Finite Group Databasebeta

← #36363525

Subgroups — #36363525

Order 22144.

386 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012064C211yesmaximal, characteristic
2#26011972C211yesmaximal, characteristic
3#26011835C211yesmaximal, characteristic, complemented
4#26011833C211yesmaximal, characteristic, complemented
5#26011830C211yesmaximal, characteristic, complemented
6#26011829C211yesmaximal, characteristic, complemented
7#26011826C211yesmaximal, characteristic, complemented
8#26011825C211yesmaximal, characteristic, complemented
9#26011823C211yesmaximal, characteristic, complemented
10#26011530C211yesmaximal, characteristic, complemented
11#26011526C211yesmaximal, characteristic, complemented
12#26011498C211yesmaximal, characteristic, complemented
13#103×D173C211yesmaximal, characteristic, complemented
14#26011456C211yesmaximal, characteristic
15#483×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C2×#14583809C2211yescharacteristic
19C2×C8×D173C2211yescharacteristic
20#19149678—21no—
21#19149665—21no—
22#19149664—21no—
23#19149661—21no—
24#19149659C2211yescharacteristic
25#19149658C2211yescharacteristic
26#19149655C2211yescharacteristic
27#19149644—21no—
28#19149641—21no—
29C4×C22⋊D173—21no—
30C4×C22⋊D173—21no—
31#19149638C2211yescharacteristic
32#19149637C2211yescharacteristic
33#19149637C2211yescharacteristic
34#19149637C2211yescharacteristic
35C4⋊C4×D173C2211yescharacteristic
36C4⋊C4×D173C2211yescharacteristic
37#19149613C2211yescharacteristic
38#19149612C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149611C2211yescharacteristic
42#19149610C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149603C2211yescharacteristic
47#19149602C2211yescharacteristic
48#19149602C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149591C2211yescharacteristic
52C22⋊Q8×C173C2211yescharacteristic
53C2×C4⋊C4×C173C2211yescharacteristic
54#106×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583826—22no—
60#14583826—22no—
61C2×Dic346—21no—
62C2×Dic346D411yescharacteristic
63C2×Dic346C2311yescharacteristic
64Q8×D173—22no—
65Q8×D173—22no—
66C2×D692D411yescharacteristic
67C2×C22⋊D173D411yescharacteristic
68C2×C22⋊D173D411yescharacteristic
69C2×C4×D173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173C2311yescharacteristic
73C2×C4×D173C2311yescharacteristic
74#14583809—22no—
75C2×#13063436C2311yescharacteristic
76C2×#13063436C2311yescharacteristic
77C8×D173—22no—
78C4⋊Dic173—21no—
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93C4×Dic173—21no—
94C4×Dic173—21no—
95C4×Dic173—21no—
96C4×Dic173—21no—
97C2×Q8×C173C2311yescharacteristic
98C22×C4×C173—21no—
99C22×C4×C173C2311yescharacteristic
100C2×C8×C173C2311yescharacteristic
101C2×C8×C173C2311yescharacteristic
102C4⋊C4×C173—21no—
103C4⋊C4×C173—21no—
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C4⋊C4×C173C2311yescharacteristic
107C22⋊C4×C173—21no—
108C22×D173C2×D411yescharacteristic
109C22×D173C2×D411yescharacteristic
110Dic346—22no—
111Dic346—21no—
112Dic346—21no—
113Dic346—42no—
114D692—22no—
115C22⋊D173—22no—
116C22⋊D173—22no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173C2×D411yescharacteristic
122C2×Dic173C2×D411yescharacteristic
123C4×D173—21no—
124C4×D173—21no—
125C4×D173—42no—
126C4×D173—81no—
127C4×D173—81no—
128C4×D173#4112yes—
129C4×D173#4112yes—
130#13063436—21no—
131#13063436—21no—
132C23×C173C2×D411yescharacteristic
133Q8×C173—21no—
134Q8×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—81no—
142C2×C4×C173C2×D411yescharacteristic
143C2×C4×C173C2411yescharacteristic
144C8×C173—21no—
145C8×C173—21no—
146C22×C173—21no—
147C22×C173—21no—
148C22×C173—21no—
149C22×C173C22×D411yescharacteristic
150C2×D173—21no—
151C2×D173—21no—
152C2×D173—21no—
153C2×D173—21no—
154C2×D173—21no—
155C2×D173#12312yes—
156C4×C173—21no—
157C4×C173—41no—
158C4×C173—41no—
159C4×C173—41no—
160C4×C173—41no—
161C4×C173C2×#4111yescharacteristic
162C4×C173C2×#4111yescharacteristic
163Dic173—21no—
164Dic173—41no—
165Dic173—41no—
166Dic173—41no—
167Dic173—41no—
168Dic173#12312yes—
169C2×C173—41no—
170C2×C173C2×#12311yescharacteristic
171C2×C173#57711yescharacteristic
172C2×C173#57811yescharacteristic
173D173—22no—
174D173—41no—
175C173#309711yescharacteristic, complemented, Sylow
176#3097—1731nomaximal, Sylow
177#536—1731no—
178#520—1731no—
179#485—1731no—
180#485—1731no—
181#485—1731no—
182#485—1731no—
183#484—1731no—
184#484—1731no—
185#483—1731no—
186#483D17311yescharacteristic, complemented
187#428—1731no—
188#427—1731no—
189#419—1731no—
190C2×#103—1731no—
191#408—1731no—
192C2×#41—1731no—
193C22×Q8—1731no—
194C2×M16—1731no—
195C22×C8—1731no—
196C4⋊Q8—3461no—
197#124—3461no—
198#123—3461no—
199#123—3461no—
200C22⋊Q8—1731no—
201C22⋊Q8—1731no—
202C22⋊Q8—1731no—
203C22⋊Q8C2×D17311yescharacteristic
204C4×Q8—3461no—
205C4×D4—3461no—
206C4×D4—3461no—
207C4×D4—3461no—
208#117—1731no—
209#117—1731no—
210#117—1731no—
211#117—1731no—
212C2×C4⋊C4—1731no—
213C2×C4⋊C4—1731no—
214C2×C4⋊C4C2×D17311yescharacteristic
215#107—1731no—
216#107—1731no—
217#107—1731no—
218#107—1731no—
219#106—1731no—
220#106—1731no—
221#106C2×D17311yescharacteristic
222#106C2×D17311yescharacteristic
223#103—1731no—
224#103—1731no—
225#103—1731no—
226#103—1731no—
227#103—1731no—
228#103—1731no—
229#103C2×D17311yescharacteristic
230#103C2×D17311yescharacteristic
231C22⋊C8—1731no—
232C22⋊C8—1731no—
233C22⋊C8—1731no—
234C22⋊C8C2×D17311yescharacteristic
235#41—3462no—
236#41—3462no—
237C2×Q8—3461no—
238C2×Q8—3462no—
239C2×Q8—3462no—
240C2×Q8—1731no—
241C2×Q8—1731no—
242C2×Q8C22×D17311yescharacteristic
243C2×D4—1731no—
244C2×D4—1731no—
245C2×D4—1731no—
246C22×C4—21no—
247C22×C4—3461no—
248C22×C4—3461no—
249C22×C4—3461no—
250C22×C4—1731no—
251C22×C4—1731no—
252C22×C4C22×D17311yescharacteristic
253M16—3462no—
254C2×C8—3462no—
255C2×C8—1731no—
256C2×C8—1731no—
257C2×C8C22×D17311yescharacteristic
258C2×C8C22×D17311yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—3461no—
262C4⋊C4—3461no—
263C4⋊C4—3461no—
264C4⋊C4—3461no—
265C4⋊C4—3461no—
266C4⋊C4—1731no—
267C4⋊C4—1731no—
268C4⋊C4—1731no—
269C4⋊C4C22×D17311yescharacteristic
270C4⋊C4C22×D17311yescharacteristic
271C4⋊C4C22×D17311yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—3461no—
274C22⋊C4—3461no—
275C22⋊C4—3461no—
276C22⋊C4—3461no—
277C22⋊C4—3461no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C42—3461no—
281C42—3461no—
282C42—3461no—
283C42—3461no—
284C23—1731no—
285C23—1731no—
286C23D4×D17311yescharacteristic
287Q8—21no—
288Q8—21no—
289Q8—3462no—
290Q8—3461no—
291Q8—3461no—
292Q8—6922no—
293D4—3462no—
294D4—3462no—
295D4—3462no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—3461no—
304C2×C4—3461no—
305C2×C4—3461no—
306C2×C4—3461no—
307C2×C4—3461no—
308C2×C4—3461no—
309C2×C4—13841no—
310C2×C4—13841no—
311C2×C4—1731no—
312C2×C4—1731no—
313C2×C4—1732no—
314C2×C4—1732no—
315C2×C4—6922no—
316C2×C4D4×D17311yescharacteristic
317C2×C4C23×D17311yescharacteristic
318C8—21no—
319C8—21no—
320C8—3461no—
321C8—3461no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—3461no—
326C22—3461no—
327C22—3461no—
328C22—3461no—
329C22—3461no—
330C22—1732no—
331C22C2×D4×D17311yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—3461no—
338C4—1732no—
339C4—6921no—
340C4—6921no—
341C4—6921no—
342C4—6921no—
343C4#41×D17311yescharacteristic
344C4#41×D17311yescharacteristic
345C2—41no—
346C2—3462no—
347C2—6921no—
348C2#123×D17311yescharacteristic
349C2#2601229711yescharacteristic
350C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#483×C173C2
socleC22×C173C22×D4
prime radicalC2×#14583826C22
hypercenter#483D173
hypocenterC173#3097
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. #26011835C2semidirect
4. #26011833C2semidirect
5. #26011830C2semidirect
6. #26011829C2semidirect
7. #26011826C2semidirect
8. #26011825C2semidirect
9. #26011823C2semidirect
10. #26011530C2semidirect
11. #26011526C2semidirect
12. #26011498C2semidirect
13. #103×D173C2semidirect
15. #483×C173C2semidirect
175. C173#3097semidirect
186. #483D173semidirect

Maximal subgroups

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

subgroupindex
#260120642
#260119722
#260118352
#260118332
#260118302
#260118292
#260118262
#260118252
#260118232
#260115302
#260115262
#260114982
#103×D1732
#260114562
#483×C1732
#3097173