Group #14545718 — subgroups

Subgroups — #14545718

Order 2176.

394 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 168 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	Q₈×#3624	C₂	1	2	yes	maximal, complemented
2	#11763843	C₂	1	6	yes	maximal, complemented
3	#11763828	C₂	1	3	yes	maximal, complemented
4	#11763813	C₂	1	3	yes	maximal, complemented
5	C₄×#63256	C₂	1	1	yes	maximal, characteristic
6	C₂×#63256	C₄	1	1	yes	characteristic
7	C₂×#63243	C₂²	1	6	yes	—
8	C₄×Q₈×C₁₇	C₄	1	1	yes	characteristic
9	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
10	#10587579	C₂²	1	3	yes	—
11	#10587578	C₄	1	3	yes	—
12	C₄×D₆₈	C₄	1	3	yes	—
13	C₄²×D₁₇	C₂²	1	3	yes	—
14	#10587543	C₂²	1	6	yes	—
15	C₄⋊#3624	C₂²	1	3	yes	—
16	C₄⋊#3624	C₂²	1	6	yes	—
17	C₄⋊#3624	C₂²	1	1	yes	characteristic
18	C₄×#3624	C₂²	1	6	yes	—
19	C₂×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
20	#63256	—	2	4	no	—
21	C₂×D₆₈	C₂×C₄	1	3	yes	—
22	C₂×C₄×D₁₇	C₂×C₄	1	3	yes	—
23	C₂×C₄×D₁₇	C₂×C₄	1	3	yes	—
24	C₂×C₄×D₁₇	C₂³	1	3	yes	—
25	C₂×#3624	C₂³	1	6	yes	—
26	C₂×#3624	C₂³	1	2	yes	—
27	#63243	—	4	6	no	—
28	C₄⋊C₄×C₁₇	C₂×C₄	1	3	yes	—
29	C₄²×C₁₇	C₂×C₄	1	3	yes	—
30	#63227	C₂×C₄	1	6	yes	—
31	C₄⋊Dic₁₇	C₂×C₄	1	3	yes	—
32	C₄×Dic₁₇	C₂×C₄	1	3	yes	—
33	C₄×Dic₁₇	C₂³	1	3	yes	—
34	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
35	C₂²×D₁₇	C₂²×C₄	1	3	yes	—
36	Q₈×C₁₇	M₁₆	1	4	yes	complemented
37	C₂×C₄×C₁₇	C₂²×C₄	1	3	yes	—
38	C₂×C₄×C₁₇	C₂²×C₄	1	3	yes	—
39	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₁₇	C₂²×C₄	1	3	yes	—
42	C₂×Dic₁₇	C₂⁴	1	1	yes	characteristic
43	D₆₈	—	2	6	no	—
44	C₄×D₁₇	—	2	6	no	—
45	C₄×D₁₇	—	2	6	no	—
46	C₄×D₁₇	—	2	3	no	—
47	C₄×D₁₇	—	2	3	no	—
48	#3624	—	2	6	no	—
49	#3624	—	2	2	no	—
50	#2991	—	17	1	no	maximal, Sylow
51	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
52	C₂×D₁₇	—	2	3	no	—
53	C₂×D₁₇	—	2	3	no	—
54	C₂×D₁₇	—	2	3	no	—
55	C₄×C₁₇	—	2	3	no	—
56	C₄×C₁₇	C₂×M₁₆	1	6	yes	—
57	C₄×C₁₇	#131	1	2	yes	—
58	Dic₁₇	—	2	1	no	—
59	Dic₁₇	—	2	3	no	—
60	Dic₁₇	H(5,𝔽₂)	1	1	yes	characteristic
61	Dic₁₇	2₋¹⁺⁴	1	1	yes	characteristic
62	C₄×#41	—	17	1	no	—
63	C₈×Q₈	—	17	2	no	—
64	#436	—	17	6	no	—
65	#431	—	17	3	no	—
66	#423	—	17	3	no	—
67	C₂×C₁₇	#520	1	1	yes	characteristic
68	C₂×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
69	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
70	D₁₇	—	4	3	no	—
71	C₂×#41	—	17	1	no	—
72	C₂×M₁₆	—	17	6	no	—
73	C₄×Q₈	—	17	1	no	—
74	C₄×Q₈	#595	1	1	yes	characteristic
75	C₄×D₄	—	17	3	no	—
76	C₄×D₄	—	17	3	no	—
77	#117	—	17	3	no	—
78	C₂×C₄²	—	17	3	no	—
79	C₄⋊C₈	—	17	3	no	—
80	C₄⋊C₈	—	17	6	no	—
81	C₄⋊C₈	—	17	1	no	—
82	C₂²⋊C₈	—	17	6	no	—
83	C₄×C₈	—	17	6	no	—
84	C₁₇	#2991	1	1	yes	characteristic, complemented, Sylow
85	#41	—	34	4	no	—
86	C₂×Q₈	C₂×#595	1	1	yes	characteristic
87	C₂×D₄	—	17	3	no	—
88	C₂²×C₄	—	17	3	no	—
89	C₂²×C₄	—	17	3	no	—
90	C₂²×C₄	—	17	3	no	—
91	M₁₆	—	68	6	no	—
92	C₂×C₈	—	17	2	no	—
93	C₂×C₈	—	17	6	no	—
94	C₄⋊C₄	—	17	3	no	—
95	C₄⋊C₄	C₂×#595	1	3	yes	—
96	C₂²⋊C₄	—	17	6	no	—
97	C₄²	—	17	3	no	—
98	C₄²	—	17	1	no	—
99	C₄²	—	17	3	no	—
100	C₄²	C₂×#595	1	3	yes	—
101	C₂³	—	17	3	no	—
102	Q₈	#63243	1	4	yes	complemented
103	D₄	—	34	6	no	—
104	C₂×C₄	—	17	1	no	—
105	C₂×C₄	—	17	1	no	—
106	C₂×C₄	—	17	3	no	—
107	C₂×C₄	—	34	3	no	—
108	C₂×C₄	—	34	3	no	—
109	C₂×C₄	—	34	6	no	—
110	C₂×C₄	—	34	6	no	—
111	C₂×C₄	C₂²×#595	1	3	yes	—
112	C₂×C₄	C₂²×#595	1	3	yes	—
113	C₂×C₄	C₂²×#595	1	1	yes	characteristic
114	C₈	—	34	6	no	—
115	C₈	—	34	2	no	—
116	C₂²	—	34	3	no	—
117	C₂²	—	34	3	no	—
118	C₂²	—	34	3	no	—
119	C₂²	C₂³×#595	1	1	yes	characteristic
120	C₄	—	2	3	no	—
121	C₄	—	17	1	no	—
122	C₄	—	17	1	no	—
123	C₄	—	34	1	no	—
124	C₄	—	34	3	no	—
125	C₄	C₂×#63243	1	6	yes	—
126	C₄	#10587680	1	2	yes	—
127	C₂	—	68	3	no	—
128	C₂	#11763878	1	1	yes	characteristic
129	C₂	C₂²×#63243	1	1	yes	characteristic
130	C₂	C₂×#10587680	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#3624

C₂

yes

maximal, complemented

#11763843

C₂

yes

maximal, complemented

#11763828

C₂

yes

maximal, complemented

#11763813

C₂

yes

maximal, complemented

C₄×#63256

C₂

yes

maximal, characteristic

C₂×#63256

C₄

yes

characteristic

C₂×#63243

C₂²

yes

—

C₄×Q₈×C₁₇

C₄

yes

characteristic

Q₈×Dic₁₇

C₂²

yes

characteristic

#10587579

C₂²

yes

—

#10587578

C₄

yes

—

C₄×D₆₈

C₄

yes

—

C₄²×D₁₇

C₂²

yes

—

#10587543

C₂²

yes

—

C₄⋊#3624

C₂²

yes

—

C₄⋊#3624

C₂²

yes

—

C₄⋊#3624

C₂²

yes

characteristic

C₄×#3624

C₂²

yes

—

C₂×Q₈×C₁₇

C₂×C₄

yes

characteristic

#63256

—

C₂×D₆₈

C₂×C₄

yes

—

C₂×C₄×D₁₇

C₂×C₄

yes

—

C₂×C₄×D₁₇

C₂×C₄

yes

—

C₂×C₄×D₁₇

C₂³

yes

—

C₂×#3624

C₂³

yes

—

C₂×#3624

C₂³

yes

—

#63243

—

C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

C₂×C₄

yes

—

#63227

C₂×C₄

yes

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

C₄×Dic₁₇

C₂×C₄

yes

—

C₄×Dic₁₇

C₂³

yes

—

C₄×Dic₁₇

C₂³

yes

characteristic

C₂²×D₁₇

C₂²×C₄

yes

—

Q₈×C₁₇

M₁₆

yes

complemented

C₂×C₄×C₁₇

C₂²×C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₇

C₂²×C₄

yes

—

C₂×Dic₁₇

C₂⁴

yes

characteristic

D₆₈

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

—

—

—

C₂²×C₁₇

C₂³×C₄

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×M₁₆

yes

—

C₄×C₁₇

#131

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

H(5,𝔽₂)

yes

characteristic

Dic₁₇

2₋¹⁺⁴

yes

characteristic

C₄×#41

—

C₈×Q₈

—

#436

—

#431

—

#423

—

C₂×C₁₇

#520

yes

characteristic

C₂×C₁₇

C₂²×M₁₆

yes

characteristic

C₂×C₁₇

C₂×#131

yes

characteristic

D₁₇

—

C₂×#41

—

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

#595

yes

characteristic

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₁₇

#2991

yes

characteristic, complemented, Sylow

#41

—

C₂×Q₈

C₂×#595

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

C₂×#595

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

100

C₄²

C₂×#595

yes

—

101

C₂³

—

102

Q₈

#63243

yes

complemented

103

D₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂²×#595

yes

—

112

C₂×C₄

C₂²×#595

yes

—

113

C₂×C₄

C₂²×#595

yes

characteristic

114

C₈

—

115

C₈

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

C₂³×#595

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

C₂×#63243

yes

—

126

C₄

#10587680

yes

—

127

C₂

—

128

C₂

#11763878

yes

characteristic

129

C₂

C₂²×#63243

yes

characteristic

130

C₂

C₂×#10587680

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₄×Q₈×C₁₇	C₄
prime radical	C₂×#63256	C₄
cosocle	C₂×Dic₁₇	C₂⁴
hypercenter	C₄×Q₈	#595
hypocenter	C₁₇	#2991
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#595

commutator, socle

C₂²×C₁₇

C₂³×C₄

Fitting

C₄×Q₈×C₁₇

C₄

prime radical

C₂×#63256

C₄

cosocle

C₂×Dic₁₇

C₂⁴

C₄×Q₈

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. Q₈×#3624	C₂	semidirect
2. #11763843	C₂	semidirect
3. #11763828	C₂	semidirect
4. #11763813	C₂	semidirect
36. Q₈×C₁₇	M₁₆	semidirect
84. C₁₇	#2991	semidirect
102. Q₈	#63243	semidirect

H ≅ G/N

type

1. Q₈×#3624

C₂

semidirect

2. #11763843

C₂

semidirect

3. #11763828

C₂

semidirect

4. #11763813

C₂

semidirect

36. Q₈×C₁₇

M₁₆

semidirect

84. C₁₇

#2991

semidirect

102. Q₈

#63243

semidirect

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

subgroup	index
Q₈×#3624	2
#11763843	2
#11763828	2
#11763813	2
C₄×#63256	2
#2991	17

subgroup

index

Subgroups — #14545718

Conjugacy classes of subgroups

Complements

Maximal subgroups