Group #22559736 — subgroups

Subgroups — #22559736

Order 8576.

140 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16863692	C₂	1	1	yes	maximal, characteristic
3	#16863688	C₂	1	1	yes	maximal, characteristic
4	#16863523	C₂	1	1	yes	maximal, characteristic
5	#16863146	C₂	1	1	yes	maximal, characteristic
6	#16863144	C₂	1	1	yes	maximal, characteristic
7	#333×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#14524043	C₂²	1	1	yes	characteristic
9	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
10	#14523977	—	2	1	no	—
11	#14523976	—	2	1	no	—
12	#14523976	—	2	1	no	—
13	#14523976	—	2	1	no	—
14	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
15	Dic₆₇⋊C₈	—	2	1	no	—
16	#14523968	—	2	1	no	—
17	#14523966	—	2	1	no	—
18	#14523953	—	2	1	no	—
19	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₃₄	—	2	1	no	—
24	C₂×Dic₁₃₄	D₄	1	1	yes	characteristic
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	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
31	Dic₆₇⋊C₄	—	2	1	no	—
32	C₄×Dic₆₇	—	2	1	no	—
33	C₂×Q₈×C₆₇	—	2	1	no	—
34	C₂×C₈×C₆₇	—	2	1	no	—
35	C₂×C₈×C₆₇	—	2	1	no	—
36	C₄⋊C₄×C₆₇	—	2	1	no	—
37	C₄⋊C₄×C₆₇	—	2	1	no	—
38	C₄²×C₆₇	C₂³	1	1	yes	characteristic
39	Q₈×C₆₇	—	8	1	no	—
40	C₂×C₄×C₆₇	—	4	1	no	—
41	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
44	C₂×Dic₆₇	—	2	1	no	—
45	C₂×Dic₆₇	—	2	1	no	—
46	C₂×Dic₆₇	—	4	1	no	—
47	Dic₁₃₄	—	2	1	no	—
48	Dic₁₃₄	—	2	1	no	—
49	Dic₁₃₄	—	8	1	no	—
50	C₈×C₆₇	—	4	1	no	—
51	C₈×C₆₇	—	4	1	no	—
52	C₆₇⋊C₈	—	2	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	—	8	1	no	—
58	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
59	C₄×C₆₇	#137	1	1	yes	characteristic
60	C₄×C₆₇	#137	1	1	yes	characteristic
61	C₄×C₆₇	#137	1	1	yes	characteristic
62	Dic₆₇	—	4	1	no	—
63	Dic₆₇	—	4	1	no	—
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	—	67	1	no	—
73	#446	—	67	1	no	—
74	#334	—	67	1	no	—
75	#333	—	67	1	no	—
76	#333	D₆₇	1	1	yes	characteristic
77	C₄⋊Q₈	—	67	1	no	—
78	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
79	C₄×Q₈	—	67	1	no	—
80	#107	—	134	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	—	134	1	no	—
86	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
87	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
88	#103	—	134	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	—	67	1	no	—
94	C₂×Q₈	—	134	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	67	1	no	—
98	C₂×C₈	—	67	1	no	—
99	C₂×C₈	—	134	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	67	1	no	—
103	C₄⋊C₄	—	134	1	no	—
104	C₄⋊C₄	—	134	1	no	—
105	C₄⋊C₄	—	134	1	no	—
106	C₄²	—	134	1	no	—
107	C₄²	C₂²×D₆₇	1	1	yes	characteristic
108	Q₈	—	8	1	no	—
109	Q₈	—	536	1	no	—
110	Q₈	—	134	1	no	—
111	Q₈	—	134	1	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	—	268	1	no	—
114	C₂×C₄	—	134	1	no	—
115	C₂×C₄	—	134	1	no	—
116	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
117	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
118	C₂×C₄	#11761628	1	1	yes	characteristic
119	C₈	—	4	1	no	—
120	C₈	—	4	1	no	—
121	C₈	—	268	1	no	—
122	C₈	—	67	2	no	—
123	C₈	—	134	1	no	—
124	C₂²	#14524016	1	1	yes	characteristic
125	C₄	—	2	1	no	—
126	C₄	—	8	1	no	—
127	C₄	—	268	1	no	—
128	C₄	—	268	1	no	—
129	C₄	—	536	1	no	—
130	C₄	Q₁₆×D₆₇	1	1	yes	characteristic
131	C₄	#14524072	1	1	yes	characteristic
132	C₄	#14524072	1	1	yes	characteristic
133	C₄	#14524075	1	1	yes	characteristic
134	C₂	#16863605	1	1	yes	characteristic
135	C₂	#16863653	1	1	yes	characteristic
136	C₂	#16863665	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863799

C₂

yes

maximal, characteristic

#16863692

C₂

yes

maximal, characteristic

#16863688

C₂

yes

maximal, characteristic

#16863523

C₂

yes

maximal, characteristic

#16863146

C₂

yes

maximal, characteristic

#16863144

C₂

yes

maximal, characteristic

#333×C₆₇

C₂

yes

maximal, characteristic

#14524043

C₂²

yes

characteristic

C₄×Dic₁₃₄

C₂²

yes

—

—

—

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

Dic₆₇⋊C₈

—

#14523968

—

#14523966

—

#14523953

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

D₄

yes

characteristic

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

characteristic

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

Q₈×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

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

Q₁₆

yes

—

C₂²×C₆₇

#121

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×Q₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₆₇

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₇

yes

characteristic

C₄×Q₈

—

#107

—

134

—

#107

—

134

—

#107

—

134

—

#107

—

134

—

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₄⋊C₈

C₂×D₆₇

yes

characteristic

C₄⋊C₈

C₂×D₆₇

yes

characteristic

#103

—

134

—

#103

—

134

—

#97

—

134

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

134

—

104

C₄⋊C₄

—

134

—

105

C₄⋊C₄

—

134

—

106

C₄²

—

134

—

107

C₄²

C₂²×D₆₇

yes

characteristic

108

Q₈

—

109

Q₈

—

536

—

110

Q₈

—

134

—

111

Q₈

—

134

—

112

C₂×C₄

—

113

C₂×C₄

—

268

—

114

C₂×C₄

—

134

—

115

C₂×C₄

—

134

—

116

C₂×C₄

D₄×D₆₇

yes

characteristic

117

C₂×C₄

D₄×D₆₇

yes

characteristic

118

C₂×C₄

#11761628

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

268

—

122

C₈

—

123

C₈

—

134

—

124

C₂²

#14524016

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

268

—

128

C₄

—

268

—

129

C₄

—

536

—

130

C₄

Q₁₆×D₆₇

yes

characteristic

131

C₄

#14524072

yes

characteristic

132

C₄

#14524072

yes

characteristic

133

C₄

#14524075

yes

characteristic

134

C₂

#16863605

yes

characteristic

135

C₂

#16863653

yes

characteristic

136

C₂

#16863665

yes

characteristic

	subgroup	G/H
center	C₂²	#14524016
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#333×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#333	D₆₇
hypocenter	C₆₇	#1696
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#14524016

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#333×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

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

H ≅ G/N

type

69. C₆₇

#1696

semidirect

Subgroups — #22559736

Conjugacy classes of subgroups

Complements