Group #17982360 — subgroups

Subgroups — #17982360

Order 4352.

214 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 61 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	#14545914	C₂	1	2	yes	maximal
2	#14545844	C₂	1	1	yes	maximal, characteristic
3	#14545833	C₂	1	2	yes	maximal
4	#14545803	C₂	1	1	yes	maximal, characteristic
5	#14545181	C₂	1	1	yes	maximal, characteristic
6	Q₈×#3633	—	2	2	no	—
7	#11763904	C₂²	1	1	yes	characteristic
8	#11763898	C₂²	1	2	yes	—
9	Q₈⋊#3633	—	2	2	no	—
10	#11763893	C₂²	1	1	yes	characteristic
11	Q₁₆×#595	C₂²	1	2	yes	—
12	#11763736	C₄	1	1	yes	characteristic
13	#11763722	C₄	1	2	yes	—
14	#11763696	C₂²	1	1	yes	characteristic
15	Q₃₂×D₁₇	C₄	1	1	yes	characteristic
16	Q₈×#595	—	2	2	no	—
17	C₄⋊#3633	—	2	2	no	—
18	C₄⋊#3633	—	4	2	no	—
19	C₄×#3633	—	4	2	no	—
20	C₄×#3633	D₄	1	2	yes	—
21	#10587665	—	2	1	no	—
22	Q₁₆×D₁₇	C₂×C₄	1	2	yes	—
23	#10587549	—	2	2	no	—
24	#10587534	C₂×C₄	1	2	yes	—
25	C₈×#595	C₂³	1	1	yes	characteristic
26	#10587529	C₂×C₄	1	1	yes	characteristic
27	C₁₇×Q₃₂	C₈	1	1	yes	characteristic, complemented
28	#10587500	C₈	1	2	yes	complemented
29	Dic₁₃₆	C₈	1	1	yes	characteristic, complemented
30	C₁₆×D₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂×#3633	—	2	2	no	—
32	C₂×#3633	—	4	2	no	—
33	Q₈×D₁₇	—	2	2	no	—
34	C₄⋊#595	—	2	2	no	—
35	C₄⋊#595	—	4	2	no	—
36	C₄×#595	—	4	2	no	—
37	C₄×#595	C₂×D₄	1	1	yes	characteristic
38	#63242	#41	1	1	yes	characteristic
39	#63241	—	4	1	no	—
40	Q₁₆×C₁₇	C₂×C₈	1	2	yes	—
41	#63231	—	2	2	no	—
42	Dic₆₈	C₂×C₈	1	2	yes	—
43	C₈×D₁₇	C₂²×C₄	1	1	yes	characteristic
44	#63216	—	2	1	no	—
45	C₁₆×C₁₇	C₂×C₈	1	1	yes	characteristic
46	#63214	C₂×C₈	1	1	yes	characteristic
47	#12066	—	17	1	no	maximal, Sylow
48	C₂×#595	—	4	2	no	—
49	C₂×#595	AGL(1,ℤ/8)	1	2	yes	—
50	#3633	—	4	2	no	—
51	#3633	—	8	2	no	—
52	Q₈×C₁₇	—	2	2	no	—
53	C₄×D₁₇	—	4	2	no	—
54	C₄×D₁₇	C₄×D₄	1	1	yes	characteristic
55	Dic₃₄	—	2	2	no	—
56	Dic₃₄	—	4	2	no	—
57	#3624	—	2	1	no	—
58	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
59	#3622	#131	1	1	yes	characteristic
60	#2176	—	17	1	no	—
61	#1579	—	17	2	no	—
62	#1368	—	17	1	no	—
63	#1334	—	17	2	no	—
64	#1313	—	17	1	no	—
65	C₂×D₁₇	#442	1	1	yes	characteristic
66	#595	—	2	2	no	—
67	#595	—	8	2	no	—
68	C₄×C₁₇	—	4	2	no	—
69	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
70	Dic₁₇	—	4	2	no	—
71	Dic₁₇	#444	1	1	yes	characteristic
72	C₂×Q₃₂	—	17	1	no	—
73	C₈×Q₈	—	34	2	no	—
74	C₄×Q₁₆	—	17	2	no	—
75	#367	—	17	1	no	—
76	#358	—	17	2	no	—
77	#346	—	17	1	no	—
78	#346	—	17	1	no	—
79	#334	—	17	2	no	—
80	Q₈⋊C₈	—	34	2	no	—
81	#322	—	17	1	no	—
82	C₂×C₁₇	#1580	1	1	yes	characteristic
83	D₁₇	#2180	1	2	yes	—
84	C₂×Q₁₆	—	17	2	no	—
85	C₄×Q₈	—	34	2	no	—
86	Q₃₂	—	17	2	no	—
87	Q₃₂	—	17	1	no	—
88	Q₃₂	#3633	1	1	yes	characteristic, complemented
89	C₂×C₁₆	—	17	1	no	—
90	C₂×C₁₆	—	17	1	no	—
91	C₂×C₁₆	—	34	1	no	—
92	#107	—	17	2	no	—
93	C₄⋊C₈	—	34	2	no	—
94	C₄⋊C₈	—	68	2	no	—
95	#103	—	34	2	no	—
96	C₄×C₈	—	17	1	no	—
97	C₄×C₈	—	17	2	no	—
98	C₄×C₈	—	68	2	no	—
99	C₁₇	#12066	1	1	yes	characteristic, complemented, Sylow
100	C₂×Q₈	—	34	2	no	—
101	Q₁₆	—	17	2	no	—
102	Q₁₆	—	34	2	no	—
103	Q₁₆	C₂×#3633	1	2	yes	—
104	C₂×C₈	—	17	1	no	—
105	C₂×C₈	—	17	1	no	—
106	C₂×C₈	—	34	2	no	—
107	C₂×C₈	—	68	2	no	—
108	C₄⋊C₄	—	34	2	no	—
109	C₄⋊C₄	—	68	2	no	—
110	C₄²	—	17	1	no	—
111	C₄²	—	68	2	no	—
112	C₁₆	—	17	1	no	—
113	C₁₆	—	34	1	no	—
114	C₁₆	—	68	1	no	—
115	C₁₆	C₂×#3633	1	1	yes	characteristic
116	Q₈	—	2	2	no	—
117	Q₈	—	34	2	no	—
118	Q₈	—	68	2	no	—
119	C₂×C₄	—	17	1	no	—
120	C₂×C₄	—	17	2	no	—
121	C₂×C₄	—	68	2	no	—
122	C₂×C₄	—	68	2	no	—
123	C₈	—	17	1	no	—
124	C₈	—	34	1	no	—
125	C₈	—	68	2	no	—
126	C₈	—	136	2	no	—
127	C₈	C₂²×#3633	1	1	yes	characteristic
128	C₂²	—	17	1	no	—
129	C₄	—	4	2	no	—
130	C₄	—	17	1	no	—
131	C₄	—	34	2	no	—
132	C₄	—	68	2	no	—
133	C₄	—	136	2	no	—
134	C₄	D₄×#3633	1	1	yes	characteristic
135	C₂	—	17	2	no	—
136	C₂	#14545915	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545914

C₂

yes

maximal

#14545844

C₂

yes

maximal, characteristic

#14545833

C₂

yes

maximal

#14545803

C₂

yes

maximal, characteristic

#14545181

C₂

yes

maximal, characteristic

Q₈×#3633

—

#11763904

C₂²

yes

characteristic

#11763898

C₂²

yes

—

Q₈⋊#3633

—

#11763893

C₂²

yes

characteristic

Q₁₆×#595

C₂²

yes

—

#11763736

C₄

yes

characteristic

#11763722

C₄

yes

—

#11763696

C₂²

yes

characteristic

Q₃₂×D₁₇

C₄

yes

—

—

—

—

D₄

yes

—

#10587665

—

Q₁₆×D₁₇

C₂×C₄

yes

—

#10587549

—

#10587534

C₂×C₄

yes

—

C₈×#595

C₂³

yes

characteristic

#10587529

C₂×C₄

yes

characteristic

C₁₇×Q₃₂

C₈

yes

characteristic, complemented

#10587500

C₈

yes

complemented

Dic₁₃₆

C₈

yes

characteristic, complemented

C₁₆×D₁₇

C₂×C₄

yes

characteristic

C₂×#3633

—

C₂×#3633

—

Q₈×D₁₇

—

—

—

—

C₂×D₄

yes

yes

—

Q₁₆×C₁₇

C₂×C₈

yes

—

#63231

—

Dic₆₈

C₂×C₈

yes

—

C₈×D₁₇

C₂²×C₄

yes

characteristic

#63216

—

C₁₆×C₁₇

C₂×C₈

yes

characteristic

#63214

C₂×C₈

yes

—

—

yes

—

#3633

—

#3633

—

Q₈×C₁₇

—

C₄×D₁₇

—

C₄×D₁₇

C₄×D₄

yes

characteristic

Dic₃₄

—

Dic₃₄

—

#3624

—

C₈×C₁₇

C₂²×C₈

yes

yes

—

—

—

—

—

C₂×D₁₇

yes

—

—

C₄×C₁₇

—

C₄×C₁₇

C₈×D₄

yes

characteristic

Dic₁₇

—

Dic₁₇

#444

yes

characteristic

C₂×Q₃₂

—

C₈×Q₈

—

C₄×Q₁₆

—

—

—

—

—

—

Q₈⋊C₈

—

#322

—

C₂×C₁₇

#1580

yes

characteristic

D₁₇

#2180

yes

—

C₂×Q₁₆

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

—

Q₃₂

#3633

yes

characteristic, complemented

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₁₇

#12066

yes

characteristic, complemented, Sylow

100

C₂×Q₈

—

101

Q₁₆

—

102

Q₁₆

—

103

Q₁₆

C₂×#3633

yes

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄²

—

111

C₄²

—

112

C₁₆

—

113

C₁₆

—

114

C₁₆

—

115

C₁₆

C₂×#3633

yes

characteristic

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

136

—

127

C₈

C₂²×#3633

yes

characteristic

128

C₂²

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

136

—

134

C₄

D₄×#3633

yes

characteristic

135

C₂

—

136

C₂

#14545915

yes

characteristic

	subgroup	G/H
center	C₂	#14545915
commutator	C₈×C₁₇	C₂²×C₈
Frattini	C₈	C₂²×#3633
Fitting	C₁₇×Q₃₂	C₈
socle	C₂×C₁₇	#1580
prime radical	C₂×D₁₇	#442
cosocle	C₈×#595	C₂³
hypercenter	Q₃₂	#3633
hypocenter	C₁₇	#12066
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

subgroup

G/H

center

C₂

#14545915

commutator

C₈×C₁₇

C₂²×C₈

Frattini

C₈

C₂²×#3633

Fitting

C₁₇×Q₃₂

C₈

C₂×C₁₇

C₂×D₁₇

C₂³

Q₃₂

C₁₇

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

N	H ≅ G/N	type
27. C₁₇×Q₃₂	C₈	semidirect
28. #10587500	C₈	semidirect
29. Dic₁₃₆	C₈	semidirect
88. Q₃₂	#3633	semidirect
99. C₁₇	#12066	semidirect

H ≅ G/N

type

27. C₁₇×Q₃₂

C₈

semidirect

28. #10587500

C₈

semidirect

29. Dic₁₃₆

C₈

semidirect

88. Q₃₂

#3633

semidirect

99. C₁₇

#12066

semidirect

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

subgroup	index
#14545914	2
#14545844	2
#14545833	2
#14545803	2
#14545181	2
#12066	17

subgroup

index

Subgroups — #17982360

Conjugacy classes of subgroups

Complements

Maximal subgroups