Group #14529772 — subgroups

Subgroups — #14529772

Order 2176.

260 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂×#10587555	C₂	1	1	yes	maximal, characteristic
2	C₄×#63223	C₂	1	1	yes	maximal, characteristic
3	#11762577	C₂	1	4	yes	maximal
4	C₁₇×#423	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₄	C₄	1	2	yes	—
6	C₂×M₁₆×C₁₇	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₇	C₄	1	2	yes	—
9	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
10	C₂×#63223	C₂²	1	2	yes	—
11	#10587555	—	2	1	no	—
12	#10587555	—	2	1	no	—
13	#10587555	—	2	1	no	—
14	#10587555	—	2	1	no	—
15	C₁₇×C₄⋊C₈	—	2	1	no	—
16	C₁₇×C₄⋊C₈	—	2	1	no	—
17	#10587474	—	2	1	no	—
18	C₄×#3622	—	2	1	no	—
19	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
22	C₂²×Dic₁₇	C₂×C₄	1	4	yes	—
23	C₂×Dic₃₄	—	2	2	no	—
24	C₂×Dic₃₄	—	2	2	no	—
25	C₂×Dic₃₄	—	4	4	no	—
26	M₁₆×C₁₇	—	2	4	no	—
27	C₂×C₈×C₁₇	—	2	2	no	—
28	C₄²×C₁₇	D₄	1	1	yes	characteristic
29	C₄²×C₁₇	D₄	1	1	yes	characteristic
30	C₄²×C₁₇	D₄	1	1	yes	characteristic
31	C₄²×C₁₇	D₄	1	1	yes	characteristic
32	C₄⋊Dic₁₇	—	2	2	no	—
33	C₄⋊Dic₁₇	—	2	2	no	—
34	C₄⋊Dic₁₇	—	2	2	no	—
35	C₄⋊Dic₁₇	—	2	2	no	—
36	#63223	—	2	2	no	—
37	#63223	Q₈	1	4	yes	complemented
38	C₂×#3622	—	2	2	no	—
39	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₇	—	2	2	no	—
41	C₂×C₄×C₁₇	—	2	2	no	—
42	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₁₇	#41	1	2	yes	—
50	C₂×Dic₁₇	—	2	4	no	—
51	C₂×Dic₁₇	—	4	4	no	—
52	Dic₃₄	—	4	4	no	—
53	Dic₃₄	—	4	4	no	—
54	C₈×C₁₇	—	4	2	no	—
55	#3622	—	2	4	no	—
56	#1997	—	17	1	no	maximal, Sylow
57	C₂²×C₁₇	—	2	1	no	—
58	C₂²×C₁₇	—	2	1	no	—
59	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₁₇	C₄×Q₈	1	2	yes	—
61	C₄×C₁₇	—	2	1	no	—
62	C₄×C₁₇	—	2	1	no	—
63	C₄×C₁₇	#100	1	2	yes	—
64	C₄×C₁₇	#101	1	2	yes	—
65	C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
66	C₄×C₁₇	#124	1	1	yes	characteristic
67	C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
68	Dic₁₇	—	4	4	no	—
69	C₂×C₄⋊Q₈	—	17	1	no	—
70	#423	D₁₇	1	1	yes	characteristic
71	C₄×M₁₆	—	17	1	no	—
72	#343	—	17	4	no	—
73	C₂×C₁₇	—	2	2	no	—
74	C₂×C₁₇	#391	1	1	yes	characteristic
75	C₂×C₁₇	C₂×#100	1	1	yes	characteristic
76	C₂×C₁₇	C₂×#101	1	1	yes	characteristic
77	C₂²×Q₈	—	17	2	no	—
78	C₂×M₁₆	—	17	2	no	—
79	C₂×M₁₆	C₂×D₁₇	1	2	yes	—
80	C₄⋊Q₈	—	34	1	no	—
81	C₄⋊Q₈	—	34	1	no	—
82	C₄⋊Q₈	—	34	1	no	—
83	C₄⋊Q₈	—	34	1	no	—
84	C₂×C₄⋊C₄	—	17	2	no	—
85	C₂×C₄⋊C₄	—	17	2	no	—
86	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
87	C₄⋊C₈	—	2	1	no	—
88	C₄⋊C₈	—	2	1	no	—
89	#97	—	34	1	no	—
90	C₄×C₈	—	34	1	no	—
91	C₁₇	#1997	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₈	—	34	2	no	—
93	C₂×Q₈	—	34	2	no	—
94	C₂×Q₈	—	68	4	no	—
95	C₂²×C₄	—	17	4	no	—
96	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
97	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
99	M₁₆	—	2	4	no	—
100	M₁₆	—	17	4	no	—
101	M₁₆	—	34	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	34	2	no	—
104	C₄⋊C₄	—	34	2	no	—
105	C₄⋊C₄	—	34	2	no	—
106	C₄⋊C₄	—	34	2	no	—
107	C₄⋊C₄	—	34	2	no	—
108	C₄²	D₆₈	1	1	yes	characteristic
109	C₄²	D₆₈	1	1	yes	characteristic
110	C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
111	C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
112	C₂³	C₂×C₄×D₁₇	1	1	yes	characteristic
113	Q₈	—	68	4	no	—
114	Q₈	—	68	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	34	4	no	—
118	C₂×C₄	—	68	4	no	—
119	C₂×C₄	#63227	1	1	yes	characteristic
120	C₂×C₄	#63227	1	1	yes	characteristic
121	C₂×C₄	#63227	1	1	yes	characteristic
122	C₂×C₄	#63227	1	1	yes	characteristic
123	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
124	C₂×C₄	#63254	1	2	yes	—
125	C₂×C₄	Q₈×D₁₇	1	2	yes	—
126	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₈	—	34	4	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	#10587573	1	2	yes	—
132	C₂²	C₂×#63227	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	68	4	no	—
136	C₄	#10587494	1	2	yes	—
137	C₄	#10587495	1	2	yes	—
138	C₄	#10587583	1	2	yes	—
139	C₄	#10587622	1	1	yes	characteristic
140	C₄	#10587630	1	1	yes	characteristic
141	C₂	—	2	2	no	—
142	C₂	#11762714	1	1	yes	characteristic
143	C₂	C₂×#10587494	1	1	yes	characteristic
144	C₂	C₂×#10587495	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587555

C₂

yes

maximal, characteristic

C₄×#63223

C₂

yes

maximal, characteristic

#11762577

C₂

yes

maximal

C₁₇×#423

C₂

yes

maximal, characteristic

C₂²×Dic₃₄

C₄

yes

—

C₂×M₁₆×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₄

yes

—

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×#63223

C₂²

yes

—

—

—

—

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

#10587474

—

C₄×#3622

—

C₂²×C₄×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

C₂×C₄

yes

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

D₄

yes

characteristic

C₄²×C₁₇

D₄

yes

characteristic

C₄²×C₁₇

D₄

yes

characteristic

C₄²×C₁₇

D₄

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63223

—

#63223

Q₈

yes

complemented

C₂×#3622

—

C₂³×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

—

C₂×C₄×C₁₇

#41

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

#3622

—

#1997

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₇

C₄×Q₈

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#100

yes

—

C₄×C₁₇

#101

yes

—

C₄×C₁₇

C₂²⋊Q₈

yes

—

C₄×C₁₇

#124

yes

characteristic

C₄×C₁₇

C₄⋊Q₈

yes

characteristic

Dic₁₇

—

C₂×C₄⋊Q₈

—

#423

D₁₇

yes

characteristic

C₄×M₁₆

—

#343

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₁₇

#1997

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₄×D₁₇

yes

characteristic

C₂²×C₄

C₄×D₁₇

yes

characteristic

C₂²×C₄

C₂²×D₁₇

yes

characteristic

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

D₆₈

yes

characteristic

109

C₄²

D₆₈

yes

characteristic

110

C₄²

C₂²⋊D₁₇

yes

characteristic

111

C₄²

C₂²⋊D₁₇

yes

characteristic

112

C₂³

C₂×C₄×D₁₇

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

#63227

yes

characteristic

120

C₂×C₄

#63227

yes

characteristic

121

C₂×C₄

#63227

yes

characteristic

122

C₂×C₄

#63227

yes

characteristic

123

C₂×C₄

C₂×D₆₈

yes

characteristic

124

C₂×C₄

#63254

yes

—

125

C₂×C₄

Q₈×D₁₇

yes

—

126

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

127

C₈

—

128

C₈

—

129

C₂²

—

130

C₂²

—

131

C₂²

#10587573

yes

—

132

C₂²

C₂×#63227

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#10587494

yes

—

137

C₄

#10587495

yes

—

138

C₄

#10587583

yes

—

139

C₄

#10587622

yes

characteristic

140

C₄

#10587630

yes

characteristic

141

C₂

—

142

C₂

#11762714

yes

characteristic

143

C₂

C₂×#10587494

yes

characteristic

144

C₂

C₂×#10587495

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#63227
commutator, prime radical	C₂³×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#423	C₂
socle	C₂²×C₁₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	#423	D₁₇
hypocenter	C₁₇	#1997
LCS	G ▷ C₂³×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×#63227

commutator, prime radical

C₂³×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#423

C₂

socle

C₂²×C₁₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂³×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
37. #63223	Q₈	semidirect
91. C₁₇	#1997	semidirect

H ≅ G/N

type

37. #63223

Q₈

semidirect

91. C₁₇

#1997

semidirect

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

subgroup	index
C₂×#10587555	2
C₄×#63223	2
#11762577	2
C₁₇×#423	2
#1997	17

subgroup

index

C₁₇×#423

Subgroups — #14529772

Conjugacy classes of subgroups

Complements

Maximal subgroups