Group #22568679 — subgroups

Subgroups — C₂²×#14523962 #22568679

Order 8576.

736 conjugacy classes of subgroups; only the 309 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 309 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	C₂²×#11761612	C₂	1	2	yes	maximal
2	C₂×#14523962	C₂	1	12	yes	maximal, direct product factor
3	C₂³×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#11761612	—	2	6	no	—
5	C₂×#11761612	C₂²	1	12	yes	—
6	C₂²×C₆₇⋊C₈	—	2	2	no	—
7	#14523962	C₂²	1	16	yes	direct product factor
8	C₂²×Q₈×C₆₇	—	2	2	no	—
9	C₂²×Q₈×C₆₇	—	2	6	no	—
10	C₂²×Q₈×C₆₇	C₄	1	6	yes	—
11	C₂²×Q₈×C₆₇	C₂²	1	6	yes	—
12	C₂³×C₄×C₆₇	C₄	1	2	yes	—
13	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
14	#11761612	—	2	12	no	—
15	#11761612	C₂³	1	8	yes	—
16	C₂×C₆₇⋊C₈	—	2	12	no	—
17	C₂⁴×C₆₇	C₂×C₄	1	1	yes	characteristic
18	C₂×Q₈×C₆₇	—	2	12	no	—
19	C₂×Q₈×C₆₇	—	2	12	no	—
20	C₂×Q₈×C₆₇	—	2	12	no	—
21	C₂×Q₈×C₆₇	—	2	12	no	—
22	C₂×Q₈×C₆₇	C₂×C₄	1	12	yes	—
23	C₂×Q₈×C₆₇	C₂³	1	4	yes	—
24	C₂²×C₄×C₆₇	—	2	6	no	—
25	C₂²×C₄×C₆₇	—	2	2	no	—
26	C₂²×C₄×C₆₇	C₂×C₄	1	12	yes	—
27	C₂²×C₄×C₆₇	C₂×C₄	1	3	yes	—
28	C₂²×C₄×C₆₇	D₄	1	2	yes	—
29	C₂²×C₄×C₆₇	D₄	1	6	yes	—
30	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
31	C₂³×C₆₇	—	2	4	no	—
32	C₂³×C₆₇	C₂²⋊C₄	1	3	yes	—
33	C₂³×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂³×C₆₇	C₂²×C₄	1	3	yes	—
35	Q₈×C₆₇	—	2	24	no	—
36	Q₈×C₆₇	—	2	8	no	—
37	C₂×C₄×C₆₇	—	2	12	no	—
38	C₂×C₄×C₆₇	—	2	12	no	—
39	C₂×C₄×C₆₇	C₂²⋊C₄	1	6	yes	—
40	C₂×C₄×C₆₇	C₂²⋊C₄	1	6	yes	—
41	C₂×C₄×C₆₇	C₂²×C₄	1	8	yes	—
42	C₂×C₄×C₆₇	C₂²×C₄	1	3	yes	—
43	C₂×C₄×C₆₇	C₂×D₄	1	6	yes	—
44	C₂×C₄×C₆₇	C₂×D₄	1	6	yes	—
45	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
46	C₆₇⋊C₈	—	2	8	no	—
47	C₂²×C₆₇	—	2	6	no	—
48	C₂²×C₆₇	—	2	6	no	—
49	C₂²×C₆₇	#101	1	4	yes	complemented
50	C₂²×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
51	C₂²×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
52	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
53	C₄×C₆₇	—	2	8	no	—
54	C₄×C₆₇	C₂×C₂²⋊C₄	1	6	yes	—
55	C₄×C₆₇	C₂²×D₄	1	2	yes	—
56	C₂×C₆₇	—	2	3	no	—
57	C₂×C₆₇	—	2	1	no	—
58	C₂×C₆₇	C₂×#101	1	6	yes	complemented
59	C₂×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×#101	—	67	1	no	maximal, Sylow
61	C₆₇	C₂²×#101	1	1	yes	characteristic, complemented, Sylow
62	C₂³×Q₈	D₆₇	1	1	yes	characteristic
63	C₂²×M₁₆	—	67	2	no	—
64	C₂×#101	—	67	12	no	—
65	C₂²×Q₈	—	2	6	no	—
66	C₂²×Q₈	—	2	2	no	—
67	C₂²×Q₈	Dic₆₇	1	6	yes	—
68	C₂²×Q₈	C₂×D₆₇	1	6	yes	—
69	C₂³×C₄	Dic₆₇	1	2	yes	—
70	C₂³×C₄	C₂×D₆₇	1	1	yes	characteristic
71	C₂×M₁₆	—	67	12	no	—
72	C₂×M₁₆	—	134	6	no	—
73	C₂²×C₈	—	134	2	no	—
74	#101	—	67	16	no	—
75	C₂⁴	C₂×Dic₆₇	1	1	yes	characteristic
76	C₂×Q₈	—	2	12	no	—
77	C₂×Q₈	—	2	12	no	—
78	C₂×Q₈	—	2	12	no	—
79	C₂×Q₈	—	2	12	no	—
80	C₂×Q₈	C₂×Dic₆₇	1	12	yes	—
81	C₂×Q₈	C₂²×D₆₇	1	4	yes	—
82	C₂²×C₄	—	2	6	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	C₂×Dic₆₇	1	12	yes	—
85	C₂²×C₄	C₂×Dic₆₇	1	3	yes	—
86	C₂²×C₄	C₂²⋊D₆₇	1	6	yes	—
87	C₂²×C₄	C₂²⋊D₆₇	1	2	yes	—
88	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
89	M₁₆	—	67	8	no	—
90	M₁₆	—	134	12	no	—
91	C₂×C₈	—	134	12	no	—
92	C₂³	—	2	4	no	—
93	C₂³	C₂²⋊Dic₆₇	1	3	yes	—
94	C₂³	C₂²⋊Dic₆₇	1	1	yes	characteristic
95	C₂³	C₂²×Dic₆₇	1	3	yes	—
96	Q₈	—	2	24	no	—
97	Q₈	—	2	8	no	—
98	C₂×C₄	—	2	12	no	—
99	C₂×C₄	—	2	12	no	—
100	C₂×C₄	C₂²⋊Dic₆₇	1	6	yes	—
101	C₂×C₄	C₂²⋊Dic₆₇	1	6	yes	—
102	C₂×C₄	C₂²×Dic₆₇	1	8	yes	—
103	C₂×C₄	C₂²×Dic₆₇	1	3	yes	—
104	C₂×C₄	C₂×C₂²⋊D₆₇	1	6	yes	—
105	C₂×C₄	C₂×C₂²⋊D₆₇	1	6	yes	—
106	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
107	C₈	—	134	8	no	—
108	C₂²	—	2	6	no	—
109	C₂²	—	2	6	no	—
110	C₂²	#14523962	1	4	yes	direct product factor
111	C₂²	C₂×C₂²⋊Dic₆₇	1	3	yes	—
112	C₂²	C₂×C₂²⋊Dic₆₇	1	3	yes	—
113	C₂²	C₂³×Dic₆₇	1	1	yes	characteristic
114	C₄	—	2	8	no	—
115	C₄	C₂×C₂²⋊Dic₆₇	1	6	yes	—
116	C₄	C₂²×C₂²⋊D₆₇	1	2	yes	—
117	C₂	—	2	3	no	—
118	C₂	—	2	1	no	—
119	C₂	C₂×#14523962	1	6	yes	direct product factor
120	C₂	C₂²×C₂²⋊Dic₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11761612

C₂

yes

maximal

C₂×#14523962

C₂

yes

maximal, direct product factor

C₂³×Q₈×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

—

C₂×#11761612

C₂²

yes

—

C₂²×C₆₇⋊C₈

—

#14523962

C₂²

yes

direct product factor

C₂²×Q₈×C₆₇

—

C₂²×Q₈×C₆₇

—

C₂²×Q₈×C₆₇

C₄

yes

—

C₂²×Q₈×C₆₇

C₂²

yes

—

C₂³×C₄×C₆₇

C₄

yes

—

C₂³×C₄×C₆₇

C₂²

yes

characteristic

#11761612

—

#11761612

C₂³

yes

—

C₂×C₆₇⋊C₈

—

C₂⁴×C₆₇

C₂×C₄

yes

characteristic

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂×C₄

yes

—

C₂×Q₈×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂²⋊C₄

yes

—

C₂³×C₆₇

C₂²⋊C₄

yes

characteristic

C₂³×C₆₇

C₂²×C₄

yes

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#101

yes

complemented

C₂²×C₆₇

C₂×C₂²⋊C₄

yes

—

C₂²×C₆₇

C₂×C₂²⋊C₄

yes

—

C₂²×C₆₇

C₂³×C₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₂×C₂²⋊C₄

yes

—

C₄×C₆₇

C₂²×D₄

yes

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂×#101

yes

complemented

C₂×C₆₇

C₂²×C₂²⋊C₄

yes

characteristic

C₂²×#101

—

maximal, Sylow

C₆₇

C₂²×#101

yes

characteristic, complemented, Sylow

C₂³×Q₈

D₆₇

yes

characteristic

C₂²×M₁₆

—

C₂×#101

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

Dic₆₇

yes

—

C₂²×Q₈

C₂×D₆₇

yes

—

C₂³×C₄

Dic₆₇

yes

—

C₂³×C₄

C₂×D₆₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

134

—

C₂²×C₈

—

134

—

#101

—

C₂⁴

C₂×Dic₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₆₇

yes

—

C₂×Q₈

C₂²×D₆₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₆₇

yes

—

C₂²×C₄

C₂×Dic₆₇

yes

—

C₂²×C₄

C₂²⋊D₆₇

yes

—

C₂²×C₄

C₂²⋊D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

—

M₁₆

—

M₁₆

—

134

—

C₂×C₈

—

134

—

C₂³

—

C₂³

C₂²⋊Dic₆₇

yes

—

C₂³

C₂²⋊Dic₆₇

yes

characteristic

C₂³

C₂²×Dic₆₇

yes

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₂²⋊Dic₆₇

yes

—

101

C₂×C₄

C₂²⋊Dic₆₇

yes

—

102

C₂×C₄

C₂²×Dic₆₇

yes

—

103

C₂×C₄

C₂²×Dic₆₇

yes

—

104

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

105

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

106

C₂×C₄

C₂³×D₆₇

yes

characteristic

107

C₈

—

134

—

108

C₂²

—

109

C₂²

—

110

C₂²

#14523962

yes

direct product factor

111

C₂²

C₂×C₂²⋊Dic₆₇

yes

—

112

C₂²

C₂×C₂²⋊Dic₆₇

yes

—

113

C₂²

C₂³×Dic₆₇

yes

characteristic

114

C₄

—

115

C₄

C₂×C₂²⋊Dic₆₇

yes

—

116

C₄

C₂²×C₂²⋊D₆₇

yes

—

117

C₂

—

118

C₂

—

119

C₂

C₂×#14523962

yes

direct product factor

120

C₂

C₂²×C₂²⋊Dic₆₇

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₆₇
commutator	C₂²×C₆₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂³×Q₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂²⋊C₄
hypercenter	C₂³×Q₈	D₆₇
hypocenter	C₆₇	C₂²×#101
LCS	G ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×Q₈

subgroup

G/H

center

C₂³

C₂²⋊Dic₆₇

commutator

C₂²×C₆₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₂³×Q₈×C₆₇

C₂

socle

C₂³×C₆₇

C₂²⋊C₄

hypercenter

C₂³×Q₈

D₆₇

hypocenter

C₆₇

C₂²×#101

LCS

G ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×Q₈

N	H ≅ G/N	type
2. C₂×#14523962	C₂	direct (N × H)
7. #14523962	C₂²	direct (N × H)
49. C₂²×C₆₇	#101	semidirect
58. C₂×C₆₇	C₂×#101	semidirect
61. C₆₇	C₂²×#101	semidirect
110. C₂²	#14523962	direct (N × H)
119. C₂	C₂×#14523962	direct (N × H)

H ≅ G/N

type

2. C₂×#14523962

C₂

direct (N × H)

7. #14523962

C₂²

direct (N × H)

49. C₂²×C₆₇

#101

semidirect

58. C₂×C₆₇

C₂×#101

semidirect

61. C₆₇

C₂²×#101

semidirect

110. C₂²

#14523962

direct (N × H)

119. C₂

C₂×#14523962

direct (N × H)

Subgroups — C₂²×#14523962 #22568679

Conjugacy classes of subgroups

Complements

Subgroups — C22×#14523962 #22568679

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×#14523962 #22568679