²

C₂×D₆₇

yes

characteristic

102

C₄²

C₂×D₆₇

yes

—

103

C₂³

—

104

Q₈

—

105

Q₈

—

134

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

134

—

112

C₂×C₄

C₂²×D₆₇

yes

characteristic

113

C₂×C₄

C₂²×D₆₇

yes

characteristic

114

C₂×C₄

C₂²×D₆₇

yes

—

115

C₂×C₄

C₂²×D₆₇

yes

—

116

C₂×C₄

C₂²×D₆₇

yes

characteristic

117

C₂²

—

134

—

118

C₂²

—

134

—

119

C₂²

—

134

—

120

C₂²

C₂³×D₆₇

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

134

—

126

C₄

—

134

—

127

C₄

—

134

—

128

C₄

—

134

—

129

C₄

—

134

—

130

C₄

#11761628

yes

—

131

C₂

—

268

—

132

C₂

C₂×#11761628

yes

characteristic

133

C₂

#14524090

yes

characteristic

134

C₂

#14524091

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₆₇
commutator, socle, cosocle	C₂²×C₆₇	C₂⁴
Fitting	C₄×Q₈×C₆₇	C₂
prime radical	C₂²×D₆₇	C₂³
hypercenter	C₄×Q₈	D₆₇
hypocenter	C₆₇	#541
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₆₇

commutator, socle, cosocle

C₂²×C₆₇

C₂⁴

Fitting

C₄×Q₈×C₆₇

C₂

prime radical

C₂²×D₆₇

C₂³

hypercenter

C₄×Q₈

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. #14524041	C₂	semidirect
4. #14524035	C₂	semidirect
5. #14524034	C₂	semidirect
6. #14524033	C₂	semidirect
9. C₄×Dic₁₃₄	C₂	semidirect
11. C₄×Q₈×C₆₇	C₂	semidirect
67. C₆₇	#541	semidirect
78. C₄×Q₈	D₆₇	semidirect

H ≅ G/N

type

1. #14524041

C₂

semidirect

4. #14524035

C₂

semidirect

5. #14524034

C₂

semidirect

6. #14524033

C₂

semidirect

9. C₄×Dic₁₃₄

C₂

semidirect

11. C₄×Q₈×C₆₇

C₂

semidirect

67. C₆₇

#541

semidirect

78. C₄×Q₈

D₆₇

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#14524041	2
#14524038	2
#14524037	2
#14524035	2
#14524034	2
#14524033	2
#14524023	2
#14524019	2
C₄×Dic₁₃₄	2
#14524000	2
C₄×Q₈×C₆₇	2
#541	67

subgroup

index

C₄×Dic₁₃₄

C₄×Q₈×C₆₇

Subgroups — #16864004

Conjugacy classes of subgroups

Complements

Maximal subgroups