Group #17960672 — subgroups

Subgroups — #17960672

Order 4352.

354 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 156 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	C₄×#10587538	C₂	1	2	yes	maximal
2	#14543826	C₂	1	4	yes	maximal
3	C₄×#10587486	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587538	C₂²	1	4	yes	—
5	#11763697	C₄	1	4	yes	complemented
6	C₄×#63216	C₄	1	4	yes	complemented
7	C₂×#10587486	C₄	1	2	yes	—
8	C₂×#10587486	C₂²	1	2	yes	—
9	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
10	C₄²×Dic₁₇	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
12	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₁₇	—	2	2	no	—
15	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
16	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
17	C₂²×#3622	C₂³	1	1	yes	characteristic
18	#10587538	—	2	8	no	—
19	C₂×#63216	C₂×C₄	1	8	yes	—
20	C₄×C₈×C₁₇	—	2	2	no	—
21	#10587486	—	2	4	no	—
22	#10587486	C₈	1	8	yes	—
23	C₄×#3622	C₂×C₄	1	2	yes	—
24	C₄×#3622	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₇	M₁₆	1	2	yes	—
26	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
27	C₂²×Dic₁₇	—	2	2	no	—
28	C₂×C₈×C₁₇	—	2	2	no	—
29	C₂×C₈×C₁₇	—	2	2	no	—
30	C₂×C₈×C₁₇	C₂×C₈	1	4	yes	—
31	C₄²×C₁₇	C₄²	1	1	yes	characteristic
32	C₄²×C₁₇	C₄²	1	1	yes	characteristic
33	C₄²×C₁₇	M₁₆	1	2	yes	—
34	C₄×Dic₁₇	—	2	2	no	—
35	C₄×Dic₁₇	—	4	4	no	—
36	C₄×Dic₁₇	C₂×C₈	1	4	yes	—
37	C₂×#3622	C₄²	1	2	yes	—
38	C₂×#3622	C₂×C₈	1	4	yes	—
39	C₂×#3622	C₂²×C₄	1	2	yes	—
40	C₂×#3622	#41	1	4	yes	—
41	#63216	—	2	8	no	—
42	#12594	—	17	1	no	maximal, Sylow
43	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
44	C₂×C₄×C₁₇	—	2	4	no	—
45	C₂×C₄×C₁₇	#97	1	2	yes	—
46	C₂×C₄×C₁₇	#97	1	2	yes	—
47	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₁₇	#117	1	2	yes	—
49	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
50	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
51	C₂×Dic₁₇	—	2	4	no	—
52	C₂×Dic₁₇	—	4	4	no	—
53	C₈×C₁₇	—	2	4	no	—
54	#3622	C₄×C₈	1	4	yes	—
55	#3622	#117	1	4	yes	—
56	C₄×M₃₂	—	17	2	no	—
57	C₄×#97	—	17	1	no	—
58	#1375	—	17	4	no	—
59	C₂²×C₁₇	—	2	2	no	—
60	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
61	C₂²×C₁₇	#431	1	2	yes	—
62	C₄×C₁₇	—	2	2	no	—
63	C₄×C₁₇	#347	1	4	yes	—
64	C₄×C₁₇	#376	1	1	yes	characteristic
65	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
66	C₄×C₁₇	#431	1	2	yes	—
67	Dic₁₇	—	4	4	no	—
68	C₂×M₃₂	—	17	4	no	—
69	C₂×#97	—	17	2	no	—
70	C₂×#97	—	17	2	no	—
71	C₂×C₄×C₈	—	17	1	no	—
72	C₂×C₄×C₈	#595	1	1	yes	characteristic
73	C₄³	—	17	1	no	—
74	#346	—	17	4	no	—
75	C₄×C₁₆	—	17	4	no	—
76	C₂×C₁₇	—	2	2	no	—
77	C₂×C₁₇	#1744	1	1	yes	characteristic
78	C₂×C₁₇	C₂×#347	1	2	yes	—
79	C₂²×C₈	—	17	1	no	—
80	C₂²×C₈	—	17	1	no	—
81	C₂²×C₈	C₂×#595	1	2	yes	—
82	C₂×C₄²	—	17	2	no	—
83	C₂×C₄²	—	34	2	no	—
84	C₂×C₄²	C₂×#595	1	1	yes	characteristic
85	M₃₂	—	34	8	no	—
86	C₂×C₁₆	—	17	8	no	—
87	#97	—	17	8	no	—
88	#97	—	34	4	no	—
89	C₄×C₈	—	2	2	no	—
90	C₄×C₈	—	17	2	no	—
91	C₄×C₈	—	17	2	no	—
92	C₁₇	#12594	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₄	—	34	2	no	—
94	C₂²×C₄	#63243	1	2	yes	—
95	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	17	4	no	—
99	C₂×C₈	—	17	2	no	—
100	C₂×C₈	—	17	2	no	—
101	C₂×C₈	—	17	4	no	—
102	C₂×C₈	#63242	1	4	yes	—
103	C₄²	—	17	4	no	—
104	C₄²	—	34	2	no	—
105	C₄²	—	68	4	no	—
106	C₄²	C₄×#595	1	1	yes	characteristic
107	C₄²	C₄×#595	1	1	yes	characteristic
108	C₄²	#63247	1	2	yes	—
109	C₁₆	—	34	8	no	—
110	C₂³	C₂×#63243	1	1	yes	characteristic
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	34	4	no	—
113	C₂×C₄	—	68	4	no	—
114	C₂×C₄	#10587542	1	2	yes	—
115	C₂×C₄	#10587542	1	2	yes	—
116	C₂×C₄	C₂×#63242	1	2	yes	—
117	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
118	C₂×C₄	#10587677	1	2	yes	—
119	C₂×C₄	C₂×#63247	1	1	yes	characteristic
120	C₈	—	2	4	no	—
121	C₈	—	17	4	no	—
122	C₈	—	17	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	C₂×#10587542	1	1	yes	characteristic
125	C₂²	#11763828	1	2	yes	—
126	C₄	—	2	2	no	—
127	C₄	—	68	4	no	—
128	C₄	#11763702	1	4	yes	—
129	C₄	#11763745	1	1	yes	characteristic
130	C₄	C₄×#63242	1	1	yes	characteristic
131	C₄	#11763826	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	#14544499	1	1	yes	characteristic
134	C₂	C₂×#11763702	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#10587538

C₂

yes

maximal

#14543826

C₂

yes

maximal

C₄×#10587486

C₂

yes

maximal, characteristic

C₂×#10587538

C₂²

yes

—

#11763697

C₄

yes

complemented

C₄×#63216

C₄

yes

complemented

C₂×#10587486

C₄

yes

—

C₂×#10587486

C₂²

yes

—

C₂×C₄×#3622

C₂²

yes

characteristic

C₄²×Dic₁₇

C₄

yes

characteristic

C₂×C₄×C₈×C₁₇

C₄

yes

characteristic

C₂²×C₈×C₁₇

C₂×C₄

yes

—

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₇

—

C₂×C₄×Dic₁₇

C₂×C₄

yes

—

C₂²×#3622

C₂×C₄

yes

characteristic

C₂²×#3622

C₂³

yes

characteristic

#10587538

—

C₂×#63216

C₂×C₄

yes

—

C₄×C₈×C₁₇

—

#10587486

—

#10587486

C₈

yes

—

C₄×#3622

C₂×C₄

yes

—

C₄×#3622

C₂×C₄

yes

—

C₂²×C₄×C₁₇

M₁₆

yes

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂²×Dic₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×C₈

yes

—

C₄²×C₁₇

C₄²

yes

characteristic

C₄²×C₁₇

C₄²

yes

characteristic

C₄²×C₁₇

M₁₆

yes

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

C₂×C₈

yes

—

C₂×#3622

C₄²

yes

—

C₂×#3622

C₂×C₈

yes

—

C₂×#3622

C₂²×C₄

yes

—

C₂×#3622

#41

yes

—

#63216

—

#12594

—

maximal, Sylow

C₂³×C₁₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#97

yes

—

C₂×C₄×C₁₇

#97

yes

—

C₂×C₄×C₁₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₁₇

#117

yes

—

C₂×C₄×C₁₇

C₂²×C₈

yes

—

C₂×C₄×C₁₇

C₂×M₁₆

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

#3622

C₄×C₈

yes

—

#3622

#117

yes

—

C₄×M₃₂

—

C₄×#97

—

#1375

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×#97

yes

characteristic

C₂²×C₁₇

#431

yes

—

C₄×C₁₇

—

C₄×C₁₇

#347

yes

—

C₄×C₁₇

#376

yes

characteristic

C₄×C₁₇

C₂×C₄×C₈

yes

characteristic

C₄×C₁₇

#431

yes

—

Dic₁₇

—

C₂×M₃₂

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

#595

yes

characteristic

C₄³

—

#346

—

C₄×C₁₆

—

C₂×C₁₇

—

C₂×C₁₇

#1744

yes

characteristic

C₂×C₁₇

C₂×#347

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×#595

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×#595

yes

characteristic

M₃₂

—

C₂×C₁₆

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₁₇

#12594

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

#63243

yes

—

C₂²×C₄

C₂²×#595

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

#63242

yes

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

C₄×#595

yes

characteristic

107

C₄²

C₄×#595

yes

characteristic

108

C₄²

#63247

yes

—

109

C₁₆

—

110

C₂³

C₂×#63243

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

#10587542

yes

—

115

C₂×C₄

#10587542

yes

—

116

C₂×C₄

C₂×#63242

yes

—

117

C₂×C₄

C₂×C₄×#595

yes

characteristic

118

C₂×C₄

#10587677

yes

—

119

C₂×C₄

C₂×#63247

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₂²

—

124

C₂²

C₂×#10587542

yes

characteristic

125

C₂²

#11763828

yes

—

126

C₄

—

127

C₄

—

128

C₄

#11763702

yes

—

129

C₄

#11763745

yes

characteristic

130

C₄

C₄×#63242

yes

characteristic

131

C₄

#11763826

yes

—

132

C₂

—

133

C₂

#14544499

yes

characteristic

134

C₂

C₂×#11763702

yes

—

	subgroup	G/H
center	C₄²	C₄×#595
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle	C₂²×C₁₇	C₂×#97
prime radical	C₂³×C₁₇	C₂×M₁₆
cosocle	C₂²×#3622	C₂³
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#12594
LCS	G ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₄²

C₄×#595

commutator

C₄×C₁₇

C₂×C₄×C₈

Frattini

C₂²×C₄

C₂²×#595

Fitting

C₂×C₄×C₈×C₁₇

C₄

socle

C₂²×C₁₇

C₂×#97

prime radical

C₂³×C₁₇

C₂×M₁₆

cosocle

C₂²×#3622

C₂³

C₂×C₄×C₈

C₁₇

G ▷ C₄×C₁₇ ▷ C₁₇

C₁ ◁ C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
5. #11763697	C₄	semidirect
6. C₄×#63216	C₄	semidirect
92. C₁₇	#12594	semidirect

H ≅ G/N

type

5. #11763697

C₄

semidirect

6. C₄×#63216

C₄

semidirect

92. C₁₇

#12594

semidirect

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

subgroup	index
C₄×#10587538	2
#14543826	2
C₄×#10587486	2
#12594	17

subgroup

index

Subgroups — #17960672

Conjugacy classes of subgroups

Complements

Maximal subgroups