Finite Group Databasebeta

← #17958591

Subgroups — #17958591

Order 4352.

414 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 72 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#14544032C211yesmaximal, characteristic
2C2×#11763654C211yesmaximal, characteristic
3#14543727C212yesmaximal, complemented
4#14543714C212yesmaximal, complemented
5#14533263C211yesmaximal, characteristic
6#11763779—21no—
7C2×C22⋊#3624—21no—
8#11763654—21no—
9#11763654—21no—
10#11763654C2212yes—
11#11763654C2212yes—
12C22×C4⋊Dic17C2211yescharacteristic
13#11762739—21no—
14#11762739—21no—
15#11762716C411yescharacteristic
16#11762716C2211yescharacteristic
17#11762681—21no—
18#11762681—21no—
19#11762678C411yescharacteristic
20#11762678C2211yescharacteristic
21C4×C22⋊Dic17C411yescharacteristic
22C2×C17×C22⋊Q8C411yescharacteristic
23C22×Q8×C17C2×C411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×Dic17D411yescharacteristic
26C23×Dic17D411yescharacteristic
27C2×#63247—41no—
28C22×#3624—41no—
29C17×C22⋊Q8—41no—
30C17×C22⋊Q8—41no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C22⋊C4×C17—21no—
34C2×C22⋊Dic17—21no—
35C2×C4⋊Dic17—41no—
36C2×C4⋊Dic17—41no—
37C2×C4⋊Dic17C2×C411yescharacteristic
38C2×C4⋊Dic17D411yescharacteristic
39C2×C4⋊Dic17D411yescharacteristic
40C2×C4⋊Dic17C2311yescharacteristic
41C2×#63225—21no—
42C2×#63225—21no—
43C2×#63225—21no—
44C2×#63225—21no—
45C2×C4×Dic17C2×C411yescharacteristic
46C2×C4×Dic17C2×C411yescharacteristic
47C22⋊#3624—22no—
48C22⋊#3624—22no—
49C22⋊#3624—41no—
50C22⋊#3624—41no—
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×Dic17—21no—
71C22×Dic17—21no—
72C22×Dic17—21no—
73C22×Dic17—21no—
74C22×Dic17—81no—
75C22×Dic17—81no—
76C22×Dic17C22⋊C411yescharacteristic
77C22×Dic17C22⋊C411yescharacteristic
78C22×Dic17C2×D411yescharacteristic
79C22×Dic17C2×D411yescharacteristic
80#63247—82no—
81C2×#3624—42no—
82C2×#3624—42no—
83C2×#3624—82no—
84C4⋊C4×C17—41no—
85C4⋊C4×C17—42no—
86C22⋊C4×C17—41no—
87C22⋊C4×C17—41no—
88C22⋊Dic17—24no—
89C4⋊Dic17—41no—
90C4⋊Dic17—41no—
91C4⋊Dic17—41no—
92C4⋊Dic17—41no—
93#63225—81no—
94#63225—81no—
95#63225—81no—
96#63225—81no—
97C4×Dic17—22no—
98C4×Dic17—41no—
99#11555—171nomaximal, Sylow
100C23×C17—22no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—41no—
104C23×C17#9911yescharacteristic
105C23×C17#9911yescharacteristic
106C23×C17C2×C22⋊C411yescharacteristic
107Q8×C17—82no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17#13112yes—
122C2×Dic17—22no—
123C2×Dic17—22no—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—42no—
129C2×Dic17—42no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—81no—
139C2×Dic17—81no—
140#3624—82no—
141#3624—82no—
142#2502—171no—
143#1463—171no—
144C2×#324—171no—
145#1325—172no—
146#1319—172no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17—21no—
150C22×C17—21no—
151C22×C17—21no—
152C22×C17—21no—
153C22×C17—21no—
154C22×C17—42no—
155C22×C17—42no—
156C22×C17#35612yes—
157C22×C17#40811yescharacteristic
158C22×C17C2×#9911yescharacteristic
159C22×C17#41311yescharacteristic
160C4×C17—22no—
161C4×C17—41no—
162C4×C17—81no—
163C4×C17—81no—
164Dic17—42no—
165Dic17—42no—
166Dic17—41no—
167Dic17—41no—
168Dic17—41no—
169Dic17—41no—
170C2×C22⋊Q8#59511yescharacteristic
171C22×C4⋊C4—171no—
172#407—341no—
173C2×C22⋊C8—341no—
174#398—341no—
175#398—341no—
176#391—171no—
177#391—171no—
178#385—341no—
179#385—341no—
180#384—171no—
181#384—171no—
182C4×C22⋊C4—171no—
183#324—172no—
184#324—172no—
185#324—341no—
186#324—341no—
187C2×C17—21no—
188C2×C17—21no—
189C2×C17—22no—
190C2×C17—41no—
191C2×C17#146811yescharacteristic
192C2×C17#212711yescharacteristic
193C2×C17C2×#35611yescharacteristic
194C22×Q8C2×#59511yescharacteristic
195C23×C4—171no—
196C23×C4—171no—
197C23×C4C2×#59511yescharacteristic
198C2×M16—681no—
199C22×C8—681no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—171no—
204C2×C4⋊C4—171no—
205C2×C4⋊C4—171no—
206C2×C4⋊C4—171no—
207C2×C4⋊C4—341no—
208C2×C4⋊C4—341no—
209C2×C4⋊C4—341no—
210C2×C4⋊C4—341no—
211C2×C4⋊C4—681no—
212C2×C4⋊C4—681no—
213C2×C4⋊C4C2×#59511yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—341no—
216C2×C42—171no—
217C2×C42—171no—
218C22⋊C8—342no—
219C22⋊C8—342no—
220C22⋊C8—681no—
221C22⋊C8—681no—
222#95—341no—
223#95—341no—
224#95—341no—
225#95—341no—
226#95—341no—
227C17#1155511yescharacteristic, complemented, Sylow
228C24C22⋊#59511yescharacteristic
229C2×Q8—22no—
230C2×Q8—41no—
231C2×Q8—41no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—41no—
238C22×C4—171no—
239C22×C4—171no—
240C22×C4—171no—
241C22×C4—171no—
242C22×C4—341no—
243C22×C4—341no—
244C22×C4—341no—
245C22×C4—341no—
246C22×C4—341no—
247C22×C4—341no—
248C22×C4—1361no—
249C22×C4—1361no—
250C22×C4C22⋊#59511yescharacteristic
251C22×C4C22×#59511yescharacteristic
252M16—1362no—
253C2×C8—682no—
254C2×C8—682no—
255C2×C8—1362no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—681no—
259C4⋊C4—681no—
260C4⋊C4—681no—
261C4⋊C4—681no—
262C4⋊C4—1361no—
263C4⋊C4—1361no—
264C4⋊C4—1361no—
265C4⋊C4—1361no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—344no—
269C42—342no—
270C42—681no—
271C23—22no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23#1058755111yescharacteristic
276C23#1058755111yescharacteristic
277C23C2×C22⋊#59511yescharacteristic
278Q8—82no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—342no—
293C2×C4—342no—
294C2×C4—341no—
295C2×C4—341no—
296C2×C4—341no—
297C2×C4—341no—
298C2×C4—681no—
299C2×C4—681no—
300C2×C4—682no—
301C2×C4—682no—
302C2×C4—681no—
303C2×C4—681no—
304C2×C4—681no—
305C2×C4—681no—
306C2×C4—681no—
307C2×C4—681no—
308C2×C4—1361no—
309C2×C4—1361no—
310C2×C4#1058767812yes—
311C8—1362no—
312C8—1362no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—42no—
321C22—42no—
322C22#1176371912yes—
323C22#1176378211yescharacteristic
324C22C2×#1058755111yescharacteristic
325C22#1176379611yescharacteristic
326C4—22no—
327C4—41no—
328C4—81no—
329C4—81no—
330C4—682no—
331C4—682no—
332C4—681no—
333C4—681no—
334C4—681no—
335C4—681no—
336C2—21no—
337C2—21no—
338C2—22no—
339C2—41no—
340C2#1454404911yescharacteristic
341C2#1454505911yescharacteristic
342C2C2×#1176371911yescharacteristic
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#11555
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #14543727C2semidirect
4. #14543714C2semidirect
227. C17#11555semidirect

Maximal subgroups

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

subgroupindex
#145440322
C2×#117636542
#145437272
#145437142
#145332632
#1155517