Group #30843504 — subgroups

Subgroups — C₂×#22559819 #30843504

Order 17152.

750 conjugacy classes of subgroups; only the 267 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 267 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×#16863801	C₂	1	2	yes	maximal
2	C₂×#16863747	C₂	1	4	yes	maximal
3	#22559819	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₈²×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524043	C₂²	1	4	yes	—
6	#16863801	—	2	4	no	—
7	#16863801	C₂²	1	8	yes	—
8	#16863747	—	2	8	no	—
9	#16863747	C₂²	1	16	yes	—
10	C₂×#14523977	—	2	4	no	—
11	C₂×#14523968	—	2	8	no	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₈²×C₆₇	C₂²	1	4	yes	—
15	C₂²×Dic₁₃₄	—	2	4	no	—
16	#14524043	—	2	4	no	—
17	#14524043	—	2	4	no	—
18	#14524043	—	2	4	no	—
19	#14524043	C₂³	1	8	yes	—
20	C₂×C₄⋊Dic₆₇	—	2	4	no	—
21	C₂×C₄⋊Dic₆₇	—	2	4	no	—
22	#14523977	—	2	16	no	—
23	#14523968	—	2	32	no	—
24	C₂²×C₈×C₆₇	D₄	1	4	yes	—
25	C₂²×C₈×C₆₇	D₄	1	2	yes	—
26	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
27	C₄×C₈×C₆₇	D₄	1	4	yes	—
28	C₄×C₈×C₆₇	D₄	1	2	yes	—
29	C₄×C₈×C₆₇	C₂³	1	4	yes	—
30	C₄×C₈×C₆₇	C₂³	1	2	yes	—
31	C₂×Dic₁₃₄	—	2	8	no	—
32	C₂×Dic₁₃₄	—	4	16	no	—
33	C₂×Dic₁₃₄	—	4	4	no	—
34	C₂²×Dic₆₇	—	4	4	no	—
35	C₄⋊Dic₆₇	—	2	8	no	—
36	C₄⋊Dic₆₇	—	2	8	no	—
37	C₄⋊Dic₆₇	—	4	4	no	—
38	C₄⋊Dic₆₇	—	4	4	no	—
39	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
41	C₂×C₈×C₆₇	—	2	8	no	—
42	C₂×C₈×C₆₇	Q₁₆	1	8	yes	—
43	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
44	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
45	C₄²×C₆₇	C₂×D₄	1	2	yes	—
46	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
48	C₂³×C₆₇	#127	1	1	yes	characteristic
49	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
50	C₂×C₄×C₆₇	#127	1	2	yes	—
51	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
52	C₂×C₄×C₆₇	#135	1	8	yes	—
53	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
55	C₂×Dic₆₇	—	4	8	no	—
56	C₂×Dic₆₇	—	8	8	no	—
57	Dic₁₃₄	—	4	16	no	—
58	Dic₁₃₄	—	8	8	no	—
59	C₈×C₆₇	—	2	8	no	—
60	C₈×C₆₇	C₂×Q₁₆	1	8	yes	—
61	C₂²×C₆₇	#494	1	2	yes	—
62	C₂²×C₆₇	#495	1	4	yes	—
63	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
64	C₄×C₆₇	#494	1	2	yes	—
65	C₄×C₆₇	#495	1	4	yes	—
66	C₄×C₆₇	C₂²×Q₁₆	1	2	yes	—
67	C₄×C₆₇	C₂×#135	1	4	yes	—
68	Dic₆₇	—	8	8	no	—
69	C₂×#1710	—	67	1	no	maximal, Sylow
70	C₂×C₆₇	#1710	1	4	yes	complemented
71	C₂×C₆₇	C₂×#494	1	1	yes	characteristic
72	C₂×C₆₇	C₂×#495	1	2	yes	—
73	C₂×#500	—	67	2	no	—
74	C₂×#487	—	67	4	no	—
75	#1710	—	67	8	no	—
76	C₂×C₈²	D₆₇	1	1	yes	characteristic
77	C₆₇	C₂×#1710	1	1	yes	characteristic, complemented, Sylow
78	C₂×C₄⋊Q₈	—	67	4	no	—
79	#500	—	67	8	no	—
80	#500	—	134	4	no	—
81	#487	—	67	16	no	—
82	#487	—	134	8	no	—
83	C₂×#107	—	134	4	no	—
84	C₂×#103	—	134	8	no	—
85	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
86	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
87	C₈²	C₂×D₆₇	1	4	yes	—
88	C₂²×Q₈	—	134	4	no	—
89	C₂²×C₈	D₂₆₈	1	4	yes	—
90	C₂²×C₈	D₂₆₈	1	2	yes	—
91	C₄⋊Q₈	—	67	8	no	—
92	C₄⋊Q₈	—	134	4	no	—
93	C₄⋊Q₈	—	134	4	no	—
94	C₄⋊Q₈	—	134	4	no	—
95	C₂×C₄⋊C₄	—	134	4	no	—
96	C₂×C₄⋊C₄	—	134	4	no	—
97	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
98	#107	—	134	16	no	—
99	#103	—	134	32	no	—
100	C₄×C₈	D₂₆₈	1	4	yes	—
101	C₄×C₈	D₂₆₈	1	2	yes	—
102	C₄×C₈	C₂²×D₆₇	1	4	yes	—
103	C₄×C₈	C₂²×D₆₇	1	2	yes	—
104	C₂×Q₈	—	268	4	no	—
105	C₂×Q₈	—	268	16	no	—
106	C₂×Q₈	—	134	8	no	—
107	C₂²×C₄	—	268	4	no	—
108	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
109	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
110	C₂×C₈	—	2	8	no	—
111	C₂×C₈	Dic₂₆₈	1	8	yes	—
112	C₂×C₈	C₂×D₂₆₈	1	8	yes	—
113	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
114	C₄⋊C₄	—	268	4	no	—
115	C₄⋊C₄	—	268	4	no	—
116	C₄⋊C₄	—	134	8	no	—
117	C₄⋊C₄	—	134	8	no	—
118	C₄²	C₂×D₂₆₈	1	2	yes	—
119	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
120	C₄²	C₂³×D₆₇	1	1	yes	characteristic
121	C₂³	#14524040	1	1	yes	characteristic
122	Q₈	—	268	16	no	—
123	Q₈	—	536	8	no	—
124	C₂×C₄	—	268	8	no	—
125	C₂×C₄	—	536	8	no	—
126	C₂×C₄	#14524040	1	1	yes	characteristic
127	C₂×C₄	#14524040	1	2	yes	—
128	C₂×C₄	C₂×Dic₂₆₈	1	4	yes	—
129	C₂×C₄	#14524065	1	8	yes	—
130	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
131	C₂×C₄	C₂²×D₂₆₈	1	2	yes	—
132	C₈	—	2	8	no	—
133	C₈	C₂×Dic₂₆₈	1	8	yes	—
134	C₂²	#16863776	1	2	yes	—
135	C₂²	#16863780	1	4	yes	—
136	C₂²	C₂×#14524040	1	1	yes	characteristic
137	C₄	—	536	8	no	—
138	C₄	#16863776	1	2	yes	—
139	C₄	#16863780	1	4	yes	—
140	C₄	C₂²×Dic₂₆₈	1	2	yes	—
141	C₄	C₂×#14524065	1	4	yes	—
142	C₂	#22559819	1	4	yes	direct product factor
143	C₂	C₂×#16863776	1	1	yes	characteristic
144	C₂	C₂×#16863780	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863801

C₂

yes

maximal

C₂×#16863747

C₂

yes

maximal

#22559819

C₂

yes

maximal, direct product factor

C₂×C₈²×C₆₇

C₂

yes

maximal, characteristic

C₂×#14524043

C₂²

yes

—

#16863801

—

#16863801

C₂²

yes

—

#16863747

—

#16863747

C₂²

yes

—

C₂×#14523977

—

C₂×#14523968

—

C₂×C₄×C₈×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₈²×C₆₇

C₂²

yes

—

C₂²×Dic₁₃₄

—

—

—

—

C₂³

yes

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

#14523977

—

#14523968

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

C₂³

yes

—

C₄×C₈×C₆₇

C₂³

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

Q₁₆

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂⁴

yes

C₂³×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×C₄×C₆₇

C₂×Q₁₆

yes

—

C₂×C₄×C₆₇

#135

yes

—

C₂×C₄×C₆₇

C₂²×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×D₄

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

C₂×Q₁₆

yes

—

C₂²×C₆₇

#494

yes

—

C₂²×C₆₇

#495

yes

—

C₂²×C₆₇

C₂×#127

yes

characteristic

C₄×C₆₇

#494

yes

—

C₄×C₆₇

#495

yes

—

C₄×C₆₇

C₂²×Q₁₆

yes

—

C₄×C₆₇

C₂×#135

yes

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×#500

—

C₂×#487

—

#1710

—

C₂×C₈²

D₆₇

yes

characteristic

C₆₇

C₂×#1710

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#500

—

#500

—

134

—

#487

—

#487

—

134

—

C₂×#107

—

134

—

C₂×#103

—

134

—

C₂×C₄×C₈

C₂×D₆₇

yes

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

C₈²

C₂×D₆₇

yes

—

C₂²×Q₈

—

134

—

C₂²×C₈

D₂₆₈

yes

—

C₂²×C₈

D₂₆₈

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

C₄⋊Q₈

—

134

—

C₄⋊Q₈

—

134

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

#107

—

134

—

#103

—

134

—

100

C₄×C₈

D₂₆₈

yes

—

101

C₄×C₈

D₂₆₈

yes

—

102

C₄×C₈

C₂²×D₆₇

yes

—

103

C₄×C₈

C₂²×D₆₇

yes

—

104

C₂×Q₈

—

268

—

105

C₂×Q₈

—

268

—

106

C₂×Q₈

—

134

—

107

C₂²×C₄

—

268

—

108

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

109

C₂²×C₄

C₂×D₂₆₈

yes

—

110

C₂×C₈

—

111

C₂×C₈

Dic₂₆₈

yes

—

112

C₂×C₈

C₂×D₂₆₈

yes

—

113

C₂×C₈

C₂×D₂₆₈

yes

—

114

C₄⋊C₄

—

268

—

115

C₄⋊C₄

—

268

—

116

C₄⋊C₄

—

134

—

117

C₄⋊C₄

—

134

—

118

C₄²

C₂×D₂₆₈

yes

—

119

C₄²

C₂×D₂₆₈

yes

characteristic

120

C₄²

C₂³×D₆₇

yes

characteristic

121

C₂³

#14524040

yes

characteristic

122

Q₈

—

268

—

123

Q₈

—

536

—

124

C₂×C₄

—

268

—

125

C₂×C₄

—

536

—

126

C₂×C₄

#14524040

yes

characteristic

127

C₂×C₄

#14524040

yes

—

128

C₂×C₄

C₂×Dic₂₆₈

yes

—

129

C₂×C₄

#14524065

yes

—

130

C₂×C₄

C₂²×D₂₆₈

yes

characteristic

131

C₂×C₄

C₂²×D₂₆₈

yes

—

132

C₈

—

133

C₈

C₂×Dic₂₆₈

yes

—

134

C₂²

#16863776

yes

—

135

C₂²

#16863780

yes

—

136

C₂²

C₂×#14524040

yes

characteristic

137

C₄

—

536

—

138

C₄

#16863776

yes

—

139

C₄

#16863780

yes

—

140

C₄

C₂²×Dic₂₆₈

yes

—

141

C₄

C₂×#14524065

yes

—

142

C₂

#22559819

yes

direct product factor

143

C₂

C₂×#16863776

yes

characteristic

144

C₂

C₂×#16863780

yes

—

	subgroup	G/H
center	C₂³	#14524040
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×C₈²×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	#127
hypercenter	C₂×C₈²	D₆₇
hypocenter	C₆₇	C₂×#1710
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#14524040

commutator, cosocle

C₄²×C₆₇

C₂⁴

Frattini

C₄²

C₂³×D₆₇

Fitting

C₂×C₈²×C₆₇

C₂

socle, prime radical

C₂³×C₆₇

#127

hypercenter

C₂×C₈²

D₆₇

hypocenter

C₆₇

C₂×#1710

LCS

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
3. #22559819	C₂	direct (N × H)
70. C₂×C₆₇	#1710	semidirect
77. C₆₇	C₂×#1710	semidirect
142. C₂	#22559819	direct (N × H)

H ≅ G/N

type

3. #22559819

C₂

direct (N × H)

70. C₂×C₆₇

#1710

semidirect

77. C₆₇

C₂×#1710

semidirect

142. C₂

#22559819

direct (N × H)

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

subgroup	index
C₂×#16863801	2
C₂×#16863747	2
#22559819	2
C₂×C₈²×C₆₇	2
C₂×#1710	67

subgroup

index

C₂×C₈²×C₆₇

Subgroups — C₂×#22559819 #30843504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22559819 #30843504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22559819 #30843504