Group #14529616 — subgroups

Subgroups — #14529616

Order 2176.

320 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 93 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	#11763359	C₂	1	1	yes	maximal, characteristic
2	#11762833	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10587476	C₂	1	4	yes	maximal, complemented
4	C₁₇×#376	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#63252	C₂²	1	2	yes	—
6	C₂×C₄²×C₁₇	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
8	#10587616	—	2	2	no	—
9	C₂×#63223	C₂²	1	2	yes	—
10	#10587561	C₄	1	2	yes	complemented
11	#10587560	C₄	1	1	yes	characteristic, complemented
12	C₄×D₆₈	—	2	2	no	—
13	#10587555	C₄	1	1	yes	characteristic, complemented
14	C₁₇×#95	—	2	2	no	—
15	#10587476	—	2	8	no	—
16	C₄⋊#3622	—	2	2	no	—
17	C₂²×C₄×C₁₇	—	2	2	no	—
18	C₂²×C₄×C₁₇	D₄	1	2	yes	—
19	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
20	C₂×C₂²⋊D₁₇	—	2	2	no	—
21	#63252	—	2	4	no	—
22	#63252	—	2	4	no	—
23	C₂×D₆₈	—	2	2	no	—
24	C₂×D₆₈	C₂×C₄	1	2	yes	—
25	C₂×C₄×D₁₇	—	2	2	no	—
26	C₂×Dic₃₄	C₂×C₄	1	2	yes	—
27	C₄²×C₁₇	—	2	2	no	—
28	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₇	D₄	1	2	yes	—
31	C₄²×C₁₇	D₄	1	4	yes	complemented
32	#63227	—	2	4	no	—
33	C₄⋊Dic₁₇	—	2	2	no	—
34	#63223	—	2	4	no	—
35	C₂×#3622	—	2	2	no	—
36	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×D₁₇	—	2	2	no	—
38	C₂×C₄×C₁₇	—	2	4	no	—
39	C₂×C₄×C₁₇	—	4	2	no	—
40	C₂×C₄×C₁₇	—	4	4	no	—
41	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇	#41	1	2	yes	—
47	C₂²⋊D₁₇	—	4	4	no	—
48	C₂×Dic₁₇	—	2	2	no	—
49	D₆₈	—	2	4	no	—
50	D₆₈	—	4	4	no	—
51	C₄×D₁₇	—	4	2	no	—
52	C₄×D₁₇	—	4	2	no	—
53	Dic₃₄	—	2	4	no	—
54	#3622	—	4	2	no	—
55	#1963	—	17	1	no	maximal, Sylow
56	C₂²×C₁₇	—	2	2	no	—
57	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₁₇	#121	1	2	yes	—
59	C₂×D₁₇	—	4	2	no	—
60	C₂×D₁₇	—	4	4	no	—
61	C₄×C₁₇	—	2	2	no	—
62	C₄×C₁₇	—	2	2	no	—
63	C₄×C₁₇	—	4	4	no	—
64	C₄×C₁₇	C₄×D₄	1	2	yes	—
65	C₄×C₁₇	#124	1	1	yes	characteristic
66	C₄×C₁₇	#127	1	1	yes	characteristic
67	Dic₁₇	—	4	2	no	—
68	#532	—	17	1	no	—
69	#423	—	17	1	no	—
70	C₂×#104	—	17	4	no	—
71	#376	D₁₇	1	1	yes	characteristic, complemented
72	C₂×C₁₇	—	2	2	no	—
73	C₂×C₁₇	#390	1	1	yes	characteristic
74	C₂×C₁₇	#421	1	2	yes	—
75	D₁₇	—	8	2	no	—
76	C₂×#41	—	17	2	no	—
77	C₂×M₁₆	—	17	2	no	—
78	C₄⋊Q₈	—	17	1	no	—
79	#127	—	17	1	no	—
80	#124	—	17	2	no	—
81	#121	—	34	2	no	—
82	C₄×D₄	—	34	2	no	—
83	C₂×C₄²	C₂×D₁₇	1	2	yes	—
84	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
85	C₄⋊C₈	—	34	2	no	—
86	#104	—	34	8	no	—
87	#95	—	2	2	no	—
88	C₁₇	#1963	1	1	yes	characteristic, complemented, Sylow
89	#41	—	34	4	no	—
90	#41	—	34	4	no	—
91	C₂×Q₈	—	17	2	no	—
92	C₂×D₄	—	17	2	no	—
93	C₂×D₄	—	34	2	no	—
94	C₂×D₄	—	34	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	34	2	no	—
97	C₂²×C₄	C₂²⋊D₁₇	1	2	yes	—
98	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
99	M₁₆	—	34	4	no	—
100	C₂×C₈	—	34	2	no	—
101	C₄⋊C₄	—	34	2	no	—
102	C₂²⋊C₄	—	34	4	no	—
103	C₄²	—	2	2	no	—
104	C₄²	C₄×D₁₇	1	1	yes	characteristic
105	C₄²	C₄×D₁₇	1	1	yes	characteristic
106	C₄²	D₆₈	1	2	yes	—
107	C₄²	D₆₈	1	4	yes	complemented
108	C₂³	—	34	2	no	—
109	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
110	Q₈	—	34	4	no	—
111	D₄	—	34	4	no	—
112	D₄	—	68	4	no	—
113	D₄	—	68	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	34	2	no	—
118	C₂×C₄	—	68	2	no	—
119	C₂×C₄	—	68	2	no	—
120	C₂×C₄	#63227	1	2	yes	—
121	C₂×C₄	#63227	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
123	C₂×C₄	C₂×D₆₈	1	2	yes	—
124	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
125	C₂×C₄	#63252	1	2	yes	—
126	C₈	—	68	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	68	2	no	—
129	C₂²	—	68	4	no	—
130	C₂²	C₂×#63227	1	1	yes	characteristic
131	C₂²	#10587616	1	2	yes	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	4	no	—
135	C₄	—	68	2	no	—
136	C₄	C₄×D₆₈	1	2	yes	—
137	C₄	#10587560	1	1	yes	characteristic
138	C₄	#10587561	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	136	2	no	—
141	C₂	#11762711	1	1	yes	characteristic
142	C₂	#11762827	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11763359

C₂

yes

maximal, characteristic

#11762833

C₂

yes

maximal, characteristic, complemented

C₂×#10587476

C₂

yes

maximal, complemented

C₁₇×#376

C₂

yes

maximal, characteristic, complemented

C₂×#63252

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

#10587616

—

C₂×#63223

C₂²

yes

—

#10587561

C₄

yes

complemented

#10587560

C₄

yes

characteristic, complemented

C₄×D₆₈

—

#10587555

C₄

yes

characteristic, complemented

C₁₇×#95

—

#10587476

—

C₄⋊#3622

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂×C₂²⋊D₁₇

—

#63252

—

#63252

—

C₂×D₆₈

—

C₂×D₆₈

C₂×C₄

yes

—

C₂×C₄×D₁₇

—

C₂×Dic₃₄

C₂×C₄

yes

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄²×C₁₇

D₄

yes

—

C₄²×C₁₇

D₄

yes

complemented

#63227

—

C₄⋊Dic₁₇

—

#63223

—

C₂×#3622

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂²×D₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

C₂²⋊D₁₇

—

C₂×Dic₁₇

—

D₆₈

—

D₆₈

—

C₄×D₁₇

—

C₄×D₁₇

—

Dic₃₄

—

#3622

—

#1963

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₇

#121

yes

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₄×D₄

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

—

—

—

D₁₇

yes

characteristic, complemented

C₂×C₁₇

—

C₂×C₁₇

#390

yes

characteristic

C₂×C₁₇

#421

yes

—

D₁₇

—

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₁₇

yes

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

C₄⋊C₈

—

#104

—

#95

—

C₁₇

#1963

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

M₁₆

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

C₄×D₁₇

yes

characteristic

105

C₄²

C₄×D₁₇

yes

characteristic

106

C₄²

D₆₈

yes

—

107

C₄²

D₆₈

yes

complemented

108

C₂³

—

109

C₂³

C₂×C₂²⋊D₁₇

yes

characteristic

110

Q₈

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

#63227

yes

—

121

C₂×C₄

#63227

yes

—

122

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

123

C₂×C₄

C₂×D₆₈

yes

—

124

C₂×C₄

C₂×D₆₈

yes

characteristic

125

C₂×C₄

#63252

yes

—

126

C₈

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂×#63227

yes

characteristic

131

C₂²

#10587616

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

C₄×D₆₈

yes

—

137

C₄

#10587560

yes

characteristic

138

C₄

#10587561

yes

characteristic

139

C₂

—

140

C₂

—

136

—

141

C₂

#11762711

yes

characteristic

142

C₂

#11762827

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₆₈
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#376	C₂
socle	C₂²×C₁₇	C₂×C₂²⋊C₄
prime radical	#11763359	C₂
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	#376	D₁₇
hypocenter	C₁₇	#1963
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×D₆₈

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#376

C₂

socle

C₂²×C₁₇

C₂×C₂²⋊C₄

prime radical

#11763359

C₂

cosocle

C₂²×C₄×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
2. #11762833	C₂	semidirect
3. C₂×#10587476	C₂	semidirect
4. C₁₇×#376	C₂	semidirect
10. #10587561	C₄	semidirect
11. #10587560	C₄	semidirect
13. #10587555	C₄	semidirect
31. C₄²×C₁₇	D₄	semidirect
71. #376	D₁₇	semidirect
88. C₁₇	#1963	semidirect
107. C₄²	D₆₈	semidirect

H ≅ G/N

type

2. #11762833

C₂

semidirect

3. C₂×#10587476

C₂

semidirect

4. C₁₇×#376

C₂

semidirect

10. #10587561

C₄

semidirect

11. #10587560

C₄

semidirect

13. #10587555

C₄

semidirect

31. C₄²×C₁₇

D₄

semidirect

71. #376

D₁₇

semidirect

88. C₁₇

#1963

semidirect

107. C₄²

D₆₈

semidirect

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

subgroup	index
#11763359	2
#11762833	2
C₂×#10587476	2
C₁₇×#376	2
#1963	17

subgroup

index

C₁₇×#376

Subgroups — #14529616

Conjugacy classes of subgroups

Complements

Maximal subgroups