Finite Group Databasebeta

← #36370518

Subgroups — #36370518

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#26012543C211yesmaximal, characteristic, complemented
2#26012541C211yesmaximal, characteristic, complemented
3#26012525C211yesmaximal, characteristic, complemented
4#26012524C211yesmaximal, characteristic, complemented
5#26012523C211yesmaximal, characteristic, complemented
6#26012521C211yesmaximal, characteristic, complemented
7#26012520C211yesmaximal, characteristic, complemented
8#26012520C211yesmaximal, characteristic, complemented
9#26012516C211yesmaximal, characteristic, complemented
10#26012515C211yesmaximal, characteristic, complemented
11#26012514C211yesmaximal, characteristic, complemented
12#26012498C211yesmaximal, characteristic, complemented
13#26012497C211yesmaximal, characteristic, complemented
14#26012495C211yesmaximal, characteristic, complemented
15#26012024C211yesmaximal, characteristic
16C2×#14583836C2211yescharacteristic
17C2×#14583836C2211yescharacteristic
18C2×#14583836C2211yescharacteristic
19C2×#14583835C2211yescharacteristic
20C2×D173⋊C8C2211yescharacteristic
21C22×#13063443C2211yescharacteristic
22#19149746C2211yescharacteristic
23#19149746C2211yescharacteristic
24#19149746C2211yescharacteristic
25#19149746C2211yescharacteristic
26C4⋊#13063443C2211yescharacteristic
27C4⋊#13063443C2211yescharacteristic
28C4⋊#13063443C2211yescharacteristic
29C4⋊#13063443C2211yescharacteristic
30C4⋊#13063443C2211yescharacteristic
31C4⋊#13063443C2211yescharacteristic
32#19149736C2211yescharacteristic
33#19149736C2211yescharacteristic
34#19149736C2211yescharacteristic
35#19149736C2211yescharacteristic
36C22⋊#13063443C2211yescharacteristic
37C22⋊#13063443C2211yescharacteristic
38#19149734C2211yescharacteristic
39#19149734C2211yescharacteristic
40#19149734C2211yescharacteristic
41#19149734C2211yescharacteristic
42C4×#13063443C2211yescharacteristic
43C4×#13063443C2211yescharacteristic
44#19149673C411yescharacteristic
45#19149669C411yescharacteristic
46#19149665C411yescharacteristic
47#19149664C411yescharacteristic
48#19149661C411yescharacteristic
49#19149657C411yescharacteristic
50#19149652C411yescharacteristic
51#19149644C2211yescharacteristic
52#19149641C2211yescharacteristic
53#19149639C2211yescharacteristic
54#19149639C2211yescharacteristic
55#19149637C2211yescharacteristic
56#19149635C2211yescharacteristic
57C42×D173C2211yescharacteristic
58#126×C173C411yescharacteristic
59#14583836—41no—
60#14583836—41no—
61#14583836—41no—
62#14583835—41no—
63D173⋊C8—41no—
64C2×#13063443—41no—
65C2×#13063443C2311yescharacteristic
66C2×#13063443C2311yescharacteristic
67C2×#13063443C2311yescharacteristic
68C2×#13063443C2311yescharacteristic
69C2×#13063443C2311yescharacteristic
70C2×#13063443C2311yescharacteristic
71C2×#13063443C2311yescharacteristic
72C2×#13063443C2311yescharacteristic
73C2×Dic346C2×C411yescharacteristic
74C2×D692C2×C411yescharacteristic
75C2×C22⋊D173C2×C411yescharacteristic
76C2×C22⋊D173C2×C411yescharacteristic
77C22×Dic173C2311yescharacteristic
78C2×C4×D173C2×C411yescharacteristic
79C2×C4×D173C2×C411yescharacteristic
80C2×C4×D173C2311yescharacteristic
81C2×C4×D173C2311yescharacteristic
82C4⋊Dic173C2×C411yescharacteristic
83#14583805C2×C411yescharacteristic
84#14583805C2×C411yescharacteristic
85#14583805C2×C411yescharacteristic
86#14583805C2×C411yescharacteristic
87#14583805C2×C411yescharacteristic
88#14583805C2×C411yescharacteristic
89C22⋊Dic173C2×C411yescharacteristic
90#14583803C2×C411yescharacteristic
91#14583803C2×C411yescharacteristic
92#14583803C2×C411yescharacteristic
93#14583803C2×C411yescharacteristic
94#14583803C2×C411yescharacteristic
95#14583803C2×C411yescharacteristic
96C4×Dic173C2×C411yescharacteristic
97C4×Dic173C2311yescharacteristic
98C4×Dic173C2311yescharacteristic
99C4×Dic173C2311yescharacteristic
100C4×Dic173C2311yescharacteristic
101C4⋊C4×C173C2×C411yescharacteristic
102C4⋊C4×C173C2×C411yescharacteristic
103C4⋊C4×C173C2×C411yescharacteristic
104C22⋊C4×C173C2×C411yescharacteristic
105C22⋊C4×C173C2×C411yescharacteristic
106C22⋊C4×C173C2×C411yescharacteristic
107C42×C173C2×C411yescharacteristic
108#13063443—21no—
109#13063443—21no—
110#13063443—21no—
111#13063443—21no—
112#13063443—21no—
113#13063443—21no—
114#13063443—21no—
115#13063443—21no—
116C22×D173C22×C411yescharacteristic
117C22×D173C22×C411yescharacteristic
118Dic346—41no—
119D692—41no—
120C22⋊D173—41no—
121C22⋊D173—41no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173C22×C411yescharacteristic
125C2×Dic173C22×C411yescharacteristic
126C2×Dic173C22×C411yescharacteristic
127C2×Dic173C22×C411yescharacteristic
128C2×Dic173C22×C411yescharacteristic
129C2×Dic173C2411yescharacteristic
130C4×D173—21no—
131C4×D173—21no—
132C4×D173—21no—
133C4×D173—21no—
134C4×D173—41no—
135C4×D173—41no—
136C23×C173C22×C411yescharacteristic
137C2×C4×C173C22×C411yescharacteristic
138C2×C4×C173C22×C411yescharacteristic
139C2×C4×C173C22×C411yescharacteristic
140C2×C4×C173C22×C411yescharacteristic
141C2×C4×C173C22×C411yescharacteristic
142C2×C4×C173C22×C411yescharacteristic
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×C173—21no—
159Dic173—21no—
160Dic173—21no—
161Dic173—21no—
162Dic173—21no—
163Dic173—21no—
164Dic173H(5,𝔽2)11yescharacteristic
165Dic1732−1+411yescharacteristic
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#424—1731no—
188#423—1731no—
189C2×M16—1731no—
190C2×M16—1731no—
191C2×M16—1731no—
192C2×M16—1731no—
193C22×C8—1731no—
194C22×C8—1731no—
195#126—1731no—
196#126#1062192411yescharacteristic
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#117—1731no—
209C2×C42—1731no—
210C4⋊C8—1731no—
211C4⋊C8—1731no—
212C4⋊C8—1731no—
213C4⋊C8—1731no—
214C4⋊C8—1731no—
215C4⋊C8—1731no—
216C4⋊C8—1731no—
217C4⋊C8—1731no—
218C4⋊C8—1731no—
219C4⋊C8—1731no—
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×C4—1731no—
241M16—6921no—
242M16—6921no—
243M16—6921no—
244M16—6921no—
245C2×C8—1731no—
246C2×C8—1731no—
247C2×C8—1731no—
248C2×C8—1731no—
249C2×C8—1731no—
250C2×C8—1731no—
251C2×C8—1731no—
252C2×C8—1731no—
253C2×C8—6921no—
254C2×C8—6921no—
255C4⋊C4—1731no—
256C4⋊C4—1731no—
257C4⋊C4—1731no—
258C4⋊C4—1731no—
259C4⋊C4—1731no—
260C4⋊C4—1731no—
261C4⋊C4—1731no—
262C4⋊C4C2×#1062192411yescharacteristic
263C4⋊C4C2×#1062192411yescharacteristic
264C4⋊C4C2×#1062192411yescharacteristic
265C22⋊C4—1731no—
266C22⋊C4—1731no—
267C22⋊C4—1731no—
268C22⋊C4—1731no—
269C22⋊C4—1731no—
270C22⋊C4—1731no—
271C22⋊C4—1731no—
272C22⋊C4C2×#1062192411yescharacteristic
273C22⋊C4C2×#1062192411yescharacteristic
274C22⋊C4C2×#1062192411yescharacteristic
275C42—1731no—
276C42—1731no—
277C42—1731no—
278C42—1731no—
279C42—1731no—
280C42C2×#1062192411yescharacteristic
281C23—1731no—
282C23—1731no—
283C23C22×#1062192411yescharacteristic
284Q8—6921no—
285D4—6921no—
286D4—6921no—
287D4—6921no—
288C2×C4—3461no—
289C2×C4—3461no—
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×C4C22×#1062192411yescharacteristic
303C2×C4C22×#1062192411yescharacteristic
304C2×C4C22×#1062192411yescharacteristic
305C2×C4C22×#1062192411yescharacteristic
306C2×C4C22×#1062192411yescharacteristic
307C2×C4C22×#1062192411yescharacteristic
308C8—3461no—
309C8—3461no—
310C8—3461no—
311C8—3461no—
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—
325C22C23×#1062192411yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—3461no—
333C4—3461no—
334C4—3461no—
335C4—3461no—
336C4—3461no—
337C4—1731no—
338C4—1731no—
339C2—41no—
340C2—6921no—
341C2—6921no—
342C2#2601256011yescharacteristic
343C2#2601257011yescharacteristic
344C2#2601257011yescharacteristic
subgroupG/H
center, FrattiniC22C23×#10621924
commutator, socleC22×C173C23×C4
Fitting#126×C173C4
prime radical#19149652C4
cosocleC2×Dic173C24
hypercenter#126#10621924
hypocenterC173#2995
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. #26012543C2semidirect
2. #26012541C2semidirect
3. #26012525C2semidirect
4. #26012524C2semidirect
5. #26012523C2semidirect
6. #26012521C2semidirect
7. #26012520C2semidirect
8. #26012520C2semidirect
9. #26012516C2semidirect
10. #26012515C2semidirect
11. #26012514C2semidirect
12. #26012498C2semidirect
13. #26012497C2semidirect
14. #26012495C2semidirect
172. C173#2995semidirect

Maximal subgroups

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

subgroupindex
#260125432
#260125412
#260125252
#260125242
#260125232
#260125212
#260125202
#260125162
#260125152
#260125142
#260124982
#260124972
#260124952
#260120242
#2995173