Group #30770559 — subgroups

Subgroups — #30770559

Order 17152.

360 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 99 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	#22562465	C₂	1	2	yes	maximal, complemented
2	#22562459	C₂	1	2	yes	maximal, complemented
3	#22559824	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559815	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523988	—	2	4	no	—
7	C₂×D₁₀₇₂	—	2	4	no	—
8	#16863776	C₂²	1	1	yes	characteristic
9	#16863773	—	2	2	no	—
10	#16863773	C₂²	1	1	yes	characteristic
11	#16863773	C₂²	1	2	yes	—
12	#16863743	—	2	2	no	—
13	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
14	C₈²×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₆₈	—	2	2	no	—
16	C₂×D₅₃₆	—	2	2	no	—
17	C₂×D₅₃₆	—	2	2	no	—
18	C₂×D₅₃₆	—	2	2	no	—
19	C₂×D₅₃₆	—	4	4	no	—
20	#14524043	—	2	1	no	—
21	#14524040	—	2	2	no	—
22	#14524040	—	2	1	no	—
23	#14523988	—	4	8	no	—
24	D₁₀₇₂	—	4	8	no	—
25	#14523965	—	4	4	no	—
26	C₂×C₁₆×C₆₇	D₄	1	4	yes	—
27	C₄×C₈×C₆₇	D₄	1	2	yes	—
28	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₃₄	—	4	1	no	—
30	C₂×D₂₆₈	—	4	2	no	—
31	C₂×D₂₆₈	—	4	1	no	—
32	C₂×D₂₆₈	—	4	2	no	—
33	C₂×D₂₆₈	—	4	4	no	—
34	Dic₂₆₈	—	4	4	no	—
35	D₅₃₆	—	4	4	no	—
36	D₅₃₆	—	4	4	no	—
37	D₅₃₆	—	4	4	no	—
38	D₅₃₆	—	8	8	no	—
39	C₄⋊Dic₆₇	—	4	2	no	—
40	C₂×C₈×C₆₇	D₈	1	4	yes	—
41	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
42	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₆₇	D₈	1	8	yes	complemented
44	C₂²×D₆₇	—	8	1	no	—
45	C₂²×D₆₇	—	8	2	no	—
46	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
47	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
48	C₂×Dic₆₇	—	8	1	no	—
49	D₂₆₈	—	8	2	no	—
50	D₂₆₈	—	8	2	no	—
51	D₂₆₈	—	8	2	no	—
52	D₂₆₈	—	8	8	no	—
53	D₂₆₈	—	8	4	no	—
54	Dic₁₃₄	—	8	2	no	—
55	C₈×C₆₇	—	2	4	no	—
56	C₈×C₆₇	C₂×D₈	1	4	yes	—
57	C₂²×C₆₇	#493	1	1	yes	characteristic
58	C₂×D₆₇	—	16	1	no	—
59	C₂×D₆₇	—	16	2	no	—
60	C₂×D₆₇	—	16	4	no	—
61	C₂×D₆₇	—	16	2	no	—
62	C₄×C₆₇	#493	1	2	yes	—
63	C₄×C₆₇	#509	1	4	yes	—
64	Dic₆₇	—	16	1	no	—
65	#12519	—	67	1	no	maximal, Sylow
66	C₂×C₆₇	#1712	1	1	yes	characteristic
67	C₂×C₆₇	#2250	1	2	yes	—
68	D₆₇	—	32	2	no	—
69	D₆₇	—	32	1	no	—
70	#2248	—	67	2	no	—
71	#2246	—	67	2	no	—
72	#1712	—	67	1	no	—
73	#1709	—	67	1	no	—
74	#1311	D₆₇	1	1	yes	characteristic, complemented
75	C₆₇	#12519	1	1	yes	characteristic, complemented, Sylow
76	C₂×SD₃₂	—	134	4	no	—
77	C₂×D₁₆	—	134	4	no	—
78	#494	—	67	1	no	—
79	#493	—	67	1	no	—
80	#493	—	67	2	no	—
81	#493	—	134	2	no	—
82	#486	—	134	2	no	—
83	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
84	C₈²	C₂×D₆₇	1	1	yes	characteristic
85	C₂×Q₁₆	—	134	2	no	—
86	C₂×D₈	—	268	4	no	—
87	C₂×D₈	—	134	2	no	—
88	C₂×D₈	—	134	2	no	—
89	C₂×D₈	—	134	2	no	—
90	C₄⋊Q₈	—	134	1	no	—
91	#127	—	134	1	no	—
92	#127	—	134	2	no	—
93	SD₃₂	—	268	8	no	—
94	D₁₆	—	268	8	no	—
95	C₂×C₁₆	D₂₆₈	1	4	yes	—
96	#102	—	268	4	no	—
97	C₄×C₈	D₂₆₈	1	2	yes	—
98	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
99	C₂×Q₈	—	268	1	no	—
100	C₂×D₄	—	268	1	no	—
101	C₂×D₄	—	268	2	no	—
102	C₂×D₄	—	268	2	no	—
103	C₂×D₄	—	268	4	no	—
104	Q₁₆	—	268	4	no	—
105	D₈	—	268	4	no	—
106	D₈	—	268	4	no	—
107	D₈	—	268	4	no	—
108	D₈	—	536	8	no	—
109	C₂×C₈	D₅₃₆	1	4	yes	—
110	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
111	C₄⋊C₄	—	268	2	no	—
112	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
113	C₁₆	D₅₃₆	1	8	yes	complemented
114	C₂³	—	536	1	no	—
115	C₂³	—	536	2	no	—
116	Q₈	—	536	2	no	—
117	D₄	—	536	2	no	—
118	D₄	—	536	2	no	—
119	D₄	—	536	2	no	—
120	D₄	—	536	8	no	—
121	D₄	—	536	4	no	—
122	C₂×C₄	—	536	1	no	—
123	C₂×C₄	#14524040	1	1	yes	characteristic
124	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
125	C₈	—	2	4	no	—
126	C₈	C₂×D₅₃₆	1	4	yes	—
127	C₂²	—	1072	1	no	—
128	C₂²	—	1072	2	no	—
129	C₂²	—	1072	4	no	—
130	C₂²	—	1072	2	no	—
131	C₂²	#16863773	1	1	yes	characteristic
132	C₄	—	1072	1	no	—
133	C₄	#16863773	1	2	yes	—
134	C₄	#16863834	1	4	yes	—
135	C₂	—	2144	2	no	—
136	C₂	—	2144	1	no	—
137	C₂	#22559824	1	1	yes	characteristic
138	C₂	#22562475	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22562465

C₂

yes

maximal, complemented

#22562459

C₂

yes

maximal, complemented

#22559824

C₂

yes

maximal, characteristic, complemented

#22559815

C₂

yes

maximal, characteristic, complemented

C₆₇×#1311

C₂

yes

maximal, characteristic, complemented

C₂×#14523988

—

C₂×D₁₀₇₂

—

#16863776

C₂²

yes

characteristic

#16863773

—

#16863773

C₂²

yes

characteristic

#16863773

C₂²

yes

—

#16863743

—

C₄×C₁₆×C₆₇

C₂²

yes

—

C₈²×C₆₇

C₂²

yes

characteristic

C₂×Dic₂₆₈

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

—

—

—

—

D₁₀₇₂

—

#14523965

—

C₂×C₁₆×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

Dic₂₆₈

—

D₅₃₆

—

D₅₃₆

—

D₅₃₆

—

D₅₃₆

—

C₄⋊Dic₆₇

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

D₈

yes

complemented

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

C₂×D₈

yes

—

C₂²×C₆₇

#493

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

#493

yes

—

C₄×C₆₇

#509

yes

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

D₆₇

—

D₆₇

—

—

—

—

—

D₆₇

yes

characteristic, complemented

C₆₇

#12519

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

134

—

C₂×D₁₆

—

134

—

—

—

—

—

134

—

#486

—

134

—

C₄×C₁₆

C₂×D₆₇

yes

—

C₈²

C₂×D₆₇

yes

characteristic

C₂×Q₁₆

—

134

—

C₂×D₈

—

268

—

C₂×D₈

—

134

—

C₂×D₈

—

134

—

C₂×D₈

—

134

—

C₄⋊Q₈

—

134

—

#127

—

134

—

#127

—

134

—

SD₃₂

—

268

—

D₁₆

—

268

—

C₂×C₁₆

D₂₆₈

yes

—

#102

—

268

—

C₄×C₈

D₂₆₈

yes

—

C₄×C₈

C₂²×D₆₇

yes

characteristic

C₂×Q₈

—

268

—

100

C₂×D₄

—

268

—

101

C₂×D₄

—

268

—

102

C₂×D₄

—

268

—

103

C₂×D₄

—

268

—

104

Q₁₆

—

268

—

105

D₈

—

268

—

106

D₈

—

268

—

107

D₈

—

268

—

108

D₈

—

536

—

109

C₂×C₈

D₅₃₆

yes

—

110

C₂×C₈

C₂×D₂₆₈

yes

—

111

C₄⋊C₄

—

268

—

112

C₄²

C₂×D₂₆₈

yes

characteristic

113

C₁₆

D₅₃₆

yes

complemented

114

C₂³

—

536

—

115

C₂³

—

536

—

116

Q₈

—

536

—

117

D₄

—

536

—

118

D₄

—

536

—

119

D₄

—

536

—

120

D₄

—

536

—

121

D₄

—

536

—

122

C₂×C₄

—

536

—

123

C₂×C₄

#14524040

yes

characteristic

124

C₂×C₄

C₂×D₅₃₆

yes

—

125

C₈

—

126

C₈

C₂×D₅₃₆

yes

—

127

C₂²

—

1072

—

128

C₂²

—

1072

—

129

C₂²

—

1072

—

130

C₂²

—

1072

—

131

C₂²

#16863773

yes

characteristic

132

C₄

—

1072

—

133

C₄

#16863773

yes

—

134

C₄

#16863834

yes

—

135

C₂

—

2144

—

136

C₂

—

2144

—

137

C₂

#22559824

yes

characteristic

138

C₂

#22562475

yes

—

	subgroup	G/H
center	C₂²	#16863773
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1311	C₂
socle	C₂²×C₆₇	#493
hypercenter	#1311	D₆₇
hypocenter	C₆₇	#12519
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#16863773

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#1311

C₂

C₂²×C₆₇

D₆₇

C₆₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #22562465	C₂	semidirect
2. #22562459	C₂	semidirect
3. #22559824	C₂	semidirect
4. #22559815	C₂	semidirect
5. C₆₇×#1311	C₂	semidirect
43. C₁₆×C₆₇	D₈	semidirect
74. #1311	D₆₇	semidirect
75. C₆₇	#12519	semidirect
113. C₁₆	D₅₃₆	semidirect

H ≅ G/N

type

1. #22562465

C₂

semidirect

2. #22562459

C₂

semidirect

3. #22559824

C₂

semidirect

4. #22559815

C₂

semidirect

5. C₆₇×#1311

C₂

semidirect

43. C₁₆×C₆₇

D₈

semidirect

74. #1311

D₆₇

semidirect

75. C₆₇

#12519

semidirect

113. C₁₆

D₅₃₆

semidirect

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

subgroup	index
#22562465	2
#22562459	2
#22559824	2
#22559815	2
C₆₇×#1311	2
#12519	67

subgroup

index

C₆₇×#1311

Subgroups — #30770559

Conjugacy classes of subgroups

Complements

Maximal subgroups