Group #22568565 — subgroups

Subgroups — #22568565

Order 8576.

752 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 197 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×#14524043	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14524030	C₂	1	4	yes	maximal, complemented
3	#16863299	C₂	1	8	yes	maximal, complemented
4	C₄²×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#127×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₃₄	C₂²	1	2	yes	—
7	#14524043	—	2	4	no	—
8	#14524030	—	2	16	no	—
9	C₂×C₄⋊Dic₆₇	C₂²	1	4	yes	—
10	C₂×C₂²⋊Dic₆₇	C₂²	1	16	yes	—
11	C₂×C₄×Dic₆₇	C₂²	1	4	yes	—
12	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
13	C₂²×D₄×C₆₇	C₂²	1	4	yes	—
14	C₂²×D₄×C₆₇	C₂²	1	2	yes	—
15	#127×C₆₇	—	2	4	no	—
16	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₃₄	—	2	8	no	—
18	C₂×Dic₁₃₄	—	2	4	no	—
19	C₂²×Dic₆₇	C₂³	1	4	yes	—
20	C₂²×Dic₆₇	C₂³	1	4	yes	—
21	C₄⋊Dic₆₇	—	2	8	no	—
22	C₂²⋊Dic₆₇	—	2	32	no	—
23	C₄×Dic₆₇	—	2	8	no	—
24	C₄×Dic₆₇	D₄	1	8	yes	complemented
25	C₂⁴×C₆₇	C₂³	1	4	yes	—
26	C₂×D₄×C₆₇	—	2	8	no	—
27	C₂×D₄×C₆₇	—	2	4	no	—
28	C₂×D₄×C₆₇	—	2	8	no	—
29	C₂×D₄×C₆₇	—	2	16	no	—
30	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
32	C₄²×C₆₇	D₄	1	4	yes	—
33	C₂³×C₆₇	—	2	4	no	—
34	C₂³×C₆₇	—	2	8	no	—
35	C₂³×C₆₇	—	2	16	no	—
36	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
37	D₄×C₆₇	—	4	16	no	—
38	D₄×C₆₇	—	4	8	no	—
39	C₂×C₄×C₆₇	C₂×D₄	1	4	yes	—
40	C₂×C₄×C₆₇	C₂×D₄	1	4	yes	—
41	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₆₇	#41	1	8	yes	—
43	C₂×Dic₆₇	—	2	8	no	—
44	C₂×Dic₆₇	—	2	8	no	—
45	C₂×Dic₆₇	—	2	4	no	—
46	C₂×Dic₆₇	C₂×D₄	1	8	yes	—
47	Dic₁₃₄	—	4	8	no	—
48	C₂²×C₆₇	—	4	8	no	—
49	C₂²×C₆₇	—	4	4	no	—
50	C₂²×C₆₇	—	4	16	no	—
51	C₂²×C₆₇	C₂²×D₄	1	2	yes	—
52	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
53	C₂²×C₆₇	C₂×#41	1	4	yes	—
54	C₄×C₆₇	#124	1	8	yes	complemented
55	C₄×C₆₇	#127	1	4	yes	—
56	Dic₆₇	—	2	8	no	—
57	Dic₆₇	—	4	4	no	—
58	C₂×C₆₇	—	8	4	no	—
59	C₂×C₆₇	C₂×#124	1	2	yes	—
60	C₂×C₆₇	C₂×#127	1	1	yes	characteristic
61	C₂×C₆₇	#532	1	4	yes	—
62	#2858	—	67	1	no	maximal, Sylow
63	C₆₇	#2858	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊Q₈	—	67	1	no	—
65	C₂×#127	D₆₇	1	1	yes	characteristic
66	C₂×#124	—	67	4	no	—
67	#390	—	67	8	no	—
68	C₄³	—	67	1	no	—
69	C₂²×Q₈	—	67	2	no	—
70	C₂²×D₄	C₂×D₆₇	1	2	yes	—
71	C₂²×D₄	C₂×D₆₇	1	4	yes	—
72	C₄⋊Q₈	—	134	4	no	—
73	#127	—	2	4	no	—
74	#124	—	134	16	no	—
75	C₂×C₄⋊C₄	—	67	4	no	—
76	C₂×C₂²⋊C₄	—	67	16	no	—
77	C₂×C₄²	—	67	2	no	—
78	C₂×C₄²	—	67	4	no	—
79	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
80	C₂⁴	C₂²×D₆₇	1	4	yes	—
81	C₂×Q₈	—	134	8	no	—
82	C₂×Q₈	—	134	4	no	—
83	C₂×D₄	—	2	8	no	—
84	C₂×D₄	—	2	8	no	—
85	C₂×D₄	—	2	4	no	—
86	C₂×D₄	—	2	16	no	—
87	C₂²×C₄	—	67	4	no	—
88	C₂²×C₄	—	67	4	no	—
89	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
91	C₄⋊C₄	—	134	8	no	—
92	C₂²⋊C₄	—	134	32	no	—
93	C₄²	—	67	8	no	—
94	C₄²	—	134	8	no	—
95	C₄²	C₂²⋊D₆₇	1	4	yes	—
96	C₂³	—	2	8	no	—
97	C₂³	—	2	4	no	—
98	C₂³	—	2	16	no	—
99	C₂³	C₂³×D₆₇	1	1	yes	characteristic
100	Q₈	—	268	8	no	—
101	D₄	—	4	16	no	—
102	D₄	—	4	8	no	—
103	C₂×C₄	—	67	8	no	—
104	C₂×C₄	—	134	8	no	—
105	C₂×C₄	—	134	4	no	—
106	C₂×C₄	—	134	8	no	—
107	C₂×C₄	D₄×D₆₇	1	4	yes	—
108	C₂×C₄	C₂×C₂²⋊D₆₇	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
110	C₂×C₄	#11761627	1	8	yes	—
111	C₂²	—	4	8	no	—
112	C₂²	—	4	4	no	—
113	C₂²	—	4	16	no	—
114	C₂²	C₂×D₄×D₆₇	1	2	yes	—
115	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
116	C₂²	C₂×#11761627	1	4	yes	—
117	C₄	—	268	4	no	—
118	C₄	—	134	8	no	—
119	C₄	#14524030	1	8	yes	complemented
120	C₄	#14524039	1	4	yes	—
121	C₂	—	8	4	no	—
122	C₂	C₂×#14524030	1	2	yes	—
123	C₂	C₂×#14524039	1	1	yes	characteristic
124	C₂	#16863944	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524043

C₂

yes

maximal, characteristic, complemented

C₂×#14524030

C₂

yes

maximal, complemented

#16863299

C₂

yes

maximal, complemented

C₄²×Dic₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

—

#14524043

—

#14524030

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₂²⋊Dic₆₇

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₂²×D₄×C₆₇

C₂²

yes

—

C₂²×D₄×C₆₇

C₂²

yes

—

#127×C₆₇

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

D₄

yes

complemented

C₂⁴×C₆₇

C₂³

yes

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

—

C₄²×C₆₇

D₄

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂⁴

yes

characteristic

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×D₄

yes

—

Dic₁₃₄

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×D₄

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂²×C₆₇

C₂×#41

yes

—

C₄×C₆₇

#124

yes

complemented

C₄×C₆₇

#127

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂×#124

yes

—

C₂×C₆₇

C₂×#127

yes

characteristic

C₂×C₆₇

#532

yes

—

#2858

—

maximal, Sylow

C₆₇

#2858

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#127

D₆₇

yes

characteristic

C₂×#124

—

#390

—

C₄³

—

C₂²×Q₈

—

C₂²×D₄

C₂×D₆₇

yes

—

C₂²×D₄

C₂×D₆₇

yes

—

C₄⋊Q₈

—

134

—

#127

—

#124

—

134

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₂⁴

C₂²×D₆₇

yes

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂²×C₄

C₂²×D₆₇

yes

—

C₄⋊C₄

—

134

—

C₂²⋊C₄

—

134

—

C₄²

—

C₄²

—

134

—

C₄²

C₂²⋊D₆₇

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₆₇

yes

characteristic

100

Q₈

—

268

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

134

—

105

C₂×C₄

—

134

—

106

C₂×C₄

—

134

—

107

C₂×C₄

D₄×D₆₇

yes

—

108

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

109

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

110

C₂×C₄

#11761627

yes

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×D₄×D₆₇

yes

—

115

C₂²

C₂²×C₂²⋊D₆₇

yes

characteristic

116

C₂²

C₂×#11761627

yes

—

117

C₄

—

268

—

118

C₄

—

134

—

119

C₄

#14524030

yes

complemented

120

C₄

#14524039

yes

—

121

C₂

—

122

C₂

C₂×#14524030

yes

—

123

C₂

C₂×#14524039

yes

characteristic

124

C₂

#16863944

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	C₂×#127×C₆₇	C₂
hypercenter	C₂×#127	D₆₇
hypocenter	C₆₇	#2858
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×#127

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₇

commutator, socle

C₂³×C₆₇

C₂⁴

Fitting

C₂×#127×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂³ ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#14524043	C₂	semidirect
2. C₂×#14524030	C₂	semidirect
3. #16863299	C₂	semidirect
4. C₄²×Dic₆₇	C₂	semidirect
24. C₄×Dic₆₇	D₄	semidirect
54. C₄×C₆₇	#124	semidirect
63. C₆₇	#2858	semidirect
119. C₄	#14524030	semidirect

H ≅ G/N

type

1. C₂×#14524043

C₂

semidirect

2. C₂×#14524030

C₂

semidirect

3. #16863299

C₂

semidirect

4. C₄²×Dic₆₇

C₂

semidirect

24. C₄×Dic₆₇

D₄

semidirect

54. C₄×C₆₇

#124

semidirect

63. C₆₇

#2858

semidirect

119. C₄

#14524030

semidirect

Subgroups — #22568565

Conjugacy classes of subgroups

Complements