Group #16955321 — subgroups

Subgroups — #16955321

Order 4352.

230 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 79 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×#11762649	C₂	1	4	yes	maximal
2	#14531037	C₂	1	1	yes	maximal, characteristic
3	C₄×#10587465	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#2247	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₁₃₆	—	2	1	no	—
6	C₂×Dic₁₃₆	C₂²	1	2	yes	—
7	#11763189	—	2	1	no	—
8	#11762649	—	2	8	no	—
9	C₂×#10587465	C₂²	1	2	yes	—
10	C₂×C₁₇×Q₃₂	—	2	1	no	—
11	C₂×C₁₇×Q₃₂	C₂²	1	2	yes	—
12	C₁₇×#494	—	2	1	no	—
13	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
14	C₂×Q₁₆×C₁₇	—	2	2	no	—
15	C₂×Q₁₆×C₁₇	—	4	1	no	—
16	C₁₇×C₄⋊Q₈	—	4	1	no	—
17	C₂×Dic₆₈	—	2	2	no	—
18	C₂×Dic₆₈	—	4	1	no	—
19	#10587555	—	4	1	no	—
20	C₁₇×Q₃₂	—	2	4	no	—
21	C₁₇×Q₃₂	—	4	2	no	—
22	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
25	Dic₁₃₆	—	2	4	no	—
26	Dic₁₃₆	—	4	2	no	—
27	#10587465	D₄	1	4	yes	—
28	C₂×Q₈×C₁₇	—	4	2	no	—
29	C₂×Dic₃₄	—	4	2	no	—
30	Q₁₆×C₁₇	—	4	2	no	—
31	Q₁₆×C₁₇	—	4	2	no	—
32	Q₁₆×C₁₇	—	8	2	no	—
33	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
34	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₁₇	—	8	2	no	—
36	C₄²×C₁₇	D₈	1	1	yes	characteristic
37	C₄⋊Dic₁₇	—	8	2	no	—
38	Dic₆₈	—	4	2	no	—
39	Dic₆₈	—	4	2	no	—
40	Dic₆₈	—	8	2	no	—
41	C₁₆×C₁₇	D₈	1	2	yes	—
42	C₁₆×C₁₇	C₂×D₄	1	2	yes	—
43	#13726	—	17	1	no	maximal, Sylow
44	Q₈×C₁₇	—	8	2	no	—
45	Q₈×C₁₇	—	8	2	no	—
46	C₂×C₄×C₁₇	—	8	2	no	—
47	C₂×C₄×C₁₇	D₁₆	1	2	yes	—
48	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
49	C₂×Dic₁₇	—	8	2	no	—
50	Dic₃₄	—	8	2	no	—
51	Dic₃₄	—	8	2	no	—
52	C₈×C₁₇	D₁₆	1	2	yes	—
53	C₈×C₁₇	#127	1	1	yes	characteristic
54	C₈×C₁₇	C₂×D₈	1	1	yes	characteristic
55	C₂×Q₆₄	—	17	4	no	—
56	#2247	—	17	1	no	—
57	#2247	D₁₇	1	1	yes	characteristic
58	C₄×C₃₂	—	17	1	no	—
59	C₂²×C₁₇	C₂×D₁₆	1	1	yes	characteristic
60	C₄×C₁₇	—	16	2	no	—
61	C₄×C₁₇	Q₆₄	1	4	yes	complemented
62	C₄×C₁₇	#493	1	1	yes	characteristic
63	C₄×C₁₇	C₂×D₁₆	1	1	yes	characteristic
64	Dic₁₇	—	16	2	no	—
65	C₂×Q₃₂	—	2	1	no	—
66	C₂×Q₃₂	—	17	2	no	—
67	C₂×Q₃₂	—	34	1	no	—
68	C₂×Q₃₂	C₂×D₁₇	1	2	yes	—
69	#494	—	2	1	no	—
70	#494	—	34	1	no	—
71	Q₆₄	—	34	8	no	—
72	C₂×C₃₂	—	17	2	no	—
73	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
74	C₂×C₁₇	#2246	1	1	yes	characteristic
75	C₂×C₁₇	C₂×Q₆₄	1	2	yes	—
76	C₂×Q₁₆	—	2	2	no	—
77	C₂×Q₁₆	—	4	1	no	—
78	C₂×Q₁₆	—	34	2	no	—
79	C₂×Q₁₆	—	68	1	no	—
80	C₄⋊Q₈	—	4	1	no	—
81	C₄⋊Q₈	—	68	1	no	—
82	Q₃₂	—	2	4	no	—
83	Q₃₂	—	4	2	no	—
84	Q₃₂	—	34	4	no	—
85	Q₃₂	—	68	2	no	—
86	C₂×C₁₆	C₂²⋊D₁₇	1	1	yes	characteristic
87	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
88	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
89	C₃₂	—	17	4	no	—
90	C₁₇	#13726	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₈	—	4	2	no	—
92	C₂×Q₈	—	68	2	no	—
93	Q₁₆	—	4	2	no	—
94	Q₁₆	—	4	2	no	—
95	Q₁₆	—	8	2	no	—
96	Q₁₆	—	68	2	no	—
97	Q₁₆	—	68	2	no	—
98	Q₁₆	—	136	2	no	—
99	C₂×C₈	#63228	1	1	yes	characteristic
100	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
101	C₄⋊C₄	—	8	2	no	—
102	C₄⋊C₄	—	136	2	no	—
103	C₄²	#63228	1	1	yes	characteristic
104	C₁₆	#63228	1	2	yes	—
105	C₁₆	D₄×D₁₇	1	2	yes	—
106	Q₈	—	8	2	no	—
107	Q₈	—	8	2	no	—
108	Q₈	—	136	2	no	—
109	Q₈	—	136	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	136	2	no	—
112	C₂×C₄	#10587497	1	2	yes	—
113	C₂×C₄	C₂×#63228	1	1	yes	characteristic
114	C₈	#10587497	1	2	yes	—
115	C₈	C₂×#63228	1	1	yes	characteristic
116	C₈	#10587626	1	1	yes	characteristic
117	C₂²	C₂×#10587497	1	1	yes	characteristic
118	C₄	—	16	2	no	—
119	C₄	—	272	2	no	—
120	C₄	#11762649	1	4	yes	complemented
121	C₄	#11763185	1	1	yes	characteristic
122	C₄	C₂×#10587497	1	1	yes	characteristic
123	C₂	#14531033	1	1	yes	characteristic
124	C₂	C₂×#11762649	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11762649

C₂

yes

maximal

#14531037

C₂

yes

maximal, characteristic

C₄×#10587465

C₂

yes

maximal, characteristic

C₁₇×#2247

C₂

yes

maximal, characteristic

C₂×Dic₁₃₆

—

C₂×Dic₁₃₆

C₂²

yes

—

#11763189

—

#11762649

—

C₂×#10587465

C₂²

yes

—

C₂×C₁₇×Q₃₂

—

C₂×C₁₇×Q₃₂

C₂²

yes

—

C₁₇×#494

—

C₄×C₁₆×C₁₇

C₂²

yes

characteristic

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₁₇×C₄⋊Q₈

—

C₂×Dic₆₈

—

C₂×Dic₆₈

—

#10587555

—

C₁₇×Q₃₂

—

C₁₇×Q₃₂

—

C₂×C₁₆×C₁₇

D₄

yes

characteristic

C₂×C₁₆×C₁₇

C₂³

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

Dic₁₃₆

—

Dic₁₃₆

—

#10587465

D₄

yes

—

C₂×Q₈×C₁₇

—

C₂×Dic₃₄

—

Q₁₆×C₁₇

—

Q₁₆×C₁₇

—

Q₁₆×C₁₇

—

C₂×C₈×C₁₇

D₈

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

D₈

yes

characteristic

C₄⋊Dic₁₇

—

Dic₆₈

—

Dic₆₈

—

Dic₆₈

—

C₁₆×C₁₇

D₈

yes

—

C₁₆×C₁₇

C₂×D₄

yes

—

#13726

—

maximal, Sylow

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

D₁₆

yes

—

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

D₁₆

yes

—

C₈×C₁₇

#127

yes

characteristic

C₈×C₁₇

C₂×D₈

yes

characteristic

C₂×Q₆₄

—

#2247

—

#2247

D₁₇

yes

characteristic

C₄×C₃₂

—

C₂²×C₁₇

C₂×D₁₆

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

Q₆₄

yes

complemented

C₄×C₁₇

#493

yes

characteristic

C₄×C₁₇

C₂×D₁₆

yes

characteristic

Dic₁₇

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

C₂×D₁₇

yes

—

#494

—

#494

—

Q₆₄

—

C₂×C₃₂

—

C₄×C₁₆

C₂×D₁₇

yes

characteristic

C₂×C₁₇

#2246

yes

characteristic

C₂×C₁₇

C₂×Q₆₄

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

C₂²⋊D₁₇

yes

characteristic

C₂×C₁₆

C₂²×D₁₇

yes

characteristic

C₄×C₈

C₂²⋊D₁₇

yes

characteristic

C₃₂

—

C₁₇

#13726

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

136

—

C₂×C₈

#63228

yes

characteristic

100

C₂×C₈

C₂×C₂²⋊D₁₇

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

136

—

103

C₄²

#63228

yes

characteristic

104

C₁₆

#63228

yes

—

105

C₁₆

D₄×D₁₇

yes

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

136

—

109

Q₈

—

136

—

110

C₂×C₄

—

111

C₂×C₄

—

136

—

112

C₂×C₄

#10587497

yes

—

113

C₂×C₄

C₂×#63228

yes

characteristic

114

C₈

#10587497

yes

—

115

C₈

C₂×#63228

yes

characteristic

116

C₈

#10587626

yes

characteristic

117

C₂²

C₂×#10587497

yes

characteristic

118

C₄

—

119

C₄

—

272

—

120

C₄

#11762649

yes

complemented

121

C₄

#11763185

yes

characteristic

122

C₄

C₂×#10587497

yes

characteristic

123

C₂

#14531033

yes

characteristic

124

C₂

C₂×#11762649

yes

—

	subgroup	G/H
center	C₂²	C₂×#10587497
commutator, cosocle	C₂×C₁₆×C₁₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₁₇
Fitting	C₁₇×#2247	C₂
socle, prime radical	C₂²×C₁₇	C₂×D₁₆
hypercenter	#2247	D₁₇
hypocenter	C₁₇	#13726
LCS	G ▷ C₂×C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2247

subgroup

G/H

center

C₂²

C₂×#10587497

commutator, cosocle

C₂×C₁₆×C₁₇

C₂³

Frattini

C₂×C₁₆

C₂²×D₁₇

Fitting

C₁₇×#2247

C₂

socle, prime radical

C₂²×C₁₇

C₂×D₁₆

D₁₇

C₁₇

G ▷ C₂×C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2247

N	H ≅ G/N	type
61. C₄×C₁₇	Q₆₄	semidirect
90. C₁₇	#13726	semidirect
120. C₄	#11762649	semidirect

H ≅ G/N

type

61. C₄×C₁₇

Q₆₄

semidirect

90. C₁₇

#13726

semidirect

120. C₄

#11762649

semidirect

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

subgroup	index
C₂×#11762649	2
#14531037	2
C₄×#10587465	2
C₁₇×#2247	2
#13726	17

subgroup

index

C₁₇×#2247

Subgroups — #16955321

Conjugacy classes of subgroups

Complements

Maximal subgroups