Group #22558296 — subgroups

Subgroups — #22558296

Order 8576.

132 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 73 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×#14523983	C₂	1	1	yes	maximal, characteristic
2	#16863351	C₂	1	1	yes	maximal, characteristic
3	C₂×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#11761612	C₄	1	1	yes	characteristic
5	#14524002	C₄	1	1	yes	characteristic
6	#14523983	—	2	1	no	—
7	#14523983	—	2	1	no	—
8	C₂×C₆₇⋊C₁₆	C₄	1	2	yes	complemented
9	#14523953	—	2	1	no	—
10	C₈×Dic₆₇	—	2	1	no	—
11	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	M₃₂×C₆₇	—	2	1	no	—
13	M₃₂×C₆₇	C₄	1	2	yes	complemented
14	C₂×C₁₆×C₆₇	—	2	1	no	—
15	#11761612	—	2	1	no	—
16	#11761612	C₈	1	2	yes	—
17	C₂×C₆₇⋊C₈	—	2	1	no	—
18	C₄⋊Dic₆₇	—	2	1	no	—
19	C₂²⋊Dic₆₇	C₈	1	2	yes	—
20	C₄×Dic₆₇	—	2	1	no	—
21	C₆₇⋊C₁₆	—	2	2	no	—
22	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
27	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
28	C₂×C₈×C₆₇	Q₈	1	1	yes	characteristic
29	C₁₆×C₆₇	—	2	2	no	—
30	C₂³×C₆₇	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₆₇	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₆₇	M₁₆	1	1	yes	characteristic
36	C₂×C₄×C₆₇	M₁₆	1	1	yes	characteristic
37	C₂×Dic₆₇	—	2	2	no	—
38	C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
41	C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
42	C₆₇⋊C₈	—	2	2	no	—
43	C₂²×C₆₇	—	4	1	no	—
44	C₂²×C₆₇	C₄×C₈	1	1	yes	characteristic
45	C₂²×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
46	C₂²×C₆₇	C₄⋊C₈	1	1	yes	characteristic
47	C₄×C₆₇	#95	1	1	yes	characteristic
48	C₄×C₆₇	#97	1	1	yes	characteristic
49	C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
50	C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
51	Dic₆₇	—	4	2	no	—
52	C₂×C₆₇	—	2	1	no	—
53	C₂×C₆₇	—	2	1	no	—
54	C₂×C₆₇	—	2	1	no	—
55	C₂×C₆₇	#336	1	1	yes	characteristic
56	#1378	—	67	1	no	maximal, Sylow
57	C₆₇	#1378	1	1	yes	characteristic, complemented, Sylow
58	C₂×M₃₂	—	67	1	no	—
59	C₂×M₃₂	D₆₇	1	1	yes	characteristic
60	#405	—	67	1	no	—
61	C₂×M₁₆	—	67	1	no	—
62	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
63	#117	—	67	1	no	—
64	M₃₂	—	2	1	no	—
65	M₃₂	—	134	1	no	—
66	M₃₂	—	134	1	no	—
67	M₃₂	Dic₆₇	1	2	yes	complemented
68	C₂×C₁₆	—	2	1	no	—
69	C₂×C₁₆	—	67	2	no	—
70	#97	—	134	1	no	—
71	C₄×C₈	—	134	1	no	—
72	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
73	M₁₆	—	67	2	no	—
74	M₁₆	—	134	1	no	—
75	C₂×C₈	—	134	1	no	—
76	C₂×C₈	Dic₁₃₄	1	1	yes	characteristic
77	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
78	C₂×C₈	D₂₆₈	1	1	yes	characteristic
79	C₂×C₈	C₂×Dic₆₇	1	1	yes	characteristic
80	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
81	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
82	C₄⋊C₄	—	134	1	no	—
83	C₂²⋊C₄	—	67	2	no	—
84	C₄²	—	134	1	no	—
85	C₁₆	—	2	2	no	—
86	C₁₆	—	134	2	no	—
87	C₂³	C₈×D₆₇	1	1	yes	characteristic
88	C₂×C₄	—	134	2	no	—
89	C₂×C₄	C₄×Dic₆₇	1	1	yes	characteristic
90	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
91	C₂×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
92	C₂×C₄	C₈×D₆₇	1	1	yes	characteristic
93	C₂×C₄	#11761611	1	1	yes	characteristic
94	C₂×C₄	#11761611	1	1	yes	characteristic
95	C₈	—	134	2	no	—
96	C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
97	C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
98	C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
99	C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
100	C₂²	—	4	1	no	—
101	C₂²	C₈×Dic₆₇	1	1	yes	characteristic
102	C₂²	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
103	C₂²	Dic₆₇⋊C₈	1	1	yes	characteristic
104	C₄	—	268	2	no	—
105	C₄	C₆₇⋊#95	1	1	yes	characteristic
106	C₄	#14523953	1	1	yes	characteristic
107	C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
108	C₄	Dic₆₇⋊C₈	1	1	yes	characteristic
109	C₂	—	2	1	no	—
110	C₂	—	2	1	no	—
111	C₂	—	2	1	no	—
112	C₂	#16863150	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523983

C₂

yes

maximal, characteristic

#16863351

C₂

yes

maximal, characteristic

C₂×M₃₂×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₄

yes

characteristic

#14524002

C₄

yes

characteristic

#14523983

—

#14523983

—

C₂×C₆₇⋊C₁₆

C₄

yes

complemented

#14523953

—

C₈×Dic₆₇

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

M₃₂×C₆₇

—

M₃₂×C₆₇

C₄

yes

complemented

C₂×C₁₆×C₆₇

—

#11761612

—

#11761612

C₈

yes

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

C₈

yes

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

Q₈

yes

characteristic

C₁₆×C₆₇

—

C₂³×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

C₄²

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

M₁₆

yes

characteristic

C₂×C₄×C₆₇

M₁₆

yes

characteristic

C₂×Dic₆₇

—

C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₄×C₈

yes

characteristic

C₂²×C₆₇

C₂²⋊C₈

yes

characteristic

C₂²×C₆₇

C₄⋊C₈

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

C₂²⋊C₈

yes

characteristic

C₄×C₆₇

C₄⋊C₈

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

D₆₇

yes

characteristic

#405

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

#117

—

M₃₂

—

M₃₂

—

134

—

M₃₂

—

134

—

M₃₂

Dic₆₇

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

134

—

C₄×C₈

—

134

—

C₂²×C₄

C₄×D₆₇

yes

characteristic

M₁₆

—

M₁₆

—

134

—

C₂×C₈

—

134

—

C₂×C₈

Dic₁₃₄

yes

characteristic

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

D₂₆₈

yes

characteristic

C₂×C₈

C₂×Dic₆₇

yes

characteristic

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₄⋊C₄

—

134

—

C₂²⋊C₄

—

C₄²

—

134

—

C₁₆

—

C₁₆

—

134

—

C₂³

C₈×D₆₇

yes

characteristic

C₂×C₄

—

134

—

C₂×C₄

C₄×Dic₆₇

yes

characteristic

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

C₂×C₄

Dic₆₇⋊C₄

yes

characteristic

C₂×C₄

C₈×D₆₇

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

134

—

C₈

(C₂×D₆₇)⋊C₄

yes

characteristic

C₈

C₂²⋊Dic₆₇

yes

characteristic

C₈

Dic₆₇⋊C₄

yes

characteristic

C₈

C₄⋊Dic₆₇

yes

characteristic

100

C₂²

—

101

C₂²

C₈×Dic₆₇

yes

characteristic

102

C₂²

(C₂×D₆₇)⋊C₈

yes

characteristic

103

C₂²

Dic₆₇⋊C₈

yes

characteristic

104

C₄

—

268

—

105

C₄

C₆₇⋊#95

yes

characteristic

106

C₄

#14523953

yes

characteristic

107

C₄

(C₂×D₆₇)⋊C₈

yes

characteristic

108

C₄

Dic₆₇⋊C₈

yes

characteristic

109

C₂

—

110

C₂

—

111

C₂

—

112

C₂

#16863150

yes

characteristic

	subgroup	G/H
center	C₈	(C₂×D₆₇)⋊C₄
commutator	C₂²×C₆₇	C₄×C₈
Frattini	C₂²×C₈	C₂×D₆₇
Fitting	C₂×M₃₂×C₆₇	C₂
socle	C₂×C₆₇	#336
hypercenter	C₂×M₃₂	D₆₇
hypocenter	C₆₇	#1378
LCS	G ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₈

(C₂×D₆₇)⋊C₄

commutator

C₂²×C₆₇

C₄×C₈

Frattini

C₂²×C₈

C₂×D₆₇

Fitting

C₂×M₃₂×C₆₇

C₂

socle

C₂×C₆₇

#336

hypercenter

C₂×M₃₂

D₆₇

hypocenter

C₆₇

#1378

LCS

G ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
8. C₂×C₆₇⋊C₁₆	C₄	semidirect
13. M₃₂×C₆₇	C₄	semidirect
57. C₆₇	#1378	semidirect
67. M₃₂	Dic₆₇	semidirect

H ≅ G/N

type

8. C₂×C₆₇⋊C₁₆

C₄

semidirect

13. M₃₂×C₆₇

C₄

semidirect

57. C₆₇

#1378

semidirect

67. M₃₂

Dic₆₇

semidirect

Subgroups — #22558296

Conjugacy classes of subgroups

Complements