Group #14539306 — subgroups

Subgroups — #14539306

Order 2176.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#11763366	C₂	1	1	yes	maximal, characteristic
2	#11763219	C₂	1	4	yes	maximal, complemented
3	#11763215	C₂	1	4	yes	maximal, complemented
4	#11762845	C₂	1	4	yes	maximal
5	C₂×#10587474	C₂	1	1	yes	maximal, characteristic
6	C₁₇×#533	C₂	1	1	yes	maximal, characteristic
7	C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₁₇×#125	C₂²	1	2	yes	—
9	C₁₇×C₂²⋊Q₈	—	2	2	no	—
10	C₄×Q₈×C₁₇	—	2	2	no	—
11	C₁₇×#117	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
13	#10587612	C₂²	1	2	yes	—
14	#10587610	—	2	2	no	—
15	C₂²×#3622	C₂²	1	2	yes	—
16	#10587556	C₂²	1	2	yes	—
17	#10587555	C₂²	1	2	yes	—
18	C₄×Dic₃₄	—	2	2	no	—
19	#10587479	C₂²	1	8	yes	—
20	#10587478	C₂²	1	8	yes	—
21	#10587474	C₂²	1	2	yes	—
22	#10587474	C₂²	1	2	yes	—
23	C₂×Q₈×C₁₇	—	2	2	no	—
24	C₂²×C₄×C₁₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
26	C₂×Dic₃₄	—	2	2	no	—
27	C₄⋊C₄×C₁₇	—	2	4	no	—
28	C₄⋊C₄×C₁₇	—	2	2	no	—
29	C₄⋊C₄×C₁₇	C₂³	1	4	yes	—
30	C₂²⋊C₄×C₁₇	—	2	2	no	—
31	C₄²×C₁₇	—	2	2	no	—
32	C₄²×C₁₇	D₄	1	2	yes	—
33	C₄²×C₁₇	C₂³	1	1	yes	characteristic
34	C₄²×C₁₇	C₂³	1	1	yes	characteristic
35	C₂²⋊Dic₁₇	—	2	2	no	—
36	C₄⋊Dic₁₇	—	2	2	no	—
37	C₄⋊Dic₁₇	C₂³	1	4	yes	—
38	#63225	—	2	4	no	—
39	C₄×Dic₁₇	—	2	2	no	—
40	C₂×#3622	Q₈	1	8	yes	complemented
41	C₂×#3622	C₂³	1	4	yes	—
42	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₁₇	—	4	4	no	—
44	C₂×C₄×C₁₇	—	2	4	no	—
45	C₂×C₄×C₁₇	—	4	2	no	—
46	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₂×Q₈	1	4	yes	—
50	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₁₇	—	2	4	no	—
52	Dic₃₄	—	4	4	no	—
53	#3622	C₂×Q₈	1	8	yes	—
54	#3162	—	17	1	no	maximal, Sylow
55	C₂²×C₁₇	—	2	2	no	—
56	C₂²×C₁₇	C₄⋊Q₈	1	2	yes	—
57	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₇	—	2	2	no	—
59	C₄×C₁₇	—	2	2	no	—
60	C₄×C₁₇	—	4	4	no	—
61	C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
62	C₄×C₁₇	C₂²×Q₈	1	2	yes	—
63	Dic₁₇	—	4	4	no	—
64	#533	—	17	1	no	—
65	#533	D₁₇	1	1	yes	characteristic
66	#501	—	17	4	no	—
67	#501	—	17	4	no	—
68	#427	—	17	4	no	—
69	C₂×#97	—	17	1	no	—
70	C₂×C₁₇	—	2	2	no	—
71	C₂×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₁₇	#575	1	2	yes	—
73	C₂²×C₈	—	17	2	no	—
74	C₄⋊Q₈	—	17	2	no	—
75	C₄⋊Q₈	C₂×D₁₇	1	2	yes	—
76	#125	—	17	2	no	—
77	#125	C₂×D₁₇	1	2	yes	—
78	C₂²⋊Q₈	—	2	2	no	—
79	C₂²⋊Q₈	—	34	2	no	—
80	C₄×Q₈	—	2	2	no	—
81	C₄×Q₈	—	34	2	no	—
82	#117	—	17	2	no	—
83	#117	C₂×D₁₇	1	2	yes	—
84	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
85	#107	—	17	8	no	—
86	#106	—	17	8	no	—
87	#97	—	17	2	no	—
88	#97	—	17	2	no	—
89	C₁₇	#3162	1	1	yes	characteristic, complemented, Sylow
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	34	2	no	—
92	C₂²×C₄	C₂²⋊D₁₇	1	2	yes	—
93	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
94	C₂×C₈	—	17	8	no	—
95	C₂×C₈	—	17	4	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	17	4	no	—
99	C₄⋊C₄	—	34	4	no	—
100	C₄⋊C₄	—	34	2	no	—
101	C₄⋊C₄	C₂²×D₁₇	1	4	yes	—
102	C₂²⋊C₄	—	2	2	no	—
103	C₂²⋊C₄	—	34	2	no	—
104	C₄²	—	2	2	no	—
105	C₄²	—	34	2	no	—
106	C₄²	C₂²⋊D₁₇	1	2	yes	—
107	C₄²	C₂²×D₁₇	1	1	yes	characteristic
108	C₄²	C₂²×D₁₇	1	1	yes	characteristic
109	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
110	Q₈	—	4	4	no	—
111	Q₈	—	68	4	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	34	4	no	—
115	C₂×C₄	Q₈×D₁₇	1	4	yes	—
116	C₂×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
117	C₂×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
119	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
120	C₈	—	17	8	no	—
121	C₂²	—	2	2	no	—
122	C₂²	#10587630	1	2	yes	—
123	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	—	4	4	no	—
127	C₄	—	68	4	no	—
128	C₄	#10587630	1	2	yes	—
129	C₄	C₂×Q₈×D₁₇	1	2	yes	—
130	C₂	—	2	2	no	—
131	C₂	C₂×#10587630	1	1	yes	characteristic
132	C₂	#11763607	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11763366

C₂

yes

maximal, characteristic

#11763219

C₂

yes

maximal, complemented

#11763215

C₂

yes

maximal, complemented

#11762845

C₂

yes

maximal

C₂×#10587474

C₂

yes

maximal, characteristic

C₁₇×#533

C₂

yes

maximal, characteristic

C₁₇×C₄⋊Q₈

C₂²

yes

—

C₁₇×#125

C₂²

yes

—

C₁₇×C₂²⋊Q₈

—

C₄×Q₈×C₁₇

—

C₁₇×#117

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

#10587612

C₂²

yes

—

#10587610

—

C₂²×#3622

C₂²

yes

—

#10587556

C₂²

yes

—

#10587555

C₂²

yes

—

C₄×Dic₃₄

—

#10587479

C₂²

yes

—

#10587478

C₂²

yes

—

#10587474

C₂²

yes

—

#10587474

C₂²

yes

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂×Dic₃₄

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

C₂³

yes

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

D₄

yes

—

C₄²×C₁₇

C₂³

yes

characteristic

C₄²×C₁₇

C₂³

yes

characteristic

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂³

yes

—

#63225

—

C₄×Dic₁₇

—

C₂×#3622

Q₈

yes

complemented

C₂×#3622

C₂³

yes

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

—

C₂×C₄×C₁₇

C₂⁴

yes

characteristic

C₂×Dic₁₇

—

Dic₃₄

—

#3622

C₂×Q₈

yes

—

#3162

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

C₄⋊Q₈

yes

—

C₂²×C₁₇

C₂²×D₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₄⋊Q₈

yes

—

C₄×C₁₇

C₂²×Q₈

yes

—

Dic₁₇

—

#533

—

#533

D₁₇

yes

—

—

—

C₂×#97

—

C₂×C₁₇

—

C₂×C₁₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₇

#575

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₇

yes

—

#125

—

#125

C₂×D₁₇

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

C₂×D₁₇

yes

—

C₂×C₄²

C₂×D₁₇

yes

—

—

—

—

C₁₇

#3162

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂²⋊D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

C₂²×D₁₇

yes

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₄²

—

106

C₄²

C₂²⋊D₁₇

yes

—

107

C₄²

C₂²×D₁₇

yes

characteristic

108

C₄²

C₂²×D₁₇

yes

characteristic

109

C₂³

C₂×C₂²⋊D₁₇

yes

characteristic

110

Q₈

—

111

Q₈

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

Q₈×D₁₇

yes

—

116

C₂×C₄

C₂×C₂²⋊D₁₇

yes

—

117

C₂×C₄

C₂×C₂²⋊D₁₇

yes

—

118

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

119

C₂×C₄

C₂³×D₁₇

yes

characteristic

120

C₈

—

121

C₂²

—

122

C₂²

#10587630

yes

—

123

C₂²

C₂²×C₂²⋊D₁₇

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#10587630

yes

—

129

C₄

C₂×Q₈×D₁₇

yes

—

130

C₂

—

131

C₂

C₂×#10587630

yes

characteristic

132

C₂

#11763607

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#533	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂³×C₁₇	C₂×D₄
hypercenter	#533	D₁₇
hypocenter	C₁₇	#3162
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₇

commutator, cosocle

C₂×C₄×C₁₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₇

Fitting

C₁₇×#533

C₂

socle

C₂²×C₁₇

C₂²×D₄

prime radical

C₂³×C₁₇

C₂×D₄

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #11763219	C₂	semidirect
3. #11763215	C₂	semidirect
40. C₂×#3622	Q₈	semidirect
89. C₁₇	#3162	semidirect

H ≅ G/N

type

2. #11763219

C₂

semidirect

3. #11763215

C₂

semidirect

40. C₂×#3622

Q₈

semidirect

89. C₁₇

#3162

semidirect

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

subgroup	index
#11763366	2
#11763219	2
#11763215	2
#11762845	2
C₂×#10587474	2
C₁₇×#533	2
#3162	17

subgroup

index

C₁₇×#533

Subgroups — #14539306

Conjugacy classes of subgroups

Complements

Maximal subgroups