Finite Group Databasebeta

← #16945379

Subgroups — #16945379

Order 4352.

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

normal subgroups only

This group has 476 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.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533783C211yesmaximal, characteristic
2#14528780C211yesmaximal, characteristic, complemented
3#14526788C214yesmaximal, complemented
4C17×#1461C211yesmaximal, characteristic, complemented
5C2×#10587560C411yescharacteristic
6C2×#10587633—21no—
7#11762833—21no—
8C2×#10587507—21no—
9#11762726C412yes—
10#11762726C2212yes—
11#11762711—21no—
12#11762711—21no—
13#11762710—21no—
14C4×C4⋊Dic17C411yescharacteristic
15#11762578C2212yes—
16C2×C4×Q8×C17C2211yescharacteristic
17C17×#431—21no—
18C17×#324—21no—
19C17×#324C2212yes—
20C22×Q8×C17D411yescharacteristic
21C22×D68—21no—
22C22×D68—21no—
23C22×D68C2×C412yes—
24C4×Q8×C17—41no—
25C4×Q8×C17—41no—
26C2×C4⋊C4×C17C2×C411yescharacteristic
27C2×C4⋊C4×C17D411yescharacteristic
28C2×C4⋊C4×C17C2311yescharacteristic
29C2×C42×C17C2×C411yescharacteristic
30C2×C42×C17D411yescharacteristic
31C2×C42×C17D411yescharacteristic
32#10587633—42no—
33#10587633—42no—
34C2×#63227—22no—
35C2×#63227—22no—
36C2×#63227—22no—
37C2×#63227—22no—
38C2×C4⋊Dic17—21no—
39C2×C4×Dic17—21no—
40C2×#63223—22no—
41#10587560—41no—
42#10587560—41no—
43C17×C4⋊C8—41no—
44C17×C22⋊C8—22no—
45C4×C8×C17—41no—
46#10587507—42no—
47#10587502C2×C412yes—
48C4⋊#3622—41no—
49C4⋊#3622—41no—
50C23×D17—22no—
51C2×Q8×C17—41no—
52C2×Q8×C17—41no—
53C2×Q8×C17D812yes—
54C2×Q8×C17SD1612yes—
55C22×C4×C17C22⋊C411yescharacteristic
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C22×C411yescharacteristic
60C22×C4×C17C2×D411yescharacteristic
61C22×C4×C17C2×D411yescharacteristic
62C22×Dic17—22no—
63C2×D68—42no—
64C2×D68—42no—
65C2×D68—42no—
66C2×D68—42no—
67C2×D68—42no—
68C2×D68—41no—
69C2×D68—41no—
70C2×D68—42no—
71C2×D68—82no—
72C2×C8×C17—42no—
73C4⋊C4×C17—41no—
74C4⋊C4×C17—41no—
75C4⋊C4×C17—41no—
76C42×C17—21no—
77C42×C17—21no—
78C42×C17—21no—
79C42×C17—21no—
80C42×C17—21no—
81C42×C17—21no—
82#63227—44no—
83#63227—42no—
84#63227—42no—
85#63227—82no—
86#63227—82no—
87C4⋊Dic17—42no—
88C4×Dic17—42no—
89#63223—42no—
90#63223—42no—
91C2×#3622—42no—
92#11882—171nomaximal, Sylow
93C23×C17C2×C22⋊C411yescharacteristic
94C22×D17—42no—
95C22×D17—42no—
96C22×D17—42no—
97C22×D17—42no—
98C22×D17—42no—
99C22×D17—42no—
100C22×D17—42no—
101Q8×C17—42no—
102Q8×C17—42no—
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—22no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17#9911yescharacteristic
117C2×C4×C17#9911yescharacteristic
118C2×C4×C17#10212yes—
119C2×C4×C17#10212yes—
120C2×C4×C17C2×D811yescharacteristic
121C2×C4×C17C2×SD1611yescharacteristic
122C2×Dic17—42no—
123C2×Dic17—42no—
124C2×Dic17—42no—
125D68—82no—
126D68—84no—
127D68—84no—
128D68—82no—
129C8×C17—82no—
130#3622—82no—
131#2548—171no—
132#1801—171no—
133#1461D1711yescharacteristic, complemented
134#1345—174no—
135C22×C17—21no—
136C22×C17—21no—
137C22×C17C2×#9911yescharacteristic
138C22×C17C2×#10211yescharacteristic
139C22×C17#42111yescharacteristic
140C2×D17—82no—
141C2×D17—82no—
142C2×D17—82no—
143C2×D17—82no—
144C2×D17—82no—
145C2×D17—82no—
146C2×D17—82no—
147C4×C17—21no—
148C4×C17—21no—
149C4×C17—22no—
150C4×C17—22no—
151C4×C17—21no—
152C4×C17—41no—
153C4×C17—41no—
154C4×C17—41no—
155C4×C17#35512yes—
156Dic17—82no—
157C2×#127—171no—
158C2×#124—341no—
159C2×C4×Q8C2×D1711yescharacteristic
160#431—21no—
161#423—341no—
162C2×#101—341no—
163#394—172no—
164#394—172no—
165#390—341no—
166#390—341no—
167#390—341no—
168C4×C4⋊C4—171no—
169#343—172no—
170#324—21no—
171#324C2×D1712yes—
172C2×C17—21no—
173C2×C17—21no—
174C2×C17#151911yescharacteristic
175C2×C17#212211yescharacteristic
176C2×C17C2×#35511yescharacteristic
177D17—162no—
178C22×Q8D6811yescharacteristic
179C22×D4—172no—
180C22×D4—341no—
181C22×D4—341no—
182C2×M16—342no—
183#127—681no—
184#127—681no—
185#124—682no—
186#124—682no—
187C4×Q8—41no—
188C4×Q8—41no—
189C2×C4⋊C4—341no—
190C2×C4⋊C4C4×D1711yescharacteristic
191C2×C4⋊C4C22⋊D1711yescharacteristic
192C2×C4⋊C4C22×D1711yescharacteristic
193C2×C22⋊C4—342no—
194C2×C22⋊C4—342no—
195C2×C22⋊C4—342no—
196C2×C22⋊C4—342no—
197C2×C42—341no—
198C2×C42C4×D1711yescharacteristic
199C2×C42D6811yescharacteristic
200C2×C42C22⋊D1711yescharacteristic
201C4⋊C8—41no—
202C4⋊C8—681no—
203C4⋊C8—681no—
204#101—682no—
205C22⋊C8—22no—
206C4×C8—41no—
207#95—172no—
208C17#1188211yescharacteristic, complemented, Sylow
209C24—342no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8#6321912yes—
213C2×Q8D13612yes—
214C2×D4—682no—
215C2×D4—682no—
216C2×D4—682no—
217C2×D4—682no—
218C2×D4—682no—
219C2×D4—681no—
220C2×D4—681no—
221C2×D4—682no—
222C2×D4—1362no—
223C22×C4—342no—
224C22×C4#6322711yescharacteristic
225C22×C4#6322711yescharacteristic
226C22×C4#6322711yescharacteristic
227C22×C4#6322711yescharacteristic
228C22×C4C2×C4×D1711yescharacteristic
229C22×C4C2×D6811yescharacteristic
230C22×C4C2×C22⋊D1711yescharacteristic
231M16—682no—
232M16—682no—
233C2×C8—42no—
234C2×C8—682no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—682no—
239C22⋊C4—684no—
240C22⋊C4—682no—
241C22⋊C4—682no—
242C22⋊C4—1362no—
243C22⋊C4—1362no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—682no—
251C23—682no—
252C23—682no—
253C23—682no—
254C23—682no—
255C23—682no—
256C23—682no—
257C23—682no—
258C23C2×#6322711yescharacteristic
259Q8—42no—
260Q8—42no—
261D4—1364no—
262D4—1362no—
263D4—1364no—
264D4—1362no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—682no—
279C2×C4—682no—
280C2×C4—682no—
281C2×C4#1058747711yescharacteristic
282C2×C4#1058747711yescharacteristic
283C2×C4#1058749212yes—
284C2×C4#1058749212yes—
285C2×C4C2×#6321911yescharacteristic
286C2×C4C2×D13611yescharacteristic
287C8—82no—
288C8—1362no—
289C22—21no—
290C22—21no—
291C22—1362no—
292C22—1362no—
293C22—1362no—
294C22—1362no—
295C22—1362no—
296C22—1362no—
297C22—1362no—
298C22C2×#1058747711yescharacteristic
299C22C2×#1058749211yescharacteristic
300C22#1176282511yescharacteristic
301C4—21no—
302C4—22no—
303C4—22no—
304C4—21no—
305C4—21no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—1362no—
310C4#1176260512yes—
311C2—21no—
312C2—21no—
313C2—2722no—
314C2#1452735311yescharacteristic
315C2#1453043311yescharacteristic
316C2C2×#1176260511yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1461C2
socleC22×C17C2×#99
prime radicalC2×#10587560C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1461D17
hypocenterC17#11882
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. #14528780C2semidirect
3. #14526788C2semidirect
4. C17×#1461C2semidirect
133. #1461D17semidirect
208. C17#11882semidirect

Maximal subgroups

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

subgroupindex
#145337832
#145287802
#145267882
C17×#14612
#1188217