Finite Group Databasebeta

← #16947069

Subgroups — #16947069

Order 4352.

388 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 59 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#14536821C211yesmaximal, characteristic
2#14527775C212yesmaximal
3#14526783C212yesmaximal
4#14526714C211yesmaximal, characteristic
5C17×#2030C211yesmaximal, characteristic
6C2×#10587610—21no—
7#11763137—22no—
8#11762749—22no—
9#11762734—21no—
10#11762734C2212yes—
11#11762730—22no—
12#11762546—22no—
13#11762533C2212yes—
14C17×#543C2211yescharacteristic
15C17×#381—21no—
16C17×#381C2211yescharacteristic
17C17×#342—21no—
18C17×#342C2211yescharacteristic
19C23×C4×C17D411yescharacteristic
20C22×Dic34—21no—
21C17×C22⋊Q8—22no—
22C17×C22⋊Q8—42no—
23C2×C22⋊C4×C17—21no—
24C2×C22⋊C4×C17—21no—
25C2×C22⋊C4×C17D411yescharacteristic
26C2×C22⋊C4×C17C2311yescharacteristic
27C2×C22⋊Dic17—22no—
28C2×C4⋊Dic17—21no—
29C2×C4⋊Dic17—21no—
30C2×C4⋊Dic17D412yes—
31#10587610—42no—
32#10587610—42no—
33C2×#63225—22no—
34C17×#95—22no—
35C17×#95—42no—
36#10587506—42no—
37#10587502—22no—
38#10587502—21no—
39#10587502—22no—
40#10587502D412yes—
41C22⋊#3622—22no—
42#10587479—42no—
43C24×C17C2×D411yescharacteristic
44C2×Q8×C17—22no—
45C22×C4×C17—21no—
46C22×C4×C17—21no—
47C22×C4×C17—21no—
48C22×C4×C17—22no—
49C22×C4×C17—41no—
50C22×C4×C17—41no—
51C22×C4×C17—41no—
52C22×C4×C17SD1612yes—
53C22×C4×C17C2×D412yes—
54C22×Dic17—22no—
55C22×Dic17—22no—
56C2×Dic34—42no—
57C2×Dic34—81no—
58C2×Dic34—81no—
59C4⋊C4×C17—22no—
60C4⋊C4×C17—42no—
61C22⋊C4×C17—41no—
62C22⋊C4×C17—41no—
63C22⋊C4×C17—41no—
64C22⋊C4×C17—41no—
65C22⋊C4×C17—41no—
66C22⋊C4×C17—41no—
67C22⋊Dic17—42no—
68C22⋊Dic17—42no—
69C4⋊Dic17—42no—
70C4⋊Dic17—42no—
71C4⋊Dic17—81no—
72#63225—44no—
73C2×#3622—42no—
74#12207—171nomaximal, Sylow
75C23×C17—21no—
76C23×C17—21no—
77C23×C17—42no—
78C23×C17#12011yescharacteristic
79C23×C17C2×SD1611yescharacteristic
80C23×C17AGL(1,ℤ/8)11yescharacteristic
81Q8×C17—82no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—41no—
85C2×C4×C17—41no—
86C2×C4×C17—42no—
87C2×C4×C17—42no—
88C2×C4×C17—42no—
89C2×C4×C17—41no—
90C2×C4×C17—42no—
91C2×C4×C17—81no—
92C2×C4×C17—82no—
93C2×C4×C17—81no—
94C2×C4×C17—82no—
95C2×Dic17—44no—
96C2×Dic17—42no—
97C2×Dic17—42no—
98C2×Dic17—42no—
99C2×Dic17—42no—
100Dic34—84no—
101#3622—82no—
102#2817—171no—
103#2030D1711yescharacteristic
104#1608—172no—
105#1344—172no—
106#1317—171no—
107C22×C17—22no—
108C22×C17—21no—
109C22×C17—21no—
110C22×C17—41no—
111C22×C17—41no—
112C22×C17—42no—
113C22×C17—81no—
114C22×C17#45011yescharacteristic
115C22×C17#45511yescharacteristic
116C22×C17B(4,𝔽2)11yescharacteristic
117C4×C17—42no—
118C4×C17—42no—
119C4×C17—81no—
120C4×C17—82no—
121C4×C17—161no—
122Dic17—82no—
123Dic17—82no—
124#543C2×D1711yescharacteristic
125C2×C22⋊Q8—341no—
126#483—342no—
127#400—342no—
128#397—172no—
129#397—341no—
130#395—342no—
131#381—21no—
132#381C2×D1711yescharacteristic
133#342—21no—
134#342C2×D1711yescharacteristic
135#328—342no—
136#323—172no—
137C2×C17—21no—
138C2×C17—21no—
139C2×C17—41no—
140C2×C17—41no—
141C2×C17#160111yescharacteristic
142C2×C17#220112yes—
143C22×Q8—341no—
144C23×C4C22⋊D1711yescharacteristic
145C22⋊Q8—22no—
146C22⋊Q8—42no—
147C22⋊Q8—682no—
148C22⋊Q8—682no—
149C2×C4⋊C4—172no—
150C2×C4⋊C4—342no—
151C2×C4⋊C4—341no—
152C2×C4⋊C4—341no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—342no—
156C2×C22⋊C4C22⋊D1711yescharacteristic
157C2×C22⋊C4C22×D1711yescharacteristic
158#106—682no—
159#103—682no—
160C22⋊C8—342no—
161#95—22no—
162#95—42no—
163#95—172no—
164#95—342no—
165#95—341no—
166#95—342no—
167C17#1220711yescharacteristic, complemented, Sylow
168C24C2×C22⋊D1711yescharacteristic
169C2×Q8—22no—
170C2×Q8—682no—
171C2×Q8—1361no—
172C2×Q8—1361no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—342no—
181C22×C4—342no—
182C22×C4#6323012yes—
183C22×C4D4×D1712yes—
184C2×C8—682no—
185C4⋊C4—22no—
186C4⋊C4—42no—
187C4⋊C4—682no—
188C4⋊C4—684no—
189C4⋊C4—682no—
190C4⋊C4—1361no—
191C22⋊C4—41no—
192C22⋊C4—41no—
193C22⋊C4—41no—
194C22⋊C4—41no—
195C22⋊C4—41no—
196C22⋊C4—41no—
197C22⋊C4—682no—
198C22⋊C4—682no—
199C23—21no—
200C23—21no—
201C23—42no—
202C23#1058762311yescharacteristic
203C23C2×#6323011yescharacteristic
204C23#1058763511yescharacteristic
205Q8—82no—
206Q8—1364no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—81no—
217C2×C4—82no—
218C2×C4—81no—
219C2×C4—82no—
220C2×C4—684no—
221C2×C4—682no—
222C2×C4—682no—
223C2×C4—682no—
224C2×C4—682no—
225C8—1362no—
226C22—22no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22—81no—
233C22#1176295911yescharacteristic
234C22#1176298611yescharacteristic
235C22#1176299511yescharacteristic
236C4—42no—
237C4—42no—
238C4—81no—
239C4—82no—
240C4—161no—
241C4—1362no—
242C4—1362no—
243C2—21no—
244C2—21no—
245C2—41no—
246C2—41no—
247C2#1452773511yescharacteristic
248C2#1453079512yes—
subgroupG/H
centerC22#11762995
commutator, cosocleC2×C22⋊C4×C17C23
FrattiniC2×C22⋊C4C22×D17
FittingC17×#2030C2
socleC22×C17B(4,𝔽2)
prime radicalC24×C17C2×D4
hypercenter#2030D17
hypocenterC17#12207
LCSG ▷ C2×C22⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2030

Complements

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

NH ≅ G/Ntype
167. C17#12207semidirect

Maximal subgroups

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

subgroupindex
#145368212
#145277752
#145267832
#145267142
C17×#20302
#1220717