Group #22561385 — subgroups

Subgroups — #22561385

Order 8576.

278 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 67 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×#14523957	C₂	1	3	yes	maximal
2	#16863162	C₂	1	3	yes	maximal, complemented
3	#543×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₂²⋊D₆₇	C₂²	1	1	yes	characteristic
5	C₂×C₂²⋊Dic₆₇	C₂²	1	3	yes	—
6	#14523957	—	2	6	no	—
7	C₂²⋊Q₈×C₆₇	—	2	6	no	—
8	C₂×C₂²⋊C₄×C₆₇	C₂²	1	3	yes	—
9	C₂³×D₆₇	Q₈	1	1	yes	characteristic, complemented
10	C₂×C₂²⋊D₆₇	—	2	3	no	—
11	C₂×C₂²⋊D₆₇	—	2	3	no	—
12	C₂²×Dic₆₇	—	2	3	no	—
13	C₂²×Dic₆₇	Q₈	1	1	yes	characteristic, complemented
14	C₂²⋊Dic₆₇	—	2	3	no	—
15	C₂²⋊Dic₆₇	—	2	3	no	—
16	C₂⁴×C₆₇	C₂³	1	1	yes	characteristic
17	C₂×Q₈×C₆₇	—	2	2	no	—
18	C₂²×C₄×C₆₇	—	2	3	no	—
19	C₄⋊C₄×C₆₇	—	2	6	no	—
20	C₂²⋊C₄×C₆₇	—	2	3	no	—
21	C₂²⋊C₄×C₆₇	D₄	1	6	yes	—
22	C₂³×C₆₇	—	8	1	no	—
23	C₂³×C₆₇	C₂×D₄	1	3	yes	—
24	C₂³×C₆₇	C₂×Q₈	1	1	yes	characteristic
25	C₂³×C₆₇	#41	1	3	yes	—
26	C₂²×D₆₇	—	2	3	no	—
27	C₂²×D₆₇	—	8	1	no	—
28	Q₈×C₆₇	—	8	2	no	—
29	C₂×C₄×C₆₇	—	2	6	no	—
30	C₂×C₄×C₆₇	—	8	3	no	—
31	C₂²⋊D₆₇	—	4	1	no	—
32	C₂²⋊D₆₇	—	4	3	no	—
33	C₂×Dic₆₇	—	2	3	no	—
34	C₂×Dic₆₇	—	4	3	no	—
35	C₂×Dic₆₇	—	8	3	no	—
36	C₂²×C₆₇	—	4	3	no	—
37	C₂²×C₆₇	—	4	3	no	—
38	C₂²×C₆₇	—	4	1	no	—
39	C₂²×C₆₇	#120	1	1	yes	characteristic
40	C₂²×C₆₇	C₂²⋊Q₈	1	3	yes	—
41	C₂²×C₆₇	#124	1	3	yes	—
42	C₂×D₆₇	—	4	1	no	—
43	C₂×D₆₇	—	8	3	no	—
44	C₄×C₆₇	—	4	6	no	—
45	Dic₆₇	—	4	1	no	—
46	Dic₆₇	—	8	3	no	—
47	C₂×C₆₇	—	2	3	no	—
48	C₂×C₆₇	—	2	1	no	—
49	C₂×C₆₇	—	2	3	no	—
50	C₂×C₆₇	#393	1	1	yes	characteristic
51	D₆₇	—	8	1	no	—
52	#2032	—	67	1	no	maximal, Sylow
53	C₆₇	#2032	1	1	yes	characteristic, complemented, Sylow
54	#543	D₆₇	1	1	yes	characteristic, complemented
55	C₂×#99	—	67	3	no	—
56	#342	—	67	3	no	—
57	C₂²×D₄	—	67	1	no	—
58	C₂²⋊Q₈	—	2	6	no	—
59	C₂×C₂²⋊C₄	—	67	3	no	—
60	C₂×C₂²⋊C₄	C₂×D₆₇	1	3	yes	—
61	#99	—	134	6	no	—
62	C₂⁴	—	67	1	no	—
63	C₂⁴	C₂²×D₆₇	1	1	yes	characteristic
64	C₂×Q₈	—	2	2	no	—
65	C₂×D₄	—	134	3	no	—
66	C₂×D₄	—	134	3	no	—
67	C₂²×C₄	—	2	3	no	—
68	C₂²×C₄	—	67	1	no	—
69	C₂²×C₄	—	134	3	no	—
70	C₄⋊C₄	—	2	6	no	—
71	C₂²⋊C₄	—	2	3	no	—
72	C₂²⋊C₄	—	134	3	no	—
73	C₂²⋊C₄	—	134	3	no	—
74	C₂²⋊C₄	C₂²⋊D₆₇	1	6	yes	—
75	C₂³	—	8	1	no	—
76	C₂³	—	536	1	no	—
77	C₂³	—	134	3	no	—
78	C₂³	C₂×C₂²⋊D₆₇	1	3	yes	—
79	C₂³	Q₈×D₆₇	1	1	yes	characteristic
80	C₂³	#11761629	1	3	yes	—
81	Q₈	—	8	2	no	—
82	D₄	—	268	1	no	—
83	D₄	—	268	3	no	—
84	C₂×C₄	—	2	6	no	—
85	C₂×C₄	—	8	3	no	—
86	C₂×C₄	—	268	3	no	—
87	C₂×C₄	—	536	3	no	—
88	C₂×C₄	—	134	3	no	—
89	C₂²	—	4	3	no	—
90	C₂²	—	4	3	no	—
91	C₂²	—	4	1	no	—
92	C₂²	—	268	1	no	—
93	C₂²	—	536	3	no	—
94	C₂²	#14524012	1	1	yes	characteristic
95	C₂²	#14524019	1	3	yes	—
96	C₂²	#14524032	1	3	yes	—
97	C₄	—	4	6	no	—
98	C₄	—	268	1	no	—
99	C₄	—	536	3	no	—
100	C₂	—	2	1	no	—
101	C₂	—	2	3	no	—
102	C₂	—	2	3	no	—
103	C₂	—	536	1	no	—
104	C₂	#16863307	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523957

C₂

yes

maximal

#16863162

C₂

yes

maximal, complemented

#543×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂²×C₂²⋊D₆₇

C₂²

yes

characteristic

C₂×C₂²⋊Dic₆₇

C₂²

yes

—

#14523957

—

C₂²⋊Q₈×C₆₇

—

C₂×C₂²⋊C₄×C₆₇

C₂²

yes

—

C₂³×D₆₇

Q₈

yes

characteristic, complemented

C₂×C₂²⋊D₆₇

—

C₂×C₂²⋊D₆₇

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

Q₈

yes

characteristic, complemented

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₂⁴×C₆₇

C₂³

yes

characteristic

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

D₄

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

—

C₂³×C₆₇

C₂×Q₈

yes

characteristic

C₂³×C₆₇

#41

yes

—

C₂²×D₆₇

—

C₂²×D₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂²⋊D₆₇

—

C₂²⋊D₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#120

yes

characteristic

C₂²×C₆₇

C₂²⋊Q₈

yes

—

C₂²×C₆₇

#124

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#393

yes

characteristic

D₆₇

—

#2032

—

maximal, Sylow

C₆₇

#2032

yes

characteristic, complemented, Sylow

#543

D₆₇

yes

characteristic, complemented

C₂×#99

—

#342

—

C₂²×D₄

—

C₂²⋊Q₈

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₆₇

yes

—

#99

—

134

—

C₂⁴

—

C₂⁴

C₂²×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

134

—

C₂×D₄

—

134

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

134

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

134

—

C₂²⋊C₄

—

134

—

C₂²⋊C₄

C₂²⋊D₆₇

yes

—

C₂³

—

C₂³

—

536

—

C₂³

—

134

—

C₂³

C₂×C₂²⋊D₆₇

yes

—

C₂³

Q₈×D₆₇

yes

characteristic

C₂³

#11761629

yes

—

Q₈

—

D₄

—

268

—

D₄

—

268

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

268

—

C₂×C₄

—

536

—

C₂×C₄

—

134

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

268

—

C₂²

—

536

—

C₂²

#14524012

yes

characteristic

C₂²

#14524019

yes

—

C₂²

#14524032

yes

—

C₄

—

C₄

—

268

—

C₄

—

536

—

100

C₂

—

101

C₂

—

102

C₂

—

103

C₂

—

536

—

104

C₂

#16863307

yes

characteristic

	subgroup	G/H
center	C₂	#16863307
commutator	C₂⁴×C₆₇	C₂³
Frattini	C₂⁴	C₂²×D₆₇
Fitting	#543×C₆₇	C₂
socle	C₂×C₆₇	#393
hypercenter	#543	D₆₇
hypocenter	C₆₇	#2032
LCS	G ▷ C₂⁴×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ #543

subgroup

G/H

center

C₂

#16863307

commutator

C₂⁴×C₆₇

C₂³

Frattini

C₂⁴

C₂²×D₆₇

Fitting

#543×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂⁴×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ C₂⁴ ◁ #543

N	H ≅ G/N	type
2. #16863162	C₂	semidirect
3. #543×C₆₇	C₂	semidirect
9. C₂³×D₆₇	Q₈	semidirect
13. C₂²×Dic₆₇	Q₈	semidirect
53. C₆₇	#2032	semidirect
54. #543	D₆₇	semidirect

H ≅ G/N

type

2. #16863162

C₂

semidirect

3. #543×C₆₇

C₂

semidirect

9. C₂³×D₆₇

Q₈

semidirect

13. C₂²×Dic₆₇

Q₈

semidirect

53. C₆₇

#2032

semidirect

54. #543

D₆₇

semidirect

Subgroups — #22561385

Conjugacy classes of subgroups

Complements