Finite Group Databasebeta

← #16945235

Subgroups — #16945235

Order 4352.

396 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 83 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#14533071C211yesmaximal, characteristic
2C2×#11762535C211yesmaximal, characteristic
3#14526810C212yesmaximal, complemented
4#14526789C212yesmaximal, complemented
5C17×#1821C211yesmaximal, characteristic
6C2×#10587612C411yescharacteristic
7C2×C22⋊#3622—21no—
8#11762748C411yescharacteristic
9#11762748C2211yescharacteristic
10#11762739C411yescharacteristic
11#11762739C2211yescharacteristic
12#11762690—21no—
13#11762690—21no—
14#11762682—21no—
15#11762672—21no—
16#11762672C411yescharacteristic
17#11762535—21no—
18#11762535C2212yes—
19C22×C4⋊C4×C17C2211yescharacteristic
20C17×#407—21no—
21C17×#343—21no—
22C17×#343C2212yes—
23C23×C4×C17C2×C411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C2×M16×C17—22no—
27C2×C4⋊C4×C17—41no—
28C2×C4⋊C4×C17—41no—
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C4⋊C4×C17D411yescharacteristic
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C4⋊C4×C17C2311yescharacteristic
33C2×C22⋊Dic17—21no—
34C2×C22⋊Dic17—21no—
35#10587612—41no—
36#10587612—41no—
37C2×C4⋊Dic17C2×C411yescharacteristic
38C2×C4⋊Dic17C2×C411yescharacteristic
39C2×#63225—21no—
40C2×#63225—21no—
41C2×C4×Dic17—21no—
42C22×#3622—41no—
43C17×C22⋊C8—41no—
44C17×C22⋊C8—41no—
45#10587502—21no—
46#10587502—21no—
47#10587502—21no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502C2×C411yescharacteristic
53#10587502C2×C411yescharacteristic
54C22⋊#3622—22no—
55C22⋊#3622—41no—
56C24×C17C22⋊C411yescharacteristic
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—22no—
64C22×C4×C17—41no—
65C22×C4×C17—42no—
66C22×C4×C17—42no—
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22×C411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C22×Dic17—21no—
74C22×Dic17—21no—
75C22×Dic17—21no—
76C22×Dic17—21no—
77M16×C17—42no—
78M16×C17—42no—
79C2×C8×C17—42no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—41no—
82C4⋊C4×C17—41no—
83C4⋊C4×C17—41no—
84C22⋊Dic17—42no—
85C22⋊Dic17—41no—
86C22⋊Dic17—41no—
87C4⋊Dic17—41no—
88C4⋊Dic17—41no—
89#63225—42no—
90#63225—42no—
91C4×Dic17—42no—
92C2×#3622—42no—
93C2×#3622—82no—
94#11850—171nomaximal, Sylow
95C23×C17—22no—
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—41no—
99C23×C17#9911yescharacteristic
100C23×C17#9911yescharacteristic
101C23×C17C2×C22⋊C411yescharacteristic
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—81no—
121C2×C4×C17#13512yes—
122C2×Dic17—41no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17—42no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C8×C17—82no—
134#3622—82no—
135#2484—171no—
136#1821D1711yescharacteristic
137C2×#324—171no—
138#1353—172no—
139#1346—172no—
140C22×C17—21no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17—21no—
144C22×C17—21no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17—42no—
148C22×C17—42no—
149C22×C17#35212yes—
150C22×C17C2×#9911yescharacteristic
151C22×C17#41611yescharacteristic
152C22×C17#42111yescharacteristic
153C4×C17—22no—
154C4×C17—41no—
155C4×C17—41no—
156C4×C17—41no—
157C4×C17—41no—
158C4×C17—41no—
159Dic17—81no—
160Dic17—81no—
161Dic17—81no—
162Dic17—81no—
163C2×#117—171no—
164C22×C4⋊C4C2×D1711yescharacteristic
165#407—21no—
166C2×C22⋊C8—341no—
167#400—171no—
168#400—171no—
169#398—171no—
170#398—171no—
171#387—341no—
172#387—341no—
173#385—341no—
174#381—171no—
175#381—341no—
176#343—21no—
177#343C2×D1712yes—
178#324—172no—
179#324—341no—
180C2×C17—21no—
181C2×C17—21no—
182C2×C17—22no—
183C2×C17—41no—
184C2×C17#151611yescharacteristic
185C2×C17C2×#35211yescharacteristic
186C2×C17#213311yescharacteristic
187C23×C4C4×D1711yescharacteristic
188C23×C4D6811yescharacteristic
189C23×C4C22⋊D1711yescharacteristic
190C2×M16—22no—
191C22×C8—681no—
192#117—681no—
193#117—681no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—171no—
197C2×C4⋊C4—171no—
198C2×C4⋊C4—341no—
199C2×C4⋊C4—341no—
200C2×C4⋊C4C4×D1711yescharacteristic
201C2×C4⋊C4D6811yescharacteristic
202C2×C4⋊C4C22⋊D1711yescharacteristic
203C2×C4⋊C4C22×D1711yescharacteristic
204C2×C22⋊C4—341no—
205C2×C22⋊C4—341no—
206C2×C42—341no—
207C22⋊C8—41no—
208C22⋊C8—41no—
209C22⋊C8—342no—
210C22⋊C8—681no—
211#95—171no—
212#95—171no—
213#95—341no—
214#95—341no—
215#95—341no—
216#95—341no—
217#95—341no—
218#95—341no—
219#95—341no—
220C17#1185011yescharacteristic, complemented, Sylow
221C24#6322711yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—341no—
233C22×C4—341no—
234C22×C4—341no—
235C22×C4—341no—
236C22×C4#6322711yescharacteristic
237C22×C4#6322711yescharacteristic
238C22×C4#6322711yescharacteristic
239C22×C4C2×C4×D1711yescharacteristic
240C22×C4C2×D6811yescharacteristic
241C22×C4C2×C22⋊D1711yescharacteristic
242M16—42no—
243M16—42no—
244C2×C8—42no—
245C2×C8—682no—
246C2×C8—1362no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—681no—
252C4⋊C4—681no—
253C4⋊C4—682no—
254C4⋊C4—682no—
255C22⋊C4—682no—
256C22⋊C4—681no—
257C22⋊C4—681no—
258C42—682no—
259C23—21no—
260C23—21no—
261C23—22no—
262C23—41no—
263C23#1058747711yescharacteristic
264C23#1058747711yescharacteristic
265C23C2×#6322711yescharacteristic
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—81no—
285C2×C4—681no—
286C2×C4—681no—
287C2×C4—681no—
288C2×C4—681no—
289C2×C4—682no—
290C2×C4—681no—
291C2×C4—681no—
292C2×C4—681no—
293C2×C4—681no—
294C2×C4—681no—
295C2×C4—681no—
296C2×C4#1058763612yes—
297C8—82no—
298C8—1362no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—42no—
307C22—42no—
308C22#1176260012yes—
309C22C2×#1058747711yescharacteristic
310C22#1176280311yescharacteristic
311C22#1176282711yescharacteristic
312C4—22no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—1361no—
319C4—1361no—
320C4—1361no—
321C4—1361no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—41no—
326C2#1452733811yescharacteristic
327C2C2×#1176260011yescharacteristic
328C2#1453047111yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1821C2
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1821D17
hypocenterC17#11850
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
3. #14526810C2semidirect
4. #14526789C2semidirect
220. C17#11850semidirect

Maximal subgroups

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

subgroupindex
#145330712
C2×#117625352
#145268102
#145267892
C17×#18212
#1185017