Group #14539307 — subgroups

Subgroups — #14539307

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	#11763362	C₂	1	2	yes	maximal
2	#11763217	C₂	1	4	yes	maximal, complemented
3	#11763216	C₂	1	4	yes	maximal, complemented
4	#11762847	C₂	1	1	yes	maximal, characteristic
5	#11762845	C₂	1	2	yes	maximal
6	C₂×#10587486	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#427	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
9	C₁₇×#117	C₂²	1	2	yes	—
10	#10587612	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
12	C₂²×#3622	C₂²	1	1	yes	characteristic
13	#10587576	C₂²	1	4	yes	—
14	#10587574	C₂²	1	4	yes	—
15	#10587564	—	2	4	no	—
16	C₄×Dic₃₄	—	2	4	no	—
17	C₁₇×#107	C₂²	1	2	yes	—
18	C₁₇×#106	C₂²	1	2	yes	—
19	#10587489	C₂²	1	2	yes	—
20	#10587488	C₂²	1	2	yes	—
21	#10587486	C₂²	1	2	yes	—
22	#10587486	C₂²	1	2	yes	—
23	#10587479	C₂²	1	4	yes	—
24	#10587478	C₂²	1	4	yes	—
25	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₁₇	D₄	1	2	yes	—
27	C₂×Dic₃₄	—	2	4	no	—
28	C₂×C₈×C₁₇	Q₈	1	4	yes	complemented
29	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₁₇	C₂³	1	4	yes	—
32	C₂²⋊C₄×C₁₇	—	2	2	no	—
33	C₄²×C₁₇	—	2	2	no	—
34	C₂²⋊Dic₁₇	—	2	2	no	—
35	C₄⋊Dic₁₇	—	2	4	no	—
36	C₄⋊Dic₁₇	C₂³	1	4	yes	—
37	#63225	—	2	4	no	—
38	#63225	—	2	4	no	—
39	C₄×Dic₁₇	—	2	2	no	—
40	C₄×Dic₁₇	D₄	1	2	yes	—
41	C₄×Dic₁₇	C₂³	1	2	yes	—
42	C₂×#3622	Q₈	1	4	yes	complemented
43	C₂×#3622	C₂³	1	2	yes	—
44	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇	—	2	4	no	—
46	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₁₇	—	2	4	no	—
51	C₂×Dic₁₇	—	4	2	no	—
52	C₂×Dic₁₇	C₂×D₄	1	4	yes	—
53	Dic₃₄	—	4	8	no	—
54	C₈×C₁₇	C₂×Q₈	1	2	yes	—
55	C₈×C₁₇	C₂×Q₈	1	2	yes	—
56	#3622	C₂×Q₈	1	4	yes	—
57	#3162	—	17	1	no	maximal, Sylow
58	C₂²×C₁₇	—	2	2	no	—
59	C₂²×C₁₇	C₄⋊Q₈	1	2	yes	—
60	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
61	C₄×C₁₇	—	4	4	no	—
62	C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
63	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
64	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
65	Dic₁₇	—	2	4	no	—
66	Dic₁₇	—	4	4	no	—
67	#533	—	17	2	no	—
68	#501	—	17	4	no	—
69	#501	—	17	4	no	—
70	#427	—	17	1	no	—
71	#427	—	17	2	no	—
72	#427	D₁₇	1	1	yes	characteristic
73	C₂×#97	—	17	1	no	—
74	C₂×C₁₇	—	2	2	no	—
75	C₂×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₁₇	#575	1	2	yes	—
77	C₂²×C₈	—	17	1	no	—
78	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
79	C₄⋊Q₈	—	17	4	no	—
80	#125	—	17	4	no	—
81	C₂²⋊Q₈	—	34	4	no	—
82	C₄×Q₈	—	34	4	no	—
83	#117	—	17	2	no	—
84	#117	C₂×D₁₇	1	2	yes	—
85	C₂×C₄²	—	17	1	no	—
86	#107	—	17	2	no	—
87	#107	—	17	4	no	—
88	#107	C₂×D₁₇	1	2	yes	—
89	#106	—	17	2	no	—
90	#106	—	17	4	no	—
91	#106	C₂×D₁₇	1	2	yes	—
92	#97	—	17	2	no	—
93	#97	—	17	2	no	—
94	C₁₇	#3162	1	1	yes	characteristic, complemented, Sylow
95	C₂×Q₈	—	34	4	no	—
96	C₂²×C₄	—	17	2	no	—
97	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
98	C₂×C₈	—	17	4	no	—
99	C₂×C₈	—	17	2	no	—
100	C₂×C₈	Dic₃₄	1	4	yes	complemented
101	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
103	C₄⋊C₄	—	17	4	no	—
104	C₄⋊C₄	—	34	4	no	—
105	C₄⋊C₄	—	34	4	no	—
106	C₄⋊C₄	—	34	4	no	—
107	C₄⋊C₄	C₂²×D₁₇	1	4	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₂²⋊C₄	—	34	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	17	2	no	—
112	C₄²	—	17	2	no	—
113	C₄²	—	34	2	no	—
114	C₂³	D₄×D₁₇	1	1	yes	characteristic
115	Q₈	—	68	8	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	17	4	no	—
118	C₂×C₄	—	34	4	no	—
119	C₂×C₄	—	68	2	no	—
120	C₂×C₄	C₂×Dic₃₄	1	2	yes	—
121	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
122	C₂×C₄	Q₈×D₁₇	1	2	yes	—
123	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
124	C₈	—	17	4	no	—
125	C₈	C₂×Dic₃₄	1	2	yes	—
126	C₈	C₂×Dic₃₄	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	#10587574	1	2	yes	—
129	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
130	C₄	—	4	4	no	—
131	C₄	—	34	4	no	—
132	C₄	—	68	4	no	—
133	C₄	#10587574	1	2	yes	—
134	C₄	C₂²×Dic₃₄	1	1	yes	characteristic
135	C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
136	C₂	—	2	2	no	—
137	C₂	C₂×#10587574	1	1	yes	characteristic
138	C₂	#11763606	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11763362

C₂

yes

maximal

#11763217

C₂

yes

maximal, complemented

#11763216

C₂

yes

maximal, complemented

#11762847

C₂

yes

maximal, characteristic

#11762845

C₂

yes

maximal

C₂×#10587486

C₂

yes

maximal, characteristic

C₁₇×#427

C₂

yes

maximal, characteristic

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#117

C₂²

yes

—

#10587612

C₂²

yes

—

C₂×C₄×Dic₁₇

C₂²

yes

characteristic

C₂²×#3622

C₂²

yes

characteristic

#10587576

C₂²

yes

—

#10587574

C₂²

yes

—

#10587564

—

C₄×Dic₃₄

—

C₁₇×#107

C₂²

yes

—

C₁₇×#106

C₂²

yes

—

#10587489

C₂²

yes

—

#10587488

C₂²

yes

—

#10587486

C₂²

yes

—

#10587486

C₂²

yes

—

#10587479

C₂²

yes

—

#10587478

C₂²

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

D₄

yes

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

Q₈

yes

complemented

C₂×C₈×C₁₇

C₂³

yes

characteristic

C₂×C₈×C₁₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₇

C₂³

yes

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂³

yes

—

#63225

—

#63225

—

C₄×Dic₁₇

—

C₄×Dic₁₇

D₄

yes

—

C₄×Dic₁₇

C₂³

yes

—

C₂×#3622

Q₈

yes

complemented

C₂×#3622

C₂³

yes

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

—

C₂×C₄×C₁₇

C₂×Q₈

yes

—

C₂×C₄×C₁₇

C₂⁴

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂×D₄

yes

—

Dic₃₄

—

C₈×C₁₇

C₂×Q₈

yes

—

C₈×C₁₇

C₂×Q₈

yes

—

#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⋊Q₈

yes

—

C₄×C₁₇

C₂²×Q₈

yes

characteristic

C₄×C₁₇

C₂²×Q₈

yes

characteristic

Dic₁₇

—

Dic₁₇

—

—

—

—

—

—

D₁₇

yes

characteristic

C₂×#97

—

C₂×C₁₇

—

C₂×C₁₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₇

#575

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₁₇

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

#117

C₂×D₁₇

yes

—

C₂×C₄²

—

#107

—

#107

—

#107

C₂×D₁₇

yes

—

#106

—

#106

—

#106

C₂×D₁₇

yes

—

#97

—

#97

—

C₁₇

#3162

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

Dic₃₄

yes

complemented

101

C₂×C₈

C₂²×D₁₇

yes

characteristic

102

C₂×C₈

C₂²×D₁₇

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂²×D₁₇

yes

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₂³

D₄×D₁₇

yes

characteristic

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂×Dic₃₄

yes

—

121

C₂×C₄

D₄×D₁₇

yes

characteristic

122

C₂×C₄

Q₈×D₁₇

yes

—

123

C₂×C₄

C₂³×D₁₇

yes

characteristic

124

C₈

—

125

C₈

C₂×Dic₃₄

yes

—

126

C₈

C₂×Dic₃₄

yes

—

127

C₂²

—

128

C₂²

#10587574

yes

—

129

C₂²

C₂×D₄×D₁₇

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

#10587574

yes

—

134

C₄

C₂²×Dic₃₄

yes

characteristic

135

C₄

C₂×Q₈×D₁₇

yes

characteristic

136

C₂

—

137

C₂

C₂×#10587574

yes

characteristic

138

C₂

#11763606

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#427	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂³×C₁₇	C₂×D₄
hypercenter	#427	D₁₇
hypocenter	C₁₇	#3162
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₁₇

commutator, cosocle

C₂×C₄×C₁₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₇

Fitting

C₁₇×#427

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 ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #11763217	C₂	semidirect
3. #11763216	C₂	semidirect
28. C₂×C₈×C₁₇	Q₈	semidirect
42. C₂×#3622	Q₈	semidirect
94. C₁₇	#3162	semidirect
100. C₂×C₈	Dic₃₄	semidirect

H ≅ G/N

type

2. #11763217

C₂

semidirect

3. #11763216

C₂

semidirect

28. C₂×C₈×C₁₇

Q₈

semidirect

42. C₂×#3622

Q₈

semidirect

94. C₁₇

#3162

semidirect

100. C₂×C₈

Dic₃₄

semidirect

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

subgroup	index
#11763362	2
#11763217	2
#11763216	2
#11762847	2
#11762845	2
C₂×#10587486	2
C₁₇×#427	2
#3162	17

subgroup

index

C₁₇×#427

Subgroups — #14539307

Conjugacy classes of subgroups

Complements

Maximal subgroups