Group #22568819 — subgroups

Subgroups — #22568819

Order 8576.

440 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 209 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	#16864162	C₂	1	2	yes	maximal
2	C₂×#14524051	C₂	1	1	yes	maximal, characteristic
3	#16863445	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863443	C₂	1	4	yes	maximal, complemented
5	C₂×#14523978	C₂	1	2	yes	maximal, complemented
6	#108×D₆₇	C₂	1	4	yes	maximal, complemented
7	#430×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11761628	C₂²	1	1	yes	characteristic
9	#14524051	—	2	4	no	—
10	#14524051	C₄	1	8	yes	—
11	#14524049	—	2	4	no	—
12	C₂×#11761612	C₂²	1	2	yes	—
13	C₂×#11761611	C₂²	1	2	yes	—
14	M₁₆×D₆₇	C₂²	1	8	yes	—
15	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈×D₆₇	C₂²	1	2	yes	—
17	#14523979	C₂²	1	4	yes	—
18	#14523978	C₂²	1	4	yes	—
19	#14523978	C₂²	1	4	yes	—
20	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
21	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	#108×C₆₇	C₂²	1	4	yes	—
23	#11761628	C₂×C₄	1	4	yes	—
24	#11761628	D₄	1	4	yes	—
25	C₂×Dic₁₃₄	Q₈	1	1	yes	characteristic
26	C₂×D₂₆₈	Q₈	1	1	yes	characteristic
27	C₂×C₂²⋊D₆₇	Q₈	1	2	yes	—
28	C₂×C₄×D₆₇	C₂³	1	2	yes	—
29	#11761612	—	2	2	no	—
30	#11761612	C₂³	1	4	yes	—
31	#11761611	—	2	8	no	—
32	#11761611	C₂×C₄	1	8	yes	—
33	C₂×C₆₇⋊C₈	—	2	2	no	—
34	C₂×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
35	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
36	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
37	C₈×D₆₇	—	2	8	no	—
38	C₈×D₆₇	C₂×C₄	1	8	yes	—
39	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	M₁₆×C₆₇	—	2	2	no	—
41	M₁₆×C₆₇	C₂×C₄	1	4	yes	—
42	M₁₆×C₆₇	C₂³	1	4	yes	—
43	C₂×C₈×C₆₇	—	2	2	no	—
44	C₂×C₈×C₆₇	C₂³	1	2	yes	—
45	C₂³×C₆₇	C₂×Q₈	1	1	yes	characteristic
46	C₂²×D₆₇	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
51	C₂²⋊D₆₇	C₄⋊C₄	1	8	yes	—
52	C₂×Dic₆₇	C₂×Q₈	1	2	yes	—
53	D₂₆₈	C₄⋊C₄	1	4	yes	—
54	C₄×D₆₇	C₂²×C₄	1	4	yes	—
55	C₄×D₆₇	C₂×D₄	1	4	yes	—
56	Dic₁₃₄	C₄⋊C₄	1	4	yes	—
57	C₈×C₆₇	—	2	4	no	—
58	C₈×C₆₇	C₂²×C₄	1	4	yes	—
59	C₆₇⋊C₈	—	2	4	no	—
60	C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
61	C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
62	C₂²×C₆₇	—	4	1	no	—
63	C₂²×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
64	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
65	C₂×D₆₇	—	4	2	no	—
66	C₂×D₆₇	C₂×C₄⋊C₄	1	4	yes	—
67	C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
68	C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
70	Dic₆₇	C₂×C₄⋊C₄	1	4	yes	—
71	C₂×C₆₇	—	2	1	no	—
72	C₂×C₆₇	—	2	2	no	—
73	C₂×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
74	D₆₇	—	2	4	no	—
75	#2916	—	67	1	no	maximal, Sylow
76	C₆₇	#2916	1	1	yes	characteristic, complemented, Sylow
77	#568	—	67	2	no	—
78	C₂×#131	—	67	1	no	—
79	#430	—	67	1	no	—
80	#430	—	67	4	no	—
81	#430	D₆₇	1	1	yes	characteristic, complemented
82	C₂×#108	—	67	2	no	—
83	C₂×#108	—	67	4	no	—
84	C₂×#41	—	67	1	no	—
85	#131	—	67	8	no	—
86	#131	—	134	4	no	—
87	#131	—	134	4	no	—
88	C₂×M₁₆	—	67	2	no	—
89	C₂×M₁₆	—	67	2	no	—
90	C₂×M₁₆	—	67	8	no	—
91	C₂×M₁₆	C₂×D₆₇	1	2	yes	—
92	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
93	C₂²×C₈	—	67	1	no	—
94	C₂²×C₈	—	67	2	no	—
95	#108	—	67	4	no	—
96	#108	—	67	4	no	—
97	#108	—	67	4	no	—
98	#108	C₂×D₆₇	1	4	yes	—
99	#41	—	67	4	no	—
100	#41	—	67	4	no	—
101	C₂×Q₈	—	67	1	no	—
102	C₂×D₄	—	67	2	no	—
103	C₂×D₄	—	67	1	no	—
104	C₂²×C₄	—	67	2	no	—
105	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
106	M₁₆	—	2	2	no	—
107	M₁₆	—	67	8	no	—
108	M₁₆	—	67	4	no	—
109	M₁₆	—	134	8	no	—
110	M₁₆	—	134	2	no	—
111	M₁₆	C₄×D₆₇	1	4	yes	—
112	M₁₆	C₂²×D₆₇	1	4	yes	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	67	4	no	—
115	C₂×C₈	—	67	1	no	—
116	C₂×C₈	—	67	1	no	—
117	C₂×C₈	—	67	8	no	—
118	C₂×C₈	—	134	8	no	—
119	C₂×C₈	—	134	2	no	—
120	C₂×C₈	C₂²×D₆₇	1	2	yes	—
121	C₂³	—	67	2	no	—
122	C₂³	Q₈×D₆₇	1	1	yes	characteristic
123	Q₈	—	67	4	no	—
124	D₄	—	67	8	no	—
125	D₄	—	67	4	no	—
126	C₂×C₄	—	67	2	no	—
127	C₂×C₄	—	67	4	no	—
128	C₂×C₄	—	67	4	no	—
129	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
130	C₂×C₄	D₄×D₆₇	1	2	yes	—
131	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
132	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
133	C₈	—	2	4	no	—
134	C₈	—	67	2	no	—
135	C₈	—	67	2	no	—
136	C₈	—	134	4	no	—
137	C₈	C₂×C₄×D₆₇	1	4	yes	—
138	C₂²	—	4	1	no	—
139	C₂²	—	268	2	no	—
140	C₂²	—	67	4	no	—
141	C₂²	C₄⋊C₄×D₆₇	1	2	yes	—
142	C₂²	C₂×Q₈×D₆₇	1	1	yes	characteristic
143	C₄	—	67	4	no	—
144	C₄	C₄⋊C₄×D₆₇	1	2	yes	—
145	C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
146	C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	2	no	—
149	C₂	—	134	4	no	—
150	C₂	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16864162

C₂

yes

maximal

C₂×#14524051

C₂

yes

maximal, characteristic

#16863445

C₂

yes

maximal, characteristic, complemented

#16863443

C₂

yes

maximal, complemented

C₂×#14523978

C₂

yes

maximal, complemented

#108×D₆₇

C₂

yes

maximal, complemented

#430×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761628

C₂²

yes

characteristic

#14524051

—

#14524051

C₄

yes

—

#14524049

—

C₂×#11761612

C₂²

yes

—

C₂×#11761611

C₂²

yes

—

M₁₆×D₆₇

C₂²

yes

—

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₆₇

C₂²

yes

—

#14523979

C₂²

yes

—

#14523978

C₂²

yes

—

#14523978

C₂²

yes

—

C₂×M₁₆×C₆₇

C₂²

yes

—

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

#108×C₆₇

C₂²

yes

—

#11761628

C₂×C₄

yes

—

#11761628

D₄

yes

—

C₂×Dic₁₃₄

Q₈

yes

characteristic

C₂×D₂₆₈

Q₈

yes

characteristic

C₂×C₂²⋊D₆₇

Q₈

yes

—

C₂×C₄×D₆₇

C₂³

yes

—

#11761612

—

#11761612

C₂³

yes

—

#11761611

—

#11761611

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₆₇⋊C₈

C₂³

yes

characteristic

C₂×C₆₇⋊C₈

C₂³

yes

characteristic

C₈×D₆₇

—

C₈×D₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

C₂×C₄

yes

—

M₁₆×C₆₇

C₂³

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂×Q₈

yes

characteristic

C₂²×D₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂²⋊D₆₇

C₄⋊C₄

yes

—

C₂×Dic₆₇

C₂×Q₈

yes

—

D₂₆₈

C₄⋊C₄

yes

—

C₄×D₆₇

C₂²×C₄

yes

—

C₄×D₆₇

C₂×D₄

yes

—

Dic₁₃₄

C₄⋊C₄

yes

—

C₈×C₆₇

—

C₈×C₆₇

C₂²×C₄

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

C₂²×C₄

yes

—

C₆₇⋊C₈

C₂²×C₄

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄⋊C₄

yes

—

C₂²×C₆₇

C₂²×Q₈

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

C₂×C₄⋊C₄

yes

—

C₄×C₆₇

C₂×C₄⋊C₄

yes

—

C₄×C₆₇

C₂³×C₄

yes

characteristic

C₄×C₆₇

C₂²×D₄

yes

characteristic

Dic₆₇

C₂×C₄⋊C₄

yes

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂²×C₄⋊C₄

yes

characteristic

D₆₇

—

#2916

—

maximal, Sylow

C₆₇

#2916

yes

characteristic, complemented, Sylow

—

—

—

—

D₆₇

yes

characteristic, complemented

C₂×#108

—

C₂×#108

—

C₂×#41

—

#131

—

#131

—

134

—

#131

—

134

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

—

C₂×M₁₆

C₂×D₆₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

—

—

—

C₂×D₆₇

yes

—

#41

—

100

#41

—

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂²×D₆₇

yes

characteristic

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

134

—

110

M₁₆

—

134

—

111

M₁₆

C₄×D₆₇

yes

—

112

M₁₆

C₂²×D₆₇

yes

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

134

—

119

C₂×C₈

—

134

—

120

C₂×C₈

C₂²×D₆₇

yes

—

121

C₂³

—

122

C₂³

Q₈×D₆₇

yes

characteristic

123

Q₈

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×C₄×D₆₇

yes

—

130

C₂×C₄

D₄×D₆₇

yes

—

131

C₂×C₄

Q₈×D₆₇

yes

characteristic

132

C₂×C₄

C₂³×D₆₇

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

134

—

137

C₈

C₂×C₄×D₆₇

yes

—

138

C₂²

—

139

C₂²

—

268

—

140

C₂²

—

141

C₂²

C₄⋊C₄×D₆₇

yes

—

142

C₂²

C₂×Q₈×D₆₇

yes

characteristic

143

C₄

—

144

C₄

C₄⋊C₄×D₆₇

yes

—

145

C₄

C₂²×C₄×D₆₇

yes

characteristic

146

C₄

C₂×D₄×D₆₇

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

—

134

—

150

C₂

C₂×C₄⋊C₄×D₆₇

yes

characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₆₇
commutator	C₄×C₆₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#430×C₆₇	C₂
socle	C₂×C₆₇	C₂²×C₄⋊C₄
hypercenter	#430	D₆₇
hypocenter	C₆₇	#2916
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

subgroup

G/H

center

C₄

C₂×D₄×D₆₇

commutator

C₄×C₆₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

#430×C₆₇

C₂

socle

C₂×C₆₇

C₂²×C₄⋊C₄

D₆₇

C₆₇

G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

N	H ≅ G/N	type
3. #16863445	C₂	semidirect
4. #16863443	C₂	semidirect
5. C₂×#14523978	C₂	semidirect
6. #108×D₆₇	C₂	semidirect
7. #430×C₆₇	C₂	semidirect
76. C₆₇	#2916	semidirect
81. #430	D₆₇	semidirect

H ≅ G/N

type

3. #16863445

C₂

semidirect

4. #16863443

C₂

semidirect

5. C₂×#14523978

C₂

semidirect

6. #108×D₆₇

C₂

semidirect

7. #430×C₆₇

C₂

semidirect

76. C₆₇

#2916

semidirect

81. #430

D₆₇

semidirect

Subgroups — #22568819

Conjugacy classes of subgroups

Complements