Finite Group Databasebeta

← #16943604

Subgroups — #16943604

Order 4352.

428 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C4×#10587630C211yesmaximal, characteristic
2#14527108C211yesmaximal, characteristic
3#14526729C212yesmaximal
4#14526717C212yesmaximal
5C17×#1461C211yesmaximal, characteristic
6C2×#10587630C411yescharacteristic
7C2×C4×Dic34C411yescharacteristic
8#11762862—21no—
9#11762716C411yescharacteristic
10#11762714C2211yescharacteristic
11#11762704—21no—
12#11762704—21no—
13#11762678C2211yescharacteristic
14C4×#63225—21no—
15C4×#63225—21no—
16C42×Dic17C411yescharacteristic
17#11762535—21no—
18#11762535C2212yes—
19C2×C4×Q8×C17C2211yescharacteristic
20C17×#431—21no—
21C17×#324—21no—
22C17×#324C2212yes—
23C22×Q8×C17C2×C411yescharacteristic
24C22×Dic34C2×C411yescharacteristic
25C4×Q8×C17—41no—
26C4×Q8×C17—41no—
27C2×C4⋊C4×C17D411yescharacteristic
28C2×C4⋊C4×C17D411yescharacteristic
29C2×C4⋊C4×C17C2311yescharacteristic
30C2×C42×C17C2×C411yescharacteristic
31C2×C42×C17D411yescharacteristic
32C2×C42×C17D411yescharacteristic
33#10587630—22no—
34#10587630C814yescomplemented
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×#63225—21no—
37C2×#63225—21no—
38C2×#63225—21no—
39C2×#63225—21no—
40C2×#63225—21no—
41C2×C4×Dic17—21no—
42C2×C4×Dic17—21no—
43C2×C4×Dic17—21no—
44C2×C4×Dic17C2×C411yescharacteristic
45C2×C4×Dic17C2×C411yescharacteristic
46C4×Dic34—41no—
47C4×Dic34—41no—
48C17×C4⋊C8—41no—
49C17×C22⋊C8—22no—
50C4×C8×C17—41no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55C22⋊#3622—22no—
56C4⋊#3622—41no—
57C4×#3622—41no—
58C2×Q8×C17—41no—
59C2×Q8×C17—41no—
60C2×Q8×C17C2×C812yes—
61C2×Q8×C17M1612yes—
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—
72C22×Dic17—21no—
73C2×Dic34—21no—
74C2×Dic34—41no—
75C2×Dic34—41no—
76C2×Dic34C2×C812yes—
77C2×C8×C17—42no—
78C4⋊C4×C17—41no—
79C4⋊C4×C17—41no—
80C4⋊C4×C17—41no—
81C42×C17—21no—
82C42×C17—21no—
83C42×C17—21no—
84C42×C17—21no—
85C42×C17—21no—
86C42×C17—21no—
87C4⋊Dic17—41no—
88#63225—24no—
89#63225—24no—
90#63225—42no—
91#63225—42no—
92#63225—42no—
93C4×Dic17—24no—
94C4×Dic17—21no—
95C4×Dic17—42no—
96C4×Dic17—41no—
97C4×Dic17—41no—
98C4×Dic17—41no—
99C4×Dic17C2×C812yes—
100C2×#3622—42no—
101#11478—171nomaximal, Sylow
102C23×C17C2×C22⋊C411yescharacteristic
103Q8×C17—42no—
104Q8×C17—42no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—22no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17C22⋊C812yes—
119C2×C4×C17C22⋊C812yes—
120C2×C4×C17#9911yescharacteristic
121C2×C4×C17#9911yescharacteristic
122C2×C4×C17C22×C811yescharacteristic
123C2×C4×C17C2×M1611yescharacteristic
124C2×Dic17—22no—
125C2×Dic17—22no—
126C2×Dic17—22no—
127C2×Dic17—22no—
128C2×Dic17—22no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136Dic34—42no—
137Dic34—42no—
138C8×C17—82no—
139#3622—82no—
140C4×C4⋊Q8—171no—
141#1461—171no—
142#1461D1711yescharacteristic
143#1325—172no—
144#1319—172no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17C2×C22⋊C811yescharacteristic
148C22×C17C2×#9911yescharacteristic
149C22×C17#41311yescharacteristic
150C4×C17—21no—
151C4×C17—21no—
152C4×C17—21no—
153C4×C17—22no—
154C4×C17—22no—
155C4×C17—41no—
156C4×C17—41no—
157C4×C17—41no—
158C4×C17#35612yes—
159Dic17—24no—
160Dic17—42no—
161Dic17—42no—
162Dic17—81no—
163C2×C4⋊Q8—171no—
164C2×C4×Q8—171no—
165C2×C4×Q8C2×D1711yescharacteristic
166#431—21no—
167#431—341no—
168#391—171no—
169#391—171no—
170#389—341no—
171#389—341no—
172#384—171no—
173C4×C4⋊C4—341no—
174C4×C4⋊C4—341no—
175C43—171no—
176#324—21no—
177#324—172no—
178#324—341no—
179#324C2×D1712yes—
180C2×C17—21no—
181C2×C17—21no—
182C2×C17#145911yescharacteristic
183C2×C17#212611yescharacteristic
184C2×C17C2×#35611yescharacteristic
185C22×Q8—171no—
186C22×Q8C4×D1711yescharacteristic
187C4⋊Q8—174no—
188C4⋊Q8—342no—
189C4×Q8—41no—
190C4×Q8—41no—
191C4×Q8—681no—
192C4×Q8—681no—
193C2×C4⋊C4—171no—
194C2×C4⋊C4—341no—
195C2×C4⋊C4—341no—
196C2×C4⋊C4—341no—
197C2×C4⋊C4—341no—
198C2×C4⋊C4—341no—
199C2×C4⋊C4D6811yescharacteristic
200C2×C4⋊C4C22⋊D1711yescharacteristic
201C2×C4⋊C4C22×D1711yescharacteristic
202C2×C42—171no—
203C2×C42—171no—
204C2×C42—341no—
205C2×C42—341no—
206C2×C42—341no—
207C2×C42C4×D1711yescharacteristic
208C2×C42D6811yescharacteristic
209C2×C42C22⋊D1711yescharacteristic
210C4⋊C8—41no—
211C4⋊C8—681no—
212C22⋊C8—22no—
213C22⋊C8—342no—
214C4×C8—41no—
215C4×C8—681no—
216#95—341no—
217#95—341no—
218#95—341no—
219#95—341no—
220C17#1147811yescharacteristic, complemented, Sylow
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—172no—
224C2×Q8—341no—
225C2×Q8—681no—
226C2×Q8—681no—
227C2×Q8C8×D1712yes—
228C2×Q8#6321812yes—
229C22×C4—341no—
230C22×C4—341no—
231C22×C4—341no—
232C22×C4—341no—
233C22×C4#6322711yescharacteristic
234C22×C4#6322711yescharacteristic
235C22×C4#6322711yescharacteristic
236C22×C4#6322711yescharacteristic
237C22×C4C2×C4×D1711yescharacteristic
238C22×C4C2×D6811yescharacteristic
239C22×C4C2×C22⋊D1711yescharacteristic
240C2×C8—42no—
241C2×C8—682no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—344no—
246C4⋊C4—344no—
247C4⋊C4—681no—
248C4⋊C4—682no—
249C4⋊C4—682no—
250C4⋊C4—682no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—172no—
258C42—344no—
259C42—341no—
260C42—681no—
261C42—682no—
262C42—681no—
263C42—681no—
264C23C2×#6322711yescharacteristic
265Q8—42no—
266Q8—42no—
267Q8—682no—
268Q8—682no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—342no—
283C2×C4—342no—
284C2×C4—342no—
285C2×C4—342no—
286C2×C4—342no—
287C2×C4—681no—
288C2×C4—681no—
289C2×C4—681no—
290C2×C4—681no—
291C2×C4—681no—
292C2×C4—681no—
293C2×C4—681no—
294C2×C4#1058747711yescharacteristic
295C2×C4#1058747711yescharacteristic
296C2×C4#1058749112yes—
297C2×C4#1058749112yes—
298C2×C4C2×C8×D1711yescharacteristic
299C2×C4C2×#6321811yescharacteristic
300C8—82no—
301C8—1362no—
302C22—21no—
303C22—21no—
304C22C2×#1058749111yescharacteristic
305C22C2×#1058747711yescharacteristic
306C22#1176279111yescharacteristic
307C4—22no—
308C4—22no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—344no—
316C4—682no—
317C4—682no—
318C4—1361no—
319C4#1176260712yes—
320C2—21no—
321C2—21no—
322C2#1452710011yescharacteristic
323C2#1453044711yescharacteristic
324C2C2×#1176260711yescharacteristic
subgroupG/H
centerC2×C4#10587477
commutatorC2×C4×C17C22×C8
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1461C2
socleC22×C17C2×#99
prime radicalC23×C17C2×C22⋊C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1461D17
hypocenterC17#11478
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
34. #10587630C8semidirect
220. C17#11478semidirect

Maximal subgroups

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

subgroupindex
C4×#105876302
#145271082
#145267292
#145267172
C17×#14612
#1147817