Group #22559607 — subgroups

Subgroups — #22559607

Order 8576.

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

normal subgroups only

This group has 160 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, complemented
2	#16863687	C₂	1	1	yes	maximal, characteristic
3	#16863683	C₂	1	1	yes	maximal, characteristic
4	#16863461	C₂	1	1	yes	maximal, characteristic
5	#16863148	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863144	C₂	1	1	yes	maximal, characteristic, complemented
7	#333×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₆₇	—	2	1	no	—
9	#14524043	C₂²	1	1	yes	characteristic
10	C₄×D₂₆₈	C₂²	1	1	yes	characteristic
11	#14523976	—	2	1	no	—
12	#14523976	—	2	1	no	—
13	#14523975	—	2	1	no	—
14	#14523974	—	2	1	no	—
15	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
16	#14523965	—	2	1	no	—
17	#14523963	—	2	1	no	—
18	(C₂×D₆₇)⋊C₈	—	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×D₂₆₈	D₄	1	1	yes	characteristic
25	C₂×C₄×D₆₇	—	2	1	no	—
26	C₂×C₆₇⋊C₈	—	2	1	no	—
27	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
28	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
29	C₈×D₆₇	—	4	2	no	—
30	C₄⋊Dic₆₇	—	2	1	no	—
31	C₄⋊Dic₆₇	—	2	1	no	—
32	C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
33	(C₂×D₆₇)⋊C₄	—	2	1	no	—
34	C₂×Q₈×C₆₇	—	2	1	no	—
35	C₂×C₈×C₆₇	—	2	1	no	—
36	C₂×C₈×C₆₇	—	2	1	no	—
37	C₄⋊C₄×C₆₇	—	2	1	no	—
38	C₄⋊C₄×C₆₇	—	2	1	no	—
39	C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	C₂²×D₆₇	—	2	1	no	—
41	Q₈×C₆₇	—	8	1	no	—
42	C₂×C₄×C₆₇	—	4	1	no	—
43	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
46	C₂×Dic₆₇	—	2	1	no	—
47	C₂×Dic₆₇	—	4	1	no	—
48	D₂₆₈	—	2	1	no	—
49	D₂₆₈	Q₁₆	1	2	yes	—
50	C₄×D₆₇	—	4	1	no	—
51	C₄×D₆₇	—	4	1	no	—
52	Dic₁₃₄	—	8	1	no	—
53	C₈×C₆₇	—	4	1	no	—
54	C₈×C₆₇	—	4	1	no	—
55	C₆₇⋊C₈	—	2	1	no	—
56	C₆₇⋊C₈	—	4	1	no	—
57	C₆₇⋊C₈	SD₁₆	1	2	yes	—
58	C₂²×C₆₇	#121	1	1	yes	characteristic
59	C₂×D₆₇	—	2	2	no	—
60	C₂×D₆₇	—	4	1	no	—
61	C₂×D₆₇	—	4	1	no	—
62	C₄×C₆₇	—	2	1	no	—
63	C₄×C₆₇	—	8	1	no	—
64	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
65	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
66	C₄×C₆₇	#135	1	1	yes	characteristic
67	C₄×C₆₇	#137	1	1	yes	characteristic
68	Dic₆₇	—	4	1	no	—
69	Dic₆₇	—	8	1	no	—
70	C₂×C₆₇	#461	1	1	yes	characteristic
71	C₂×C₆₇	#467	1	1	yes	characteristic
72	C₂×C₆₇	#472	1	1	yes	characteristic
73	D₆₇	—	4	2	no	—
74	#1675	—	67	1	no	maximal, Sylow
75	C₆₇	#1675	1	1	yes	characteristic, complemented, Sylow
76	#500	—	67	1	no	—
77	#476	—	67	1	no	—
78	#476	—	67	1	no	—
79	C₈×D₄	—	67	1	no	—
80	#335	—	67	1	no	—
81	#333	—	67	1	no	—
82	#333	D₆₇	1	1	yes	characteristic, complemented
83	C₂²×C₈	—	134	1	no	—
84	C₄⋊Q₈	—	67	1	no	—
85	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
86	C₄×D₄	—	67	1	no	—
87	#107	—	134	1	no	—
88	#107	—	134	1	no	—
89	#106	—	134	1	no	—
90	#106	—	134	1	no	—
91	C₄⋊C₈	—	67	1	no	—
92	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
93	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
94	#102	—	134	1	no	—
95	#102	—	134	1	no	—
96	C₂²⋊C₈	—	134	1	no	—
97	C₄×C₈	—	67	1	no	—
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	—	134	1	no	—
100	C₂×D₄	—	67	1	no	—
101	C₂²×C₄	—	134	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	268	2	no	—
105	C₂×C₈	—	67	1	no	—
106	C₂×C₈	—	67	1	no	—
107	C₂×C₈	—	134	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	67	1	no	—
111	C₄⋊C₄	—	134	1	no	—
112	C₄⋊C₄	—	134	1	no	—
113	C₂²⋊C₄	—	134	1	no	—
114	C₄²	C₂²×D₆₇	1	1	yes	characteristic
115	C₂³	—	134	1	no	—
116	Q₈	—	8	1	no	—
117	Q₈	—	536	1	no	—
118	D₄	—	67	2	no	—
119	D₄	—	134	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	268	1	no	—
122	C₂×C₄	—	268	1	no	—
123	C₂×C₄	—	268	1	no	—
124	C₂×C₄	—	134	1	no	—
125	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
126	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
127	C₂×C₄	#11761628	1	1	yes	characteristic
128	C₈	—	4	1	no	—
129	C₈	—	4	1	no	—
130	C₈	—	268	1	no	—
131	C₈	—	67	2	no	—
132	C₈	—	134	1	no	—
133	C₂²	—	268	1	no	—
134	C₂²	—	268	1	no	—
135	C₂²	—	134	2	no	—
136	C₂²	#14524016	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	8	1	no	—
139	C₄	—	268	1	no	—
140	C₄	—	536	1	no	—
141	C₄	SD₁₆×D₆₇	1	1	yes	characteristic
142	C₄	Q₁₆×D₆₇	1	1	yes	characteristic
143	C₄	#14524064	1	1	yes	characteristic
144	C₄	#14524075	1	1	yes	characteristic
145	C₂	—	268	2	no	—
146	C₂	#16863600	1	1	yes	characteristic
147	C₂	#16863636	1	1	yes	characteristic
148	C₂	#16863665	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863799

C₂

yes

maximal, characteristic, complemented

#16863687

C₂

yes

maximal, characteristic

#16863683

C₂

yes

maximal, characteristic

#16863461

C₂

yes

maximal, characteristic

#16863148

C₂

yes

maximal, characteristic, complemented

#16863144

C₂

yes

maximal, characteristic, complemented

#333×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₆₇

—

#14524043

C₂²

yes

characteristic

C₄×D₂₆₈

C₂²

yes

—

—

—

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

#14523965

—

#14523963

—

(C₂×D₆₇)⋊C₈

—

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×D₂₆₈

D₄

yes

characteristic

C₂×C₄×D₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₈×D₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

characteristic

(C₂×D₆₇)⋊C₄

—

C₂×Q₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂²×D₆₇

—

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

—

D₂₆₈

—

D₂₆₈

Q₁₆

yes

—

C₄×D₆₇

—

C₄×D₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

SD₁₆

yes

—

C₂²×C₆₇

#121

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₄×C₆₇

C₂×Q₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

#1675

—

maximal, Sylow

C₆₇

#1675

yes

characteristic, complemented, Sylow

#500

—

#476

—

#476

—

C₈×D₄

—

#335

—

#333

—

#333

D₆₇

yes

characteristic, complemented

C₂²×C₈

—

134

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₇

yes

characteristic

C₄×D₄

—

#107

—

134

—

#107

—

134

—

#106

—

134

—

#106

—

134

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₇

yes

characteristic

C₄⋊C₈

C₂×D₆₇

yes

characteristic

#102

—

134

—

#102

—

134

—

C₂²⋊C₈

—

134

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

134

—

100

C₂×D₄

—

101

C₂²×C₄

—

134

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

268

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

134

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

134

—

112

C₄⋊C₄

—

134

—

113

C₂²⋊C₄

—

134

—

114

C₄²

C₂²×D₆₇

yes

characteristic

115

C₂³

—

134

—

116

Q₈

—

117

Q₈

—

536

—

118

D₄

—

119

D₄

—

134

—

120

C₂×C₄

—

121

C₂×C₄

—

268

—

122

C₂×C₄

—

268

—

123

C₂×C₄

—

268

—

124

C₂×C₄

—

134

—

125

C₂×C₄

D₄×D₆₇

yes

characteristic

126

C₂×C₄

D₄×D₆₇

yes

characteristic

127

C₂×C₄

#11761628

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

268

—

131

C₈

—

132

C₈

—

134

—

133

C₂²

—

268

—

134

C₂²

—

268

—

135

C₂²

—

134

—

136

C₂²

#14524016

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

268

—

140

C₄

—

536

—

141

C₄

SD₁₆×D₆₇

yes

characteristic

142

C₄

Q₁₆×D₆₇

yes

characteristic

143

C₄

#14524064

yes

characteristic

144

C₄

#14524075

yes

characteristic

145

C₂

—

268

—

146

C₂

#16863600

yes

characteristic

147

C₂

#16863636

yes

characteristic

148

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	#1675
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
1. #16863799	C₂	semidirect
5. #16863148	C₂	semidirect
6. #16863144	C₂	semidirect
7. #333×C₆₇	C₂	semidirect
75. C₆₇	#1675	semidirect
82. #333	D₆₇	semidirect

H ≅ G/N

type

1. #16863799

C₂

semidirect

5. #16863148

C₂

semidirect

6. #16863144

C₂

semidirect

7. #333×C₆₇

C₂

semidirect

75. C₆₇

#1675

semidirect

82. #333

D₆₇

semidirect

Subgroups — #22559607

Conjugacy classes of subgroups

Complements