Group #17961626 — subgroups

Subgroups — #17961626

Order 4352.

302 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 120 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	#14545116	C₂	1	2	yes	maximal
2	#14543828	C₂	1	4	yes	maximal
3	#14528471	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587538	C₂²	1	4	yes	—
5	C₄⋊#63216	—	2	4	no	—
6	C₂×#10587474	C₄	1	2	yes	—
7	C₂×#10587474	C₄	1	1	yes	characteristic
8	C₂×C₄×#3622	C₂²	1	2	yes	—
9	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
10	C₁₇×#376	C₄	1	1	yes	characteristic
11	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
12	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
13	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
14	C₂²×#3622	C₂×C₄	1	2	yes	—
15	C₂²×#3622	C₂³	1	1	yes	characteristic
16	#10587538	—	2	8	no	—
17	C₂×#63216	—	2	4	no	—
18	C₁₇×#95	—	2	2	no	—
19	#10587474	—	2	4	no	—
20	#10587474	C₈	1	4	yes	—
21	C₄×#3622	—	2	4	no	—
22	C₄×#3622	D₄	1	2	yes	—
23	C₄×#3622	Q₈	1	2	yes	—
24	C₂²×C₄×C₁₇	—	2	2	no	—
25	C₂²×C₄×C₁₇	M₁₆	1	2	yes	—
26	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
27	C₄²×C₁₇	—	2	2	no	—
28	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
29	C₄²×C₁₇	C₂×C₈	1	2	yes	—
30	C₄²×C₁₇	M₁₆	1	4	yes	—
31	C₂×#3622	—	2	2	no	—
32	C₂×#3622	—	2	2	no	—
33	C₂×#3622	—	2	2	no	—
34	C₂×#3622	C₄⋊C₄	1	2	yes	—
35	C₂×#3622	C₂×C₈	1	4	yes	—
36	C₂×#3622	C₂×D₄	1	1	yes	characteristic
37	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
38	C₂×#3622	#41	1	4	yes	—
39	#63216	—	4	4	no	—
40	#12884	—	17	1	no	maximal, Sylow
41	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
42	C₂×C₄×C₁₇	—	2	4	no	—
43	C₂×C₄×C₁₇	—	4	2	no	—
44	C₂×C₄×C₁₇	—	4	4	no	—
45	C₂×C₄×C₁₇	C₄⋊C₈	1	4	yes	—
46	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇	#117	1	2	yes	—
48	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
50	#3622	—	2	2	no	—
51	#3622	—	2	4	no	—
52	#3622	#124	1	2	yes	—
53	#3622	#125	1	2	yes	—
54	#2150	—	17	2	no	—
55	#1726	—	17	1	no	—
56	#1376	—	17	4	no	—
57	C₂²×C₁₇	—	2	2	no	—
58	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
59	C₂²×C₁₇	#432	1	2	yes	—
60	C₄×C₁₇	—	2	4	no	—
61	C₄×C₁₇	—	4	4	no	—
62	C₄×C₁₇	#382	1	1	yes	characteristic
63	C₄×C₁₇	#423	1	1	yes	characteristic
64	C₄×C₁₇	#431	1	2	yes	—
65	C₂×M₃₂	—	17	4	no	—
66	C₂×#97	—	17	1	no	—
67	C₂×#97	—	17	2	no	—
68	C₂×C₄×C₈	—	17	1	no	—
69	C₂×C₄×C₈	—	17	2	no	—
70	#376	#595	1	1	yes	characteristic
71	C₄⋊C₁₆	—	34	4	no	—
72	C₂×C₁₇	—	2	2	no	—
73	C₂×C₁₇	#1831	1	1	yes	characteristic
74	C₂×C₁₇	#2153	1	2	yes	—
75	C₂²×C₈	—	17	1	no	—
76	C₂²×C₈	—	17	2	no	—
77	C₂²×C₈	—	17	1	no	—
78	C₂×C₄²	C₂×#595	1	2	yes	—
79	C₂×C₄²	C₂×#595	1	1	yes	characteristic
80	M₃₂	—	34	8	no	—
81	C₂×C₁₆	—	34	4	no	—
82	#97	—	17	4	no	—
83	#97	—	34	4	no	—
84	C₄×C₈	—	17	2	no	—
85	C₄×C₈	—	17	2	no	—
86	C₄×C₈	—	34	4	no	—
87	#95	—	2	2	no	—
88	C₁₇	#12884	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	#63247	1	2	yes	—
91	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
92	C₂×C₈	—	17	1	no	—
93	C₂×C₈	—	17	1	no	—
94	C₂×C₈	—	17	4	no	—
95	C₂×C₈	—	17	4	no	—
96	C₂×C₈	—	17	2	no	—
97	C₂×C₈	—	34	2	no	—
98	C₂×C₈	—	34	2	no	—
99	C₂×C₈	—	34	2	no	—
100	C₄²	—	2	2	no	—
101	C₄²	#63242	1	2	yes	—
102	C₄²	#63243	1	4	yes	—
103	C₄²	C₄⋊#595	1	2	yes	—
104	C₁₆	—	68	4	no	—
105	C₂³	C₂×#63247	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	#10587544	1	4	yes	—
110	C₂×C₄	C₂×#63242	1	1	yes	characteristic
111	C₂×C₄	C₂×#63243	1	2	yes	—
112	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
113	C₂×C₄	#10587677	1	2	yes	—
114	C₈	—	17	2	no	—
115	C₈	—	17	2	no	—
116	C₈	—	34	2	no	—
117	C₈	—	34	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	C₂×#10587544	1	1	yes	characteristic
120	C₂²	#11763830	1	2	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	4	4	no	—
123	C₄	#11763753	1	1	yes	characteristic
124	C₄	#11763813	1	1	yes	characteristic
125	C₄	#11763828	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	#14544701	1	1	yes	characteristic
128	C₂	#14545127	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14545116

C₂

yes

maximal

#14543828

C₂

yes

maximal

#14528471

C₂

yes

maximal, characteristic

C₂×#10587538

C₂²

yes

—

C₄⋊#63216

—

C₂×#10587474

C₄

yes

—

C₂×#10587474

C₄

yes

characteristic

C₂×C₄×#3622

C₂²

yes

—

C₂×C₄×#3622

C₂²

yes

characteristic

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

—

C₄×#3622

D₄

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

C₂×C₈

yes

—

C₄²×C₁₇

M₁₆

yes

—

—

—

—

C₄⋊C₄

yes

—

C₂×#3622

C₂×C₈

yes

—

C₂×#3622

C₂×D₄

yes

characteristic

C₂×#3622

C₂×Q₈

yes

characteristic

C₂×#3622

#41

yes

—

#63216

—

#12884

—

maximal, Sylow

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

characteristic

C₂×C₄×C₁₇

#117

yes

—

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₇

C₂×M₁₆

yes

—

—

—

yes

—

#3622

#125

yes

—

#2150

—

#1726

—

#1376

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₁₇

#432

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

C₂×M₃₂

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#376

#595

yes

characteristic

C₄⋊C₁₆

—

C₂×C₁₇

—

C₂×C₁₇

#1831

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

—

#95

—

C₁₇

#12884

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

#63247

yes

—

C₂²×C₄

C₂²×#595

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₄²

—

101

C₄²

#63242

yes

—

102

C₄²

#63243

yes

—

103

C₄²

C₄⋊#595

yes

—

104

C₁₆

—

105

C₂³

C₂×#63247

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#10587544

yes

—

110

C₂×C₄

C₂×#63242

yes

characteristic

111

C₂×C₄

C₂×#63243

yes

—

112

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

113

C₂×C₄

#10587677

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₂²

—

119

C₂²

C₂×#10587544

yes

characteristic

120

C₂²

#11763830

yes

—

121

C₄

—

122

C₄

—

123

C₄

#11763753

yes

characteristic

124

C₄

#11763813

yes

characteristic

125

C₄

#11763828

yes

—

126

C₂

—

127

C₂

#14544701

yes

characteristic

128

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	#12884
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
88. C₁₇	#12884	semidirect

H ≅ G/N

type

88. C₁₇

#12884

semidirect

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

subgroup	index
#14545116	2
#14543828	2
#14528471	2
#12884	17

subgroup

index

Subgroups — #17961626

Conjugacy classes of subgroups

Complements

Maximal subgroups