Group #22559828 — subgroups

Subgroups — #22559828

Order 8576.

198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16863801	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863792	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863768	C₂	1	1	yes	maximal, characteristic
4	#16863748	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863742	C₂	1	1	yes	maximal, characteristic
6	#16863117	C₂	1	1	yes	maximal, characteristic, complemented
7	#492×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#11761616	—	2	2	no	—
9	#14524043	C₂²	1	1	yes	characteristic
10	#14524043	C₂²	1	1	yes	characteristic
11	#14523977	—	2	2	no	—
12	#14523974	—	2	2	no	—
13	#14523966	—	2	2	no	—
14	#14523964	—	2	2	no	—
15	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₆₇	—	2	2	no	—
18	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	#127×C₆₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₃₄	—	2	1	no	—
22	C₂×Dic₁₃₄	—	2	1	no	—
23	#11761616	—	4	4	no	—
24	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
25	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
26	C₄⋊Dic₆₇	—	2	1	no	—
27	C₄⋊Dic₆₇	—	2	1	no	—
28	C₄⋊Dic₆₇	—	2	1	no	—
29	C₄⋊Dic₆₇	—	2	1	no	—
30	C₂×Q₈×C₆₇	—	2	1	no	—
31	C₂×D₄×C₆₇	—	2	1	no	—
32	C₂×D₄×C₆₇	—	2	1	no	—
33	C₂×D₄×C₆₇	—	2	1	no	—
34	SD₁₆×C₆₇	—	4	4	no	—
35	C₂×C₈×C₆₇	D₄	1	2	yes	—
36	C₄⋊C₄×C₆₇	—	2	1	no	—
37	C₄⋊C₄×C₆₇	—	2	1	no	—
38	C₄²×C₆₇	C₂³	1	1	yes	characteristic
39	C₂³×C₆₇	—	4	1	no	—
40	Q₈×C₆₇	—	4	2	no	—
41	D₄×C₆₇	—	4	2	no	—
42	D₄×C₆₇	—	4	2	no	—
43	D₄×C₆₇	—	4	2	no	—
44	C₂×C₄×C₆₇	—	4	1	no	—
45	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₆₇	—	4	1	no	—
49	C₂×Dic₆₇	—	4	1	no	—
50	Dic₁₃₄	—	4	2	no	—
51	Dic₁₃₄	—	4	2	no	—
52	C₈×C₆₇	SD₁₆	1	4	yes	complemented
53	C₆₇⋊C₈	—	2	2	no	—
54	C₆₇⋊C₈	—	2	2	no	—
55	C₂²×C₆₇	—	8	1	no	—
56	C₂²×C₆₇	—	8	1	no	—
57	C₂²×C₆₇	—	8	1	no	—
58	C₂²×C₆₇	#127	1	1	yes	characteristic
59	C₄×C₆₇	—	8	1	no	—
60	C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
61	C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₆₇	#137	1	2	yes	—
63	Dic₆₇	—	8	1	no	—
64	Dic₆₇	—	8	1	no	—
65	C₂×C₆₇	—	16	1	no	—
66	C₂×C₆₇	#492	1	1	yes	characteristic
67	C₂×C₆₇	#496	1	1	yes	characteristic
68	C₂×C₆₇	#497	1	1	yes	characteristic
69	#1714	—	67	1	no	maximal, Sylow
70	C₆₇	#1714	1	1	yes	characteristic, complemented, Sylow
71	#500	—	67	1	no	—
72	#498	—	67	1	no	—
73	#492	—	67	1	no	—
74	#492	D₆₇	1	1	yes	characteristic
75	#487	—	67	1	no	—
76	#486	—	67	1	no	—
77	#322	—	67	1	no	—
78	C₂×SD₁₆	—	2	2	no	—
79	C₂×SD₁₆	—	134	2	no	—
80	C₄⋊Q₈	—	67	1	no	—
81	C₄⋊Q₈	—	67	1	no	—
82	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
83	#127	C₂×D₆₇	1	1	yes	characteristic
84	#107	—	134	2	no	—
85	#106	—	134	2	no	—
86	#103	—	134	2	no	—
87	#102	—	134	2	no	—
88	C₄×C₈	—	67	1	no	—
89	C₄×C₈	—	67	1	no	—
90	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	134	1	no	—
93	C₂×Q₈	—	134	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	2	1	no	—
97	SD₁₆	—	4	4	no	—
98	SD₁₆	—	268	4	no	—
99	C₂×C₈	—	67	2	no	—
100	C₂×C₈	—	67	2	no	—
101	C₂×C₈	C₂²⋊D₆₇	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	134	1	no	—
105	C₄⋊C₄	—	134	1	no	—
106	C₄⋊C₄	—	134	1	no	—
107	C₄⋊C₄	—	134	1	no	—
108	C₄²	C₂²×D₆₇	1	1	yes	characteristic
109	C₂³	—	4	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	268	2	no	—
112	Q₈	—	268	2	no	—
113	D₄	—	4	2	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	268	1	no	—
118	C₂×C₄	—	268	1	no	—
119	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
120	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
122	C₈	—	134	2	no	—
123	C₈	—	134	2	no	—
124	C₈	#11761616	1	4	yes	complemented
125	C₂²	—	8	1	no	—
126	C₂²	—	8	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	#14524039	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	536	1	no	—
131	C₄	—	536	1	no	—
132	C₄	C₂×#11761616	1	2	yes	—
133	C₄	#14524070	1	2	yes	—
134	C₄	#14524072	1	2	yes	—
135	C₂	—	16	1	no	—
136	C₂	#16863768	1	1	yes	characteristic
137	C₂	#16863782	1	1	yes	characteristic
138	C₂	#16863787	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863801

C₂

yes

maximal, characteristic, complemented

#16863792

C₂

yes

maximal, characteristic, complemented

#16863768

C₂

yes

maximal, characteristic

#16863748

C₂

yes

maximal, characteristic, complemented

#16863742

C₂

yes

maximal, characteristic

#16863117

C₂

yes

maximal, characteristic, complemented

#492×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761616

—

#14524043

C₂²

yes

characteristic

#14524043

C₂²

yes

—

—

—

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₆₇

—

C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

#127×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

#11761616

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

SD₁₆×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

SD₁₆

yes

complemented

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#127

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₂×SD₁₆

yes

—

C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₄×C₆₇

#137

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

D₆₇

yes

—

—

—

C₂×SD₁₆

—

C₂×SD₁₆

—

134

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₇

yes

characteristic

#127

C₂×D₆₇

yes

characteristic

#107

—

134

—

#106

—

134

—

#103

—

134

—

#102

—

134

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

268

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂²⋊D₆₇

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

134

—

105

C₄⋊C₄

—

134

—

106

C₄⋊C₄

—

134

—

107

C₄⋊C₄

—

134

—

108

C₄²

C₂²×D₆₇

yes

characteristic

109

C₂³

—

110

Q₈

—

111

Q₈

—

268

—

112

Q₈

—

268

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

268

—

118

C₂×C₄

—

268

—

119

C₂×C₄

D₄×D₆₇

yes

characteristic

120

C₂×C₄

D₄×D₆₇

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

122

C₈

—

134

—

123

C₈

—

134

—

124

C₈

#11761616

yes

complemented

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#14524039

yes

characteristic

129

C₄

—

130

C₄

—

536

—

131

C₄

—

536

—

132

C₄

C₂×#11761616

yes

—

133

C₄

#14524070

yes

—

134

C₄

#14524072

yes

—

135

C₂

—

136

C₂

#16863768

yes

characteristic

137

C₂

#16863782

yes

characteristic

138

C₂

#16863787

yes

characteristic

	subgroup	G/H
center	C₂²	#14524039
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#492×C₆₇	C₂
socle	C₂²×C₆₇	#127
hypercenter	#492	D₆₇
hypocenter	C₆₇	#1714
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#14524039

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#492×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #16863801	C₂	semidirect
2. #16863792	C₂	semidirect
4. #16863748	C₂	semidirect
6. #16863117	C₂	semidirect
52. C₈×C₆₇	SD₁₆	semidirect
70. C₆₇	#1714	semidirect
124. C₈	#11761616	semidirect

H ≅ G/N

type

1. #16863801

C₂

semidirect

2. #16863792

C₂

semidirect

4. #16863748

C₂

semidirect

6. #16863117

C₂

semidirect

52. C₈×C₆₇

SD₁₆

semidirect

70. C₆₇

#1714

semidirect

124. C₈

#11761616

semidirect

Subgroups — #22559828

Conjugacy classes of subgroups

Complements