Group #22562378 — subgroups

Subgroups — #22562378

Order 8576.

126 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 45 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	#16863513	C₂	1	1	yes	maximal, characteristic
3	#16863219	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863218	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863216	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863206	C₂	1	1	yes	maximal, characteristic
7	#360×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#14524050	—	2	1	no	—
9	#14524042	—	2	1	no	—
10	#14523983	C₂²	1	1	yes	characteristic
11	#14523979	C₂²	1	1	yes	characteristic
12	#14523975	C₂²	1	1	yes	characteristic
13	#14523974	—	2	1	no	—
14	#14523970	—	2	1	no	—
15	C₈×Dic₆₇	C₂²	1	1	yes	characteristic
16	#135×C₆₇	C₂²	1	1	yes	characteristic
17	M₃₂×C₆₇	C₂²	1	1	yes	characteristic
18	#106×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₃₄	—	4	1	no	—
20	#11761612	—	2	1	no	—
21	#11761612	—	4	1	no	—
22	C₂×C₆₇⋊C₈	—	4	1	no	—
23	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
24	C₄⋊Dic₆₇	—	2	1	no	—
25	Dic₆₇⋊C₄	—	4	1	no	—
26	C₄×Dic₆₇	D₄	1	1	yes	characteristic
27	C₆₇⋊C₁₆	—	2	1	no	—
28	#41×C₆₇	—	2	1	no	—
29	Q₁₆×C₆₇	Q₈	1	1	yes	characteristic, complemented
30	SD₁₆×C₆₇	—	2	1	no	—
31	D₈×C₆₇	Q₈	1	1	yes	characteristic, complemented
32	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₆₇	—	2	1	no	—
34	C₁₆×C₆₇	—	2	1	no	—
35	Q₈×C₆₇	—	2	1	no	—
36	D₄×C₆₇	—	2	1	no	—
37	D₄×C₆₇	—	4	1	no	—
38	C₂×C₄×C₆₇	—	4	1	no	—
39	C₂×C₄×C₆₇	—	4	1	no	—
40	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₆₇	—	2	1	no	—
42	C₂×Dic₆₇	—	4	1	no	—
43	Dic₁₃₄	—	8	2	no	—
44	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₆₇	#41	1	1	yes	characteristic
46	C₆₇⋊C₈	—	4	1	no	—
47	C₆₇⋊C₈	SD₁₆	1	1	yes	characteristic
48	C₆₇⋊C₈	SD₁₆	1	1	yes	characteristic
49	C₂²×C₆₇	—	4	1	no	—
50	C₂²×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
51	C₄×C₆₇	—	4	1	no	—
52	C₄×C₆₇	—	8	1	no	—
53	C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
54	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
55	Dic₆₇	—	2	2	no	—
56	Dic₆₇	—	8	1	no	—
57	C₂×C₆₇	—	2	1	no	—
58	C₂×C₆₇	—	8	1	no	—
59	C₂×C₆₇	#476	1	1	yes	characteristic
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	—	67	1	no	—
66	#364	—	67	1	no	—
67	#360	—	67	1	no	—
68	#360	D₆₇	1	1	yes	characteristic
69	#135	C₂×D₆₇	1	1	yes	characteristic
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	—	67	1	no	—
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	—	2	1	no	—
81	C₂×Q₈	—	268	1	no	—
82	Q₁₆	Dic₁₃₄	1	1	yes	characteristic, complemented
83	SD₁₆	—	2	1	no	—
84	D₈	Dic₁₃₄	1	1	yes	characteristic, complemented
85	M₁₆	—	268	1	no	—
86	M₁₆	—	134	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	—	2	1	no	—
97	Q₈	—	536	2	no	—
98	D₄	—	2	1	no	—
99	D₄	—	4	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	4	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	—	268	1	no	—
106	C₈	—	67	1	no	—
107	C₈	—	67	1	no	—
108	C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
109	C₈	#11761627	1	1	yes	characteristic
110	C₂²	—	4	1	no	—
111	C₂²	#14524067	1	1	yes	characteristic
112	C₄	—	4	1	no	—
113	C₄	—	8	1	no	—
114	C₄	—	536	1	no	—
115	C₄	—	134	2	no	—
116	C₄	#14524020	1	1	yes	characteristic
117	C₄	SD₁₆×D₆₇	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	8	1	no	—
120	C₂	#16863684	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863793

C₂

yes

maximal, characteristic, complemented

#16863513

C₂

yes

maximal, characteristic

#16863219

C₂

yes

maximal, characteristic, complemented

#16863218

C₂

yes

maximal, characteristic, complemented

#16863216

C₂

yes

maximal, characteristic, complemented

#16863206

C₂

yes

maximal, characteristic

#360×C₆₇

C₂

yes

maximal, characteristic

#14524050

—

#14524042

—

#14523983

C₂²

yes

characteristic

#14523979

C₂²

yes

characteristic

#14523975

C₂²

yes

characteristic

#14523974

—

#14523970

—

C₈×Dic₆₇

C₂²

yes

characteristic

#135×C₆₇

C₂²

yes

characteristic

M₃₂×C₆₇

C₂²

yes

characteristic

#106×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

D₄

yes

characteristic

C₆₇⋊C₁₆

—

#41×C₆₇

—

Q₁₆×C₆₇

Q₈

yes

characteristic, complemented

SD₁₆×C₆₇

—

D₈×C₆₇

Q₈

yes

characteristic, complemented

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₁₆×C₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

C₂×Q₈

yes

characteristic

C₈×C₆₇

#41

yes

characteristic

C₆₇⋊C₈

—

C₆₇⋊C₈

SD₁₆

yes

characteristic

C₆₇⋊C₈

SD₁₆

yes

characteristic

C₂²×C₆₇

—

C₂²×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₄×C₆₇

C₂×SD₁₆

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₆₇

yes

characteristic

#135

C₂×D₆₇

yes

characteristic

#131

—

134

—

C₄⋊Q₈

—

134

—

M₃₂

—

M₃₂

C₂×D₆₇

yes

—

—

—

134

—

#106

C₂×D₆₇

yes

characteristic

#104

—

134

—

C₄×C₈

—

#41

—

C₂×Q₈

—

268

—

Q₁₆

Dic₁₃₄

yes

characteristic, complemented

SD₁₆

—

D₈

Dic₁₃₄

yes

characteristic, complemented

M₁₆

—

268

—

M₁₆

—

134

—

C₂×C₈

—

268

—

C₂×C₈

—

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

134

—

C₄²

—

C₁₆

—

C₁₆

—

134

—

Q₈

—

Q₈

—

536

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

268

—

103

C₂×C₄

—

134

—

104

C₂×C₄

D₄×D₆₇

yes

characteristic

105

C₈

—

268

—

106

C₈

—

107

C₈

—

108

C₈

C₂×Dic₁₃₄

yes

characteristic

109

C₈

#11761627

yes

characteristic

110

C₂²

—

111

C₂²

#14524067

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

536

—

115

C₄

—

134

—

116

C₄

#14524020

yes

characteristic

117

C₄

SD₁₆×D₆₇

yes

characteristic

118

C₂

—

119

C₂

—

120

C₂

#16863684

yes

characteristic

	subgroup	G/H
center	C₂	#16863684
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#360×C₆₇	C₂
socle	C₂×C₆₇	#476
hypercenter	#360	D₆₇
hypocenter	C₆₇	#2230
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #360

subgroup

G/H

center

C₂

#16863684

commutator

C₂×C₈×C₆₇

C₂³

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

#360×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #360

N	H ≅ G/N	type
1. #16863793	C₂	semidirect
3. #16863219	C₂	semidirect
4. #16863218	C₂	semidirect
5. #16863216	C₂	semidirect
29. Q₁₆×C₆₇	Q₈	semidirect
31. D₈×C₆₇	Q₈	semidirect
61. C₆₇	#2230	semidirect
82. Q₁₆	Dic₁₃₄	semidirect
84. D₈	Dic₁₃₄	semidirect

H ≅ G/N

type

1. #16863793

C₂

semidirect

3. #16863219

C₂

semidirect

4. #16863218

C₂

semidirect

5. #16863216

C₂

semidirect

29. Q₁₆×C₆₇

Q₈

semidirect

31. D₈×C₆₇

Q₈

semidirect

61. C₆₇

#2230

semidirect

82. Q₁₆

Dic₁₃₄

semidirect

84. D₈

Dic₁₃₄

semidirect

Subgroups — #22562378

Conjugacy classes of subgroups

Complements