Finite Group Databasebeta

← #16945148

Subgroups — #16945148

Order 4352.

382 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 79 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#14534644C211yesmaximal, characteristic
2#14527128C211yesmaximal, characteristic, complemented
3#14526813C212yesmaximal, complemented
4#14526786C212yesmaximal
5C17×#1801C211yesmaximal, characteristic, complemented
6C2×#10587561C411yescharacteristic
7#11762866—21no—
8#11762744—21no—
9#11762729C411yescharacteristic
10#11762729C2211yescharacteristic
11#11762726C411yescharacteristic
12#11762726C2211yescharacteristic
13#11762720—21no—
14#11762700—21no—
15#11762700—21no—
16#11762677C411yescharacteristic
17#11762535—21no—
18#11762535C2212yes—
19C2×C4×Q8×C17C2211yescharacteristic
20C17×#423—21no—
21C2×C17×#101—21no—
22C17×#343C2212yes—
23C22×Q8×C17D411yescharacteristic
24C22×D68C2×C411yescharacteristic
25C22×Dic34C2×C411yescharacteristic
26C2×M16×C17—22no—
27C4×Q8×C17—41no—
28C4×Q8×C17—41no—
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C4⋊C4×C17D411yescharacteristic
31C2×C4⋊C4×C17C2311yescharacteristic
32C2×C42×C17C2×C411yescharacteristic
33C2×C42×C17D411yescharacteristic
34C2×C42×C17D411yescharacteristic
35C2×#63227—21no—
36C2×#63227—21no—
37C2×#63227—21no—
38C2×#63227—21no—
39C2×#63227—21no—
40C2×#63227—21no—
41C2×#63225—21no—
42C2×#63225—21no—
43#10587561—41no—
44#10587561—41no—
45C17×C4⋊C8—41no—
46C17×C4⋊C8—41no—
47C17×#101—42no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502C2×C411yescharacteristic
53#10587502C2×C411yescharacteristic
54C22⋊#3622—22no—
55C4⋊#3622—41no—
56#10587474—41no—
57C23×D17—21no—
58C2×Q8×C17—21no—
59C2×Q8×C17—21no—
60C2×Q8×C17—41no—
61C2×Q8×C17—41no—
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22×C411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×C4×C17C2×D411yescharacteristic
69C22×Dic17—21no—
70C22×Dic17—21no—
71C22×Dic17—21no—
72C2×D68—41no—
73C2×D68—81no—
74C2×Dic34—41no—
75C2×Dic34—81no—
76M16×C17—42no—
77M16×C17—42no—
78C2×C8×C17—42no—
79C4⋊C4×C17—41no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—41no—
82C42×C17—21no—
83C42×C17—21no—
84C42×C17—21no—
85C42×C17—21no—
86C42×C17—21no—
87C42×C17—21no—
88#63227—42no—
89#63227—42no—
90#63227—41no—
91#63227—41no—
92#63227—41no—
93#63227—41no—
94#63227—81no—
95#63227—81no—
96#63225—81no—
97#63225—81no—
98C2×#3622—42no—
99#11833—171nomaximal, Sylow
100C23×C17C2×C22⋊C411yescharacteristic
101C22×D17—41no—
102C22×D17—41no—
103C22×D17—41no—
104C22×D17—41no—
105C22×D17—41no—
106C22×D17—41no—
107C22×D17—41no—
108Q8×C17—42no—
109Q8×C17—42no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17#9911yescharacteristic
126C2×C4×C17#9911yescharacteristic
127C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
128C2×C4×C17#13711yescharacteristic
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—
138D68—82no—
139Dic34—82no—
140C8×C17—82no—
141#3622—82no—
142#2632—171no—
143#1801D1711yescharacteristic, complemented
144#1467—171no—
145#1354—172no—
146#1345—172no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17C2×#9911yescharacteristic
150C22×C17#41711yescharacteristic
151C22×C17#42111yescharacteristic
152C2×D17—81no—
153C2×D17—81no—
154C2×D17—81no—
155C2×D17—81no—
156C2×D17—81no—
157C2×D17—81no—
158C2×D17—81no—
159C4×C17—21no—
160C4×C17—21no—
161C4×C17—21no—
162C4×C17—21no—
163C4×C17—41no—
164C4×C17—41no—
165C4×C17—41no—
166C4×C17—41no—
167C4×C17—41no—
168Dic17—81no—
169Dic17—81no—
170Dic17—81no—
171C2×#124—171no—
172C2×C4×Q8C2×D1711yescharacteristic
173#432—341no—
174#423—21no—
175C2×#101—21no—
176#399—341no—
177#395—171no—
178#395—171no—
179#394—171no—
180#394—171no—
181#393—341no—
182#388—341no—
183#388—341no—
184#383—171no—
185#343C2×D1712yes—
186#324—172no—
187#324—341no—
188C2×C17—21no—
189C2×C17—21no—
190C2×C17#151311yescharacteristic
191C2×C17#212211yescharacteristic
192C2×C17#213311yescharacteristic
193D17—161no—
194C22×Q8—171no—
195C22×Q8C22⋊D1711yescharacteristic
196C22×D4—171no—
197C2×M16—22no—
198#124—681no—
199#124—681no—
200C4×Q8—41no—
201C4×Q8—41no—
202C2×C4⋊C4—341no—
203C2×C4⋊C4—341no—
204C2×C4⋊C4C4×D1711yescharacteristic
205C2×C4⋊C4D6811yescharacteristic
206C2×C4⋊C4C22×D1711yescharacteristic
207C2×C22⋊C4—341no—
208C2×C22⋊C4—341no—
209C2×C22⋊C4—341no—
210C2×C22⋊C4—341no—
211C2×C22⋊C4—341no—
212C2×C22⋊C4—341no—
213C2×C42C4×D1711yescharacteristic
214C2×C42D6811yescharacteristic
215C2×C42C22⋊D1711yescharacteristic
216C4⋊C8—41no—
217C4⋊C8—41no—
218C4⋊C8—681no—
219#101—42no—
220C22⋊C8—342no—
221#97—681no—
222#95—171no—
223#95—171no—
224#95—341no—
225#95—341no—
226#95—341no—
227#95—341no—
228C17#1183311yescharacteristic, complemented, Sylow
229C24—341no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—681no—
235C2×Q8—1361no—
236C2×D4—681no—
237C2×D4—1361no—
238C22×C4—341no—
239C22×C4—341no—
240C22×C4—341no—
241C22×C4#6322711yescharacteristic
242C22×C4#6322711yescharacteristic
243C22×C4#6322711yescharacteristic
244C22×C4#6322711yescharacteristic
245C22×C4C2×C4×D1711yescharacteristic
246C22×C4C2×D6811yescharacteristic
247C22×C4C2×C22⋊D1711yescharacteristic
248M16—42no—
249M16—42no—
250C2×C8—42no—
251C2×C8—682no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—1361no—
256C4⋊C4—1361no—
257C22⋊C4—682no—
258C22⋊C4—682no—
259C22⋊C4—681no—
260C22⋊C4—681no—
261C22⋊C4—681no—
262C22⋊C4—681no—
263C22⋊C4—1361no—
264C22⋊C4—1361no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C23—681no—
272C23—681no—
273C23—681no—
274C23—681no—
275C23—681no—
276C23—681no—
277C23—681no—
278C23C2×#6322711yescharacteristic
279Q8—42no—
280Q8—42no—
281Q8—1362no—
282D4—1362no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
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#1058747711yescharacteristic
308C2×C4#1058747711yescharacteristic
309C2×C4#1058761811yescharacteristic
310C2×C4#1058762811yescharacteristic
311C8—82no—
312C8—1362no—
313C22—21no—
314C22—21no—
315C22—1361no—
316C22—1361no—
317C22—1361no—
318C22—1361no—
319C22—1361no—
320C22—1361no—
321C22—1361no—
322C22C2×#1058747711yescharacteristic
323C22#1176280811yescharacteristic
324C22#1176282711yescharacteristic
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—1361no—
335C4—1361no—
336C4—1361no—
337C2—21no—
338C2—21no—
339C2—2721no—
340C2#1452732511yescharacteristic
341C2#1453043511yescharacteristic
342C2#1453047111yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1801C2
socleC22×C17C2×#99
prime radicalC22×D68C2×C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1801D17
hypocenterC17#11833
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. #14527128C2semidirect
3. #14526813C2semidirect
5. C17×#1801C2semidirect
143. #1801D17semidirect
228. C17#11833semidirect

Maximal subgroups

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

subgroupindex
#145346442
#145271282
#145268132
#145267862
C17×#18012
#1183317