Finite Group Databasebeta

← #36365661

Subgroups — #36365661

Order 22144.

382 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012181C211yesmaximal, characteristic, complemented
2#26012146C211yesmaximal, characteristic
3#26011860C211yesmaximal, characteristic, complemented
4#26011833C211yesmaximal, characteristic, complemented
5#26011827C211yesmaximal, characteristic, complemented
6#26011792C211yesmaximal, characteristic, complemented
7#26011749C211yesmaximal, characteristic, complemented
8#26011732C211yesmaximal, characteristic, complemented
9#26011702C211yesmaximal, characteristic, complemented
10#26011655C211yesmaximal, characteristic, complemented
11#26011655C211yesmaximal, characteristic, complemented
12#26011593C211yesmaximal, characteristic, complemented
13#26011493C211yesmaximal, characteristic, complemented
14#26011488C211yesmaximal, characteristic, complemented
15#435×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583826C2211yescharacteristic
18C2×Dic692C2211yescharacteristic
19C2×Dic692C2211yescharacteristic
20C2×#14583815C2211yescharacteristic
21#19149679C2211yescharacteristic
22#19149672—21no—
23#19149671C2211yescharacteristic
24#19149669—21no—
25#19149661—21no—
26#19149661—21no—
27#19149660—21no—
28#19149658C2211yescharacteristic
29#19149658C2211yescharacteristic
30#19149658C2211yescharacteristic
31#19149658C2211yescharacteristic
32#19149650—21no—
33C4×Dic346C2211yescharacteristic
34C4×D692C2211yescharacteristic
35#19149638C2211yescharacteristic
36#19149635—21no—
37C2×C4⋊Dic173C2211yescharacteristic
38C4⋊C4×D173—21no—
39#19149613C2211yescharacteristic
40#19149613C2211yescharacteristic
41#19149611C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149604C2211yescharacteristic
46#19149604C2211yescharacteristic
47#19149604C2211yescharacteristic
48#19149604C2211yescharacteristic
49#19149601C2211yescharacteristic
50#19149601C2211yescharacteristic
51#19149601C2211yescharacteristic
52C2×M16×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic
54C4×D4×C173C2211yescharacteristic
55C4⋊C8×C173C2211yescharacteristic
56C22⋊C8×C173C2211yescharacteristic
57C22⋊C8×C173C2211yescharacteristic
58#97×C173C2211yescharacteristic
59#14583826—22no—
60#14583826—22no—
61#14583826—41no—
62#14583826—41no—
63C2×Dic346C2311yescharacteristic
64C2×Dic346C2311yescharacteristic
65C2×Dic346C2311yescharacteristic
66C2×D692C2311yescharacteristic
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C22×Dic173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173—21no—
73Dic692—41no—
74Dic692—41no—
75#14583815—22no—
76C4⋊Dic173—21no—
77C4⋊Dic173—21no—
78C4⋊Dic173—21no—
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89#14583805—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93C22⋊Dic173—21no—
94C22⋊Dic173—21no—
95#14583803—21no—
96C4×Dic173—21no—
97C4×Dic173—21no—
98C2×D4×C173D411yescharacteristic
99C22×C4×C173C2311yescharacteristic
100C22×C4×C173C2311yescharacteristic
101M16×C173—41no—
102C2×C8×C173—41no—
103C2×C8×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C4⋊C4×C173D411yescharacteristic
108C22⋊C4×C173D411yescharacteristic
109C22⋊C4×C173D411yescharacteristic
110C42×C173C2311yescharacteristic
111C22×D173—21no—
112Dic346—21no—
113Dic346—21no—
114Dic346—21no—
115Dic346—21no—
116Dic346—21no—
117Dic346—21no—
118D692—21no—
119D692#4112yes—
120C22⋊D173—42no—
121C22⋊D173—42no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—21no—
129C2×Dic173—42no—
130C4×D173—42no—
131C4×D173—42no—
132C4×D173—42no—
133C23×C173C2×D411yescharacteristic
134C23×C173C2×D411yescharacteristic
135D4×C173—41no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173C2×D411yescharacteristic
141C2×C4×C173C2×D411yescharacteristic
142C2×C4×C173C2×D411yescharacteristic
143C2×C4×C173C2×D411yescharacteristic
144C2×C4×C173C2411yescharacteristic
145C8×C173—21no—
146C8×C173—21no—
147C8×C173—21no—
148C8×C173—21no—
149C22×C173—21no—
150C22×C173—21no—
151C22×C173—21no—
152C22×C173—21no—
153C22×C173—21no—
154C22×C173—21no—
155C22×C173C22×D411yescharacteristic
156C2×D173—22no—
157C2×D173—41no—
158C2×D173—41no—
159C4×C173—21no—
160C4×C173—21no—
161C4×C173—21no—
162C4×C173—21no—
163C4×C173C2×#4111yescharacteristic
164C4×C173H(5,𝔽2)11yescharacteristic
165Dic173—41no—
166Dic173—41no—
167Dic173—41no—
168Dic173—41no—
169Dic173—41no—
170Dic173—41no—
171Dic173—41no—
172C2×C173—41no—
173C2×C173—41no—
174C2×C173#54611yescharacteristic
175C2×C173#57511yescharacteristic
176C2×C173#57811yescharacteristic
177D173—42no—
178C173#331511yescharacteristic, complemented, Sylow
179#3315—1731nomaximal, Sylow
180#552—1731no—
181#547—1731no—
182#491—1731no—
183#485—1731no—
184#484—1731no—
185#478—1731no—
186#470—1731no—
187#467—1731no—
188#462—1731no—
189#452—1731no—
190#452—1731no—
191#440—1731no—
192#435D17311yescharacteristic, complemented
193#417—1731no—
194#416—1731no—
195C2×#41—1731no—
196C2×#41—1731no—
197C2×Q16—1731no—
198C2×Q16—1731no—
199C2×SD16—1731no—
200C2×M16C2×D17311yescharacteristic
201C22×C8C2×D17311yescharacteristic
202C4⋊Q8—1731no—
203#126—3461no—
204#125—3461no—
205#125—1731no—
206#123—3461no—
207#123—3461no—
208#123—3461no—
209C22⋊Q8—1731no—
210C22⋊Q8—1731no—
211C22⋊Q8—1731no—
212C22⋊Q8—1731no—
213#121—3461no—
214C4×Q8—1731no—
215C4×D4—1731no—
216C4×D4C2×D17311yescharacteristic
217#117—3461no—
218#117—1731no—
219C2×C4⋊C4—3461no—
220C2×C4⋊C4—1731no—
221#107—1731no—
222#107—1731no—
223#106—1731no—
224C4⋊C8C2×D17311yescharacteristic
225#103—1731no—
226#103—1731no—
227#103—1731no—
228#103—1731no—
229#103—1731no—
230#103—1731no—
231#103—1731no—
232#102—1731no—
233#102—1731no—
234#102—1731no—
235C22⋊C8C2×D17311yescharacteristic
236C22⋊C8C2×D17311yescharacteristic
237#97C2×D17311yescharacteristic
238#41—3462no—
239#41—3462no—
240#41—6921no—
241#41—6921no—
242C2×Q8—1731no—
243C2×Q8—1731no—
244C2×Q8—1731no—
245C2×D4—3461no—
246C2×D4—3461no—
247C2×D4—1731no—
248C2×D4D69211yescharacteristic
249C22×C4—3461no—
250C22×C4—3461no—
251C22×C4—3461no—
252C22×C4—3461no—
253C22×C4C22×D17311yescharacteristic
254C22×C4C22×D17311yescharacteristic
255Q16—6921no—
256Q16—6921no—
257SD16—3462no—
258M16—41no—
259C2×C8—41no—
260C2×C8C22×D17311yescharacteristic
261C2×C8C22×D17311yescharacteristic
262C2×C8C22×D17311yescharacteristic
263C2×C8C22×D17311yescharacteristic
264C4⋊C4—3461no—
265C4⋊C4—3461no—
266C4⋊C4—3461no—
267C4⋊C4—3461no—
268C4⋊C4—3461no—
269C4⋊C4—3461no—
270C4⋊C4—3461no—
271C4⋊C4—3461no—
272C4⋊C4—3461no—
273C4⋊C4—3461no—
274C4⋊C4—1731no—
275C4⋊C4—1731no—
276C4⋊C4—1731no—
277C4⋊C4—1731no—
278C4⋊C4D69211yescharacteristic
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—3461no—
284C22⋊C4—3461no—
285C22⋊C4D69211yescharacteristic
286C22⋊C4D69211yescharacteristic
287C42—3461no—
288C42—3461no—
289C42C22×D17311yescharacteristic
290C23—3461no—
291C23C2×D69211yescharacteristic
292C23C2×D69211yescharacteristic
293Q8—3461no—
294Q8—3461no—
295Q8—3461no—
296Q8—3461no—
297Q8—3461no—
298Q8—3461no—
299D4—41no—
300D4—3461no—
301D4—1732no—
302D4—6922no—
303D4—6922no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—3461no—
309C2×C4—3461no—
310C2×C4—3461no—
311C2×C4—3461no—
312C2×C4—3461no—
313C2×C4—3461no—
314C2×C4—3461no—
315C2×C4—6922no—
316C2×C4—6922no—
317C2×C4—6922no—
318C2×C4—6922no—
319C2×C4C2×D69211yescharacteristic
320C2×C4C2×D69211yescharacteristic
321C2×C4C2×D69211yescharacteristic
322C2×C4C2×D69211yescharacteristic
323C2×C4C23×D17311yescharacteristic
324C8—21no—
325C8—21no—
326C8—21no—
327C8—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—3462no—
335C22—6921no—
336C22—6921no—
337C22C22×D69211yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—6921no—
343C4—6921no—
344C4—6921no—
345C4—6921no—
346C4—6921no—
347C4—6921no—
348C4—6921no—
349C4#41×D17311yescharacteristic
350C4#1914972411yescharacteristic
351C2—41no—
352C2—41no—
353C2—6922no—
354C2#2601213611yescharacteristic
355C2#2601229111yescharacteristic
356C2#2601230811yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#435×C173C2
socleC22×C173C22×D4
prime radical#26012146C2
hypercenter#435D173
hypocenterC173#3315
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #26012181C2semidirect
3. #26011860C2semidirect
4. #26011833C2semidirect
5. #26011827C2semidirect
6. #26011792C2semidirect
7. #26011749C2semidirect
8. #26011732C2semidirect
9. #26011702C2semidirect
10. #26011655C2semidirect
11. #26011655C2semidirect
12. #26011593C2semidirect
13. #26011493C2semidirect
14. #26011488C2semidirect
15. #435×C173C2semidirect
178. C173#3315semidirect
192. #435D173semidirect

Maximal subgroups

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

subgroupindex
#260121812
#260121462
#260118602
#260118332
#260118272
#260117922
#260117492
#260117322
#260117022
#260116552
#260115932
#260114932
#260114882
#435×C1732
#3315173