Group #22562560 — subgroups

Subgroups — C₄×Q₈×Dic₆₇ #22562560

Order 8576.

684 conjugacy classes of subgroups; only the 425 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 425 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×Q₈×Dic₆₇	C₂	1	2	yes	maximal
2	C₄⋊C₄×Dic₆₇	C₂	1	6	yes	maximal
3	C₄×C₄⋊Dic₆₇	C₂	1	3	yes	maximal
4	C₄²×Dic₆₇	C₂	1	3	yes	maximal
5	C₂×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₆₇	C₄	1	16	yes	direct product factor
7	C₂×C₄⋊Dic₆₇	C₂²	1	6	yes	—
8	C₂×C₄×Dic₆₇	C₂²	1	3	yes	—
9	C₂×C₄×Dic₆₇	C₂²	1	6	yes	—
10	C₂×C₄×Dic₆₇	C₂²	1	6	yes	—
11	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
12	C₆₇⋊#95	C₂²	1	6	yes	—
13	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₆₇	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
16	C₂×C₄²×C₆₇	C₂²	1	3	yes	—
17	C₂²×Dic₆₇	C₂³	1	6	yes	—
18	C₂²×Dic₆₇	C₂³	1	2	yes	—
19	C₄⋊Dic₆₇	C₂×C₄	1	24	yes	—
20	C₄×Dic₆₇	—	2	6	no	—
21	C₄×Dic₆₇	—	2	12	no	—
22	C₄×Dic₆₇	C₂×C₄	1	24	yes	—
23	C₄×Dic₆₇	Q₈	1	4	yes	direct product factor
24	C₂×Q₈×C₆₇	C₂×C₄	1	8	yes	—
25	C₂×Q₈×C₆₇	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
27	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
28	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₆₇	C₂×C₄	1	12	yes	—
30	C₄²×C₆₇	C₂×C₄	1	12	yes	—
31	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
32	Q₈×C₆₇	C₄²	1	16	yes	complemented
33	C₂×C₄×C₆₇	—	2	6	no	—
34	C₂×C₄×C₆₇	C₂²×C₄	1	12	yes	—
35	C₂×C₄×C₆₇	C₂²×C₄	1	6	yes	—
36	C₂×C₄×C₆₇	C₂²×C₄	1	6	yes	—
37	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₆₇	#41	1	2	yes	—
40	C₂×Dic₆₇	—	2	6	no	—
41	C₂×Dic₆₇	—	2	6	no	—
42	C₂×Dic₆₇	C₂²×C₄	1	12	yes	—
43	C₂×Dic₆₇	C₂²×C₄	1	4	yes	—
44	C₂×Dic₆₇	C₂×Q₈	1	4	yes	—
45	C₂×Dic₆₇	#41	1	4	yes	—
46	C₂²×C₆₇	C₂³×C₄	1	2	yes	—
47	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
48	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
49	C₂²×C₆₇	C₂×#41	1	2	yes	—
50	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
51	C₄×C₆₇	—	2	6	no	—
52	C₄×C₆₇	C₂×C₄²	1	12	yes	—
53	C₄×C₆₇	C₄×Q₈	1	4	yes	complemented
54	Dic₆₇	—	2	12	no	—
55	Dic₆₇	C₄×Q₈	1	8	yes	direct product factor
56	C₂×C₆₇	C₂²×C₄²	1	1	yes	characteristic
57	C₂×C₆₇	C₂×C₄×Q₈	1	2	yes	—
58	C₂×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
59	C₂×C₆₇	C₄×#41	1	2	yes	—
60	C₂×C₆₇	C₄×#41	1	1	yes	characteristic
61	C₄²×Q₈	—	67	1	no	maximal, Sylow
62	C₆₇	C₄²×Q₈	1	1	yes	characteristic, complemented, Sylow
63	C₂×C₄×Q₈	—	67	2	no	—
64	C₂×C₄×Q₈	D₆₇	1	1	yes	characteristic
65	C₄×C₄⋊C₄	—	67	6	no	—
66	C₄×C₄⋊C₄	—	67	3	no	—
67	C₄³	—	67	3	no	—
68	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
69	C₄×Q₈	—	67	16	no	—
70	C₄×Q₈	Dic₆₇	1	8	yes	direct product factor
71	C₂×C₄⋊C₄	—	67	6	no	—
72	C₂×C₄⋊C₄	C₂×D₆₇	1	3	yes	—
73	C₂×C₄²	—	67	6	no	—
74	C₂×C₄²	—	67	1	no	—
75	C₂×C₄²	—	67	6	no	—
76	C₂×C₄²	—	67	3	no	—
77	C₂×C₄²	C₂×D₆₇	1	3	yes	—
78	#95	—	67	6	no	—
79	C₂×Q₈	C₄×D₆₇	1	8	yes	—
80	C₂×Q₈	C₂×Dic₆₇	1	4	yes	—
81	C₂²×C₄	—	67	2	no	—
82	C₂²×C₄	—	67	6	no	—
83	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
84	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
85	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
86	C₄⋊C₄	—	67	24	no	—
87	C₄⋊C₄	C₂×Dic₆₇	1	12	yes	—
88	C₄²	—	67	24	no	—
89	C₄²	—	67	4	no	—
90	C₄²	—	134	12	no	—
91	C₄²	—	134	6	no	—
92	C₄²	C₂×Dic₆₇	1	12	yes	—
93	C₂³	C₂³×D₆₇	1	1	yes	characteristic
94	Q₈	C₄×Dic₆₇	1	16	yes	direct product factor
95	C₂×C₄	—	2	6	no	—
96	C₂×C₄	—	67	4	no	—
97	C₂×C₄	—	67	4	no	—
98	C₂×C₄	—	67	4	no	—
99	C₂×C₄	—	67	12	no	—
100	C₂×C₄	—	134	6	no	—
101	C₂×C₄	—	134	6	no	—
102	C₂×C₄	C₂×C₄×D₆₇	1	12	yes	—
103	C₂×C₄	C₂²×Dic₆₇	1	6	yes	—
104	C₂×C₄	C₂²×Dic₆₇	1	6	yes	—
105	C₂×C₄	C₂²×Dic₆₇	1	2	yes	—
106	C₂×C₄	Q₈×D₆₇	1	2	yes	—
107	C₂×C₄	#11761629	1	2	yes	—
108	C₂²	C₂²×C₄×D₆₇	1	2	yes	—
109	C₂²	C₂³×Dic₆₇	1	1	yes	characteristic
110	C₂²	C₂×Q₈×D₆₇	1	1	yes	characteristic
111	C₂²	#41×D₆₇	1	2	yes	—
112	C₂²	C₂×#11761629	1	1	yes	characteristic
113	C₄	—	2	6	no	—
114	C₄	—	67	8	no	—
115	C₄	—	134	12	no	—
116	C₄	C₂×C₄×Dic₆₇	1	12	yes	—
117	C₄	Q₈×Dic₆₇	1	4	yes	direct product factor
118	C₂	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
119	C₂	C₄×Q₈×D₆₇	1	2	yes	—
120	C₂	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
121	C₂	C₄×#11761629	1	2	yes	—
122	C₂	#41×Dic₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₆₇

C₂

yes

maximal

C₄⋊C₄×Dic₆₇

C₂

yes

maximal

C₄×C₄⋊Dic₆₇

C₂

yes

maximal

C₄²×Dic₆₇

C₂

yes

maximal

C₂×C₄×Q₈×C₆₇

C₂

yes

maximal, characteristic

Q₈×Dic₆₇

C₄

yes

direct product factor

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

C₆₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₆₇

C₂²

yes

characteristic

C₄×Q₈×C₆₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

C₂×C₄

yes

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

C₂×C₄

yes

—

C₄×Dic₆₇

Q₈

yes

direct product factor

C₂×Q₈×C₆₇

C₂×C₄

yes

—

C₂×Q₈×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

C₂×C₄

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

Q₈×C₆₇

C₄²

yes

complemented

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×Q₈

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂²×C₄

yes

—

C₂×Dic₆₇

C₂²×C₄

yes

—

C₂×Dic₆₇

C₂×Q₈

yes

—

C₂×Dic₆₇

#41

yes

—

C₂²×C₆₇

C₂³×C₄

yes

—

C₂²×C₆₇

C₂³×C₄

yes

characteristic

C₂²×C₆₇

C₂²×Q₈

yes

characteristic

C₂²×C₆₇

C₂×#41

yes

—

C₂²×C₆₇

C₂×#41

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₂×C₄²

yes

—

C₄×C₆₇

C₄×Q₈

yes

complemented

Dic₆₇

—

Dic₆₇

C₄×Q₈

yes

direct product factor

C₂×C₆₇

C₂²×C₄²

yes

characteristic

C₂×C₆₇

C₂×C₄×Q₈

yes

—

C₂×C₆₇

C₂×C₄×Q₈

yes

characteristic

C₂×C₆₇

C₄×#41

yes

—

C₂×C₆₇

C₄×#41

yes

characteristic

C₄²×Q₈

—

maximal, Sylow

C₆₇

C₄²×Q₈

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

C₂×C₄×Q₈

D₆₇

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

C₄³

—

C₂²×Q₈

C₂×D₆₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

Dic₆₇

yes

direct product factor

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₇

yes

—

#95

—

C₂×Q₈

C₄×D₆₇

yes

—

C₂×Q₈

C₂×Dic₆₇

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

C₂×Dic₆₇

yes

—

C₄²

—

C₄²

—

C₄²

—

134

—

C₄²

—

134

—

C₄²

C₂×Dic₆₇

yes

—

C₂³

C₂³×D₆₇

yes

characteristic

Q₈

C₄×Dic₆₇

yes

direct product factor

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

134

—

101

C₂×C₄

—

134

—

102

C₂×C₄

C₂×C₄×D₆₇

yes

—

103

C₂×C₄

C₂²×Dic₆₇

yes

—

104

C₂×C₄

C₂²×Dic₆₇

yes

—

105

C₂×C₄

C₂²×Dic₆₇

yes

—

106

C₂×C₄

Q₈×D₆₇

yes

—

107

C₂×C₄

#11761629

yes

—

108

C₂²

C₂²×C₄×D₆₇

yes

—

109

C₂²

C₂³×Dic₆₇

yes

characteristic

110

C₂²

C₂×Q₈×D₆₇

yes

characteristic

111

C₂²

#41×D₆₇

yes

—

112

C₂²

C₂×#11761629

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

134

—

116

C₄

C₂×C₄×Dic₆₇

yes

—

117

C₄

Q₈×Dic₆₇

yes

direct product factor

118

C₂

C₂²×C₄×Dic₆₇

yes

characteristic

119

C₂

C₄×Q₈×D₆₇

yes

—

120

C₂

C₂×Q₈×Dic₆₇

yes

characteristic

121

C₂

C₄×#11761629

yes

—

122

C₂

#41×Dic₆₇

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₆₇
commutator	C₂×C₆₇	C₂²×C₄²
Frattini	C₂³	C₂³×D₆₇
Fitting	C₂×C₄×Q₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁴
hypercenter	C₂×C₄×Q₈	D₆₇
hypocenter	C₆₇	C₄²×Q₈
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂²×C₄

C₂²×D₆₇

commutator

C₂×C₆₇

C₂²×C₄²

Frattini

C₂³

C₂³×D₆₇

Fitting

C₂×C₄×Q₈×C₆₇

C₂

socle

C₂³×C₆₇

C₂⁴

hypercenter

C₂×C₄×Q₈

D₆₇

hypocenter

C₆₇

C₄²×Q₈

LCS

G ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
6. Q₈×Dic₆₇	C₄	direct (N × H)
14. C₄×Q₈×C₆₇	C₄	semidirect
23. C₄×Dic₆₇	Q₈	direct (N × H)
32. Q₈×C₆₇	C₄²	semidirect
53. C₄×C₆₇	C₄×Q₈	semidirect
55. Dic₆₇	C₄×Q₈	direct (N × H)
62. C₆₇	C₄²×Q₈	semidirect
70. C₄×Q₈	Dic₆₇	direct (N × H)
94. Q₈	C₄×Dic₆₇	direct (N × H)
117. C₄	Q₈×Dic₆₇	direct (N × H)

H ≅ G/N

type

6. Q₈×Dic₆₇

C₄

direct (N × H)

14. C₄×Q₈×C₆₇

C₄

semidirect

23. C₄×Dic₆₇

Q₈

direct (N × H)

32. Q₈×C₆₇

C₄²

semidirect

53. C₄×C₆₇

C₄×Q₈

semidirect

55. Dic₆₇

C₄×Q₈

direct (N × H)

62. C₆₇

C₄²×Q₈

semidirect

70. C₄×Q₈

Dic₆₇

direct (N × H)

94. Q₈

C₄×Dic₆₇

direct (N × H)

117. C₄

Q₈×Dic₆₇

direct (N × H)

Subgroups — C₄×Q₈×Dic₆₇ #22562560

Conjugacy classes of subgroups

Complements

Subgroups — C4×Q8×Dic67 #22562560

Conjugacy classes of subgroups

Complements

Subgroups — C₄×Q₈×Dic₆₇ #22562560