Group #17948525 — subgroups

Subgroups — #17948525

Order 4352.

266 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 84 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	#14545145	C₂	1	2	yes	maximal
2	C₄×#10587474	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#63216	C₄	1	4	yes	complemented
4	C₂²⋊#63216	—	2	2	no	—
5	#11763697	—	2	2	no	—
6	C₂×#10587474	—	2	2	no	—
7	C₂×C₄×#3622	C₄	1	1	yes	characteristic
8	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
9	C₄³×C₁₇	C₄	1	1	yes	characteristic
10	C₂×C₄²×C₁₇	—	2	1	no	—
11	C₂×C₄²×C₁₇	—	2	1	no	—
12	C₂×C₄²×C₁₇	C₈	1	2	yes	—
13	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×#3622	—	2	1	no	—
15	C₂²×#3622	D₄	1	2	yes	—
16	C₂×#63216	—	2	4	no	—
17	#10587474	—	2	8	no	—
18	C₄×#3622	—	2	1	no	—
19	C₄×#3622	C₈	1	2	yes	—
20	C₄×#3622	C₂×C₄	1	2	yes	—
21	C₄×#3622	D₄	1	1	yes	characteristic
22	C₄×#3622	Q₈	1	1	yes	characteristic
23	C₂²×C₄×C₁₇	—	2	2	no	—
24	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
27	C₄²×C₁₇	—	2	4	no	—
28	C₄²×C₁₇	—	4	2	no	—
29	C₄²×C₁₇	—	4	2	no	—
30	C₄²×C₁₇	C₄²	1	1	yes	characteristic
31	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
32	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
33	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
34	C₂×#3622	—	2	4	no	—
35	C₂×#3622	—	2	2	no	—
36	C₂×#3622	—	2	4	no	—
37	C₂×#3622	C₂²⋊C₄	1	2	yes	—
38	C₂×#3622	C₄⋊C₄	1	2	yes	—
39	#63216	—	4	4	no	—
40	#7149	—	17	1	no	maximal, Sylow
41	C₂³×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
42	C₂×C₄×C₁₇	—	2	2	no	—
43	C₂×C₄×C₁₇	—	2	2	no	—
44	C₂×C₄×C₁₇	—	2	2	no	—
45	C₂×C₄×C₁₇	—	2	4	no	—
46	C₂×C₄×C₁₇	—	4	2	no	—
47	C₂×C₄×C₁₇	—	4	2	no	—
48	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
50	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
54	#3622	—	2	4	no	—
55	#3622	—	2	2	no	—
56	#3622	#104	1	4	yes	—
57	#2163	—	17	2	no	—
58	C₄×#97	—	17	1	no	—
59	C₂²×C₁₇	—	2	1	no	—
60	C₂²×C₁₇	—	2	1	no	—
61	C₂²×C₁₇	#336	1	1	yes	characteristic
62	C₂²×C₁₇	#349	1	2	yes	—
63	C₄×C₁₇	—	2	1	no	—
64	C₄×C₁₇	—	2	1	no	—
65	C₄×C₁₇	—	2	1	no	—
66	C₄×C₁₇	—	2	1	no	—
67	C₄×C₁₇	—	4	4	no	—
68	C₄×C₁₇	#339	1	2	yes	—
69	C₄×C₁₇	#347	1	2	yes	—
70	C₂×#97	—	34	2	no	—
71	C₂×C₄×C₈	—	17	1	no	—
72	C₂×C₄×C₈	—	17	1	no	—
73	C₄³	#595	1	1	yes	characteristic
74	C₄⋊C₁₆	—	17	4	no	—
75	C₂²⋊C₁₆	—	34	2	no	—
76	#346	—	34	2	no	—
77	C₂×C₁₇	—	2	2	no	—
78	C₂×C₁₇	#1276	1	1	yes	characteristic
79	C₂×C₁₇	#1375	1	1	yes	characteristic
80	C₂×C₁₇	#1377	1	1	yes	characteristic
81	C₂²×C₈	—	17	2	no	—
82	C₂²×C₈	—	34	1	no	—
83	C₂×C₄²	—	2	1	no	—
84	C₂×C₄²	—	2	1	no	—
85	C₂×C₄²	#3624	1	2	yes	—
86	C₂×C₄²	C₂×#595	1	1	yes	characteristic
87	C₂×C₁₆	—	34	4	no	—
88	#97	—	34	8	no	—
89	C₄×C₈	—	17	2	no	—
90	C₄×C₈	—	17	2	no	—
91	C₄×C₈	—	17	1	no	—
92	C₄×C₈	—	17	1	no	—
93	C₄×C₈	—	34	1	no	—
94	C₁₇	#7149	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
97	C₂²×C₄	#63247	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊#595	1	1	yes	characteristic
99	C₂×C₈	—	17	2	no	—
100	C₂×C₈	—	17	2	no	—
101	C₂×C₈	—	34	4	no	—
102	C₂×C₈	—	34	4	no	—
103	C₂×C₈	—	34	2	no	—
104	C₄²	—	2	4	no	—
105	C₄²	—	4	2	no	—
106	C₄²	—	4	2	no	—
107	C₄²	#63242	1	1	yes	characteristic
108	C₄²	#63243	1	1	yes	characteristic
109	C₄²	C₄×#595	1	1	yes	characteristic
110	C₄²	C₄⋊#595	1	1	yes	characteristic
111	C₁₆	—	68	4	no	—
112	C₂³	C₂²⋊#3624	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	C₄×#3624	1	1	yes	characteristic
120	C₂×C₄	C₄⋊#3624	1	1	yes	characteristic
121	C₂×C₄	#10587542	1	1	yes	characteristic
122	C₂×C₄	#10587543	1	1	yes	characteristic
123	C₂×C₄	#10587544	1	1	yes	characteristic
124	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
125	C₈	—	17	4	no	—
126	C₈	—	34	4	no	—
127	C₈	—	34	2	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	#11763675	1	1	yes	characteristic
131	C₂²	#11763705	1	2	yes	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	4	4	no	—
137	C₄	#11763680	1	2	yes	—
138	C₄	#11763701	1	2	yes	—
139	C₂	—	2	2	no	—
140	C₂	#14543612	1	1	yes	characteristic
141	C₂	#14543825	1	1	yes	characteristic
142	C₂	#14543831	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545145

C₂

yes

maximal

C₄×#10587474

C₂

yes

maximal, characteristic

C₄⋊#63216

C₄

yes

—

—

—

C₂×C₄×#3622

C₄

yes

characteristic

C₂×C₄×#3622

C₂²

yes

characteristic

C₄³×C₁₇

C₄

yes

characteristic

C₂×C₄²×C₁₇

—

C₂×C₄²×C₁₇

—

C₂×C₄²×C₁₇

C₈

yes

—

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂²×#3622

—

C₂²×#3622

D₄

yes

—

—

—

—

C₈

yes

—

C₄×#3622

C₂×C₄

yes

—

C₄×#3622

D₄

yes

characteristic

C₄×#3622

Q₈

yes

characteristic

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₇

M₁₆

yes

characteristic

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₄²

yes

characteristic

C₄²×C₁₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₇

C₂×C₈

yes

characteristic

C₄²×C₁₇

M₁₆

yes

—

—

—

C₂²⋊C₄

yes

—

C₂×#3622

C₄⋊C₄

yes

—

#63216

—

#7149

—

maximal, Sylow

C₂³×C₁₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#95

yes

characteristic

C₂×C₄×C₁₇

C₄×C₈

yes

characteristic

C₂×C₄×C₁₇

#97

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₇

C₄⋊C₈

yes

characteristic

C₂×C₄×C₁₇

C₄⋊C₈

yes

—

—

yes

—

#2163

—

C₄×#97

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#336

yes

characteristic

C₂²×C₁₇

#349

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#339

yes

—

C₄×C₁₇

#347

yes

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

#595

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

#3624

yes

—

C₂×C₄²

C₂×#595

yes

characteristic

C₂×C₁₆

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₁₇

#7149

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

C₂²⋊#595

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

#63242

yes

characteristic

108

C₄²

#63243

yes

characteristic

109

C₄²

C₄×#595

yes

characteristic

110

C₄²

C₄⋊#595

yes

characteristic

111

C₁₆

—

112

C₂³

C₂²⋊#3624

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₄×#3624

yes

characteristic

120

C₂×C₄

C₄⋊#3624

yes

characteristic

121

C₂×C₄

#10587542

yes

characteristic

122

C₂×C₄

#10587543

yes

characteristic

123

C₂×C₄

#10587544

yes

characteristic

124

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

#11763675

yes

characteristic

131

C₂²

#11763705

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

#11763680

yes

—

138

C₄

#11763701

yes

—

139

C₂

—

140

C₂

#14543612

yes

characteristic

141

C₂

#14543825

yes

characteristic

142

C₂

#14543831

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×#595)⋊C₄
commutator	C₂×C₄×C₁₇	C₄×C₈
Frattini	C₂×C₄²	C₂×#595
Fitting	C₄³×C₁₇	C₄
socle	C₂²×C₁₇	#336
prime radical	C₂³×C₁₇	C₂²⋊C₈
cosocle	C₂×C₄×#3622	C₂²
hypercenter	C₄³	#595
hypocenter	C₁₇	#7149
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

subgroup

G/H

center

C₂×C₄

(C₂×#595)⋊C₄

commutator

C₂×C₄×C₁₇

C₄×C₈

Frattini

C₂×C₄²

C₂×#595

Fitting

C₄³×C₁₇

C₄

socle

C₂²×C₁₇

#336

prime radical

C₂³×C₁₇

C₂²⋊C₈

cosocle

C₂×C₄×#3622

C₂²

C₄³

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

N	H ≅ G/N	type
3. C₄⋊#63216	C₄	semidirect
94. C₁₇	#7149	semidirect

H ≅ G/N

type

3. C₄⋊#63216

C₄

semidirect

94. C₁₇

#7149

semidirect

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

subgroup	index
#14545145	2
C₄×#10587474	2
#7149	17

subgroup

index

#14545145

C₄×#10587474

#7149

Subgroups — #17948525

Conjugacy classes of subgroups

Complements

Maximal subgroups