Group #17948677 — subgroups

Subgroups — #17948677

Order 4352.

210 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 80 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	C₂×#11762531	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#63216	—	2	2	no	—
4	C₂²⋊#63216	C₄	1	4	yes	—
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	C₄×C₈×C₁₇	—	2	1	no	—
16	C₄×C₈×C₁₇	C₈	1	2	yes	—
17	C₄×#3622	—	2	1	no	—
18	C₄×#3622	C₈	1	2	yes	—
19	C₄×#3622	D₄	1	2	yes	—
20	C₄×#3622	D₄	1	1	yes	characteristic
21	C₄×#3622	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
23	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
24	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
25	C₂×C₈×C₁₇	—	2	2	no	—
26	C₂×C₈×C₁₇	—	4	2	no	—
27	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
29	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
30	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
31	C₂×#3622	—	2	4	no	—
32	C₂×#3622	—	2	2	no	—
33	C₂×#3622	—	4	2	no	—
34	C₂×#3622	C₂²⋊C₄	1	2	yes	—
35	C₂×#3622	C₄⋊C₄	1	2	yes	—
36	#63216	—	4	4	no	—
37	#7196	—	17	1	no	maximal, Sylow
38	C₂³×C₁₇	#97	1	1	yes	characteristic
39	C₂×C₄×C₁₇	—	2	2	no	—
40	C₂×C₄×C₁₇	—	2	2	no	—
41	C₂×C₄×C₁₇	—	2	2	no	—
42	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
43	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
47	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
48	C₈×C₁₇	—	4	2	no	—
49	#3622	—	2	4	no	—
50	#3622	—	2	2	no	—
51	#3622	—	2	2	no	—
52	#2163	—	17	2	no	—
53	C₂×#322	—	17	1	no	—
54	C₂²×C₁₇	—	2	1	no	—
55	C₂²×C₁₇	—	2	1	no	—
56	C₂²×C₁₇	#336	1	1	yes	characteristic
57	C₂²×C₁₇	#347	1	2	yes	—
58	C₄×C₁₇	—	2	1	no	—
59	C₄×C₁₇	—	2	1	no	—
60	C₄×C₁₇	—	2	1	no	—
61	C₄×C₁₇	—	2	1	no	—
62	C₄×C₁₇	#344	1	2	yes	—
63	C₄×C₁₇	#364	1	2	yes	—
64	C₂×C₄×C₈	—	17	1	no	—
65	C₂×C₄×C₈	—	17	1	no	—
66	C₂×C₄×C₈	#595	1	1	yes	characteristic
67	C₄⋊C₁₆	—	34	2	no	—
68	C₂²⋊C₁₆	—	17	4	no	—
69	#346	—	34	2	no	—
70	#322	—	34	2	no	—
71	C₂×C₁₇	—	2	2	no	—
72	C₂×C₁₇	#1281	1	1	yes	characteristic
73	C₂×C₁₇	#1375	1	1	yes	characteristic
74	C₂×C₁₇	#1376	1	1	yes	characteristic
75	C₂²×C₈	—	2	1	no	—
76	C₂²×C₈	—	17	2	no	—
77	C₂²×C₈	—	34	1	no	—
78	C₂×C₄²	C₂×#595	1	1	yes	characteristic
79	C₂×C₁₆	—	34	4	no	—
80	C₄×C₈	—	2	1	no	—
81	C₄×C₈	—	17	2	no	—
82	C₄×C₈	—	17	1	no	—
83	C₄×C₈	—	17	1	no	—
84	C₄×C₈	—	17	2	no	—
85	C₄×C₈	—	34	1	no	—
86	C₄×C₈	#3624	1	2	yes	—
87	C₁₇	#7196	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₄	#63243	1	1	yes	characteristic
89	C₂²×C₄	C₄×#595	1	1	yes	characteristic
90	C₂²×C₄	#63247	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	4	2	no	—
93	C₂×C₈	—	17	2	no	—
94	C₂×C₈	—	17	2	no	—
95	C₂×C₈	—	34	4	no	—
96	C₂×C₈	—	34	2	no	—
97	C₂×C₈	—	68	2	no	—
98	C₄²	#63242	1	1	yes	characteristic
99	C₄²	C₄⋊#595	1	1	yes	characteristic
100	C₄²	C₂×#3624	1	1	yes	characteristic
101	C₄²	C₂²⋊#595	1	1	yes	characteristic
102	C₁₆	—	68	4	no	—
103	C₂³	#10587542	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	C₄×#3624	1	1	yes	characteristic
108	C₂×C₄	C₄⋊#3624	1	1	yes	characteristic
109	C₂×C₄	#10587543	1	1	yes	characteristic
110	C₂×C₄	#10587544	1	1	yes	characteristic
111	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
112	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
113	C₈	—	4	2	no	—
114	C₈	—	34	4	no	—
115	C₈	—	34	2	no	—
116	C₈	—	34	2	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	#11763675	1	1	yes	characteristic
120	C₂²	#11763701	1	2	yes	—
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	#11763691	1	2	yes	—
126	C₄	#11763731	1	2	yes	—
127	C₂	—	2	2	no	—
128	C₂	#14543624	1	1	yes	characteristic
129	C₂	#14543825	1	1	yes	characteristic
130	C₂	#14543829	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545145

C₂

yes

maximal

C₂×#11762531

C₂

yes

maximal, characteristic

C₄⋊#63216

—

C₂²⋊#63216

C₄

yes

—

#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

—

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

—

—

—

C₂³×C₁₇

yes

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

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

—

—

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#336

yes

characteristic

C₂²×C₁₇

#347

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#344

yes

—

C₄×C₁₇

#364

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

#595

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#322

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×#595

yes

characteristic

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#3624

yes

—

C₁₇

#7196

yes

characteristic, complemented, Sylow

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

yes

C₄²

yes

100

C₄²

C₂×#3624

yes

characteristic

101

C₄²

C₂²⋊#595

yes

characteristic

102

C₁₆

—

103

C₂³

#10587542

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₄×#3624

yes

characteristic

108

C₂×C₄

C₄⋊#3624

yes

characteristic

109

C₂×C₄

#10587543

yes

characteristic

110

C₂×C₄

#10587544

yes

characteristic

111

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

112

C₂×C₄

C₂²⋊#3624

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₂²

—

118

C₂²

—

119

C₂²

#11763675

yes

characteristic

120

C₂²

#11763701

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#11763691

yes

—

126

C₄

#11763731

yes

—

127

C₂

—

128

C₂

#14543624

yes

characteristic

129

C₂

#14543825

yes

characteristic

130

C₂

#14543829

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×#595)⋊C₄
commutator	C₂×C₄×C₁₇	C₄×C₈
Frattini	C₂×C₄²	C₂×#595
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle	C₂²×C₁₇	#336
prime radical	C₂³×C₁₇	#97
cosocle	C₂×C₄×#3622	C₂²
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#7196
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

(C₂×#595)⋊C₄

commutator

C₂×C₄×C₁₇

C₄×C₈

Frattini

C₂×C₄²

C₂×#595

Fitting

C₂×C₄×C₈×C₁₇

C₄

C₂²×C₁₇

C₂³×C₁₇

C₂×C₄×#3622

C₂²

C₂×C₄×C₈

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
87. C₁₇	#7196	semidirect

H ≅ G/N

type

87. C₁₇

#7196

semidirect

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

subgroup	index
#14545145	2
C₂×#11762531	2
#7196	17

subgroup

index

#14545145

C₂×#11762531

#7196

Subgroups — #17948677

Conjugacy classes of subgroups

Complements

Maximal subgroups