Group #17948447 — subgroups

Subgroups — #17948447

Order 4352.

230 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 62 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	#14543825	C₂	1	2	yes	maximal
2	#14529605	C₂	1	1	yes	maximal, characteristic
3	C₂×#10587538	—	2	2	no	—
4	C₂×#10587506	—	2	2	no	—
5	C₂×#10587474	C₂²	1	1	yes	characteristic
6	#11762675	C₄	1	1	yes	characteristic
7	C₂×C₁₇×C₄⋊Q₈	C₄	1	1	yes	characteristic
8	C₂²×Q₈×C₁₇	—	2	1	no	—
9	C₁₇×C₄⋊Q₈	—	2	1	no	—
10	C₁₇×C₄⋊Q₈	—	4	1	no	—
11	C₁₇×C₄⋊Q₈	C₈	1	2	yes	—
12	C₂×C₄⋊C₄×C₁₇	—	4	1	no	—
13	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₇	—	2	1	no	—
15	C₂²×#3622	D₄	1	2	yes	—
16	#10587538	—	4	2	no	—
17	#10587538	—	4	2	no	—
18	C₂×#63216	—	2	4	no	—
19	#10587506	—	4	2	no	—
20	#10587506	—	4	2	no	—
21	#10587502	—	4	1	no	—
22	#10587474	—	4	1	no	—
23	C₂×Q₈×C₁₇	—	2	2	no	—
24	C₂×Q₈×C₁₇	—	4	1	no	—
25	C₂×Q₈×C₁₇	—	4	2	no	—
26	C₂×Q₈×C₁₇	—	4	2	no	—
27	C₂²×C₄×C₁₇	—	2	1	no	—
28	C₂²×C₄×C₁₇	—	4	1	no	—
29	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×Dic₁₇	—	4	1	no	—
31	C₄⋊C₄×C₁₇	—	4	2	no	—
32	C₄⋊C₄×C₁₇	—	4	2	no	—
33	C₄²×C₁₇	—	2	1	no	—
34	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
35	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
36	C₄⋊Dic₁₇	—	4	2	no	—
37	C₂×#3622	—	2	2	no	—
38	C₂×#3622	—	2	2	no	—
39	C₂×#3622	D₈	1	2	yes	—
40	C₂×#3622	SD₁₆	1	2	yes	—
41	#63216	—	4	4	no	—
42	#7117	—	17	1	no	maximal, Sylow
43	C₂³×C₁₇	#100	1	1	yes	characteristic
44	Q₈×C₁₇	—	4	4	no	—
45	Q₈×C₁₇	—	8	2	no	—
46	C₂×C₄×C₁₇	—	2	1	no	—
47	C₂×C₄×C₁₇	—	2	1	no	—
48	C₂×C₄×C₁₇	—	4	2	no	—
49	C₂×C₄×C₁₇	—	4	2	no	—
50	C₂×C₄×C₁₇	—	8	1	no	—
51	C₂×C₄×C₁₇	—	8	1	no	—
52	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₇	#99	1	1	yes	characteristic
54	C₂×C₄×C₁₇	#102	1	2	yes	—
55	C₂×C₄×C₁₇	#104	1	2	yes	—
56	C₂×Dic₁₇	—	8	2	no	—
57	C₂×Dic₁₇	—	8	1	no	—
58	#3622	—	2	2	no	—
59	#3622	—	2	2	no	—
60	#1960	—	17	1	no	—
61	#1375	—	17	2	no	—
62	C₂²×C₁₇	—	2	2	no	—
63	C₂²×C₁₇	#323	1	1	yes	characteristic
64	C₂²×C₁₇	#329	1	1	yes	characteristic
65	C₂²×C₁₇	#331	1	1	yes	characteristic
66	C₄×C₁₇	—	4	1	no	—
67	C₄×C₁₇	—	4	1	no	—
68	C₄×C₁₇	—	8	2	no	—
69	C₄×C₁₇	D₄⋊C₈	1	2	yes	—
70	C₄×C₁₇	#327	1	2	yes	—
71	Dic₁₇	—	16	1	no	—
72	C₂×C₄⋊Q₈	#595	1	1	yes	characteristic
73	C₂×M₃₂	—	34	2	no	—
74	C₂×#103	—	34	2	no	—
75	C₂×#97	—	17	1	no	—
76	#382	—	17	1	no	—
77	C₂×C₁₇	—	2	1	no	—
78	C₂×C₁₇	—	2	1	no	—
79	C₂×C₁₇	#1270	1	1	yes	characteristic
80	C₂×C₁₇	#1356	1	2	yes	—
81	C₂²×Q₈	—	2	1	no	—
82	C₂²×C₈	—	17	2	no	—
83	C₄⋊Q₈	—	2	1	no	—
84	C₄⋊Q₈	—	4	1	no	—
85	C₄⋊Q₈	#3624	1	2	yes	—
86	C₂×C₄⋊C₄	—	4	1	no	—
87	C₂×C₄⋊C₄	—	34	1	no	—
88	C₂×C₄²	C₂×#595	1	1	yes	characteristic
89	M₃₂	—	68	2	no	—
90	M₃₂	—	68	2	no	—
91	C₂×C₁₆	—	34	4	no	—
92	#103	—	68	2	no	—
93	#103	—	68	2	no	—
94	#97	—	68	1	no	—
95	#95	—	68	1	no	—
96	C₁₇	#7117	1	1	yes	characteristic, complemented, Sylow
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	4	1	no	—
99	C₂×Q₈	—	4	2	no	—
100	C₂×Q₈	—	4	2	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	4	1	no	—
103	C₂²×C₄	—	68	1	no	—
104	C₂²×C₄	C₂²⋊#595	1	1	yes	characteristic
105	C₂×C₈	—	17	2	no	—
106	C₂×C₈	—	17	2	no	—
107	C₂×C₈	—	34	2	no	—
108	C₂×C₈	—	34	2	no	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	68	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	C₂×#3624	1	1	yes	characteristic
114	C₄²	#63247	1	1	yes	characteristic
115	C₁₆	—	68	4	no	—
116	C₂³	#10587553	1	1	yes	characteristic
117	Q₈	—	4	4	no	—
118	Q₈	—	8	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	8	1	no	—
124	C₂×C₄	—	8	1	no	—
125	C₂×C₄	—	136	2	no	—
126	C₂×C₄	—	136	1	no	—
127	C₂×C₄	#10587547	1	2	yes	—
128	C₂×C₄	#10587548	1	2	yes	—
129	C₂×C₄	#10587551	1	1	yes	characteristic
130	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
131	C₈	—	34	2	no	—
132	C₈	—	34	2	no	—
133	C₂²	—	2	2	no	—
134	C₂²	#11763652	1	1	yes	characteristic
135	C₂²	#11763664	1	1	yes	characteristic
136	C₂²	#11763667	1	1	yes	characteristic
137	C₄	—	4	1	no	—
138	C₄	—	4	1	no	—
139	C₄	—	8	2	no	—
140	C₄	—	272	1	no	—
141	C₄	D₄⋊#3624	1	2	yes	—
142	C₄	#11763660	1	2	yes	—
143	C₂	—	2	1	no	—
144	C₂	—	2	1	no	—
145	C₂	#14543600	1	1	yes	characteristic
146	C₂	#14543789	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14543825

C₂

yes

maximal

#14529605

C₂

yes

maximal, characteristic

C₂×#10587538

—

C₂×#10587506

—

C₂×#10587474

C₂²

yes

characteristic

#11762675

C₄

yes

characteristic

C₂×C₁₇×C₄⋊Q₈

C₄

yes

characteristic

C₂²×Q₈×C₁₇

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

C₈

yes

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂²×#3622

D₄

yes

—

—

—

—

—

—

—

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂²×Dic₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂×C₈

yes

characteristic

C₄²×C₁₇

M₁₆

yes

characteristic

C₄⋊Dic₁₇

—

C₂×#3622

—

C₂×#3622

—

C₂×#3622

D₈

yes

—

C₂×#3622

SD₁₆

yes

—

—

—

C₂³×C₁₇

yes

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₈

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

—

C₂×C₄×C₁₇

#104

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

D₄⋊C₈

yes

—

C₄×C₁₇

#327

yes

—

Dic₁₇

—

C₂×C₄⋊Q₈

#595

yes

characteristic

C₂×M₃₂

—

C₂×#103

—

C₂×#97

—

#382

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#1270

yes

characteristic

C₂×C₁₇

#1356

yes

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#3624

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×#595

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

—

—

—

—

C₁₇

#7117

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂²⋊#595

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

C₂×#3624

yes

characteristic

114

C₄²

#63247

yes

characteristic

115

C₁₆

—

116

C₂³

#10587553

yes

characteristic

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

136

—

126

C₂×C₄

—

136

—

127

C₂×C₄

#10587547

yes

—

128

C₂×C₄

#10587548

yes

—

129

C₂×C₄

#10587551

yes

characteristic

130

C₂×C₄

C₂²⋊#3624

yes

characteristic

131

C₈

—

132

C₈

—

133

C₂²

—

134

C₂²

#11763652

yes

characteristic

135

C₂²

#11763664

yes

characteristic

136

C₂²

#11763667

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

272

—

141

C₄

D₄⋊#3624

yes

—

142

C₄

#11763660

yes

—

143

C₂

—

144

C₂

—

145

C₂

#14543600

yes

characteristic

146

C₂

#14543789

yes

—

	subgroup	G/H
center	C₂²	#11763664
commutator	C₄²×C₁₇	C₂×C₈
Frattini	C₂×C₄²	C₂×#595
Fitting	C₂×C₁₇×C₄⋊Q₈	C₄
socle	C₂²×C₁₇	#329
prime radical	C₂³×C₁₇	#100
cosocle	C₂×#10587474	C₂²
hypercenter	C₂×C₄⋊Q₈	#595
hypocenter	C₁₇	#7117
LCS	G ▷ C₄²×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

#11763664

commutator

C₄²×C₁₇

C₂×C₈

Frattini

C₂×C₄²

C₂×#595

Fitting

C₂×C₁₇×C₄⋊Q₈

C₄

C₂²×C₁₇

C₂³×C₁₇

C₂²

C₂×C₄⋊Q₈

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
96. C₁₇	#7117	semidirect

H ≅ G/N

type

96. C₁₇

#7117

semidirect

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

subgroup	index
#14543825	2
#14529605	2
#7117	17

subgroup

index

#14543825

#14529605

#7117

Subgroups — #17948447

Conjugacy classes of subgroups

Complements

Maximal subgroups