Finite Group Databasebeta

← #17958627

Subgroups — #17958627

Order 4352.

412 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 70 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#14544046C211yesmaximal, characteristic, complemented
2#14544026C211yesmaximal, characteristic, complemented
3#14543715C212yesmaximal, complemented
4#14543713C212yesmaximal
5#14533764C211yesmaximal, characteristic
6#11763830—21no—
7#11763826—21no—
8#11763655—21no—
9#11763655—21no—
10#11763654C2212yes—
11#11763654C2212yes—
12C2×#10587555C411yescharacteristic
13C2×#10587561C411yescharacteristic
14C2×#10587561C411yescharacteristic
15C2×C4×Dic34C2211yescharacteristic
16#11762731—21no—
17#11762731—21no—
18#11762715C2211yescharacteristic
19#11762711C2211yescharacteristic
20#11762701—21no—
21#11762701—21no—
22C4×C4⋊C4×C17C411yescharacteristic
23C22×D68C2×C411yescharacteristic
24C22×Dic34C2×C411yescharacteristic
25C22×Dic34C2×C411yescharacteristic
26C2×C4⋊C4×C17—21no—
27C2×C42×C17C2×C411yescharacteristic
28C2×C42×C17C2×C411yescharacteristic
29C2×C42×C17C2×C411yescharacteristic
30C2×#63227—21no—
31C2×#63227—21no—
32C2×#63227—21no—
33C2×#63227—21no—
34C2×#63227—21no—
35C2×#63227—21no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17C2311yescharacteristic
39C2×#63225D411yescharacteristic
40C2×#63225D411yescharacteristic
41C2×C4×Dic17D411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43#10587561—41no—
44#10587561—41no—
45#10587561—42no—
46#10587555—42no—
47C4×Dic34—41no—
48C4×Dic34—41no—
49C22⋊#3624—22no—
50C22⋊#3624—22no—
51#10587544—41no—
52#10587542—41no—
53C4⋊#3624—41no—
54C4×#3624—41no—
55C17×#95—21no—
56#10587502—21no—
57#10587502—21no—
58C23×D17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22×C411yescharacteristic
64C22×Dic17—21no—
65C22×Dic17C22⋊C411yescharacteristic
66C22×Dic17C22⋊C411yescharacteristic
67C22×Dic17C2×D411yescharacteristic
68C22×Dic17C2×D411yescharacteristic
69C2×D68—22no—
70C2×D68—41no—
71C2×D68—41no—
72C2×Dic34—22no—
73C2×Dic34—22no—
74C2×Dic34—22no—
75C2×Dic34—41no—
76C2×Dic34—41no—
77C2×Dic34—41no—
78C2×#3624—42no—
79C2×#3624—42no—
80C4⋊C4×C17—42no—
81C42×C17—21no—
82C42×C17—21no—
83C42×C17—41no—
84C42×C17—41no—
85#63227—42no—
86#63227—42no—
87#63227—42no—
88#63227—42no—
89#63227—41no—
90#63227—41no—
91#63227—42no—
92C4⋊Dic17—41no—
93C4⋊Dic17—42no—
94C4⋊Dic17—42no—
95#63225—41no—
96#63225—41no—
97C4×Dic17—21no—
98C4×Dic17—21no—
99C4×Dic17—21no—
100C4×Dic17—21no—
101#11565—171nomaximal, Sylow
102C23×C17C2×C22⋊C411yescharacteristic
103C22×D17—41no—
104C22×D17—41no—
105C22×D17—41no—
106C22×D17—41no—
107C22×D17—41no—
108C22×D17—41no—
109C22×D17—41no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—22no—
113C2×C4×C17—21no—
114C2×C4×C17—41no—
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#9911yescharacteristic
122C2×C4×C17#9911yescharacteristic
123C2×C4×C17#13112yes—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—21no—
130C2×Dic17—21no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17#10012yes—
137D68—82no—
138Dic34—44no—
139Dic34—82no—
140#3624—82no—
141#3624—82no—
142#2545—171no—
143#1467—171no—
144#1461—171no—
145#1319—172no—
146#1318—172no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17#40811yescharacteristic
150C22×C17C2×#9911yescharacteristic
151C22×C17C2×#10011yescharacteristic
152C2×D17—81no—
153C2×D17—81no—
154C2×D17—81no—
155C2×D17—81no—
156C2×D17—81no—
157C2×D17—81no—
158C2×D17—81no—
159C4×C17—21no—
160C4×C17—21no—
161C4×C17—22no—
162C4×C17—42no—
163C4×C17—41no—
164C4×C17—81no—
165Dic17—22no—
166Dic17—41no—
167Dic17—41no—
168Dic17—41no—
169Dic17—81no—
170C2×C4⋊Q8—171no—
171C2×#124—171no—
172C2×#124—171no—
173C2×C4×Q8—171no—
174#432—341no—
175#431—341no—
176#396—341no—
177#396—341no—
178#391—171no—
179#390—171no—
180#388—341no—
181#388—341no—
182C4×C4⋊C4#59511yescharacteristic
183#324—172no—
184#324—172no—
185#324—341no—
186#324—341no—
187C2×C17—21no—
188C2×C17—21no—
189C2×C17#147111yescharacteristic
190C2×C17#212611yescharacteristic
191C2×C17#213311yescharacteristic
192D17—161no—
193C22×Q8—171no—
194C22×Q8—171no—
195C22×D4—171no—
196C4⋊Q8—682no—
197#124—681no—
198#124—681no—
199#124—682no—
200C4×Q8—681no—
201C4×Q8—681no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—171no—
204C2×C4⋊C4—171no—
205C2×C4⋊C4—171no—
206C2×C4⋊C4—341no—
207C2×C4⋊C4—341no—
208C2×C22⋊C4—341no—
209C2×C22⋊C4—341no—
210C2×C22⋊C4—341no—
211C2×C22⋊C4—341no—
212C2×C22⋊C4—341no—
213C2×C22⋊C4—341no—
214C2×C42—171no—
215C2×C42—171no—
216C2×C42C2×#59511yescharacteristic
217C2×C42C2×#59511yescharacteristic
218C2×C42C2×#59511yescharacteristic
219C4⋊C8—681no—
220C4⋊C8—681no—
221C22⋊C8—342no—
222C22⋊C8—342no—
223#97—681no—
224C4×C8—681no—
225#95—21no—
226#95—341no—
227#95—341no—
228C17#1156511yescharacteristic, complemented, Sylow
229C24—341no—
230C2×Q8—342no—
231C2×Q8—342no—
232C2×Q8—342no—
233C2×Q8—681no—
234C2×Q8—681no—
235C2×Q8—681no—
236C2×D4—342no—
237C2×D4—681no—
238C2×D4—681no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—171no—
242C22×C4—171no—
243C22×C4—171no—
244C22×C4—171no—
245C22×C4—341no—
246C22×C4C22⋊#59511yescharacteristic
247C22×C4C22⋊#59511yescharacteristic
248C22×C4C22×#59511yescharacteristic
249C2×C8—682no—
250C2×C8—682no—
251C4⋊C4—42no—
252C4⋊C4—681no—
253C4⋊C4—681no—
254C4⋊C4—681no—
255C4⋊C4—682no—
256C4⋊C4—682no—
257C22⋊C4—682no—
258C22⋊C4—682no—
259C22⋊C4—682no—
260C22⋊C4—682no—
261C22⋊C4—682no—
262C22⋊C4—681no—
263C22⋊C4—681no—
264C42—21no—
265C42—21no—
266C42—41no—
267C42—41no—
268C42—341no—
269C42—341no—
270C42—341no—
271C42—341no—
272C23—681no—
273C23—681no—
274C23—681no—
275C23—681no—
276C23—681no—
277C23—681no—
278C23—681no—
279C23C2×C22⋊#59511yescharacteristic
280Q8—684no—
281Q8—1362no—
282D4—1362no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—172no—
295C2×C4—341no—
296C2×C4—341no—
297C2×C4—341no—
298C2×C4—341no—
299C2×C4—341no—
300C2×C4—341no—
301C2×C4—341no—
302C2×C4—681no—
303C2×C4—681no—
304C2×C4—681no—
305C2×C4—681no—
306C2×C4—681no—
307C2×C4#1058753911yescharacteristic
308C2×C4#1058753911yescharacteristic
309C2×C4#1058768012yes—
310C8—1362no—
311C8—1362no—
312C22—21no—
313C22—21no—
314C22—1361no—
315C22—1361no—
316C22—1361no—
317C22—1361no—
318C22—1361no—
319C22—1361no—
320C22—1361no—
321C22#1176378411yescharacteristic
322C22C2×#1058753911yescharacteristic
323C22D17⋊#10011yescharacteristic
324C4—21no—
325C4—21no—
326C4—22no—
327C4—42no—
328C4—41no—
329C4—81no—
330C4—342no—
331C4—681no—
332C4—681no—
333C4—681no—
334C4—1361no—
335C2—21no—
336C2—21no—
337C2—2721no—
338C2#1454406111yescharacteristic
339C2#1454505711yescharacteristic
340C2#1454508111yescharacteristic
subgroupG/H
centerC22C2×#10587539
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC4×C4⋊C4×C17C4
socleC22×C17C2×#99
prime radicalC22×D68C2×C4
cosocleC2×C4⋊Dic17C23
hypercenterC4×C4⋊C4#595
hypocenterC17#11565
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #14544046C2semidirect
2. #14544026C2semidirect
3. #14543715C2semidirect
228. C17#11565semidirect

Maximal subgroups

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

subgroupindex
#145440462
#145440262
#145437152
#145437132
#145337642
#1156517