Finite Group Databasebeta

← #16945378

Subgroups — #16945378

Order 4352.

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

normal subgroups only

This group has 476 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
1#14528780C211yesmaximal, characteristic, complemented
2#14527108C211yesmaximal, characteristic, complemented
3#14526787C214yesmaximal, complemented
4C17×#2548C211yesmaximal, characteristic
5#11762862—21no—
6#11762833—21no—
7C2×#10587507—21no—
8#11762578C2212yes—
9#11762535—21no—
10#11762535C2212yes—
11C2×C17×#127C411yescharacteristic
12C2×C17×#124—21no—
13C2×C4×Q8×C17C2211yescharacteristic
14C17×#394C412yes—
15C17×#394C2212yes—
16C17×#390—21no—
17C17×#390—21no—
18C17×#390—21no—
19C4×C4⋊C4×C17C411yescharacteristic
20C22×Q8×C17D411yescharacteristic
21C22×D4×C17—21no—
22C22×D4×C17—21no—
23C22×D4×C17C2×C412yes—
24C17×#127—41no—
25C17×#127—41no—
26C17×#124—42no—
27C17×#124—42no—
28C4×Q8×C17—41no—
29C4×Q8×C17—41no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C4⋊C4×C17C2311yescharacteristic
34C2×C22⋊C4×C17—22no—
35C2×C22⋊C4×C17—22no—
36C2×C22⋊C4×C17—22no—
37C2×C22⋊C4×C17—22no—
38C2×C42×C17—21no—
39C2×C42×C17C2×C411yescharacteristic
40C2×C42×C17D411yescharacteristic
41C2×C42×C17D411yescharacteristic
42C2×#63223—22no—
43C17×#95C2×C412yes—
44#10587507—42no—
45C22⋊#3622—22no—
46C4⋊#3622—41no—
47C4⋊#3622—41no—
48C4⋊#3622—41no—
49C4×#3622—41no—
50C24×C17—22no—
51C2×Q8×C17—41no—
52C2×Q8×C17—41no—
53C2×Q8×C17D812yes—
54C2×Q8×C17SD1612yes—
55C2×D4×C17—42no—
56C2×D4×C17—42no—
57C2×D4×C17—42no—
58C2×D4×C17—42no—
59C2×D4×C17—42no—
60C2×D4×C17—41no—
61C2×D4×C17—41no—
62C2×D4×C17—42no—
63C2×D4×C17—82no—
64C22×C4×C17—22no—
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22×C411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C4⋊C4×C17—41no—
73C4⋊C4×C17—41no—
74C4⋊C4×C17—41no—
75C4⋊C4×C17—42no—
76C22⋊C4×C17—44no—
77C22⋊C4×C17—42no—
78C22⋊C4×C17—42no—
79C22⋊C4×C17—82no—
80C22⋊C4×C17—82no—
81C42×C17—21no—
82C42×C17—21no—
83C42×C17—21no—
84C42×C17—21no—
85C42×C17—21no—
86C42×C17—21no—
87C42×C17—42no—
88#63223—42no—
89#63223—42no—
90C2×#3622—42no—
91C2×#3622—42no—
92#11882—171nomaximal, Sylow
93C23×C17—42no—
94C23×C17—42no—
95C23×C17—42no—
96C23×C17—42no—
97C23×C17—42no—
98C23×C17—42no—
99C23×C17—42no—
100C23×C17C2×C22⋊C411yescharacteristic
101Q8×C17—42no—
102Q8×C17—42no—
103D4×C17—82no—
104D4×C17—84no—
105D4×C17—84no—
106D4×C17—82no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—22no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—42no—
121C2×C4×C17—42no—
122C2×C4×C17—42no—
123C2×C4×C17#9911yescharacteristic
124C2×C4×C17#9911yescharacteristic
125C2×C4×C17#10212yes—
126C2×C4×C17#10212yes—
127C2×C4×C17C2×D811yescharacteristic
128C2×C4×C17C2×SD1611yescharacteristic
129#3622—82no—
130#3622—82no—
131#2548D1711yescharacteristic
132#1801—171no—
133#1461—171no—
134#1345—174no—
135C22×C17—21no—
136C22×C17—21no—
137C22×C17—82no—
138C22×C17—82no—
139C22×C17—82no—
140C22×C17—82no—
141C22×C17—82no—
142C22×C17—82no—
143C22×C17—82no—
144C22×C17C2×#9911yescharacteristic
145C22×C17C2×#10211yescharacteristic
146C22×C17#42111yescharacteristic
147C4×C17—21no—
148C4×C17—21no—
149C4×C17—22no—
150C4×C17—22no—
151C4×C17—21no—
152C4×C17—41no—
153C4×C17—41no—
154C4×C17—41no—
155C4×C17—82no—
156C4×C17#35512yes—
157C2×#127Dic1711yescharacteristic
158C2×#124—21no—
159C2×C4×Q8C2×D1711yescharacteristic
160#431—341no—
161#423—341no—
162C2×#101—341no—
163#394Dic1712yes—
164#394C2×D1712yes—
165#390—21no—
166#390—21no—
167#390—21no—
168C4×C4⋊C4Dic1711yescharacteristic
169#343—172no—
170#324—172no—
171#324—341no—
172C2×C17—21no—
173C2×C17—21no—
174C2×C17—162no—
175C2×C17#151911yescharacteristic
176C2×C17#212211yescharacteristic
177C2×C17C2×#35511yescharacteristic
178C22×Q8C22⋊D1711yescharacteristic
179C22×D4—21no—
180C22×D4—21no—
181C22×D4C2×Dic1712yes—
182C2×M16—342no—
183#127—41no—
184#127—41no—
185#124—42no—
186#124—42no—
187C4×Q8—41no—
188C4×Q8—41no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4C2×Dic1711yescharacteristic
191C2×C4⋊C4C22⋊D1711yescharacteristic
192C2×C4⋊C4C22×D1711yescharacteristic
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C42—21no—
198C2×C42C2×Dic1711yescharacteristic
199C2×C42C22⋊D1711yescharacteristic
200C2×C42C22⋊D1711yescharacteristic
201C4⋊C8—681no—
202C4⋊C8—681no—
203C4⋊C8—681no—
204#101—682no—
205C22⋊C8—342no—
206C4×C8—681no—
207#95C2×Dic1712yes—
208C17#1188211yescharacteristic, complemented, Sylow
209C24—22no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8#6322812yes—
213C2×Q8#6322912yes—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—82no—
223C22×C4—22no—
224C22×C4C22⋊Dic1711yescharacteristic
225C22×C4C22⋊Dic1711yescharacteristic
226C22×C4C22⋊Dic1711yescharacteristic
227C22×C4C22⋊Dic1711yescharacteristic
228C22×C4C22×Dic1711yescharacteristic
229C22×C4C2×C22⋊D1711yescharacteristic
230C22×C4C2×C22⋊D1711yescharacteristic
231M16—682no—
232M16—682no—
233C2×C8—682no—
234C2×C8—682no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C22⋊C4—44no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—82no—
243C22⋊C4—82no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23C2×C22⋊Dic1711yescharacteristic
259Q8—42no—
260Q8—42no—
261D4—84no—
262D4—82no—
263D4—84no—
264D4—82no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4#1058750312yes—
282C2×C4#1058750312yes—
283C2×C4#1058750511yescharacteristic
284C2×C4#1058750511yescharacteristic
285C2×C4C2×#6322811yescharacteristic
286C2×C4C2×#6322911yescharacteristic
287C8—1362no—
288C8—1362no—
289C22—21no—
290C22—21no—
291C22—82no—
292C22—82no—
293C22—82no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—82no—
298C22C2×#1058750511yescharacteristic
299C22C2×#1058750311yescharacteristic
300C22#1176282611yescharacteristic
301C4—21no—
302C4—22no—
303C4—22no—
304C4—21no—
305C4—21no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—82no—
310C4#1176260612yes—
311C2—21no—
312C2—21no—
313C2—162no—
314C2#1452735211yescharacteristic
315C2#1453043411yescharacteristic
316C2C2×#1176260611yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2548C2
socleC22×C17C2×#99
prime radicalC2×C17×#127C4
cosocleC2×C4⋊C4×C17C23
hypercenter#2548D17
hypocenterC17#11882
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2548

Complements

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

NH ≅ G/Ntype
1. #14528780C2semidirect
2. #14527108C2semidirect
3. #14526787C2semidirect
208. C17#11882semidirect

Maximal subgroups

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

subgroupindex
#145287802
#145271082
#145267872
C17×#25482
#1188217