Finite Group Databasebeta

← #17959125

Subgroups — #17959125

Order 4352.

468 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 88 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#14544483C211yesmaximal, characteristic
2#14544017C211yesmaximal, characteristic
3#14543781C212yesmaximal
4#14543779C212yesmaximal
5#14532054C211yesmaximal, characteristic
6C2×#10587677—21no—
7#11763825—21no—
8#11763687—21no—
9#11763687C2212yes—
10#11763655—21no—
11#11763653C2212yes—
12C2×#10587558C2211yescharacteristic
13#11762747C411yescharacteristic, complemented
14#11762744C2211yescharacteristic
15#11762742—21no—
16#11762731—21no—
17#11762728C411yescharacteristic, complemented
18#11762720C2211yescharacteristic
19C4×#63225—21no—
20C4×C4⋊Dic17C411yescharacteristic, complemented
21C4×#63227—21no—
22C2×C4×Q8×C17C411yescharacteristic, complemented
23C22×Q8×C17C2×C411yescharacteristic
24C22×C4×D17D411yescharacteristic
25C2×C22⋊#595—22no—
26#10587677—24no—
27#10587677—42no—
28C2×C4⋊#595—41no—
29C2×C4×#595—41no—
30C4×Q8×C17—41no—
31C4×Q8×C17—41no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C42×C17—21no—
35C2×C42×C17C2×C411yescharacteristic
36C2×#63227—21no—
37C2×#63227—21no—
38C2×#63227D411yescharacteristic
39C2×#63227C2311yescharacteristic
40C2×C4⋊Dic17—21no—
41C2×#63225—21no—
42C2×#63225—21no—
43C2×#63225C2×C411yescharacteristic
44C2×#63225D411yescharacteristic
45C2×C4×Dic17—21no—
46C2×C4×Dic17D411yescharacteristic
47#10587558—41no—
48#10587558—41no—
49C22⋊#3624—22no—
50C4⋊#3624—41no—
51C4×#3624—41no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502C2×C411yescharacteristic
57#10587502C2×C411yescharacteristic
58C23×D17C2×D411yescharacteristic
59C22×#595—42no—
60C2×Q8×C17—42no—
61C2×Q8×C17—41no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22×C411yescharacteristic
67C22×Dic17—21no—
68C22×Dic17—21no—
69C22×Dic17C22⋊C411yescharacteristic
70C22×Dic17C22⋊C411yescharacteristic
71C22×Dic17C2×D411yescharacteristic
72C2×C4×D17—21no—
73C2×C4×D17—21no—
74C2×C4×D17—22no—
75C2×C4×D17SD1612yes—
76C2×C4×D17Q1612yes—
77C22⋊#595—24no—
78C22⋊#595—42no—
79C2×#3624—42no—
80C4⋊#595—44no—
81C4×#595—44no—
82C4⋊C4×C17—41no—
83C4⋊C4×C17—42no—
84C42×C17—21no—
85C42×C17—21no—
86C42×C17—41no—
87C42×C17—41no—
88#63227—41no—
89#63227—41no—
90#63227—42no—
91#63227—42no—
92C4⋊Dic17—42no—
93#63225—41no—
94#63225—41no—
95#63225—42no—
96#63225—42no—
97C4×Dic17—21no—
98C4×Dic17—21no—
99C4×Dic17—42no—
100#11715—171nomaximal, Sylow
101C23×C17C2×C22⋊C411yescharacteristic
102C22×D17—21no—
103C22×D17—21no—
104C22×D17—41no—
105C22×D17—41no—
106C22×D17C2×SD1611yescharacteristic
107C22×D17C2×Q1611yescharacteristic
108C2×#595—44no—
109C2×#595—82no—
110C2×#595—82no—
111Q8×C17—42no—
112Q8×C17—42no—
113C2×C4×C17—22no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17#9911yescharacteristic
124C2×C4×C17#9911yescharacteristic
125C2×C4×C17#10312yes—
126C2×C4×C17#10412yes—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—22no—
130C2×Dic17—21no—
131C2×Dic17—21no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17#10312yes—
140C4×D17—22no—
141C4×D17—22no—
142C4×D17—22no—
143C4×D17—22no—
144#3624—82no—
145#2396—171no—
146#1740—171no—
147#1458—171no—
148#1352—172no—
149#1351—172no—
150C22×C17—21no—
151C22×C17—21no—
152C22×C17C2×#9911yescharacteristic
153C22×C17C2×#10311yescharacteristic
154C22×C17C2×#10411yescharacteristic
155C2×D17—21no—
156C2×D17—21no—
157C2×D17—41no—
158C2×D17—41no—
159C2×D17—41no—
160C2×D17—41no—
161C2×D17—81no—
162#595—84no—
163C4×C17—42no—
164C4×C17—41no—
165C4×C17—41no—
166C4×C17—81no—
167C4×C17#32812yes—
168C4×C17#32812yes—
169Dic17—22no—
170Dic17—42no—
171Dic17—41no—
172Dic17—41no—
173Dic17—81no—
174C2×C4×Q8#59511yescharacteristic, complemented
175C2×#117—171no—
176C2×#117—341no—
177#431—341no—
178#400—171no—
179#399—171no—
180#398—341no—
181#396—341no—
182#395—171no—
183#393—171no—
184C4×C4⋊C4—171no—
185C4×C4⋊C4—341no—
186C4×C22⋊C4—341no—
187#342—172no—
188#342—341no—
189#324—341no—
190#323—172no—
191C2×C17—21no—
192C2×C17—21no—
193C2×C17C2×#32811yescharacteristic
194C2×C17#212011yescharacteristic
195C2×C17#212611yescharacteristic
196D17—41no—
197D17—41no—
198C22×Q8C2×#59511yescharacteristic
199C23×C4—171no—
200C4×Q8—41no—
201C4×Q8—41no—
202#117—344no—
203#117—681no—
204#117—681no—
205#117—682no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—171no—
208C2×C4⋊C4—171no—
209C2×C4⋊C4—341no—
210C2×C4⋊C4—341no—
211C2×C4⋊C4—341no—
212C2×C4⋊C4—681no—
213C2×C4⋊C4C2×#59511yescharacteristic
214C2×C22⋊C4—171no—
215C2×C22⋊C4—171no—
216C2×C22⋊C4—342no—
217C2×C22⋊C4—341no—
218C2×C22⋊C4—341no—
219C2×C42—21no—
220C2×C42—171no—
221C2×C42—341no—
222C2×C42—681no—
223C2×C42C2×#59511yescharacteristic
224C4⋊C8—681no—
225C22⋊C8—342no—
226C4×C8—681no—
227#95—171no—
228#95—171no—
229#95—341no—
230#95—341no—
231#95—341no—
232#95—341no—
233C17#1171511yescharacteristic, complemented, Sylow
234C24—171no—
235C2×Q8—42no—
236C2×Q8—41no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—171no—
240C22×C4—172no—
241C22×C4—172no—
242C22×C4—171no—
243C22×C4—171no—
244C22×C4—342no—
245C22×C4—341no—
246C22×C4—341no—
247C22×C4—341no—
248C22×C4—341no—
249C22×C4—682no—
250C22×C4C22⋊#59511yescharacteristic
251C22×C4C22⋊#59511yescharacteristic
252C22×C4C22×#59511yescharacteristic
253C2×C8—682no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—681no—
257C4⋊C4—681no—
258C4⋊C4—682no—
259C4⋊C4—682no—
260C4⋊C4—682no—
261C4⋊C4—684no—
262C22⋊C4—344no—
263C22⋊C4—681no—
264C22⋊C4—681no—
265C22⋊C4—682no—
266C22⋊C4—682no—
267C22⋊C4—682no—
268C42—21no—
269C42—21no—
270C42—41no—
271C42—41no—
272C42—341no—
273C42—341no—
274C42—684no—
275C42—682no—
276C23—171no—
277C23—171no—
278C23—341no—
279C23—341no—
280C23—681no—
281C23—681no—
282C23C2×C22⋊#59511yescharacteristic
283Q8—42no—
284Q8—42no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—172no—
296C2×C4—341no—
297C2×C4—341no—
298C2×C4—342no—
299C2×C4—342no—
300C2×C4—342no—
301C2×C4—342no—
302C2×C4—342no—
303C2×C4—341no—
304C2×C4—341no—
305C2×C4—681no—
306C2×C4—681no—
307C2×C4—681no—
308C2×C4—681no—
309C2×C4—681no—
310C2×C4—681no—
311C2×C4—684no—
312C2×C4—681no—
313C2×C4—1362no—
314C2×C4—1362no—
315C2×C4#1058754912yes—
316C2×C4#1058755012yes—
317C2×C4#1058755111yescharacteristic
318C2×C4#1058755111yescharacteristic
319C8—1362no—
320C22—21no—
321C22—21no—
322C22—341no—
323C22—341no—
324C22—681no—
325C22—681no—
326C22—681no—
327C22—681no—
328C22—1361no—
329C22C2×#1058755111yescharacteristic
330C22C2×#1058754911yescharacteristic
331C22C2×#1058755011yescharacteristic
332C4—42no—
333C4—41no—
334C4—41no—
335C4—81no—
336C4—342no—
337C4—682no—
338C4—681no—
339C4—681no—
340C4—1361no—
341C4—1364no—
342C4#1176366312yes—
343C4#1176366312yes—
344C2—21no—
345C2—21no—
346C2—681no—
347C2—681no—
348C2C2×#1176366311yescharacteristic
349C2#1454503511yescharacteristic
350C2#1454505511yescharacteristic
subgroupG/H
centerC2×C4#10587551
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C4×Q8×C17C4
socleC22×C17C2×#99
prime radicalC23×D17C2×D4
cosocleC2×#63227C23
hypercenterC2×C4×Q8#595
hypocenterC17#11715
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
13. #11762747C4semidirect
17. #11762728C4semidirect
20. C4×C4⋊Dic17C4semidirect
22. C2×C4×Q8×C17C4semidirect
174. C2×C4×Q8#595semidirect
233. C17#11715semidirect

Maximal subgroups

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

subgroupindex
#145444832
#145440172
#145437812
#145437792
#145320542
#1171517