Group #16946163 — subgroups

Subgroups — #16946163

Order 4352.

212 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 117 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	Q₈×#63214	C₂	1	2	yes	maximal
2	#14526848	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#63214	C₂	1	2	yes	maximal
4	#14526702	C₂	1	1	yes	maximal, characteristic
5	C₈×Q₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#63214	—	2	2	no	—
7	C₄⋊#63214	C₂²	1	2	yes	—
8	C₄×#63214	—	2	2	no	—
9	C₄×#63214	C₂²	1	1	yes	characteristic
10	C₄×#63214	C₂²	1	1	yes	characteristic
11	C₈×Q₈×C₁₇	C₂²	1	2	yes	—
12	C₄×Q₁₆×C₁₇	C₄	1	1	yes	characteristic
13	C₁₇×#335	C₄	1	1	yes	characteristic
14	C₁₇×Q₈⋊C₈	C₄	1	2	yes	—
15	C₈²×C₁₇	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₁₇	C₈	1	1	yes	characteristic
17	C₄×Q₈×C₁₇	C₂×C₄	1	2	yes	—
18	C₁₇×#107	C₈	1	1	yes	characteristic
19	C₁₇×C₄⋊C₈	—	2	2	no	—
20	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
21	C₁₇×#103	C₈	1	2	yes	—
22	C₄×C₈×C₁₇	—	2	2	no	—
23	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
26	C₂×#63214	—	2	2	no	—
27	C₂×#63214	—	2	1	no	—
28	C₂×#63214	D₄	1	2	yes	—
29	C₂×Q₈×C₁₇	C₂×C₈	1	2	yes	—
30	Q₁₆×C₁₇	C₁₆	1	4	yes	complemented
31	C₂×C₈×C₁₇	—	2	2	no	—
32	C₂×C₈×C₁₇	—	2	1	no	—
33	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
37	C₄⋊C₄×C₁₇	—	2	2	no	—
38	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
39	C₄²×C₁₇	—	2	2	no	—
40	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
41	#63214	—	2	2	no	—
42	#63214	—	4	1	no	—
43	#63214	—	4	2	no	—
44	#12059	—	17	1	no	maximal, Sylow
45	Q₈×C₁₇	—	2	2	no	—
46	Q₈×C₁₇	C₂×C₁₆	1	4	yes	—
47	C₂×C₄×C₁₇	—	2	2	no	—
48	C₂×C₄×C₁₇	—	2	2	no	—
49	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
52	C₈×C₁₇	—	2	1	no	—
53	C₈×C₁₇	—	2	1	no	—
54	C₈×C₁₇	—	4	1	no	—
55	C₈×C₁₇	—	4	2	no	—
56	C₈×C₁₇	C₂×C₁₆	1	2	yes	—
57	C₈×C₁₇	#137	1	2	yes	—
58	Q₈×C₁₆	—	17	2	no	—
59	C₈×Q₁₆	D₁₇	1	1	yes	characteristic
60	#1367	—	17	1	no	—
61	Q₈⋊C₁₆	—	17	2	no	—
62	#1312	—	17	1	no	—
63	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
64	C₄×C₁₇	—	2	4	no	—
65	C₄×C₁₇	—	2	1	no	—
66	C₄×C₁₇	—	4	2	no	—
67	C₄×C₁₇	#441	1	1	yes	characteristic
68	C₄×C₁₇	#444	1	1	yes	characteristic
69	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
70	C₄×C₁₇	#504	1	1	yes	characteristic
71	C₈×Q₈	C₂×D₁₇	1	2	yes	—
72	C₄×Q₁₆	Dic₁₇	1	1	yes	characteristic
73	C₄⋊C₁₆	—	17	2	no	—
74	C₄⋊C₁₆	—	34	2	no	—
75	C₄×C₁₆	—	17	1	no	—
76	C₄×C₁₆	—	17	1	no	—
77	C₄×C₁₆	—	34	2	no	—
78	#335	Dic₁₇	1	1	yes	characteristic
79	Q₈⋊C₈	Dic₁₇	1	2	yes	—
80	C₈²	C₂×D₁₇	1	1	yes	characteristic
81	C₂×C₁₇	#1579	1	1	yes	characteristic
82	C₂×C₁₇	D₄×C₁₆	1	1	yes	characteristic
83	C₂×C₁₇	#2171	1	1	yes	characteristic
84	C₂×Q₁₆	#3622	1	1	yes	characteristic
85	C₄×Q₈	C₂×Dic₁₇	1	2	yes	—
86	C₂×C₁₆	—	17	2	no	—
87	C₂×C₁₆	—	34	2	no	—
88	C₂×C₁₆	—	34	1	no	—
89	#107	#3622	1	1	yes	characteristic
90	C₄⋊C₈	—	2	2	no	—
91	C₄⋊C₈	C₂×Dic₁₇	1	2	yes	—
92	#103	#3622	1	2	yes	—
93	C₄×C₈	—	2	2	no	—
94	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
95	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
96	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
97	C₁₇	#12059	1	1	yes	characteristic, complemented, Sylow
98	C₂×Q₈	C₂×#3622	1	2	yes	—
99	Q₁₆	#63214	1	4	yes	complemented
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	C₂×#3622	1	1	yes	characteristic
103	C₂×C₈	C₂×#3622	1	1	yes	characteristic
104	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
105	C₂×C₈	#63254	1	1	yes	characteristic
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	C₂×#3622	1	2	yes	—
108	C₄²	—	2	2	no	—
109	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
110	C₁₆	—	34	2	no	—
111	C₁₆	—	68	1	no	—
112	C₁₆	—	68	2	no	—
113	Q₈	—	2	2	no	—
114	Q₈	C₂×#63214	1	4	yes	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
118	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
119	C₂×C₄	#10587634	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	2	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	4	2	no	—
124	C₈	C₂×#63214	1	2	yes	—
125	C₈	#10587604	1	2	yes	—
126	C₂²	D₄×#3622	1	1	yes	characteristic
127	C₄	—	2	4	no	—
128	C₄	—	2	1	no	—
129	C₄	—	4	2	no	—
130	C₄	#11762916	1	1	yes	characteristic
131	C₄	#11762931	1	1	yes	characteristic
132	C₄	C₂²×#63214	1	1	yes	characteristic
133	C₄	#11763226	1	1	yes	characteristic
134	C₂	#14527603	1	1	yes	characteristic
135	C₂	D₄×#63214	1	1	yes	characteristic
136	C₂	#14530626	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#63214

C₂

yes

maximal

#14526848

C₂

yes

maximal, characteristic

Q₈⋊#63214

C₂

yes

maximal

#14526702

C₂

yes

maximal, characteristic

C₈×Q₁₆×C₁₇

C₂

yes

maximal, characteristic

C₄⋊#63214

—

C₄⋊#63214

C₂²

yes

—

C₄×#63214

—

C₄×#63214

C₂²

yes

characteristic

C₄×#63214

C₂²

yes

characteristic

C₈×Q₈×C₁₇

C₂²

yes

—

C₄×Q₁₆×C₁₇

C₄

yes

characteristic

C₁₇×#335

C₄

yes

characteristic

C₁₇×Q₈⋊C₈

C₄

yes

—

C₈²×C₁₇

C₂²

yes

characteristic

C₂×Q₁₆×C₁₇

C₈

yes

characteristic

C₄×Q₈×C₁₇

C₂×C₄

yes

—

C₁₇×#107

C₈

yes

characteristic

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

C₂×C₄

yes

—

C₁₇×#103

C₈

yes

—

C₄×C₈×C₁₇

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂³

yes

—

—

D₄

yes

—

C₂×Q₈×C₁₇

C₂×C₈

yes

—

Q₁₆×C₁₇

C₁₆

yes

complemented

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

#41

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

C₂×C₈

yes

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂²×C₄

yes

—

—

—

—

Q₈×C₁₇

—

Q₈×C₁₇

C₂×C₁₆

yes

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₇

#131

yes

characteristic

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₂×C₁₆

yes

—

C₈×C₁₇

#137

yes

—

Q₈×C₁₆

—

C₈×Q₁₆

D₁₇

yes

characteristic

#1367

—

Q₈⋊C₁₆

—

#1312

—

C₂²×C₁₇

C₈×D₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

C₂²×C₁₆

yes

characteristic

C₄×C₁₇

#504

yes

characteristic

C₈×Q₈

C₂×D₁₇

yes

—

C₄×Q₁₆

Dic₁₇

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

Dic₁₇

yes

characteristic

Q₈⋊C₈

Dic₁₇

yes

—

C₈²

C₂×D₁₇

yes

characteristic

C₂×C₁₇

#1579

yes

characteristic

C₂×C₁₇

D₄×C₁₆

yes

C₂×C₁₇

yes

C₂×Q₁₆

yes

C₄×Q₈

C₂×Dic₁₇

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

#3622

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×Dic₁₇

yes

—

#103

#3622

yes

—

C₄×C₈

—

C₄×C₈

C₂×Dic₁₇

yes

characteristic

C₄×C₈

C₂×Dic₁₇

yes

characteristic

C₄×C₈

C₂²×D₁₇

yes

characteristic

C₁₇

#12059

yes

characteristic, complemented, Sylow

C₂×Q₈

C₂×#3622

yes

—

Q₁₆

#63214

yes

complemented

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₂×#3622

yes

characteristic

103

C₂×C₈

C₂×#3622

yes

characteristic

104

C₂×C₈

D₄×D₁₇

yes

characteristic

105

C₂×C₈

#63254

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂×#3622

yes

—

108

C₄²

—

109

C₄²

C₂²×Dic₁₇

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

113

Q₈

—

114

Q₈

C₂×#63214

yes

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂²×#3622

yes

characteristic

118

C₂×C₄

D₄×Dic₁₇

yes

characteristic

119

C₂×C₄

#10587634

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

C₂×#63214

yes

—

125

C₈

#10587604

yes

—

126

C₂²

D₄×#3622

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#11762916

yes

characteristic

131

C₄

#11762931

yes

characteristic

132

C₄

C₂²×#63214

yes

characteristic

133

C₄

#11763226

yes

characteristic

134

C₂

#14527603

yes

characteristic

135

C₂

D₄×#63214

yes

characteristic

136

C₂

#14530626

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₇
commutator	C₄×C₁₇	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₈×Q₁₆×C₁₇	C₂
socle, prime radical	C₂²×C₁₇	C₈×D₄
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	C₈×Q₁₆	D₁₇
hypocenter	C₁₇	#12059
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₁₆

subgroup

G/H

center

C₂×C₈

D₄×D₁₇

commutator

C₄×C₁₇

C₂²×C₁₆

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₈×Q₁₆×C₁₇

C₂

socle, prime radical

C₂²×C₁₇

C₈×D₄

cosocle

C₄×C₈×C₁₇

C₂³

hypercenter

C₈×Q₁₆

D₁₇

hypocenter

C₁₇

#12059

LCS

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₁₆

N	H ≅ G/N	type
30. Q₁₆×C₁₇	C₁₆	semidirect
97. C₁₇	#12059	semidirect
99. Q₁₆	#63214	semidirect

H ≅ G/N

type

30. Q₁₆×C₁₇

C₁₆

semidirect

97. C₁₇

#12059

semidirect

99. Q₁₆

#63214

semidirect

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

subgroup	index
Q₈×#63214	2
#14526848	2
Q₈⋊#63214	2
#14526702	2
C₈×Q₁₆×C₁₇	2
#12059	17

subgroup

index

C₈×Q₁₆×C₁₇

Subgroups — #16946163

Conjugacy classes of subgroups

Complements

Maximal subgroups