Group #30770485 — subgroups

Subgroups — #30770485

Order 17152.

306 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22562459	C₂	1	1	yes	maximal, characteristic
2	#22562434	C₂	1	2	yes	maximal, complemented
3	#22559802	C₂	1	2	yes	maximal, complemented
4	C₁₆×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2252	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₀₇₂	—	2	2	no	—
7	#16863773	C₂²	1	2	yes	—
8	#16863770	—	2	2	no	—
9	#16863744	—	2	2	no	—
10	#16863196	—	2	4	no	—
11	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	#500×C₆₇	C₂²	1	2	yes	—
14	#366×C₆₇	—	2	2	no	—
15	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#11761615	—	4	4	no	—
17	C₂×D₅₃₆	—	2	2	no	—
18	C₂×D₅₃₆	—	2	2	no	—
19	#14524040	—	2	2	no	—
20	D₁₀₇₂	—	4	4	no	—
21	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
22	#14523963	—	4	4	no	—
23	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
24	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
25	C₄⋊Q₈×C₆₇	—	2	2	no	—
26	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
27	#107×C₆₇	—	2	2	no	—
28	#107×C₆₇	—	2	2	no	—
29	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
30	C₂×D₂₆₈	—	4	2	no	—
31	C₂×D₂₆₈	—	4	2	no	—
32	C₂×D₂₆₈	—	4	2	no	—
33	#11761615	—	8	8	no	—
34	D₅₃₆	—	4	4	no	—
35	D₅₃₆	—	4	4	no	—
36	C₂×C₆₇⋊C₈	D₈	1	2	yes	—
37	C₂×C₆₇⋊C₈	D₈	1	2	yes	—
38	C₆₇⋊C₁₆	—	2	2	no	—
39	C₂×Q₈×C₆₇	—	4	2	no	—
40	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₄⋊C₄×C₆₇	—	4	2	no	—
43	C₄⋊C₄×C₆₇	—	4	2	no	—
44	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₁₆×C₆₇	SD₁₆	1	4	yes	complemented
46	C₂²×D₆₇	—	8	2	no	—
47	Q₈×C₆₇	—	8	4	no	—
48	C₂×C₄×C₆₇	—	8	2	no	—
49	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
52	D₂₆₈	—	8	2	no	—
53	D₂₆₈	—	8	2	no	—
54	D₂₆₈	—	8	4	no	—
55	D₂₆₈	—	8	4	no	—
56	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
57	C₈×C₆₇	#135	1	2	yes	—
58	C₆₇⋊C₈	—	2	2	no	—
59	C₆₇⋊C₈	D₁₆	1	4	yes	complemented
60	C₂²×C₆₇	#493	1	1	yes	characteristic
61	C₂×D₆₇	—	16	2	no	—
62	C₂×D₆₇	—	16	2	no	—
63	C₂×D₆₇	—	16	2	no	—
64	C₄×C₆₇	—	16	2	no	—
65	C₄×C₆₇	#492	1	1	yes	characteristic
66	C₄×C₆₇	#495	1	1	yes	characteristic
67	C₄×C₆₇	C₂×D₁₆	1	2	yes	—
68	C₄×C₆₇	#508	1	2	yes	—
69	#12503	—	67	1	no	maximal, Sylow
70	C₂×C₆₇	#1706	1	1	yes	characteristic
71	C₂×C₆₇	#2246	1	1	yes	characteristic
72	C₂×C₆₇	#2249	1	1	yes	characteristic
73	D₆₇	—	32	2	no	—
74	#2252	D₆₇	1	1	yes	characteristic, complemented
75	#2246	—	67	1	no	—
76	#2240	—	67	2	no	—
77	#1706	—	67	2	no	—
78	C₈×C₁₆	—	67	1	no	—
79	C₆₇	#12503	1	1	yes	characteristic, complemented, Sylow
80	C₂×D₁₆	—	134	2	no	—
81	#500	C₂×D₆₇	1	2	yes	—
82	#493	—	67	2	no	—
83	#492	—	134	2	no	—
84	#486	—	134	2	no	—
85	#366	—	2	2	no	—
86	#357	—	134	4	no	—
87	C₄×C₁₆	—	67	1	no	—
88	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
89	C₈²	—	67	1	no	—
90	C₂×SD₁₆	—	268	4	no	—
91	C₂×D₈	—	134	2	no	—
92	C₂×D₈	—	134	2	no	—
93	C₄⋊Q₈	—	2	2	no	—
94	#127	—	134	2	no	—
95	D₁₆	—	268	4	no	—
96	C₂×C₁₆	—	67	2	no	—
97	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
98	#107	—	2	2	no	—
99	#107	—	2	2	no	—
100	#102	—	268	4	no	—
101	C₄×C₈	—	67	1	no	—
102	C₄×C₈	—	67	1	no	—
103	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
104	C₂×Q₈	—	4	2	no	—
105	C₂×D₄	—	268	2	no	—
106	C₂×D₄	—	268	2	no	—
107	C₂×D₄	—	268	2	no	—
108	SD₁₆	—	536	8	no	—
109	D₈	—	268	4	no	—
110	D₈	—	268	4	no	—
111	C₂×C₈	—	67	2	no	—
112	C₂×C₈	—	67	2	no	—
113	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
114	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	—	4	2	no	—
117	C₄²	D₄×D₆₇	1	1	yes	characteristic
118	C₁₆	—	134	2	no	—
119	C₁₆	#11761615	1	4	yes	complemented
120	C₂³	—	536	2	no	—
121	Q₈	—	8	4	no	—
122	D₄	—	536	2	no	—
123	D₄	—	536	2	no	—
124	D₄	—	536	4	no	—
125	D₄	—	536	4	no	—
126	C₂×C₄	—	8	2	no	—
127	C₂×C₄	#14524039	1	1	yes	characteristic
128	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
129	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
130	C₈	—	67	4	no	—
131	C₈	—	134	2	no	—
132	C₈	C₂×#11761615	1	2	yes	—
133	C₈	#14524066	1	2	yes	—
134	C₂²	—	1072	2	no	—
135	C₂²	—	1072	2	no	—
136	C₂²	—	1072	2	no	—
137	C₂²	#16863771	1	1	yes	characteristic
138	C₄	—	16	2	no	—
139	C₄	#16863770	1	1	yes	characteristic
140	C₄	#16863781	1	1	yes	characteristic
141	C₄	D₁₆×D₆₇	1	2	yes	—
142	C₄	#16863829	1	2	yes	—
143	C₂	—	2144	2	no	—
144	C₂	#22559802	1	1	yes	characteristic
145	C₂	#22562457	1	1	yes	characteristic
146	C₂	#22562471	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562459

C₂

yes

maximal, characteristic

#22562434

C₂

yes

maximal, complemented

#22559802

C₂

yes

maximal, complemented

C₁₆×C₆₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₆₇×#2252

C₂

yes

maximal, characteristic, complemented

C₂×D₁₀₇₂

—

#16863773

C₂²

yes

—

#16863770

—

#16863744

—

#16863196

—

C₄×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₈×C₆₇⋊C₈

C₂²

yes

characteristic

#500×C₆₇

C₂²

yes

—

#366×C₆₇

—

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×#11761615

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

#14524040

—

D₁₀₇₂

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

#14523963

—

C₄×C₆₇⋊C₈

D₄

yes

characteristic

C₄×C₆₇⋊C₈

D₄

yes

characteristic

C₄⋊Q₈×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

—

#107×C₆₇

—

#107×C₆₇

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

#11761615

—

D₅₃₆

—

D₅₃₆

—

C₂×C₆₇⋊C₈

D₈

yes

—

C₂×C₆₇⋊C₈

D₈

yes

—

C₆₇⋊C₁₆

—

C₂×Q₈×C₆₇

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

SD₁₆

yes

complemented

C₂²×D₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

C₂×SD₁₆

yes

—

C₈×C₆₇

#135

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

D₁₆

yes

complemented

C₂²×C₆₇

#493

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

C₂×D₁₆

yes

—

C₄×C₆₇

#508

yes

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

#2252

D₆₇

yes

characteristic, complemented

#2246

—

#2240

—

#1706

—

C₈×C₁₆

—

C₆₇

#12503

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

134

—

#500

C₂×D₆₇

yes

—

#493

—

#492

—

134

—

#486

—

134

—

#366

—

#357

—

134

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₇

yes

characteristic

C₈²

—

C₂×SD₁₆

—

268

—

C₂×D₈

—

134

—

C₂×D₈

—

134

—

C₄⋊Q₈

—

#127

—

134

—

D₁₆

—

268

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₆₇

yes

—

#107

—

#107

—

100

#102

—

268

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₂²×D₆₇

yes

characteristic

104

C₂×Q₈

—

105

C₂×D₄

—

268

—

106

C₂×D₄

—

268

—

107

C₂×D₄

—

268

—

108

SD₁₆

—

536

—

109

D₈

—

268

—

110

D₈

—

268

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

D₄×D₆₇

yes

characteristic

114

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

D₄×D₆₇

yes

characteristic

118

C₁₆

—

134

—

119

C₁₆

#11761615

yes

complemented

120

C₂³

—

536

—

121

Q₈

—

122

D₄

—

536

—

123

D₄

—

536

—

124

D₄

—

536

—

125

D₄

—

536

—

126

C₂×C₄

—

127

C₂×C₄

#14524039

yes

characteristic

128

C₂×C₄

D₈×D₆₇

yes

characteristic

129

C₂×C₄

D₈×D₆₇

yes

characteristic

130

C₈

—

131

C₈

—

134

—

132

C₈

C₂×#11761615

yes

—

133

C₈

#14524066

yes

—

134

C₂²

—

1072

—

135

C₂²

—

1072

—

136

C₂²

—

1072

—

137

C₂²

#16863771

yes

characteristic

138

C₄

—

139

C₄

#16863770

yes

characteristic

140

C₄

#16863781

yes

characteristic

141

C₄

D₁₆×D₆₇

yes

—

142

C₄

#16863829

yes

—

143

C₂

—

2144

—

144

C₂

#22559802

yes

characteristic

145

C₂

#22562457

yes

characteristic

146

C₂

#22562471

yes

characteristic

	subgroup	G/H
center	C₂²	#16863771
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#2252	C₂
socle	C₂²×C₆₇	#493
prime radical	#22562459	C₂
hypercenter	#2252	D₆₇
hypocenter	C₆₇	#12503
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#16863771

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#2252

C₂

C₂²×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
2. #22562434	C₂	semidirect
3. #22559802	C₂	semidirect
4. C₁₆×C₆₇⋊C₈	C₂	semidirect
5. C₆₇×#2252	C₂	semidirect
45. C₁₆×C₆₇	SD₁₆	semidirect
59. C₆₇⋊C₈	D₁₆	semidirect
74. #2252	D₆₇	semidirect
79. C₆₇	#12503	semidirect
119. C₁₆	#11761615	semidirect

H ≅ G/N

type

2. #22562434

C₂

semidirect

3. #22559802

C₂

semidirect

4. C₁₆×C₆₇⋊C₈

C₂

semidirect

5. C₆₇×#2252

C₂

semidirect

45. C₁₆×C₆₇

SD₁₆

semidirect

59. C₆₇⋊C₈

D₁₆

semidirect

74. #2252

D₆₇

semidirect

79. C₆₇

#12503

semidirect

119. C₁₆

#11761615

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22562459	2
#22562434	2
#22559802	2
C₁₆×C₆₇⋊C₈	2
C₆₇×#2252	2
#12503	67

subgroup

index

#22562459

#22562434

#22559802

C₁₆×C₆₇⋊C₈

C₆₇×#2252

#12503

Subgroups — #30770485

Conjugacy classes of subgroups

Complements

Maximal subgroups