Group #14528821 — subgroups

Subgroups — #14528821

Order 2176.

292 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 89 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×C₈×D₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587493	C₂	1	2	yes	maximal, complemented
3	#11762772	C₂	1	2	yes	maximal
4	#11762574	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#341	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₁₇	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₁₇	C₂²	1	2	yes	—
8	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×#63223	C₂²	1	2	yes	—
10	C₂²×#3622	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₁₇	—	2	4	no	—
12	C₂×C₈×D₁₇	C₄	1	4	yes	—
13	#10587493	—	2	4	no	—
14	#10587491	—	2	4	no	—
15	C₂³×D₁₇	Q₈	1	1	yes	characteristic
16	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
17	C₂²×Dic₁₇	Q₈	1	1	yes	characteristic
18	C₂×C₄×D₁₇	—	2	2	no	—
19	C₂×C₄×D₁₇	—	2	2	no	—
20	C₂×C₄×D₁₇	C₂×C₄	1	2	yes	—
21	C₂×C₄×D₁₇	D₄	1	2	yes	—
22	M₁₆×C₁₇	—	2	2	no	—
23	M₁₆×C₁₇	—	2	2	no	—
24	C₂×C₈×C₁₇	—	2	2	no	—
25	C₂×C₈×C₁₇	—	2	2	no	—
26	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
27	#63223	—	2	2	no	—
28	#63223	—	2	2	no	—
29	C₂×#3622	—	2	2	no	—
30	C₂×#3622	C₂×C₄	1	2	yes	—
31	C₂×#3622	D₄	1	4	yes	—
32	C₈×D₁₇	—	2	8	no	—
33	C₂³×C₁₇	C₂×Q₈	1	1	yes	characteristic
34	C₂²×D₁₇	—	2	2	no	—
35	C₂²×D₁₇	—	2	4	no	—
36	C₂²×D₁₇	C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₇	#41	1	2	yes	—
41	C₂×Dic₁₇	—	2	2	no	—
42	C₂×Dic₁₇	C₄⋊C₄	1	2	yes	—
43	C₄×D₁₇	—	2	2	no	—
44	C₄×D₁₇	—	2	2	no	—
45	C₄×D₁₇	—	2	4	no	—
46	C₈×C₁₇	—	2	2	no	—
47	C₈×C₁₇	—	4	2	no	—
48	#3622	—	4	2	no	—
49	#3622	C₂²⋊C₄	1	4	yes	—
50	#1810	—	17	1	no	maximal, Sylow
51	C₂²×C₁₇	—	2	2	no	—
52	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₁₇	C₂²⋊Q₈	1	2	yes	—
54	C₂×D₁₇	—	2	8	no	—
55	C₂×D₁₇	—	2	2	no	—
56	C₂×D₁₇	—	2	2	no	—
57	C₂×D₁₇	#108	1	4	yes	complemented
58	C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₄×C₁₇	#117	1	1	yes	characteristic
60	C₄×C₁₇	#121	1	2	yes	—
61	Dic₁₇	—	2	2	no	—
62	C₂³×C₈	—	17	1	no	—
63	C₂×#108	—	17	2	no	—
64	#407	—	17	2	no	—
65	#341	—	17	1	no	—
66	#341	D₁₇	1	1	yes	characteristic, complemented
67	C₂×C₁₇	—	2	2	no	—
68	C₂×C₁₇	#380	1	1	yes	characteristic
69	C₂×C₁₇	C₂×#108	1	2	yes	—
70	D₁₇	—	2	4	no	—
71	C₂³×C₄	—	17	1	no	—
72	C₂×M₁₆	—	17	2	no	—
73	C₂×M₁₆	C₂×D₁₇	1	2	yes	—
74	C₂²×C₈	—	17	1	no	—
75	C₂²×C₈	—	17	4	no	—
76	C₂²×C₈	—	34	4	no	—
77	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
78	#108	—	34	4	no	—
79	C₂²⋊C₈	—	34	4	no	—
80	C₁₇	#1810	1	1	yes	characteristic, complemented, Sylow
81	C₂⁴	—	17	1	no	—
82	C₂²×C₄	—	17	1	no	—
83	C₂²×C₄	—	17	2	no	—
84	C₂²×C₄	—	17	2	no	—
85	C₂²×C₄	—	34	2	no	—
86	C₂²×C₄	—	34	2	no	—
87	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	34	2	no	—
91	M₁₆	—	34	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	17	2	no	—
95	C₂×C₈	—	17	4	no	—
96	C₂×C₈	—	34	8	no	—
97	C₂×C₈	—	34	2	no	—
98	C₂×C₈	C₄×D₁₇	1	2	yes	—
99	C₂³	—	17	2	no	—
100	C₂³	—	34	4	no	—
101	C₂³	—	34	2	no	—
102	C₂³	Q₈×D₁₇	1	1	yes	characteristic
103	C₂×C₄	—	17	2	no	—
104	C₂×C₄	—	34	2	no	—
105	C₂×C₄	—	34	2	no	—
106	C₂×C₄	—	34	2	no	—
107	C₂×C₄	—	34	4	no	—
108	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
109	C₂×C₄	#63252	1	2	yes	—
110	C₂×C₄	D₄×D₁₇	1	2	yes	—
111	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	4	2	no	—
114	C₈	—	17	4	no	—
115	C₈	—	68	2	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	17	4	no	—
118	C₂²	—	34	8	no	—
119	C₂²	—	34	2	no	—
120	C₂²	—	34	2	no	—
121	C₂²	C₄⋊C₄×D₁₇	1	1	yes	characteristic
122	C₂²	#10587582	1	2	yes	—
123	C₄	—	34	2	no	—
124	C₄	#10587558	1	1	yes	characteristic
125	C₄	C₂²⋊C₄×D₁₇	1	1	yes	characteristic
126	C₄	#10587570	1	2	yes	—
127	C₂	—	2	2	no	—
128	C₂	—	34	4	no	—
129	C₂	#11762668	1	1	yes	characteristic
130	C₂	#108×D₁₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₁₇

C₂

yes

maximal, characteristic

C₂×#10587493

C₂

yes

maximal, complemented

#11762772

C₂

yes

maximal

#11762574

C₂

yes

maximal, characteristic, complemented

C₁₇×#341

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₇

C₂²

yes

—

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₂×#63223

C₂²

yes

—

C₂²×#3622

C₂²

yes

characteristic

C₂×C₈×D₁₇

—

C₂×C₈×D₁₇

C₄

yes

—

#10587493

—

#10587491

—

C₂³×D₁₇

Q₈

yes

characteristic

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

Q₈

yes

characteristic

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

C₂×C₄

yes

—

C₂×C₄×D₁₇

D₄

yes

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#3622

D₄

yes

—

C₈×D₁₇

—

C₂³×C₁₇

C₂×Q₈

yes

characteristic

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂²×D₁₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

C₄⋊C₄

yes

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

#3622

—

#3622

C₂²⋊C₄

yes

—

#1810

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₇

C₂²⋊Q₈

yes

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

#108

yes

complemented

C₄×C₁₇

C₂×C₂²⋊C₄

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

Dic₁₇

—

C₂³×C₈

—

—

—

—

D₁₇

yes

characteristic, complemented

C₂×C₁₇

—

C₂×C₁₇

#380

yes

characteristic

C₂×C₁₇

C₂×#108

yes

—

D₁₇

—

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₇

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₁₇

yes

characteristic

#108

—

C₂²⋊C₈

—

C₁₇

#1810

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₁₇

yes

—

C₂³

—

100

C₂³

—

101

C₂³

—

102

C₂³

Q₈×D₁₇

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

109

C₂×C₄

#63252

yes

—

110

C₂×C₄

D₄×D₁₇

yes

—

111

C₂×C₄

D₄×D₁₇

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₄⋊C₄×D₁₇

yes

characteristic

122

C₂²

#10587582

yes

—

123

C₄

—

124

C₄

#10587558

yes

characteristic

125

C₄

C₂²⋊C₄×D₁₇

yes

characteristic

126

C₄

#10587570

yes

—

127

C₂

—

128

C₂

—

129

C₂

#11762668

yes

characteristic

130

C₂

#108×D₁₇

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#341	C₂
socle	C₂²×C₁₇	C₂×C₄⋊C₄
prime radical	C₂³×D₁₇	Q₈
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	#341	D₁₇
hypocenter	C₁₇	#1810
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₇

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#341

C₂

socle

C₂²×C₁₇

C₂×C₄⋊C₄

prime radical

C₂³×D₁₇

Q₈

cosocle

C₂²×C₄×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
2. C₂×#10587493	C₂	semidirect
4. #11762574	C₂	semidirect
5. C₁₇×#341	C₂	semidirect
57. C₂×D₁₇	#108	semidirect
66. #341	D₁₇	semidirect
80. C₁₇	#1810	semidirect

H ≅ G/N

type

2. C₂×#10587493

C₂

semidirect

4. #11762574

C₂

semidirect

5. C₁₇×#341

C₂

semidirect

57. C₂×D₁₇

#108

semidirect

66. #341

D₁₇

semidirect

80. C₁₇

#1810

semidirect

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

subgroup	index
C₂²×C₈×D₁₇	2
C₂×#10587493	2
#11762772	2
#11762574	2
C₁₇×#341	2
#1810	17

subgroup

index

C₂²×C₈×D₁₇

C₁₇×#341

Subgroups — #14528821

Conjugacy classes of subgroups

Complements

Maximal subgroups