Group #16864130 — subgroups

Subgroups — #16864130

Order 4288.

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

normal subgroups only

This group has 236 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	#14524039	C₂	1	1	yes	maximal, characteristic
2	#14524029	C₂	1	2	yes	maximal, complemented
3	#14524029	C₂	1	2	yes	maximal, complemented
4	#14524028	C₂	1	2	yes	maximal, complemented
5	#14524024	C₂	1	1	yes	maximal, characteristic, complemented
6	#14524008	C₂	1	2	yes	maximal, complemented
7	C₄×C₂²⋊D₆₇	C₂	1	2	yes	maximal, complemented
8	(C₂²⋊D₆₇)⋊C₄	C₂	1	2	yes	maximal, complemented
9	#123×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₁₃₄	C₂²	1	2	yes	—
11	C₂×D₂₆₈	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₆₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₆₇	C₂²	1	2	yes	—
14	C₂²×Dic₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₆₇	C₂²	1	2	yes	—
16	Dic₆₇⋊C₄	C₂²	1	2	yes	—
17	Dic₆₇⋊C₄	C₂²	1	2	yes	—
18	C₂²⋊Dic₆₇	C₂²	1	2	yes	—
19	C₂²⋊Dic₆₇	C₂²	1	1	yes	characteristic
20	(C₂×D₆₇)⋊C₄	C₂²	1	2	yes	—
21	(C₂×D₆₇)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₆₇)⋊C₄	C₂²	1	2	yes	—
23	C₄×Dic₆₇	C₂²	1	2	yes	—
24	C₄×Dic₆₇	C₂²	1	2	yes	—
25	C₄×Dic₆₇	C₂²	1	1	yes	characteristic
26	C₂×D₄×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄×C₆₇	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
30	C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
31	C₂³×C₆₇	C₂³	1	1	yes	characteristic
32	C₂³×C₆₇	C₂³	1	1	yes	characteristic
33	C₂²×D₆₇	C₂³	1	2	yes	—
34	D₄×C₆₇	—	4	1	no	—
35	C₂×C₄×C₆₇	—	4	1	no	—
36	C₂×C₄×C₆₇	C₂³	1	2	yes	—
37	C₂×C₄×C₆₇	C₂³	1	2	yes	—
38	C₂×C₄×C₆₇	C₂³	1	1	yes	characteristic
39	C₂²⋊D₆₇	—	2	4	no	—
40	C₂²⋊D₆₇	—	2	4	no	—
41	C₂×Dic₆₇	—	4	1	no	—
42	C₂×Dic₆₇	C₂³	1	2	yes	—
43	C₂×Dic₆₇	C₂³	1	2	yes	—
44	C₂×Dic₆₇	C₂³	1	2	yes	—
45	D₂₆₈	—	4	1	no	—
46	C₄×D₆₇	—	4	2	no	—
47	Dic₁₃₄	—	2	4	no	—
48	C₂²×C₆₇	—	2	2	no	—
49	C₂²×C₆₇	—	2	2	no	—
50	C₂²×C₆₇	—	2	1	no	—
51	C₂²×C₆₇	—	2	1	no	—
52	C₂²×C₆₇	C₂⁴	1	1	yes	characteristic
53	C₂×D₆₇	—	2	2	no	—
54	C₂×D₆₇	—	2	2	no	—
55	C₂×D₆₇	—	2	2	no	—
56	C₄×C₆₇	—	2	2	no	—
57	C₄×C₆₇	—	2	2	no	—
58	C₄×C₆₇	—	2	1	no	—
59	Dic₆₇	—	2	2	no	—
60	Dic₆₇	—	2	2	no	—
61	Dic₆₇	#41	1	4	yes	—
62	C₂×C₆₇	—	4	1	no	—
63	C₂×C₆₇	—	4	1	no	—
64	C₂×C₆₇	C₂×#41	1	2	yes	—
65	C₂×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
66	D₆₇	—	4	2	no	—
67	C₆₇	#559	1	1	yes	characteristic, complemented, Sylow
68	#559	—	67	1	no	maximal, Sylow
69	#127	—	67	1	no	—
70	#124	—	67	2	no	—
71	#124	—	67	2	no	—
72	#123	—	67	1	no	—
73	#123	—	67	2	no	—
74	#123	D₆₇	1	1	yes	characteristic, complemented
75	C₄×Q₈	—	67	2	no	—
76	C₄×D₄	—	67	2	no	—
77	C₄×D₄	—	67	2	no	—
78	C₂×Q₈	—	67	2	no	—
79	C₂×D₄	—	67	2	no	—
80	C₂×D₄	—	67	2	no	—
81	C₂×D₄	—	67	1	no	—
82	C₂×D₄	C₂×D₆₇	1	1	yes	characteristic
83	C₂²×C₄	—	67	1	no	—
84	C₂²×C₄	—	67	2	no	—
85	C₂²×C₄	C₂×D₆₇	1	1	yes	characteristic
86	C₄⋊C₄	—	67	2	no	—
87	C₄⋊C₄	—	67	2	no	—
88	C₄⋊C₄	C₂×D₆₇	1	2	yes	—
89	C₂²⋊C₄	—	67	2	no	—
90	C₂²⋊C₄	—	67	1	no	—
91	C₂²⋊C₄	—	67	2	no	—
92	C₂²⋊C₄	—	67	2	no	—
93	C₂²⋊C₄	—	67	2	no	—
94	C₂²⋊C₄	C₂×D₆₇	1	2	yes	—
95	C₂²⋊C₄	C₂×D₆₇	1	1	yes	characteristic
96	C₄²	—	67	2	no	—
97	C₄²	—	67	2	no	—
98	C₄²	—	67	1	no	—
99	C₂³	—	67	2	no	—
100	C₂³	C₂²×D₆₇	1	1	yes	characteristic
101	C₂³	C₂²×D₆₇	1	1	yes	characteristic
102	Q₈	—	134	4	no	—
103	D₄	—	4	1	no	—
104	D₄	—	268	1	no	—
105	D₄	—	134	4	no	—
106	D₄	—	134	4	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	268	2	no	—
109	C₂×C₄	—	268	1	no	—
110	C₂×C₄	—	67	2	no	—
111	C₂×C₄	—	67	2	no	—
112	C₂×C₄	—	67	2	no	—
113	C₂×C₄	C₂²×D₆₇	1	2	yes	—
114	C₂×C₄	C₂²×D₆₇	1	2	yes	—
115	C₂×C₄	C₂²×D₆₇	1	1	yes	characteristic
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	—	134	2	no	—
121	C₂²	—	134	2	no	—
122	C₂²	—	134	2	no	—
123	C₂²	C₂³×D₆₇	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	67	4	no	—
128	C₄	—	134	2	no	—
129	C₄	—	134	2	no	—
130	C₂	—	4	1	no	—
131	C₂	—	4	1	no	—
132	C₂	—	268	2	no	—
133	C₂	#41×D₆₇	1	2	yes	—
134	C₂	#14524088	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14524039

C₂

yes

maximal, characteristic

#14524029

C₂

yes

maximal, complemented

#14524029

C₂

yes

maximal, complemented

#14524028

C₂

yes

maximal, complemented

#14524024

C₂

yes

maximal, characteristic, complemented

#14524008

C₂

yes

maximal, complemented

C₄×C₂²⋊D₆₇

C₂

yes

maximal, complemented

(C₂²⋊D₆₇)⋊C₄

C₂

yes

maximal, complemented

#123×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₃₄

C₂²

yes

—

C₂×D₂₆₈

C₂²

yes

characteristic

C₂×C₂²⋊D₆₇

C₂²

yes

—

C₂×C₂²⋊D₆₇

C₂²

yes

—

C₂²×Dic₆₇

C₂²

yes

characteristic

C₂×C₄×D₆₇

C₂²

yes

—

Dic₆₇⋊C₄

C₂²

yes

—

Dic₆₇⋊C₄

C₂²

yes

—

C₂²⋊Dic₆₇

C₂²

yes

—

C₂²⋊Dic₆₇

C₂²

yes

characteristic

(C₂×D₆₇)⋊C₄

C₂²

yes

—

(C₂×D₆₇)⋊C₄

C₂²

yes

—

(C₂×D₆₇)⋊C₄

C₂²

yes

—

C₄×Dic₆₇

C₂²

yes

—

C₄×Dic₆₇

C₂²

yes

—

C₄×Dic₆₇

C₂²

yes

characteristic

C₂×D₄×C₆₇

C₂²

yes

characteristic

C₂²×C₄×C₆₇

C₂²

yes

characteristic

C₄⋊C₄×C₆₇

C₂²

yes

—

C₂²⋊C₄×C₆₇

C₂²

yes

—

C₂²⋊C₄×C₆₇

C₂²

yes

characteristic

C₂³×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

C₂³

yes

characteristic

C₂²×D₆₇

C₂³

yes

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂³

yes

—

C₂×C₄×C₆₇

C₂³

yes

—

C₂×C₄×C₆₇

C₂³

yes

characteristic

C₂²⋊D₆₇

—

C₂²⋊D₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂³

yes

—

C₂×Dic₆₇

C₂³

yes

—

C₂×Dic₆₇

C₂³

yes

—

D₂₆₈

—

C₄×D₆₇

—

Dic₁₃₄

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂⁴

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

#41

yes

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂×#41

yes

—

C₂×C₆₇

H(5,𝔽₂)

yes

characteristic

D₆₇

—

C₆₇

#559

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₆₇

yes

characteristic, complemented

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₆₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₆₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₆₇

yes

—

C₂²⋊C₄

C₂×D₆₇

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₂³

—

100

C₂³

C₂²×D₆₇

yes

characteristic

101

C₂³

C₂²×D₆₇

yes

characteristic

102

Q₈

—

134

—

103

D₄

—

104

D₄

—

268

—

105

D₄

—

134

—

106

D₄

—

134

—

107

C₂×C₄

—

108

C₂×C₄

—

268

—

109

C₂×C₄

—

268

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²×D₆₇

yes

—

114

C₂×C₄

C₂²×D₆₇

yes

—

115

C₂×C₄

C₂²×D₆₇

yes

characteristic

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

134

—

121

C₂²

—

134

—

122

C₂²

—

134

—

123

C₂²

C₂³×D₆₇

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

134

—

129

C₄

—

134

—

130

C₂

—

131

C₂

—

132

C₂

—

268

—

133

C₂

#41×D₆₇

yes

—

134

C₂

#14524088

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₆₇
commutator, socle, cosocle	C₂²×C₆₇	C₂⁴
Fitting	#123×C₆₇	C₂
prime radical	#14524039	C₂
hypercenter	#123	D₆₇
hypocenter	C₆₇	#559
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ #123

subgroup

G/H

center, Frattini

C₂²

C₂³×D₆₇

commutator, socle, cosocle

C₂²×C₆₇

C₂⁴

Fitting

#123×C₆₇

C₂

C₂

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂² ◁ #123

N	H ≅ G/N	type
2. #14524029	C₂	semidirect
3. #14524029	C₂	semidirect
4. #14524028	C₂	semidirect
5. #14524024	C₂	semidirect
6. #14524008	C₂	semidirect
7. C₄×C₂²⋊D₆₇	C₂	semidirect
8. (C₂²⋊D₆₇)⋊C₄	C₂	semidirect
9. #123×C₆₇	C₂	semidirect
67. C₆₇	#559	semidirect
74. #123	D₆₇	semidirect

H ≅ G/N

type

2. #14524029

C₂

semidirect

3. #14524029

C₂

semidirect

4. #14524028

C₂

semidirect

5. #14524024

C₂

semidirect

6. #14524008

C₂

semidirect

7. C₄×C₂²⋊D₆₇

C₂

semidirect

8. (C₂²⋊D₆₇)⋊C₄

C₂

semidirect

9. #123×C₆₇

C₂

semidirect

67. C₆₇

#559

semidirect

74. #123

D₆₇

semidirect

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

subgroup	index
#14524039	2
#14524029	2
#14524028	2
#14524024	2
#14524008	2
C₄×C₂²⋊D₆₇	2
(C₂²⋊D₆₇)⋊C₄	2
#123×C₆₇	2
#559	67

subgroup

index

C₄×C₂²⋊D₆₇

(C₂²⋊D₆₇)⋊C₄

#123×C₆₇

#559

Subgroups — #16864130

Conjugacy classes of subgroups

Complements

Maximal subgroups