Group #14544911 — subgroups

Subgroups — #14544911

Order 2176.

332 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 98 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	C₂×#10587549	C₂	1	4	yes	maximal
2	C₄×#63242	C₂	1	1	yes	maximal, characteristic
3	#11763755	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×D₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈×D₁₇	C₂²	1	2	yes	—
6	C₂×C₄⋊#595	—	2	2	no	—
7	C₂×C₄⋊#595	C₂²	1	2	yes	—
8	C₂×#63242	C₂²	1	2	yes	—
9	C₁₇×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
10	#10587630	C₄	1	2	yes	complemented
11	C₄⋊C₄×D₁₇	—	2	2	no	—
12	#10587574	—	2	2	no	—
13	C₄²×D₁₇	C₂²	1	1	yes	characteristic
14	#10587555	C₄	1	1	yes	characteristic, complemented
15	#10587549	—	2	8	no	—
16	C₄×#3624	—	2	2	no	—
17	C₂²×#595	—	2	2	no	—
18	C₂×Q₈×C₁₇	C₂×C₄	1	2	yes	—
19	Q₈×D₁₇	—	2	4	no	—
20	Q₈×D₁₇	—	2	4	no	—
21	C₂×C₄×D₁₇	—	2	2	no	—
22	C₂×C₄×D₁₇	D₄	1	2	yes	—
23	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
24	C₂×Dic₃₄	—	2	2	no	—
25	C₂×Dic₃₄	C₂×C₄	1	2	yes	—
26	C₂×#3624	—	2	2	no	—
27	C₄⋊#595	—	2	4	no	—
28	C₄⋊#595	—	4	4	no	—
29	#63242	—	2	2	no	—
30	#63242	D₄	1	4	yes	—
31	C₄⋊C₄×C₁₇	—	2	2	no	—
32	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₄⋊Dic₁₇	—	2	2	no	—
34	#63225	—	2	4	no	—
35	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₁₇	D₄	1	2	yes	—
37	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂×#595	—	4	2	no	—
39	C₂×#595	—	4	4	no	—
40	Q₈×C₁₇	—	2	4	no	—
41	C₂×C₄×C₁₇	—	2	2	no	—
42	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×Dic₁₇	—	2	2	no	—
45	C₂×Dic₁₇	C₂²⋊C₄	1	2	yes	—
46	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
47	C₄×D₁₇	—	4	2	no	—
48	C₄×D₁₇	—	4	2	no	—
49	C₄×D₁₇	SD₁₆	1	4	yes	—
50	C₄×D₁₇	Q₁₆	1	4	yes	—
51	C₄×D₁₇	C₂×D₄	1	2	yes	—
52	C₄×D₁₇	#41	1	2	yes	—
53	Dic₃₄	—	2	4	no	—
54	Dic₃₄	—	4	4	no	—
55	#3624	—	2	4	no	—
56	#1956	—	17	1	no	maximal, Sylow
57	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×D₁₇	#124	1	1	yes	characteristic
59	C₂×D₁₇	#127	1	1	yes	characteristic
60	C₂×D₁₇	C₂×SD₁₆	1	2	yes	—
61	C₂×D₁₇	C₂×Q₁₆	1	2	yes	—
62	#595	—	8	2	no	—
63	C₄×C₁₇	—	4	2	no	—
64	C₄×C₁₇	#103	1	4	yes	complemented
65	C₄×C₁₇	C₄×D₄	1	2	yes	—
66	Dic₁₇	—	4	2	no	—
67	Dic₁₇	#103	1	4	yes	complemented
68	Dic₁₇	#121	1	2	yes	—
69	C₂×C₄⋊Q₈	—	17	1	no	—
70	C₂×#103	—	17	4	no	—
71	C₂×C₄×C₈	—	17	1	no	—
72	#384	—	17	1	no	—
73	C₂×C₁₇	#390	1	1	yes	characteristic
74	C₂×C₁₇	C₂×#103	1	2	yes	—
75	D₁₇	#487	1	2	yes	—
76	D₁₇	#492	1	1	yes	characteristic
77	D₁₇	#494	1	1	yes	characteristic
78	C₂²×Q₈	—	17	2	no	—
79	C₂²×C₈	—	17	2	no	—
80	C₄⋊Q₈	—	17	2	no	—
81	C₄⋊Q₈	—	17	1	no	—
82	C₄⋊Q₈	—	34	2	no	—
83	C₄⋊Q₈	#595	1	1	yes	characteristic, complemented
84	C₂×C₄⋊C₄	—	17	2	no	—
85	C₂×C₄⋊C₄	—	34	2	no	—
86	C₂×C₄⋊C₄	—	34	2	no	—
87	C₂×C₄²	—	17	1	no	—
88	#103	—	34	8	no	—
89	C₄×C₈	—	34	2	no	—
90	C₁₇	#1956	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₈	—	17	2	no	—
92	C₂×Q₈	—	34	2	no	—
93	C₂×Q₈	—	34	4	no	—
94	C₂×Q₈	—	34	4	no	—
95	C₂×Q₈	C₂×#595	1	2	yes	—
96	C₂²×C₄	—	17	2	no	—
97	C₂²×C₄	—	17	1	no	—
98	C₂²×C₄	—	34	2	no	—
99	C₂²×C₄	—	34	2	no	—
100	C₂×C₈	—	17	4	no	—
101	C₂×C₈	—	34	2	no	—
102	C₂×C₈	—	34	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	34	4	no	—
105	C₄⋊C₄	—	34	4	no	—
106	C₄⋊C₄	—	34	2	no	—
107	C₄⋊C₄	—	68	4	no	—
108	C₄²	—	17	2	no	—
109	C₄²	—	17	1	no	—
110	C₄²	C₂×#595	1	1	yes	characteristic
111	C₂³	—	17	1	no	—
112	Q₈	—	2	4	no	—
113	Q₈	—	34	4	no	—
114	Q₈	—	68	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	17	2	no	—
117	C₂×C₄	—	17	1	no	—
118	C₂×C₄	—	17	2	no	—
119	C₂×C₄	—	17	2	no	—
120	C₂×C₄	—	17	4	no	—
121	C₂×C₄	—	17	4	no	—
122	C₂×C₄	—	34	2	no	—
123	C₂×C₄	—	68	2	no	—
124	C₂×C₄	—	68	2	no	—
125	C₂×C₄	—	68	2	no	—
126	C₂×C₄	—	68	4	no	—
127	C₂×C₄	C₂²⋊#595	1	2	yes	—
128	C₂×C₄	C₂²×#595	1	1	yes	characteristic
129	C₈	—	34	4	no	—
130	C₂²	—	17	1	no	—
131	C₂²	—	17	1	no	—
132	C₂²	—	17	2	no	—
133	C₂²	—	17	2	no	—
134	C₂²	C₂×C₂²⋊#595	1	1	yes	characteristic
135	C₄	—	4	2	no	—
136	C₄	—	17	2	no	—
137	C₄	—	17	4	no	—
138	C₄	—	68	2	no	—
139	C₄	—	136	2	no	—
140	C₄	#10587549	1	4	yes	complemented
141	C₄	D₄×#595	1	2	yes	—
142	C₂	—	17	1	no	—
143	C₂	—	17	2	no	—
144	C₂	—	17	1	no	—
145	C₂	#11763761	1	1	yes	characteristic
146	C₂	C₂×#10587549	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587549

C₂

yes

maximal

C₄×#63242

C₂

yes

maximal, characteristic

#11763755

C₂

yes

maximal, characteristic

C₄⋊Q₈×D₁₇

C₂

yes

maximal, characteristic

C₂×Q₈×D₁₇

C₂²

yes

—

C₂×C₄⋊#595

—

C₂×C₄⋊#595

C₂²

yes

—

C₂×#63242

C₂²

yes

—

C₁₇×C₄⋊Q₈

C₄

yes

characteristic, complemented

#10587630

C₄

yes

complemented

C₄⋊C₄×D₁₇

—

#10587574

—

C₄²×D₁₇

C₂²

yes

characteristic

#10587555

C₄

yes

characteristic, complemented

#10587549

—

C₄×#3624

—

C₂²×#595

—

C₂×Q₈×C₁₇

C₂×C₄

yes

—

Q₈×D₁₇

—

Q₈×D₁₇

—

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

D₄

yes

—

C₂×C₄×D₁₇

C₂³

yes

characteristic

C₂×Dic₃₄

—

C₂×Dic₃₄

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄⋊Dic₁₇

—

#63225

—

C₄×Dic₁₇

C₂×C₄

yes

characteristic

C₄×Dic₁₇

D₄

yes

—

C₂²×D₁₇

C₂×D₄

yes

characteristic

C₂×#595

—

C₂×#595

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂²⋊C₄

yes

—

C₂×Dic₁₇

C₂×D₄

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

SD₁₆

yes

—

C₄×D₁₇

Q₁₆

yes

—

C₄×D₁₇

C₂×D₄

yes

—

C₄×D₁₇

#41

yes

—

Dic₃₄

—

Dic₃₄

—

#3624

—

#1956

—

maximal, Sylow

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

C₂×D₁₇

yes

C₂×D₁₇

yes

C₂×D₁₇

C₂×SD₁₆

yes

—

C₂×D₁₇

C₂×Q₁₆

yes

—

#595

—

C₄×C₁₇

—

C₄×C₁₇

#103

yes

complemented

C₄×C₁₇

C₄×D₄

yes

—

Dic₁₇

—

Dic₁₇

#103

yes

complemented

Dic₁₇

#121

yes

—

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×C₄×C₈

—

#384

—

C₂×C₁₇

#390

yes

characteristic

C₂×C₁₇

C₂×#103

yes

—

D₁₇

#487

yes

—

D₁₇

yes

D₁₇

yes

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#595

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

#103

—

C₄×C₈

—

C₁₇

#1956

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×#595

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

—

109

C₄²

—

110

C₄²

C₂×#595

yes

characteristic

111

C₂³

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

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

C₂²⋊#595

yes

—

128

C₂×C₄

C₂²×#595

yes

characteristic

129

C₈

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

C₂×C₂²⋊#595

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

136

—

140

C₄

#10587549

yes

complemented

141

C₄

D₄×#595

yes

—

142

C₂

—

143

C₂

—

144

C₂

—

145

C₂

#11763761

yes

characteristic

146

C₂

C₂×#10587549

yes

—

	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	#1956
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
9. C₁₇×C₄⋊Q₈	C₄	semidirect
10. #10587630	C₄	semidirect
14. #10587555	C₄	semidirect
64. C₄×C₁₇	#103	semidirect
67. Dic₁₇	#103	semidirect
83. C₄⋊Q₈	#595	semidirect
90. C₁₇	#1956	semidirect
140. C₄	#10587549	semidirect

H ≅ G/N

type

9. C₁₇×C₄⋊Q₈

C₄

semidirect

10. #10587630

C₄

semidirect

14. #10587555

C₄

semidirect

64. C₄×C₁₇

#103

semidirect

67. Dic₁₇

#103

semidirect

83. C₄⋊Q₈

#595

semidirect

90. C₁₇

#1956

semidirect

140. C₄

#10587549

semidirect

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

subgroup	index
C₂×#10587549	2
C₄×#63242	2
#11763755	2
C₄⋊Q₈×D₁₇	2
#1956	17

subgroup

index

C₄⋊Q₈×D₁₇

Subgroups — #14544911

Conjugacy classes of subgroups

Complements

Maximal subgroups