Finite Group Databasebeta

← #17950221

Subgroups — #17950221

Order 4352.

388 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 96 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#14544712C211yesmaximal, characteristic
2#14544699C211yesmaximal, characteristic
3#14543603C212yesmaximal
4#14543602C212yesmaximal
5#14537143C211yesmaximal, characteristic
6C2×C4⋊#3624—21no—
7#11763675—21no—
8#11763675—21no—
9#11763675—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×#10587575—21no—
13#11762708C2211yescharacteristic
14#11762707C411yescharacteristic
15#11762706—21no—
16#11762705C2211yescharacteristic
17#11762704—21no—
18#11762678C411yescharacteristic
19#11762678C411yescharacteristic
20#11762673—21no—
21C42×Dic17C2211yescharacteristic
22C17×#389C411yescharacteristic
23C22×#3624—22no—
24C22×#3624—22no—
25C2×C4⋊C4×C17—21no—
26C2×C4⋊C4×C17C2×C411yescharacteristic
27C2×C4⋊C4×C17C2×C411yescharacteristic
28C2×C42×C17C2×C411yescharacteristic
29C2×C4⋊Dic17—21no—
30C2×C4⋊Dic17C2×C411yescharacteristic
31C2×C4⋊Dic17C2×C411yescharacteristic
32C2×#63225—21no—
33C2×#63225—21no—
34C2×#63225—21no—
35C2×#63225—21no—
36C2×#63225—21no—
37C2×#63225—21no—
38C2×C4×Dic17C2×C411yescharacteristic
39C2×C4×Dic17D411yescharacteristic
40C2×C4×Dic17D411yescharacteristic
41C2×C4×Dic17D411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43C2×C4×Dic17C2311yescharacteristic
44#10587575—42no—
45#10587575—42no—
46C4⋊#3624—42no—
47C17×#95—21no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51C22×C4×C17—21no—
52C22×C4×C17—21no—
53C22×C4×C17C22⋊C411yescharacteristic
54C22×C4×C17C22⋊C411yescharacteristic
55C22×C4×C17C22×C411yescharacteristic
56C22×Dic17—21no—
57C22×Dic17—21no—
58C22×Dic17C22⋊C411yescharacteristic
59C22×Dic17C22⋊C411yescharacteristic
60C22×Dic17C2×D411yescharacteristic
61C22×Dic17C2×D411yescharacteristic
62C2×#3624—42no—
63C2×#3624—42no—
64C2×#3624—42no—
65C2×#3624—42no—
66C2×#3624—42no—
67C2×#3624—42no—
68C4⋊C4×C17—22no—
69C4⋊C4×C17—41no—
70C4⋊C4×C17—42no—
71C42×C17—22no—
72C4⋊Dic17—22no—
73C4⋊Dic17—41no—
74C4⋊Dic17—41no—
75C4⋊Dic17—41no—
76#63225—42no—
77#63225—42no—
78#63225—42no—
79#63225—42no—
80#63225—42no—
81#63225—41no—
82#63225—41no—
83C4×Dic17—22no—
84C4×Dic17—22no—
85C4×Dic17—41no—
86C4×Dic17—41no—
87C4×Dic17—41no—
88C4×Dic17D812yes—
89C4×Dic17SD1612yes—
90#7723—171nomaximal, Sylow
91C23×C17C2×C22⋊C411yescharacteristic
92C2×C4×C17—21no—
93C2×C4×C17—21no—
94C2×C4×C17—21no—
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—21no—
99C2×C4×C17—41no—
100C2×C4×C17—41no—
101C2×C4×C17—41no—
102C2×C4×C17—41no—
103C2×C4×C17—41no—
104C2×C4×C17—41no—
105C2×C4×C17#10212yes—
106C2×C4×C17#13112yes—
107C2×Dic17—21no—
108C2×Dic17—21no—
109C2×Dic17—21no—
110C2×Dic17—21no—
111C2×Dic17—21no—
112C2×Dic17—21no—
113C2×Dic17—21no—
114C2×Dic17—21no—
115C2×Dic17—41no—
116C2×Dic17—41no—
117C2×Dic17—41no—
118C2×Dic17—41no—
119C2×Dic17—41no—
120C2×Dic17—41no—
121C2×Dic17#10112yes—
122C2×Dic17#10212yes—
123C2×Dic17C2×D811yescharacteristic
124C2×Dic17C2×SD1611yescharacteristic
125C2×Dic17#13712yes—
126#3624—82no—
127#3624—82no—
128#2859—171no—
129#1839—171no—
130#1831—171no—
131#1271—172no—
132#1271—172no—
133C22×C17#40811yescharacteristic
134C22×C17#41011yescharacteristic
135C22×C17C2×#10111yescharacteristic
136C22×C17C2×#10211yescharacteristic
137C22×C17#41911yescharacteristic
138C22×C17#42111yescharacteristic
139C22×C17#42111yescharacteristic
140C4×C17—22no—
141C4×C17—22no—
142C4×C17—41no—
143C4×C17—81no—
144C4×C17—81no—
145Dic17—22no—
146Dic17—22no—
147Dic17—21no—
148Dic17—21no—
149Dic17—22no—
150Dic17—81no—
151Dic17—81no—
152C2×#125—341no—
153C2×C4⋊C8—341no—
154#389—171no—
155#389—171no—
156#389—171no—
157#389—341no—
158#389—341no—
159#389#59511yescharacteristic
160#384—171no—
161#384—171no—
162#382—341no—
163C43—171no—
164#336—172no—
165#336—172no—
166#336—341no—
167#336—341no—
168#336—341no—
169C2×C17#147011yescharacteristic
170C2×C17#148011yescharacteristic
171C2×C17#149711yescharacteristic
172C2×C17#151511yescharacteristic
173C2×C17#151711yescharacteristic
174C2×C17#153411yescharacteristic
175C2×C17#154611yescharacteristic
176C22×C8—342no—
177C22×C8—342no—
178#125—682no—
179#125—682no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—171no—
182C2×C4⋊C4—171no—
183C2×C4⋊C4—341no—
184C2×C4⋊C4—341no—
185C2×C4⋊C4—341no—
186C2×C4⋊C4—341no—
187C2×C4⋊C4—341no—
188C2×C4⋊C4—341no—
189C2×C4⋊C4—341no—
190C2×C4⋊C4C2×#59511yescharacteristic
191C2×C4⋊C4C2×#59511yescharacteristic
192C2×C42—171no—
193C2×C42—171no—
194C2×C42—171no—
195C2×C42—171no—
196C2×C42—171no—
197C2×C42—171no—
198C2×C42C2×#59511yescharacteristic
199C4⋊C8—682no—
200#95—21no—
201#95—341no—
202#95—341no—
203#95—341no—
204C17#772311yescharacteristic, complemented, Sylow
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—171no—
208C22×C4—171no—
209C22×C4—171no—
210C22×C4—171no—
211C22×C4—341no—
212C22×C4—341no—
213C22×C4C22⋊#59511yescharacteristic
214C22×C4C22⋊#59511yescharacteristic
215C22×C4C22×#59511yescharacteristic
216C2×C8—682no—
217C2×C8—682no—
218C2×C8—682no—
219C2×C8—682no—
220C2×C8—682no—
221C2×C8—682no—
222C4⋊C4—22no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C4⋊C4—342no—
226C4⋊C4—681no—
227C4⋊C4—681no—
228C4⋊C4—682no—
229C4⋊C4—682no—
230C4⋊C4—681no—
231C4⋊C4—681no—
232C4⋊C4—681no—
233C4⋊C4—682no—
234C4⋊C4—682no—
235C4⋊C4—682no—
236C42—22no—
237C42—172no—
238C42—172no—
239C42—342no—
240C42—342no—
241C42—681no—
242C42—681no—
243C42—681no—
244C23C2×C22⋊#59511yescharacteristic
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—172no—
259C2×C4—171no—
260C2×C4—172no—
261C2×C4—171no—
262C2×C4—172no—
263C2×C4—341no—
264C2×C4—341no—
265C2×C4—341no—
266C2×C4—341no—
267C2×C4—341no—
268C2×C4—341no—
269C2×C4—341no—
270C2×C4—341no—
271C2×C4—681no—
272C2×C4—681no—
273C2×C4—681no—
274C2×C4—681no—
275C2×C4—681no—
276C2×C4—681no—
277C2×C4#1058754712yes—
278C2×C4#1058767812yes—
279C8—1362no—
280C8—1362no—
281C22#1176378211yescharacteristic
282C22#1176378811yescharacteristic
283C22D17⋊#10111yescharacteristic
284C22C2×#1058754711yescharacteristic
285C22#1176380211yescharacteristic
286C22#1176380611yescharacteristic
287C22#1176380611yescharacteristic
288C4—22no—
289C4—22no—
290C4—41no—
291C4—81no—
292C4—81no—
293C4—342no—
294C4—341no—
295C4—341no—
296C4—342no—
297C4—342no—
298C4—1361no—
299C4—1361no—
300C2#1454405611yescharacteristic
301C2#1454408911yescharacteristic
302C2#1454414611yescharacteristic
303C2#1454420611yescharacteristic
304C2#1454421411yescharacteristic
305C2#1454426711yescharacteristic
306C2#1454429411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#389C4
socle, prime radicalC23×C17C2×C22⋊C4
cosocleC2×C4×Dic17C23
hypercenter#389#595
hypocenterC17#7723
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
204. C17#7723semidirect

Maximal subgroups

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

subgroupindex
#145447122
#145446992
#145436032
#145436022
#145371432
#772317