Group #16946328 — subgroups

Subgroups — #16946328

Order 4352.

206 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 79 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×Dic₁₃₆	C₂	1	1	yes	maximal, characteristic
2	#14527682	C₂	1	2	yes	maximal
3	#14526858	C₂	1	1	yes	maximal, characteristic
4	#14526749	C₂	1	2	yes	maximal
5	C₁₇×#1311	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₃₆	C₄	1	1	yes	characteristic
7	#11762939	—	2	2	no	—
8	C₄×Dic₆₈	C₂²	1	2	yes	—
9	#11762635	C₄	1	1	yes	characteristic
10	#11762614	C₄	1	2	yes	—
11	#11762562	C₂²	1	2	yes	—
12	#11762541	—	2	2	no	—
13	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
15	C₈²×C₁₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₆₈	C₂×C₄	1	2	yes	—
17	C₄×Dic₃₄	—	2	2	no	—
18	C₂×C₁₆×C₁₇	—	2	1	no	—
19	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
22	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
23	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
24	#10587489	C₂×C₄	1	2	yes	—
25	#10587487	—	2	2	no	—
26	#10587486	—	4	2	no	—
27	#10587485	—	4	2	no	—
28	C₄⋊#3622	—	2	2	no	—
29	Dic₁₃₆	—	2	2	no	—
30	C₂×Dic₃₄	—	2	2	no	—
31	C₂×C₈×C₁₇	—	2	1	no	—
32	C₂×C₈×C₁₇	D₈	1	2	yes	—
33	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
35	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
36	C₄⋊Dic₁₇	—	2	2	no	—
37	#63225	—	4	2	no	—
38	C₄×Dic₁₇	—	4	2	no	—
39	C₂×#3622	—	4	2	no	—
40	Dic₆₈	—	2	2	no	—
41	Dic₆₈	—	2	2	no	—
42	C₁₆×C₁₇	—	2	2	no	—
43	C₁₆×C₁₇	—	2	1	no	—
44	C₁₆×C₁₇	M₁₆	1	2	yes	—
45	#12085	—	17	1	no	maximal, Sylow
46	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
49	C₂×Dic₁₇	—	4	2	no	—
50	C₂×Dic₁₇	—	4	2	no	—
51	Dic₃₄	—	4	2	no	—
52	Dic₃₄	—	4	2	no	—
53	C₈×C₁₇	—	2	1	no	—
54	C₈×C₁₇	—	2	2	no	—
55	C₈×C₁₇	—	4	1	no	—
56	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
57	C₈×C₁₇	#131	1	1	yes	characteristic
58	#3622	—	8	2	no	—
59	C₄×Q₃₂	—	17	1	no	—
60	#1591	—	17	2	no	—
61	#1372	—	17	1	no	—
62	#1332	—	17	2	no	—
63	#1311	D₁₇	1	1	yes	characteristic
64	C₂²×C₁₇	C₄×D₈	1	1	yes	characteristic
65	C₄×C₁₇	—	2	1	no	—
66	C₄×C₁₇	#436	1	1	yes	characteristic
67	C₄×C₁₇	#443	1	1	yes	characteristic
68	C₄×C₁₇	#510	1	2	yes	—
69	Dic₁₇	—	8	2	no	—
70	Dic₁₇	—	8	2	no	—
71	C₂×Q₃₂	—	17	1	no	—
72	#446	—	34	2	no	—
73	C₄×Q₁₆	—	17	2	no	—
74	#366	—	17	1	no	—
75	#358	—	17	2	no	—
76	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
77	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
78	#335	—	17	2	no	—
79	Q₈⋊C₈	—	34	2	no	—
80	C₈²	C₂×D₁₇	1	1	yes	characteristic
81	C₂×C₁₇	#1589	1	1	yes	characteristic
82	C₂×C₁₇	#2176	1	1	yes	characteristic
83	C₂×C₁₇	#2179	1	1	yes	characteristic
84	C₂×Q₁₆	—	17	2	no	—
85	C₄×Q₈	—	34	2	no	—
86	Q₃₂	—	34	2	no	—
87	C₂×C₁₆	—	2	1	no	—
88	C₂×C₁₆	C₄×D₁₇	1	1	yes	characteristic
89	C₂×C₁₆	C₄×D₁₇	1	1	yes	characteristic
90	#107	—	17	2	no	—
91	C₄⋊C₈	—	34	2	no	—
92	C₄⋊C₈	—	68	2	no	—
93	#103	—	34	2	no	—
94	#97	—	68	2	no	—
95	C₄×C₈	D₆₈	1	1	yes	characteristic
96	C₄×C₈	D₆₈	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
98	C₁₇	#12085	1	1	yes	characteristic, complemented, Sylow
99	C₂×Q₈	—	34	2	no	—
100	Q₁₆	—	34	2	no	—
101	Q₁₆	—	34	2	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	68	2	no	—
104	C₂×C₈	D₁₃₆	1	2	yes	—
105	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
106	C₂×C₈	#63252	1	1	yes	characteristic
107	C₄⋊C₄	—	34	2	no	—
108	C₄⋊C₄	—	68	2	no	—
109	C₄²	—	68	2	no	—
110	C₄²	C₂×D₆₈	1	1	yes	characteristic
111	C₁₆	—	2	2	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	#63218	1	2	yes	—
114	Q₈	—	68	2	no	—
115	Q₈	—	68	2	no	—
116	C₂×C₄	—	68	2	no	—
117	C₂×C₄	—	68	2	no	—
118	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
119	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
120	C₂×C₄	#10587590	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	—	2	1	no	—
123	C₈	—	4	1	no	—
124	C₈	—	136	2	no	—
125	C₈	C₂×#63218	1	1	yes	characteristic
126	C₈	#10587587	1	1	yes	characteristic
127	C₂²	C₄×D₁₃₆	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	136	2	no	—
130	C₄	—	136	2	no	—
131	C₄	#11762888	1	1	yes	characteristic
132	C₄	#11762927	1	1	yes	characteristic
133	C₄	#11763252	1	2	yes	—
134	C₂	#14527670	1	1	yes	characteristic
135	C₂	#14530656	1	1	yes	characteristic
136	C₂	#14530671	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₁₃₆

C₂

yes

maximal, characteristic

#14527682

C₂

yes

maximal

#14526858

C₂

yes

maximal, characteristic

#14526749

C₂

yes

maximal

C₁₇×#1311

C₂

yes

maximal, characteristic

C₂×Dic₁₃₆

C₄

yes

characteristic

#11762939

—

C₄×Dic₆₈

C₂²

yes

—

#11762635

C₄

yes

characteristic

#11762614

C₄

yes

—

#11762562

C₂²

yes

—

#11762541

—

C₄×C₁₆×C₁₇

C₂²

yes

characteristic

C₄×C₁₆×C₁₇

C₂²

yes

characteristic

C₈²×C₁₇

C₂²

yes

characteristic

C₂×Dic₆₈

C₂×C₄

yes

—

C₄×Dic₃₄

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

C₂³

yes

characteristic

#10587489

C₂×C₄

yes

—

—

—

—

—

Dic₁₃₆

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₇

#41

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄⋊Dic₁₇

—

#63225

—

C₄×Dic₁₇

—

C₂×#3622

—

Dic₆₈

—

Dic₆₈

—

C₁₆×C₁₇

—

C₁₆×C₁₇

—

C₁₆×C₁₇

M₁₆

yes

—

#12085

—

maximal, Sylow

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

#135

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₂×M₁₆

yes

C₈×C₁₇

yes

—

C₄×Q₃₂

—

—

—

—

D₁₇

yes

characteristic

C₂²×C₁₇

C₄×D₈

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

Dic₁₇

—

Dic₁₇

—

C₂×Q₃₂

—

#446

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

C₂×D₁₇

yes

characteristic

C₄×C₁₆

C₂×D₁₇

yes

characteristic

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₁₇

yes

characteristic

C₂×C₁₆

C₄×D₁₇

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#97

—

C₄×C₈

D₆₈

yes

characteristic

C₄×C₈

D₆₈

yes

characteristic

C₄×C₈

C₂²×D₁₇

yes

characteristic

C₁₇

#12085

yes

characteristic, complemented, Sylow

C₂×Q₈

—

100

Q₁₆

—

101

Q₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

D₁₃₆

yes

—

105

C₂×C₈

C₂×C₄×D₁₇

yes

characteristic

106

C₂×C₈

#63252

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

—

110

C₄²

C₂×D₆₈

yes

characteristic

111

C₁₆

—

112

C₁₆

—

113

C₁₆

#63218

yes

—

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₄×D₆₈

yes

characteristic

119

C₂×C₄

C₂×D₁₃₆

yes

characteristic

120

C₂×C₄

#10587590

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

136

—

125

C₈

C₂×#63218

yes

characteristic

126

C₈

#10587587

yes

characteristic

127

C₂²

C₄×D₁₃₆

yes

characteristic

128

C₄

—

129

C₄

—

136

—

130

C₄

—

136

—

131

C₄

#11762888

yes

characteristic

132

C₄

#11762927

yes

characteristic

133

C₄

#11763252

yes

—

134

C₂

#14527670

yes

characteristic

135

C₂

#14530656

yes

characteristic

136

C₂

#14530671

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₃₆
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1311	C₂
socle, prime radical	C₂²×C₁₇	C₄×D₈
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	#1311	D₁₇
hypocenter	C₁₇	#12085
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂×C₄

C₂×D₁₃₆

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#1311

C₂

socle, prime radical

C₂²×C₁₇

C₄×D₈

cosocle

C₄×C₈×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

N	H ≅ G/N	type
98. C₁₇	#12085	semidirect

H ≅ G/N

type

98. C₁₇

#12085

semidirect

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

subgroup	index
C₄×Dic₁₃₆	2
#14527682	2
#14526858	2
#14526749	2
C₁₇×#1311	2
#12085	17

subgroup

index

C₄×Dic₁₃₆

C₁₇×#1311

Subgroups — #16946328

Conjugacy classes of subgroups

Complements

Maximal subgroups