Group #16949072 — subgroups

Subgroups — #16949072

Order 4352.

246 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 99 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	#14531066	C₂	1	2	yes	maximal
2	#14531059	C₂	1	2	yes	maximal
3	#14528402	C₂	1	1	yes	maximal, characteristic
4	#14528394	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1311	C₂	1	1	yes	maximal, characteristic
6	#11763214	—	2	1	no	—
7	#11763214	C₂²	1	1	yes	characteristic
8	#11763214	C₂²	1	1	yes	characteristic
9	#11763214	C₂²	1	2	yes	—
10	#11763189	—	2	1	no	—
11	#11763160	—	2	1	no	—
12	#11763160	—	2	1	no	—
13	#11762637	—	2	4	no	—
14	#11762635	—	2	4	no	—
15	C₄×C₁₆×C₁₇	C₂²	1	2	yes	—
16	C₈²×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₆₈	—	4	2	no	—
18	#10587555	—	2	2	no	—
19	#10587555	—	2	1	no	—
20	#10587555	—	2	1	no	—
21	C₂×C₁₆×C₁₇	D₄	1	4	yes	—
22	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
23	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
25	#10587489	—	2	2	no	—
26	#10587489	—	2	2	no	—
27	#10587489	—	2	2	no	—
28	#10587489	—	2	2	no	—
29	#10587489	—	4	2	no	—
30	#10587487	—	4	2	no	—
31	#10587487	—	4	2	no	—
32	C₂×Dic₃₄	—	4	1	no	—
33	C₂×Dic₃₄	—	4	1	no	—
34	C₂×Dic₃₄	—	4	2	no	—
35	C₂×C₈×C₁₇	D₈	1	2	yes	—
36	C₂×C₈×C₁₇	D₈	1	2	yes	—
37	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
38	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
39	C₄⋊Dic₁₇	—	4	1	no	—
40	C₄⋊Dic₁₇	—	4	2	no	—
41	C₄⋊Dic₁₇	—	4	1	no	—
42	C₄⋊Dic₁₇	—	4	1	no	—
43	C₄⋊Dic₁₇	—	4	1	no	—
44	C₄⋊Dic₁₇	—	4	2	no	—
45	Dic₆₈	—	8	4	no	—
46	C₁₆×C₁₇	Q₁₆	1	8	yes	complemented
47	#12524	—	17	1	no	maximal, Sylow
48	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
51	C₂×Dic₁₇	—	8	1	no	—
52	C₂×Dic₁₇	—	8	2	no	—
53	C₂×Dic₁₇	—	8	1	no	—
54	Dic₃₄	—	8	2	no	—
55	Dic₃₄	—	8	4	no	—
56	Dic₃₄	—	8	2	no	—
57	C₈×C₁₇	—	2	2	no	—
58	C₈×C₁₇	—	2	2	no	—
59	C₈×C₁₇	C₂×Q₁₆	1	4	yes	—
60	#2254	—	17	2	no	—
61	#2252	—	17	2	no	—
62	#1713	—	17	1	no	—
63	#1710	—	17	1	no	—
64	#1311	D₁₇	1	1	yes	characteristic
65	C₂²×C₁₇	#493	1	1	yes	characteristic
66	C₄×C₁₇	#494	1	2	yes	—
67	C₄×C₁₇	#509	1	2	yes	—
68	C₄×C₁₇	#510	1	2	yes	—
69	Dic₁₇	—	16	1	no	—
70	Dic₁₇	—	16	2	no	—
71	Dic₁₇	—	16	1	no	—
72	#500	—	17	1	no	—
73	#500	—	17	1	no	—
74	#500	—	17	2	no	—
75	#500	—	34	1	no	—
76	#494	—	34	1	no	—
77	#487	—	34	1	no	—
78	#487	—	34	1	no	—
79	#367	—	34	4	no	—
80	#366	—	34	4	no	—
81	C₄×C₁₆	C₂×D₁₇	1	2	yes	—
82	C₈²	C₂×D₁₇	1	1	yes	characteristic
83	C₂×C₁₇	#1713	1	1	yes	characteristic
84	C₂×C₁₇	#2250	1	1	yes	characteristic
85	C₂×C₁₇	#2251	1	1	yes	characteristic
86	C₂×Q₁₆	—	68	2	no	—
87	C₄⋊Q₈	—	34	1	no	—
88	C₄⋊Q₈	—	34	1	no	—
89	C₄⋊Q₈	—	34	2	no	—
90	C₂×C₁₆	D₆₈	1	4	yes	—
91	#107	—	34	2	no	—
92	#107	—	34	2	no	—
93	#107	—	34	2	no	—
94	#107	—	34	2	no	—
95	#107	—	68	2	no	—
96	#103	—	68	2	no	—
97	#103	—	68	2	no	—
98	C₄×C₈	D₆₈	1	1	yes	characteristic
99	C₄×C₈	D₆₈	1	1	yes	characteristic
100	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
101	C₁₇	#12524	1	1	yes	characteristic, complemented, Sylow
102	C₂×Q₈	—	68	1	no	—
103	C₂×Q₈	—	68	1	no	—
104	C₂×Q₈	—	68	2	no	—
105	Q₁₆	—	136	4	no	—
106	C₂×C₈	D₁₃₆	1	2	yes	—
107	C₂×C₈	D₁₃₆	1	2	yes	—
108	C₂×C₈	C₂×D₆₈	1	2	yes	—
109	C₄⋊C₄	—	68	1	no	—
110	C₄⋊C₄	—	68	2	no	—
111	C₄⋊C₄	—	68	1	no	—
112	C₄⋊C₄	—	68	1	no	—
113	C₄⋊C₄	—	68	1	no	—
114	C₄⋊C₄	—	68	2	no	—
115	C₄²	C₂×D₆₈	1	1	yes	characteristic
116	C₁₆	Dic₆₈	1	8	yes	complemented
117	Q₈	—	136	4	no	—
118	Q₈	—	136	2	no	—
119	Q₈	—	136	2	no	—
120	C₂×C₄	—	136	1	no	—
121	C₂×C₄	—	136	2	no	—
122	C₂×C₄	—	136	1	no	—
123	C₂×C₄	#10587560	1	1	yes	characteristic
124	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
125	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	—	2	2	no	—
128	C₈	C₂×Dic₆₈	1	4	yes	—
129	C₂²	#11763186	1	1	yes	characteristic
130	C₄	—	272	2	no	—
131	C₄	—	272	1	no	—
132	C₄	—	272	1	no	—
133	C₄	#11763189	1	2	yes	—
134	C₄	#11763247	1	2	yes	—
135	C₄	#11763252	1	2	yes	—
136	C₂	#14528402	1	1	yes	characteristic
137	C₂	#14531050	1	1	yes	characteristic
138	C₂	#14531055	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14531066

C₂

yes

maximal

#14531059

C₂

yes

maximal

#14528402

C₂

yes

maximal, characteristic

#14528394

C₂

yes

maximal, characteristic

C₁₇×#1311

C₂

yes

maximal, characteristic

#11763214

—

#11763214

C₂²

yes

characteristic

#11763214

C₂²

yes

characteristic

#11763214

C₂²

yes

—

—

—

—

—

—

C₄×C₁₆×C₁₇

C₂²

yes

—

C₈²×C₁₇

C₂²

yes

characteristic

C₂×Dic₆₈

—

#10587555

—

#10587555

—

#10587555

—

C₂×C₁₆×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

C₂³

yes

—

—

—

—

—

—

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

Dic₆₈

—

C₁₆×C₁₇

Q₁₆

yes

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₂×Q₁₆

yes

—

—

—

—

—

D₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

yes

—

C₄×C₁₇

#509

yes

—

C₄×C₁₇

#510

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₁₇

yes

—

C₈²

C₂×D₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

D₆₈

yes

—

—

—

—

—

—

—

—

C₄×C₈

D₆₈

yes

characteristic

C₄×C₈

D₆₈

yes

characteristic

100

C₄×C₈

C₂²×D₁₇

yes

characteristic

101

C₁₇

#12524

yes

characteristic, complemented, Sylow

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

Q₁₆

—

136

—

106

C₂×C₈

D₁₃₆

yes

—

107

C₂×C₈

D₁₃₆

yes

—

108

C₂×C₈

C₂×D₆₈

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

C₂×D₆₈

yes

characteristic

116

C₁₆

Dic₆₈

yes

complemented

117

Q₈

—

136

—

118

Q₈

—

136

—

119

Q₈

—

136

—

120

C₂×C₄

—

136

—

121

C₂×C₄

—

136

—

122

C₂×C₄

—

136

—

123

C₂×C₄

#10587560

yes

characteristic

124

C₂×C₄

C₂×D₁₃₆

yes

characteristic

125

C₂×C₄

C₂×D₁₃₆

yes

characteristic

126

C₈

—

127

C₈

—

128

C₈

C₂×Dic₆₈

yes

—

129

C₂²

#11763186

yes

characteristic

130

C₄

—

272

—

131

C₄

—

272

—

132

C₄

—

272

—

133

C₄

#11763189

yes

—

134

C₄

#11763247

yes

—

135

C₄

#11763252

yes

—

136

C₂

#14528402

yes

characteristic

137

C₂

#14531050

yes

characteristic

138

C₂

#14531055

yes

characteristic

	subgroup	G/H
center	C₂²	#11763186
commutator, cosocle	C₄×C₈×C₁₇	C₂³
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1311	C₂
socle, prime radical	C₂²×C₁₇	#493
hypercenter	#1311	D₁₇
hypocenter	C₁₇	#12524
LCS	G ▷ C₄×C₈×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#11763186

commutator, cosocle

C₄×C₈×C₁₇

C₂³

Frattini

C₄×C₈

C₂²×D₁₇

Fitting

C₁₇×#1311

C₂

C₂²×C₁₇

D₁₇

C₁₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
46. C₁₆×C₁₇	Q₁₆	semidirect
101. C₁₇	#12524	semidirect
116. C₁₆	Dic₆₈	semidirect

H ≅ G/N

type

46. C₁₆×C₁₇

Q₁₆

semidirect

101. C₁₇

#12524

semidirect

116. C₁₆

Dic₆₈

semidirect

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

subgroup	index
#14531066	2
#14531059	2
#14528402	2
#14528394	2
C₁₇×#1311	2
#12524	17

subgroup

index

C₁₇×#1311

Subgroups — #16949072

Conjugacy classes of subgroups

Complements

Maximal subgroups