Finite Group Databasebeta

← #36364227

Subgroups — #36364227

Order 22144.

350 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#26012106C211yesmaximal, characteristic
2#26012099C211yesmaximal, characteristic, complemented
3#26011898C211yesmaximal, characteristic, complemented
4#26011882C211yesmaximal, characteristic, complemented
5#26011844C211yesmaximal, characteristic, complemented
6#26011840C211yesmaximal, characteristic, complemented
7#26011750C211yesmaximal, characteristic, complemented
8#26011734C211yesmaximal, characteristic
9#26011731C211yesmaximal, characteristic, complemented
10#26011719C211yesmaximal, characteristic
11#26011596C211yesmaximal, characteristic, complemented
12#26011591C211yesmaximal, characteristic, complemented
13C4×Dic692C211yesmaximal, characteristic, complemented
14#26011445C211yesmaximal, characteristic, complemented
15C4×SD16×C173C211yesmaximal, characteristic, complemented
16C2×Dic692C2211yescharacteristic
17C2×#14583816C2211yescharacteristic
18C2×#14583814C2211yescharacteristic
19C2×#14583809C2211yescharacteristic
20C2×C8×D173C2211yescharacteristic
21#19149678C2211yescharacteristic
22#19149677C2211yescharacteristic
23#19149674—21no—
24#19149673—21no—
25#19149672—21no—
26#19149671—21no—
27#19149670C2211yescharacteristic
28#19149669—21no—
29#19149666C2211yescharacteristic
30#19149665—21no—
31#19149662—21no—
32#19149659C2211yescharacteristic
33#19149659C2211yescharacteristic
34#19149658—21no—
35#19149655C2211yescharacteristic
36#19149650C2211yescharacteristic
37C4×Dic346C2211yescharacteristic
38C4×Dic346C2211yescharacteristic
39#19149636C2211yescharacteristic
40#19149613C2211yescharacteristic
41#19149612C2211yescharacteristic
42#19149610C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149602C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149590C2211yescharacteristic
50#19149589C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×SD16×C173C2211yescharacteristic
53C4×Q8×C173C2211yescharacteristic
54C4×D4×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C4×C8×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346C2311yescharacteristic
61C2×Dic346C2311yescharacteristic
62C2×C22⋊D173—21no—
63C2×C4×D173C2311yescharacteristic
64Dic692—22no—
65#14583816—22no—
66#14583814—22no—
67#14583809—22no—
68C2×#13063436C2311yescharacteristic
69C2×#13063436C2311yescharacteristic
70C8×D173—22no—
71C4⋊Dic173—21no—
72C4⋊Dic173—21no—
73C4⋊Dic173—21no—
74C4⋊Dic173C2311yescharacteristic
75C4⋊Dic173C2311yescharacteristic
76#14583805—21no—
77#14583805—21no—
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805D411yescharacteristic
84#14583805D411yescharacteristic
85C22⋊Dic173—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803D411yescharacteristic
92#14583803D411yescharacteristic
93C4×Dic173—21no—
94C4×Dic173—21no—
95C4×Dic173C2311yescharacteristic
96C2×Q8×C173C2311yescharacteristic
97C2×D4×C173C2311yescharacteristic
98C22×C4×C173—21no—
99SD16×C173—22no—
100C2×C8×C173C2311yescharacteristic
101C2×C8×C173C2311yescharacteristic
102C4⋊C4×C173—21no—
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C22⋊C4×C173—21no—
106C42×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173C2×D411yescharacteristic
109Dic346—21no—
110Dic346—21no—
111Dic346—21no—
112Dic346—21no—
113Dic346—81no—
114C22⋊D173—81no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173C2×D411yescharacteristic
120C2×Dic173C2×D411yescharacteristic
121C2×Dic173C2×D411yescharacteristic
122C4×D173—21no—
123C4×D173—21no—
124#13063436—21no—
125#13063436#4112yes—
126C23×C173—21no—
127Q8×C173—21no—
128Q8×C173—21no—
129D4×C173—21no—
130D4×C173—21no—
131C2×C4×C173—21no—
132C2×C4×C173—21no—
133C2×C4×C173—21no—
134C2×C4×C173—81no—
135C2×C4×C173C2×D411yescharacteristic
136C2×C4×C173C2×D411yescharacteristic
137C2×C4×C173C2411yescharacteristic
138C8×C173—21no—
139C8×C173#4112yes—
140C22×C173—41no—
141C22×C173—41no—
142C22×C173—41no—
143C22×C173C22×D411yescharacteristic
144C2×D173—21no—
145C2×D173—21no—
146C2×D173—21no—
147C4×C173—21no—
148C4×C173—21no—
149C4×C173—41no—
150C4×C173—41no—
151C4×C173—41no—
152C4×C173C2×#4111yescharacteristic
153C4×C173C2×#4111yescharacteristic
154Dic173—21no—
155Dic173—21no—
156Dic173—21no—
157Dic173—41no—
158Dic173—41no—
159Dic173—41no—
160Dic173—41no—
161C2×C173—81no—
162C2×C173#53211yescharacteristic
163C2×C173#57711yescharacteristic
164C2×C173#57811yescharacteristic
165D173—41no—
166C173#317811yescharacteristic, complemented, Sylow
167#3178—1731nomaximal, Sylow
168#542—1731no—
169#541—1731no—
170#501—1731no—
171#497—1731no—
172#488—1731no—
173#487—1731no—
174#470—1731no—
175#468—1731no—
176#467—1731no—
177#465—1731no—
178#441—1731no—
179#440—1731no—
180C4×Q16—1731no—
181C4×SD16D17311yescharacteristic, complemented
182#405—1731no—
183C2×Q16—1731no—
184C2×Q16—1731no—
185C2×SD16—1731no—
186C2×SD16C2×D17311yescharacteristic
187C2×M16—1731no—
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⋊Q8—1731no—
204#121—1731no—
205C4×Q8—1731no—
206C4×Q8—1731no—
207C4×Q8C2×D17311yescharacteristic
208C4×D4C2×D17311yescharacteristic
209#117—1731no—
210#107—1731no—
211#107—1731no—
212#106—1731no—
213#106C2×D17311yescharacteristic
214#103—1731no—
215#103—1731no—
216#103—1731no—
217#103—1731no—
218#103—1731no—
219#103C2×D17311yescharacteristic
220#102—1731no—
221#102C2×D17311yescharacteristic
222#97—1731no—
223#97—1731no—
224C4×C8—1731no—
225C4×C8C2×D17311yescharacteristic
226C2×Q8—3461no—
227C2×Q8—1731no—
228C2×Q8—1731no—
229C2×Q8C22×D17311yescharacteristic
230C2×D4—3461no—
231C2×D4C22×D17311yescharacteristic
232C22×C4—21no—
233C22×C4—1731no—
234Q16—3462no—
235Q16—3462no—
236SD16—22no—
237SD16—3462no—
238M16—3462no—
239C2×C8—3462no—
240C2×C8—1731no—
241C2×C8—1731no—
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—3461no—
263C22⋊C4—3461no—
264C22⋊C4—3461no—
265C22⋊C4—3461no—
266C22⋊C4—3461no—
267C22⋊C4—3461no—
268C22⋊C4—1731no—
269C22⋊C4—1731no—
270C42—21no—
271C42—3461no—
272C42—3461no—
273C42—1731no—
274C42C22×D17311yescharacteristic
275C23—21no—
276C23—1731no—
277Q8—21no—
278Q8—21no—
279Q8—3461no—
280Q8—3461no—
281Q8—3461no—
282Q8—3461no—
283Q8—13841no—
284D4—21no—
285D4—21no—
286D4—13841no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—81no—
291C2×C4—3461no—
292C2×C4—3461no—
293C2×C4—3461no—
294C2×C4—3461no—
295C2×C4—3461no—
296C2×C4—3461no—
297C2×C4—1731no—
298C2×C4—1731no—
299C2×C4—1731no—
300C2×C4D4×D17311yescharacteristic
301C2×C4D4×D17311yescharacteristic
302C2×C4C23×D17311yescharacteristic
303C8—21no—
304C8—3461no—
305C8—1732no—
306C8#1458382612yes—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—3461no—
311C22—3461no—
312C22—3461no—
313C22C2×D4×D17311yescharacteristic
314C4—21no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—3461no—
320C4—3461no—
321C4—3461no—
322C4—6921no—
323C4—6921no—
324C4—6921no—
325C4—6921no—
326C4#41×D17311yescharacteristic
327C4C2×#1458382611yescharacteristic
328C2—81no—
329C2—6921no—
330C2#2601203811yescharacteristic
331C2#2601229711yescharacteristic
332C2#2601230511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC4×SD16×C173C2
socleC22×C173C22×D4
prime radical#19149650C22
hypercenterC4×SD16D173
hypocenterC173#3178
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #26012099C2semidirect
3. #26011898C2semidirect
4. #26011882C2semidirect
5. #26011844C2semidirect
6. #26011840C2semidirect
7. #26011750C2semidirect
9. #26011731C2semidirect
11. #26011596C2semidirect
12. #26011591C2semidirect
13. C4×Dic692C2semidirect
14. #26011445C2semidirect
15. C4×SD16×C173C2semidirect
166. C173#3178semidirect
181. C4×SD16D173semidirect

Maximal subgroups

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

subgroupindex
#260121062
#260120992
#260118982
#260118822
#260118442
#260118402
#260117502
#260117342
#260117312
#260117192
#260115962
#260115912
C4×Dic6922
#260114452
C4×SD16×C1732
#3178173