Group #14530040 — subgroups

Subgroups — #14530040

Order 2176.

296 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂²×D₁₃₆	C₂	1	1	yes	maximal, characteristic
2	#11762813	C₂	1	2	yes	maximal, complemented
3	C₂×#10587494	C₂	1	2	yes	maximal, complemented
4	#11762574	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#424	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₆₈	C₂²	1	2	yes	—
7	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
9	C₁₇×#117	C₂²	1	1	yes	characteristic
10	C₂×#63223	C₂²	1	2	yes	—
11	C₂×D₁₃₆	—	2	4	no	—
12	C₂×D₁₃₆	—	2	2	no	—
13	C₁₇×C₄⋊C₈	—	2	2	no	—
14	#10587494	—	2	4	no	—
15	#10587480	—	2	4	no	—
16	C₂³×D₁₇	—	2	2	no	—
17	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
18	C₂×D₆₈	—	2	2	no	—
19	C₂×D₆₈	—	2	2	no	—
20	C₂×D₆₈	—	2	2	no	—
21	C₂×D₆₈	—	2	2	no	—
22	C₂×D₆₈	—	2	2	no	—
23	C₂×D₆₈	D₄	1	4	yes	—
24	M₁₆×C₁₇	—	2	2	no	—
25	C₂×C₈×C₁₇	—	2	2	no	—
26	C₂×C₈×C₁₇	—	2	1	no	—
27	C₂×C₈×C₁₇	D₄	1	2	yes	—
28	C₄⋊C₄×C₁₇	D₄	1	2	yes	—
29	C₂²⋊C₄×C₁₇	—	2	1	no	—
30	C₄²×C₁₇	—	2	1	no	—
31	#63223	—	2	2	no	—
32	#63223	—	2	2	no	—
33	C₂×#3622	—	2	2	no	—
34	D₁₃₆	—	2	8	no	—
35	C₂³×C₁₇	#41	1	1	yes	characteristic
36	C₂²×D₁₇	—	2	4	no	—
37	C₂²×D₁₇	—	4	2	no	—
38	C₂²×D₁₇	—	4	2	no	—
39	C₂²×D₁₇	—	8	4	no	—
40	C₂×C₄×C₁₇	—	2	2	no	—
41	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₇	#41	1	2	yes	—
45	D₆₈	—	2	4	no	—
46	D₆₈	—	2	4	no	—
47	D₆₈	—	4	2	no	—
48	D₆₈	—	4	2	no	—
49	C₈×C₁₇	—	2	2	no	—
50	C₈×C₁₇	—	4	1	no	—
51	#3622	—	4	2	no	—
52	#2045	—	17	1	no	maximal, Sylow
53	C₂²×C₁₇	—	2	2	no	—
54	C₂²×C₁₇	#121	1	1	yes	characteristic
55	C₂²×C₁₇	#123	1	2	yes	—
56	C₂×D₁₇	—	4	4	no	—
57	C₂×D₁₇	—	8	4	no	—
58	C₂×D₁₇	—	8	4	no	—
59	C₂×D₁₇	—	8	4	no	—
60	C₄×C₁₇	—	4	2	no	—
61	C₄×C₁₇	#120	1	1	yes	characteristic
62	C₄×C₁₇	#121	1	2	yes	—
63	C₄×C₁₇	#124	1	1	yes	characteristic
64	C₂²×D₈	—	17	1	no	—
65	#424	D₁₇	1	1	yes	characteristic, complemented
66	#418	—	17	2	no	—
67	C₂×#100	—	17	2	no	—
68	#341	—	17	1	no	—
69	C₂×C₁₇	—	2	2	no	—
70	C₂×C₁₇	#394	1	1	yes	characteristic
71	C₂×C₁₇	#472	1	2	yes	—
72	D₁₇	—	8	4	no	—
73	C₂²×D₄	—	17	2	no	—
74	C₂×D₈	—	34	4	no	—
75	C₂×D₈	—	34	2	no	—
76	C₂×M₁₆	—	17	2	no	—
77	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
78	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
79	#117	C₂×D₁₇	1	1	yes	characteristic
80	C₄⋊C₈	—	2	2	no	—
81	#102	—	34	4	no	—
82	#100	—	34	4	no	—
83	C₁₇	#2045	1	1	yes	characteristic, complemented, Sylow
84	C₂⁴	—	34	2	no	—
85	C₂×D₄	—	17	4	no	—
86	C₂×D₄	—	34	2	no	—
87	C₂×D₄	—	34	2	no	—
88	C₂×D₄	—	34	2	no	—
89	C₂×D₄	—	34	2	no	—
90	C₂×D₄	—	34	2	no	—
91	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
92	D₈	—	34	8	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	34	2	no	—
95	M₁₆	—	34	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	34	2	no	—
99	C₂×C₈	C₂²⋊D₁₇	1	2	yes	—
100	C₄⋊C₄	D₆₈	1	2	yes	—
101	C₂²⋊C₄	—	2	1	no	—
102	C₄²	—	2	1	no	—
103	C₂³	—	34	4	no	—
104	C₂³	—	68	2	no	—
105	C₂³	—	68	2	no	—
106	C₂³	—	136	4	no	—
107	C₂³	#63252	1	1	yes	characteristic
108	D₄	—	34	4	no	—
109	D₄	—	34	4	no	—
110	D₄	—	68	2	no	—
111	D₄	—	68	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
114	C₂×C₄	D₄×D₁₇	1	2	yes	—
115	C₂×C₄	#63256	1	2	yes	—
116	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	4	1	no	—
119	C₈	—	68	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	68	4	no	—
122	C₂²	—	136	4	no	—
123	C₂²	—	136	4	no	—
124	C₂²	—	136	4	no	—
125	C₂²	#10587580	1	2	yes	—
126	C₂²	#10587616	1	1	yes	characteristic
127	C₄	—	4	2	no	—
128	C₄	#10587581	1	2	yes	—
129	C₄	#10587623	1	1	yes	characteristic
130	C₄	#10587633	1	1	yes	characteristic
131	C₂	—	2	2	no	—
132	C₂	—	136	4	no	—
133	C₂	#11762726	1	1	yes	characteristic
134	C₂	#11763079	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₁₃₆

C₂

yes

maximal, characteristic

#11762813

C₂

yes

maximal, complemented

C₂×#10587494

C₂

yes

maximal, complemented

#11762574

C₂

yes

maximal, characteristic, complemented

C₁₇×#424

C₂

yes

maximal, characteristic, complemented

C₂²×D₆₈

C₂²

yes

—

C₂×M₁₆×C₁₇

C₂²

yes

characteristic

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#117

C₂²

yes

characteristic

C₂×#63223

C₂²

yes

—

C₂×D₁₃₆

—

C₂×D₁₃₆

—

C₁₇×C₄⋊C₈

—

#10587494

—

#10587480

—

C₂³×D₁₇

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

D₄

yes

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

—

C₄⋊C₄×C₁₇

D₄

yes

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

#63223

—

#63223

—

C₂×#3622

—

D₁₃₆

—

C₂³×C₁₇

#41

yes

characteristic

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

D₆₈

—

D₆₈

—

D₆₈

—

D₆₈

—

C₈×C₁₇

—

C₈×C₁₇

—

#3622

—

#2045

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

#121

yes

characteristic

C₂²×C₁₇

#123

yes

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#120

yes

characteristic

C₄×C₁₇

#121

yes

—

C₄×C₁₇

#124

yes

characteristic

C₂²×D₈

—

#424

D₁₇

yes

characteristic, complemented

#418

—

C₂×#100

—

#341

—

C₂×C₁₇

—

C₂×C₁₇

#394

yes

characteristic

C₂×C₁₇

#472

yes

—

D₁₇

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₇

yes

characteristic

C₂²×C₈

C₂×D₁₇

yes

characteristic

#117

C₂×D₁₇

yes

characteristic

C₄⋊C₈

—

#102

—

#100

—

C₁₇

#2045

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

D₈

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₁₇

yes

—

100

C₄⋊C₄

D₆₈

yes

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

136

—

107

C₂³

#63252

yes

characteristic

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×D₆₈

yes

characteristic

114

C₂×C₄

D₄×D₁₇

yes

—

115

C₂×C₄

#63256

yes

—

116

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

136

—

123

C₂²

—

136

—

124

C₂²

—

136

—

125

C₂²

#10587580

yes

—

126

C₂²

#10587616

yes

characteristic

127

C₄

—

128

C₄

#10587581

yes

—

129

C₄

#10587623

yes

characteristic

130

C₄

#10587633

yes

characteristic

131

C₂

—

132

C₂

—

136

—

133

C₂

#11762726

yes

characteristic

134

C₂

#11763079

yes

—

	subgroup	G/H
center	C₂²	#10587616
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#424	C₂
socle	C₂²×C₁₇	#121
prime radical	C₂²×D₁₃₆	C₂
hypercenter	#424	D₁₇
hypocenter	C₁₇	#2045
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#10587616

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#424

C₂

socle

C₂²×C₁₇

#121

prime radical

C₂²×D₁₃₆

C₂

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
2. #11762813	C₂	semidirect
3. C₂×#10587494	C₂	semidirect
4. #11762574	C₂	semidirect
5. C₁₇×#424	C₂	semidirect
65. #424	D₁₇	semidirect
83. C₁₇	#2045	semidirect

H ≅ G/N

type

2. #11762813

C₂

semidirect

3. C₂×#10587494

C₂

semidirect

4. #11762574

C₂

semidirect

5. C₁₇×#424

C₂

semidirect

65. #424

D₁₇

semidirect

83. C₁₇

#2045

semidirect

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

subgroup	index
C₂²×D₁₃₆	2
#11762813	2
C₂×#10587494	2
#11762574	2
C₁₇×#424	2
#2045	17

subgroup

index

C₂²×D₁₃₆

C₁₇×#424

Subgroups — #14530040

Conjugacy classes of subgroups

Complements

Maximal subgroups