Group #22562379 — subgroups

Subgroups — #22562379

Order 8576.

126 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 43 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	#16863793	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863512	C₂	1	1	yes	maximal, characteristic
3	#16863220	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863216	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863207	C₂	1	1	yes	maximal, characteristic
6	#16863205	C₂	1	1	yes	maximal, characteristic
7	#365×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14524065	C₂²	1	1	yes	characteristic
9	#14524051	—	2	1	no	—
10	#14524042	—	2	1	no	—
11	#14523983	C₂²	1	1	yes	characteristic
12	#14523975	C₂²	1	1	yes	characteristic
13	#14523974	—	2	1	no	—
14	#14523969	—	2	1	no	—
15	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
16	M₃₂×C₆₇	C₂²	1	1	yes	characteristic
17	#108×C₆₇	C₂²	1	1	yes	characteristic
18	#106×C₆₇	C₂²	1	1	yes	characteristic
19	#11761628	—	2	1	no	—
20	C₂×Dic₁₃₄	—	4	1	no	—
21	Dic₂₆₈	Q₈	1	1	yes	characteristic, complemented
22	#11761617	—	2	1	no	—
23	D₅₃₆	Q₈	1	1	yes	characteristic, complemented
24	#11761611	—	4	1	no	—
25	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
26	C₈×D₆₇	—	4	1	no	—
27	C₄⋊Dic₆₇	—	2	1	no	—
28	Dic₆₇⋊C₄	—	4	1	no	—
29	C₄×Dic₆₇	D₄	1	1	yes	characteristic
30	C₆₇⋊C₁₆	—	2	1	no	—
31	M₁₆×C₆₇	—	2	1	no	—
32	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₆₇	—	2	1	no	—
34	C₁₆×C₆₇	—	2	1	no	—
35	C₂×C₄×C₆₇	—	4	1	no	—
36	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
37	C₂²⋊D₆₇	—	4	1	no	—
38	C₂×Dic₆₇	—	2	1	no	—
39	C₂×Dic₆₇	—	4	1	no	—
40	D₂₆₈	—	2	1	no	—
41	C₄×D₆₇	—	4	1	no	—
42	Dic₁₃₄	—	2	1	no	—
43	Dic₁₃₄	—	8	2	no	—
44	C₈×C₆₇	—	4	1	no	—
45	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
46	C₈×C₆₇	#41	1	1	yes	characteristic
47	C₆₇⋊C₈	SD₁₆	1	1	yes	characteristic
48	C₆₇⋊C₈	SD₁₆	1	1	yes	characteristic
49	C₂²×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₆₇	—	4	1	no	—
51	C₄×C₆₇	—	8	1	no	—
52	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
53	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
54	Dic₆₇	—	2	2	no	—
55	Dic₆₇	—	4	1	no	—
56	Dic₆₇	—	8	1	no	—
57	C₂×C₆₇	—	2	1	no	—
58	C₂×C₆₇	#476	1	1	yes	characteristic
59	D₆₇	—	8	1	no	—
60	#2230	—	67	1	no	maximal, Sylow
61	C₆₇	#2230	1	1	yes	characteristic, complemented, Sylow
62	#498	—	67	1	no	—
63	#443	—	67	1	no	—
64	#365	—	67	1	no	—
65	#365	D₆₇	1	1	yes	characteristic, complemented
66	#364	—	67	1	no	—
67	#360	—	67	1	no	—
68	#360	—	67	1	no	—
69	#135	—	67	1	no	—
70	#131	—	134	1	no	—
71	C₄⋊Q₈	—	134	1	no	—
72	M₃₂	—	67	1	no	—
73	M₃₂	C₂×D₆₇	1	1	yes	characteristic
74	#108	C₂×D₆₇	1	1	yes	characteristic
75	#106	—	67	1	no	—
76	#106	—	134	1	no	—
77	#106	C₂×D₆₇	1	1	yes	characteristic
78	#104	—	134	1	no	—
79	C₄×C₈	—	67	1	no	—
80	#41	—	134	1	no	—
81	C₂×Q₈	—	268	1	no	—
82	Q₁₆	—	67	1	no	—
83	SD₁₆	—	134	1	no	—
84	D₈	—	67	1	no	—
85	M₁₆	—	2	1	no	—
86	M₁₆	—	268	1	no	—
87	C₂×C₈	—	268	1	no	—
88	C₂×C₈	—	67	1	no	—
89	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	268	1	no	—
92	C₄⋊C₄	—	134	1	no	—
93	C₄²	—	67	1	no	—
94	C₁₆	—	2	1	no	—
95	C₁₆	—	134	1	no	—
96	Q₈	—	536	2	no	—
97	Q₈	—	134	1	no	—
98	D₄	—	268	1	no	—
99	D₄	—	134	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	268	1	no	—
102	C₂×C₄	—	268	1	no	—
103	C₂×C₄	—	134	1	no	—
104	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
105	C₈	—	4	1	no	—
106	C₈	—	67	1	no	—
107	C₈	—	67	1	no	—
108	C₈	Q₈×D₆₇	1	1	yes	characteristic
109	C₈	#11761628	1	1	yes	characteristic
110	C₂²	—	268	1	no	—
111	C₂²	#14524070	1	1	yes	characteristic
112	C₄	—	8	1	no	—
113	C₄	—	268	1	no	—
114	C₄	—	536	1	no	—
115	C₄	—	134	2	no	—
116	C₄	#14524021	1	1	yes	characteristic
117	C₄	SD₁₆×D₆₇	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	536	1	no	—
120	C₂	#16863685	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863793

C₂

yes

maximal, characteristic, complemented

#16863512

C₂

yes

maximal, characteristic

#16863220

C₂

yes

maximal, characteristic, complemented

#16863216

C₂

yes

maximal, characteristic, complemented

#16863207

C₂

yes

maximal, characteristic

#16863205

C₂

yes

maximal, characteristic

#365×C₆₇

C₂

yes

maximal, characteristic, complemented

#14524065

C₂²

yes

—

—

C₂²

yes

characteristic

#14523975

C₂²

yes

characteristic

#14523974

—

#14523969

—

C₈×Dic₆₇

C₂²

yes

characteristic

M₃₂×C₆₇

C₂²

yes

characteristic

#108×C₆₇

C₂²

yes

characteristic

#106×C₆₇

C₂²

yes

characteristic

#11761628

—

C₂×Dic₁₃₄

—

Dic₂₆₈

Q₈

yes

characteristic, complemented

#11761617

—

D₅₃₆

Q₈

yes

characteristic, complemented

#11761611

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₈×D₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

D₄

yes

characteristic

C₆₇⋊C₁₆

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₁₆×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²⋊D₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

C₄×D₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

C₂×Q₈

yes

characteristic

C₈×C₆₇

#41

yes

characteristic

C₆₇⋊C₈

SD₁₆

yes

characteristic

C₆₇⋊C₈

SD₁₆

yes

characteristic

C₂²×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₄×C₆₇

C₂×SD₁₆

yes

characteristic

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#476

yes

characteristic

D₆₇

—

#2230

—

maximal, Sylow

C₆₇

#2230

yes

characteristic, complemented, Sylow

—

—

—

D₆₇

yes

characteristic, complemented

—

—

—

—

—

134

—

C₄⋊Q₈

—

134

—

M₃₂

—

M₃₂

C₂×D₆₇

yes

characteristic

#108

C₂×D₆₇

yes

characteristic

#106

—

#106

—

134

—

#106

C₂×D₆₇

yes

characteristic

#104

—

134

—

C₄×C₈

—

#41

—

134

—

C₂×Q₈

—

268

—

Q₁₆

—

SD₁₆

—

134

—

D₈

—

M₁₆

—

M₁₆

—

268

—

C₂×C₈

—

268

—

C₂×C₈

—

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

134

—

C₄²

—

C₁₆

—

C₁₆

—

134

—

Q₈

—

536

—

Q₈

—

134

—

D₄

—

268

—

D₄

—

134

—

100

C₂×C₄

—

101

C₂×C₄

—

268

—

102

C₂×C₄

—

268

—

103

C₂×C₄

—

134

—

104

C₂×C₄

D₄×D₆₇

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

Q₈×D₆₇

yes

characteristic

109

C₈

#11761628

yes

characteristic

110

C₂²

—

268

—

111

C₂²

#14524070

yes

characteristic

112

C₄

—

113

C₄

—

268

—

114

C₄

—

536

—

115

C₄

—

134

—

116

C₄

#14524021

yes

characteristic

117

C₄

SD₁₆×D₆₇

yes

characteristic

118

C₂

—

119

C₂

—

536

—

120

C₂

#16863685

yes

characteristic

	subgroup	G/H
center	C₂	#16863685
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#365×C₆₇	C₂
socle	C₂×C₆₇	#476
hypercenter	#365	D₆₇
hypocenter	C₆₇	#2230
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

subgroup

G/H

center

C₂

#16863685

commutator

C₂×C₈×C₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

#365×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

N	H ≅ G/N	type
1. #16863793	C₂	semidirect
3. #16863220	C₂	semidirect
4. #16863216	C₂	semidirect
7. #365×C₆₇	C₂	semidirect
21. Dic₂₆₈	Q₈	semidirect
23. D₅₃₆	Q₈	semidirect
61. C₆₇	#2230	semidirect
65. #365	D₆₇	semidirect

H ≅ G/N

type

1. #16863793

C₂

semidirect

3. #16863220

C₂

semidirect

4. #16863216

C₂

semidirect

7. #365×C₆₇

C₂

semidirect

21. Dic₂₆₈

Q₈

semidirect

23. D₅₃₆

Q₈

semidirect

61. C₆₇

#2230

semidirect

65. #365

D₆₇

semidirect

Subgroups — #22562379

Conjugacy classes of subgroups

Complements