Finite Group Databasebeta

← #36362731

Subgroups — #36362731

Order 22144.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012019C211yesmaximal, characteristic
2#26011620C211yesmaximal, characteristic, complemented
3#26011618C211yesmaximal, characteristic, complemented
4#26011568C211yesmaximal, characteristic, complemented
5#26011567C211yesmaximal, characteristic, complemented
6#26011562C211yesmaximal, characteristic, complemented
7#26011562C211yesmaximal, characteristic, complemented
8#26011561C211yesmaximal, characteristic, complemented
9#26011560C211yesmaximal, characteristic, complemented
10#26011552C211yesmaximal, characteristic, complemented
11#26011551C211yesmaximal, characteristic, complemented
12#26011547C211yesmaximal, characteristic, complemented
13#26011516C211yesmaximal, characteristic, complemented
14#26011513C211yesmaximal, characteristic, complemented
15#424×C173C211yesmaximal, characteristic, complemented
16C2×#14583809C2211yescharacteristic
17C2×#14583809C2211yescharacteristic
18C2×#14583809C2211yescharacteristic
19C2×C8×D173C2211yescharacteristic
20#19149674C411yescharacteristic
21#19149672C411yescharacteristic
22#19149669C411yescharacteristic
23#19149665C411yescharacteristic
24#19149664C411yescharacteristic
25#19149661C411yescharacteristic
26#19149657C411yescharacteristic
27#19149652C411yescharacteristic
28C4×Dic346C2211yescharacteristic
29C4×D692C2211yescharacteristic
30C4×C22⋊D173C2211yescharacteristic
31C4×C22⋊D173C2211yescharacteristic
32#19149636C2211yescharacteristic
33C42×D173C2211yescharacteristic
34C4⋊#13063436C2211yescharacteristic
35C4⋊#13063436C2211yescharacteristic
36#19149608C2211yescharacteristic
37#19149608C2211yescharacteristic
38#19149608C2211yescharacteristic
39#19149608C2211yescharacteristic
40C22⋊#13063436C2211yescharacteristic
41C22⋊#13063436C2211yescharacteristic
42#19149591C2211yescharacteristic
43#19149591C2211yescharacteristic
44#19149591C2211yescharacteristic
45#19149591C2211yescharacteristic
46#19149590C2211yescharacteristic
47#19149589C2211yescharacteristic
48#19149589C2211yescharacteristic
49#19149589C2211yescharacteristic
50C4×#13063436C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×M16×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic
54#117×C173C2211yescharacteristic
55C4⋊C8×C173C2211yescharacteristic
56C4⋊C8×C173C2211yescharacteristic
57C4⋊C8×C173C2211yescharacteristic
58C4⋊C8×C173C2211yescharacteristic
59C2×Dic346C2×C411yescharacteristic
60C2×D692C2×C411yescharacteristic
61C2×C22⋊D173C2×C411yescharacteristic
62C2×C22⋊D173C2×C411yescharacteristic
63C2×C4×D173C2×C411yescharacteristic
64C2×C4×D173C2×C411yescharacteristic
65C2×C4×D173C2311yescharacteristic
66C2×C4×D173C2311yescharacteristic
67#14583809—41no—
68#14583809—41no—
69#14583809—41no—
70C2×#13063436C2311yescharacteristic
71C2×#13063436C2311yescharacteristic
72C2×#13063436C2311yescharacteristic
73C2×#13063436C2311yescharacteristic
74C8×D173—41no—
75C4⋊Dic173C2×C411yescharacteristic
76C4⋊Dic173C2×C411yescharacteristic
77#14583805C2×C411yescharacteristic
78#14583805C2×C411yescharacteristic
79#14583805C2×C411yescharacteristic
80#14583805C2×C411yescharacteristic
81#14583805C2×C411yescharacteristic
82#14583805C2×C411yescharacteristic
83C22⋊Dic173C2×C411yescharacteristic
84C22⋊Dic173C2×C411yescharacteristic
85#14583803C2×C411yescharacteristic
86#14583803C2×C411yescharacteristic
87#14583803C2×C411yescharacteristic
88#14583803C2×C411yescharacteristic
89#14583803C2×C411yescharacteristic
90#14583803C2×C411yescharacteristic
91C4×Dic173C2×C411yescharacteristic
92C4×Dic173C2×C411yescharacteristic
93C4×Dic173C2311yescharacteristic
94C4×Dic173C2311yescharacteristic
95C22×C4×C173C2311yescharacteristic
96M16×C173—41no—
97C2×C8×C173—41no—
98C2×C8×C173C2311yescharacteristic
99C2×C8×C173C2311yescharacteristic
100C2×C8×C173C2311yescharacteristic
101C2×C8×C173C2311yescharacteristic
102C4⋊C4×C173C2×C411yescharacteristic
103C4⋊C4×C173C2×C411yescharacteristic
104C22⋊C4×C173C2×C411yescharacteristic
105C22⋊C4×C173C2×C411yescharacteristic
106C42×C173C2311yescharacteristic
107C42×C173C2311yescharacteristic
108C22×D173C22×C411yescharacteristic
109C22×D173C22×C411yescharacteristic
110Dic346—41no—
111D692—41no—
112C22⋊D173—41no—
113C22⋊D173—41no—
114C2×Dic173C22×C411yescharacteristic
115C2×Dic173C22×C411yescharacteristic
116C2×Dic173C22×C411yescharacteristic
117C2×Dic173C22×C411yescharacteristic
118C2×Dic173C22×C411yescharacteristic
119C2×Dic173C22×C411yescharacteristic
120C4×D173—21no—
121C4×D173—21no—
122C4×D173—21no—
123C4×D173—21no—
124C4×D173—41no—
125C4×D173—41no—
126#13063436—21no—
127#13063436—21no—
128#13063436—21no—
129#13063436—21no—
130C23×C173C22×C411yescharacteristic
131C2×C4×C173—21no—
132C2×C4×C173—21no—
133C2×C4×C173C22×C411yescharacteristic
134C2×C4×C173C22×C411yescharacteristic
135C2×C4×C173C22×C411yescharacteristic
136C2×C4×C173C22×C411yescharacteristic
137C2×C4×C173C22×C411yescharacteristic
138C2×C4×C173C2411yescharacteristic
139C8×C173—21no—
140C8×C173—21no—
141C8×C173—21no—
142C8×C173—21no—
143C22×C173—21no—
144C22×C173—21no—
145C22×C173—21no—
146C22×C173C23×C411yescharacteristic
147C2×D173—21no—
148C2×D173—21no—
149C2×D173—21no—
150C2×D173—21no—
151C2×D173—21no—
152C2×D173—21no—
153C4×C173—21no—
154C4×C173—21no—
155C4×C173—21no—
156C4×C173—21no—
157C4×C173—21no—
158C4×C173H(5,𝔽2)11yescharacteristic
159C4×C1732−1+411yescharacteristic
160Dic173—21no—
161Dic173—21no—
162Dic173—21no—
163Dic173—21no—
164Dic173—21no—
165Dic173—21no—
166C2×C173—41no—
167C2×C173#52011yescharacteristic
168C2×C173#56811yescharacteristic
169C2×C173#56811yescharacteristic
170D173—41no—
171D173—41no—
172C173#299511yescharacteristic, complemented, Sylow
173#2995—1731nomaximal, Sylow
174#529—1731no—
175#446—1731no—
176#446—1731no—
177#436—1731no—
178#436—1731no—
179#435—1731no—
180#435—1731no—
181#435—1731no—
182#435—1731no—
183#433—1731no—
184#433—1731no—
185#432—1731no—
186#424—1731no—
187#424D17311yescharacteristic, complemented
188#423—1731no—
189C2×M16—1731no—
190C2×M16—1731no—
191C2×M16—1731no—
192C2×M16C2×D17311yescharacteristic
193C22×C8—1731no—
194C22×C8C2×D17311yescharacteristic
195#126—1731no—
196#126—1731no—
197#125—1731no—
198#124—1731no—
199#123—1731no—
200#123—1731no—
201C22⋊Q8—1731no—
202#121—1731no—
203C4×Q8—1731no—
204C4×D4—1731no—
205C4×D4—1731no—
206C4×D4—1731no—
207#117—1731no—
208#117C2×D17311yescharacteristic
209C2×C42—1731no—
210C4⋊C8—1731no—
211C4⋊C8—1731no—
212C4⋊C8—1731no—
213C4⋊C8—1731no—
214C4⋊C8—1731no—
215C4⋊C8—1731no—
216C4⋊C8C2×D17311yescharacteristic
217C4⋊C8C2×D17311yescharacteristic
218C4⋊C8C2×D17311yescharacteristic
219C4⋊C8C2×D17311yescharacteristic
220C22⋊C8—1731no—
221C22⋊C8—1731no—
222C22⋊C8—1731no—
223C22⋊C8—1731no—
224C22⋊C8—1731no—
225C22⋊C8—1731no—
226#97—1731no—
227#97—1731no—
228#97—1731no—
229#97—1731no—
230C4×C8—1731no—
231C4×C8—1731no—
232C2×Q8—1731no—
233C2×D4—1731no—
234C2×D4—1731no—
235C2×D4—1731no—
236C22×C4—1731no—
237C22×C4—1731no—
238C22×C4—1731no—
239C22×C4—1731no—
240C22×C4C22×D17311yescharacteristic
241M16—41no—
242M16—6921no—
243M16—6921no—
244M16—6921no—
245C2×C8—41no—
246C2×C8—1731no—
247C2×C8—1731no—
248C2×C8—1731no—
249C2×C8—1731no—
250C2×C8—6921no—
251C2×C8C22×D17311yescharacteristic
252C2×C8C22×D17311yescharacteristic
253C2×C8C22×D17311yescharacteristic
254C2×C8C22×D17311yescharacteristic
255C4⋊C4—1731no—
256C4⋊C4—1731no—
257C4⋊C4—1731no—
258C4⋊C4—1731no—
259C4⋊C4—1731no—
260C4⋊C4—1731no—
261C4⋊C4—1731no—
262C4⋊C4—1731no—
263C4⋊C4C4×D17311yescharacteristic
264C4⋊C4C4×D17311yescharacteristic
265C22⋊C4—1731no—
266C22⋊C4—1731no—
267C22⋊C4—1731no—
268C22⋊C4—1731no—
269C22⋊C4—1731no—
270C22⋊C4—1731no—
271C22⋊C4—1731no—
272C22⋊C4—1731no—
273C22⋊C4C4×D17311yescharacteristic
274C22⋊C4C4×D17311yescharacteristic
275C42—1731no—
276C42—1731no—
277C42—1731no—
278C42—1731no—
279C42C22×D17311yescharacteristic
280C42C22×D17311yescharacteristic
281C23—1731no—
282C23—1731no—
283C23C2×C4×D17311yescharacteristic
284Q8—6921no—
285D4—6921no—
286D4—6921no—
287D4—6921no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—3461no—
291C2×C4—3461no—
292C2×C4—3461no—
293C2×C4—3461no—
294C2×C4—1731no—
295C2×C4—1731no—
296C2×C4—1731no—
297C2×C4—1731no—
298C2×C4—1731no—
299C2×C4—1731no—
300C2×C4—6921no—
301C2×C4—6921no—
302C2×C4C2×C4×D17311yescharacteristic
303C2×C4C2×C4×D17311yescharacteristic
304C2×C4C2×C4×D17311yescharacteristic
305C2×C4C2×C4×D17311yescharacteristic
306C2×C4C2×C4×D17311yescharacteristic
307C2×C4C23×D17311yescharacteristic
308C8—21no—
309C8—21no—
310C8—21no—
311C8—21no—
312C8—3461no—
313C8—3461no—
314C8—3461no—
315C8—3461no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—3461no—
320C22—3461no—
321C22—3461no—
322C22—3461no—
323C22—3461no—
324C22—3461no—
325C22C22×C4×D17311yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—3461no—
332C4—3461no—
333C4—3461no—
334C4—3461no—
335C4—3461no—
336C4—3461no—
337C4#1914972411yescharacteristic
338C4#1914972711yescharacteristic
339C2—41no—
340C2—6921no—
341C2—6921no—
342C2#2601197211yescharacteristic
343C2#2601225611yescharacteristic
344C2#2601225611yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D173
commutator, socleC22×C173C23×C4
Fitting#424×C173C2
prime radical#19149652C4
cosocleC2×C4×C173C24
hypercenter#424D173
hypocenterC173#2995
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ #424

Complements

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

NH ≅ G/Ntype
2. #26011620C2semidirect
3. #26011618C2semidirect
4. #26011568C2semidirect
5. #26011567C2semidirect
6. #26011562C2semidirect
7. #26011562C2semidirect
8. #26011561C2semidirect
9. #26011560C2semidirect
10. #26011552C2semidirect
11. #26011551C2semidirect
12. #26011547C2semidirect
13. #26011516C2semidirect
14. #26011513C2semidirect
15. #424×C173C2semidirect
172. C173#2995semidirect
187. #424D173semidirect

Maximal subgroups

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

subgroupindex
#260120192
#260116202
#260116182
#260115682
#260115672
#260115622
#260115612
#260115602
#260115522
#260115512
#260115472
#260115162
#260115132
#424×C1732
#2995173