Group #22565928 — subgroups

Subgroups — #22565928

Order 8576.

782 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 181 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×#14524026	C₂	1	3	yes	maximal
2	#16863315	C₂	1	4	yes	maximal, complemented
3	#16863307	C₂	1	4	yes	maximal, complemented
4	#16863259	C₂	1	3	yes	maximal, complemented
5	C₂³×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₂²⋊D₆₇	C₂²	1	1	yes	characteristic
7	#14524026	—	2	12	no	—
8	C₂×Dic₆₇⋊C₄	C₂²	1	6	yes	—
9	C₂×C₂²⋊Dic₆₇	C₂²	1	3	yes	—
10	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	6	yes	—
11	C₆₇⋊#95	C₂²	1	12	yes	—
12	C₂²×Q₈×C₆₇	—	2	12	no	—
13	C₂²×Q₈×C₆₇	C₂²	1	4	yes	—
14	C₂³×C₄×C₆₇	C₂²	1	3	yes	—
15	C₂³×D₆₇	C₂³	1	1	yes	characteristic
16	C₂×C₂²⋊D₆₇	—	2	6	no	—
17	C₂²×Dic₆₇	C₂³	1	6	yes	—
18	C₂²×Dic₆₇	C₂³	1	1	yes	characteristic
19	Dic₆₇⋊C₄	—	2	12	no	—
20	C₂²⋊Dic₆₇	—	2	6	no	—
21	(C₂×D₆₇)⋊C₄	—	2	12	no	—
22	C₂⁴×C₆₇	C₂³	1	1	yes	characteristic
23	C₂×Q₈×C₆₇	—	2	32	no	—
24	C₂×Q₈×C₆₇	—	2	24	no	—
25	C₂²×C₄×C₆₇	—	2	12	no	—
26	C₂²×C₄×C₆₇	D₄	1	12	yes	—
27	C₂²×C₄×C₆₇	C₂³	1	6	yes	—
28	C₂³×C₆₇	—	2	4	no	—
29	C₂³×C₆₇	C₂×D₄	1	6	yes	—
30	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
31	C₂²×D₆₇	—	2	3	no	—
32	C₂²×D₆₇	—	2	4	no	—
33	Q₈×C₆₇	—	2	32	no	—
34	C₂×C₄×C₆₇	—	2	24	no	—
35	C₂×C₄×C₆₇	—	2	12	no	—
36	C₂×C₄×C₆₇	C₂×D₄	1	12	yes	—
37	C₂²⋊D₆₇	—	4	4	no	—
38	C₂×Dic₆₇	—	2	6	no	—
39	C₂×Dic₆₇	—	2	3	no	—
40	C₂×Dic₆₇	—	2	12	no	—
41	C₂²×C₆₇	—	2	12	no	—
42	C₂²×C₆₇	#120	1	4	yes	—
43	C₂²×C₆₇	C₂²×D₄	1	3	yes	—
44	C₂²×C₆₇	2₋¹⁺⁴	1	4	yes	—
45	C₂×D₆₇	—	4	1	no	—
46	C₂×D₆₇	—	4	6	no	—
47	C₄×C₆₇	—	2	12	no	—
48	Dic₆₇	—	4	1	no	—
49	Dic₆₇	—	4	6	no	—
50	C₂×C₆₇	—	2	4	no	—
51	C₂×C₆₇	C₂×#120	1	1	yes	characteristic
52	C₂×C₆₇	#536	1	6	yes	—
53	D₆₇	—	8	1	no	—
54	#2614	—	67	1	no	maximal, Sylow
55	C₆₇	#2614	1	1	yes	characteristic, complemented, Sylow
56	C₂³×Q₈	D₆₇	1	1	yes	characteristic, complemented
57	C₂×#123	—	67	3	no	—
58	#395	—	67	4	no	—
59	#393	—	67	4	no	—
60	#381	—	67	3	no	—
61	C₂²×Q₈	—	2	12	no	—
62	C₂²×Q₈	C₂×D₆₇	1	4	yes	—
63	C₂²×D₄	—	67	1	no	—
64	C₂³×C₄	C₂×D₆₇	1	3	yes	—
65	#123	—	134	12	no	—
66	C₂×C₄⋊C₄	—	67	6	no	—
67	C₂×C₂²⋊C₄	—	67	3	no	—
68	C₂×C₂²⋊C₄	—	67	6	no	—
69	#95	—	67	12	no	—
70	C₂⁴	—	67	1	no	—
71	C₂⁴	C₂²×D₆₇	1	1	yes	characteristic
72	C₂×Q₈	—	2	32	no	—
73	C₂×Q₈	—	2	24	no	—
74	C₂×D₄	—	134	6	no	—
75	C₂²×C₄	—	2	12	no	—
76	C₂²×C₄	—	67	6	no	—
77	C₂²×C₄	—	67	1	no	—
78	C₂²×C₄	C₂²⋊D₆₇	1	12	yes	—
79	C₂²×C₄	C₂²×D₆₇	1	6	yes	—
80	C₄⋊C₄	—	134	12	no	—
81	C₂²⋊C₄	—	134	6	no	—
82	C₂²⋊C₄	—	134	12	no	—
83	C₂³	—	2	4	no	—
84	C₂³	—	134	3	no	—
85	C₂³	—	134	4	no	—
86	C₂³	C₂×C₂²⋊D₆₇	1	6	yes	—
87	C₂³	C₂³×D₆₇	1	1	yes	characteristic
88	Q₈	—	2	32	no	—
89	D₄	—	268	4	no	—
90	C₂×C₄	—	2	24	no	—
91	C₂×C₄	—	2	12	no	—
92	C₂×C₄	—	134	6	no	—
93	C₂×C₄	—	134	3	no	—
94	C₂×C₄	—	134	12	no	—
95	C₂×C₄	C₂×C₂²⋊D₆₇	1	12	yes	—
96	C₂²	—	2	12	no	—
97	C₂²	—	268	1	no	—
98	C₂²	—	268	6	no	—
99	C₂²	#14524012	1	4	yes	—
100	C₂²	C₂²×C₂²⋊D₆₇	1	3	yes	—
101	C₂²	#14524091	1	4	yes	—
102	C₄	—	2	12	no	—
103	C₄	—	268	1	no	—
104	C₄	—	268	6	no	—
105	C₂	—	2	4	no	—
106	C₂	—	536	1	no	—
107	C₂	C₂×#14524012	1	1	yes	characteristic
108	C₂	#16863970	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524026

C₂

yes

maximal

#16863315

C₂

yes

maximal, complemented

#16863307

C₂

yes

maximal, complemented

#16863259

C₂

yes

maximal, complemented

C₂³×Q₈×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂²×C₂²⋊D₆₇

C₂²

yes

characteristic

#14524026

—

C₂×Dic₆₇⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₆₇

C₂²

yes

—

C₂×(C₂×D₆₇)⋊C₄

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₆₇

—

C₂²×Q₈×C₆₇

C₂²

yes

—

C₂³×C₄×C₆₇

C₂²

yes

—

C₂³×D₆₇

C₂³

yes

characteristic

C₂×C₂²⋊D₆₇

—

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

characteristic

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

(C₂×D₆₇)⋊C₄

—

C₂⁴×C₆₇

C₂³

yes

characteristic

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂²⋊D₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#120

yes

—

C₂²×C₆₇

C₂²×D₄

yes

—

C₂²×C₆₇

2₋¹⁺⁴

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂×#120

yes

characteristic

C₂×C₆₇

#536

yes

—

D₆₇

—

#2614

—

maximal, Sylow

C₆₇

#2614

yes

characteristic, complemented, Sylow

C₂³×Q₈

D₆₇

yes

characteristic, complemented

—

—

—

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₆₇

yes

—

C₂²×D₄

—

C₂³×C₄

C₂×D₆₇

yes

—

#123

—

134

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

#95

—

C₂⁴

—

C₂⁴

C₂²×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

134

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

—

C₄⋊C₄

—

134

—

C₂²⋊C₄

—

134

—

C₂²⋊C₄

—

134

—

C₂³

—

C₂³

—

134

—

C₂³

—

134

—

C₂³

C₂×C₂²⋊D₆₇

yes

—

C₂³

C₂³×D₆₇

yes

characteristic

Q₈

—

D₄

—

268

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

C₂²

—

C₂²

—

268

—

C₂²

—

268

—

C₂²

#14524012

yes

—

100

C₂²

C₂²×C₂²⋊D₆₇

yes

—

101

C₂²

#14524091

yes

—

102

C₄

—

103

C₄

—

268

—

104

C₄

—

268

—

105

C₂

—

106

C₂

—

536

—

107

C₂

C₂×#14524012

yes

characteristic

108

C₂

#16863970

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	C₂³×Q₈×C₆₇	C₂
hypercenter	C₂³×Q₈	D₆₇
hypocenter	C₆₇	#2614
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₇

commutator, socle

C₂³×C₆₇

C₂⁴

Fitting

C₂³×Q₈×C₆₇

C₂

hypercenter

C₂³×Q₈

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂³ ◁ C₂³×Q₈

N	H ≅ G/N	type
2. #16863315	C₂	semidirect
3. #16863307	C₂	semidirect
4. #16863259	C₂	semidirect
5. C₂³×Q₈×C₆₇	C₂	semidirect
55. C₆₇	#2614	semidirect
56. C₂³×Q₈	D₆₇	semidirect

H ≅ G/N

type

2. #16863315

C₂

semidirect

3. #16863307

C₂

semidirect

4. #16863259

C₂

semidirect

5. C₂³×Q₈×C₆₇

C₂

semidirect

55. C₆₇

#2614

semidirect

56. C₂³×Q₈

D₆₇

semidirect

Subgroups — #22565928

Conjugacy classes of subgroups

Complements