Group #22567917 — subgroups

Subgroups — #22567917

Order 8576.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16863313	C₂	1	3	yes	maximal
2	#16863307	C₂	1	2	yes	maximal, complemented
3	#16863288	C₂	1	6	yes	maximal, complemented
4	#16863261	C₂	1	3	yes	maximal, complemented
5	#395×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₆₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₆₇	C₂²	1	3	yes	—
8	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	6	yes	—
9	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	6	yes	—
10	C₂×C₄×Dic₆₇	C₂²	1	3	yes	—
11	C₆₇⋊#95	C₂²	1	3	yes	—
12	C₆₇⋊#95	C₂²	1	6	yes	—
13	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
15	#95×C₆₇	C₂²	1	3	yes	—
16	C₂³×D₆₇	C₂³	1	2	yes	—
17	C₂×D₂₆₈	—	4	3	no	—
18	C₂²×Dic₆₇	C₂³	1	6	yes	—
19	C₄⋊Dic₆₇	—	4	3	no	—
20	(C₂×D₆₇)⋊C₄	—	2	12	no	—
21	(C₂×D₆₇)⋊C₄	—	4	6	no	—
22	C₄×Dic₆₇	—	2	6	no	—
23	C₂×Q₈×C₆₇	—	4	3	no	—
24	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
26	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
27	C₄⋊C₄×C₆₇	—	4	3	no	—
28	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
29	C₂²×D₆₇	—	2	6	no	—
30	C₂²×D₆₇	—	2	6	no	—
31	C₂²×D₆₇	—	2	2	no	—
32	Q₈×C₆₇	—	4	4	no	—
33	C₂×C₄×C₆₇	—	2	3	no	—
34	C₂×C₄×C₆₇	—	2	3	no	—
35	C₂×C₄×C₆₇	—	2	3	no	—
36	C₂×C₄×C₆₇	—	2	3	no	—
37	C₂×C₄×C₆₇	—	2	6	no	—
38	C₂×C₄×C₆₇	#41	1	6	yes	—
39	C₂×Dic₆₇	—	2	6	no	—
40	C₂×Dic₆₇	—	2	6	no	—
41	C₂×Dic₆₇	—	2	6	no	—
42	D₂₆₈	—	4	4	no	—
43	C₂²×C₆₇	C₂×#41	1	3	yes	—
44	C₂²×C₆₇	H(5,𝔽₂)	1	3	yes	—
45	C₂²×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
46	C₂×D₆₇	—	4	6	no	—
47	C₂×D₆₇	—	4	6	no	—
48	C₂×D₆₇	—	4	2	no	—
49	C₄×C₆₇	—	2	6	no	—
50	C₄×C₆₇	—	4	1	no	—
51	C₄×C₆₇	—	4	3	no	—
52	Dic₆₇	—	4	6	no	—
53	C₂×C₆₇	#539	1	3	yes	—
54	C₂×C₆₇	#555	1	3	yes	—
55	C₂×C₆₇	H(3,𝔽₄)	1	1	yes	characteristic
56	D₆₇	—	8	2	no	—
57	#2785	—	67	1	no	maximal, Sylow
58	C₆₇	#2785	1	1	yes	characteristic, complemented, Sylow
59	#395	D₆₇	1	1	yes	characteristic, complemented
60	#394	—	67	3	no	—
61	#393	—	67	2	no	—
62	#388	—	67	6	no	—
63	#382	—	67	3	no	—
64	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
65	C₂²×D₄	—	67	1	no	—
66	C₂×C₄⋊C₄	—	67	3	no	—
67	C₂×C₄⋊C₄	C₂×D₆₇	1	3	yes	—
68	C₂×C₂²⋊C₄	—	67	6	no	—
69	C₂×C₂²⋊C₄	—	67	6	no	—
70	C₂×C₄²	—	67	3	no	—
71	#95	—	67	3	no	—
72	#95	—	67	6	no	—
73	#95	C₂×D₆₇	1	3	yes	—
74	C₂⁴	—	67	2	no	—
75	C₂×Q₈	—	4	3	no	—
76	C₂×D₄	—	268	3	no	—
77	C₂²×C₄	—	67	6	no	—
78	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
79	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
80	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
81	C₄⋊C₄	—	4	3	no	—
82	C₄⋊C₄	—	268	3	no	—
83	C₂²⋊C₄	—	268	6	no	—
84	C₂²⋊C₄	—	134	12	no	—
85	C₄²	—	134	6	no	—
86	C₂³	—	134	6	no	—
87	C₂³	—	134	6	no	—
88	C₂³	—	134	2	no	—
89	C₂³	C₂³×D₆₇	1	1	yes	characteristic
90	Q₈	—	4	4	no	—
91	D₄	—	268	4	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	2	3	no	—
94	C₂×C₄	—	2	6	no	—
95	C₂×C₄	—	2	3	no	—
96	C₂×C₄	—	2	3	no	—
97	C₂×C₄	—	134	6	no	—
98	C₂×C₄	—	134	6	no	—
99	C₂×C₄	—	134	6	no	—
100	C₂×C₄	#11761629	1	6	yes	—
101	C₂²	—	268	6	no	—
102	C₂²	—	268	2	no	—
103	C₂²	—	268	6	no	—
104	C₂²	C₂×#11761629	1	3	yes	—
105	C₂²	#14524088	1	3	yes	—
106	C₂²	#14524090	1	1	yes	characteristic
107	C₄	—	2	6	no	—
108	C₄	—	4	1	no	—
109	C₄	—	4	3	no	—
110	C₄	—	268	6	no	—
111	C₂	—	536	2	no	—
112	C₂	#16863995	1	3	yes	—
113	C₂	#16864112	1	3	yes	—
114	C₂	#16864139	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863313

C₂

yes

maximal

#16863307

C₂

yes

maximal, complemented

#16863288

C₂

yes

maximal, complemented

#16863261

C₂

yes

maximal, complemented

#395×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₆₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×(C₂×D₆₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₆₇)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

#95×C₆₇

C₂²

yes

—

C₂³×D₆₇

C₂³

yes

—

C₂×D₂₆₈

—

C₂²×Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

—

(C₂×D₆₇)⋊C₄

—

(C₂×D₆₇)⋊C₄

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

—

C₂³×C₆₇

C₂⁴

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

C₂²×C₆₇

C₂×#41

yes

—

C₂²×C₆₇

H(5,𝔽₂)

yes

—

C₂²×C₆₇

2₋¹⁺⁴

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

Dic₆₇

—

C₂×C₆₇

#539

yes

—

C₂×C₆₇

#555

yes

—

C₂×C₆₇

H(3,𝔽₄)

yes

characteristic

D₆₇

—

#2785

—

maximal, Sylow

C₆₇

#2785

yes

characteristic, complemented, Sylow

#395

D₆₇

yes

characteristic, complemented

—

—

—

—

C₂²×Q₈

C₂×D₆₇

yes

characteristic

C₂²×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

#95

—

#95

C₂×D₆₇

yes

—

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

268

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂²×C₄

C₂²×D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

268

—

C₂²⋊C₄

—

268

—

C₂²⋊C₄

—

134

—

C₄²

—

134

—

C₂³

—

134

—

C₂³

—

134

—

C₂³

—

134

—

C₂³

C₂³×D₆₇

yes

characteristic

Q₈

—

D₄

—

268

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

100

C₂×C₄

#11761629

yes

—

101

C₂²

—

268

—

102

C₂²

—

268

—

103

C₂²

—

268

—

104

C₂²

C₂×#11761629

yes

—

105

C₂²

#14524088

yes

—

106

C₂²

#14524090

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

268

—

111

C₂

—

536

—

112

C₂

#16863995

yes

—

113

C₂

#16864112

yes

—

114

C₂

#16864139

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	#395×C₆₇	C₂
hypercenter	#395	D₆₇
hypocenter	C₆₇	#2785
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ #395

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₇

commutator, socle

C₂³×C₆₇

C₂⁴

Fitting

#395×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂³ ◁ #395

N	H ≅ G/N	type
2. #16863307	C₂	semidirect
3. #16863288	C₂	semidirect
4. #16863261	C₂	semidirect
5. #395×C₆₇	C₂	semidirect
58. C₆₇	#2785	semidirect
59. #395	D₆₇	semidirect

H ≅ G/N

type

2. #16863307

C₂

semidirect

3. #16863288

C₂

semidirect

4. #16863261

C₂

semidirect

5. #395×C₆₇

C₂

semidirect

58. C₆₇

#2785

semidirect

59. #395

D₆₇

semidirect

Subgroups — #22567917

Conjugacy classes of subgroups

Complements