Group #22568249 — subgroups

Subgroups — #22568249

Order 8576.

630 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14524040	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863313	C₂	1	6	yes	maximal, complemented
3	#16863305	C₂	1	6	yes	maximal, complemented
4	#16863264	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#126×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₆₈	C₂²	1	6	yes	—
7	C₂²×C₂²⋊D₆₇	C₂²	1	4	yes	—
8	#14524040	—	2	4	no	—
9	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	12	yes	—
10	C₆₇⋊#95	C₂²	1	6	yes	—
11	#126×C₆₇	—	2	4	no	—
12	C₂×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
13	C₂×C₂²⋊C₄×C₆₇	C₂²	1	3	yes	—
14	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×D₆₇	C₂³	1	4	yes	—
16	C₂×D₂₆₈	—	2	12	no	—
17	C₂×D₂₆₈	—	4	12	no	—
18	C₂×C₂²⋊D₆₇	—	4	12	no	—
19	C₂²×Dic₆₇	C₂³	1	4	yes	—
20	(C₂×D₆₇)⋊C₄	—	4	12	no	—
21	C₂⁴×C₆₇	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
23	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
24	C₄⋊C₄×C₆₇	—	2	6	no	—
25	C₂²⋊C₄×C₆₇	—	2	6	no	—
26	C₄²×C₆₇	D₄	1	4	yes	complemented
27	C₂³×C₆₇	—	2	1	no	—
28	C₂³×C₆₇	—	2	6	no	—
29	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
30	C₂²×D₆₇	—	2	4	no	—
31	C₂²×D₆₇	—	2	12	no	—
32	C₂²×D₆₇	—	2	12	no	—
33	C₂×C₄×C₆₇	—	2	6	no	—
34	C₂×C₄×C₆₇	—	2	3	no	—
35	C₂×C₄×C₆₇	—	2	6	no	—
36	C₂×C₄×C₆₇	C₂×D₄	1	6	yes	—
37	C₂²⋊D₆₇	—	4	16	no	—
38	C₂×Dic₆₇	—	2	12	no	—
39	D₂₆₈	—	4	24	no	—
40	C₂²×C₆₇	—	4	3	no	—
41	C₂²×C₆₇	—	4	4	no	—
42	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
43	C₂²×C₆₇	H(5,𝔽₂)	1	6	yes	—
44	C₂×D₆₇	—	4	12	no	—
45	C₂×D₆₇	—	4	4	no	—
46	C₂×D₆₇	—	4	12	no	—
47	C₄×C₆₇	—	2	6	no	—
48	C₄×C₆₇	—	4	3	no	—
49	Dic₆₇	—	4	4	no	—
50	C₂×C₆₇	—	8	1	no	—
51	C₂×C₆₇	#535	1	3	yes	—
52	C₂×C₆₇	#560	1	4	yes	—
53	D₆₇	—	8	4	no	—
54	#2818	—	67	1	no	maximal, Sylow
55	C₆₇	#2818	1	1	yes	characteristic, complemented, Sylow
56	C₂×#127	—	67	1	no	—
57	C₂×#126	D₆₇	1	1	yes	characteristic, complemented
58	#394	—	67	6	no	—
59	#392	—	67	6	no	—
60	#383	—	67	1	no	—
61	C₂²×D₄	—	67	4	no	—
62	C₂²×D₄	—	67	6	no	—
63	#127	—	134	4	no	—
64	#126	—	2	4	no	—
65	C₂×C₄⋊C₄	C₂×D₆₇	1	3	yes	—
66	C₂×C₂²⋊C₄	—	67	12	no	—
67	C₂×C₂²⋊C₄	C₂×D₆₇	1	3	yes	—
68	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
69	#95	—	67	6	no	—
70	C₂⁴	—	67	4	no	—
71	C₂⁴	C₂²×D₆₇	1	1	yes	characteristic
72	C₂×D₄	—	268	12	no	—
73	C₂×D₄	—	268	12	no	—
74	C₂×D₄	—	134	12	no	—
75	C₂²×C₄	—	67	4	no	—
76	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
77	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
78	C₄⋊C₄	—	2	6	no	—
79	C₂²⋊C₄	—	2	6	no	—
80	C₂²⋊C₄	—	268	12	no	—
81	C₄²	C₂²⋊D₆₇	1	4	yes	complemented
82	C₂³	—	2	1	no	—
83	C₂³	—	2	6	no	—
84	C₂³	—	134	4	no	—
85	C₂³	—	134	12	no	—
86	C₂³	—	134	12	no	—
87	C₂³	C₂³×D₆₇	1	1	yes	characteristic
88	D₄	—	268	24	no	—
89	D₄	—	268	16	no	—
90	C₂×C₄	—	2	6	no	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	2	6	no	—
93	C₂×C₄	—	134	12	no	—
94	C₂×C₄	C₂×C₂²⋊D₆₇	1	6	yes	—
95	C₂²	—	4	4	no	—
96	C₂²	—	4	3	no	—
97	C₂²	—	268	4	no	—
98	C₂²	—	268	12	no	—
99	C₂²	—	268	12	no	—
100	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
101	C₂²	#14524089	1	6	yes	—
102	C₄	—	2	6	no	—
103	C₄	—	4	3	no	—
104	C₄	—	268	4	no	—
105	C₂	—	8	1	no	—
106	C₂	—	536	4	no	—
107	C₂	#16863960	1	3	yes	—
108	C₂	#16864137	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524040

C₂

yes

maximal, characteristic, complemented

#16863313

C₂

yes

maximal, complemented

#16863305

C₂

yes

maximal, complemented

#16863264

C₂

yes

maximal, characteristic, complemented

C₂×#126×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₆₈

C₂²

yes

—

C₂²×C₂²⋊D₆₇

C₂²

yes

—

#14524040

—

C₂×(C₂×D₆₇)⋊C₄

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

#126×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂³×D₆₇

C₂³

yes

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×C₂²⋊D₆₇

—

C₂²×Dic₆₇

C₂³

yes

—

(C₂×D₆₇)⋊C₄

—

C₂⁴×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

D₄

yes

complemented

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂⁴

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂²⋊D₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂²×C₆₇

H(5,𝔽₂)

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#535

yes

—

C₂×C₆₇

#560

yes

—

D₆₇

—

#2818

—

maximal, Sylow

C₆₇

#2818

yes

characteristic, complemented, Sylow

C₂×#127

—

C₂×#126

D₆₇

yes

characteristic, complemented

#394

—

#392

—

#383

—

C₂²×D₄

—

C₂²×D₄

—

#127

—

134

—

#126

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₆₇

yes

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

#95

—

C₂⁴

—

C₂⁴

C₂²×D₆₇

yes

characteristic

C₂×D₄

—

268

—

C₂×D₄

—

268

—

C₂×D₄

—

134

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

268

—

C₄²

C₂²⋊D₆₇

yes

complemented

C₂³

—

C₂³

—

C₂³

—

134

—

C₂³

—

134

—

C₂³

—

134

—

C₂³

C₂³×D₆₇

yes

characteristic

D₄

—

268

—

D₄

—

268

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

134

—

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

C₂²

—

C₂²

—

C₂²

—

268

—

C₂²

—

268

—

C₂²

—

268

—

100

C₂²

C₂²×C₂²⋊D₆₇

yes

characteristic

101

C₂²

#14524089

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

268

—

105

C₂

—

106

C₂

—

536

—

107

C₂

#16863960

yes

—

108

C₂

#16864137

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	C₂×#126×C₆₇	C₂
hypercenter	C₂×#126	D₆₇
hypocenter	C₆₇	#2818
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×#126

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₇

commutator, socle

C₂³×C₆₇

C₂⁴

Fitting

C₂×#126×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂³ ◁ C₂×#126

N	H ≅ G/N	type
1. C₂×#14524040	C₂	semidirect
2. #16863313	C₂	semidirect
3. #16863305	C₂	semidirect
4. #16863264	C₂	semidirect
5. C₂×#126×C₆₇	C₂	semidirect
26. C₄²×C₆₇	D₄	semidirect
55. C₆₇	#2818	semidirect
57. C₂×#126	D₆₇	semidirect
81. C₄²	C₂²⋊D₆₇	semidirect

H ≅ G/N

type

1. C₂×#14524040

C₂

semidirect

2. #16863313

C₂

semidirect

3. #16863305

C₂

semidirect

4. #16863264

C₂

semidirect

5. C₂×#126×C₆₇

C₂

semidirect

26. C₄²×C₆₇

D₄

semidirect

55. C₆₇

#2818

semidirect

57. C₂×#126

D₆₇

semidirect

81. C₄²

C₂²⋊D₆₇

semidirect

Subgroups — #22568249

Conjugacy classes of subgroups

Complements