Group #22573782 — subgroups

Subgroups — C₂×#16864125 #22573782

Order 8576.

1048 conjugacy classes of subgroups; only the 581 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 581 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	#16864125	C₂	1	16	yes	maximal, direct product factor
2	C₂×#14524043	C₂	1	3	yes	maximal
3	C₂×#14524034	C₂	1	6	yes	maximal
4	C₂×Q₈×Dic₆₇	C₂	1	2	yes	maximal
5	C₂×C₄×Dic₁₃₄	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₃₄	C₂²	1	3	yes	—
8	#14524043	C₂²	1	24	yes	—
9	#14524034	C₂²	1	48	yes	—
10	Q₈×Dic₆₇	C₂²	1	16	yes	—
11	C₄×Dic₁₃₄	C₂²	1	24	yes	—
12	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₆₇	C₂²	1	6	yes	—
14	C₂×C₄⋊Dic₆₇	C₂²	1	6	yes	—
15	C₂×Dic₆₇⋊C₄	C₂²	1	6	yes	—
16	C₂×C₄×Dic₆₇	C₂²	1	6	yes	—
17	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₆₇	C₂²	1	8	yes	—
19	C₂×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
20	C₂×C₄²×C₆₇	C₂²	1	3	yes	—
21	C₂×Dic₁₃₄	—	2	12	no	—
22	C₂×Dic₁₃₄	C₂³	1	12	yes	—
23	C₂²×Dic₆₇	C₂³	1	2	yes	—
24	C₂²×Dic₆₇	C₂³	1	6	yes	—
25	C₄⋊Dic₆₇	C₂³	1	4	yes	—
26	C₄⋊Dic₆₇	C₂³	1	24	yes	—
27	C₄⋊Dic₆₇	C₂³	1	24	yes	—
28	Dic₆₇⋊C₄	C₂³	1	24	yes	—
29	C₄×Dic₆₇	C₂³	1	24	yes	—
30	C₂×Q₈×C₆₇	Q₈	1	8	yes	complemented
31	C₂×Q₈×C₆₇	C₂³	1	4	yes	—
32	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
33	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
34	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
36	C₄²×C₆₇	C₂³	1	12	yes	—
37	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
38	Q₈×C₆₇	C₂×Q₈	1	16	yes	complemented
39	C₂×C₄×C₆₇	—	2	6	no	—
40	C₂×C₄×C₆₇	C₂×Q₈	1	12	yes	—
41	C₂×C₄×C₆₇	#41	1	4	yes	—
42	C₂×C₄×C₆₇	C₂⁴	1	6	yes	—
43	C₂×C₄×C₆₇	C₂⁴	1	6	yes	—
44	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
45	C₂×Dic₆₇	—	2	12	no	—
46	C₂×Dic₆₇	—	2	4	no	—
47	C₂×Dic₆₇	C₂⁴	1	4	yes	—
48	C₂×Dic₆₇	C₂⁴	1	12	yes	—
49	Dic₁₃₄	—	2	24	no	—
50	C₂²×C₆₇	C₂²×Q₈	1	2	yes	—
51	C₂²×C₆₇	C₂×#41	1	2	yes	—
52	C₂²×C₆₇	2₋¹⁺⁴	1	2	yes	—
53	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
54	C₄×C₆₇	—	2	6	no	—
55	C₄×C₆₇	C₂²×Q₈	1	12	yes	—
56	C₄×C₆₇	C₂×#41	1	4	yes	—
57	Dic₆₇	—	2	12	no	—
58	Dic₆₇	—	2	4	no	—
59	C₂×C₆₇	#557	1	4	yes	complemented
60	C₂×C₆₇	C₂³×Q₈	1	1	yes	characteristic
61	C₂×C₆₇	C₂²×#41	1	1	yes	characteristic
62	C₂×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
63	C₂×#557	—	67	1	no	maximal, Sylow
64	C₆₇	C₂×#557	1	1	yes	characteristic, complemented, Sylow
65	#557	—	67	16	no	—
66	C₂×C₄⋊Q₈	—	67	3	no	—
67	C₂×#125	—	67	6	no	—
68	C₂×C₄×Q₈	—	67	2	no	—
69	C₂×C₄×Q₈	—	67	3	no	—
70	C₂×C₄×Q₈	D₆₇	1	1	yes	characteristic
71	C₂²×Q₈	—	67	3	no	—
72	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
73	C₄⋊Q₈	—	67	24	no	—
74	#125	—	67	48	no	—
75	C₄×Q₈	—	67	16	no	—
76	C₄×Q₈	—	67	24	no	—
77	C₄×Q₈	C₂×D₆₇	1	8	yes	—
78	C₂×C₄⋊C₄	—	67	1	no	—
79	C₂×C₄⋊C₄	—	67	6	no	—
80	C₂×C₄⋊C₄	—	67	6	no	—
81	C₂×C₄⋊C₄	—	67	6	no	—
82	C₂×C₄⋊C₄	C₂×D₆₇	1	3	yes	—
83	C₂×C₄²	—	67	6	no	—
84	C₂×C₄²	C₂×D₆₇	1	3	yes	—
85	C₂×Q₈	—	67	12	no	—
86	C₂×Q₈	—	134	12	no	—
87	C₂×Q₈	Dic₁₃₄	1	8	yes	complemented
88	C₂×Q₈	C₂²×D₆₇	1	4	yes	—
89	C₂²×C₄	—	67	2	no	—
90	C₂²×C₄	—	67	6	no	—
91	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
92	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
93	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
94	C₄⋊C₄	—	67	4	no	—
95	C₄⋊C₄	—	67	24	no	—
96	C₄⋊C₄	—	67	24	no	—
97	C₄⋊C₄	—	67	24	no	—
98	C₄⋊C₄	C₂²×D₆₇	1	12	yes	—
99	C₄²	—	67	24	no	—
100	C₄²	C₂²×D₆₇	1	12	yes	—
101	C₂³	C₂³×D₆₇	1	1	yes	characteristic
102	Q₈	—	134	24	no	—
103	Q₈	C₂×Dic₁₃₄	1	16	yes	complemented
104	C₂×C₄	—	2	6	no	—
105	C₂×C₄	—	67	4	no	—
106	C₂×C₄	—	67	12	no	—
107	C₂×C₄	—	134	12	no	—
108	C₂×C₄	—	134	4	no	—
109	C₂×C₄	C₂×Dic₁₃₄	1	12	yes	—
110	C₂×C₄	#11761629	1	4	yes	—
111	C₂×C₄	C₂³×D₆₇	1	6	yes	—
112	C₂×C₄	C₂³×D₆₇	1	6	yes	—
113	C₂×C₄	C₂³×D₆₇	1	2	yes	—
114	C₂²	C₂²×Dic₁₃₄	1	2	yes	—
115	C₂²	C₂×#11761629	1	2	yes	—
116	C₂²	#14524090	1	2	yes	—
117	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
118	C₄	—	2	6	no	—
119	C₄	—	134	12	no	—
120	C₄	—	134	4	no	—
121	C₄	C₂²×Dic₁₃₄	1	12	yes	—
122	C₄	C₂×#11761629	1	4	yes	—
123	C₂	#16864125	1	4	yes	direct product factor
124	C₂	C₂³×Dic₁₃₄	1	1	yes	characteristic
125	C₂	C₂²×#11761629	1	1	yes	characteristic
126	C₂	C₂×#14524090	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16864125

C₂

yes

maximal, direct product factor

C₂×#14524043

C₂

yes

maximal

C₂×#14524034

C₂

yes

maximal

C₂×Q₈×Dic₆₇

C₂

yes

maximal

C₂×C₄×Dic₁₃₄

C₂

yes

maximal

C₂×C₄×Q₈×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

—

#14524043

C₂²

yes

—

#14524034

C₂²

yes

—

Q₈×Dic₆₇

C₂²

yes

—

C₄×Dic₁₃₄

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×Dic₆₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₂²×Q₈×C₆₇

C₂²

yes

characteristic

C₄×Q₈×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

C₂³

yes

—

C₄×Dic₆₇

C₂³

yes

—

C₂×Q₈×C₆₇

Q₈

yes

complemented

C₂×Q₈×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

C₂³

yes

—

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

Q₈×C₆₇

C₂×Q₈

yes

complemented

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂⁴

yes

—

C₂×Dic₆₇

C₂⁴

yes

—

Dic₁₃₄

—

C₂²×C₆₇

C₂²×Q₈

yes

—

C₂²×C₆₇

C₂×#41

yes

—

C₂²×C₆₇

2₋¹⁺⁴

yes

—

C₂²×C₆₇

C₂⁵

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₂²×Q₈

yes

—

C₄×C₆₇

C₂×#41

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#557

yes

complemented

C₂×C₆₇

C₂³×Q₈

yes

characteristic

C₂×C₆₇

C₂²×#41

yes

characteristic

C₂×C₆₇

C₂×2₋¹⁺⁴

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#557

—

C₂×C₄⋊Q₈

—

C₂×#125

—

C₂×C₄×Q₈

—

C₂×C₄×Q₈

—

C₂×C₄×Q₈

D₆₇

yes

characteristic

C₂²×Q₈

—

C₂²×Q₈

C₂×D₆₇

yes

characteristic

C₄⋊Q₈

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₆₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×Q₈

Dic₁₃₄

yes

complemented

C₂×Q₈

C₂²×D₆₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₆₇

yes

—

C₄²

—

100

C₄²

C₂²×D₆₇

yes

—

101

C₂³

C₂³×D₆₇

yes

characteristic

102

Q₈

—

134

—

103

Q₈

C₂×Dic₁₃₄

yes

complemented

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

134

—

108

C₂×C₄

—

134

—

109

C₂×C₄

C₂×Dic₁₃₄

yes

—

110

C₂×C₄

#11761629

yes

—

111

C₂×C₄

C₂³×D₆₇

yes

—

112

C₂×C₄

C₂³×D₆₇

yes

—

113

C₂×C₄

C₂³×D₆₇

yes

—

114

C₂²

C₂²×Dic₁₃₄

yes

—

115

C₂²

C₂×#11761629

yes

—

116

C₂²

#14524090

yes

—

117

C₂²

C₂⁴×D₆₇

yes

characteristic

118

C₄

—

119

C₄

—

134

—

120

C₄

—

134

—

121

C₄

C₂²×Dic₁₃₄

yes

—

122

C₄

C₂×#11761629

yes

—

123

C₂

#16864125

yes

direct product factor

124

C₂

C₂³×Dic₁₃₄

yes

characteristic

125

C₂

C₂²×#11761629

yes

characteristic

126

C₂

C₂×#14524090

yes

characteristic

	subgroup	G/H
center	C₂³	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂⁵
Frattini	C₂²	C₂⁴×D₆₇
Fitting	C₂×C₄×Q₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁴
hypercenter	C₂×C₄×Q₈	D₆₇
hypocenter	C₆₇	C₂×#557
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂³

C₂³×D₆₇

commutator

C₂²×C₆₇

C₂⁵

Frattini

C₂²

C₂⁴×D₆₇

Fitting

C₂×C₄×Q₈×C₆₇

C₂

socle

C₂³×C₆₇

C₂⁴

hypercenter

C₂×C₄×Q₈

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
1. #16864125	C₂	direct (N × H)
30. C₂×Q₈×C₆₇	Q₈	semidirect
38. Q₈×C₆₇	C₂×Q₈	semidirect
59. C₂×C₆₇	#557	semidirect
64. C₆₇	C₂×#557	semidirect
87. C₂×Q₈	Dic₁₃₄	semidirect
103. Q₈	C₂×Dic₁₃₄	semidirect
123. C₂	#16864125	direct (N × H)

H ≅ G/N

type

1. #16864125

C₂

direct (N × H)

30. C₂×Q₈×C₆₇

Q₈

semidirect

38. Q₈×C₆₇

C₂×Q₈

semidirect

59. C₂×C₆₇

#557

semidirect

64. C₆₇

C₂×#557

semidirect

87. C₂×Q₈

Dic₁₃₄

semidirect

103. Q₈

C₂×Dic₁₃₄

semidirect

123. C₂

#16864125

direct (N × H)

Subgroups — C₂×#16864125 #22573782

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16864125 #22573782

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16864125 #22573782