Group #22568802 — subgroups

Subgroups — #22568802

Order 8576.

476 conjugacy classes of subgroups; only the 253 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 253 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	C₂×#14524050	C₂	1	1	yes	maximal, characteristic
2	#16863879	C₂	1	1	yes	maximal, characteristic
3	#16863439	C₂	1	3	yes	maximal, complemented
4	#16863436	C₂	1	3	yes	maximal, complemented
5	#16863432	C₂	1	3	yes	maximal, complemented
6	C₂×#14523974	C₂	1	3	yes	maximal, complemented
7	#520×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#14524050	C₄	1	8	yes	complemented
9	C₂×#11761612	C₂²	1	3	yes	—
10	C₂²×C₆₇⋊C₈	C₂²	1	3	yes	—
11	Q₈×Dic₆₇	—	2	1	no	—
12	D₄×Dic₆₇	—	2	3	no	—
13	#14524002	C₂²	1	3	yes	—
14	C₂×C₄⋊Dic₆₇	C₂²	1	3	yes	—
15	#14523976	C₂²	1	3	yes	—
16	#14523976	C₂²	1	3	yes	—
17	#14523974	C₂²	1	3	yes	—
18	#14523974	C₂²	1	6	yes	—
19	#14523974	C₂²	1	1	yes	characteristic
20	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₆₇	—	2	1	no	—
22	C₄×D₄×C₆₇	—	2	3	no	—
23	#117×C₆₇	C₂²	1	3	yes	—
24	C₂×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
25	C₂²×Dic₆₇	—	2	3	no	—
26	#11761612	C₂×C₄	1	12	yes	complemented
27	C₂×C₆₇⋊C₈	C₂×C₄	1	12	yes	complemented
28	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
29	C₂×C₆₇⋊C₈	C₂³	1	3	yes	—
30	C₄⋊Dic₆₇	—	2	3	no	—
31	C₄⋊Dic₆₇	C₂³	1	3	yes	—
32	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
33	C₂²⋊Dic₆₇	—	2	3	no	—
34	C₄×Dic₆₇	—	2	3	no	—
35	#41×C₆₇	C₂×C₄	1	4	yes	—
36	#41×C₆₇	Q₈	1	4	yes	—
37	C₂×Q₈×C₆₇	D₄	1	1	yes	characteristic
38	C₂×D₄×C₆₇	D₄	1	3	yes	—
39	C₂²×C₄×C₆₇	—	2	3	no	—
40	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
41	C₄⋊C₄×C₆₇	—	2	3	no	—
42	C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
43	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂²⋊C₄×C₆₇	—	2	3	no	—
45	C₄²×C₆₇	—	2	3	no	—
46	C₂³×C₆₇	C₂×D₄	1	3	yes	—
47	Q₈×C₆₇	C₄⋊C₄	1	4	yes	—
48	D₄×C₆₇	C₄⋊C₄	1	12	yes	—
49	C₂×C₄×C₆₇	—	2	1	no	—
50	C₂×C₄×C₆₇	—	2	3	no	—
51	C₂×C₄×C₆₇	—	8	3	no	—
52	C₂×C₄×C₆₇	C₂²×C₄	1	6	yes	—
53	C₂×C₄×C₆₇	C₂×D₄	1	3	yes	—
54	C₂×C₄×C₆₇	C₂×Q₈	1	6	yes	—
55	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₆₇	—	2	1	no	—
57	C₂×Dic₆₇	—	2	3	no	—
58	C₂×Dic₆₇	—	8	3	no	—
59	C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
60	C₆₇⋊C₈	C₂²×C₄	1	6	yes	—
61	C₂²×C₆₇	—	2	3	no	—
62	C₂²×C₆₇	—	2	3	no	—
63	C₂²×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
64	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₆₇	—	4	1	no	—
66	C₄×C₆₇	—	4	3	no	—
67	C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
68	C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
70	Dic₆₇	—	4	1	no	—
71	Dic₆₇	—	4	3	no	—
72	C₂×C₆₇	—	2	6	no	—
73	C₂×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₆₇	#577	1	1	yes	characteristic
75	C₂×C₆₇	#577	1	1	yes	characteristic
76	#2912	—	67	1	no	maximal, Sylow
77	C₆₇	#2912	1	1	yes	characteristic, complemented, Sylow
78	C₂×#131	—	67	1	no	—
79	#520	—	67	1	no	—
80	#520	D₆₇	1	1	yes	characteristic
81	#428	—	67	3	no	—
82	#428	—	67	3	no	—
83	#427	—	67	3	no	—
84	C₂×#106	—	67	3	no	—
85	C₂×#41	C₂×D₆₇	1	1	yes	characteristic
86	#131	—	67	8	no	—
87	C₂×M₁₆	—	67	3	no	—
88	C₂²×C₈	—	67	3	no	—
89	C₄×Q₈	—	2	1	no	—
90	C₄×Q₈	—	134	1	no	—
91	C₄×D₄	—	2	3	no	—
92	C₄×D₄	—	134	3	no	—
93	#117	—	67	3	no	—
94	#117	C₂×D₆₇	1	3	yes	—
95	C₂×C₄⋊C₄	—	67	3	no	—
96	C₂×C₄⋊C₄	C₂×D₆₇	1	3	yes	—
97	#107	—	67	3	no	—
98	#107	—	67	3	no	—
99	#106	—	67	3	no	—
100	#106	—	67	6	no	—
101	#106	—	67	1	no	—
102	#41	Dic₁₃₄	1	4	yes	—
103	#41	C₄×D₆₇	1	4	yes	—
104	C₂×Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
105	C₂×D₄	C₂²⋊D₆₇	1	3	yes	—
106	C₂²×C₄	—	2	3	no	—
107	C₂²×C₄	—	134	3	no	—
108	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
109	M₁₆	—	67	12	no	—
110	C₂×C₈	—	67	12	no	—
111	C₂×C₈	—	67	1	no	—
112	C₂×C₈	—	67	3	no	—
113	C₄⋊C₄	—	2	3	no	—
114	C₄⋊C₄	—	67	3	no	—
115	C₄⋊C₄	—	67	1	no	—
116	C₄⋊C₄	—	134	3	no	—
117	C₄⋊C₄	C₂²×D₆₇	1	3	yes	—
118	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
119	C₂²⋊C₄	—	2	3	no	—
120	C₂²⋊C₄	—	134	3	no	—
121	C₄²	—	2	3	no	—
122	C₄²	—	134	3	no	—
123	C₂³	C₂×C₂²⋊D₆₇	1	3	yes	—
124	Q₈	Dic₆₇⋊C₄	1	4	yes	—
125	D₄	Dic₆₇⋊C₄	1	12	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	3	no	—
128	C₂×C₄	—	8	3	no	—
129	C₂×C₄	—	536	3	no	—
130	C₂×C₄	—	134	1	no	—
131	C₂×C₄	—	134	3	no	—
132	C₂×C₄	C₂×C₄×D₆₇	1	6	yes	—
133	C₂×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
134	C₂×C₄	C₂×Dic₁₃₄	1	6	yes	—
135	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
136	C₈	—	67	2	no	—
137	C₈	—	67	6	no	—
138	C₂²	—	2	3	no	—
139	C₂²	—	2	3	no	—
140	C₂²	C₂×Dic₆₇⋊C₄	1	6	yes	—
141	C₂²	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
142	C₄	—	4	1	no	—
143	C₄	—	4	3	no	—
144	C₄	—	268	3	no	—
145	C₄	—	268	1	no	—
146	C₄	C₂×Dic₆₇⋊C₄	1	6	yes	—
147	C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
148	C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
149	C₂	—	2	6	no	—
150	C₂	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
151	C₂	#16864204	1	1	yes	characteristic
152	C₂	#16864207	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524050

C₂

yes

maximal, characteristic

#16863879

C₂

yes

maximal, characteristic

#16863439

C₂

yes

maximal, complemented

#16863436

C₂

yes

maximal, complemented

#16863432

C₂

yes

maximal, complemented

C₂×#14523974

C₂

yes

maximal, complemented

#520×C₆₇

C₂

yes

maximal, characteristic

#14524050

C₄

yes

complemented

C₂×#11761612

C₂²

yes

—

C₂²×C₆₇⋊C₈

C₂²

yes

—

Q₈×Dic₆₇

—

D₄×Dic₆₇

—

#14524002

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

#14523976

C₂²

yes

—

#14523976

C₂²

yes

—

#14523974

C₂²

yes

—

#14523974

C₂²

yes

—

#14523974

C₂²

yes

characteristic

C₂×#41×C₆₇

C₂²

yes

characteristic

C₄×Q₈×C₆₇

—

C₄×D₄×C₆₇

—

#117×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂²×Dic₆₇

—

#11761612

C₂×C₄

yes

complemented

C₂×C₆₇⋊C₈

C₂×C₄

yes

complemented

C₂×C₆₇⋊C₈

C₂³

yes

characteristic

C₂×C₆₇⋊C₈

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

C₂³

yes

characteristic

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

#41×C₆₇

C₂×C₄

yes

—

#41×C₆₇

Q₈

yes

—

C₂×Q₈×C₆₇

D₄

yes

characteristic

C₂×D₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

C₂³

yes

characteristic

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

—

Q₈×C₆₇

C₄⋊C₄

yes

—

D₄×C₆₇

C₄⋊C₄

yes

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₆₇⋊C₈

C₂²×C₄

yes

—

C₆₇⋊C₈

C₂²×C₄

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×C₄⋊C₄

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₄⋊C₄

yes

—

C₄×C₆₇

C₂³×C₄

yes

characteristic

C₄×C₆₇

C₂²×Q₈

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₂²×C₄⋊C₄

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×#131

—

#520

—

#520

D₆₇

yes

—

—

—

—

C₂×#41

C₂×D₆₇

yes

characteristic

#131

—

C₂×M₁₆

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

134

—

C₄×D₄

—

C₄×D₄

—

134

—

#117

—

#117

C₂×D₆₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

#107

—

#107

—

#106

—

100

#106

—

101

#106

—

102

#41

Dic₁₃₄

yes

—

103

#41

C₄×D₆₇

yes

—

104

C₂×Q₈

C₂²⋊D₆₇

yes

characteristic

105

C₂×D₄

C₂²⋊D₆₇

yes

—

106

C₂²×C₄

—

107

C₂²×C₄

—

134

—

108

C₂²×C₄

C₂²×D₆₇

yes

—

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

134

—

117

C₄⋊C₄

C₂²×D₆₇

yes

—

118

C₄⋊C₄

C₂²×D₆₇

yes

characteristic

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

134

—

121

C₄²

—

122

C₄²

—

134

—

123

C₂³

C₂×C₂²⋊D₆₇

yes

—

124

Q₈

Dic₆₇⋊C₄

yes

—

125

D₄

Dic₆₇⋊C₄

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

536

—

130

C₂×C₄

—

134

—

131

C₂×C₄

—

134

—

132

C₂×C₄

C₂×C₄×D₆₇

yes

—

133

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

134

C₂×C₄

C₂×Dic₁₃₄

yes

—

135

C₂×C₄

C₂³×D₆₇

yes

characteristic

136

C₈

—

137

C₈

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂×Dic₆₇⋊C₄

yes

—

141

C₂²

C₂²×C₂²⋊D₆₇

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

268

—

145

C₄

—

268

—

146

C₄

C₂×Dic₆₇⋊C₄

yes

—

147

C₄

C₂²×C₄×D₆₇

yes

characteristic

148

C₄

C₂²×Dic₁₃₄

yes

characteristic

149

C₂

—

150

C₂

C₂²×Dic₆₇⋊C₄

yes

characteristic

151

C₂

#16864204

yes

characteristic

152

C₂

#16864207

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₆₇
commutator	C₄×C₆₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#520×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#520	D₆₇
hypocenter	C₆₇	#2912
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ #520

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₆₇

commutator

C₄×C₆₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

#520×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

D₆₇

C₆₇

G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×#41 ◁ #520

N	H ≅ G/N	type
3. #16863439	C₂	semidirect
4. #16863436	C₂	semidirect
5. #16863432	C₂	semidirect
6. C₂×#14523974	C₂	semidirect
8. #14524050	C₄	semidirect
26. #11761612	C₂×C₄	semidirect
27. C₂×C₆₇⋊C₈	C₂×C₄	semidirect
77. C₆₇	#2912	semidirect

H ≅ G/N

type

3. #16863439

C₂

semidirect

4. #16863436

C₂

semidirect

5. #16863432

C₂

semidirect

6. C₂×#14523974

C₂

semidirect

8. #14524050

C₄

semidirect

26. #11761612

C₂×C₄

semidirect

27. C₂×C₆₇⋊C₈

C₂×C₄

semidirect

77. C₆₇

#2912

semidirect

Subgroups — #22568802

Conjugacy classes of subgroups

Complements