Group #14544314 — subgroups

Subgroups — #14544314

Order 2176.

228 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 60 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	#11763832	C₂	1	1	yes	maximal, characteristic
2	#11763832	C₂	1	1	yes	maximal, characteristic
3	#11763668	C₂	1	4	yes	maximal, complemented
4	C₄⋊Q₈×D₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈×D₁₇	—	2	1	no	—
6	C₁₇×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	#10587630	—	2	1	no	—
8	C₄⋊C₄×D₁₇	—	2	1	no	—
9	C₄⋊C₄×D₁₇	C₄	1	2	yes	—
10	#10587574	—	2	1	no	—
11	#10587574	C₂²	1	2	yes	—
12	C₄²×D₁₇	C₂²	1	1	yes	characteristic
13	#10587555	C₄	1	1	yes	characteristic
14	#10587544	C₂²	1	2	yes	—
15	#10587544	C₂²	1	2	yes	—
16	#10587543	—	2	1	no	—
17	#10587543	—	2	1	no	—
18	#10587542	—	2	1	no	—
19	#10587542	—	2	1	no	—
20	C₂×Q₈×C₁₇	—	2	1	no	—
21	Q₈×D₁₇	—	4	4	no	—
22	C₂×C₄×D₁₇	—	2	2	no	—
23	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
25	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
26	C₂×Dic₃₄	—	2	1	no	—
27	C₂×Dic₃₄	—	2	2	no	—
28	C₂×#3624	—	2	2	no	—
29	C₂×#3624	—	2	2	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	C₄⋊Dic₁₇	—	2	1	no	—
34	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
35	#63225	—	2	2	no	—
36	#63225	—	2	2	no	—
37	C₄×Dic₁₇	D₄	1	1	yes	characteristic
38	C₄×Dic₁₇	D₄	1	1	yes	characteristic
39	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
40	C₂²×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
41	Q₈×C₁₇	—	4	2	no	—
42	C₂×C₄×C₁₇	—	2	2	no	—
43	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×Dic₁₇	—	2	2	no	—
47	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
50	C₄×D₁₇	—	2	2	no	—
51	C₄×D₁₇	—	2	2	no	—
52	C₄×D₁₇	—	2	2	no	—
53	C₄×D₁₇	—	4	4	no	—
54	Dic₃₄	—	4	2	no	—
55	Dic₃₄	—	4	4	no	—
56	#3624	—	4	2	no	—
57	#3624	—	4	2	no	—
58	#1556	—	17	1	no	maximal, Sylow
59	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×D₁₇	—	2	1	no	—
61	C₂×D₁₇	—	2	1	no	—
62	C₂×D₁₇	#101	1	2	yes	—
63	C₄×C₁₇	—	2	1	no	—
64	C₄×C₁₇	—	2	1	no	—
65	C₄×C₁₇	—	2	1	no	—
66	C₄×C₁₇	—	4	2	no	—
67	Dic₁₇	—	2	1	no	—
68	Dic₁₇	—	4	2	no	—
69	Dic₁₇	AGL(1,ℤ/8)	1	2	yes	—
70	Dic₁₇	#137	1	2	yes	—
71	C₂×C₄⋊Q₈	—	17	1	no	—
72	#432	—	17	1	no	—
73	#432	—	17	1	no	—
74	#332	—	17	4	no	—
75	C₂×C₁₇	C₂×#101	1	1	yes	characteristic
76	C₂×C₁₇	#418	1	1	yes	characteristic
77	C₂×C₁₇	#419	1	1	yes	characteristic
78	D₁₇	—	2	2	no	—
79	C₂²×Q₈	—	34	1	no	—
80	C₄⋊Q₈	—	17	2	no	—
81	C₄⋊Q₈	—	17	1	no	—
82	C₄⋊Q₈	—	34	1	no	—
83	C₄⋊Q₈	—	34	1	no	—
84	C₄⋊Q₈	#595	1	1	yes	characteristic
85	C₂×C₄⋊C₄	—	17	2	no	—
86	C₂×C₄⋊C₄	—	34	1	no	—
87	C₂×C₄²	—	17	1	no	—
88	C₄⋊C₈	—	17	2	no	—
89	C₄⋊C₈	—	17	2	no	—
90	C₂²⋊C₈	—	34	1	no	—
91	C₂²⋊C₈	—	34	1	no	—
92	#97	—	34	1	no	—
93	#97	—	34	1	no	—
94	C₁₇	#1556	1	1	yes	characteristic, complemented, Sylow
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	34	2	no	—
97	C₂×Q₈	—	34	1	no	—
98	C₂×Q₈	—	68	4	no	—
99	C₂²×C₄	—	17	1	no	—
100	C₂²×C₄	—	17	1	no	—
101	C₂²×C₄	—	17	1	no	—
102	C₂²×C₄	—	34	2	no	—
103	C₂×C₈	—	34	2	no	—
104	C₂×C₈	—	34	2	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	17	2	no	—
107	C₄⋊C₄	—	34	2	no	—
108	C₄⋊C₄	—	34	2	no	—
109	C₄⋊C₄	—	34	1	no	—
110	C₄⋊C₄	C₂×#595	1	2	yes	—
111	C₄²	—	17	1	no	—
112	C₄²	—	17	1	no	—
113	C₄²	—	17	1	no	—
114	C₄²	C₂×#595	1	1	yes	characteristic
115	C₂³	—	17	1	no	—
116	Q₈	—	4	2	no	—
117	Q₈	—	68	2	no	—
118	Q₈	—	68	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	17	1	no	—
121	C₂×C₄	—	17	1	no	—
122	C₂×C₄	—	17	1	no	—
123	C₂×C₄	—	34	2	no	—
124	C₂×C₄	—	34	2	no	—
125	C₂×C₄	—	34	2	no	—
126	C₂×C₄	—	34	2	no	—
127	C₂×C₄	—	68	4	no	—
128	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
129	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
130	C₂×C₄	C₂²×#595	1	1	yes	characteristic
131	C₈	—	68	2	no	—
132	C₈	—	68	2	no	—
133	C₂²	—	17	2	no	—
134	C₂²	—	34	1	no	—
135	C₂²	—	34	1	no	—
136	C₂²	C₂×C₂²⋊#595	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	2	no	—
141	C₄	—	17	2	no	—
142	C₄	—	17	2	no	—
143	C₄	—	34	1	no	—
144	C₄	—	68	2	no	—
145	C₂	—	34	2	no	—
146	C₂	D₁₇⋊#101	1	1	yes	characteristic
147	C₂	#11763801	1	1	yes	characteristic
148	C₂	#11763802	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763832

C₂

yes

maximal, characteristic

#11763832

C₂

yes

maximal, characteristic

#11763668

C₂

yes

maximal, complemented

C₄⋊Q₈×D₁₇

C₂

yes

maximal, characteristic

C₂×Q₈×D₁₇

—

C₁₇×C₄⋊Q₈

C₄

yes

characteristic

#10587630

—

C₄⋊C₄×D₁₇

—

C₄⋊C₄×D₁₇

C₄

yes

—

#10587574

—

#10587574

C₂²

yes

—

C₄²×D₁₇

C₂²

yes

characteristic

#10587555

C₄

yes

characteristic

#10587544

C₂²

yes

—

#10587544

C₂²

yes

—

—

—

—

—

C₂×Q₈×C₁₇

—

Q₈×D₁₇

—

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

C₂×C₄

yes

characteristic

C₂×C₄×D₁₇

D₄

yes

characteristic

C₂×C₄×D₁₇

D₄

yes

characteristic

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×#3624

—

C₂×#3624

—

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

D₄

yes

characteristic

C₄×Dic₁₇

D₄

yes

characteristic

C₄×Dic₁₇

C₂³

yes

characteristic

C₂²×D₁₇

C₂²⋊C₄

yes

characteristic

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

C₂×D₄

yes

characteristic

C₂×Dic₁₇

C₂×D₄

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

Dic₃₄

—

Dic₃₄

—

—

—

—

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

#101

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

AGL(1,ℤ/8)

yes

—

Dic₁₇

#137

yes

—

C₂×C₄⋊Q₈

—

#432

—

#432

—

#332

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

D₁₇

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#595

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₁₇

#1556

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

C₂×#595

yes

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₂×#595

yes

characteristic

115

C₂³

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂²⋊#595

yes

characteristic

129

C₂×C₄

C₂²⋊#595

yes

characteristic

130

C₂×C₄

C₂²×#595

yes

characteristic

131

C₈

—

132

C₈

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

C₂×C₂²⋊#595

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₂

—

146

C₂

D₁₇⋊#101

yes

characteristic

147

C₂

#11763801

yes

characteristic

148

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	C₂²⋊C₄
cosocle	C₄×Dic₁₇	C₂³
hypercenter	C₄⋊Q₈	#595
hypocenter	C₁₇	#1556
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×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

C₂²⋊C₄

cosocle

C₄×Dic₁₇

C₂³

C₄⋊Q₈

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
3. #11763668	C₂	semidirect
94. C₁₇	#1556	semidirect

H ≅ G/N

type

3. #11763668

C₂

semidirect

94. C₁₇

#1556

semidirect

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

subgroup	index
#11763832	2
#11763668	2
C₄⋊Q₈×D₁₇	2
#1556	17

subgroup

index

#11763832

#11763668

C₄⋊Q₈×D₁₇

#1556

Subgroups — #14544314

Conjugacy classes of subgroups

Complements

Maximal subgroups