Group #22558039 — subgroups

Subgroups — #22558039

Order 8576.

152 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 77 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	#16863449	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523954	C₂	1	1	yes	maximal, characteristic
3	#431×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₆₇⋊C₈	C₄	1	2	yes	—
5	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
6	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
7	#14523954	—	2	1	no	—
8	#14523954	—	2	1	no	—
9	C₄×C₆₇⋊C₈	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	C₄×C₈×C₆₇	C₄	1	2	yes	—
14	C₂×C₆₇⋊C₈	—	2	2	no	—
15	C₂×C₆₇⋊C₈	—	2	2	no	—
16	C₂×C₆₇⋊C₈	—	2	2	no	—
17	C₂×C₆₇⋊C₈	C₈	1	4	yes	complemented
18	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₆₇	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₆₇	—	2	2	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	D₄	1	1	yes	characteristic
26	C₂³×C₆₇	C₄⋊C₄	1	1	yes	characteristic
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	C₂²⋊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×C₄×C₆₇	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₆₇	M₁₆	1	2	yes	—
37	C₈×C₆₇	—	4	2	no	—
38	C₆₇⋊C₈	—	2	4	no	—
39	C₆₇⋊C₈	—	4	2	no	—
40	C₂²×C₆₇	—	2	1	no	—
41	C₂²×C₆₇	—	2	1	no	—
42	C₂²×C₆₇	#95	1	1	yes	characteristic
43	C₂²×C₆₇	C₄⋊C₈	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	C₄×C₈	1	1	yes	characteristic
49	C₄×C₆₇	#97	1	1	yes	characteristic
50	C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
51	C₂×C₆₇	—	2	2	no	—
52	C₂×C₆₇	#336	1	1	yes	characteristic
53	C₂×C₆₇	H(3,ℤ/4)	1	1	yes	characteristic
54	C₂×C₆₇	#338	1	1	yes	characteristic
55	#1275	—	67	1	no	maximal, Sylow
56	C₆₇	#1275	1	1	yes	characteristic, complemented, Sylow
57	#431	—	67	1	no	—
58	#431	D₆₇	1	1	yes	characteristic
59	C₂×#97	—	67	1	no	—
60	C₂²×C₈	—	67	2	no	—
61	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
62	C₄⋊C₈	—	2	1	no	—
63	C₄⋊C₈	—	134	1	no	—
64	C₂²⋊C₈	—	2	1	no	—
65	C₂²⋊C₈	—	134	1	no	—
66	#97	—	134	1	no	—
67	#97	—	134	1	no	—
68	C₄×C₈	—	67	2	no	—
69	C₄×C₈	Dic₆₇	1	2	yes	—
70	C₂²×C₄	Dic₁₃₄	1	1	yes	characteristic
71	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
72	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	67	4	no	—
75	C₂×C₈	—	134	2	no	—
76	C₂×C₈	—	134	2	no	—
77	C₂×C₈	—	134	2	no	—
78	C₄²	C₄×D₆₇	1	1	yes	characteristic
79	C₄²	D₂₆₈	1	1	yes	characteristic
80	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
81	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
82	C₂³	Dic₆₇⋊C₄	1	1	yes	characteristic
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	C₄×Dic₆₇	1	1	yes	characteristic
86	C₂×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
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	C₄⋊Dic₆₇	1	1	yes	characteristic
91	C₂×C₄	C₈×D₆₇	1	2	yes	—
92	C₂×C₄	#11761611	1	2	yes	—
93	C₈	—	4	2	no	—
94	C₈	—	268	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	Dic₆₇⋊C₈	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	C₈×Dic₆₇	1	1	yes	characteristic
105	C₄	#14523953	1	1	yes	characteristic
106	C₄	(C₂×D₆₇)⋊C₈	1	2	yes	—
107	C₂	—	2	2	no	—
108	C₂	#16863150	1	1	yes	characteristic
109	C₂	#16863152	1	1	yes	characteristic
110	C₂	C₆₇⋊#338	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863449

C₂

yes

maximal, characteristic

C₂×#14523954

C₂

yes

maximal, characteristic

#431×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₄

yes

—

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

#14523954

—

#14523954

—

C₄×C₆₇⋊C₈

C₄

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₄

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₈

yes

complemented

C₂²×C₄×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

Q₈

yes

characteristic

C₂×C₈×C₆₇

—

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

C₂×C₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₂³×C₆₇

C₄⋊C₄

yes

characteristic

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

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₈

yes

—

C₂×C₄×C₆₇

M₁₆

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#95

yes

characteristic

C₂²×C₆₇

C₄⋊C₈

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₄×C₈

yes

characteristic

C₄×C₆₇

#97

yes

characteristic

C₄×C₆₇

C₂²⋊C₈

yes

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#431

—

#431

D₆₇

yes

characteristic

C₂×#97

—

C₂²×C₈

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₂²⋊C₈

—

C₂²⋊C₈

—

134

—

#97

—

134

—

#97

—

134

—

C₄×C₈

—

C₄×C₈

Dic₆₇

yes

—

C₂²×C₄

Dic₁₃₄

yes

characteristic

C₂²×C₄

C₄×D₆₇

yes

characteristic

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₄²

C₄×D₆₇

yes

characteristic

C₄²

D₂₆₈

yes

characteristic

C₄²

C₂×Dic₆₇

yes

characteristic

C₄²

C₂²⋊D₆₇

yes

characteristic

C₂³

Dic₆₇⋊C₄

yes

characteristic

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

C₂²⋊Dic₆₇

yes

characteristic

C₂×C₄

Dic₆₇⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₆₇

yes

characteristic

C₂×C₄

C₈×D₆₇

yes

—

C₂×C₄

#11761611

yes

—

C₈

—

C₈

—

268

—

C₈

—

134

—

C₂²

—

C₂²

—

C₂²

C₆₇⋊#95

yes

characteristic

C₂²

Dic₆₇⋊C₈

yes

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

C₈×Dic₆₇

yes

characteristic

105

C₄

#14523953

yes

characteristic

106

C₄

(C₂×D₆₇)⋊C₈

yes

—

107

C₂

—

108

C₂

#16863150

yes

characteristic

109

C₂

#16863152

yes

characteristic

110

C₂

C₆₇⋊#338

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄×Dic₆₇
commutator	C₄×C₆₇	C₄×C₈
Frattini	C₂×C₄²	C₂×D₆₇
Fitting	#431×C₆₇	C₂
socle	C₂²×C₆₇	#95
hypercenter	#431	D₆₇
hypocenter	C₆₇	#1275
LCS	G ▷ C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center

C₂×C₄

C₄×Dic₆₇

commutator

C₄×C₆₇

C₄×C₈

Frattini

C₂×C₄²

C₂×D₆₇

Fitting

#431×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
17. C₂×C₆₇⋊C₈	C₈	semidirect
56. C₆₇	#1275	semidirect

H ≅ G/N

type

17. C₂×C₆₇⋊C₈

C₈

semidirect

56. C₆₇

#1275

semidirect

Subgroups — #22558039

Conjugacy classes of subgroups

Complements