Group #22558075 — subgroups

Subgroups — #22558075

Order 8576.

152 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₄×#11761612	C₂	1	1	yes	maximal, characteristic
2	#16863264	C₂	1	1	yes	maximal, characteristic
3	#432×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#11761612	—	2	1	no	—
5	#14523954	—	2	1	no	—
6	C₄×C₆₇⋊C₈	C₄	1	2	yes	complemented
7	C₆₇⋊#95	—	2	1	no	—
8	C₆₇⋊#95	—	2	1	no	—
9	C₆₇⋊#95	C₄	1	2	yes	—
10	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₆₇	—	2	1	no	—
12	C₂²⋊C₈×C₆₇	—	2	1	no	—
13	#97×C₆₇	C₄	1	2	yes	complemented
14	C₂²×Dic₆₇	—	2	2	no	—
15	#11761612	—	2	4	no	—
16	C₂×C₆₇⋊C₈	—	2	2	no	—
17	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
20	C₂×C₈×C₆₇	—	2	2	no	—
21	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₆₇	D₄	1	1	yes	characteristic
24	C₄²×C₆₇	Q₈	1	1	yes	characteristic
25	C₂³×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₆₇	—	2	2	no	—
27	C₂×C₄×C₆₇	—	4	1	no	—
28	C₂×C₄×C₆₇	—	4	1	no	—
29	C₂×C₄×C₆₇	C₄²	1	1	yes	characteristic
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	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×Dic₆₇	—	4	2	no	—
36	C₂×Dic₆₇	—	4	2	no	—
37	C₂×Dic₆₇	—	4	2	no	—
38	C₈×C₆₇	—	4	2	no	—
39	C₆₇⋊C₈	—	2	4	no	—
40	C₂²×C₆₇	—	2	1	no	—
41	C₂²×C₆₇	—	2	1	no	—
42	C₂²×C₆₇	#95	1	1	yes	characteristic
43	C₂²×C₆₇	#104	1	2	yes	—
44	C₄×C₆₇	—	2	1	no	—
45	C₄×C₆₇	—	2	1	no	—
46	C₄×C₆₇	—	2	1	no	—
47	C₄×C₆₇	—	2	1	no	—
48	C₄×C₆₇	—	2	1	no	—
49	C₄×C₆₇	—	2	1	no	—
50	Dic₆₇	—	8	2	no	—
51	C₂×C₆₇	—	2	2	no	—
52	C₂×C₆₇	H(3,ℤ/4)	1	1	yes	characteristic
53	C₂×C₆₇	#339	1	1	yes	characteristic
54	C₂×C₆₇	#344	1	1	yes	characteristic
55	#1288	—	67	1	no	maximal, Sylow
56	C₆₇	#1288	1	1	yes	characteristic, complemented, Sylow
57	#432	D₆₇	1	1	yes	characteristic
58	C₄×M₁₆	—	67	1	no	—
59	#383	—	67	1	no	—
60	C₂×M₁₆	—	134	1	no	—
61	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
62	C₄⋊C₈	—	2	1	no	—
63	C₂²⋊C₈	—	2	1	no	—
64	#97	—	134	1	no	—
65	#97	Dic₆₇	1	2	yes	complemented
66	C₄×C₈	—	67	2	no	—
67	#95	—	67	2	no	—
68	#95	—	134	1	no	—
69	#95	—	134	1	no	—
70	C₂²×C₄	—	134	2	no	—
71	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
72	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
73	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
74	M₁₆	—	134	4	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	134	2	no	—
77	C₄²	Dic₁₃₄	1	1	yes	characteristic
78	C₄²	C₄×D₆₇	1	1	yes	characteristic
79	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
80	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
81	C₂³	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	—	268	2	no	—
86	C₂×C₄	—	268	2	no	—
87	C₂×C₄	—	268	2	no	—
88	C₂×C₄	C₄×Dic₆₇	1	1	yes	characteristic
89	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
90	C₂×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
91	C₂×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
92	C₂×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
93	C₂×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
94	C₈	—	4	2	no	—
95	C₈	—	134	4	no	—
96	C₂²	—	2	1	no	—
97	C₂²	—	2	1	no	—
98	C₂²	C₆₇⋊#95	1	1	yes	characteristic
99	C₂²	#14523969	1	2	yes	—
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	536	2	no	—
107	C₂	—	2	2	no	—
108	C₂	#16863152	1	1	yes	characteristic
109	C₂	#16863157	1	1	yes	characteristic
110	C₂	#16863168	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#11761612

C₂

yes

maximal, characteristic

#16863264

C₂

yes

maximal, characteristic

#432×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

—

#14523954

—

C₄×C₆₇⋊C₈

C₄

yes

complemented

C₆₇⋊#95

—

C₆₇⋊#95

—

C₆₇⋊#95

C₄

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

#97×C₆₇

C₄

yes

complemented

C₂²×Dic₆₇

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

Q₈

yes

characteristic

C₂³×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄²

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

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#95

yes

characteristic

C₂²×C₆₇

#104

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#432

D₆₇

yes

characteristic

C₄×M₁₆

—

#383

—

C₂×M₁₆

—

134

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

134

—

#97

Dic₆₇

yes

complemented

C₄×C₈

—

#95

—

#95

—

134

—

#95

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

C₄×D₆₇

yes

characteristic

C₂²×C₄

D₂₆₈

yes

characteristic

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

M₁₆

—

134

—

C₂×C₈

—

C₂×C₈

—

134

—

C₄²

Dic₁₃₄

yes

characteristic

C₄²

C₄×D₆₇

yes

characteristic

C₄²

C₂×Dic₆₇

yes

characteristic

C₄²

C₂²⋊D₆₇

yes

characteristic

C₂³

(C₂×D₆₇)⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

268

—

C₂×C₄

—

268

—

C₂×C₄

—

268

—

C₂×C₄

C₄×Dic₆₇

yes

characteristic

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₆₇

yes

characteristic

C₂×C₄

Dic₆₇⋊C₄

yes

characteristic

C₂×C₄

Dic₆₇⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₆₇

yes

characteristic

C₈

—

C₈

—

134

—

C₂²

—

C₂²

—

C₂²

C₆₇⋊#95

yes

characteristic

C₂²

#14523969

yes

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

536

—

107

C₂

—

108

C₂

#16863152

yes

characteristic

109

C₂

#16863157

yes

characteristic

110

C₂

#16863168

yes

characteristic

	subgroup	G/H
center	C₂²	C₆₇⋊#95
commutator	C₂×C₄×C₆₇	C₄²
Frattini	C₂×C₄²	C₂×D₆₇
Fitting	#432×C₆₇	C₂
socle	C₂²×C₆₇	#95
hypercenter	#432	D₆₇
hypocenter	C₆₇	#1288
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

C₆₇⋊#95

commutator

C₂×C₄×C₆₇

C₄²

Frattini

C₂×C₄²

C₂×D₆₇

Fitting

#432×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
6. C₄×C₆₇⋊C₈	C₄	semidirect
13. #97×C₆₇	C₄	semidirect
56. C₆₇	#1288	semidirect
65. #97	Dic₆₇	semidirect

H ≅ G/N

type

6. C₄×C₆₇⋊C₈

C₄

semidirect

13. #97×C₆₇

C₄

semidirect

56. C₆₇

#1288

semidirect

65. #97

Dic₆₇

semidirect

Subgroups — #22558075

Conjugacy classes of subgroups

Complements