Group #14528366 — subgroups

Subgroups — #14528366

Order 2176.

140 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 53 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	#11763160	C₂	1	1	yes	maximal, characteristic
2	#11763104	C₂	1	1	yes	maximal, characteristic
3	#11763094	C₂	1	1	yes	maximal, characteristic
4	#11762560	C₂	1	1	yes	maximal, characteristic
5	#11762557	C₂	1	1	yes	maximal, characteristic
6	#11762556	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#446	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
9	#10587555	C₂²	1	1	yes	characteristic
10	#10587555	C₂²	1	1	yes	characteristic
11	C₁₇×C₄⋊C₈	—	2	1	no	—
12	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₁₇×#97	—	2	1	no	—
14	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	#10587506	—	2	1	no	—
16	#10587506	—	2	1	no	—
17	#10587487	—	2	1	no	—
18	#10587487	—	2	1	no	—
19	#10587479	—	2	1	no	—
20	#10587478	—	2	1	no	—
21	C₄⋊#3622	C₂²	1	1	yes	characteristic
22	C₄⋊#3622	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₁₇	D₄	1	1	yes	characteristic
24	C₂×Dic₃₄	—	2	1	no	—
25	C₂×Dic₃₄	—	2	1	no	—
26	C₂×C₈×C₁₇	—	2	1	no	—
27	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
28	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
29	C₄⋊C₄×C₁₇	—	2	1	no	—
30	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
31	C₄²×C₁₇	—	2	1	no	—
32	C₄²×C₁₇	C₂³	1	1	yes	characteristic
33	C₄⋊Dic₁₇	—	2	1	no	—
34	C₄⋊Dic₁₇	—	2	1	no	—
35	C₄⋊Dic₁₇	—	2	1	no	—
36	C₄⋊Dic₁₇	—	2	1	no	—
37	C₂×#3622	—	2	1	no	—
38	C₂×#3622	—	2	1	no	—
39	Q₈×C₁₇	—	2	1	no	—
40	Q₈×C₁₇	—	2	1	no	—
41	C₂×C₄×C₁₇	—	2	1	no	—
42	C₂×C₄×C₁₇	—	2	1	no	—
43	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
46	C₂×Dic₁₇	—	4	1	no	—
47	C₂×Dic₁₇	—	4	1	no	—
48	Dic₃₄	—	4	1	no	—
49	Dic₃₄	—	4	1	no	—
50	Dic₃₄	—	8	1	no	—
51	C₈×C₁₇	—	2	1	no	—
52	C₈×C₁₇	—	2	1	no	—
53	C₈×C₁₇	—	4	1	no	—
54	#3622	—	4	1	no	—
55	#3622	—	4	1	no	—
56	#1704	—	17	1	no	maximal, Sylow
57	C₂²×C₁₇	#121	1	1	yes	characteristic
58	C₄×C₁₇	—	2	1	no	—
59	C₄×C₁₇	—	4	1	no	—
60	C₄×C₁₇	—	4	1	no	—
61	C₄×C₁₇	#135	1	1	yes	characteristic
62	C₄×C₁₇	#137	1	1	yes	characteristic
63	C₄×C₁₇	#137	1	1	yes	characteristic
64	C₄×C₁₇	#137	1	1	yes	characteristic
65	Dic₁₇	—	8	1	no	—
66	Dic₁₇	—	8	1	no	—
67	#487	—	17	1	no	—
68	#477	—	17	1	no	—
69	#475	—	17	1	no	—
70	#446	D₁₇	1	1	yes	characteristic
71	#334	—	17	1	no	—
72	#333	—	17	1	no	—
73	#332	—	17	1	no	—
74	C₂×C₁₇	#464	1	1	yes	characteristic
75	C₂×C₁₇	#470	1	1	yes	characteristic
76	C₂×C₁₇	#471	1	1	yes	characteristic
77	C₄⋊Q₈	—	17	1	no	—
78	C₄⋊Q₈	—	17	1	no	—
79	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
80	#107	—	34	1	no	—
81	#106	—	34	1	no	—
82	C₄⋊C₈	—	2	1	no	—
83	C₄⋊C₈	—	17	1	no	—
84	C₄⋊C₈	—	17	1	no	—
85	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
86	#103	—	34	1	no	—
87	#103	—	34	1	no	—
88	#103	—	34	1	no	—
89	#103	—	34	1	no	—
90	#97	—	2	1	no	—
91	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
92	C₁₇	#1704	1	1	yes	characteristic, complemented, Sylow
93	C₂×Q₈	—	34	1	no	—
94	C₂×Q₈	—	34	1	no	—
95	C₂×Q₈	D₆₈	1	1	yes	characteristic
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	34	1	no	—
98	C₂×C₈	—	34	1	no	—
99	C₂×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
100	C₂×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	34	1	no	—
103	C₄⋊C₄	—	34	1	no	—
104	C₄⋊C₄	—	34	1	no	—
105	C₄⋊C₄	—	34	1	no	—
106	C₄⋊C₄	D₆₈	1	1	yes	characteristic
107	C₄²	—	2	1	no	—
108	C₄²	C₂²×D₁₇	1	1	yes	characteristic
109	Q₈	—	2	1	no	—
110	Q₈	—	2	1	no	—
111	Q₈	—	68	1	no	—
112	Q₈	—	68	1	no	—
113	Q₈	—	136	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	68	1	no	—
117	C₂×C₄	—	68	1	no	—
118	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
119	C₂×C₄	#63252	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	2	1	no	—
123	C₈	—	4	1	no	—
124	C₈	—	68	1	no	—
125	C₈	—	68	1	no	—
126	C₂²	#10587616	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	4	1	no	—
129	C₄	—	4	1	no	—
130	C₄	—	136	1	no	—
131	C₄	—	136	1	no	—
132	C₄	#10587595	1	1	yes	characteristic
133	C₄	#10587595	1	1	yes	characteristic
134	C₄	#10587628	1	1	yes	characteristic
135	C₄	#10587636	1	1	yes	characteristic
136	C₂	#11763031	1	1	yes	characteristic
137	C₂	#11763067	1	1	yes	characteristic
138	C₂	#11763072	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763160

C₂

yes

maximal, characteristic

#11763104

C₂

yes

maximal, characteristic

#11763094

C₂

yes

maximal, characteristic

#11762560

C₂

yes

maximal, characteristic

#11762557

C₂

yes

maximal, characteristic

#11762556

C₂

yes

maximal, characteristic

C₁₇×#446

C₂

yes

maximal, characteristic

C₄×Q₈×C₁₇

C₂²

yes

characteristic

#10587555

C₂²

yes

characteristic

#10587555

C₂²

yes

characteristic

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

C₂²

yes

characteristic

C₁₇×#97

—

C₄×C₈×C₁₇

C₂²

yes

—

—

—

—

—

—

C₂²

yes

characteristic

C₄⋊#3622

C₂²

yes

characteristic

C₂×Q₈×C₁₇

D₄

yes

characteristic

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

characteristic

C₂×C₈×C₁₇

D₄

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

D₄

yes

characteristic

C₄²×C₁₇

—

C₄²×C₁₇

C₂³

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₂×#3622

—

C₂×#3622

—

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

—

—

—

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

—

—

—

D₁₇

yes

—

—

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

C₂×D₁₇

yes

characteristic

#107

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₁₇

yes

—

—

—

—

—

C₄×C₈

C₂×D₁₇

yes

characteristic

C₁₇

#1704

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₆₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₁₇

yes

characteristic

100

C₂×C₈

C₂²⋊D₁₇

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

D₆₈

yes

characteristic

107

C₄²

—

108

C₄²

C₂²×D₁₇

yes

characteristic

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

136

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₂×D₆₈

yes

characteristic

119

C₂×C₄

#63252

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₂²

#10587616

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

136

—

131

C₄

—

136

—

132

C₄

#10587595

yes

characteristic

133

C₄

#10587595

yes

characteristic

134

C₄

#10587628

yes

characteristic

135

C₄

#10587636

yes

characteristic

136

C₂

#11763031

yes

characteristic

137

C₂

#11763067

yes

characteristic

138

C₂

#11763072

yes

characteristic

	subgroup	G/H
center	C₂²	#10587616
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#446	C₂
socle, prime radical	C₂²×C₁₇	#121
hypercenter	#446	D₁₇
hypocenter	C₁₇	#1704
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #446

subgroup

G/H

center

C₂²

#10587616

commutator, cosocle

C₄²×C₁₇

C₂³

Frattini

C₄²

C₂²×D₁₇

Fitting

C₁₇×#446

C₂

C₂²×C₁₇

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #446

N	H ≅ G/N	type
92. C₁₇	#1704	semidirect

H ≅ G/N

type

92. C₁₇

#1704

semidirect

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

subgroup	index
#11763160	2
#11763104	2
#11763094	2
#11762560	2
#11762557	2
#11762556	2
C₁₇×#446	2
#1704	17

subgroup

index

C₁₇×#446

Subgroups — #14528366

Conjugacy classes of subgroups

Complements

Maximal subgroups