Group #17948354 — subgroups

Subgroups — #17948354

Order 4352.

146 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 58 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	#14543813	C₂	1	1	yes	maximal, characteristic
2	C₈×#63216	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×#3622	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#63216	—	2	1	no	—
5	C₄×#63216	—	2	1	no	—
6	Q₈×#3622	—	2	1	no	—
7	#11762562	C₄	1	1	yes	characteristic
8	Q₈⋊#3622	—	2	1	no	—
9	C₈×#3622	C₂²	1	1	yes	characteristic
10	C₄×Q₁₆×C₁₇	C₄	1	1	yes	characteristic
11	C₂×Q₁₆×C₁₇	C₈	1	1	yes	characteristic
12	C₄×Q₈×C₁₇	—	2	1	no	—
13	C₂×#63216	—	2	2	no	—
14	C₂×#63216	—	4	1	no	—
15	C₁₇×#107	C₈	1	1	yes	characteristic
16	C₁₇×#103	—	2	1	no	—
17	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₄⋊#3622	—	2	1	no	—
19	C₄⋊#3622	—	4	1	no	—
20	C₄×#3622	—	4	1	no	—
21	C₄×#3622	D₄	1	1	yes	characteristic
22	C₄×#3622	D₄	1	1	yes	characteristic
23	C₂×Q₈×C₁₇	—	2	1	no	—
24	Q₁₆×C₁₇	—	2	1	no	—
25	Q₁₆×C₁₇	C₁₆	1	2	yes	complemented
26	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈×C₁₇	M₁₆	1	1	yes	characteristic
28	C₄⋊C₄×C₁₇	—	2	1	no	—
29	C₄⋊C₄×C₁₇	—	4	1	no	—
30	C₄²×C₁₇	—	4	1	no	—
31	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×#3622	—	2	1	no	—
33	C₂×#3622	—	4	1	no	—
34	C₂×#3622	D₈	1	1	yes	characteristic
35	C₂×#3622	SD₁₆	1	1	yes	characteristic
36	#63216	—	2	4	no	—
37	#63216	—	8	1	no	—
38	#7069	—	17	1	no	maximal, Sylow
39	Q₈×C₁₇	—	2	2	no	—
40	Q₈×C₁₇	—	4	1	no	—
41	C₂×C₄×C₁₇	—	4	1	no	—
42	C₂×C₄×C₁₇	—	4	1	no	—
43	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
44	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
45	C₂×C₄×C₁₇	#104	1	1	yes	characteristic
46	C₈×C₁₇	—	2	1	no	—
47	C₈×C₁₇	C₂×C₁₆	1	1	yes	characteristic
48	C₈×C₁₇	M₃₂	1	1	yes	characteristic
49	#3622	—	2	2	no	—
50	#3622	—	2	1	no	—
51	#3622	—	8	1	no	—
52	#3622	SD₃₂	1	1	yes	characteristic
53	#3622	Q₃₂	1	1	yes	characteristic
54	C₈×Q₁₆	—	17	1	no	—
55	#1370	—	17	1	no	—
56	C₈×C₁₆	—	17	1	no	—
57	C₂²×C₁₇	D₄⋊C₈	1	1	yes	characteristic
58	C₄×C₁₇	—	2	1	no	—
59	C₄×C₁₇	—	4	2	no	—
60	C₄×C₁₇	—	8	1	no	—
61	C₄×C₁₇	C₂²⋊C₁₆	1	1	yes	characteristic
62	C₄×C₁₇	#350	1	1	yes	characteristic
63	C₄×C₁₇	#358	1	1	yes	characteristic
64	C₄×C₁₇	#359	1	1	yes	characteristic
65	C₈×Q₈	—	34	1	no	—
66	C₄×Q₁₆	#595	1	1	yes	characteristic
67	C₄⋊C₁₆	—	34	1	no	—
68	C₄×C₁₆	—	34	1	no	—
69	#335	—	17	1	no	—
70	Q₈⋊C₈	—	34	1	no	—
71	C₈²	—	17	1	no	—
72	C₂×C₁₇	D₄⋊C₁₆	1	1	yes	characteristic
73	C₂×C₁₇	#1332	1	1	yes	characteristic
74	C₂×C₁₇	#1335	1	1	yes	characteristic
75	C₂×Q₁₆	#3624	1	1	yes	characteristic
76	C₄×Q₈	—	2	1	no	—
77	C₂×C₁₆	—	34	2	no	—
78	C₂×C₁₆	—	68	1	no	—
79	#107	#3624	1	1	yes	characteristic
80	C₄⋊C₈	—	34	1	no	—
81	C₄⋊C₈	—	68	1	no	—
82	#103	—	2	1	no	—
83	C₄×C₈	—	17	1	no	—
84	C₄×C₈	—	17	1	no	—
85	C₄×C₈	—	68	1	no	—
86	C₄×C₈	C₂×#595	1	1	yes	characteristic
87	C₁₇	#7069	1	1	yes	characteristic, complemented, Sylow
88	C₂×Q₈	—	2	1	no	—
89	Q₁₆	—	2	1	no	—
90	Q₁₆	#63216	1	2	yes	complemented
91	C₂×C₈	—	17	1	no	—
92	C₂×C₈	—	17	1	no	—
93	C₂×C₈	—	34	1	no	—
94	C₂×C₈	—	68	1	no	—
95	C₂×C₈	C₂×#3624	1	1	yes	characteristic
96	C₂×C₈	#63247	1	1	yes	characteristic
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	4	1	no	—
99	C₄²	—	4	1	no	—
100	C₄²	C₂²⋊#595	1	1	yes	characteristic
101	C₁₆	—	34	4	no	—
102	C₁₆	—	136	1	no	—
103	Q₈	—	2	2	no	—
104	Q₈	—	4	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	#10587547	1	1	yes	characteristic
108	C₂×C₄	#10587548	1	1	yes	characteristic
109	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
110	C₈	—	2	1	no	—
111	C₈	—	17	1	no	—
112	C₈	—	17	1	no	—
113	C₈	—	34	2	no	—
114	C₈	—	34	1	no	—
115	C₈	—	136	1	no	—
116	C₈	C₂×#63216	1	1	yes	characteristic
117	C₈	#10587538	1	1	yes	characteristic
118	C₂²	D₄⋊#3624	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	4	2	no	—
121	C₄	—	8	1	no	—
122	C₄	C₂²⋊#63216	1	1	yes	characteristic
123	C₄	#11763707	1	1	yes	characteristic
124	C₄	#11763722	1	1	yes	characteristic
125	C₄	#11763724	1	1	yes	characteristic
126	C₂	D₄⋊#63216	1	1	yes	characteristic
127	C₂	#14543740	1	1	yes	characteristic
128	C₂	#14543746	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14543813

C₂

yes

maximal, characteristic

C₈×#63216

C₂

yes

maximal, characteristic

Q₁₆×#3622

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

characteristic

Q₈⋊#3622

—

C₈×#3622

C₂²

yes

characteristic

C₄×Q₁₆×C₁₇

C₄

yes

characteristic

C₂×Q₁₆×C₁₇

C₈

yes

characteristic

C₄×Q₈×C₁₇

—

C₂×#63216

—

C₂×#63216

—

C₁₇×#107

C₈

yes

characteristic

C₁₇×#103

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

—

—

D₄

yes

characteristic

C₄×#3622

D₄

yes

characteristic

C₂×Q₈×C₁₇

—

Q₁₆×C₁₇

—

Q₁₆×C₁₇

C₁₆

yes

complemented

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₂×C₈×C₁₇

M₁₆

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂²⋊C₄

yes

—

—

D₈

yes

characteristic

C₂×#3622

SD₁₆

yes

—

—

—

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₈

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₈×C₁₇

—

C₈×C₁₇

C₂×C₁₆

yes

characteristic

C₈×C₁₇

M₃₂

yes

—

—

—

SD₃₂

yes

characteristic

#3622

Q₃₂

yes

characteristic

C₈×Q₁₆

—

#1370

—

C₈×C₁₆

—

C₂²×C₁₇

D₄⋊C₈

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂²⋊C₁₆

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₈×Q₈

—

C₄×Q₁₆

#595

yes

characteristic

C₄⋊C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

—

C₂×C₁₇

D₄⋊C₁₆

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

yes

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

#3624

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#595

yes

characteristic

C₁₇

#7069

yes

characteristic, complemented, Sylow

C₂×Q₈

—

Q₁₆

—

Q₁₆

#63216

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

100

C₄²

C₂²⋊#595

yes

characteristic

101

C₁₆

—

102

C₁₆

—

136

—

103

Q₈

—

104

Q₈

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#10587547

yes

characteristic

108

C₂×C₄

#10587548

yes

characteristic

109

C₂×C₄

C₂²⋊#3624

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

136

—

116

C₈

C₂×#63216

yes

characteristic

117

C₈

#10587538

yes

characteristic

118

C₂²

D₄⋊#3624

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

C₂²⋊#63216

yes

characteristic

123

C₄

#11763707

yes

characteristic

124

C₄

#11763722

yes

characteristic

125

C₄

#11763724

yes

characteristic

126

C₂

D₄⋊#63216

yes

characteristic

127

C₂

#14543740

yes

characteristic

128

C₂

#14543746

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10587547
commutator	C₈×C₁₇	C₂×C₁₆
Frattini	C₄×C₈	C₂×#595
Fitting	C₄×Q₁₆×C₁₇	C₄
socle, prime radical	C₂²×C₁₇	D₄⋊C₈
cosocle	C₈×#3622	C₂²
hypercenter	C₄×Q₁₆	#595
hypocenter	C₁₇	#7069
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₁₆

subgroup

G/H

center

C₂×C₄

#10587547

commutator

C₈×C₁₇

C₂×C₁₆

Frattini

C₄×C₈

C₂×#595

Fitting

C₄×Q₁₆×C₁₇

C₄

socle, prime radical

C₂²×C₁₇

D₄⋊C₈

C₂²

C₄×Q₁₆

C₁₇

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₁₆

N	H ≅ G/N	type
25. Q₁₆×C₁₇	C₁₆	semidirect
87. C₁₇	#7069	semidirect
90. Q₁₆	#63216	semidirect

H ≅ G/N

type

25. Q₁₆×C₁₇

C₁₆

semidirect

87. C₁₇

#7069

semidirect

90. Q₁₆

#63216

semidirect

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

subgroup	index
#14543813	2
C₈×#63216	2
Q₁₆×#3622	2
#7069	17

subgroup

index

#14543813

C₈×#63216

Q₁₆×#3622

#7069

Subgroups — #17948354

Conjugacy classes of subgroups

Complements

Maximal subgroups