Finite Group Databasebeta

← #36364990

Subgroups — #36364990

Order 22144.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012108C211yesmaximal, characteristic
2#26012101C211yesmaximal, characteristic, complemented
3#26011797C211yesmaximal, characteristic, complemented
4#26011793C211yesmaximal, characteristic, complemented
5#26011785C211yesmaximal, characteristic, complemented
6#26011774C211yesmaximal, characteristic, complemented
7#26011749C211yesmaximal, characteristic, complemented
8#26011730C211yesmaximal, characteristic, complemented
9#26011718C211yesmaximal, characteristic
10#26011712C211yesmaximal, characteristic, complemented
11#26011711C211yesmaximal, characteristic, complemented
12#26011700C211yesmaximal, characteristic, complemented
13#26011694C211yesmaximal, characteristic, complemented
14#26011519C211yesmaximal, characteristic, complemented
15#468×C173C211yesmaximal, characteristic
16C2×Dic692C2211yescharacteristic
17C2×#14583816C2211yescharacteristic
18C2×#14583814C2211yescharacteristic
19C22×#13063436C2211yescharacteristic
20#19149678C2211yescharacteristic
21#19149677C2211yescharacteristic
22#19149672—21no—
23#19149672—21no—
24#19149672—21no—
25#19149670—21no—
26#19149670C2211yescharacteristic
27#19149669—21no—
28#19149666—21no—
29#19149666C2211yescharacteristic
30#19149660—21no—
31#19149658C2211yescharacteristic
32#19149658C2211yescharacteristic
33#19149656—21no—
34Q8×Dic173C2211yescharacteristic
35#19149644C2211yescharacteristic
36#19149644C2211yescharacteristic
37D4×Dic173C2211yescharacteristic
38#19149638C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149611C2211yescharacteristic
41#19149610C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149608C2211yescharacteristic
44#19149608C2211yescharacteristic
45#19149608C2211yescharacteristic
46#19149604C2211yescharacteristic
47#19149604C2211yescharacteristic
48#19149603C2211yescharacteristic
49#19149602C2211yescharacteristic
50#19149602C2211yescharacteristic
51#19149600C2211yescharacteristic
52C2×SD16×C173C2211yescharacteristic
53C2×M16×C173C2211yescharacteristic
54C22⋊Q8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#107×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346C2311yescharacteristic
61C2×Dic346C2311yescharacteristic
62C22×Dic173—21no—
63Dic692—41no—
64#14583816—41no—
65#14583814—41no—
66C2×#13063436—41no—
67C2×#13063436C2311yescharacteristic
68C2×#13063436C2311yescharacteristic
69C4⋊Dic173—21no—
70C4⋊Dic173—21no—
71C4⋊Dic173D411yescharacteristic
72C4⋊Dic173D411yescharacteristic
73C4⋊Dic173C2311yescharacteristic
74C4⋊Dic173C2311yescharacteristic
75#14583805—21no—
76#14583805—21no—
77#14583805—21no—
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83C22⋊Dic173—21no—
84C22⋊Dic173—21no—
85C22⋊Dic173—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88C22⋊Dic173D411yescharacteristic
89C22⋊Dic173D411yescharacteristic
90C4×Dic173—21no—
91C4×Dic173—21no—
92C4×Dic173—21no—
93C4×Dic173C2311yescharacteristic
94C4×Dic173C2311yescharacteristic
95C2×Q8×C173C2311yescharacteristic
96C2×D4×C173—21no—
97C2×D4×C173C2311yescharacteristic
98C22×C4×C173C2311yescharacteristic
99SD16×C173—41no—
100M16×C173—41no—
101C2×C8×C173C2311yescharacteristic
102C2×C8×C173C2311yescharacteristic
103C4⋊C4×C173—21no—
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C22⋊C4×C173—21no—
107C22⋊C4×C173—21no—
108Dic346—21no—
109Dic346—21no—
110Dic346—21no—
111Dic346—21no—
112Dic346—81no—
113C2×Dic173—21no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—81no—
118C2×Dic173C2×D411yescharacteristic
119C2×Dic173C2×D411yescharacteristic
120C2×Dic173C2×D411yescharacteristic
121C2×Dic173C2×D411yescharacteristic
122#13063436—21no—
123#13063436—21no—
124C23×C173—21no—
125C23×C173C2×D411yescharacteristic
126Q8×C173—21no—
127Q8×C173—21no—
128D4×C173—21no—
129D4×C173—21no—
130D4×C173—81no—
131C2×C4×C173—21no—
132C2×C4×C173—21no—
133C2×C4×C173—21no—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173C2×D411yescharacteristic
137C2×C4×C173C2411yescharacteristic
138C8×C173—21no—
139C8×C173—21no—
140C22×C173—21no—
141C22×C173—21no—
142C22×C173—21no—
143C22×C173—41no—
144C22×C173—41no—
145C22×C173—41no—
146C22×C173C22×D411yescharacteristic
147C4×C173—21no—
148C4×C173—41no—
149C4×C173—41no—
150C4×C173—41no—
151C4×C173H(5,𝔽2)11yescharacteristic
152C4×C1732−1+411yescharacteristic
153Dic173—21no—
154Dic173—21no—
155Dic173—21no—
156Dic173—21no—
157Dic173—41no—
158Dic173—41no—
159Dic173—41no—
160Dic173—41no—
161C2×C173—41no—
162C2×C173—81no—
163C2×C173#53711yescharacteristic
164C2×C173#57711yescharacteristic
165C2×C173#57811yescharacteristic
166C173#325811yescharacteristic, complemented, Sylow
167#3258—1731nomaximal, Sylow
168#542—1731no—
169#541—1731no—
170#479—1731no—
171#478—1731no—
172#477—1731no—
173#475—1731no—
174#470—1731no—
175#468D17311yescharacteristic
176#467—1731no—
177#465—1731no—
178#464—1731no—
179#464—1731no—
180#462—1731no—
181#461—1731no—
182#424—1731no—
183C2×Q16—1731no—
184C2×Q16—1731no—
185C2×SD16—1731no—
186C2×SD16C2×D17311yescharacteristic
187C2×M16C2×D17311yescharacteristic
188C22×C8—1731no—
189C4⋊Q8—1731no—
190C4⋊Q8—1731no—
191#126—3461no—
192#126—3461no—
193#126—3461no—
194#125—3461no—
195#125—3461no—
196#125—1731no—
197#124—3461no—
198#124—1731no—
199#123—3461no—
200C22⋊Q8—3461no—
201C22⋊Q8—1731no—
202C22⋊Q8—1731no—
203C22⋊Q8C2×D17311yescharacteristic
204#121C2×D17311yescharacteristic
205C4×Q8—1731no—
206C4×Q8—1731no—
207C4×Q8—1731no—
208C4×D4—1731no—
209#117—1731no—
210#107—1731no—
211#107C2×D17311yescharacteristic
212#106—1731no—
213#106—1731no—
214C4⋊C8—1731no—
215C4⋊C8—1731no—
216C4⋊C8—1731no—
217C4⋊C8—1731no—
218#103—1731no—
219#103—1731no—
220#103—1731no—
221#103—1731no—
222#103—1731no—
223#103C2×D17311yescharacteristic
224#102—1731no—
225#102C2×D17311yescharacteristic
226C2×Q8—3461no—
227C2×Q8—1731no—
228C2×Q8—1731no—
229C2×Q8C22×D17311yescharacteristic
230C2×D4—21no—
231C2×D4C22×D17311yescharacteristic
232C22×C4—3461no—
233C22×C4C22×D17311yescharacteristic
234Q16—6921no—
235Q16—6921no—
236SD16—41no—
237SD16—6921no—
238M16—41no—
239C2×C8—1731no—
240C2×C8—1731no—
241C2×C8—6921no—
242C2×C8C22×D17311yescharacteristic
243C2×C8C22×D17311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—3461no—
246C4⋊C4—3461no—
247C4⋊C4—3461no—
248C4⋊C4—3461no—
249C4⋊C4—3461no—
250C4⋊C4—3461no—
251C4⋊C4—3461no—
252C4⋊C4—3461no—
253C4⋊C4—3461no—
254C4⋊C4—3461no—
255C4⋊C4—1731no—
256C4⋊C4—1731no—
257C4⋊C4—1731no—
258C4⋊C4—1731no—
259C4⋊C4C22×D17311yescharacteristic
260C4⋊C4C22×D17311yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—3461no—
264C22⋊C4—3461no—
265C22⋊C4—3461no—
266C22⋊C4—3461no—
267C22⋊C4—3461no—
268C22⋊C4—1731no—
269C22⋊C4—1731no—
270C42—3461no—
271C42—3461no—
272C42—3461no—
273C42—1731no—
274C42—1731no—
275C23—21no—
276C23D4×D17311yescharacteristic
277Q8—21no—
278Q8—21no—
279Q8—3461no—
280Q8—3461no—
281Q8—3461no—
282Q8—3461no—
283Q8—13841no—
284D4—21no—
285D4—21no—
286D4—81no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—3461no—
293C2×C4—3461no—
294C2×C4—3461no—
295C2×C4—3461no—
296C2×C4—13841no—
297C2×C4—1731no—
298C2×C4—1731no—
299C2×C4—1731no—
300C2×C4—1731no—
301C2×C4D4×D17311yescharacteristic
302C2×C4C23×D17311yescharacteristic
303C8—21no—
304C8—21no—
305C8—3461no—
306C8—3461no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22C2×D4×D17311yescharacteristic
314C4—21no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—3461no—
319C4—3461no—
320C4—3461no—
321C4—3461no—
322C4—6921no—
323C4—6921no—
324C4—6921no—
325C4—6921no—
326C4#1914972411yescharacteristic
327C4#1914972611yescharacteristic
328C2—41no—
329C2—81no—
330C2#2601206811yescharacteristic
331C2#2601230111yescharacteristic
332C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#468×C173C2
socleC22×C173C22×D4
prime radical#121×C173C22
hypercenter#468D173
hypocenterC173#3258
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
2. #26012101C2semidirect
3. #26011797C2semidirect
4. #26011793C2semidirect
5. #26011785C2semidirect
6. #26011774C2semidirect
7. #26011749C2semidirect
8. #26011730C2semidirect
10. #26011712C2semidirect
11. #26011711C2semidirect
12. #26011700C2semidirect
13. #26011694C2semidirect
14. #26011519C2semidirect
166. C173#3258semidirect

Maximal subgroups

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

subgroupindex
#260121082
#260121012
#260117972
#260117932
#260117852
#260117742
#260117492
#260117302
#260117182
#260117122
#260117112
#260117002
#260116942
#260115192
#468×C1732
#3258173