Group #14541787 — subgroups

Subgroups — #14541787

Order 2176.

348 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 139 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	#11763538	C₂	1	1	yes	maximal, characteristic
2	#11763188	C₂	1	3	yes	maximal
3	#11763019	C₂	1	3	yes	maximal
4	#11763016	C₂	1	3	yes	maximal
5	Q₈×#3622	C₂	1	1	yes	maximal, characteristic
6	C₄×#63231	C₂	1	3	yes	maximal
7	C₁₇×#557	C₂	1	1	yes	maximal, characteristic
8	C₁₇×C₄⋊Q₈	C₂²	1	3	yes	—
9	C₁₇×#125	—	2	3	no	—
10	C₄×Q₈×C₁₇	—	2	1	no	—
11	C₄×Q₈×C₁₇	C₂²	1	3	yes	—
12	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
13	Q₈×Dic₁₇	—	2	1	no	—
14	C₂×#63231	C₂²	1	3	yes	—
15	C₂×#63231	C₂²	1	6	yes	—
16	#10587576	—	2	3	no	—
17	#10587555	C₂²	1	3	yes	—
18	C₄×Dic₃₄	C₂²	1	3	yes	—
19	#10587506	C₂²	1	3	yes	—
20	#10587481	C₂²	1	3	yes	—
21	#10587478	C₂²	1	1	yes	characteristic
22	C₄⋊#3622	C₂²	1	3	yes	—
23	C₄×#3622	C₂²	1	3	yes	—
24	C₂×Q₈×C₁₇	D₄	1	1	yes	characteristic
25	C₂×Q₈×C₁₇	C₂³	1	3	yes	—
26	C₂×Dic₃₄	C₂³	1	3	yes	—
27	C₄⋊C₄×C₁₇	—	2	3	no	—
28	C₄⋊C₄×C₁₇	—	2	3	no	—
29	C₄⋊C₄×C₁₇	—	2	3	no	—
30	C₄⋊C₄×C₁₇	D₄	1	3	yes	—
31	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
32	C₄²×C₁₇	—	2	3	no	—
33	C₄²×C₁₇	C₂³	1	3	yes	—
34	#63231	—	2	6	no	—
35	#63231	—	4	6	no	—
36	C₄⋊Dic₁₇	—	2	3	no	—
37	C₄⋊Dic₁₇	—	2	3	no	—
38	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
39	#63225	—	2	3	no	—
40	C₄×Dic₁₇	—	2	3	no	—
41	C₂×#3622	C₂³	1	3	yes	—
42	C₂×#3622	C₂³	1	1	yes	characteristic
43	Q₈×C₁₇	—	2	3	no	—
44	Q₈×C₁₇	—	2	3	no	—
45	Q₈×C₁₇	Q₁₆	1	4	yes	complemented
46	C₂×C₄×C₁₇	—	2	1	no	—
47	C₂×C₄×C₁₇	—	2	3	no	—
48	C₂×C₄×C₁₇	C₂×D₄	1	3	yes	—
49	C₂×C₄×C₁₇	C₂×D₄	1	3	yes	—
50	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₁₇	—	2	3	no	—
52	C₂×Dic₁₇	—	2	1	no	—
53	Dic₃₄	—	2	3	no	—
54	Dic₃₄	—	2	3	no	—
55	#3622	—	2	3	no	—
56	#3622	#41	1	2	yes	—
57	#3395	—	17	1	no	maximal, Sylow
58	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₁₇	—	2	3	no	—
60	C₄×C₁₇	—	4	3	no	—
61	C₄×C₁₇	—	4	1	no	—
62	C₄×C₁₇	C₂×Q₁₆	1	6	yes	—
63	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
64	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
65	Dic₁₇	—	4	3	no	—
66	Dic₁₇	—	4	1	no	—
67	#557	—	17	1	no	—
68	#557	D₁₇	1	1	yes	characteristic
69	#494	—	17	3	no	—
70	#462	—	17	3	no	—
71	#462	—	17	3	no	—
72	C₈×Q₈	—	17	1	no	—
73	C₄×Q₁₆	—	17	3	no	—
74	C₂×C₁₇	#550	1	1	yes	characteristic
75	C₂×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
76	C₂×C₁₇	#578	1	1	yes	characteristic
77	C₂×Q₁₆	—	17	3	no	—
78	C₂×Q₁₆	—	17	6	no	—
79	C₄⋊Q₈	—	17	3	no	—
80	C₄⋊Q₈	C₂×D₁₇	1	3	yes	—
81	#125	—	2	3	no	—
82	#125	—	34	3	no	—
83	C₄×Q₈	—	2	1	no	—
84	C₄×Q₈	—	17	3	no	—
85	C₄×Q₈	—	34	1	no	—
86	C₄×Q₈	C₂×D₁₇	1	3	yes	—
87	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
88	#107	—	17	1	no	—
89	C₄⋊C₈	—	17	3	no	—
90	#103	—	17	3	no	—
91	#103	—	17	3	no	—
92	C₄×C₈	—	17	3	no	—
93	C₁₇	#3395	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₈	—	17	3	no	—
95	C₂×Q₈	C₂²⋊D₁₇	1	1	yes	characteristic
96	C₂×Q₈	C₂²×D₁₇	1	3	yes	—
97	Q₁₆	—	34	6	no	—
98	Q₁₆	—	68	6	no	—
99	C₂×C₈	—	17	1	no	—
100	C₂×C₈	—	17	3	no	—
101	C₄⋊C₄	—	2	3	no	—
102	C₄⋊C₄	—	2	3	no	—
103	C₄⋊C₄	—	2	3	no	—
104	C₄⋊C₄	—	17	1	no	—
105	C₄⋊C₄	—	34	3	no	—
106	C₄⋊C₄	—	34	3	no	—
107	C₄⋊C₄	—	34	3	no	—
108	C₄⋊C₄	C₂²⋊D₁₇	1	3	yes	—
109	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
110	C₄²	—	2	3	no	—
111	C₄²	—	34	3	no	—
112	C₄²	C₂²×D₁₇	1	3	yes	—
113	Q₈	—	2	3	no	—
114	Q₈	—	2	3	no	—
115	Q₈	—	34	3	no	—
116	Q₈	—	34	3	no	—
117	Q₈	#63231	1	4	yes	complemented
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	34	1	no	—
121	C₂×C₄	—	34	3	no	—
122	C₂×C₄	C₂×C₂²⋊D₁₇	1	3	yes	—
123	C₂×C₄	C₂×C₂²⋊D₁₇	1	3	yes	—
124	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
125	C₈	—	17	2	no	—
126	C₈	—	34	3	no	—
127	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
128	C₄	—	2	3	no	—
129	C₄	—	4	3	no	—
130	C₄	—	4	1	no	—
131	C₄	—	68	3	no	—
132	C₄	—	68	1	no	—
133	C₄	C₂×#63231	1	6	yes	—
134	C₄	#41×D₁₇	1	1	yes	characteristic
135	C₄	#10587696	1	1	yes	characteristic
136	C₂	#11763478	1	1	yes	characteristic
137	C₂	C₂²×#63231	1	1	yes	characteristic
138	C₂	#11763624	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763538

C₂

yes

maximal, characteristic

#11763188

C₂

yes

maximal

#11763019

C₂

yes

maximal

#11763016

C₂

yes

maximal

Q₈×#3622

C₂

yes

maximal, characteristic

C₄×#63231

C₂

yes

maximal

C₁₇×#557

C₂

yes

maximal, characteristic

C₁₇×C₄⋊Q₈

C₂²

yes

—

C₁₇×#125

—

C₄×Q₈×C₁₇

—

C₄×Q₈×C₁₇

C₂²

yes

—

C₄×Q₈×C₁₇

C₂²

yes

characteristic

Q₈×Dic₁₇

—

C₂×#63231

C₂²

yes

—

C₂×#63231

C₂²

yes

—

#10587576

—

#10587555

C₂²

yes

—

C₄×Dic₃₄

C₂²

yes

—

#10587506

C₂²

yes

—

#10587481

C₂²

yes

—

#10587478

C₂²

yes

characteristic

C₄⋊#3622

C₂²

yes

—

C₄×#3622

C₂²

yes

—

C₂×Q₈×C₁₇

D₄

yes

characteristic

C₂×Q₈×C₁₇

C₂³

yes

—

C₂×Dic₃₄

C₂³

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

D₄

yes

—

C₄⋊C₄×C₁₇

C₂³

yes

characteristic

C₄²×C₁₇

—

C₄²×C₁₇

C₂³

yes

—

#63231

—

#63231

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂³

yes

characteristic

#63225

—

C₄×Dic₁₇

—

C₂×#3622

C₂³

yes

—

C₂×#3622

C₂³

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

—

Q₈×C₁₇

Q₁₆

yes

complemented

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂⁴

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

#3622

—

#3622

#41

yes

—

#3395

—

maximal, Sylow

C₂²×C₁₇

C₂²×D₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×Q₁₆

yes

—

C₄×C₁₇

C₂×#41

yes

characteristic

C₄×C₁₇

H(5,𝔽₂)

yes

characteristic

Dic₁₇

—

Dic₁₇

—

#557

—

#557

D₁₇

yes

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₂×C₁₇

#550

yes

characteristic

C₂×C₁₇

C₂²×Q₁₆

yes

characteristic

C₂×C₁₇

#578

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₇

yes

—

#125

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₁₇

yes

—

C₄×Q₈

C₂×D₁₇

yes

characteristic

#107

—

C₄⋊C₈

—

#103

—

#103

—

C₄×C₈

—

C₁₇

#3395

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₁₇

yes

characteristic

C₂×Q₈

C₂²×D₁₇

yes

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

C₂²⋊D₁₇

yes

—

109

C₄⋊C₄

C₂²×D₁₇

yes

characteristic

110

C₄²

—

111

C₄²

—

112

C₄²

C₂²×D₁₇

yes

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

Q₈

#63231

yes

complemented

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂×C₂²⋊D₁₇

yes

—

123

C₂×C₄

C₂×C₂²⋊D₁₇

yes

—

124

C₂×C₄

C₂³×D₁₇

yes

characteristic

125

C₈

—

126

C₈

—

127

C₂²

C₂²×C₂²⋊D₁₇

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

C₂×#63231

yes

—

134

C₄

#41×D₁₇

yes

characteristic

135

C₄

#10587696

yes

characteristic

136

C₂

#11763478

yes

characteristic

137

C₂

C₂²×#63231

yes

characteristic

138

C₂

#11763624

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#557	C₂
socle, prime radical	C₂²×C₁₇	C₂²×D₄
hypercenter	#557	D₁₇
hypocenter	C₁₇	#3395
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₇

commutator, cosocle

C₂×C₄×C₁₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₇

Fitting

C₁₇×#557

C₂

socle, prime radical

C₂²×C₁₇

C₂²×D₄

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

N	H ≅ G/N	type
45. Q₈×C₁₇	Q₁₆	semidirect
93. C₁₇	#3395	semidirect
117. Q₈	#63231	semidirect

H ≅ G/N

type

45. Q₈×C₁₇

Q₁₆

semidirect

93. C₁₇

#3395

semidirect

117. Q₈

#63231

semidirect

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

subgroup	index
#11763538	2
#11763188	2
#11763019	2
#11763016	2
Q₈×#3622	2
C₄×#63231	2
C₁₇×#557	2
#3395	17

subgroup

index

C₁₇×#557

Subgroups — #14541787

Conjugacy classes of subgroups

Complements

Maximal subgroups