Group #17961663 — subgroups

Subgroups — #17961663

Order 4352.

342 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 144 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×#11763732	C₂	1	4	yes	maximal
2	#14545116	C₂	1	2	yes	maximal
3	#14528471	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587538	C₂²	1	4	yes	—
5	#11763732	—	2	8	no	—
6	C₄⋊#63216	—	2	4	no	—
7	C₂×#10587474	C₄	1	2	yes	—
8	C₂×#10587474	C₄	1	1	yes	characteristic
9	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
10	C₂×C₄×#3622	C₂²	1	2	yes	—
11	C₁₇×#376	C₄	1	1	yes	characteristic
12	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
15	C₂²×#3622	C₂×C₄	1	2	yes	—
16	C₂²×#3622	C₂³	1	1	yes	characteristic
17	#10587538	—	2	8	no	—
18	C₂×#63216	—	2	4	no	—
19	C₁₇×#95	—	2	2	no	—
20	#10587474	—	2	4	no	—
21	#10587474	C₈	1	4	yes	—
22	C₄×#3622	D₄	1	2	yes	—
23	C₄×#3622	D₄	1	4	yes	—
24	C₄×#3622	Q₈	1	4	yes	—
25	C₄×#3622	Q₈	1	2	yes	—
26	C₂²×C₄×C₁₇	—	2	2	no	—
27	C₂²×C₄×C₁₇	M₁₆	1	2	yes	—
28	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
29	C₄²×C₁₇	—	2	2	no	—
30	C₄²×C₁₇	C₄⋊C₄	1	4	yes	—
31	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
32	C₄²×C₁₇	C₂×C₈	1	2	yes	—
33	C₂×#3622	—	2	2	no	—
34	C₂×#3622	C₄⋊C₄	1	2	yes	—
35	C₂×#3622	C₄⋊C₄	1	4	yes	—
36	C₂×#3622	C₂×C₈	1	4	yes	—
37	C₂×#3622	M₁₆	1	4	yes	—
38	C₂×#3622	C₂×D₄	1	1	yes	characteristic
39	C₂×#3622	C₂×D₄	1	2	yes	—
40	C₂×#3622	C₂×Q₈	1	2	yes	—
41	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
42	#63216	—	4	4	no	—
43	#12901	—	17	1	no	maximal, Sylow
44	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
45	C₂×C₄×C₁₇	—	2	4	no	—
46	C₂×C₄×C₁₇	—	4	2	no	—
47	C₂×C₄×C₁₇	—	4	4	no	—
48	C₂×C₄×C₁₇	C₄⋊C₈	1	4	yes	—
49	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
50	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
53	#3622	—	2	2	no	—
54	#3622	C₄⋊C₈	1	8	yes	—
55	#3622	#127	1	1	yes	characteristic
56	#3622	C₄⋊Q₈	1	1	yes	characteristic
57	#3622	C₄⋊Q₈	1	2	yes	—
58	C₂×#364	—	17	4	no	—
59	#2150	—	17	2	no	—
60	#1726	—	17	1	no	—
61	C₂²×C₁₇	—	2	2	no	—
62	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
63	C₂²×C₁₇	#423	1	2	yes	—
64	C₄×C₁₇	—	2	4	no	—
65	C₄×C₁₇	—	4	4	no	—
66	C₄×C₁₇	#384	1	1	yes	characteristic
67	C₄×C₁₇	C₂×C₄⋊C₈	1	2	yes	—
68	C₄×C₁₇	#423	1	1	yes	characteristic
69	C₂×M₃₂	—	17	4	no	—
70	C₂×#97	—	17	1	no	—
71	C₂×#97	—	17	2	no	—
72	C₂×C₄×C₈	—	17	2	no	—
73	C₂×C₄×C₈	—	17	1	no	—
74	#376	#595	1	1	yes	characteristic
75	#364	—	34	8	no	—
76	C₄⋊C₁₆	—	34	4	no	—
77	C₂×C₁₇	—	2	2	no	—
78	C₂×C₁₇	#1842	1	1	yes	characteristic
79	C₂×C₁₇	#2153	1	2	yes	—
80	C₂²×C₈	—	17	1	no	—
81	C₂²×C₈	—	17	2	no	—
82	C₂²×C₈	—	17	1	no	—
83	C₂×C₄²	C₂×#595	1	2	yes	—
84	C₂×C₄²	C₂×#595	1	1	yes	characteristic
85	M₃₂	—	34	8	no	—
86	C₂×C₁₆	—	34	4	no	—
87	#97	—	17	4	no	—
88	#97	—	34	4	no	—
89	C₄×C₈	—	17	2	no	—
90	C₄×C₈	—	17	4	no	—
91	C₄×C₈	—	17	4	no	—
92	C₄×C₈	—	17	2	no	—
93	#95	—	2	2	no	—
94	C₁₇	#12901	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	#63247	1	2	yes	—
97	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
98	C₂×C₈	—	17	4	no	—
99	C₂×C₈	—	17	4	no	—
100	C₂×C₈	—	17	2	no	—
101	C₂×C₈	—	17	4	no	—
102	C₂×C₈	—	17	2	no	—
103	C₂×C₈	—	17	1	no	—
104	C₂×C₈	—	17	1	no	—
105	C₂×C₈	—	17	2	no	—
106	C₂×C₈	—	34	2	no	—
107	C₄²	—	2	2	no	—
108	C₄²	#63242	1	2	yes	—
109	C₄²	C₄⋊#595	1	4	yes	—
110	C₄²	C₄⋊#595	1	2	yes	—
111	C₁₆	—	68	4	no	—
112	C₂³	C₂×#63247	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	#10587544	1	4	yes	—
117	C₂×C₄	C₂×#63242	1	1	yes	characteristic
118	C₂×C₄	D₁₇⋊M₁₆	1	2	yes	—
119	C₂×C₄	C₂×C₄⋊#595	1	2	yes	—
120	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
121	C₈	—	17	1	no	—
122	C₈	—	17	2	no	—
123	C₈	—	17	1	no	—
124	C₈	—	17	8	no	—
125	C₈	—	34	2	no	—
126	C₂²	—	2	2	no	—
127	C₂²	C₂×#10587544	1	1	yes	characteristic
128	C₂²	#11763812	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	—	4	4	no	—
131	C₄	#11763755	1	1	yes	characteristic
132	C₄	#11763810	1	2	yes	—
133	C₄	#11763814	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	#14544717	1	1	yes	characteristic
136	C₂	#14545127	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763732

C₂

yes

maximal

#14545116

C₂

yes

maximal

#14528471

C₂

yes

maximal, characteristic

C₂×#10587538

C₂²

yes

—

#11763732

—

C₄⋊#63216

—

C₂×#10587474

C₄

yes

—

C₂×#10587474

C₄

yes

characteristic

C₂×C₄×#3622

C₂²

yes

characteristic

C₂×C₄×#3622

C₂²

yes

—

C₁₇×#376

C₄

yes

characteristic

C₂×C₄²×C₁₇

C₂×C₄

yes

—

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂²×#3622

C₂×C₄

yes

characteristic

C₂²×#3622

C₂×C₄

yes

—

C₂²×#3622

C₂³

yes

characteristic

#10587538

—

C₂×#63216

—

C₁₇×#95

—

#10587474

—

#10587474

C₈

yes

—

C₄×#3622

D₄

yes

—

C₄×#3622

D₄

yes

—

C₄×#3622

Q₈

yes

—

C₄×#3622

Q₈

yes

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

M₁₆

yes

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

—

C₄²×C₁₇

C₄⋊C₄

yes

—

C₄²×C₁₇

C₄⋊C₄

yes

—

C₄²×C₁₇

C₂×C₈

yes

—

C₂×#3622

—

C₂×#3622

C₄⋊C₄

yes

—

C₂×#3622

C₄⋊C₄

yes

—

C₂×#3622

C₂×C₈

yes

—

C₂×#3622

M₁₆

yes

—

C₂×#3622

C₂×D₄

yes

characteristic

C₂×#3622

C₂×D₄

yes

—

C₂×#3622

C₂×Q₈

yes

—

C₂×#3622

C₂×Q₈

yes

—

—

C₂³×C₁₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄⋊C₈

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₇

C₂×M₁₆

yes

—

#3622

—

#3622

C₄⋊C₈

yes

—

yes

C₄⋊Q₈

yes

characteristic

#3622

C₄⋊Q₈

yes

—

C₂×#364

—

#2150

—

#1726

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₁₇

#423

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#384

yes

characteristic

C₄×C₁₇

C₂×C₄⋊C₈

yes

—

C₄×C₁₇

#423

yes

characteristic

C₂×M₃₂

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

yes

—

C₄⋊C₁₆

—

C₂×C₁₇

—

C₂×C₁₇

#1842

yes

characteristic

C₂×C₁₇

#2153

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×#595

yes

—

C₂×C₄²

C₂×#595

yes

characteristic

M₃₂

—

C₂×C₁₆

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

#95

—

C₁₇

#12901

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

#63247

yes

—

C₂²×C₄

C₂²×#595

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₄²

—

108

C₄²

#63242

yes

—

109

C₄²

C₄⋊#595

yes

—

110

C₄²

C₄⋊#595

yes

—

111

C₁₆

—

112

C₂³

C₂×#63247

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

#10587544

yes

—

117

C₂×C₄

C₂×#63242

yes

characteristic

118

C₂×C₄

D₁₇⋊M₁₆

yes

—

119

C₂×C₄

C₂×C₄⋊#595

yes

—

120

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₂²

—

127

C₂²

C₂×#10587544

yes

characteristic

128

C₂²

#11763812

yes

—

129

C₄

—

130

C₄

—

131

C₄

#11763755

yes

characteristic

132

C₄

#11763810

yes

—

133

C₄

#11763814

yes

characteristic

134

C₂

—

135

C₂

#14544717

yes

characteristic

136

C₂

#14545127

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#595
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₁₇×#376	C₄
socle	C₂²×C₁₇	C₂×C₄⋊C₈
prime radical	C₂³×C₁₇	C₂×M₁₆
cosocle	C₂²×#3622	C₂³
hypercenter	#376	#595
hypocenter	C₁₇	#12901
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×C₄⋊#595

commutator

C₂×C₄×C₁₇

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#595

Fitting

C₁₇×#376

C₄

socle

C₂²×C₁₇

C₂×C₄⋊C₈

prime radical

C₂³×C₁₇

C₂×M₁₆

cosocle

C₂²×#3622

C₂³

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
94. C₁₇	#12901	semidirect

H ≅ G/N

type

94. C₁₇

#12901

semidirect

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

subgroup	index
C₂×#11763732	2
#14545116	2
#14528471	2
#12901	17

subgroup

index

Subgroups — #17961663

Conjugacy classes of subgroups

Complements

Maximal subgroups