Group #22558102 — subgroups

Subgroups — #22558102

Order 8576.

172 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16863453	C₂	1	1	yes	maximal, characteristic
2	C₄×#11761612	C₂	1	1	yes	maximal, characteristic
3	#376×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#11761612	—	2	1	no	—
5	C₄⋊(C₆₇⋊C₈)	C₄	1	2	yes	complemented
6	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
7	#14523954	—	2	1	no	—
8	#14523954	C₄	1	2	yes	complemented
9	C₄×C₆₇⋊C₈	—	2	1	no	—
10	C₂×C₄²×C₆₇	—	2	1	no	—
11	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
12	#95×C₆₇	—	2	1	no	—
13	#95×C₆₇	C₄	1	2	yes	—
14	#11761612	—	2	4	no	—
15	C₂×C₆₇⋊C₈	—	2	2	no	—
16	C₂×C₆₇⋊C₈	—	2	2	no	—
17	C₂²×C₄×C₆₇	—	2	2	no	—
18	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
21	C₄²×C₆₇	—	2	4	no	—
22	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₆₇	D₄	1	1	yes	characteristic
25	C₄²×C₆₇	Q₈	1	1	yes	characteristic
26	C₂³×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₆₇	—	2	2	no	—
28	C₂×C₄×C₆₇	—	2	2	no	—
29	C₂×C₄×C₆₇	—	2	4	no	—
30	C₂×C₄×C₆₇	—	4	1	no	—
31	C₂×C₄×C₆₇	—	4	2	no	—
32	C₂×C₄×C₆₇	—	4	2	no	—
33	C₂×C₄×C₆₇	C₄²	1	1	yes	characteristic
34	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
39	C₆₇⋊C₈	—	2	4	no	—
40	C₆₇⋊C₈	—	4	2	no	—
41	C₂²×C₆₇	—	2	1	no	—
42	C₂²×C₆₇	—	2	1	no	—
43	C₂²×C₆₇	#95	1	1	yes	characteristic
44	C₂²×C₆₇	#104	1	2	yes	—
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	C₄×C₆₇	—	4	4	no	—
51	C₄×C₆₇	#104	1	2	yes	—
52	C₂×C₆₇	—	2	2	no	—
53	C₂×C₆₇	#339	1	1	yes	characteristic
54	C₂×C₆₇	#339	1	1	yes	characteristic
55	C₂×C₆₇	#344	1	1	yes	characteristic
56	#1300	—	67	1	no	maximal, Sylow
57	C₆₇	#1300	1	1	yes	characteristic, complemented, Sylow
58	#432	—	67	1	no	—
59	C₄×M₁₆	—	67	1	no	—
60	#376	D₆₇	1	1	yes	characteristic
61	C₂×M₁₆	—	134	1	no	—
62	C₂×C₄²	—	2	1	no	—
63	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
64	C₄⋊C₈	—	67	2	no	—
65	C₂²⋊C₈	—	134	1	no	—
66	#97	—	67	2	no	—
67	#97	—	134	1	no	—
68	C₄×C₈	—	134	1	no	—
69	#95	—	2	1	no	—
70	#95	Dic₆₇	1	2	yes	—
71	C₂²×C₄	—	2	2	no	—
72	C₂²×C₄	C₂×Dic₆₇	1	1	yes	characteristic
73	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
74	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
75	M₁₆	—	134	4	no	—
76	C₂×C₈	—	134	2	no	—
77	C₂×C₈	—	134	2	no	—
78	C₄²	—	2	4	no	—
79	C₄²	Dic₁₃₄	1	1	yes	characteristic
80	C₄²	C₄×D₆₇	1	1	yes	characteristic
81	C₄²	C₄×D₆₇	1	1	yes	characteristic
82	C₄²	D₂₆₈	1	1	yes	characteristic
83	C₂³	C₂²⋊Dic₆₇	1	1	yes	characteristic
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	4	1	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	C₄×Dic₆₇	1	1	yes	characteristic
91	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
92	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
93	C₂×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
94	C₂×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
95	C₂×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
96	C₈	—	268	2	no	—
97	C₈	—	134	4	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	C₆₇⋊#95	1	1	yes	characteristic
101	C₂²	#14523970	1	2	yes	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	—	4	4	no	—
108	C₄	#14523971	1	2	yes	—
109	C₂	—	2	2	no	—
110	C₂	#16863155	1	1	yes	characteristic
111	C₂	#16863156	1	1	yes	characteristic
112	C₂	#16863166	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863453

C₂

yes

maximal, characteristic

C₄×#11761612

C₂

yes

maximal, characteristic

#376×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

—

C₄⋊(C₆₇⋊C₈)

C₄

yes

complemented

C₂²⋊(C₆₇⋊C₈)

—

#14523954

—

#14523954

C₄

yes

complemented

C₄×C₆₇⋊C₈

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

#95×C₆₇

—

#95×C₆₇

C₄

yes

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₄²×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

—

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

C₆₇⋊C₈

—

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

—

C₄×C₆₇

#104

yes

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#432

—

C₄×M₁₆

—

#376

D₆₇

yes

characteristic

C₂×M₁₆

—

134

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

134

—

#97

—

#97

—

134

—

C₄×C₈

—

134

—

#95

—

#95

Dic₆₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₆₇

yes

characteristic

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

M₁₆

—

134

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₄²

—

C₄²

Dic₁₃₄

yes

characteristic

C₄²

C₄×D₆₇

yes

characteristic

C₄²

C₄×D₆₇

yes

characteristic

C₄²

D₂₆₈

yes

characteristic

C₂³

C₂²⋊Dic₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₆₇

yes

characteristic

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

characteristic

C₂×C₄

Dic₆₇⋊C₄

yes

characteristic

C₂×C₄

Dic₆₇⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₆₇

yes

characteristic

C₈

—

268

—

C₈

—

134

—

C₂²

—

C₂²

—

100

C₂²

C₆₇⋊#95

yes

characteristic

101

C₂²

#14523970

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

#14523971

yes

—

109

C₂

—

110

C₂

#16863155

yes

characteristic

111

C₂

#16863156

yes

characteristic

112

C₂

#16863166

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₆₇)⋊C₄
commutator	C₂×C₄×C₆₇	C₄²
Frattini	C₂×C₄²	C₂×D₆₇
Fitting	#376×C₆₇	C₂
socle	C₂²×C₆₇	#95
hypercenter	#376	D₆₇
hypocenter	C₆₇	#1300
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

(C₂×D₆₇)⋊C₄

commutator

C₂×C₄×C₆₇

C₄²

Frattini

C₂×C₄²

C₂×D₆₇

Fitting

#376×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
5. C₄⋊(C₆₇⋊C₈)	C₄	semidirect
8. #14523954	C₄	semidirect
57. C₆₇	#1300	semidirect

H ≅ G/N

type

5. C₄⋊(C₆₇⋊C₈)

C₄

semidirect

8. #14523954

C₄

semidirect

57. C₆₇

#1300

semidirect

Subgroups — #22558102

Conjugacy classes of subgroups

Complements