Finite Group Databasebeta

← #36365452

Subgroups — #36365452

Order 22144.

386 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#26012150C211yesmaximal, characteristic
3#26011846C211yesmaximal, characteristic, complemented
4#26011833C211yesmaximal, characteristic, complemented
5#26011805C211yesmaximal, characteristic, complemented
6#26011799C211yesmaximal, characteristic, complemented
7#26011732C211yesmaximal, characteristic, complemented
8#26011720C211yesmaximal, characteristic, complemented
9#26011714C211yesmaximal, characteristic, complemented
10#26011655C211yesmaximal, characteristic, complemented
11#26011649C211yesmaximal, characteristic, complemented
12C4×Dic692C211yesmaximal, characteristic, complemented
13#26011488C211yesmaximal, characteristic, complemented
14C2×#19149604C211yesmaximal, characteristic, complemented
15C8×D4×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C22×Dic346C2211yescharacteristic
18C2×Dic692C2211yescharacteristic
19C2×Dic692C2211yescharacteristic
20C2×#14583815C2211yescharacteristic
21#19149672—21no—
22#19149671C2211yescharacteristic
23#19149669—21no—
24#19149667C2211yescharacteristic
25#19149665—21no—
26#19149660—21no—
27#19149658C2211yescharacteristic
28#19149658C2211yescharacteristic
29#19149658C2211yescharacteristic
30#19149657—21no—
31#19149653C2211yescharacteristic
32#19149651—21no—
33C4×Dic346C2211yescharacteristic
34C4×Dic346C2211yescharacteristic
35#19149639—21no—
36#19149638C2211yescharacteristic
37#19149635—21no—
38C2×C4⋊Dic173C2211yescharacteristic
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
52C22×C8×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic
54C4×D4×C173C2211yescharacteristic
55C4⋊C8×C173C2211yescharacteristic
56C22⋊C8×C173C2211yescharacteristic
57C22⋊C8×C173C2211yescharacteristic
58C4×C8×C173C2211yescharacteristic
59#14583826—22no—
60#14583826—41no—
61C2×Dic346—22no—
62C2×Dic346—21no—
63C2×Dic346—41no—
64C2×Dic346C2311yescharacteristic
65C2×Dic346C2311yescharacteristic
66C2×Dic346C2311yescharacteristic
67C2×D692C2311yescharacteristic
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C22×Dic173—21no—
71C22×Dic173—21no—
72C2×C4×D173—21no—
73Dic692—22no—
74Dic692—41no—
75#14583815—41no—
76C4⋊Dic173—21no—
77C4⋊Dic173C2311yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95C4×Dic173—21no—
96C4×Dic173—21no—
97C4×Dic173—21no—
98C2×D4×C173D411yescharacteristic
99C22×C4×C173C2311yescharacteristic
100C22×C4×C173C2311yescharacteristic
101C2×C8×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—42no—
118Dic346#4112yes—
119D692—21no—
120D692—21no—
121C22⋊D173—42no—
122C22⋊D173—42no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173—42no—
131C2×Dic173—42no—
132C4×D173—42no—
133C23×C173C2×D411yescharacteristic
134C23×C173C2×D411yescharacteristic
135D4×C173—22no—
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#13512yes—
155C22×C173C22×D411yescharacteristic
156C2×D173—41no—
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—22no—
166Dic173—41no—
167Dic173—41no—
168Dic173—41no—
169Dic173—41no—
170Dic173—41no—
171Dic173—41no—
172C2×C173—22no—
173C2×C173—41no—
174C2×C173#54611yescharacteristic
175C2×C173C2×#13511yescharacteristic
176C2×C173#57811yescharacteristic
177D173—81no—
178C173#330011yescharacteristic, complemented, Sylow
179#3300—1731nomaximal, Sylow
180#552—1731no—
181#548—1731no—
182#488—1731no—
183#485—1731no—
184#480—1731no—
185#479—1731no—
186#467—1731no—
187#465—1731no—
188#464—1731no—
189#452—1731no—
190#451—1731no—
191C4×Q16—1731no—
192C8×D4D17311yescharacteristic, complemented
193#416—1731no—
194C2×#103—1731no—
195C2×#41—1731no—
196C22×Q8—1731no—
197C2×Q16—1731no—
198C2×Q16—1731no—
199C2×SD16—1731no—
200C22×C8C2×D17311yescharacteristic
201C22×C8C2×D17311yescharacteristic
202#126—3461no—
203#125—3461no—
204#125—1731no—
205#124—3461no—
206#124—1731no—
207#123—3461no—
208C22⋊Q8—3461no—
209C22⋊Q8—1731no—
210C22⋊Q8—1731no—
211C22⋊Q8—1731no—
212#121—3461no—
213#121—1731no—
214C4×Q8—1731no—
215C4×Q8—1731no—
216C4×D4—3461no—
217C4×D4C2×D17311yescharacteristic
218#117—3461no—
219#117—1731no—
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
237C4×C8C2×D17311yescharacteristic
238#41—3462no—
239#41—6921no—
240C2×Q8—3462no—
241C2×Q8—3461no—
242C2×Q8—1731no—
243C2×Q8—1731no—
244C2×Q8—1731no—
245C2×Q8—6921no—
246C2×D4—3461no—
247C2×D4—3461no—
248C2×D4—1731no—
249C2×D4D69211yescharacteristic
250C22×C4—3461no—
251C22×C4—3461no—
252C22×C4—3461no—
253C22×C4C22×D17311yescharacteristic
254C22×C4C22×D17311yescharacteristic
255Q16—3462no—
256Q16—6921no—
257SD16—6921no—
258C2×C8—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—1731no—
273C4⋊C4—1731no—
274C4⋊C4—1731no—
275C4⋊C4—1731no—
276C4⋊C4D69211yescharacteristic
277C22⋊C4—3461no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—3461no—
284C22⋊C4D69211yescharacteristic
285C22⋊C4D69211yescharacteristic
286C42—3461no—
287C42—3461no—
288C42—3461no—
289C42C22×D17311yescharacteristic
290C23—3461no—
291C23C2×D69211yescharacteristic
292C23C2×D69211yescharacteristic
293Q8—3461no—
294Q8—3461no—
295Q8—3461no—
296Q8—3461no—
297Q8—3461no—
298Q8—1732no—
299Q8—6922no—
300D4—22no—
301D4—3461no—
302D4—3461no—
303D4—6922no—
304D4—6922no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—3461no—
310C2×C4—3461no—
311C2×C4—3461no—
312C2×C4—3461no—
313C2×C4—3461no—
314C2×C4—3461no—
315C2×C4—3461no—
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—6921no—
334C22—6921no—
335C22—6921no—
336C22#1914970112yes—
337C22C22×D69211yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—3462no—
343C4—6921no—
344C4—6921no—
345C4—6921no—
346C4—6921no—
347C4—6921no—
348C4—6921no—
349C4#41×D17311yescharacteristic
350C4#1914972411yescharacteristic
351C2—22no—
352C2—41no—
353C2—13841no—
354C2#2601213611yescharacteristic
355C2C2×#1914970111yescharacteristic
356C2#2601230811yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC8×D4×C173C2
socleC22×C173C22×D4
prime radical#26012150C2
hypercenterC8×D4D173
hypocenterC173#3300
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #26012181C2semidirect
3. #26011846C2semidirect
4. #26011833C2semidirect
5. #26011805C2semidirect
6. #26011799C2semidirect
7. #26011732C2semidirect
8. #26011720C2semidirect
9. #26011714C2semidirect
10. #26011655C2semidirect
11. #26011649C2semidirect
12. C4×Dic692C2semidirect
13. #26011488C2semidirect
14. C2×#19149604C2semidirect
15. C8×D4×C173C2semidirect
178. C173#3300semidirect
192. C8×D4D173semidirect

Maximal subgroups

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

subgroupindex
#260121812
#260121502
#260118462
#260118332
#260118052
#260117992
#260117322
#260117202
#260117142
#260116552
#260116492
C4×Dic6922
#260114882
C2×#191496042
C8×D4×C1732
#3300173