Group #14544969 — subgroups

Subgroups — #14544969

Order 2176.

284 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 78 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	#11763813	C₂	1	1	yes	maximal, characteristic
2	C₄²×#595	C₂	1	1	yes	maximal, characteristic
3	#11763679	C₂	1	4	yes	maximal
4	#11763364	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×#595	—	2	1	no	—
6	C₂×C₄×#595	—	2	1	no	—
7	C₂×C₄×#595	C₂²	1	2	yes	—
8	C₂×#63243	C₂²	1	2	yes	—
9	C₁₇×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
10	#10587632	—	2	1	no	—
11	Q₈×Dic₁₇	—	2	1	no	—
12	#10587583	—	2	1	no	—
13	#10587578	C₂²	1	2	yes	—
14	#10587576	C₄	1	2	yes	complemented
15	#10587573	—	2	1	no	—
16	C₄²×D₁₇	C₂²	1	1	yes	characteristic
17	#10587555	C₄	1	1	yes	characteristic, complemented
18	C₄⋊#3624	—	2	1	no	—
19	C₄⋊#3624	—	2	1	no	—
20	C₂²×#595	—	2	2	no	—
21	C₂×Q₈×C₁₇	—	2	1	no	—
22	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
23	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
24	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
25	C₂×Dic₃₄	—	2	1	no	—
26	C₂×#3624	—	2	2	no	—
27	C₄×#595	—	2	4	no	—
28	C₄×#595	—	2	4	no	—
29	C₄×#595	—	2	2	no	—
30	C₄×#595	Q₈	1	4	yes	complemented
31	#63243	—	2	4	no	—
32	C₄⋊C₄×C₁₇	—	2	1	no	—
33	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
34	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
35	#63227	—	2	2	no	—
36	C₄⋊Dic₁₇	—	2	1	no	—
37	C₄⋊Dic₁₇	—	2	2	no	—
38	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
39	#63225	—	2	2	no	—
40	C₄×Dic₁₇	—	2	2	no	—
41	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₁₇	D₄	1	1	yes	characteristic
43	C₄×Dic₁₇	D₄	1	1	yes	characteristic
44	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂×#595	—	2	4	no	—
46	C₂×#595	—	2	4	no	—
47	C₂×#595	—	2	4	no	—
48	Q₈×C₁₇	—	4	2	no	—
49	C₂×C₄×C₁₇	—	2	2	no	—
50	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×Dic₁₇	—	2	2	no	—
54	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
57	C₄×D₁₇	—	2	2	no	—
58	C₄×D₁₇	—	2	2	no	—
59	C₄×D₁₇	C₂×Q₈	1	2	yes	—
60	C₄×D₁₇	#41	1	2	yes	—
61	Dic₃₄	—	4	2	no	—
62	#3624	—	4	2	no	—
63	#1994	—	17	1	no	maximal, Sylow
64	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×D₁₇	—	2	1	no	—
66	C₂×D₁₇	—	2	1	no	—
67	C₂×D₁₇	C₂²⋊Q₈	1	2	yes	—
68	#595	—	2	8	no	—
69	C₄×C₁₇	—	4	2	no	—
70	C₄×C₁₇	#104	1	2	yes	—
71	C₄×C₁₇	#104	1	2	yes	—
72	C₄×C₁₇	C₄×Q₈	1	2	yes	—
73	Dic₁₇	—	2	1	no	—
74	Dic₁₇	—	2	1	no	—
75	Dic₁₇	—	4	2	no	—
76	Dic₁₇	#124	1	1	yes	characteristic
77	Dic₁₇	C₄⋊Q₈	1	1	yes	characteristic
78	#533	—	17	1	no	—
79	#423	—	17	1	no	—
80	C₄³	—	17	1	no	—
81	#339	—	17	4	no	—
82	C₂×C₁₇	#391	1	1	yes	characteristic
83	C₂×C₁₇	C₂×#104	1	1	yes	characteristic
84	C₂×C₁₇	C₂×#104	1	1	yes	characteristic
85	D₁₇	—	2	2	no	—
86	C₂×M₁₆	—	17	2	no	—
87	C₄⋊Q₈	—	17	1	no	—
88	C₄⋊Q₈	#595	1	1	yes	characteristic, complemented
89	#125	—	17	2	no	—
90	C₂²⋊Q₈	—	34	1	no	—
91	C₂²⋊Q₈	—	34	1	no	—
92	C₄×Q₈	—	34	1	no	—
93	C₄×Q₈	—	34	1	no	—
94	#117	—	17	2	no	—
95	C₂×C₄²	—	17	1	no	—
96	C₂×C₄²	—	17	2	no	—
97	C₂×C₄²	—	34	1	no	—
98	C₂×C₄²	—	34	1	no	—
99	C₄⋊C₈	—	34	1	no	—
100	C₄⋊C₈	—	34	1	no	—
101	C₁₇	#1994	1	1	yes	characteristic, complemented, Sylow
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	34	1	no	—
104	C₂²×C₄	—	17	1	no	—
105	C₂²×C₄	—	17	1	no	—
106	C₂²×C₄	—	17	1	no	—
107	C₂²×C₄	—	34	2	no	—
108	M₁₆	—	34	4	no	—
109	C₂×C₈	—	34	2	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	17	2	no	—
112	C₄⋊C₄	—	34	2	no	—
113	C₄⋊C₄	—	34	2	no	—
114	C₄⋊C₄	—	34	1	no	—
115	C₄⋊C₄	C₂×#595	1	2	yes	—
116	C₂²⋊C₄	—	34	2	no	—
117	C₄²	—	17	1	no	—
118	C₄²	—	17	1	no	—
119	C₄²	—	17	1	no	—
120	C₄²	—	17	4	no	—
121	C₄²	—	34	2	no	—
122	C₄²	—	34	2	no	—
123	C₄²	—	34	4	no	—
124	C₄²	—	34	4	no	—
125	C₄²	C₂×#595	1	1	yes	characteristic
126	C₂³	—	17	1	no	—
127	Q₈	—	4	2	no	—
128	Q₈	—	68	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	17	1	no	—
131	C₂×C₄	—	17	2	no	—
132	C₂×C₄	—	17	2	no	—
133	C₂×C₄	—	17	1	no	—
134	C₂×C₄	—	17	1	no	—
135	C₂×C₄	—	34	4	no	—
136	C₂×C₄	—	34	4	no	—
137	C₂×C₄	—	34	4	no	—
138	C₂×C₄	—	34	2	no	—
139	C₂×C₄	—	34	2	no	—
140	C₂×C₄	—	34	2	no	—
141	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
142	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
143	C₂×C₄	C₂²×#595	1	1	yes	characteristic
144	C₈	—	68	2	no	—
145	C₂²	—	17	2	no	—
146	C₂²	—	34	1	no	—
147	C₂²	—	34	1	no	—
148	C₂²	C₂×C₂²⋊#595	1	1	yes	characteristic
149	C₄	—	4	2	no	—
150	C₄	—	17	1	no	—
151	C₄	—	17	1	no	—
152	C₄	—	34	8	no	—
153	C₄	—	34	1	no	—
154	C₄	—	34	1	no	—
155	C₄	—	68	2	no	—
156	C₄	#10587548	1	2	yes	—
157	C₄	#10587550	1	2	yes	—
158	C₄	Q₈×#595	1	2	yes	—
159	C₂	—	34	2	no	—
160	C₂	(C₂×#595)⋊Q₈	1	1	yes	characteristic
161	C₂	C₂×#10587548	1	1	yes	characteristic
162	C₂	C₂×#10587550	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763813

C₂

yes

maximal, characteristic

C₄²×#595

C₂

yes

maximal, characteristic

#11763679

C₂

yes

maximal

#11763364

C₂

yes

maximal, characteristic

C₂×C₄×#595

—

C₂×C₄×#595

—

C₂×C₄×#595

C₂²

yes

—

C₂×#63243

C₂²

yes

—

C₁₇×C₄⋊Q₈

C₄

yes

characteristic, complemented

#10587632

—

Q₈×Dic₁₇

—

#10587583

—

#10587578

C₂²

yes

—

#10587576

C₄

yes

complemented

#10587573

—

C₄²×D₁₇

C₂²

yes

characteristic

#10587555

C₄

yes

characteristic, complemented

C₄⋊#3624

—

C₄⋊#3624

—

C₂²×#595

—

C₂×Q₈×C₁₇

—

C₂×C₄×D₁₇

D₄

yes

characteristic

C₂×C₄×D₁₇

D₄

yes

characteristic

C₂×C₄×D₁₇

C₂³

yes

characteristic

C₂×Dic₃₄

—

—

—

—

—

Q₈

yes

complemented

#63243

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

C₂×C₄

yes

characteristic

#63227

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

#63225

—

C₄×Dic₁₇

—

C₄×Dic₁₇

C₂×C₄

yes

characteristic

C₄×Dic₁₇

D₄

yes

characteristic

C₄×Dic₁₇

D₄

yes

characteristic

C₂²×D₁₇

C₂×D₄

yes

—

—

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₇

C₂×D₄

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

C₂×Q₈

yes

—

C₄×D₁₇

#41

yes

—

Dic₃₄

—

#3624

—

#1994

—

maximal, Sylow

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

C₂²⋊Q₈

yes

—

#595

—

C₄×C₁₇

—

C₄×C₁₇

#104

yes

—

C₄×C₁₇

#104

yes

—

C₄×C₁₇

C₄×Q₈

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

#124

yes

characteristic

Dic₁₇

C₄⋊Q₈

yes

characteristic

#533

—

#423

—

C₄³

—

#339

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

#595

yes

characteristic, complemented

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₁₇

#1994

yes

characteristic, complemented, Sylow

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

M₁₆

—

109

C₂×C₈

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

C₂×#595

yes

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

C₂×#595

yes

characteristic

126

C₂³

—

127

Q₈

—

128

Q₈

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

C₂²⋊#595

yes

characteristic

142

C₂×C₄

C₂²⋊#595

yes

characteristic

143

C₂×C₄

C₂²×#595

yes

characteristic

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

C₂×C₂²⋊#595

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#10587548

yes

—

157

C₄

#10587550

yes

—

158

C₄

Q₈×#595

yes

—

159

C₂

—

160

C₂

(C₂×#595)⋊Q₈

yes

characteristic

161

C₂

C₂×#10587548

yes

characteristic

162

C₂

C₂×#10587550

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#595
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#595
Fitting	C₁₇×C₄⋊Q₈	C₄
socle	C₂²×C₁₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₇	C₂×D₄
cosocle	C₂×C₄×D₁₇	C₂³
hypercenter	C₄⋊Q₈	#595
hypocenter	C₁₇	#1994
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#595

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#595

Fitting

C₁₇×C₄⋊Q₈

C₄

socle

C₂²×C₁₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₇

C₂×D₄

cosocle

C₂×C₄×D₁₇

C₂³

C₄⋊Q₈

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
9. C₁₇×C₄⋊Q₈	C₄	semidirect
14. #10587576	C₄	semidirect
17. #10587555	C₄	semidirect
30. C₄×#595	Q₈	semidirect
88. C₄⋊Q₈	#595	semidirect
101. C₁₇	#1994	semidirect

H ≅ G/N

type

9. C₁₇×C₄⋊Q₈

C₄

semidirect

14. #10587576

C₄

semidirect

17. #10587555

C₄

semidirect

30. C₄×#595

Q₈

semidirect

88. C₄⋊Q₈

#595

semidirect

101. C₁₇

#1994

semidirect

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

subgroup	index
#11763813	2
C₄²×#595	2
#11763679	2
#11763364	2
#1994	17

subgroup

index

C₄²×#595

Subgroups — #14544969

Conjugacy classes of subgroups

Complements

Maximal subgroups