Group #22559331 — subgroups

Subgroups — #22559331

Order 8576.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16864132	C₂	1	1	yes	maximal, characteristic
2	#16863768	C₂	1	1	yes	maximal, characteristic
3	#16863692	C₂	1	2	yes	maximal, complemented
4	#16863126	C₂	1	2	yes	maximal, complemented
5	#331×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#11761616	—	2	2	no	—
7	#14524043	C₂²	1	1	yes	characteristic
8	#14524030	—	2	2	no	—
9	#14524024	—	2	2	no	—
10	C₄×Dic₁₃₄	C₂²	1	2	yes	—
11	D₄×Dic₆₇	—	2	2	no	—
12	#14523977	—	2	2	no	—
13	#14523968	—	2	2	no	—
14	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
15	#127×C₆₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
17	C₂×Dic₁₃₄	—	2	1	no	—
18	C₂×Dic₁₃₄	D₄	1	2	yes	—
19	C₂²×Dic₆₇	—	2	2	no	—
20	#11761616	—	4	4	no	—
21	C₂×C₆₇⋊C₈	—	2	1	no	—
22	C₄⋊Dic₆₇	—	2	2	no	—
23	C₄⋊Dic₆₇	D₄	1	2	yes	—
24	Dic₆₇⋊C₄	—	2	2	no	—
25	C₂²⋊Dic₆₇	—	2	4	no	—
26	C₂²⋊Dic₆₇	—	2	2	no	—
27	C₄×Dic₆₇	—	2	2	no	—
28	C₂×D₄×C₆₇	—	2	2	no	—
29	C₂×D₄×C₆₇	D₄	1	2	yes	—
30	C₂×C₈×C₆₇	—	2	2	no	—
31	C₄²×C₆₇	C₂³	1	1	yes	characteristic
32	C₂³×C₆₇	—	2	2	no	—
33	D₄×C₆₇	—	4	2	no	—
34	D₄×C₆₇	—	4	2	no	—
35	D₄×C₆₇	—	4	2	no	—
36	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₆₇	—	2	2	no	—
39	C₂×Dic₆₇	—	2	2	no	—
40	C₂×Dic₆₇	—	4	1	no	—
41	C₂×Dic₆₇	—	8	2	no	—
42	Dic₁₃₄	—	2	2	no	—
43	Dic₁₃₄	—	2	2	no	—
44	Dic₁₃₄	—	4	2	no	—
45	C₈×C₆₇	—	4	2	no	—
46	C₆₇⋊C₈	—	2	2	no	—
47	C₂²×C₆₇	—	4	4	no	—
48	C₂²×C₆₇	—	4	2	no	—
49	C₂²×C₆₇	#120	1	1	yes	characteristic
50	C₄×C₆₇	—	2	1	no	—
51	C₄×C₆₇	#135	1	2	yes	—
52	C₄×C₆₇	#137	1	2	yes	—
53	Dic₆₇	—	4	2	no	—
54	Dic₆₇	—	4	2	no	—
55	Dic₆₇	—	8	1	no	—
56	C₂×C₆₇	—	8	2	no	—
57	C₂×C₆₇	#452	1	2	yes	—
58	C₂×C₆₇	#453	1	1	yes	characteristic
59	#1630	—	67	1	no	maximal, Sylow
60	C₆₇	#1630	1	1	yes	characteristic, complemented, Sylow
61	#559	—	67	1	no	—
62	#492	—	67	1	no	—
63	#477	—	67	2	no	—
64	#331	D₆₇	1	1	yes	characteristic
65	Q₈⋊C₈	—	67	2	no	—
66	C₂×SD₁₆	—	134	2	no	—
67	C₄⋊Q₈	—	67	1	no	—
68	#127	C₂×D₆₇	1	1	yes	characteristic
69	#124	—	134	2	no	—
70	#123	—	134	2	no	—
71	C₄×Q₈	—	67	2	no	—
72	C₄×D₄	—	134	2	no	—
73	#107	—	134	2	no	—
74	C₄⋊C₈	C₂×D₆₇	1	2	yes	—
75	#103	—	134	2	no	—
76	C₄×C₈	—	67	1	no	—
77	C₂×Q₈	—	67	2	no	—
78	C₂×Q₈	—	134	1	no	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	D₂₆₈	1	2	yes	—
81	C₂²×C₄	—	134	2	no	—
82	SD₁₆	—	268	4	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	134	1	no	—
85	C₄⋊C₄	—	67	2	no	—
86	C₄⋊C₄	—	134	2	no	—
87	C₄⋊C₄	—	134	2	no	—
88	C₂²⋊C₄	—	134	4	no	—
89	C₂²⋊C₄	—	134	2	no	—
90	C₄²	—	134	2	no	—
91	C₄²	C₂²×D₆₇	1	1	yes	characteristic
92	C₂³	—	2	2	no	—
93	Q₈	—	268	2	no	—
94	Q₈	—	134	2	no	—
95	Q₈	—	134	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	C₂×C₄	—	268	1	no	—
100	C₂×C₄	—	536	2	no	—
101	C₂×C₄	—	134	2	no	—
102	C₂×C₄	—	134	2	no	—
103	C₂×C₄	D₄×D₆₇	1	2	yes	—
104	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
105	C₈	—	4	2	no	—
106	C₈	—	134	2	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	4	2	no	—
109	C₂²	#14524013	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	268	2	no	—
112	C₄	—	268	2	no	—
113	C₄	—	536	1	no	—
114	C₄	#14524062	1	2	yes	—
115	C₄	#14524072	1	2	yes	—
116	C₂	—	8	2	no	—
117	C₂	#16863559	1	2	yes	—
118	C₂	#16863566	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16864132

C₂

yes

maximal, characteristic

#16863768

C₂

yes

maximal, characteristic

#16863692

C₂

yes

maximal, complemented

#16863126

C₂

yes

maximal, complemented

#331×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761616

—

#14524043

C₂²

yes

characteristic

#14524030

—

#14524024

—

C₄×Dic₁₃₄

C₂²

yes

—

D₄×Dic₆₇

—

#14523977

—

#14523968

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

#127×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

D₄

yes

—

C₂²×Dic₆₇

—

#11761616

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#120

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

#135

yes

—

C₄×C₆₇

#137

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#452

yes

—

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

D₆₇

yes

characteristic

Q₈⋊C₈

—

C₂×SD₁₆

—

134

—

C₄⋊Q₈

—

#127

C₂×D₆₇

yes

characteristic

#124

—

134

—

#123

—

134

—

C₄×Q₈

—

C₄×D₄

—

134

—

#107

—

134

—

C₄⋊C₈

C₂×D₆₇

yes

—

#103

—

134

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×D₄

—

C₂×D₄

D₂₆₈

yes

—

C₂²×C₄

—

134

—

SD₁₆

—

268

—

C₂×C₈

—

C₂×C₈

—

134

—

C₄⋊C₄

—

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₂²⋊C₄

—

134

—

C₂²⋊C₄

—

134

—

C₄²

—

134

—

C₄²

C₂²×D₆₇

yes

characteristic

C₂³

—

Q₈

—

268

—

Q₈

—

134

—

Q₈

—

134

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

268

—

100

C₂×C₄

—

536

—

101

C₂×C₄

—

134

—

102

C₂×C₄

—

134

—

103

C₂×C₄

D₄×D₆₇

yes

—

104

C₂×C₄

C₂×D₂₆₈

yes

characteristic

105

C₈

—

106

C₈

—

134

—

107

C₂²

—

108

C₂²

—

109

C₂²

#14524013

yes

characteristic

110

C₄

—

111

C₄

—

268

—

112

C₄

—

268

—

113

C₄

—

536

—

114

C₄

#14524062

yes

—

115

C₄

#14524072

yes

—

116

C₂

—

117

C₂

#16863559

yes

—

118

C₂

#16863566

yes

characteristic

	subgroup	G/H
center	C₂²	#14524013
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#331×C₆₇	C₂
socle	C₂²×C₆₇	#120
hypercenter	#331	D₆₇
hypocenter	C₆₇	#1630
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#14524013

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#331×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
3. #16863692	C₂	semidirect
4. #16863126	C₂	semidirect
60. C₆₇	#1630	semidirect

H ≅ G/N

type

3. #16863692

C₂

semidirect

4. #16863126

C₂

semidirect

60. C₆₇

#1630

semidirect

Subgroups — #22559331

Conjugacy classes of subgroups

Complements