Group #22558310 — subgroups

Subgroups — #22558310

Order 8576.

132 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 69 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	#16863434	C₂	1	1	yes	maximal, characteristic
3	#427×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#14524002	—	2	1	no	—
5	#14523983	—	2	1	no	—
6	#14523983	—	2	1	no	—
7	C₂×C₆₇⋊C₁₆	C₄	1	2	yes	complemented
8	#14523977	—	2	1	no	—
9	#14523975	C₄	1	2	yes	complemented
10	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	#117×C₆₇	—	2	1	no	—
12	#107×C₆₇	—	2	1	no	—
13	#106×C₆₇	C₄	1	2	yes	complemented
14	C₄⋊Dic₆₇	—	2	2	no	—
15	C₂²⋊Dic₆₇	—	4	1	no	—
16	C₄×Dic₆₇	—	4	1	no	—
17	C₆₇⋊C₁₆	—	2	2	no	—
18	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
19	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
23	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
24	C₂×C₈×C₆₇	Q₈	1	1	yes	characteristic
25	C₄⋊C₄×C₆₇	—	2	2	no	—
26	C₂²⋊C₄×C₆₇	—	4	1	no	—
27	C₄²×C₆₇	—	4	1	no	—
28	C₂³×C₆₇	D₈	1	1	yes	characteristic
29	C₂×C₄×C₆₇	—	4	2	no	—
30	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
35	C₂×C₄×C₆₇	Q₁₆	1	1	yes	characteristic
36	C₂×Dic₆₇	—	4	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	C₂²×C₆₇	—	4	1	no	—
42	C₂²×C₆₇	#102	1	1	yes	characteristic
43	C₂²×C₆₇	#102	1	1	yes	characteristic
44	C₂²×C₆₇	#107	1	1	yes	characteristic
45	C₄×C₆₇	—	8	2	no	—
46	C₄×C₆₇	#95	1	1	yes	characteristic
47	C₄×C₆₇	#103	1	1	yes	characteristic
48	C₄×C₆₇	#103	1	1	yes	characteristic
49	C₄×C₆₇	#106	1	1	yes	characteristic
50	Dic₆₇	—	8	2	no	—
51	C₂×C₆₇	—	2	1	no	—
52	C₂×C₆₇	—	2	1	no	—
53	C₂×C₆₇	—	2	1	no	—
54	C₂×C₆₇	#340	1	1	yes	characteristic
55	#1384	—	67	1	no	maximal, Sylow
56	C₆₇	#1384	1	1	yes	characteristic, complemented, Sylow
57	C₂×M₃₂	—	67	1	no	—
58	#427	—	67	1	no	—
59	#427	D₆₇	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
61	#117	—	2	1	no	—
62	#117	—	134	1	no	—
63	M₃₂	—	134	1	no	—
64	M₃₂	—	134	1	no	—
65	C₂×C₁₆	—	67	2	no	—
66	#107	—	2	1	no	—
67	#107	—	134	1	no	—
68	#106	—	67	2	no	—
69	#106	Dic₆₇	1	2	yes	complemented
70	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
71	C₂×C₈	Dic₁₃₄	1	1	yes	characteristic
72	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
73	C₂×C₈	C₄×D₆₇	1	1	yes	characteristic
74	C₂×C₈	D₂₆₈	1	1	yes	characteristic
75	C₂×C₈	C₂×Dic₆₇	1	1	yes	characteristic
76	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	134	2	no	—
79	C₂²⋊C₄	—	4	1	no	—
80	C₂²⋊C₄	—	268	1	no	—
81	C₄²	—	4	1	no	—
82	C₄²	—	268	1	no	—
83	C₁₆	—	134	2	no	—
84	C₂³	#11761613	1	1	yes	characteristic
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	—	268	2	no	—
87	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
88	C₂×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
89	C₂×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
90	C₂×C₄	#11761615	1	1	yes	characteristic
91	C₂×C₄	#11761616	1	1	yes	characteristic
92	C₂×C₄	#11761618	1	1	yes	characteristic
93	C₈	C₄×Dic₆₇	1	1	yes	characteristic
94	C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
95	C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
96	C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
97	C₂²	—	4	1	no	—
98	C₂²	#14523963	1	1	yes	characteristic
99	C₂²	#14523964	1	1	yes	characteristic
100	C₂²	#14523976	1	1	yes	characteristic
101	C₄	—	8	2	no	—
102	C₄	—	536	2	no	—
103	C₄	C₆₇⋊#95	1	1	yes	characteristic
104	C₄	#14523966	1	1	yes	characteristic
105	C₄	#14523967	1	1	yes	characteristic
106	C₄	#14523974	1	1	yes	characteristic
107	C₂	—	2	1	no	—
108	C₂	—	2	1	no	—
109	C₂	—	2	1	no	—
110	C₂	#16863158	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523983

C₂

yes

maximal, characteristic

#16863434

C₂

yes

maximal, characteristic

#427×C₆₇

C₂

yes

maximal, characteristic

#14524002

—

#14523983

—

#14523983

—

C₂×C₆₇⋊C₁₆

C₄

yes

complemented

#14523977

—

#14523975

C₄

yes

complemented

C₂²×C₈×C₆₇

C₂²

yes

characteristic

#117×C₆₇

—

#107×C₆₇

—

#106×C₆₇

C₄

yes

complemented

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

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

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

D₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

SD₁₆

yes

characteristic

C₂×C₄×C₆₇

SD₁₆

yes

characteristic

C₂×C₄×C₆₇

Q₁₆

yes

characteristic

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

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

#427

—

#427

D₆₇

yes

characteristic

C₂²×C₈

C₂×D₆₇

yes

characteristic

#117

—

#117

—

134

—

M₃₂

—

134

—

M₃₂

—

134

—

C₂×C₁₆

—

#107

—

#107

—

134

—

#106

—

#106

Dic₆₇

yes

complemented

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

Dic₁₃₄

yes

characteristic

C₂×C₈

C₄×D₆₇

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

C₄⋊C₄

—

C₄⋊C₄

—

134

—

C₂²⋊C₄

—

C₂²⋊C₄

—

268

—

C₄²

—

C₄²

—

268

—

C₁₆

—

134

—

C₂³

#11761613

yes

characteristic

C₂×C₄

—

C₂×C₄

—

268

—

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₆₇

yes

characteristic

C₂×C₄

Dic₆₇⋊C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

C₄×Dic₆₇

yes

characteristic

C₈

(C₂×D₆₇)⋊C₄

yes

characteristic

C₈

Dic₆₇⋊C₄

yes

characteristic

C₈

C₄⋊Dic₆₇

yes

characteristic

C₂²

—

C₂²

yes

C₂²

yes

100

C₂²

#14523976

yes

characteristic

101

C₄

—

102

C₄

—

536

—

103

C₄

C₆₇⋊#95

yes

characteristic

104

C₄

#14523966

yes

characteristic

105

C₄

#14523967

yes

characteristic

106

C₄

#14523974

yes

characteristic

107

C₂

—

108

C₂

—

109

C₂

—

110

C₂

#16863158

yes

characteristic

	subgroup	G/H
center	C₄	#14523974
commutator	C₈×C₆₇	C₄²
Frattini	C₂²×C₈	C₂×D₆₇
Fitting	#427×C₆₇	C₂
socle	C₂×C₆₇	#340
hypercenter	#427	D₆₇
hypocenter	C₆₇	#1384
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

subgroup

G/H

center

C₄

#14523974

commutator

C₈×C₆₇

C₄²

Frattini

C₂²×C₈

C₂×D₆₇

Fitting

#427×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

N	H ≅ G/N	type
7. C₂×C₆₇⋊C₁₆	C₄	semidirect
9. #14523975	C₄	semidirect
13. #106×C₆₇	C₄	semidirect
56. C₆₇	#1384	semidirect
69. #106	Dic₆₇	semidirect

H ≅ G/N

type

7. C₂×C₆₇⋊C₁₆

C₄

semidirect

9. #14523975

C₄

semidirect

13. #106×C₆₇

C₄

semidirect

56. C₆₇

#1384

semidirect

69. #106

Dic₆₇

semidirect

Subgroups — #22558310

Conjugacy classes of subgroups

Complements