Group #14544972 — subgroups

Subgroups — #14544972

Order 2176.

244 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 70 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	#11763814	C₂	1	1	yes	maximal, characteristic
2	#11763745	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	C₂²	1	2	yes	—
6	C₂×#63243	C₂²	1	2	yes	—
7	C₁₇×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
8	#10587632	—	2	1	no	—
9	Q₈×Dic₁₇	—	2	1	no	—
10	#10587583	—	2	1	no	—
11	#10587578	C₂²	1	2	yes	—
12	#10587576	C₄	1	2	yes	complemented
13	#10587573	—	2	1	no	—
14	C₄²×D₁₇	C₂²	1	1	yes	characteristic
15	#10587555	C₄	1	1	yes	characteristic, complemented
16	(C₂×#595)⋊C₄	—	2	1	no	—
17	(C₂×#595)⋊C₄	—	2	1	no	—
18	#10587544	—	2	1	no	—
19	#10587544	—	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	2	no	—
28	C₄×#595	Q₈	1	4	yes	complemented
29	#63243	—	2	4	no	—
30	C₄⋊C₄×C₁₇	—	2	1	no	—
31	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
32	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
33	#63227	—	2	2	no	—
34	C₄⋊Dic₁₇	—	2	1	no	—
35	C₄⋊Dic₁₇	—	2	2	no	—
36	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
37	#63225	—	2	2	no	—
38	C₄×Dic₁₇	—	2	2	no	—
39	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₁₇	D₄	1	1	yes	characteristic
41	C₄×Dic₁₇	D₄	1	1	yes	characteristic
42	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
43	C₂×#595	—	2	4	no	—
44	C₂×#595	—	4	2	no	—
45	C₂×#595	—	4	2	no	—
46	Q₈×C₁₇	—	4	2	no	—
47	C₂×C₄×C₁₇	—	2	2	no	—
48	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×Dic₁₇	—	2	2	no	—
52	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
55	C₄×D₁₇	—	4	1	no	—
56	C₄×D₁₇	—	4	1	no	—
57	C₄×D₁₇	C₂×Q₈	1	2	yes	—
58	C₄×D₁₇	#41	1	2	yes	—
59	Dic₃₄	—	4	2	no	—
60	#3624	—	4	2	no	—
61	#1995	—	17	1	no	maximal, Sylow
62	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×D₁₇	—	2	1	no	—
64	C₂×D₁₇	—	2	1	no	—
65	C₂×D₁₇	C₂²⋊Q₈	1	2	yes	—
66	#595	—	4	4	no	—
67	C₄×C₁₇	—	2	1	no	—
68	C₄×C₁₇	—	2	1	no	—
69	C₄×C₁₇	—	4	2	no	—
70	C₄×C₁₇	C₄×Q₈	1	2	yes	—
71	Dic₁₇	—	2	1	no	—
72	Dic₁₇	—	2	1	no	—
73	Dic₁₇	—	4	2	no	—
74	Dic₁₇	#124	1	1	yes	characteristic
75	Dic₁₇	C₄⋊Q₈	1	1	yes	characteristic
76	#533	—	17	1	no	—
77	#423	—	17	1	no	—
78	#376	—	17	1	no	—
79	#339	—	17	4	no	—
80	C₂×C₁₇	#391	1	1	yes	characteristic
81	C₂×C₁₇	#421	1	1	yes	characteristic
82	C₂×C₁₇	#421	1	1	yes	characteristic
83	D₁₇	—	2	2	no	—
84	C₂×M₁₆	—	17	2	no	—
85	C₄⋊Q₈	—	17	1	no	—
86	C₄⋊Q₈	#595	1	1	yes	characteristic, complemented
87	#125	—	17	2	no	—
88	C₂²⋊Q₈	—	34	1	no	—
89	C₂²⋊Q₈	—	34	1	no	—
90	C₄×Q₈	—	34	1	no	—
91	C₄×Q₈	—	34	1	no	—
92	#117	—	17	2	no	—
93	C₂×C₄²	—	17	1	no	—
94	C₂×C₄²	—	17	2	no	—
95	C₄⋊C₈	—	34	1	no	—
96	C₄⋊C₈	—	34	1	no	—
97	#95	—	34	1	no	—
98	#95	—	34	1	no	—
99	C₁₇	#1995	1	1	yes	characteristic, complemented, Sylow
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	—	34	1	no	—
102	C₂²×C₄	—	17	1	no	—
103	C₂²×C₄	—	17	1	no	—
104	C₂²×C₄	—	17	1	no	—
105	C₂²×C₄	—	34	2	no	—
106	M₁₆	—	34	4	no	—
107	C₂×C₈	—	34	2	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	17	2	no	—
110	C₄⋊C₄	—	34	2	no	—
111	C₄⋊C₄	—	34	2	no	—
112	C₄⋊C₄	—	34	1	no	—
113	C₄⋊C₄	C₂×#595	1	2	yes	—
114	C₂²⋊C₄	—	34	2	no	—
115	C₄²	—	17	1	no	—
116	C₄²	—	17	1	no	—
117	C₄²	—	17	1	no	—
118	C₄²	—	17	4	no	—
119	C₄²	—	34	2	no	—
120	C₄²	—	34	2	no	—
121	C₄²	C₂×#595	1	1	yes	characteristic
122	C₂³	—	17	1	no	—
123	Q₈	—	4	2	no	—
124	Q₈	—	68	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	17	1	no	—
127	C₂×C₄	—	17	2	no	—
128	C₂×C₄	—	17	2	no	—
129	C₂×C₄	—	17	1	no	—
130	C₂×C₄	—	17	1	no	—
131	C₂×C₄	—	34	4	no	—
132	C₂×C₄	—	34	2	no	—
133	C₂×C₄	—	68	1	no	—
134	C₂×C₄	—	68	1	no	—
135	C₂×C₄	—	68	2	no	—
136	C₂×C₄	—	68	2	no	—
137	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
138	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
139	C₂×C₄	C₂²×#595	1	1	yes	characteristic
140	C₈	—	68	2	no	—
141	C₂²	—	17	2	no	—
142	C₂²	—	34	1	no	—
143	C₂²	—	34	1	no	—
144	C₂²	C₂×C₂²⋊#595	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	4	2	no	—
148	C₄	—	17	1	no	—
149	C₄	—	17	1	no	—
150	C₄	—	34	1	no	—
151	C₄	—	34	1	no	—
152	C₄	—	68	4	no	—
153	C₄	—	68	2	no	—
154	C₄	Q₈×#595	1	2	yes	—
155	C₂	—	34	2	no	—
156	C₂	(C₂×#595)⋊Q₈	1	1	yes	characteristic
157	C₂	#11763807	1	1	yes	characteristic
158	C₂	#11763808	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763814

C₂

yes

maximal, characteristic

#11763745

C₂

yes

maximal, characteristic

#11763679

C₂

yes

maximal

#11763364

C₂

yes

maximal, characteristic

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

(C₂×#595)⋊C₄

—

(C₂×#595)⋊C₄

—

#10587544

—

#10587544

—

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

—

C₂×#3624

—

C₄×#595

—

C₄×#595

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

—

#1995

—

maximal, Sylow

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

C₂²⋊Q₈

yes

—

#595

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₄×Q₈

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

#124

yes

characteristic

Dic₁₇

C₄⋊Q₈

yes

—

—

—

—

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

—

C₄⋊C₈

—

C₄⋊C₈

—

#95

—

#95

—

C₁₇

#1995

yes

characteristic, complemented, Sylow

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

M₁₆

—

107

C₂×C₈

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

C₂×#595

yes

—

114

C₂²⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

C₂×#595

yes

characteristic

122

C₂³

—

123

Q₈

—

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

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

C₂²⋊#595

yes

characteristic

138

C₂×C₄

C₂²⋊#595

yes

characteristic

139

C₂×C₄

C₂²×#595

yes

characteristic

140

C₈

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

C₂×C₂²⋊#595

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

Q₈×#595

yes

—

155

C₂

—

156

C₂

(C₂×#595)⋊Q₈

yes

characteristic

157

C₂

#11763807

yes

characteristic

158

C₂

#11763808

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	#1995
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
7. C₁₇×C₄⋊Q₈	C₄	semidirect
12. #10587576	C₄	semidirect
15. #10587555	C₄	semidirect
28. C₄×#595	Q₈	semidirect
86. C₄⋊Q₈	#595	semidirect
99. C₁₇	#1995	semidirect

H ≅ G/N

type

7. C₁₇×C₄⋊Q₈

C₄

semidirect

12. #10587576

C₄

semidirect

15. #10587555

C₄

semidirect

28. C₄×#595

Q₈

semidirect

86. C₄⋊Q₈

#595

semidirect

99. C₁₇

#1995

semidirect

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

subgroup	index
#11763814	2
#11763745	2
#11763679	2
#11763364	2
#1995	17

subgroup

index

Subgroups — #14544972

Conjugacy classes of subgroups

Complements

Maximal subgroups