Group #22562285 — subgroups

Subgroups — #22562285

Order 8576.

170 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 49 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	#16863831	C₂	1	2	yes	maximal, complemented
2	#16863773	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863217	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863208	C₂	1	2	yes	maximal, complemented
5	#443×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₅₃₆	—	2	1	no	—
7	C₂×D₅₃₆	C₂²	1	2	yes	—
8	#14524040	—	2	1	no	—
9	#14523986	—	2	2	no	—
10	#14523984	—	2	2	no	—
11	#14523983	C₂²	1	2	yes	—
12	#135×C₆₇	C₂²	1	1	yes	characteristic
13	#131×C₆₇	—	2	1	no	—
14	#108×C₆₇	C₂²	1	1	yes	characteristic
15	#104×C₆₇	—	2	1	no	—
16	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×D₂₆₈	—	2	2	no	—
18	C₂×D₂₆₈	—	4	2	no	—
19	D₅₃₆	—	2	2	no	—
20	D₅₃₆	—	2	2	no	—
21	D₅₃₆	—	4	1	no	—
22	D₅₃₆	—	4	1	no	—
23	C₆₇⋊C₁₆	—	2	2	no	—
24	#41×C₆₇	—	2	1	no	—
25	Q₁₆×C₆₇	D₄	1	1	yes	characteristic, complemented
26	SD₁₆×C₆₇	—	2	1	no	—
27	D₈×C₆₇	D₄	1	1	yes	characteristic, complemented
28	M₁₆×C₆₇	—	2	1	no	—
29	M₁₆×C₆₇	—	4	1	no	—
30	C₂×C₈×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	C₂²×D₆₇	—	4	2	no	—
35	Q₈×C₆₇	—	2	1	no	—
36	D₄×C₆₇	—	2	1	no	—
37	D₄×C₆₇	—	4	1	no	—
38	C₂×C₄×C₆₇	—	2	1	no	—
39	C₂×C₄×C₆₇	—	4	1	no	—
40	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
41	D₂₆₈	—	4	2	no	—
42	D₂₆₈	—	4	2	no	—
43	D₂₆₈	—	8	4	no	—
44	C₈×C₆₇	—	4	1	no	—
45	C₈×C₆₇	D₈	1	1	yes	characteristic
46	C₈×C₆₇	D₈	1	1	yes	characteristic
47	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₈×C₆₇	#41	1	1	yes	characteristic
49	C₂²×C₆₇	—	4	1	no	—
50	C₂²×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
51	C₂×D₆₇	—	8	4	no	—
52	C₂×D₆₇	—	8	2	no	—
53	C₄×C₆₇	—	2	2	no	—
54	C₄×C₆₇	—	4	1	no	—
55	C₄×C₆₇	#121	1	1	yes	characteristic
56	C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
57	C₂×C₆₇	—	2	1	no	—
58	C₂×C₆₇	—	8	1	no	—
59	C₂×C₆₇	#459	1	1	yes	characteristic
60	D₆₇	—	16	2	no	—
61	#2213	—	67	1	no	maximal, Sylow
62	C₆₇	#2213	1	1	yes	characteristic, complemented, Sylow
63	#509	—	67	2	no	—
64	#493	—	67	1	no	—
65	#443	D₆₇	1	1	yes	characteristic, complemented
66	#364	—	67	1	no	—
67	#361	—	67	2	no	—
68	#135	C₂×D₆₇	1	1	yes	characteristic
69	C₂×D₈	—	67	2	no	—
70	C₂×D₈	—	134	1	no	—
71	#131	—	2	1	no	—
72	#127	—	134	1	no	—
73	SD₃₂	—	134	2	no	—
74	D₁₆	—	134	2	no	—
75	M₃₂	—	67	2	no	—
76	#108	C₂×D₆₇	1	1	yes	characteristic
77	#104	—	2	1	no	—
78	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
79	#41	—	2	1	no	—
80	C₂×D₄	—	268	2	no	—
81	C₂×D₄	—	134	2	no	—
82	Q₁₆	D₂₆₈	1	1	yes	characteristic, complemented
83	SD₁₆	—	2	1	no	—
84	D₈	—	268	1	no	—
85	D₈	—	268	1	no	—
86	D₈	—	134	2	no	—
87	D₈	—	134	2	no	—
88	D₈	D₂₆₈	1	1	yes	characteristic, complemented
89	M₁₆	—	2	1	no	—
90	M₁₆	—	4	1	no	—
91	C₂×C₈	—	4	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	—	268	2	no	—
97	Q₈	—	2	1	no	—
98	D₄	—	2	1	no	—
99	D₄	—	4	1	no	—
100	D₄	—	268	2	no	—
101	D₄	—	268	2	no	—
102	D₄	—	536	4	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
106	C₈	—	4	1	no	—
107	C₈	#11761613	1	1	yes	characteristic
108	C₈	#11761613	1	1	yes	characteristic
109	C₈	C₂×D₂₆₈	1	1	yes	characteristic
110	C₈	#11761628	1	1	yes	characteristic
111	C₂²	—	4	1	no	—
112	C₂²	—	536	4	no	—
113	C₂²	—	536	2	no	—
114	C₂²	#14524068	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	4	1	no	—
117	C₄	#14524017	1	1	yes	characteristic
118	C₄	C₂×#11761613	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	8	1	no	—
121	C₂	—	1072	2	no	—
122	C₂	#16863591	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863831

C₂

yes

maximal, complemented

#16863773

C₂

yes

maximal, characteristic, complemented

#16863217

C₂

yes

maximal, characteristic, complemented

#16863208

C₂

yes

maximal, complemented

#443×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×D₅₃₆

—

C₂×D₅₃₆

C₂²

yes

—

—

—

—

C₂²

yes

—

#135×C₆₇

C₂²

yes

characteristic

#131×C₆₇

—

#108×C₆₇

C₂²

yes

characteristic

#104×C₆₇

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×D₂₆₈

—

C₂×D₂₆₈

—

D₅₃₆

—

D₅₃₆

—

D₅₃₆

—

D₅₃₆

—

C₆₇⋊C₁₆

—

#41×C₆₇

—

Q₁₆×C₆₇

D₄

yes

characteristic, complemented

SD₁₆×C₆₇

—

D₈×C₆₇

D₄

yes

characteristic, complemented

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₂²×D₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

—

C₈×C₆₇

D₈

yes

characteristic

C₈×C₆₇

D₈

yes

characteristic

C₈×C₆₇

C₂×D₄

yes

characteristic

C₈×C₆₇

#41

yes

characteristic

C₂²×C₆₇

—

C₂²×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#121

yes

characteristic

C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#459

yes

characteristic

D₆₇

—

#2213

—

maximal, Sylow

C₆₇

#2213

yes

characteristic, complemented, Sylow

#509

—

#493

—

#443

D₆₇

yes

characteristic, complemented

#364

—

#361

—

#135

C₂×D₆₇

yes

characteristic

C₂×D₈

—

C₂×D₈

—

134

—

#131

—

#127

—

134

—

SD₃₂

—

134

—

D₁₆

—

134

—

M₃₂

—

#108

C₂×D₆₇

yes

characteristic

#104

—

C₄×C₈

C₂×D₆₇

yes

characteristic

#41

—

C₂×D₄

—

268

—

C₂×D₄

—

134

—

Q₁₆

D₂₆₈

yes

characteristic, complemented

SD₁₆

—

D₈

—

268

—

D₈

—

268

—

D₈

—

134

—

D₈

—

134

—

D₈

D₂₆₈

yes

characteristic, complemented

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄²

C₂²⋊D₆₇

yes

characteristic

C₁₆

—

134

—

C₂³

—

268

—

Q₈

—

D₄

—

D₄

—

100

D₄

—

268

—

101

D₄

—

268

—

102

D₄

—

536

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

106

C₈

—

107

C₈

#11761613

yes

characteristic

108

C₈

#11761613

yes

characteristic

109

C₈

C₂×D₂₆₈

yes

characteristic

110

C₈

#11761628

yes

characteristic

111

C₂²

—

112

C₂²

—

536

—

113

C₂²

—

536

—

114

C₂²

#14524068

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

#14524017

yes

characteristic

118

C₄

C₂×#11761613

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

—

1072

—

122

C₂

#16863591

yes

characteristic

	subgroup	G/H
center	C₂	#16863591
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#443×C₆₇	C₂
socle	C₂×C₆₇	#459
hypercenter	#443	D₆₇
hypocenter	C₆₇	#2213
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #443

subgroup

G/H

center

C₂

#16863591

commutator

C₂×C₈×C₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

#443×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #443

N	H ≅ G/N	type
1. #16863831	C₂	semidirect
2. #16863773	C₂	semidirect
3. #16863217	C₂	semidirect
4. #16863208	C₂	semidirect
5. #443×C₆₇	C₂	semidirect
25. Q₁₆×C₆₇	D₄	semidirect
27. D₈×C₆₇	D₄	semidirect
62. C₆₇	#2213	semidirect
65. #443	D₆₇	semidirect
82. Q₁₆	D₂₆₈	semidirect
88. D₈	D₂₆₈	semidirect

H ≅ G/N

type

1. #16863831

C₂

semidirect

2. #16863773

C₂

semidirect

3. #16863217

C₂

semidirect

4. #16863208

C₂

semidirect

5. #443×C₆₇

C₂

semidirect

25. Q₁₆×C₆₇

D₄

semidirect

27. D₈×C₆₇

D₄

semidirect

62. C₆₇

#2213

semidirect

65. #443

D₆₇

semidirect

82. Q₁₆

D₂₆₈

semidirect

88. D₈

D₂₆₈

semidirect

Subgroups — #22562285

Conjugacy classes of subgroups

Complements