Group #14537142 — subgroups

Subgroups — #14537142

Order 2176.

752 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 197 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×#10587560	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587633	C₂	1	4	yes	maximal, complemented
3	#11762710	C₂	1	8	yes	maximal, complemented
4	C₄²×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₈×C₁₇	C₂²	1	2	yes	—
7	C₂²×D₆₈	C₂²	1	4	yes	—
8	C₂²×D₆₈	C₂²	1	2	yes	—
9	C₁₇×C₄⋊Q₈	—	2	4	no	—
10	C₂×C₄⋊C₄×C₁₇	C₂²	1	4	yes	—
11	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
12	#10587633	—	2	16	no	—
13	C₂×#63227	C₂²	1	16	yes	—
14	C₂×C₄×Dic₁₇	C₂²	1	4	yes	—
15	C₂×C₄×Dic₁₇	C₂²	1	2	yes	—
16	#10587560	—	2	4	no	—
17	C₂³×D₁₇	C₂³	1	4	yes	—
18	C₂×Q₈×C₁₇	—	2	4	no	—
19	C₂×Q₈×C₁₇	—	2	8	no	—
20	C₂²×C₄×C₁₇	C₂³	1	4	yes	—
21	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₁₇	C₂³	1	2	yes	—
23	C₂²×Dic₁₇	C₂³	1	4	yes	—
24	C₂×D₆₈	—	2	8	no	—
25	C₂×D₆₈	—	2	4	no	—
26	C₂×D₆₈	—	2	8	no	—
27	C₂×D₆₈	—	2	16	no	—
28	C₄⋊C₄×C₁₇	—	2	8	no	—
29	C₄²×C₁₇	D₄	1	4	yes	—
30	#63227	—	2	32	no	—
31	C₄×Dic₁₇	—	2	8	no	—
32	C₄×Dic₁₇	D₄	1	8	yes	complemented
33	C₂³×C₁₇	C₂⁴	1	1	yes	characteristic
34	C₂²×D₁₇	—	2	4	no	—
35	C₂²×D₁₇	—	2	8	no	—
36	C₂²×D₁₇	—	2	16	no	—
37	Q₈×C₁₇	—	4	8	no	—
38	C₂×C₄×C₁₇	—	2	4	no	—
39	C₂×C₄×C₁₇	—	2	8	no	—
40	C₂×C₄×C₁₇	C₂×D₄	1	4	yes	—
41	C₂×C₄×C₁₇	C₂×D₄	1	4	yes	—
42	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₇	#41	1	8	yes	—
44	C₂×Dic₁₇	—	2	8	no	—
45	C₂×Dic₁₇	C₂×D₄	1	8	yes	—
46	D₆₈	—	4	16	no	—
47	D₆₈	—	4	8	no	—
48	#2858	—	17	1	no	maximal, Sylow
49	C₂²×C₁₇	C₂²×D₄	1	2	yes	—
50	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
51	C₂²×C₁₇	C₂×#41	1	4	yes	—
52	C₂×D₁₇	—	4	8	no	—
53	C₂×D₁₇	—	4	4	no	—
54	C₂×D₁₇	—	4	16	no	—
55	C₄×C₁₇	—	4	4	no	—
56	C₄×C₁₇	#124	1	8	yes	complemented
57	C₄×C₁₇	#127	1	4	yes	—
58	Dic₁₇	—	2	8	no	—
59	C₂×C₄⋊Q₈	D₁₇	1	1	yes	characteristic, complemented
60	C₂×#127	—	17	1	no	—
61	C₂×#124	—	17	4	no	—
62	#390	—	17	8	no	—
63	C₄³	—	17	1	no	—
64	C₂×C₁₇	C₂×#124	1	2	yes	—
65	C₂×C₁₇	C₂×#127	1	1	yes	characteristic
66	C₂×C₁₇	#532	1	4	yes	—
67	D₁₇	—	8	4	no	—
68	C₂²×Q₈	C₂×D₁₇	1	2	yes	—
69	C₂²×D₄	—	17	2	no	—
70	C₂²×D₄	—	17	4	no	—
71	C₄⋊Q₈	—	2	4	no	—
72	#127	—	34	4	no	—
73	#124	—	34	16	no	—
74	C₂×C₄⋊C₄	C₂×D₁₇	1	4	yes	—
75	C₂×C₂²⋊C₄	—	17	16	no	—
76	C₂×C₄²	—	17	2	no	—
77	C₂×C₄²	—	17	4	no	—
78	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
79	C₁₇	#2858	1	1	yes	characteristic, complemented, Sylow
80	C₂⁴	—	17	4	no	—
81	C₂×Q₈	—	2	8	no	—
82	C₂×Q₈	—	2	4	no	—
83	C₂×D₄	—	34	8	no	—
84	C₂×D₄	—	34	16	no	—
85	C₂×D₄	—	34	4	no	—
86	C₂×D₄	—	34	8	no	—
87	C₂²×C₄	—	17	4	no	—
88	C₂²×C₄	C₂²×D₁₇	1	4	yes	—
89	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₁₇	1	2	yes	—
91	C₄⋊C₄	—	2	8	no	—
92	C₂²⋊C₄	—	34	32	no	—
93	C₄²	—	17	8	no	—
94	C₄²	—	34	8	no	—
95	C₄²	C₂²⋊D₁₇	1	4	yes	—
96	C₂³	—	34	8	no	—
97	C₂³	—	34	16	no	—
98	C₂³	—	34	4	no	—
99	C₂³	C₂³×D₁₇	1	1	yes	characteristic
100	Q₈	—	4	8	no	—
101	D₄	—	68	8	no	—
102	D₄	—	68	16	no	—
103	C₂×C₄	—	2	8	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	17	8	no	—
106	C₂×C₄	—	34	8	no	—
107	C₂×C₄	D₄×D₁₇	1	4	yes	—
108	C₂×C₄	#63256	1	8	yes	—
109	C₂×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
110	C₂×C₄	C₂×C₂²⋊D₁₇	1	4	yes	—
111	C₂²	—	68	8	no	—
112	C₂²	—	68	4	no	—
113	C₂²	—	68	16	no	—
114	C₂²	C₂×D₄×D₁₇	1	2	yes	—
115	C₂²	C₂×#63256	1	4	yes	—
116	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
117	C₄	—	4	4	no	—
118	C₄	—	34	8	no	—
119	C₄	#10587626	1	4	yes	—
120	C₄	#10587633	1	8	yes	complemented
121	C₂	—	136	4	no	—
122	C₂	C₂×#10587633	1	2	yes	—
123	C₂	C₂×#10587626	1	1	yes	characteristic
124	C₂	#11763360	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587560

C₂

yes

maximal, characteristic

C₂×#10587633

C₂

yes

maximal, complemented

#11762710

C₂

yes

maximal, complemented

C₄²×Dic₁₇

C₂

yes

maximal, characteristic, complemented

C₂×C₁₇×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₇

C₂²

yes

—

C₂²×D₆₈

C₂²

yes

—

C₂²×D₆₈

C₂²

yes

—

C₁₇×C₄⋊Q₈

—

C₂×C₄⋊C₄×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

#10587633

—

C₂×#63227

C₂²

yes

—

C₂×C₄×Dic₁₇

C₂²

yes

—

C₂×C₄×Dic₁₇

C₂²

yes

—

#10587560

—

C₂³×D₁₇

C₂³

yes

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

C₂³

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×C₄×C₁₇

C₂³

yes

—

C₂²×Dic₁₇

C₂³

yes

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₂×D₆₈

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

D₄

yes

—

#63227

—

C₄×Dic₁₇

—

C₄×Dic₁₇

D₄

yes

complemented

C₂³×C₁₇

C₂⁴

yes

characteristic

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂²×D₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

#41

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂×D₄

yes

—

D₆₈

—

D₆₈

—

#2858

—

maximal, Sylow

C₂²×C₁₇

C₂²×D₄

yes

—

C₂²×C₁₇

C₂²×D₄

yes

characteristic

C₂²×C₁₇

C₂×#41

yes

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#124

yes

complemented

C₄×C₁₇

#127

yes

—

Dic₁₇

—

C₂×C₄⋊Q₈

D₁₇

yes

characteristic, complemented

C₂×#127

—

C₂×#124

—

#390

—

C₄³

—

C₂×C₁₇

C₂×#124

yes

—

C₂×C₁₇

C₂×#127

yes

characteristic

C₂×C₁₇

#532

yes

—

D₁₇

—

C₂²×Q₈

C₂×D₁₇

yes

—

C₂²×D₄

—

C₂²×D₄

—

C₄⋊Q₈

—

#127

—

#124

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

C₁₇

#2858

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂²×C₄

C₂²×D₁₇

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₁₇

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₁₇

yes

characteristic

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

D₄×D₁₇

yes

—

108

C₂×C₄

#63256

yes

—

109

C₂×C₄

C₂×C₂²⋊D₁₇

yes

—

110

C₂×C₄

C₂×C₂²⋊D₁₇

yes

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×D₄×D₁₇

yes

—

115

C₂²

C₂×#63256

yes

—

116

C₂²

C₂²×C₂²⋊D₁₇

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

#10587626

yes

—

120

C₄

#10587633

yes

complemented

121

C₂

—

136

—

122

C₂

C₂×#10587633

yes

—

123

C₂

C₂×#10587626

yes

characteristic

124

C₂

#11763360

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₇
commutator, socle, cosocle	C₂³×C₁₇	C₂⁴
Fitting	C₂×C₁₇×C₄⋊Q₈	C₂
prime radical	C₂×#10587560	C₂
hypercenter	C₂×C₄⋊Q₈	D₁₇
hypocenter	C₁₇	#2858
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₇

commutator, socle, cosocle

C₂³×C₁₇

C₂⁴

Fitting

C₂×C₁₇×C₄⋊Q₈

C₂

prime radical

C₂×#10587560

C₂

hypercenter

C₂×C₄⋊Q₈

D₁₇

C₁₇

G ▷ C₂³×C₁₇ ▷ C₁₇

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. C₂×#10587633	C₂	semidirect
3. #11762710	C₂	semidirect
4. C₄²×Dic₁₇	C₂	semidirect
5. C₂×C₁₇×C₄⋊Q₈	C₂	semidirect
32. C₄×Dic₁₇	D₄	semidirect
56. C₄×C₁₇	#124	semidirect
59. C₂×C₄⋊Q₈	D₁₇	semidirect
79. C₁₇	#2858	semidirect
120. C₄	#10587633	semidirect

H ≅ G/N

type

2. C₂×#10587633

C₂

semidirect

3. #11762710

C₂

semidirect

4. C₄²×Dic₁₇

C₂

semidirect

5. C₂×C₁₇×C₄⋊Q₈

C₂

semidirect

32. C₄×Dic₁₇

D₄

semidirect

56. C₄×C₁₇

#124

semidirect

59. C₂×C₄⋊Q₈

D₁₇

semidirect

79. C₁₇

#2858

semidirect

120. C₄

#10587633

semidirect

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

subgroup	index
C₂×#10587560	2
C₂×#10587633	2
#11762710	2
C₄²×Dic₁₇	2
C₂×C₁₇×C₄⋊Q₈	2
#2858	17

subgroup

index

C₂×#10587560

C₂×#10587633

#11762710

C₄²×Dic₁₇

C₂×C₁₇×C₄⋊Q₈

#2858

Subgroups — #14537142

Conjugacy classes of subgroups

Complements

Maximal subgroups