Finite Group Databasebeta

← #17958579

Subgroups — #17958579

Order 4352.

366 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 68 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#14544035C211yesmaximal, characteristic
2#14544015C211yesmaximal, characteristic
3#14543727C212yesmaximal, complemented
4#14543714C212yesmaximal, complemented
5#14533238C211yesmaximal, characteristic
6#11763829—21no—
7#11763825—21no—
8#11763654C2212yes—
9#11763654C2212yes—
10#11763652—21no—
11#11763652—21no—
12C2×#10587612C2211yescharacteristic
13#11762737—21no—
14#11762737—21no—
15#11762716C411yescharacteristic
16#11762716C2211yescharacteristic
17#11762690—21no—
18#11762690—21no—
19#11762678C2211yescharacteristic
20#11762674C411yescharacteristic
21C4×C22⋊Dic17C411yescharacteristic
22C2×C17×C22⋊Q8C411yescharacteristic
23C22×Q8×C17C2×C411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C17×C22⋊Q8—41no—
26C17×C22⋊Q8—41no—
27C2×C4⋊C4×C17—21no—
28C2×C4⋊C4×C17C2×C411yescharacteristic
29C2×C22⋊C4×C17—21no—
30C2×C22⋊Dic17—21no—
31C2×C22⋊Dic17D411yescharacteristic
32C2×C22⋊Dic17D411yescharacteristic
33#10587612—41no—
34#10587612—41no—
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×C4⋊Dic17C2311yescharacteristic
37C2×#63225—21no—
38C2×#63225—21no—
39C2×C4×Dic17C2×C411yescharacteristic
40C2×C4×Dic17C2×C411yescharacteristic
41C2×C4×Dic17D411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43C22⋊#3624—22no—
44C22⋊#3624—22no—
45#10587544—41no—
46#10587542—41no—
47C4⋊#3624—41no—
48C4×#3624—41no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56C24×C17C22⋊C411yescharacteristic
57C2×Q8×C17—22no—
58C2×Q8×C17—41no—
59C2×Q8×C17—41no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—22no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—41no—
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22×C411yescharacteristic
68C22×Dic17—21no—
69C22×Dic17—21no—
70C22×Dic17C22⋊C411yescharacteristic
71C22×Dic17C22⋊C411yescharacteristic
72C22×Dic17C2×D411yescharacteristic
73C22×Dic17C2×D411yescharacteristic
74C2×#3624—42no—
75C2×#3624—42no—
76C4⋊C4×C17—41no—
77C4⋊C4×C17—41no—
78C4⋊C4×C17—41no—
79C22⋊C4×C17—42no—
80C22⋊Dic17—41no—
81C22⋊Dic17—41no—
82C22⋊Dic17—42no—
83C4⋊Dic17—41no—
84C4⋊Dic17—41no—
85#63225—42no—
86#63225—42no—
87C4×Dic17—21no—
88C4×Dic17—21no—
89C4×Dic17—21no—
90C4×Dic17—21no—
91C4×Dic17—22no—
92C4×Dic17—41no—
93#11552—171nomaximal, Sylow
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—41no—
98C23×C17—41no—
99C23×C17C2×C22⋊C411yescharacteristic
100Q8×C17—82no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—42no—
106C2×C4×C17—42no—
107C2×C4×C17—41no—
108C2×C4×C17—41no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17#9911yescharacteristic
114C2×C4×C17#9911yescharacteristic
115C2×C4×C17#13112yes—
116C2×Dic17—22no—
117C2×Dic17—21no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—41no—
122C2×Dic17—41no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130#3624—82no—
131#3624—82no—
132#2499—171no—
133#1464—171no—
134#1458—171no—
135#1325—172no—
136#1319—172no—
137C22×C17—21no—
138C22×C17—21no—
139C22×C17—41no—
140C22×C17—41no—
141C22×C17—41no—
142C22×C17—81no—
143C22×C17—81no—
144C22×C17#40811yescharacteristic
145C22×C17C2×#9911yescharacteristic
146C22×C17#41311yescharacteristic
147C4×C17—21no—
148C4×C17—21no—
149C4×C17—22no—
150C4×C17—81no—
151C4×C17—81no—
152Dic17—42no—
153Dic17—41no—
154Dic17—41no—
155Dic17—41no—
156Dic17—41no—
157Dic17—81no—
158C2×C22⋊Q8#59511yescharacteristic
159C2×#117—171no—
160#432—341no—
161#431—341no—
162#397—341no—
163#397—341no—
164#391—171no—
165#391—171no—
166#387—341no—
167#387—341no—
168#384—171no—
169#382—171no—
170C4×C22⋊C4—171no—
171#324—172no—
172#324—172no—
173#323—341no—
174#323—341no—
175C2×C17—21no—
176C2×C17—21no—
177C2×C17—81no—
178C2×C17#146811yescharacteristic
179C2×C17#212611yescharacteristic
180C2×C17#213511yescharacteristic
181C22×Q8C2×#59511yescharacteristic
182C23×C4C2×#59511yescharacteristic
183C22⋊Q8—41no—
184C22⋊Q8—41no—
185#117—681no—
186#117—681no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—171no—
189C2×C4⋊C4—171no—
190C2×C4⋊C4—341no—
191C2×C4⋊C4—341no—
192C2×C4⋊C4C2×#59511yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—171no—
195C2×C22⋊C4—171no—
196C2×C22⋊C4—341no—
197C2×C42—171no—
198C2×C42—171no—
199C2×C42—171no—
200C2×C42—171no—
201C4⋊C8—681no—
202C4⋊C8—681no—
203C22⋊C8—342no—
204C22⋊C8—342no—
205#97—681no—
206C4×C8—681no—
207#95—341no—
208#95—341no—
209#95—341no—
210#95—341no—
211#95—341no—
212#95—341no—
213#95—341no—
214C17#1155211yescharacteristic, complemented, Sylow
215C24C22⋊#59511yescharacteristic
216C2×Q8—22no—
217C2×Q8—41no—
218C2×Q8—41no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—41no—
225C22×C4—171no—
226C22×C4—171no—
227C22×C4—171no—
228C22×C4—171no—
229C22×C4—341no—
230C22×C4—341no—
231C22×C4C22⋊#59511yescharacteristic
232C22×C4C22×#59511yescharacteristic
233C2×C8—682no—
234C2×C8—682no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—681no—
239C4⋊C4—681no—
240C4⋊C4—682no—
241C4⋊C4—682no—
242C22⋊C4—42no—
243C22⋊C4—681no—
244C22⋊C4—681no—
245C22⋊C4—682no—
246C42—341no—
247C42—341no—
248C42—341no—
249C42—341no—
250C42—342no—
251C42—681no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23C2×C22⋊#59511yescharacteristic
258Q8—82no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—342no—
272C2×C4—341no—
273C2×C4—341no—
274C2×C4—341no—
275C2×C4—341no—
276C2×C4—681no—
277C2×C4—681no—
278C2×C4—681no—
279C2×C4—681no—
280C2×C4—681no—
281C2×C4—681no—
282C2×C4—681no—
283C2×C4—681no—
284C2×C4—681no—
285C2×C4#1058755111yescharacteristic
286C2×C4#1058755111yescharacteristic
287C2×C4#1058767812yes—
288C8—1362no—
289C8—1362no—
290C22—21no—
291C22—21no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22#1176378211yescharacteristic
298C22C2×#1058755111yescharacteristic
299C22#1176379611yescharacteristic
300C4—22no—
301C4—21no—
302C4—21no—
303C4—81no—
304C4—81no—
305C4—682no—
306C4—681no—
307C4—681no—
308C4—681no—
309C4—681no—
310C4—1361no—
311C2—21no—
312C2—21no—
313C2—81no—
314C2#1454404911yescharacteristic
315C2#1454505511yescharacteristic
316C2#1454508611yescharacteristic
subgroupG/H
centerC22C2×#10587551
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×C22⋊Q8C4
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC2×C4⋊Dic17C23
hypercenterC2×C22⋊Q8#595
hypocenterC17#11552
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #14543727C2semidirect
4. #14543714C2semidirect
214. C17#11552semidirect

Maximal subgroups

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

subgroupindex
#145440352
#145440152
#145437272
#145437142
#145332382
#1155217