Group #16955288 — subgroups

Subgroups — #16955288

Order 4352.

160 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 55 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	#14530892	C₂	1	1	yes	maximal, characteristic
2	#14526991	C₂	1	2	yes	maximal, complemented
3	#14526976	C₂	1	2	yes	maximal, complemented
4	#14526924	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#2161	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₃₆	C₂²	1	1	yes	characteristic
7	#11763050	—	2	1	no	—
8	#11762643	C₂²	1	2	yes	—
9	#11762637	C₂²	1	1	yes	characteristic
10	#11762614	—	2	1	no	—
11	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₁₇×#108	—	2	1	no	—
13	C₁₇×#368	—	2	1	no	—
14	C₁₇×#368	C₂²	1	2	yes	—
15	C₂×M₁₆×C₁₇	—	4	1	no	—
16	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
17	#10587610	—	4	1	no	—
18	C₂×Dic₆₈	—	2	1	no	—
19	C₁₇×M₃₂	—	2	2	no	—
20	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
22	C₁₇×#108	—	2	2	no	—
23	C₁₇×#108	—	4	1	no	—
24	#10587489	—	2	1	no	—
25	#10587487	—	4	1	no	—
26	Dic₁₃₆	—	2	2	no	—
27	#10587465	—	2	2	no	—
28	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
29	C₂×Dic₃₄	—	4	1	no	—
30	M₁₆×C₁₇	—	4	2	no	—
31	M₁₆×C₁₇	—	8	1	no	—
32	C₂×C₈×C₁₇	—	2	1	no	—
33	C₂×C₈×C₁₇	—	2	1	no	—
34	C₂×C₈×C₁₇	—	8	1	no	—
35	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
36	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
37	C₂²⋊Dic₁₇	—	8	1	no	—
38	C₄⋊Dic₁₇	—	4	1	no	—
39	#63225	—	8	1	no	—
40	Dic₆₈	—	4	1	no	—
41	Dic₆₈	—	4	1	no	—
42	C₁₆×C₁₇	—	2	1	no	—
43	C₁₆×C₁₇	#41	1	2	yes	—
44	#13718	—	17	1	no	maximal, Sylow
45	C₂³×C₁₇	SD₃₂	1	1	yes	characteristic
46	C₂×C₄×C₁₇	—	2	1	no	—
47	C₂×C₄×C₁₇	—	2	1	no	—
48	C₂×C₄×C₁₇	SD₃₂	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
50	C₂×Dic₁₇	—	8	1	no	—
51	C₂×Dic₁₇	—	8	1	no	—
52	Dic₃₄	—	8	1	no	—
53	Dic₃₄	—	8	1	no	—
54	C₈×C₁₇	—	2	1	no	—
55	C₈×C₁₇	—	8	2	no	—
56	C₈×C₁₇	#123	1	1	yes	characteristic
57	C₈×C₁₇	#137	1	1	yes	characteristic
58	#2219	—	17	1	no	—
59	#2161	D₁₇	1	1	yes	characteristic
60	#1426	—	17	2	no	—
61	#1420	—	17	2	no	—
62	#1400	—	17	1	no	—
63	C₂²×C₁₇	—	2	1	no	—
64	C₂²×C₁₇	—	4	1	no	—
65	C₂²×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
66	C₄×C₁₇	—	2	1	no	—
67	C₄×C₁₇	#480	1	1	yes	characteristic
68	C₄×C₁₇	#509	1	1	yes	characteristic
69	Dic₁₇	—	16	1	no	—
70	Dic₁₇	—	16	1	no	—
71	C₂×Q₃₂	—	17	1	no	—
72	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
73	#467	—	34	1	no	—
74	C₂×#108	—	2	1	no	—
75	M₆₄	—	17	2	no	—
76	#368	—	2	1	no	—
77	#368	C₂×D₁₇	1	2	yes	—
78	#367	—	17	1	no	—
79	#358	—	34	1	no	—
80	C₂×C₁₇	—	2	1	no	—
81	C₂×C₁₇	—	4	1	no	—
82	C₂×C₁₇	#2233	1	1	yes	characteristic
83	C₂×Q₁₆	—	34	1	no	—
84	C₂×M₁₆	—	4	1	no	—
85	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
86	C₂²⋊Q₈	—	68	1	no	—
87	Q₃₂	—	34	2	no	—
88	M₃₂	—	2	2	no	—
89	C₂×C₁₆	C₂²⋊D₁₇	1	1	yes	characteristic
90	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
91	#108	—	2	2	no	—
92	#108	—	4	1	no	—
93	#107	—	34	1	no	—
94	#103	—	68	1	no	—
95	C₃₂	—	34	2	no	—
96	C₁₇	#13718	1	1	yes	characteristic, complemented, Sylow
97	C₂×Q₈	—	68	1	no	—
98	C₂²×C₄	#63228	1	1	yes	characteristic
99	Q₁₆	—	68	1	no	—
100	Q₁₆	—	68	1	no	—
101	M₁₆	—	4	2	no	—
102	M₁₆	—	8	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	8	1	no	—
106	C₂×C₈	#63228	1	1	yes	characteristic
107	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
108	C₄⋊C₄	—	68	1	no	—
109	C₄⋊C₄	—	136	1	no	—
110	C₂²⋊C₄	—	136	1	no	—
111	C₁₆	—	2	1	no	—
112	C₁₆	#63252	1	2	yes	—
113	C₂³	#10587499	1	1	yes	characteristic
114	Q₈	—	136	1	no	—
115	Q₈	—	136	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	136	1	no	—
119	C₂×C₄	—	136	1	no	—
120	C₂×C₄	#10587499	1	1	yes	characteristic
121	C₂×C₄	C₂×#63228	1	1	yes	characteristic
122	C₈	—	2	1	no	—
123	C₈	—	8	2	no	—
124	C₈	#10587615	1	1	yes	characteristic
125	C₈	#10587628	1	1	yes	characteristic
126	C₂²	—	2	1	no	—
127	C₂²	—	4	1	no	—
128	C₂²	C₂×#10587499	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	272	1	no	—
131	C₄	—	272	1	no	—
132	C₄	#11763122	1	1	yes	characteristic
133	C₄	#11763245	1	1	yes	characteristic
134	C₂	—	2	1	no	—
135	C₂	—	4	1	no	—
136	C₂	#14530969	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14530892

C₂

yes

maximal, characteristic

#14526991

C₂

yes

maximal, complemented

#14526976

C₂

yes

maximal, complemented

#14526924

C₂

yes

maximal, characteristic

C₁₇×#2161

C₂

yes

maximal, characteristic

C₂×Dic₁₃₆

C₂²

yes

characteristic

#11763050

—

#11762643

C₂²

yes

—

#11762637

C₂²

yes

characteristic

#11762614

—

C₂×C₁₇×M₃₂

C₂²

yes

characteristic

C₂×C₁₇×#108

—

C₁₇×#368

—

C₁₇×#368

C₂²

yes

—

C₂×M₁₆×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

characteristic

#10587610

—

C₂×Dic₆₈

—

C₁₇×M₃₂

—

C₂×C₁₆×C₁₇

D₄

yes

characteristic

C₂×C₁₆×C₁₇

C₂³

yes

characteristic

C₁₇×#108

—

C₁₇×#108

—

#10587489

—

#10587487

—

Dic₁₃₆

—

#10587465

—

C₂²×C₄×C₁₇

D₈

yes

characteristic

C₂×Dic₃₄

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₈

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63225

—

Dic₆₈

—

Dic₆₈

—

C₁₆×C₁₇

—

C₁₆×C₁₇

#41

yes

—

#13718

—

maximal, Sylow

C₂³×C₁₇

SD₃₂

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

SD₃₂

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

yes

C₈×C₁₇

yes

—

D₁₇

yes

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×SD₃₂

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

C₂×Q₃₂

—

C₂×M₃₂

C₂×D₁₇

yes

characteristic

#467

—

C₂×#108

—

M₆₄

—

#368

—

#368

C₂×D₁₇

yes

—

#367

—

#358

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#2233

yes

characteristic

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₁₇

yes

characteristic

C₂²⋊Q₈

—

Q₃₂

—

M₃₂

—

C₂×C₁₆

C₂²⋊D₁₇

yes

characteristic

C₂×C₁₆

C₂²×D₁₇

yes

—

—

—

—

C₃₂

—

C₁₇

#13718

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

#63228

yes

characteristic

Q₁₆

—

100

Q₁₆

—

101

M₁₆

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

#63228

yes

characteristic

107

C₂×C₈

C₂×C₂²⋊D₁₇

yes

characteristic

108

C₄⋊C₄

—

109

C₄⋊C₄

—

136

—

110

C₂²⋊C₄

—

136

—

111

C₁₆

—

112

C₁₆

#63252

yes

—

113

C₂³

#10587499

yes

characteristic

114

Q₈

—

136

—

115

Q₈

—

136

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

136

—

119

C₂×C₄

—

136

—

120

C₂×C₄

#10587499

yes

characteristic

121

C₂×C₄

C₂×#63228

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

#10587615

yes

characteristic

125

C₈

#10587628

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

C₂×#10587499

yes

characteristic

129

C₄

—

130

C₄

—

272

—

131

C₄

—

272

—

132

C₄

#11763122

yes

characteristic

133

C₄

#11763245

yes

characteristic

134

C₂

—

135

C₂

—

136

C₂

#14530969

yes

characteristic

	subgroup	G/H
center	C₂	#14530969
commutator, cosocle	C₂×C₁₆×C₁₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₁₇
Fitting	C₁₇×#2161	C₂
socle	C₂×C₁₇	#2233
prime radical	C₂³×C₁₇	SD₃₂
hypercenter	#2161	D₁₇
hypocenter	C₁₇	#13718
LCS	G ▷ C₂×C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂ ◁ #2161

subgroup

G/H

center

C₂

#14530969

commutator, cosocle

C₂×C₁₆×C₁₇

C₂³

Frattini

C₂×C₁₆

C₂²×D₁₇

Fitting

C₁₇×#2161

C₂

socle

C₂×C₁₇

#2233

prime radical

C₂³×C₁₇

SD₃₂

D₁₇

C₁₇

G ▷ C₂×C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂ ◁ #2161

N	H ≅ G/N	type
2. #14526991	C₂	semidirect
3. #14526976	C₂	semidirect
96. C₁₇	#13718	semidirect

H ≅ G/N

type

2. #14526991

C₂

semidirect

3. #14526976

C₂

semidirect

96. C₁₇

#13718

semidirect

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

subgroup	index
#14530892	2
#14526991	2
#14526976	2
#14526924	2
C₁₇×#2161	2
#13718	17

subgroup

index

C₁₇×#2161

Subgroups — #16955288

Conjugacy classes of subgroups

Complements

Maximal subgroups