Group #22566159 — subgroups

Subgroups — #22566159

Order 8576.

416 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14524035	C₂	1	1	yes	maximal, characteristic
2	#16863334	C₂	1	8	yes	maximal
3	#16863303	C₂	1	4	yes	maximal
4	#16863242	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#14524035	—	2	4	no	—
7	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
8	C₂×Dic₆₇⋊C₄	C₂²	1	4	yes	—
9	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
10	C₆₇⋊#95	C₂²	1	4	yes	—
11	C₆₇⋊#95	C₂²	1	16	yes	—
12	C₂²×Q₈×C₆₇	C₂²	1	2	yes	—
13	C₄⋊Q₈×C₆₇	—	2	4	no	—
14	C₂×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
15	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×Dic₆₇	C₂³	1	4	yes	—
17	C₂²×Dic₆₇	C₂³	1	4	yes	—
18	C₄⋊Dic₆₇	—	2	4	no	—
19	Dic₆₇⋊C₄	—	2	8	no	—
20	C₄×Dic₆₇	—	2	4	no	—
21	C₂×Q₈×C₆₇	—	2	4	no	—
22	C₂×Q₈×C₆₇	—	2	8	no	—
23	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
24	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
25	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₆₇	—	2	8	no	—
27	C₄²×C₆₇	D₄	1	4	yes	—
28	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
29	Q₈×C₆₇	—	4	8	no	—
30	C₂×C₄×C₆₇	—	2	4	no	—
31	C₂×C₄×C₆₇	—	2	4	no	—
32	C₂×C₄×C₆₇	—	2	8	no	—
33	C₂×C₄×C₆₇	C₂×D₄	1	4	yes	—
34	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
35	C₂×C₄×C₆₇	#41	1	4	yes	—
36	C₂×Dic₆₇	—	2	4	no	—
37	C₂×Dic₆₇	—	2	4	no	—
38	C₂×Dic₆₇	—	2	8	no	—
39	C₂×Dic₆₇	—	2	8	no	—
40	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
41	C₂²×C₆₇	C₂×#41	1	2	yes	—
42	C₂²×C₆₇	2₋¹⁺⁴	1	4	yes	—
43	C₄×C₆₇	—	2	4	no	—
44	C₄×C₆₇	—	4	4	no	—
45	C₄×C₆₇	#124	1	4	yes	—
46	Dic₆₇	—	4	4	no	—
47	Dic₆₇	—	4	4	no	—
48	C₂×C₆₇	C₂×#124	1	1	yes	characteristic
49	C₂×C₆₇	#536	1	2	yes	—
50	C₂×C₆₇	#541	1	4	yes	—
51	#2642	—	67	1	no	maximal, Sylow
52	C₆₇	#2642	1	1	yes	characteristic, complemented, Sylow
53	C₂×C₄⋊Q₈	D₆₇	1	1	yes	characteristic
54	C₂×#125	—	67	1	no	—
55	#400	—	67	8	no	—
56	#391	—	67	4	no	—
57	#376	—	67	1	no	—
58	C₂²×Q₈	C₂×D₆₇	1	2	yes	—
59	C₄⋊Q₈	—	2	4	no	—
60	#125	—	134	4	no	—
61	C₂×C₄⋊C₄	—	67	4	no	—
62	C₂×C₄⋊C₄	—	67	2	no	—
63	C₂×C₄⋊C₄	C₂×D₆₇	1	4	yes	—
64	C₂×C₄²	—	67	2	no	—
65	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
66	#95	—	67	4	no	—
67	#95	—	67	16	no	—
68	C₂×Q₈	—	2	4	no	—
69	C₂×Q₈	—	2	8	no	—
70	C₂²×C₄	—	67	4	no	—
71	C₂²×C₄	—	67	4	no	—
72	C₂²×C₄	C₂²×D₆₇	1	4	yes	—
73	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
74	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
75	C₄⋊C₄	—	2	8	no	—
76	C₄⋊C₄	—	134	8	no	—
77	C₄⋊C₄	—	134	4	no	—
78	C₄²	—	134	4	no	—
79	C₄²	C₂²⋊D₆₇	1	4	yes	—
80	C₂³	C₂³×D₆₇	1	1	yes	characteristic
81	Q₈	—	4	8	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	2	8	no	—
85	C₂×C₄	—	134	4	no	—
86	C₂×C₄	—	134	4	no	—
87	C₂×C₄	—	134	8	no	—
88	C₂×C₄	—	134	8	no	—
89	C₂×C₄	C₂×C₂²⋊D₆₇	1	4	yes	—
90	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
91	C₂×C₄	#11761627	1	4	yes	—
92	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
93	C₂²	C₂×#11761627	1	2	yes	—
94	C₂²	#14524091	1	4	yes	—
95	C₄	—	2	4	no	—
96	C₄	—	4	4	no	—
97	C₄	—	268	4	no	—
98	C₄	—	268	4	no	—
99	C₄	#14524030	1	4	yes	—
100	C₂	C₂×#14524030	1	1	yes	characteristic
101	C₂	#16863970	1	2	yes	—
102	C₂	#16864007	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524035

C₂

yes

maximal, characteristic

#16863334

C₂

yes

maximal

#16863303

C₂

yes

maximal

#16863242

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₆₇

C₂

yes

maximal, characteristic

#14524035

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×Dic₆₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₆₇

C₂²

yes

—

C₄⋊Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

D₄

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

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

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂²×C₆₇

C₂×#41

yes

—

C₂²×C₆₇

2₋¹⁺⁴

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#124

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

C₂×#124

yes

characteristic

C₂×C₆₇

#536

yes

—

C₂×C₆₇

#541

yes

—

#2642

—

maximal, Sylow

C₆₇

#2642

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₆₇

yes

—

—

—

—

C₂²×Q₈

C₂×D₆₇

yes

—

C₄⋊Q₈

—

#125

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄²

—

134

—

C₄²

C₂²⋊D₆₇

yes

—

C₂³

C₂³×D₆₇

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

C₂×C₄

#11761627

yes

—

C₂²

C₂²×C₂²⋊D₆₇

yes

characteristic

C₂²

C₂×#11761627

yes

—

C₂²

#14524091

yes

—

C₄

—

C₄

—

C₄

—

268

—

C₄

—

268

—

C₄

#14524030

yes

—

100

C₂

C₂×#14524030

yes

characteristic

101

C₂

#16863970

yes

—

102

C₂

#16864007

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₆₇	C₂
hypercenter	C₂×C₄⋊Q₈	D₆₇
hypocenter	C₆₇	#2642
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₇

commutator, socle

C₂³×C₆₇

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₆₇

C₂

hypercenter

C₂×C₄⋊Q₈

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
52. C₆₇	#2642	semidirect

H ≅ G/N

type

52. C₆₇

#2642

semidirect

Subgroups — #22566159

Conjugacy classes of subgroups

Complements