Group #22559735 — subgroups

Subgroups — #22559735

Order 8576.

140 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16863799	C₂	1	1	yes	maximal, characteristic
2	#16863691	C₂	1	1	yes	maximal, characteristic
3	#16863527	C₂	1	1	yes	maximal, characteristic
4	#16863146	C₂	1	1	yes	maximal, characteristic
5	#16863145	C₂	1	1	yes	maximal, characteristic
6	#16863144	C₂	1	1	yes	maximal, characteristic
7	#477×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#14524043	C₂²	1	1	yes	characteristic
9	#14523976	—	2	1	no	—
10	#14523976	—	2	1	no	—
11	#14523976	—	2	1	no	—
12	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
13	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
15	#14523967	—	2	1	no	—
16	#14523954	—	2	1	no	—
17	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	#107×C₆₇	—	2	1	no	—
21	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	#103×C₆₇	—	2	1	no	—
23	C₂×Dic₁₃₄	—	2	1	no	—
24	C₂×C₆₇⋊C₈	—	2	1	no	—
25	C₂×C₆₇⋊C₈	—	2	1	no	—
26	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
27	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
28	C₄⋊Dic₆₇	—	2	1	no	—
29	C₄⋊Dic₆₇	—	2	1	no	—
30	C₂×Q₈×C₆₇	—	2	1	no	—
31	C₂×Q₈×C₆₇	D₄	1	1	yes	characteristic
32	C₂×C₈×C₆₇	—	2	1	no	—
33	C₄⋊C₄×C₆₇	—	2	1	no	—
34	C₄⋊C₄×C₆₇	—	2	1	no	—
35	C₄⋊C₄×C₆₇	—	2	1	no	—
36	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
37	C₄²×C₆₇	—	2	1	no	—
38	C₄²×C₆₇	C₂³	1	1	yes	characteristic
39	Q₈×C₆₇	—	2	1	no	—
40	Q₈×C₆₇	—	2	1	no	—
41	Q₈×C₆₇	—	8	1	no	—
42	C₂×C₄×C₆₇	—	2	1	no	—
43	C₂×C₄×C₆₇	—	2	1	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	#41	1	1	yes	characteristic
48	C₂×Dic₆₇	—	4	1	no	—
49	Dic₁₃₄	—	8	1	no	—
50	C₈×C₆₇	—	4	1	no	—
51	C₆₇⋊C₈	—	2	1	no	—
52	C₆₇⋊C₈	—	4	1	no	—
53	C₆₇⋊C₈	—	4	1	no	—
54	C₆₇⋊C₈	Q₁₆	1	2	yes	—
55	C₂²×C₆₇	#121	1	1	yes	characteristic
56	C₄×C₆₇	—	2	1	no	—
57	C₄×C₆₇	—	4	1	no	—
58	C₄×C₆₇	—	4	1	no	—
59	C₄×C₆₇	—	8	1	no	—
60	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
61	C₄×C₆₇	#137	1	1	yes	characteristic
62	C₄×C₆₇	#137	1	1	yes	characteristic
63	C₄×C₆₇	#137	1	1	yes	characteristic
64	Dic₆₇	—	8	1	no	—
65	C₂×C₆₇	#462	1	1	yes	characteristic
66	C₂×C₆₇	#470	1	1	yes	characteristic
67	C₂×C₆₇	#472	1	1	yes	characteristic
68	#1696	—	67	1	no	maximal, Sylow
69	C₆₇	#1696	1	1	yes	characteristic, complemented, Sylow
70	#500	—	67	1	no	—
71	#477	—	67	1	no	—
72	#477	D₆₇	1	1	yes	characteristic
73	#446	—	67	1	no	—
74	#334	—	67	1	no	—
75	#333	—	67	1	no	—
76	#333	—	67	1	no	—
77	C₄⋊Q₈	—	67	1	no	—
78	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
79	C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
80	#107	—	2	1	no	—
81	#107	—	134	1	no	—
82	#107	—	134	1	no	—
83	#107	—	134	1	no	—
84	C₄⋊C₈	—	67	1	no	—
85	C₄⋊C₈	—	67	1	no	—
86	C₄⋊C₈	—	134	1	no	—
87	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
88	#103	—	2	1	no	—
89	#103	—	134	1	no	—
90	#97	—	134	1	no	—
91	C₄×C₈	—	67	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	134	1	no	—
94	C₂×Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	67	1	no	—
97	C₂×C₈	—	67	1	no	—
98	C₂×C₈	—	134	1	no	—
99	C₂×C₈	—	134	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	134	1	no	—
104	C₄⋊C₄	—	134	1	no	—
105	C₄⋊C₄	C₂²⋊D₆₇	1	1	yes	characteristic
106	C₄²	—	2	1	no	—
107	C₄²	C₂²×D₆₇	1	1	yes	characteristic
108	Q₈	—	2	1	no	—
109	Q₈	—	2	1	no	—
110	Q₈	—	8	1	no	—
111	Q₈	—	536	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	268	1	no	—
116	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
118	C₂×C₄	#11761627	1	1	yes	characteristic
119	C₈	—	4	1	no	—
120	C₈	—	268	1	no	—
121	C₈	—	268	1	no	—
122	C₈	—	67	2	no	—
123	C₈	—	134	1	no	—
124	C₂²	#14524015	1	1	yes	characteristic
125	C₄	—	2	1	no	—
126	C₄	—	4	1	no	—
127	C₄	—	4	1	no	—
128	C₄	—	8	1	no	—
129	C₄	—	536	1	no	—
130	C₄	Q₁₆×D₆₇	1	1	yes	characteristic
131	C₄	#14524073	1	1	yes	characteristic
132	C₄	#14524074	1	1	yes	characteristic
133	C₄	#14524075	1	1	yes	characteristic
134	C₂	#16863604	1	1	yes	characteristic
135	C₂	#16863652	1	1	yes	characteristic
136	C₂	#16863664	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863799

C₂

yes

maximal, characteristic

#16863691

C₂

yes

maximal, characteristic

#16863527

C₂

yes

maximal, characteristic

#16863146

C₂

yes

maximal, characteristic

#16863145

C₂

yes

maximal, characteristic

#16863144

C₂

yes

maximal, characteristic

#477×C₆₇

C₂

yes

maximal, characteristic

#14524043

C₂²

yes

—

—

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

#14523967

—

#14523954

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

C₄×Q₈×C₆₇

C₂²

yes

characteristic

#107×C₆₇

—

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

#103×C₆₇

—

C₂×Dic₁₃₄

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

Q₁₆

yes

—

C₂²×C₆₇

#121

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×Q₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#500

—

#477

—

#477

D₆₇

yes

—

—

—

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₇

yes

characteristic

C₄×Q₈

C₂×D₆₇

yes

characteristic

#107

—

#107

—

134

—

#107

—

134

—

#107

—

134

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₄⋊C₈

C₂×D₆₇

yes

characteristic

#103

—

#103

—

134

—

#97

—

134

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×Q₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

134

—

104

C₄⋊C₄

—

134

—

105

C₄⋊C₄

C₂²⋊D₆₇

yes

characteristic

106

C₄²

—

107

C₄²

C₂²×D₆₇

yes

characteristic

108

Q₈

—

109

Q₈

—

110

Q₈

—

111

Q₈

—

536

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

268

—

116

C₂×C₄

D₄×D₆₇

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

118

C₂×C₄

#11761627

yes

characteristic

119

C₈

—

120

C₈

—

268

—

121

C₈

—

268

—

122

C₈

—

123

C₈

—

134

—

124

C₂²

#14524015

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

536

—

130

C₄

Q₁₆×D₆₇

yes

characteristic

131

C₄

#14524073

yes

characteristic

132

C₄

#14524074

yes

characteristic

133

C₄

#14524075

yes

characteristic

134

C₂

#16863604

yes

characteristic

135

C₂

#16863652

yes

characteristic

136

C₂

#16863664

yes

characteristic

	subgroup	G/H
center	C₂²	#14524015
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#477×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#477	D₆₇
hypocenter	C₆₇	#1696
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

subgroup

G/H

center

C₂²

#14524015

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#477×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #477

N	H ≅ G/N	type
69. C₆₇	#1696	semidirect

H ≅ G/N

type

69. C₆₇

#1696

semidirect

Subgroups — #22559735

Conjugacy classes of subgroups

Complements