Finite Group Databasebeta

← #16944719

Subgroups — #16944719

Order 4352.

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

normal subgroups only

This group has 420 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#14534701C211yesmaximal, characteristic
2C4×#10587504C211yesmaximal, characteristic
3#14526815C212yesmaximal, complemented
4#14526791C212yesmaximal, complemented
5C17×#1459C211yesmaximal, characteristic
6C2×#10587610C411yescharacteristic
7C2×#10587504—21no—
8C4×#63223—21no—
9#11762739C2211yescharacteristic
10#11762737—21no—
11#11762734C411yescharacteristic
12#11762729C2211yescharacteristic
13#11762723—21no—
14#11762721C411yescharacteristic
15#11762693—21no—
16#11762682—21no—
17#11762672C411yescharacteristic
18#11762578C2212yes—
19C2×C4×D4×C17C2211yescharacteristic
20C2×C17×C22⋊C8—21no—
21C17×#324C2212yes—
22C17×#323—21no—
23C22×D4×C17D411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C22×Dic34C2×C411yescharacteristic
27C22×C8×C17—41no—
28C4×D4×C17—41no—
29C4×D4×C17—41no—
30C2×C4⋊C4×C17C2311yescharacteristic
31C2×C22⋊C4×C17C2×C411yescharacteristic
32C2×C22⋊C4×C17D411yescharacteristic
33C2×C42×C17D411yescharacteristic
34C2×C22⋊Dic17—21no—
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17—21no—
38C2×C4⋊Dic17C2×C411yescharacteristic
39#10587610—41no—
40#10587610—41no—
41C2×#63225—21no—
42C2×#63225—21no—
43C2×#63225—21no—
44C2×#63223—22no—
45C17×C22⋊C8—22no—
46C17×C22⋊C8—41no—
47#10587504—24no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502C2×C411yescharacteristic
54#10587502C2×C411yescharacteristic
55#10587474—41no—
56C4×#3622—41no—
57C24×C17C22⋊C411yescharacteristic
58C24×C17C22⋊C411yescharacteristic
59C2×D4×C17—22no—
60C2×D4×C17—22no—
61C2×D4×C17—41no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—81no—
67C22×C4×C17—81no—
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—
77C2×Dic34—41no—
78C2×Dic34—81no—
79C2×C8×C17—42no—
80C2×C8×C17—82no—
81C4⋊C4×C17—41no—
82C22⋊C4×C17—41no—
83C22⋊C4×C17—41no—
84C42×C17—21no—
85C42×C17—21no—
86C22⋊Dic17—42no—
87C22⋊Dic17—42no—
88C22⋊Dic17—81no—
89C22⋊Dic17—81no—
90C4⋊Dic17—41no—
91#63225—41no—
92#63225—41no—
93#63225—42no—
94#63225—81no—
95#63223—24no—
96#63223—42no—
97C2×#3622—42no—
98#11739—171nomaximal, Sylow
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—41no—
105C23×C17—41no—
106C23×C17—41no—
107C23×C17—41no—
108C23×C17#10412yes—
109C23×C17C2×C22⋊C411yescharacteristic
110D4×C17—44no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—41no—
117C2×C4×C17—42no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—81no—
122C2×C4×C17—81no—
123C2×C4×C17—81no—
124C2×C4×C17#9911yescharacteristic
125C2×C4×C17#9911yescharacteristic
126C2×C4×C17#13512yes—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139Dic34—82no—
140C8×C17—82no—
141#3622—44no—
142#2638—171no—
143C4×#100—171no—
144#1459D1711yescharacteristic
145#1354—172no—
146#1346—172no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17—22no—
150C22×C17—42no—
151C22×C17—41no—
152C22×C17—41no—
153C22×C17—41no—
154C22×C17—41no—
155C22×C17—41no—
156C22×C17—81no—
157C22×C17—81no—
158C22×C17—81no—
159C22×C17C2×#9911yescharacteristic
160C22×C17#41611yescharacteristic
161C22×C17C2×#10411yescharacteristic
162C4×C17—21no—
163C4×C17—21no—
164C4×C17—22no—
165C4×C17—41no—
166C4×C17—41no—
167C4×C17—41no—
168Dic17—81no—
169Dic17—81no—
170Dic17—81no—
171Dic17—81no—
172C2×C22⋊Q8—171no—
173C2×C4×D4C2×D1711yescharacteristic
174C2×#100—341no—
175C2×C22⋊C8—21no—
176C4×M16—341no—
177#398—171no—
178#397—171no—
179#397—341no—
180#395—171no—
181#394—341no—
182#393—171no—
183#387—341no—
184#385—341no—
185#381—171no—
186#343—172no—
187#324C2×D1712yes—
188#323—21no—
189C2×C17—21no—
190C2×C17—21no—
191C2×C17—42no—
192C2×C17—81no—
193C2×C17#150111yescharacteristic
194C2×C17#212111yescharacteristic
195C2×C17#213511yescharacteristic
196C22×Q8—171no—
197C22×D4D6811yescharacteristic
198C23×C4C4×D1711yescharacteristic
199C23×C4C22⋊D1711yescharacteristic
200C2×M16—342no—
201C22×C8—41no—
202C22⋊Q8—681no—
203C22⋊Q8—681no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—171no—
207C2×C4⋊C4—341no—
208C2×C4⋊C4—341no—
209C2×C4⋊C4—341no—
210C2×C4⋊C4C22×D1711yescharacteristic
211C2×C22⋊C4—341no—
212C2×C22⋊C4—341no—
213C2×C22⋊C4—341no—
214C2×C22⋊C4—341no—
215C2×C22⋊C4C4×D1711yescharacteristic
216C2×C22⋊C4C22⋊D1711yescharacteristic
217C2×C42D6811yescharacteristic
218#100—344no—
219C22⋊C8—22no—
220C22⋊C8—41no—
221#97—681no—
222C4×C8—681no—
223#95—171no—
224#95—171no—
225#95—341no—
226#95—341no—
227#95—341no—
228#95—341no—
229#95—341no—
230C17#1173911yescharacteristic, complemented, Sylow
231C24#6322711yescharacteristic
232C24#6322711yescharacteristic
233C2×Q8—681no—
234C2×Q8—1361no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—41no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—341no—
245C22×C4—341no—
246C22×C4—341no—
247C22×C4—341no—
248C22×C4#6322711yescharacteristic
249C22×C4#6322711yescharacteristic
250C22×C4C2×C4×D1711yescharacteristic
251C22×C4C2×D6811yescharacteristic
252C22×C4C2×C22⋊D1711yescharacteristic
253M16—344no—
254M16—682no—
255C2×C8—42no—
256C2×C8—82no—
257C2×C8—682no—
258C4⋊C4—41no—
259C4⋊C4—681no—
260C4⋊C4—681no—
261C4⋊C4—681no—
262C4⋊C4—682no—
263C4⋊C4—1361no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—682no—
267C22⋊C4—682no—
268C22⋊C4—1361no—
269C22⋊C4—1361no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23#1058749612yes—
282C23C2×#6322711yescharacteristic
283Q8—1362no—
284D4—44no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—681no—
299C2×C4—681no—
300C2×C4—681no—
301C2×C4—681no—
302C2×C4—681no—
303C2×C4—681no—
304C2×C4—681no—
305C2×C4—681no—
306C2×C4—681no—
307C2×C4—681no—
308C2×C4—681no—
309C2×C4—681no—
310C2×C4#1058747711yescharacteristic
311C2×C4#1058747711yescharacteristic
312C2×C4#1058759012yes—
313C8—82no—
314C8—684no—
315C22—21no—
316C22—21no—
317C22—22no—
318C22—42no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22C2×#1058747711yescharacteristic
328C22#1176280611yescharacteristic
329C22C2×#1058749611yescharacteristic
330C4—22no—
331C4—21no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—1361no—
337C4—1361no—
338C4—1361no—
339C4—1361no—
340C2—21no—
341C2—21no—
342C2—42no—
343C2—81no—
344C2#1452727611yescharacteristic
345C2#1453042811yescharacteristic
346C2#1453047811yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1459C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C4⋊C4×C17C23
hypercenter#1459D17
hypocenterC17#11739
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #14526815C2semidirect
4. #14526791C2semidirect
230. C17#11739semidirect

Maximal subgroups

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

subgroupindex
#145347012
C4×#105875042
#145268152
#145267912
C17×#14592
#1173917