Finite Group Databasebeta

← #16921909

Subgroups — #16921909

Order 4352.

226 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14526909C211yesmaximal, characteristic
2#14526716C211yesmaximal, characteristic
3C17×#2025C211yesmaximal, characteristic
4C2×#10587505—21no—
5#11762535—21no—
6C2×C17×#125C411yescharacteristic, complemented
7C2×C17×#102—21no—
8C17×#390C2211yescharacteristic
9C17×#340—21no—
10C17×#336C411yescharacteristic, complemented
11C22×D4×C17D411yescharacteristic
12C22×C8×C17—21no—
13C17×#125—42no—
14C2×C4⋊C4×C17—21no—
15C2×C4⋊C4×C17—41no—
16C2×C4⋊C4×C17D411yescharacteristic
17C2×C22⋊C4×C17—41no—
18C2×C42×C17C2×C411yescharacteristic
19C2×C22⋊Dic17—41no—
20C17×#102—42no—
21#10587505—22no—
22#10587505—41no—
23C22⋊#3622—41no—
24C24×C17—21no—
25C2×D4×C17—21no—
26C2×D4×C17—41no—
27C2×D4×C17—41no—
28C2×D4×C17SD1611yescharacteristic
29C2×D4×C17Q1611yescharacteristic
30C22×C4×C17—21no—
31C22×C4×C17—21no—
32C22×C4×C17—41no—
33C22×C4×C17C22⋊C411yescharacteristic
34C22×Dic17—81no—
35C2×C8×C17—41no—
36C2×C8×C17—41no—
37C2×C8×C17—41no—
38C4⋊C4×C17—22no—
39C4⋊C4×C17—41no—
40C4⋊C4×C17—41no—
41C4⋊C4×C17—42no—
42C4⋊C4×C17—42no—
43C22⋊C4×C17—82no—
44C42×C17—41no—
45C22⋊Dic17—42no—
46C22⋊Dic17—81no—
47C2×#3622—81no—
48#7428—171nomaximal, Sylow
49C23×C17—21no—
50C23×C17—21no—
51C23×C17—41no—
52C23×C17—41no—
53C23×C17—81no—
54C23×C17—81no—
55C23×C17#9911yescharacteristic
56D4×C17—41no—
57D4×C17—41no—
58D4×C17—81no—
59C2×C4×C17—21no—
60C2×C4×C17—21no—
61C2×C4×C17—41no—
62C2×C4×C17—41no—
63C2×C4×C17—41no—
64C2×C4×C17—42no—
65C2×C4×C17—41no—
66C2×C4×C17—81no—
67C2×C4×C17—81no—
68C2×C4×C17—81no—
69C2×C4×C17#10311yescharacteristic
70C2×C4×C17#10411yescharacteristic
71C2×Dic17—82no—
72C2×Dic17—161no—
73C2×Dic17—161no—
74C8×C17—81no—
75#3622—161no—
76#2025D1711yescharacteristic
77#1393—171no—
78#1318—171no—
79C22×C17—21no—
80C22×C17—21no—
81C22×C17—41no—
82C22×C17—41no—
83C22×C17—81no—
84C22×C17—81no—
85C22×C17—81no—
86C22×C17—81no—
87C22×C17—161no—
88C22×C17#32811yescharacteristic
89C22×C17AΣL(1,𝔽16)11yescharacteristic
90C22×C17#35411yescharacteristic
91C4×C17—21no—
92C4×C17—21no—
93C4×C17—81no—
94C4×C17—82no—
95C4×C17—81no—
96Dic17—162no—
97C2×#125Dic1711yescharacteristic, complemented
98C2×#102—21no—
99C2×#99—341no—
100#390C2×D1711yescharacteristic
101#340—21no—
102#336Dic1711yescharacteristic, complemented
103#324—341no—
104C2×C17—21no—
105C2×C17—21no—
106C2×C17—81no—
107C2×C17—81no—
108C2×C17#135111yescharacteristic
109C2×C17#141211yescharacteristic
110C2×C17#141411yescharacteristic
111C22×D4C22⋊D1711yescharacteristic
112C22×C8—21no—
113#125—42no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4C22⋊D1711yescharacteristic
117C2×C22⋊C4—41no—
118C2×C22⋊C4—681no—
119C2×C42C2×Dic1711yescharacteristic
120#102—42no—
121#99—342no—
122#99—681no—
123C22⋊C8—681no—
124C17#742811yescharacteristic, complemented, Sylow
125C24—21no—
126C2×D4—21no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4#6323011yescharacteristic
130C2×D4#6323111yescharacteristic
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—41no—
134C22×C4—1361no—
135C22×C4C22⋊Dic1711yescharacteristic
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—1361no—
140C4⋊C4—22no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C22⋊C4—82no—
146C22⋊C4—682no—
147C22⋊C4—1361no—
148C42—41no—
149C23—21no—
150C23—21no—
151C23—41no—
152C23—41no—
153C23—81no—
154C23—81no—
155C23#1058750511yescharacteristic
156D4—41no—
157D4—41no—
158D4—81no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4—81no—
169C2×C4—2721no—
170C2×C4—2721no—
171C2×C4—1362no—
172C2×C4#1058750611yescharacteristic
173C2×C4#1058750811yescharacteristic
174C8—81no—
175C8—2721no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22—81no—
184C22—161no—
185C22#1176254611yescharacteristic
186C22#1176259611yescharacteristic
187C22#1176260411yescharacteristic
188C4—21no—
189C4—21no—
190C4—81no—
191C4—82no—
192C4—81no—
193C4—2722no—
194C2—21no—
195C2—21no—
196C2—81no—
197C2—81no—
198C2#1452680511yescharacteristic
199C2#1452695511yescharacteristic
200C2#1452695911yescharacteristic
subgroupG/H
centerC22#11762604
commutatorC2×C42×C17C2×C4
Frattini#390C2×D17
FittingC17×#2025C2
socleC22×C17#354
prime radicalC22×D4×C17D4
cosocleC17×#390C22
hypercenter#2025D17
hypocenterC17#7428
LCSG ▷ C2×C42×C17 ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ #2025

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. C2×C17×#125C4semidirect
10. C17×#336C4semidirect
97. C2×#125Dic17semidirect
102. #336Dic17semidirect
124. C17#7428semidirect

Maximal subgroups

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

subgroupindex
#145269092
#145267162
C17×#20252
#742817