Finite Group Databasebeta

← #16944831

Subgroups — #16944831

Order 4352.

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

normal subgroups only

This group has 428 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#14534361C211yesmaximal, characteristic
2#14528523C211yesmaximal, characteristic
3#14526815C212yesmaximal, complemented
4#14526791C212yesmaximal, complemented
5C17×#1464C211yesmaximal, characteristic
6C22×#63223—21no—
7C2×C4×Dic34C411yescharacteristic
8#11762739C2211yescharacteristic
9#11762734C411yescharacteristic
10#11762729C2211yescharacteristic
11#11762721C411yescharacteristic
12#11762697—21no—
13#11762693—21no—
14#11762690—21no—
15#11762682—21no—
16#11762677C411yescharacteristic
17#11762578—21no—
18#11762578C2212yes—
19C2×C17×#117C2211yescharacteristic
20C17×#432—21no—
21C17×#324C2212yes—
22C17×#323—21no—
23C23×C4×C17D411yescharacteristic
24C22×Dic34C2×C411yescharacteristic
25C17×#117—41no—
26C17×#117—41no—
27C2×C4⋊C4×C17D411yescharacteristic
28C2×C4⋊C4×C17C2311yescharacteristic
29C2×C22⋊C4×C17C2×C411yescharacteristic
30C2×C22⋊C4×C17D411yescharacteristic
31C2×C42×C17C2×C411yescharacteristic
32C2×C42×C17D411yescharacteristic
33C2×C22⋊Dic17—21no—
34C2×C22⋊Dic17—21no—
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×#63225—21no—
37C2×#63225—21no—
38C2×#63225—21no—
39C2×C4×Dic17—21no—
40C2×#63223—24no—
41C2×#63223—22no—
42C2×#63223—41no—
43C2×#63223—42no—
44C22×#3622—41no—
45C4×Dic34—41no—
46C4×Dic34—41no—
47C17×C4⋊C8—41no—
48C17×C22⋊C8—22no—
49C17×#97—41no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502C2×C411yescharacteristic
57#10587502C2×C411yescharacteristic
58C24×C17C22⋊C411yescharacteristic
59C22×C4×C17—22no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22×C411yescharacteristic
68C22×C4×C17C2×D411yescharacteristic
69C22×C4×C17C2×D411yescharacteristic
70C22×Dic17—21no—
71C22×Dic17—21no—
72C22×Dic17—21no—
73C22×Dic17—21no—
74C2×Dic34—41no—
75C2×Dic34—42no—
76C2×C8×C17—42no—
77C4⋊C4×C17—41no—
78C4⋊C4×C17—41no—
79C22⋊C4×C17—41no—
80C22⋊C4×C17—41no—
81C42×C17—21no—
82C42×C17—21no—
83C42×C17—21no—
84C42×C17—21no—
85C22⋊Dic17—42no—
86C22⋊Dic17—42no—
87C4⋊Dic17—41no—
88#63225—42no—
89#63225—41no—
90#63225—41no—
91#63225—41no—
92#63225—41no—
93C4×Dic17—42no—
94#63223—24no—
95#63223—44no—
96#63223—42no—
97C2×#3622—44no—
98C2×#3622—42no—
99#11763—171nomaximal, Sylow
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—41no—
103C23×C17—41no—
104C23×C17#10412yes—
105C23×C17C2×C22⋊C411yescharacteristic
106C2×C4×C17—22no—
107C2×C4×C17—22no—
108C2×C4×C17—22no—
109C2×C4×C17—22no—
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—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17#9911yescharacteristic
123C2×C4×C17#9911yescharacteristic
124C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
125C2×C4×C17#13711yescharacteristic
126C2×Dic17—22no—
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—
138Dic34—42no—
139Dic34—42no—
140C8×C17—82no—
141#3622—44no—
142#2599—171no—
143#1742—171no—
144#1464D1711yescharacteristic
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—81no—
154C22×C17C2×#9911yescharacteristic
155C22×C17#41711yescharacteristic
156C22×C17C2×#10411yescharacteristic
157C4×C17—21no—
158C4×C17—21no—
159C4×C17—21no—
160C4×C17—21no—
161C4×C17—41no—
162C4×C17—41no—
163C4×C17—41no—
164C4×C17—41no—
165Dic17—42no—
166Dic17—81no—
167Dic17—81no—
168Dic17—81no—
169C22×M16—341no—
170C2×C4×Q8—171no—
171C2×#117C2×D1711yescharacteristic
172#432—21no—
173#398—171no—
174#397—171no—
175#395—171no—
176#393—171no—
177#388—341no—
178#387—341no—
179#387—341no—
180#385—341no—
181#383—171no—
182#343—172no—
183#343—341no—
184#324C2×D1712yes—
185#323—21no—
186C2×C17—21no—
187C2×C17—21no—
188C2×C17—42no—
189C2×C17#150411yescharacteristic
190C2×C17#212011yescharacteristic
191C2×C17#213511yescharacteristic
192C22×Q8—171no—
193C23×C4D6811yescharacteristic
194C2×M16—344no—
195C2×M16—342no—
196C2×M16—681no—
197C2×M16—682no—
198C22×C8—681no—
199C4×Q8—681no—
200C4×Q8—681no—
201#117—41no—
202#117—41no—
203C2×C4⋊C4—171no—
204C2×C4⋊C4—341no—
205C2×C4⋊C4—341no—
206C2×C4⋊C4—341no—
207C2×C4⋊C4D6811yescharacteristic
208C2×C4⋊C4C22×D1711yescharacteristic
209C2×C22⋊C4—341no—
210C2×C22⋊C4—341no—
211C2×C22⋊C4C4×D1711yescharacteristic
212C2×C22⋊C4C22⋊D1711yescharacteristic
213C2×C42—341no—
214C2×C42C4×D1711yescharacteristic
215C2×C42C22⋊D1711yescharacteristic
216C4⋊C8—41no—
217C22⋊C8—22no—
218#97—41no—
219#95—171no—
220#95—171no—
221#95—341no—
222#95—341no—
223#95—341no—
224#95—341no—
225#95—341no—
226#95—341no—
227C17#1176311yescharacteristic, complemented, Sylow
228C24#6322711yescharacteristic
229C2×Q8—681no—
230C2×Q8—682no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—341no—
237C22×C4—341no—
238C22×C4—341no—
239C22×C4—341no—
240C22×C4#6322711yescharacteristic
241C22×C4#6322711yescharacteristic
242C22×C4#6322711yescharacteristic
243C22×C4C2×C4×D1711yescharacteristic
244C22×C4C2×D6811yescharacteristic
245C22×C4C2×C22⋊D1711yescharacteristic
246M16—344no—
247M16—684no—
248M16—682no—
249C2×C8—42no—
250C2×C8—684no—
251C2×C8—682no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—681no—
255C4⋊C4—682no—
256C4⋊C4—681no—
257C4⋊C4—681no—
258C4⋊C4—681no—
259C4⋊C4—681no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—682no—
263C22⋊C4—682no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—682no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23#1058749612yes—
274C23C2×#6322711yescharacteristic
275Q8—682no—
276Q8—682no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—342no—
294C2×C4—681no—
295C2×C4—681no—
296C2×C4—681no—
297C2×C4—681no—
298C2×C4—681no—
299C2×C4—681no—
300C2×C4—681no—
301C2×C4—681no—
302C2×C4—681no—
303C2×C4—681no—
304C2×C4—681no—
305C2×C4#1058747711yescharacteristic
306C2×C4#1058747711yescharacteristic
307C2×C4#1058759411yescharacteristic
308C2×C4#1058759511yescharacteristic
309C8—82no—
310C8—684no—
311C22—21no—
312C22—21no—
313C22—22no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22C2×#1058747711yescharacteristic
319C22#1176281111yescharacteristic
320C22C2×#1058749611yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—682no—
330C4—1361no—
331C4—1361no—
332C4—1361no—
333C2—21no—
334C2—21no—
335C2—42no—
336C2#1452728911yescharacteristic
337C2#1453042311yescharacteristic
338C2#1453047811yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1464C2
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1464D17
hypocenterC17#11763
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
3. #14526815C2semidirect
4. #14526791C2semidirect
227. C17#11763semidirect

Maximal subgroups

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

subgroupindex
#145343612
#145285232
#145268152
#145267912
C17×#14642
#1176317