Group #17948490 — subgroups

Subgroups — #17948490

Order 4352.

266 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 128 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×#11763697	C₂	1	2	yes	maximal
2	C₂×#11762531	C₂	1	1	yes	maximal, characteristic
3	C₂²×#63216	C₄	1	4	yes	—
4	#11763697	—	2	2	no	—
5	#11763697	—	2	2	no	—
6	C₂×C₄×#3622	C₄	1	1	yes	characteristic
7	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
8	#11762531	—	2	2	no	—
9	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
10	C₂²×C₈×C₁₇	—	2	1	no	—
11	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
12	C₂²×#3622	—	2	1	no	—
13	C₂²×#3622	C₂×C₄	1	2	yes	—
14	C₂×#63216	—	2	4	no	—
15	C₂×#63216	—	2	4	no	—
16	C₂×#63216	C₈	1	8	yes	complemented
17	C₄×C₈×C₁₇	—	2	1	no	—
18	C₄×C₈×C₁₇	C₈	1	2	yes	—
19	C₄×#3622	—	2	1	no	—
20	C₄×#3622	C₈	1	2	yes	—
21	C₄×#3622	D₄	1	2	yes	—
22	C₄×#3622	D₄	1	1	yes	characteristic
23	C₄×#3622	Q₈	1	1	yes	characteristic
24	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
25	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
26	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
27	C₂×C₈×C₁₇	—	2	2	no	—
28	C₂×C₈×C₁₇	—	4	2	no	—
29	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
31	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
32	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
33	C₂×#3622	—	2	2	no	—
34	C₂×#3622	—	4	2	no	—
35	C₂×#3622	C₂²⋊C₄	1	2	yes	—
36	C₂×#3622	C₄⋊C₄	1	2	yes	—
37	C₂×#3622	C₂×C₈	1	4	yes	—
38	C₂×#3622	M₁₆	1	4	yes	—
39	#63216	—	2	8	no	—
40	#7137	—	17	1	no	maximal, Sylow
41	C₂³×C₁₇	#97	1	1	yes	characteristic
42	C₂×C₄×C₁₇	—	2	2	no	—
43	C₂×C₄×C₁₇	—	2	2	no	—
44	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
45	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₁₇	#97	1	2	yes	—
48	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
52	C₈×C₁₇	—	4	2	no	—
53	#3622	—	4	2	no	—
54	#3622	C₂²⋊C₈	1	4	yes	—
55	#3622	C₄⋊C₈	1	4	yes	—
56	C₂×#346	—	17	2	no	—
57	C₂×#322	—	17	1	no	—
58	C₂²×C₁₇	#322	1	2	yes	—
59	C₂²×C₁₇	#336	1	1	yes	characteristic
60	C₂²×C₁₇	#347	1	2	yes	—
61	C₂²×C₁₇	#347	1	2	yes	—
62	C₄×C₁₇	—	2	1	no	—
63	C₄×C₁₇	—	2	1	no	—
64	C₄×C₁₇	—	2	1	no	—
65	C₄×C₁₇	—	2	1	no	—
66	C₄×C₁₇	C₈²	1	1	yes	characteristic
67	C₄×C₁₇	#322	1	1	yes	characteristic
68	C₄×C₁₇	#336	1	2	yes	—
69	C₂²×C₁₆	—	17	4	no	—
70	C₂×C₄×C₈	—	17	1	no	—
71	C₂×C₄×C₈	—	17	1	no	—
72	C₂×C₄×C₈	#595	1	1	yes	characteristic
73	#346	—	34	2	no	—
74	#346	—	34	2	no	—
75	#322	—	34	2	no	—
76	C₂×C₁₇	#1273	1	1	yes	characteristic
77	C₂×C₁₇	#1313	1	4	yes	—
78	C₂×C₁₇	#1375	1	1	yes	characteristic
79	C₂×C₁₇	#1375	1	1	yes	characteristic
80	C₂²×C₈	—	2	1	no	—
81	C₂²×C₈	—	17	2	no	—
82	C₂²×C₈	—	34	1	no	—
83	C₂×C₄²	C₂×#595	1	1	yes	characteristic
84	C₂×C₁₆	—	17	8	no	—
85	C₂×C₁₆	—	34	4	no	—
86	C₂×C₁₆	—	34	4	no	—
87	C₄×C₈	—	2	1	no	—
88	C₄×C₈	—	17	2	no	—
89	C₄×C₈	—	17	1	no	—
90	C₄×C₈	—	17	1	no	—
91	C₄×C₈	—	17	2	no	—
92	C₄×C₈	—	34	1	no	—
93	C₄×C₈	#3624	1	2	yes	—
94	C₁₇	#7137	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₄	#63243	1	1	yes	characteristic
96	C₂²×C₄	C₄×#595	1	1	yes	characteristic
97	C₂²×C₄	#63247	1	1	yes	characteristic
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	4	2	no	—
100	C₂×C₈	—	17	4	no	—
101	C₂×C₈	—	17	4	no	—
102	C₂×C₈	—	17	2	no	—
103	C₂×C₈	—	17	2	no	—
104	C₂×C₈	—	34	2	no	—
105	C₂×C₈	—	68	2	no	—
106	C₄²	#63242	1	1	yes	characteristic
107	C₄²	C₄⋊#595	1	1	yes	characteristic
108	C₄²	C₂×#3624	1	1	yes	characteristic
109	C₄²	C₂²⋊#595	1	1	yes	characteristic
110	C₁₆	—	34	8	no	—
111	C₂³	#10587542	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	C₈×#595	1	2	yes	—
115	C₂×C₄	#10587532	1	2	yes	—
116	C₂×C₄	C₄×#3624	1	1	yes	characteristic
117	C₂×C₄	C₄⋊#3624	1	1	yes	characteristic
118	C₂×C₄	#10587543	1	1	yes	characteristic
119	C₂×C₄	#10587544	1	1	yes	characteristic
120	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
121	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
122	C₈	—	4	2	no	—
123	C₈	—	17	4	no	—
124	C₈	—	17	4	no	—
125	C₈	—	68	2	no	—
126	C₂²	#11763651	1	2	yes	—
127	C₂²	#11763675	1	1	yes	characteristic
128	C₂²	#11763701	1	2	yes	—
129	C₂²	#11763702	1	2	yes	—
130	C₄	—	2	1	no	—
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	C₈×#3624	1	1	yes	characteristic
135	C₄	#11763650	1	1	yes	characteristic
136	C₄	#11763674	1	2	yes	—
137	C₂	#14543606	1	1	yes	characteristic
138	C₂	#14543702	1	4	yes	—
139	C₂	#14543825	1	1	yes	characteristic
140	C₂	#14543826	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763697

C₂

yes

maximal

C₂×#11762531

C₂

yes

maximal, characteristic

C₂²×#63216

C₄

yes

—

#11763697

—

#11763697

—

C₂×C₄×#3622

C₄

yes

characteristic

C₂×C₄×#3622

C₂²

yes

characteristic

#11762531

—

C₂×C₄×C₈×C₁₇

C₄

yes

characteristic

C₂²×C₈×C₁₇

—

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂²×#3622

—

C₂²×#3622

C₂×C₄

yes

—

C₂×#63216

—

C₂×#63216

—

C₂×#63216

C₈

yes

complemented

C₄×C₈×C₁₇

—

C₄×C₈×C₁₇

C₈

yes

—

C₄×#3622

—

C₄×#3622

C₈

yes

—

C₄×#3622

D₄

yes

—

C₄×#3622

D₄

yes

characteristic

C₄×#3622

Q₈

yes

characteristic

C₂²×C₄×C₁₇

C₄²

yes

characteristic

C₂²×C₄×C₁₇

M₁₆

yes

characteristic

C₂²×C₄×C₁₇

M₁₆

yes

characteristic

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

C₂²⋊C₄

yes

characteristic

C₄²×C₁₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₇

C₂×C₈

yes

characteristic

C₄²×C₁₇

C₂×C₈

yes

—

—

C₂²⋊C₄

yes

—

C₂×#3622

C₄⋊C₄

yes

—

C₂×#3622

C₂×C₈

yes

—

C₂×#3622

M₁₆

yes

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#95

yes

characteristic

C₂×C₄×C₁₇

C₄×C₈

yes

—

C₂×C₄×C₁₇

C₄×C₈

yes

characteristic

C₂×C₄×C₁₇

#97

yes

—

C₂×C₄×C₁₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₇

C₄⋊C₈

yes

characteristic

C₂×C₄×C₁₇

C₄⋊C₈

yes

characteristic

C₈×C₁₇

—

#3622

—

#3622

C₂²⋊C₈

yes

—

#3622

C₄⋊C₈

yes

—

C₂×#346

—

C₂×#322

—

C₂²×C₁₇

#322

yes

—

C₂²×C₁₇

#336

yes

characteristic

C₂²×C₁₇

#347

yes

—

C₂²×C₁₇

#347

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₈²

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

C₂²×C₁₆

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

yes

—

—

—

C₂×C₁₇

#1273

yes

characteristic

C₂×C₁₇

#1313

yes

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×#595

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#3624

yes

—

C₁₇

#7137

yes

characteristic, complemented, Sylow

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄²

#63242

yes

characteristic

107

C₄²

C₄⋊#595

yes

characteristic

108

C₄²

C₂×#3624

yes

characteristic

109

C₄²

C₂²⋊#595

yes

characteristic

110

C₁₆

—

111

C₂³

#10587542

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₈×#595

yes

—

115

C₂×C₄

#10587532

yes

—

116

C₂×C₄

C₄×#3624

yes

characteristic

117

C₂×C₄

C₄⋊#3624

yes

characteristic

118

C₂×C₄

#10587543

yes

characteristic

119

C₂×C₄

#10587544

yes

characteristic

120

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

121

C₂×C₄

C₂²⋊#3624

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₂²

#11763651

yes

—

127

C₂²

#11763675

yes

characteristic

128

C₂²

#11763701

yes

—

129

C₂²

#11763702

yes

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

C₈×#3624

yes

characteristic

135

C₄

#11763650

yes

characteristic

136

C₄

#11763674

yes

—

137

C₂

#14543606

yes

characteristic

138

C₂

#14543702

yes

—

139

C₂

#14543825

yes

characteristic

140

C₂

#14543826

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₄×#595
commutator	C₄×C₁₇	C₈²
Frattini	C₂×C₄²	C₂×#595
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	#97
cosocle	C₂×C₄×#3622	C₂²
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#7137
LCS	G ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₄

C₄×#595

commutator

C₄×C₁₇

C₈²

Frattini

C₂×C₄²

C₂×#595

Fitting

C₂×C₄×C₈×C₁₇

C₄

socle, prime radical

C₂³×C₁₇

#97

cosocle

C₂×C₄×#3622

C₂²

C₂×C₄×C₈

C₁₇

G ▷ C₄×C₁₇ ▷ C₁₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
16. C₂×#63216	C₈	semidirect
94. C₁₇	#7137	semidirect

H ≅ G/N

type

16. C₂×#63216

C₈

semidirect

94. C₁₇

#7137

semidirect

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

subgroup	index
C₂×#11763697	2
C₂×#11762531	2
#7137	17

subgroup

index

C₂×#11763697

C₂×#11762531

#7137

Subgroups — #17948490

Conjugacy classes of subgroups

Complements

Maximal subgroups