Group #14544722 — subgroups

Subgroups — #14544722

Order 2176.

256 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 94 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×#10587534	C₂	1	2	yes	maximal
2	C₂×#10587533	C₂	1	2	yes	maximal
3	#11763755	C₂	1	1	yes	maximal, characteristic
4	#11763753	C₂	1	1	yes	maximal, characteristic
5	#97×D₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#595	—	2	2	no	—
7	C₂×C₄⋊#595	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#595	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊#595	C₂²	1	2	yes	—
10	C₂×C₈×D₁₇	C₂²	1	2	yes	—
11	C₄²×D₁₇	C₂²	1	1	yes	characteristic
12	(C₂×#595)⋊C₄	—	2	2	no	—
13	#10587534	—	2	4	no	—
14	#10587533	—	2	4	no	—
15	C₁₇×#97	C₄	1	1	yes	characteristic, complemented
16	#10587486	C₄	1	2	yes	complemented
17	#10587474	C₄	1	1	yes	characteristic, complemented
18	C₂²×#595	—	2	2	no	—
19	C₂²×#595	—	2	1	no	—
20	C₂²×#595	—	2	1	no	—
21	C₂×C₄×D₁₇	D₄	1	2	yes	—
22	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
23	C₄⋊#595	—	2	2	no	—
24	C₄⋊#595	—	2	2	no	—
25	C₄⋊#595	—	2	2	no	—
26	C₄⋊#595	—	2	2	no	—
27	C₄⋊#595	—	4	4	no	—
28	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
29	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₇	Q₈	1	2	yes	—
32	C₂×#3622	C₂×C₄	1	2	yes	—
33	C₈×D₁₇	D₄	1	4	yes	—
34	C₈×D₁₇	Q₈	1	4	yes	—
35	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
36	C₂×#595	—	4	2	no	—
37	C₂×#595	—	4	1	no	—
38	C₂×#595	—	4	1	no	—
39	C₂×#595	—	4	4	no	—
40	C₂×#595	—	4	2	no	—
41	C₂×#595	—	4	2	no	—
42	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×Dic₁₇	C₄⋊C₄	1	2	yes	—
45	C₂×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
46	C₄×D₁₇	—	2	2	no	—
47	C₄×D₁₇	—	2	2	no	—
48	C₄×D₁₇	C₂×D₄	1	2	yes	—
49	C₄×D₁₇	C₂×Q₈	1	2	yes	—
50	C₈×C₁₇	C₄⋊C₄	1	4	yes	complemented
51	#3622	C₄⋊C₄	1	4	yes	complemented
52	#1847	—	17	1	no	maximal, Sylow
53	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×D₁₇	#127	1	1	yes	characteristic
55	C₂×D₁₇	C₄⋊Q₈	1	1	yes	characteristic
56	C₂×D₁₇	AGL(1,ℤ/8)	1	2	yes	—
57	C₂×D₁₇	#137	1	2	yes	—
58	#595	—	8	2	no	—
59	#595	—	8	1	no	—
60	#595	—	8	1	no	—
61	C₄×C₁₇	—	2	2	no	—
62	C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
63	Dic₁₇	—	2	2	no	—
64	Dic₁₇	C₄⋊Q₈	1	2	yes	—
65	C₂×#107	—	17	2	no	—
66	C₂×#106	—	17	2	no	—
67	C₂×#97	—	17	1	no	—
68	#384	—	17	1	no	—
69	#382	—	17	1	no	—
70	C₂×C₁₇	#384	1	1	yes	characteristic
71	C₂×C₁₇	#428	1	2	yes	—
72	D₁₇	#496	1	1	yes	characteristic
73	D₁₇	#497	1	1	yes	characteristic
74	D₁₇	#501	1	2	yes	—
75	C₂²×C₈	—	17	2	no	—
76	C₂×C₄⋊C₄	—	17	1	no	—
77	C₂×C₄⋊C₄	—	17	1	no	—
78	C₂×C₄⋊C₄	—	17	2	no	—
79	C₂×C₄⋊C₄	—	34	2	no	—
80	C₂×C₄²	—	17	1	no	—
81	#107	—	34	4	no	—
82	#106	—	34	4	no	—
83	#97	—	17	2	no	—
84	#97	—	17	1	no	—
85	#97	#595	1	1	yes	characteristic, complemented
86	#95	—	34	2	no	—
87	C₁₇	#1847	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₄	—	17	1	no	—
89	C₂²×C₄	—	17	2	no	—
90	C₂²×C₄	—	34	1	no	—
91	C₂²×C₄	—	34	1	no	—
92	C₂²×C₄	—	34	2	no	—
93	C₂×C₈	—	17	2	no	—
94	C₂×C₈	—	17	4	no	—
95	C₂×C₈	—	17	4	no	—
96	C₂×C₈	C₂×#595	1	2	yes	—
97	C₄⋊C₄	—	34	2	no	—
98	C₄⋊C₄	—	34	2	no	—
99	C₄⋊C₄	—	34	2	no	—
100	C₄⋊C₄	—	34	2	no	—
101	C₄⋊C₄	—	68	4	no	—
102	C₄²	—	17	2	no	—
103	C₄²	—	17	1	no	—
104	C₄²	C₂×#595	1	1	yes	characteristic
105	C₂³	—	17	1	no	—
106	C₂×C₄	—	17	1	no	—
107	C₂×C₄	—	17	2	no	—
108	C₂×C₄	—	17	2	no	—
109	C₂×C₄	—	17	2	no	—
110	C₂×C₄	—	34	2	no	—
111	C₂×C₄	—	34	2	no	—
112	C₂×C₄	—	68	1	no	—
113	C₂×C₄	—	68	2	no	—
114	C₂×C₄	—	68	1	no	—
115	C₂×C₄	—	68	4	no	—
116	C₂×C₄	—	68	2	no	—
117	C₂×C₄	—	68	2	no	—
118	C₂×C₄	C₄⋊#595	1	2	yes	—
119	C₂×C₄	C₂²×#595	1	1	yes	characteristic
120	C₈	—	17	4	no	—
121	C₈	C₄⋊#595	1	4	yes	complemented
122	C₂²	—	17	1	no	—
123	C₂²	—	17	1	no	—
124	C₂²	—	17	2	no	—
125	C₂²	—	17	2	no	—
126	C₂²	C₂×C₄⋊#595	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	17	2	no	—
129	C₄	—	34	2	no	—
130	C₄	—	136	2	no	—
131	C₄	—	136	1	no	—
132	C₄	—	136	1	no	—
133	C₄	C₂×C₄⋊#595	1	2	yes	—
134	C₂	—	17	1	no	—
135	C₂	—	17	1	no	—
136	C₂	—	17	2	no	—
137	C₂	#11763755	1	1	yes	characteristic
138	C₂	#11763820	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587534

C₂

yes

maximal

C₂×#10587533

C₂

yes

maximal

#11763755

C₂

yes

maximal, characteristic

#11763753

C₂

yes

maximal, characteristic

#97×D₁₇

C₂

yes

maximal, characteristic

C₂×C₄⋊#595

—

C₂×C₄⋊#595

C₂²

yes

characteristic

C₂×C₄⋊#595

C₂²

yes

characteristic

C₂×C₄⋊#595

C₂²

yes

—

C₂×C₈×D₁₇

C₂²

yes

—

C₄²×D₁₇

C₂²

yes

characteristic

(C₂×#595)⋊C₄

—

#10587534

—

#10587533

—

C₁₇×#97

C₄

yes

characteristic, complemented

#10587486

C₄

yes

complemented

#10587474

C₄

yes

characteristic, complemented

C₂²×#595

—

C₂²×#595

—

C₂²×#595

—

C₂×C₄×D₁₇

D₄

yes

—

C₂×C₄×D₁₇

C₂³

yes

—

—

—

—

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄×Dic₁₇

C₂×C₄

yes

characteristic

C₄×Dic₁₇

Q₈

yes

—

C₂×#3622

C₂×C₄

yes

—

C₈×D₁₇

D₄

yes

—

C₈×D₁₇

Q₈

yes

—

C₂²×D₁₇

C₂×D₄

yes

—

—

—

—

—

—

C₂×C₄×C₁₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₇

C₄⋊C₄

yes

—

C₂×Dic₁₇

C₂×Q₈

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

C₂×D₄

yes

—

C₄×D₁₇

C₂×Q₈

yes

—

C₈×C₁₇

C₄⋊C₄

yes

complemented

#3622

C₄⋊C₄

yes

complemented

#1847

—

maximal, Sylow

C₂²×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₇

#127

yes

characteristic

C₂×D₁₇

C₄⋊Q₈

yes

characteristic

C₂×D₁₇

AGL(1,ℤ/8)

yes

—

C₂×D₁₇

#137

yes

—

#595

—

#595

—

#595

—

C₄×C₁₇

—

C₄×C₁₇

C₂×C₄⋊C₄

yes

—

Dic₁₇

—

Dic₁₇

C₄⋊Q₈

yes

—

C₂×#107

—

C₂×#106

—

C₂×#97

—

#384

—

#382

—

C₂×C₁₇

#384

yes

characteristic

C₂×C₁₇

#428

yes

—

D₁₇

yes

D₁₇

yes

D₁₇

yes

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

—

—

—

—

yes

characteristic, complemented

#95

—

C₁₇

#1847

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#595

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₄²

C₂×#595

yes

characteristic

105

C₂³

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₄⋊#595

yes

—

119

C₂×C₄

C₂²×#595

yes

characteristic

120

C₈

—

121

C₈

C₄⋊#595

yes

complemented

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

C₂×C₄⋊#595

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

136

—

131

C₄

—

136

—

132

C₄

—

136

—

133

C₄

C₂×C₄⋊#595

yes

—

134

C₂

—

135

C₂

—

136

C₂

—

137

C₂

#11763755

yes

characteristic

138

C₂

#11763820

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#595
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#595
Fitting	C₁₇×#97	C₄
socle	C₂²×C₁₇	C₂×C₄⋊C₄
prime radical	C₂²×D₁₇	C₂×D₄
cosocle	C₂×C₄×D₁₇	C₂³
hypercenter	#97	#595
hypocenter	C₁₇	#1847
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊#595

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#595

Fitting

C₁₇×#97

C₄

socle

C₂²×C₁₇

C₂×C₄⋊C₄

prime radical

C₂²×D₁₇

C₂×D₄

cosocle

C₂×C₄×D₁₇

C₂³

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
15. C₁₇×#97	C₄	semidirect
16. #10587486	C₄	semidirect
17. #10587474	C₄	semidirect
50. C₈×C₁₇	C₄⋊C₄	semidirect
51. #3622	C₄⋊C₄	semidirect
85. #97	#595	semidirect
87. C₁₇	#1847	semidirect
121. C₈	C₄⋊#595	semidirect

H ≅ G/N

type

15. C₁₇×#97

C₄

semidirect

16. #10587486

C₄

semidirect

17. #10587474

C₄

semidirect

50. C₈×C₁₇

C₄⋊C₄

semidirect

51. #3622

C₄⋊C₄

semidirect

85. #97

#595

semidirect

87. C₁₇

#1847

semidirect

121. C₈

C₄⋊#595

semidirect

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

subgroup	index
C₂×#10587534	2
C₂×#10587533	2
#11763755	2
#11763753	2
#97×D₁₇	2
#1847	17

subgroup

index

#97×D₁₇

Subgroups — #14544722

Conjugacy classes of subgroups

Complements

Maximal subgroups