Group #22558308 — subgroups

Subgroups — #22558308

Order 8576.

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

normal subgroups only

This group has 136 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×#14523979	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523975	C₂	1	1	yes	maximal, characteristic
3	C₂×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#11761612	—	2	1	no	—
5	C₂×C₄⋊Dic₆₇	—	2	1	no	—
6	#14523979	—	2	1	no	—
7	#14523979	C₄	1	2	yes	complemented
8	#14523975	—	2	1	no	—
9	#14523975	C₄	1	2	yes	—
10	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	M₃₂×C₆₇	—	2	1	no	—
12	M₃₂×C₆₇	—	2	1	no	—
13	C₂×C₁₆×C₆₇	C₄	1	2	yes	complemented
14	C₂²×Dic₆₇	—	4	1	no	—
15	#11761612	—	2	2	no	—
16	#11761612	—	4	1	no	—
17	C₂×C₆₇⋊C₈	—	4	1	no	—
18	C₄⋊Dic₆₇	—	2	2	no	—
19	C₄⋊Dic₆₇	—	4	1	no	—
20	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
21	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
25	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂×C₈×C₆₇	Q₈	1	1	yes	characteristic
27	C₁₆×C₆₇	—	2	2	no	—
28	C₂³×C₆₇	SD₁₆	1	1	yes	characteristic
29	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₆₇	D₈	1	1	yes	characteristic
33	C₂×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×C₆₇	Q₁₆	1	1	yes	characteristic
35	C₂×Dic₆₇	—	4	2	no	—
36	C₂×Dic₆₇	—	8	2	no	—
37	C₈×C₆₇	C₄²	1	1	yes	characteristic
38	C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
40	C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
41	C₆₇⋊C₈	—	4	2	no	—
42	C₂²×C₆₇	—	2	2	no	—
43	C₂²×C₆₇	#102	1	1	yes	characteristic
44	C₂²×C₆₇	#103	1	1	yes	characteristic
45	C₂²×C₆₇	#106	1	1	yes	characteristic
46	C₄×C₆₇	#95	1	1	yes	characteristic
47	C₄×C₆₇	#102	1	1	yes	characteristic
48	C₄×C₆₇	#103	1	1	yes	characteristic
49	C₄×C₆₇	#107	1	1	yes	characteristic
50	Dic₆₇	—	8	2	no	—
51	C₂×C₆₇	—	2	1	no	—
52	C₂×C₆₇	—	2	1	no	—
53	C₂×C₆₇	#340	1	1	yes	characteristic
54	C₂×C₆₇	#365	1	2	yes	—
55	#1383	—	67	1	no	maximal, Sylow
56	C₆₇	#1383	1	1	yes	characteristic, complemented, Sylow
57	C₂×M₃₂	D₆₇	1	1	yes	characteristic
58	C₂×#108	—	67	1	no	—
59	C₂×#106	—	67	1	no	—
60	C₂×M₁₆	—	134	1	no	—
61	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
62	C₂×C₄⋊C₄	—	134	1	no	—
63	M₃₂	—	2	1	no	—
64	M₃₂	—	2	1	no	—
65	C₂×C₁₆	Dic₆₇	1	2	yes	complemented
66	#108	—	67	2	no	—
67	#108	—	134	1	no	—
68	#106	—	67	2	no	—
69	#106	—	134	1	no	—
70	C₂²×C₄	—	268	1	no	—
71	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
72	M₁₆	—	268	1	no	—
73	M₁₆	—	134	2	no	—
74	C₂×C₈	—	268	1	no	—
75	C₂×C₈	Dic₁₃₄	1	1	yes	characteristic
76	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
77	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
78	C₂×C₈	C₂×Dic₆₇	1	1	yes	characteristic
79	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
80	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
81	C₄⋊C₄	—	268	1	no	—
82	C₄⋊C₄	—	134	2	no	—
83	C₁₆	—	2	2	no	—
84	C₂³	#11761617	1	1	yes	characteristic
85	C₂×C₄	—	268	2	no	—
86	C₂×C₄	—	536	2	no	—
87	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
88	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
89	C₂×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
90	C₂×C₄	D₅₃₆	1	1	yes	characteristic
91	C₂×C₄	#11761617	1	1	yes	characteristic
92	C₂×C₄	Dic₂₆₈	1	1	yes	characteristic
93	C₈	—	268	2	no	—
94	C₈	C₄×Dic₆₇	1	1	yes	characteristic
95	C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
96	C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
97	C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
98	C₂²	—	2	2	no	—
99	C₂²	#14523965	1	1	yes	characteristic
100	C₂²	#14523968	1	1	yes	characteristic
101	C₂²	#14523975	1	1	yes	characteristic
102	C₄	—	536	2	no	—
103	C₄	C₆₇⋊#95	1	1	yes	characteristic
104	C₄	#14523965	1	1	yes	characteristic
105	C₄	#14523968	1	1	yes	characteristic
106	C₄	#14523977	1	1	yes	characteristic
107	C₂	—	2	1	no	—
108	C₂	—	2	1	no	—
109	C₂	#16863159	1	1	yes	characteristic
110	C₂	#16863219	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523979

C₂

yes

maximal, characteristic

C₂×#14523975

C₂

yes

maximal, characteristic

C₂×M₃₂×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

—

C₂×C₄⋊Dic₆₇

—

#14523979

—

#14523979

C₄

yes

complemented

#14523975

—

#14523975

C₄

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

M₃₂×C₆₇

—

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

C₄

yes

complemented

C₂²×Dic₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂×C₈×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

Q₈

yes

characteristic

C₁₆×C₆₇

—

C₂³×C₆₇

SD₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

D₈

yes

characteristic

C₂×C₄×C₆₇

SD₁₆

yes

characteristic

C₂×C₄×C₆₇

Q₁₆

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

C₄²

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

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#340

yes

characteristic

C₂×C₆₇

#365

yes

—

#1383

—

maximal, Sylow

C₆₇

#1383

yes

characteristic, complemented, Sylow

C₂×M₃₂

D₆₇

yes

characteristic

C₂×#108

—

C₂×#106

—

C₂×M₁₆

—

134

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₂×C₄⋊C₄

—

134

—

M₃₂

—

M₃₂

—

C₂×C₁₆

Dic₆₇

yes

complemented

#108

—

#108

—

134

—

#106

—

#106

—

134

—

C₂²×C₄

—

268

—

C₂²×C₄

D₂₆₈

yes

characteristic

M₁₆

—

268

—

M₁₆

—

134

—

C₂×C₈

—

268

—

C₂×C₈

Dic₁₃₄

yes

characteristic

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₄×D₆₇

yes

characteristic

C₂×C₈

C₂×Dic₆₇

yes

characteristic

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₄⋊C₄

—

268

—

C₄⋊C₄

—

134

—

C₁₆

—

C₂³

#11761617

yes

characteristic

C₂×C₄

—

268

—

C₂×C₄

—

536

—

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₆₇

yes

characteristic

C₂×C₄

D₅₃₆

yes

characteristic

C₂×C₄

#11761617

yes

characteristic

C₂×C₄

Dic₂₆₈

yes

characteristic

C₈

—

268

—

C₈

C₄×Dic₆₇

yes

characteristic

C₈

C₂²⋊Dic₆₇

yes

characteristic

C₈

Dic₆₇⋊C₄

yes

characteristic

C₈

Dic₆₇⋊C₄

yes

characteristic

C₂²

—

C₂²

#14523965

yes

characteristic

100

C₂²

#14523968

yes

characteristic

101

C₂²

#14523975

yes

characteristic

102

C₄

—

536

—

103

C₄

C₆₇⋊#95

yes

characteristic

104

C₄

#14523965

yes

characteristic

105

C₄

#14523968

yes

characteristic

106

C₄

#14523977

yes

characteristic

107

C₂

—

108

C₂

—

109

C₂

#16863159

yes

characteristic

110

C₂

#16863219

yes

—

	subgroup	G/H
center	C₂²	#14523975
commutator	C₈×C₆₇	C₄²
Frattini	C₂²×C₈	C₂×D₆₇
Fitting	C₂×M₃₂×C₆₇	C₂
socle	C₂²×C₆₇	#106
hypercenter	C₂×M₃₂	D₆₇
hypocenter	C₆₇	#1383
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂²

#14523975

commutator

C₈×C₆₇

C₄²

Frattini

C₂²×C₈

C₂×D₆₇

Fitting

C₂×M₃₂×C₆₇

C₂

socle

C₂²×C₆₇

#106

hypercenter

C₂×M₃₂

D₆₇

hypocenter

C₆₇

#1383

LCS

G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
7. #14523979	C₄	semidirect
13. C₂×C₁₆×C₆₇	C₄	semidirect
56. C₆₇	#1383	semidirect
65. C₂×C₁₆	Dic₆₇	semidirect

H ≅ G/N

type

7. #14523979

C₄

semidirect

13. C₂×C₁₆×C₆₇

C₄

semidirect

56. C₆₇

#1383

semidirect

65. C₂×C₁₆

Dic₆₇

semidirect

Subgroups — #22558308

Conjugacy classes of subgroups

Complements