Group #14544968 — subgroups

Subgroups — #14544968

Order 2176.

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

normal subgroups only

This group has 276 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	#11763768	C₂	1	1	yes	maximal, characteristic
2	#11763753	C₂	1	1	yes	maximal, characteristic
3	#11763682	C₂	1	4	yes	maximal
4	C₄⋊Q₈×D₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈×D₁₇	—	2	1	no	—
6	C₂×C₄⋊#595	C₂²	1	2	yes	—
7	C₂×#63242	C₂²	1	2	yes	—
8	C₁₇×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
9	#10587630	—	2	1	no	—
10	C₄⋊C₄×D₁₇	—	2	1	no	—
11	C₄⋊C₄×D₁₇	C₂²	1	2	yes	—
12	#10587574	—	2	1	no	—
13	#10587574	C₄	1	2	yes	complemented
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	#10587542	—	2	1	no	—
19	#10587542	—	2	1	no	—
20	C₂²×#595	—	2	2	no	—
21	C₂×Q₈×C₁₇	—	2	1	no	—
22	Q₈×D₁₇	—	4	4	no	—
23	C₂×C₄×D₁₇	—	2	2	no	—
24	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
25	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
26	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
27	C₂×Dic₃₄	—	2	1	no	—
28	C₂×Dic₃₄	—	2	2	no	—
29	C₂×#3624	—	2	2	no	—
30	C₄⋊#595	—	2	4	no	—
31	#63242	—	2	2	no	—
32	#63242	Q₈	1	4	yes	complemented
33	C₄⋊C₄×C₁₇	—	2	1	no	—
34	C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
35	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
36	C₄⋊Dic₁₇	—	2	1	no	—
37	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
38	#63225	—	2	2	no	—
39	#63225	—	2	2	no	—
40	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
41	C₄×Dic₁₇	D₄	1	1	yes	characteristic
42	C₄×Dic₁₇	D₄	1	1	yes	characteristic
43	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂×#595	—	4	2	no	—
45	C₂×#595	—	4	2	no	—
46	C₂×#595	—	4	2	no	—
47	Q₈×C₁₇	—	4	2	no	—
48	C₂×C₄×C₁₇	—	2	2	no	—
49	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×Dic₁₇	—	2	2	no	—
53	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
56	C₄×D₁₇	—	2	2	no	—
57	C₄×D₁₇	—	2	2	no	—
58	C₄×D₁₇	—	4	4	no	—
59	C₄×D₁₇	C₂×Q₈	1	2	yes	—
60	C₄×D₁₇	#41	1	2	yes	—
61	Dic₃₄	—	4	4	no	—
62	Dic₃₄	—	4	2	no	—
63	#3624	—	2	4	no	—
64	#1993	—	17	1	no	maximal, Sylow
65	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×D₁₇	#124	1	1	yes	characteristic
67	C₂×D₁₇	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×D₁₇	AGL(1,ℤ/8)	1	2	yes	—
69	C₂×D₁₇	#137	1	2	yes	—
70	#595	—	8	2	no	—
71	C₄×C₁₇	—	2	1	no	—
72	C₄×C₁₇	—	2	1	no	—
73	C₄×C₁₇	—	4	2	no	—
74	C₄×C₁₇	C₄×Q₈	1	2	yes	—
75	Dic₁₇	—	2	1	no	—
76	Dic₁₇	—	2	1	no	—
77	Dic₁₇	—	4	2	no	—
78	Dic₁₇	C₂²⋊Q₈	1	2	yes	—
79	C₂×C₄⋊Q₈	—	17	1	no	—
80	C₂×#97	—	17	1	no	—
81	#382	—	17	1	no	—
82	#340	—	17	4	no	—
83	C₂×C₁₇	#391	1	1	yes	characteristic
84	C₂×C₁₇	#418	1	1	yes	characteristic
85	C₂×C₁₇	#419	1	1	yes	characteristic
86	D₁₇	#489	1	2	yes	—
87	D₁₇	#501	1	2	yes	—
88	C₂²×Q₈	—	34	1	no	—
89	C₂²×C₈	—	17	2	no	—
90	C₄⋊Q₈	—	17	2	no	—
91	C₄⋊Q₈	—	17	1	no	—
92	C₄⋊Q₈	—	34	1	no	—
93	C₄⋊Q₈	—	34	1	no	—
94	C₄⋊Q₈	#595	1	1	yes	characteristic, complemented
95	C₂×C₄⋊C₄	—	17	2	no	—
96	C₂×C₄⋊C₄	—	17	2	no	—
97	C₂×C₄⋊C₄	—	34	1	no	—
98	C₂×C₄²	—	17	1	no	—
99	#97	—	34	1	no	—
100	#97	—	34	1	no	—
101	#95	—	34	1	no	—
102	#95	—	34	1	no	—
103	C₁₇	#1993	1	1	yes	characteristic, complemented, Sylow
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	34	2	no	—
106	C₂×Q₈	—	34	1	no	—
107	C₂×Q₈	—	68	4	no	—
108	C₂²×C₄	—	17	1	no	—
109	C₂²×C₄	—	17	1	no	—
110	C₂²×C₄	—	17	1	no	—
111	C₂²×C₄	—	34	2	no	—
112	C₂²×C₄	—	34	2	no	—
113	C₂×C₈	—	17	4	no	—
114	C₂×C₈	—	34	2	no	—
115	C₂×C₈	—	34	2	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	17	2	no	—
118	C₄⋊C₄	—	34	2	no	—
119	C₄⋊C₄	—	34	2	no	—
120	C₄⋊C₄	—	34	4	no	—
121	C₄⋊C₄	—	34	1	no	—
122	C₄⋊C₄	C₂×#595	1	2	yes	—
123	C₄²	—	17	1	no	—
124	C₄²	—	17	1	no	—
125	C₄²	—	17	1	no	—
126	C₄²	C₂×#595	1	1	yes	characteristic
127	C₂³	—	17	1	no	—
128	Q₈	—	4	2	no	—
129	Q₈	—	68	2	no	—
130	Q₈	—	68	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	17	1	no	—
133	C₂×C₄	—	17	2	no	—
134	C₂×C₄	—	17	2	no	—
135	C₂×C₄	—	17	1	no	—
136	C₂×C₄	—	17	1	no	—
137	C₂×C₄	—	34	2	no	—
138	C₂×C₄	—	34	2	no	—
139	C₂×C₄	—	34	2	no	—
140	C₂×C₄	—	68	4	no	—
141	C₂×C₄	—	68	2	no	—
142	C₂×C₄	—	68	2	no	—
143	C₂×C₄	—	68	2	no	—
144	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
145	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
146	C₂×C₄	C₂²×#595	1	1	yes	characteristic
147	C₈	—	34	4	no	—
148	C₂²	—	17	1	no	—
149	C₂²	—	17	1	no	—
150	C₂²	—	17	2	no	—
151	C₂²	—	17	2	no	—
152	C₂²	C₂×C₂²⋊#595	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	2	no	—
156	C₄	—	17	2	no	—
157	C₄	—	34	1	no	—
158	C₄	—	34	1	no	—
159	C₄	—	68	2	no	—
160	C₄	—	136	2	no	—
161	C₄	Q₈×#595	1	2	yes	—
162	C₂	—	17	2	no	—
163	C₂	—	17	2	no	—
164	C₂	(C₂×#595)⋊Q₈	1	1	yes	characteristic
165	C₂	#11763801	1	1	yes	characteristic
166	C₂	#11763802	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763768

C₂

yes

maximal, characteristic

#11763753

C₂

yes

maximal, characteristic

#11763682

C₂

yes

maximal

C₄⋊Q₈×D₁₇

C₂

yes

maximal, characteristic

C₂×Q₈×D₁₇

—

C₂×C₄⋊#595

C₂²

yes

—

C₂×#63242

C₂²

yes

—

C₁₇×C₄⋊Q₈

C₄

yes

characteristic, complemented

#10587630

—

C₄⋊C₄×D₁₇

—

C₄⋊C₄×D₁₇

C₂²

yes

—

#10587574

—

#10587574

C₄

yes

complemented

C₄²×D₁₇

C₂²

yes

characteristic

#10587555

C₄

yes

characteristic, complemented

(C₂×#595)⋊C₄

—

(C₂×#595)⋊C₄

—

#10587542

—

#10587542

—

C₂²×#595

—

C₂×Q₈×C₁₇

—

Q₈×D₁₇

—

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

D₄

yes

characteristic

C₂×C₄×D₁₇

D₄

yes

characteristic

C₂×C₄×D₁₇

C₂³

yes

characteristic

C₂×Dic₃₄

—

C₂×Dic₃₄

—

—

—

—

Q₈

yes

complemented

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

#63225

—

#63225

—

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

—

C₄×D₁₇

C₂×Q₈

yes

—

C₄×D₁₇

#41

yes

—

Dic₃₄

—

Dic₃₄

—

#3624

—

#1993

—

maximal, Sylow

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₇

#124

yes

characteristic

C₂×D₁₇

C₄⋊Q₈

yes

characteristic

C₂×D₁₇

AGL(1,ℤ/8)

yes

—

C₂×D₁₇

#137

yes

—

#595

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₄×Q₈

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

C₂²⋊Q₈

yes

—

C₂×C₄⋊Q₈

—

C₂×#97

—

#382

—

#340

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

yes

—

D₁₇

#501

yes

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#595

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

#97

—

100

#97

—

101

#95

—

102

#95

—

103

C₁₇

#1993

yes

characteristic, complemented, Sylow

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

C₂×#595

yes

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

C₂×#595

yes

characteristic

127

C₂³

—

128

Q₈

—

129

Q₈

—

130

Q₈

—

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

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

C₂²⋊#595

yes

characteristic

145

C₂×C₄

C₂²⋊#595

yes

characteristic

146

C₂×C₄

C₂²×#595

yes

characteristic

147

C₈

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂×C₂²⋊#595

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

136

—

161

C₄

Q₈×#595

yes

—

162

C₂

—

163

C₂

—

164

C₂

(C₂×#595)⋊Q₈

yes

characteristic

165

C₂

#11763801

yes

characteristic

166

C₂

#11763802

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	#1993
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
8. C₁₇×C₄⋊Q₈	C₄	semidirect
13. #10587574	C₄	semidirect
15. #10587555	C₄	semidirect
32. #63242	Q₈	semidirect
94. C₄⋊Q₈	#595	semidirect
103. C₁₇	#1993	semidirect

H ≅ G/N

type

8. C₁₇×C₄⋊Q₈

C₄

semidirect

13. #10587574

C₄

semidirect

15. #10587555

C₄

semidirect

32. #63242

Q₈

semidirect

94. C₄⋊Q₈

#595

semidirect

103. C₁₇

#1993

semidirect

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

subgroup	index
#11763768	2
#11763753	2
#11763682	2
C₄⋊Q₈×D₁₇	2
#1993	17

subgroup

index

C₄⋊Q₈×D₁₇

Subgroups — #14544968

Conjugacy classes of subgroups

Complements

Maximal subgroups