Group #14545446 — subgroups

Subgroups — #14545446

Order 2176.

440 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 198 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	#11763888	C₂	1	2	yes	maximal
2	#11763824	C₂	1	6	yes	maximal, complemented
3	C₂×#10587536	C₂	1	3	yes	maximal, complemented
4	D₁₇⋊#108	C₂	1	3	yes	maximal, complemented
5	#131×D₁₇	C₂	1	1	yes	maximal, characteristic
6	#41×D₁₇	C₂²	1	1	yes	characteristic
7	#10587680	—	2	2	no	—
8	#10587678	—	2	6	no	—
9	D₁₇⋊M₁₆	C₂²	1	6	yes	—
10	C₂×#63243	C₂²	1	6	yes	—
11	C₁₇×#131	C₄	1	1	yes	characteristic
12	#10587634	C₄	1	1	yes	characteristic
13	#10587593	C₄	1	3	yes	—
14	M₁₆×D₁₇	C₂²	1	3	yes	—
15	#10587587	C₄	1	3	yes	—
16	C₂×C₈×D₁₇	C₂²	1	3	yes	—
17	#10587536	C₂²	1	1	yes	characteristic
18	#10587536	C₂²	1	3	yes	—
19	#10587536	C₂²	1	6	yes	—
20	#10587535	C₂²	1	6	yes	—
21	#41×C₁₇	C₂×C₄	1	1	yes	characteristic
22	#63256	D₄	1	1	yes	characteristic
23	Q₈×D₁₇	Q₈	1	1	yes	characteristic
24	#63254	D₄	1	3	yes	—
25	D₄×D₁₇	Q₈	1	3	yes	—
26	#63252	C₂×C₄	1	3	yes	—
27	C₂×C₄×D₁₇	C₂³	1	3	yes	—
28	#63247	—	2	6	no	—
29	C₂×#3624	—	2	6	no	—
30	#63243	—	2	6	no	—
31	#63243	C₂³	1	6	yes	—
32	#63243	C₂³	1	2	yes	—
33	#63242	—	2	6	no	—
34	M₁₆×C₁₇	C₂×C₄	1	3	yes	—
35	C₂×C₈×C₁₇	C₂×C₄	1	3	yes	—
36	#63223	C₂×C₄	1	3	yes	—
37	C₂×#3622	C₂×C₄	1	3	yes	—
38	#63218	C₂×C₄	1	6	yes	—
39	C₈×D₁₇	C₂×C₄	1	3	yes	—
40	C₈×D₁₇	C₂×C₄	1	3	yes	—
41	C₈×D₁₇	C₂³	1	3	yes	—
42	C₈×D₁₇	C₂³	1	1	yes	characteristic
43	C₂²×D₁₇	C₂×Q₈	1	3	yes	—
44	Q₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
45	D₄×C₁₇	C₄⋊C₄	1	3	yes	—
46	C₂×C₄×C₁₇	C₂²×C₄	1	3	yes	—
47	C₂²⋊D₁₇	C₄⋊C₄	1	6	yes	—
48	C₂×Dic₁₇	C₂×D₄	1	3	yes	—
49	D₆₈	C₄⋊C₄	1	3	yes	—
50	C₄×D₁₇	C₂²×C₄	1	3	yes	—
51	C₄×D₁₇	C₂×D₄	1	3	yes	—
52	C₄×D₁₇	C₂×Q₈	1	3	yes	—
53	C₄×D₁₇	C₂⁴	1	1	yes	characteristic
54	Dic₃₄	C₄⋊C₄	1	3	yes	—
55	#3624	—	2	2	no	—
56	#3624	—	2	6	no	—
57	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
58	C₈×C₁₇	C₂²×C₄	1	3	yes	—
59	#3622	C₂²×C₄	1	1	yes	characteristic
60	#3622	C₂²×C₄	1	3	yes	—
61	#2916	—	17	1	no	maximal, Sylow
62	C₂²×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
63	C₂×D₁₇	—	4	3	no	—
64	C₂×D₁₇	C₂×C₄⋊C₄	1	3	yes	—
65	C₂×D₁₇	C₂²×Q₈	1	1	yes	characteristic
66	C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
67	C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
68	Dic₁₇	C₂×C₄⋊C₄	1	3	yes	—
69	Dic₁₇	C₂²×D₄	1	1	yes	characteristic
70	#568	—	17	2	no	—
71	C₂×#131	—	17	1	no	—
72	#430	—	17	6	no	—
73	C₂×#108	—	17	3	no	—
74	C₂×#108	—	17	3	no	—
75	C₂×C₁₇	—	2	3	no	—
76	C₂×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
77	D₁₇	—	2	1	no	—
78	D₁₇	—	2	3	no	—
79	C₂×#41	—	17	1	no	—
80	#131	—	17	1	no	—
81	#131	—	17	3	no	—
82	#131	—	17	3	no	—
83	#131	—	34	6	no	—
84	#131	—	34	2	no	—
85	#131	#595	1	1	yes	characteristic
86	C₂×M₁₆	—	17	3	no	—
87	C₂×M₁₆	—	17	6	no	—
88	C₂×M₁₆	—	17	6	no	—
89	C₂²×C₈	—	17	3	no	—
90	#108	—	17	6	no	—
91	#108	—	17	1	no	—
92	#108	—	17	3	no	—
93	#108	—	17	6	no	—
94	C₁₇	#2916	1	1	yes	characteristic, complemented, Sylow
95	#41	—	17	1	no	—
96	#41	—	17	3	no	—
97	#41	—	17	3	no	—
98	#41	C₂×#595	1	1	yes	characteristic
99	C₂×Q₈	—	17	1	no	—
100	C₂×D₄	—	17	3	no	—
101	C₂²×C₄	—	17	3	no	—
102	M₁₆	—	17	3	no	—
103	M₁₆	—	17	6	no	—
104	M₁₆	—	17	6	no	—
105	M₁₆	—	17	2	no	—
106	M₁₆	—	34	6	no	—
107	M₁₆	—	34	6	no	—
108	M₁₆	C₂×#595	1	3	yes	—
109	C₂×C₈	—	17	3	no	—
110	C₂×C₈	—	17	1	no	—
111	C₂×C₈	—	17	3	no	—
112	C₂×C₈	—	17	3	no	—
113	C₂×C₈	—	17	3	no	—
114	C₂×C₈	—	34	6	no	—
115	C₂×C₈	—	34	6	no	—
116	C₂×C₈	C₂×#595	1	3	yes	—
117	C₂³	—	17	3	no	—
118	Q₈	—	17	3	no	—
119	Q₈	C₄⋊#595	1	1	yes	characteristic
120	D₄	—	17	6	no	—
121	D₄	—	17	3	no	—
122	D₄	C₄⋊#595	1	3	yes	—
123	C₂×C₄	—	17	3	no	—
124	C₂×C₄	—	17	3	no	—
125	C₂×C₄	—	17	3	no	—
126	C₂×C₄	—	17	1	no	—
127	C₂×C₄	—	17	3	no	—
128	C₂×C₄	C₂²×#595	1	3	yes	—
129	C₈	—	17	1	no	—
130	C₈	—	17	3	no	—
131	C₈	—	34	2	no	—
132	C₈	—	34	6	no	—
133	C₈	C₂²×#595	1	1	yes	characteristic
134	C₈	C₂²×#595	1	3	yes	—
135	C₂²	—	17	1	no	—
136	C₂²	—	17	3	no	—
137	C₂²	—	68	3	no	—
138	C₂²	C₂×C₄⋊#595	1	3	yes	—
139	C₄	—	17	1	no	—
140	C₄	—	17	3	no	—
141	C₄	C₂×C₄⋊#595	1	3	yes	—
142	C₄	C₂³×#595	1	1	yes	characteristic
143	C₂	—	2	3	no	—
144	C₂	—	34	1	no	—
145	C₂	—	34	3	no	—
146	C₂	C₂²×C₄⋊#595	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763888

C₂

yes

maximal

#11763824

C₂

yes

maximal, complemented

C₂×#10587536

C₂

yes

maximal, complemented

D₁₇⋊#108

C₂

yes

maximal, complemented

#131×D₁₇

C₂

yes

maximal, characteristic

#41×D₁₇

C₂²

yes

characteristic

#10587680

—

#10587678

—

D₁₇⋊M₁₆

C₂²

yes

—

C₂×#63243

C₂²

yes

—

C₁₇×#131

C₄

yes

characteristic

#10587634

C₄

yes

characteristic

#10587593

C₄

yes

—

M₁₆×D₁₇

C₂²

yes

—

#10587587

C₄

yes

—

C₂×C₈×D₁₇

C₂²

yes

—

#10587536

C₂²

yes

characteristic

#10587536

C₂²

yes

—

#10587536

C₂²

yes

—

#10587535

C₂²

yes

—

#41×C₁₇

C₂×C₄

yes

characteristic

#63256

D₄

yes

characteristic

Q₈×D₁₇

Q₈

yes

characteristic

#63254

D₄

yes

—

D₄×D₁₇

Q₈

yes

—

#63252

C₂×C₄

yes

—

C₂×C₄×D₁₇

C₂³

yes

—

—

—

—

C₂³

yes

—

#63243

C₂³

yes

—

#63242

—

M₁₆×C₁₇

C₂×C₄

yes

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

#63223

C₂×C₄

yes

—

C₂×#3622

C₂×C₄

yes

—

#63218

C₂×C₄

yes

—

C₈×D₁₇

C₂×C₄

yes

—

C₈×D₁₇

C₂×C₄

yes

—

C₈×D₁₇

C₂³

yes

—

C₈×D₁₇

C₂³

yes

characteristic

C₂²×D₁₇

C₂×Q₈

yes

—

Q₈×C₁₇

C₄⋊C₄

yes

characteristic

D₄×C₁₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

—

C₂²⋊D₁₇

C₄⋊C₄

yes

—

C₂×Dic₁₇

C₂×D₄

yes

—

D₆₈

C₄⋊C₄

yes

—

C₄×D₁₇

C₂²×C₄

yes

—

C₄×D₁₇

C₂×D₄

yes

—

C₄×D₁₇

C₂×Q₈

yes

—

C₄×D₁₇

C₂⁴

yes

characteristic

Dic₃₄

C₄⋊C₄

yes

—

#3624

—

#3624

—

C₈×C₁₇

C₂²×C₄

yes

characteristic

C₈×C₁₇

C₂²×C₄

yes

—

#3622

C₂²×C₄

yes

characteristic

#3622

C₂²×C₄

yes

—

#2916

—

maximal, Sylow

C₂²×C₁₇

C₂×C₄⋊C₄

yes

—

C₂×D₁₇

—

C₂×D₁₇

C₂×C₄⋊C₄

yes

—

C₂×D₁₇

C₂²×Q₈

yes

characteristic

C₄×C₁₇

C₂×C₄⋊C₄

yes

—

C₄×C₁₇

C₂³×C₄

yes

characteristic

Dic₁₇

C₂×C₄⋊C₄

yes

—

Dic₁₇

C₂²×D₄

yes

—

—

—

—

—

C₂×C₁₇

—

C₂×C₁₇

C₂²×C₄⋊C₄

yes

characteristic

D₁₇

—

D₁₇

—

C₂×#41

—

—

—

—

—

—

yes

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

—

—

—

—

C₁₇

#2916

yes

characteristic, complemented, Sylow

—

—

—

yes

C₂×Q₈

—

100

C₂×D₄

—

101

C₂²×C₄

—

102

M₁₆

—

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

C₂×#595

yes

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₂×#595

yes

—

117

C₂³

—

118

Q₈

—

119

Q₈

C₄⋊#595

yes

characteristic

120

D₄

—

121

D₄

—

122

D₄

C₄⋊#595

yes

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂²×#595

yes

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

C₂²×#595

yes

characteristic

134

C₈

C₂²×#595

yes

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₂×C₄⋊#595

yes

—

139

C₄

—

140

C₄

—

141

C₄

C₂×C₄⋊#595

yes

—

142

C₄

C₂³×#595

yes

characteristic

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

C₂²×C₄⋊#595

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₄⋊#595
commutator	C₄×C₁₇	C₂³×C₄
Frattini	C₄	C₂³×#595
Fitting	C₁₇×#131	C₄
socle	C₂×C₁₇	C₂²×C₄⋊C₄
prime radical	#41×D₁₇	C₂²
cosocle	C₄×D₁₇	C₂⁴
hypercenter	#131	#595
hypocenter	C₁₇	#2916
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

subgroup

G/H

center

C₂

C₂²×C₄⋊#595

commutator

C₄×C₁₇

C₂³×C₄

Frattini

C₄

C₂³×#595

Fitting

C₁₇×#131

C₄

socle

C₂×C₁₇

C₂²×C₄⋊C₄

prime radical

#41×D₁₇

C₂²

cosocle

C₄×D₁₇

C₂⁴

C₁₇

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131

N	H ≅ G/N	type
2. #11763824	C₂	semidirect
3. C₂×#10587536	C₂	semidirect
4. D₁₇⋊#108	C₂	semidirect
94. C₁₇	#2916	semidirect

H ≅ G/N

type

2. #11763824

C₂

semidirect

3. C₂×#10587536

C₂

semidirect

4. D₁₇⋊#108

C₂

semidirect

94. C₁₇

#2916

semidirect

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

subgroup	index
#11763888	2
#11763824	2
C₂×#10587536	2
D₁₇⋊#108	2
#131×D₁₇	2
#2916	17

subgroup

index

#11763888

#11763824

C₂×#10587536

D₁₇⋊#108

#131×D₁₇

#2916

Subgroups — #14545446

Conjugacy classes of subgroups

Complements

Maximal subgroups