Group #22562282 — subgroups

Subgroups — #22562282

Order 8576.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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	#16863832	C₂	1	2	yes	maximal, complemented
2	#16863514	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863217	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863209	C₂	1	2	yes	maximal, complemented
5	#493×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14524065	C₂²	1	1	yes	characteristic
7	#14524049	—	2	1	no	—
8	#14523987	—	2	2	no	—
9	#14523984	—	2	2	no	—
10	#14523983	C₂²	1	2	yes	—
11	#14523979	C₂²	1	1	yes	characteristic
12	#14523971	—	2	1	no	—
13	C₂×D₈×C₆₇	—	2	1	no	—
14	C₂×D₈×C₆₇	C₂²	1	2	yes	—
15	#127×C₆₇	—	2	1	no	—
16	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#11761628	—	2	1	no	—
18	Dic₂₆₈	D₄	1	1	yes	characteristic, complemented
19	#11761617	—	2	1	no	—
20	D₅₃₆	D₄	1	1	yes	characteristic, complemented
21	#11761612	—	2	1	no	—
22	#11761611	—	4	1	no	—
23	C₈×D₆₇	—	4	1	no	—
24	C₆₇⋊C₁₆	—	2	2	no	—
25	C₂×D₄×C₆₇	—	2	2	no	—
26	C₂×D₄×C₆₇	—	4	2	no	—
27	D₈×C₆₇	—	2	2	no	—
28	D₈×C₆₇	—	2	2	no	—
29	D₈×C₆₇	—	4	1	no	—
30	D₈×C₆₇	—	4	1	no	—
31	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₄²×C₆₇	D₄	1	1	yes	characteristic
34	C₂³×C₆₇	—	4	2	no	—
35	D₄×C₆₇	—	4	2	no	—
36	D₄×C₆₇	—	4	2	no	—
37	D₄×C₆₇	—	8	4	no	—
38	C₂×C₄×C₆₇	—	2	1	no	—
39	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₂²⋊D₆₇	—	4	1	no	—
41	D₂₆₈	—	2	1	no	—
42	C₄×D₆₇	—	4	1	no	—
43	Dic₁₃₄	—	2	1	no	—
44	C₈×C₆₇	D₈	1	1	yes	characteristic
45	C₈×C₆₇	D₈	1	1	yes	characteristic
46	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₈×C₆₇	#41	1	1	yes	characteristic
48	C₆₇⋊C₈	—	4	1	no	—
49	C₂²×C₆₇	—	8	4	no	—
50	C₂²×C₆₇	—	8	2	no	—
51	C₂²×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
52	C₂×D₆₇	—	4	1	no	—
53	C₄×C₆₇	—	2	2	no	—
54	C₄×C₆₇	#121	1	1	yes	characteristic
55	C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
56	Dic₆₇	—	4	1	no	—
57	C₂×C₆₇	—	2	1	no	—
58	C₂×C₆₇	—	16	2	no	—
59	C₂×C₆₇	#459	1	1	yes	characteristic
60	D₆₇	—	8	1	no	—
61	#2213	—	67	1	no	maximal, Sylow
62	C₆₇	#2213	1	1	yes	characteristic, complemented, Sylow
63	#509	—	67	2	no	—
64	#493	D₆₇	1	1	yes	characteristic, complemented
65	#443	—	67	1	no	—
66	#364	—	67	1	no	—
67	#361	—	67	2	no	—
68	#135	—	67	1	no	—
69	C₂×D₈	—	2	1	no	—
70	C₂×D₈	C₂×D₆₇	1	2	yes	—
71	#131	—	134	1	no	—
72	#127	—	2	1	no	—
73	SD₃₂	—	134	2	no	—
74	D₁₆	—	134	2	no	—
75	M₃₂	—	67	2	no	—
76	#108	—	67	1	no	—
77	#104	—	134	1	no	—
78	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
79	#41	—	134	1	no	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	—	4	2	no	—
82	Q₁₆	—	67	1	no	—
83	SD₁₆	—	134	1	no	—
84	D₈	—	2	2	no	—
85	D₈	—	2	2	no	—
86	D₈	—	4	1	no	—
87	D₈	—	4	1	no	—
88	D₈	—	67	1	no	—
89	M₁₆	—	268	1	no	—
90	M₁₆	—	134	1	no	—
91	C₂×C₈	—	268	1	no	—
92	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
94	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
95	C₁₆	—	134	2	no	—
96	C₂³	—	4	2	no	—
97	Q₈	—	134	1	no	—
98	D₄	—	4	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	8	4	no	—
101	D₄	—	268	1	no	—
102	D₄	—	134	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	268	1	no	—
105	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
106	C₈	—	268	1	no	—
107	C₈	#11761613	1	1	yes	characteristic
108	C₈	#11761613	1	1	yes	characteristic
109	C₈	D₄×D₆₇	1	1	yes	characteristic
110	C₈	#11761627	1	1	yes	characteristic
111	C₂²	—	8	4	no	—
112	C₂²	—	8	2	no	—
113	C₂²	—	268	1	no	—
114	C₂²	#14524069	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	268	1	no	—
117	C₄	#14524014	1	1	yes	characteristic
118	C₄	C₂×#11761613	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	16	2	no	—
121	C₂	—	536	1	no	—
122	C₂	#16863588	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863832

C₂

yes

maximal, complemented

#16863514

C₂

yes

maximal, characteristic, complemented

#16863217

C₂

yes

maximal, characteristic, complemented

#16863209

C₂

yes

maximal, complemented

#493×C₆₇

C₂

yes

maximal, characteristic, complemented

#14524065

C₂²

yes

—

—

—

C₂²

yes

—

#14523979

C₂²

yes

characteristic

#14523971

—

C₂×D₈×C₆₇

—

C₂×D₈×C₆₇

C₂²

yes

—

#127×C₆₇

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

#11761628

—

Dic₂₆₈

D₄

yes

characteristic, complemented

#11761617

—

D₅₃₆

D₄

yes

characteristic, complemented

#11761612

—

#11761611

—

C₈×D₆₇

—

C₆₇⋊C₁₆

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

D₈×C₆₇

—

D₈×C₆₇

—

D₈×C₆₇

—

D₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₂³×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²⋊D₆₇

—

D₂₆₈

—

C₄×D₆₇

—

Dic₁₃₄

—

C₈×C₆₇

D₈

yes

characteristic

C₈×C₆₇

D₈

yes

characteristic

C₈×C₆₇

C₂×D₄

yes

characteristic

C₈×C₆₇

#41

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#121

yes

characteristic

C₄×C₆₇

C₂×D₈

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#459

yes

characteristic

D₆₇

—

#2213

—

maximal, Sylow

C₆₇

#2213

yes

characteristic, complemented, Sylow

#509

—

#493

D₆₇

yes

characteristic, complemented

—

—

—

—

C₂×D₈

—

C₂×D₈

C₂×D₆₇

yes

—

#131

—

134

—

#127

—

SD₃₂

—

134

—

D₁₆

—

134

—

M₃₂

—

#108

—

#104

—

134

—

C₄×C₈

C₂×D₆₇

yes

characteristic

#41

—

134

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

SD₁₆

—

134

—

D₈

—

D₈

—

D₈

—

D₈

—

D₈

—

M₁₆

—

268

—

M₁₆

—

134

—

C₂×C₈

—

268

—

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄²

C₂²⋊D₆₇

yes

characteristic

C₁₆

—

134

—

C₂³

—

Q₈

—

134

—

D₄

—

D₄

—

100

D₄

—

101

D₄

—

268

—

102

D₄

—

134

—

103

C₂×C₄

—

104

C₂×C₄

—

268

—

105

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

106

C₈

—

268

—

107

C₈

#11761613

yes

characteristic

108

C₈

#11761613

yes

characteristic

109

C₈

D₄×D₆₇

yes

characteristic

110

C₈

#11761627

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

—

268

—

114

C₂²

#14524069

yes

characteristic

115

C₄

—

116

C₄

—

268

—

117

C₄

#14524014

yes

characteristic

118

C₄

C₂×#11761613

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

—

536

—

122

C₂

#16863588

yes

characteristic

	subgroup	G/H
center	C₂	#16863588
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#493×C₆₇	C₂
socle	C₂×C₆₇	#459
hypercenter	#493	D₆₇
hypocenter	C₆₇	#2213
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #493

subgroup

G/H

center

C₂

#16863588

commutator

C₂×C₈×C₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

#493×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #493

N	H ≅ G/N	type
1. #16863832	C₂	semidirect
2. #16863514	C₂	semidirect
3. #16863217	C₂	semidirect
4. #16863209	C₂	semidirect
5. #493×C₆₇	C₂	semidirect
18. Dic₂₆₈	D₄	semidirect
20. D₅₃₆	D₄	semidirect
62. C₆₇	#2213	semidirect
64. #493	D₆₇	semidirect

H ≅ G/N

type

1. #16863832

C₂

semidirect

2. #16863514

C₂

semidirect

3. #16863217

C₂

semidirect

4. #16863209

C₂

semidirect

5. #493×C₆₇

C₂

semidirect

18. Dic₂₆₈

D₄

semidirect

20. D₅₃₆

D₄

semidirect

62. C₆₇

#2213

semidirect

64. #493

D₆₇

semidirect

Subgroups — #22562282

Conjugacy classes of subgroups

Complements