Finite Group Databasebeta

← #36367179

Subgroups — #36367179

Order 22144.

1094 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012217C211yesmaximal, characteristic, complemented
2#26012210C213yesmaximal, complemented
3#26012181C213yesmaximal, complemented
4#26012175C213yesmaximal, complemented
5Q8×C22⋊D173C211yesmaximal, characteristic, complemented
6#26012148C211yesmaximal, characteristic, complemented
7#26012141C213yesmaximal, complemented
8#26012043C213yesmaximal, complemented
9#26012022C213yesmaximal, complemented
10#26012020C213yesmaximal, complemented
11#41×Dic173C211yesmaximal, characteristic, complemented
12C4×#14583825C213yesmaximal, complemented
13C2×Q8×Dic173C211yesmaximal, characteristic, complemented
14C4×Q8×D173C211yesmaximal, characteristic, complemented
15#548×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2213yes—
17C2×Q8×D173C2211yescharacteristic
18#19149678C2213yes—
19#19149677C2213yes—
20#19149672C2216yes—
21#19149672C2216yes—
22#19149671C2213yes—
23#19149670C2213yes—
24#19149669C2216yes—
25#19149666C2213yes—
26#19149661C2216yes—
27#19149660C2216yes—
28#19149658C2213yes—
29#19149658C2213yes—
30#19149656C2216yes—
31#19149655C2213yes—
32#19149650C2213yes—
33Q8×Dic173C2213yes—
34Q8×Dic173C2211yescharacteristic
35Q8×Dic173C2211yescharacteristic
36Q8×Dic173C2213yes—
37Q8×Dic173C2211yescharacteristic
38C4×Dic346C2213yes—
39C4×Dic346C2213yes—
40#19149644C2213yes—
41#19149644C2213yes—
42D4×Dic173C2213yes—
43D4×Dic173C2213yes—
44D4×Dic173C2213yes—
45C4×C22⋊D173C2213yes—
46C4×C22⋊D173C2213yes—
47#19149639C2211yescharacteristic
48#19149639C2213yes—
49#19149638C2213yes—
50#19149637C2213yes—
51#19149635C2213yes—
52#19149635C2213yes—
53#19149635C2216yes—
54C2×C4⋊Dic173C2213yes—
55C4⋊C4×D173C2213yes—
56C2×C4×Dic173C2213yes—
57C2×C4×Dic173C2213yes—
58C42×D173C2213yes—
59C2×#41×C173C2211yescharacteristic
60C22×Q8×C173C2211yescharacteristic
61#124×C173C2213yes—
62C22⋊Q8×C173C2213yes—
63#121×C173C2213yes—
64C4×Q8×C173C2211yescharacteristic
65C4×D4×C173C2213yes—
66#14583825—212no—
67C2×Dic346C2313yes—
68C2×Dic346C2313yes—
69Q8×D173—24no—
70C2×C22⋊D173C2313yes—
71C2×C22⋊D173C2311yescharacteristic
72C22×Dic173C2311yescharacteristic
73C22×Dic173C2313yes—
74C22×Dic173C2313yes—
75C22×Dic173C2313yes—
76C2×C4×D173C2311yescharacteristic
77C2×C4×D173C2313yes—
78C2×C4×D173C2313yes—
79C4⋊Dic173C2313yes—
80C4⋊Dic173C2313yes—
81C4⋊Dic173C2316yes—
82C4⋊Dic173C2313yes—
83C4⋊Dic173C2313yes—
84#14583805C2316yes—
85#14583805C2311yescharacteristic
86#14583805C2313yes—
87#14583805C2313yes—
88#14583805C2313yes—
89#14583805C2316yes—
90#14583805C2316yes—
91C22⋊Dic173C2313yes—
92C22⋊Dic173C2313yes—
93C22⋊Dic173C2316yes—
94C22⋊Dic173C2316yes—
95#14583803C2311yescharacteristic
96#14583803C2313yes—
97C4×Dic173C2313yes—
98C4×Dic173C2313yes—
99C4×Dic173C2313yes—
100C4×Dic173C2311yescharacteristic
101C4×Dic173C2313yes—
102C4×Dic173C2313yes—
103C4×Dic173C2313yes—
104C4×Dic173C2313yes—
105C4×Dic173C2313yes—
106#41×C173—24no—
107C2×Q8×C173—24no—
108C2×Q8×C173C2313yes—
109C2×Q8×C173C2311yescharacteristic
110C2×D4×C173C2313yes—
111C2×D4×C173C2313yes—
112C22×C4×C173C2313yes—
113C22×C4×C173C2313yes—
114C4⋊C4×C173C2313yes—
115C4⋊C4×C173C2313yes—
116C22⋊C4×C173C2311yescharacteristic
117C22⋊C4×C173C2313yes—
118C22⋊C4×C173C2316yes—
119C42×C173C2313yes—
120C22×D173C2411yescharacteristic
121Dic346—26no—
122Dic346—26no—
123C22⋊D173—26no—
124C22⋊D173#4114yescomplemented
125C2×Dic173—26no—
126C2×Dic173—26no—
127C2×Dic173—22no—
128C2×Dic173—26no—
129C2×Dic173C2413yes—
130C2×Dic173C2413yes—
131C2×Dic173C2413yes—
132C2×Dic173C2411yescharacteristic
133C2×Dic173C2413yes—
134C2×Dic173C2411yescharacteristic
135C2×Dic173C2411yescharacteristic
136C4×D173—26no—
137C4×D173—26no—
138C4×D173—22no—
139C23×C173C2413yes—
140C23×C173C2411yescharacteristic
141Q8×C173—26no—
142Q8×C173#4114yescomplemented
143D4×C173—26no—
144D4×C173—26no—
145C2×C4×C173—26no—
146C2×C4×C173—26no—
147C2×C4×C173C2413yes—
148C2×C4×C173C2411yescharacteristic
149C2×C4×C173C2413yes—
150C2×C4×C173C2411yescharacteristic
151C2×C4×C173C2413yes—
152C22×C173—21no—
153C22×C173—21no—
154C22×C173—23no—
155C22×C173—23no—
156C22×C173—23no—
157C22×C173C2×#4112yes—
158C22×C173C2511yescharacteristic
159C2×D173—21no—
160C2×D173—21no—
161C2×D173C2×#4112yes—
162C4×C173—21no—
163C4×C173—23no—
164C4×C173—23no—
165C4×C173—21no—
166C4×C173C2×#4116yes—
167Dic173—23no—
168Dic173—23no—
169Dic173—23no—
170Dic173—21no—
171Dic173—23no—
172Dic173C2×#4112yes—
173Dic1732−1+412yes—
174C2×C173—22no—
175C2×C173—43no—
176C2×C173C22×#4111yescharacteristic
177C2×C173C22×#4111yescharacteristic
178C2×C173C2×2−1+411yescharacteristic
179D173—22no—
180C173#348211yescharacteristic, complemented, Sylow
181#3482—1731nomaximal, Sylow
182#557—1731no—
183#556—1733no—
184#552—1733no—
185#552—1733no—
186D4×Q8—1731no—
187#548—1731no—
188#548D17311yescharacteristic, complemented
189#547—1733no—
190#533—1733no—
191#529—1733no—
192#529—1733no—
193C4×#41—1731no—
194C4×#41—1733no—
195C2×C4×Q8—1731no—
196C2×C4×Q8—1731no—
197C2×#41—1733no—
198C2×#41C2×D17311yescharacteristic
199C22×Q8—1731no—
200C22×Q8C2×D17311yescharacteristic
201C4⋊Q8—1733no—
202C4⋊Q8—1733no—
203#126—1736no—
204#126—1736no—
205#125—1733no—
206#125—1736no—
207#125—1733no—
208#124—1733no—
209#124C2×D17313yes—
210#123—1736no—
211#123—1736no—
212C22⋊Q8—1733no—
213C22⋊Q8—1733no—
214C22⋊Q8—1733no—
215C22⋊Q8—1736no—
216C22⋊Q8C2×D17313yes—
217#121—1733no—
218#121C2×D17313yes—
219C4×Q8—1731no—
220C4×Q8—1733no—
221C4×Q8—1731no—
222C4×Q8—1733no—
223C4×Q8—1731no—
224C4×Q8—1733no—
225C4×Q8—1733no—
226C4×Q8—1733no—
227C4×Q8—1733no—
228C4×Q8C2×D17311yescharacteristic
229C4×D4—1733no—
230C4×D4—1733no—
231C4×D4—1733no—
232C4×D4—1733no—
233C4×D4—1733no—
234C4×D4—1731no—
235C4×D4—1733no—
236C4×D4C2×D17313yes—
237#117—1733no—
238#117—1733no—
239#117—1733no—
240#117—1733no—
241#117—1736no—
242C2×C4⋊C4—1733no—
243C2×C4⋊C4—1733no—
244C2×C42—1733no—
245C2×C42—1733no—
246C2×C42—1733no—
247#41—24no—
248#41—34612no—
249C2×Q8—24no—
250C2×Q8—3464no—
251C2×Q8—1733no—
252C2×Q8—1733no—
253C2×Q8C22×D17313yes—
254C2×Q8C22×D17311yescharacteristic
255C2×D4—1733no—
256C2×D4—1731no—
257C2×D4C22×D17313yes—
258C2×D4C22×D17313yes—
259C22×C4—1731no—
260C22×C4—1733no—
261C22×C4—1733no—
262C22×C4—1733no—
263C22×C4—1731no—
264C22×C4—1733no—
265C22×C4—1733no—
266C22×C4C22×D17313yes—
267C22×C4C22×D17313yes—
268C4⋊C4—1733no—
269C4⋊C4—1733no—
270C4⋊C4—1736no—
271C4⋊C4—1736no—
272C4⋊C4—1731no—
273C4⋊C4—1733no—
274C4⋊C4—1733no—
275C4⋊C4—1733no—
276C4⋊C4—1733no—
277C4⋊C4—1733no—
278C4⋊C4—1736no—
279C4⋊C4—1736no—
280C4⋊C4C22×D17313yes—
281C4⋊C4C22×D17313yes—
282C22⋊C4—1733no—
283C22⋊C4—1736no—
284C22⋊C4—1733no—
285C22⋊C4—1736no—
286C22⋊C4—1731no—
287C22⋊C4—1733no—
288C22⋊C4C22×D17311yescharacteristic
289C22⋊C4C22×D17313yes—
290C22⋊C4C22×D17316yes—
291C42—1733no—
292C42—1731no—
293C42—1733no—
294C42—1733no—
295C42—1733no—
296C42—1733no—
297C42—1733no—
298C42—1733no—
299C42—1733no—
300C42C22×D17313yes—
301C23—1731no—
302C23C23×D17313yes—
303C23C23×D17311yescharacteristic
304Q8—26no—
305Q8—3466no—
306Q8—3466no—
307Q8#1458382514yescomplemented
308D4—26no—
309D4—26no—
310D4—3466no—
311D4—1734no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—3466no—
315C2×C4—3462no—
316C2×C4—3466no—
317C2×C4—3462no—
318C2×C4—3466no—
319C2×C4—3466no—
320C2×C4—3466no—
321C2×C4—1731no—
322C2×C4—1733no—
323C2×C4—1733no—
324C2×C4—1733no—
325C2×C4—1733no—
326C2×C4—1731no—
327C2×C4—1731no—
328C2×C4C23×D17311yescharacteristic
329C2×C4C23×D17311yescharacteristic
330C2×C4C23×D17313yes—
331C2×C4C23×D17313yes—
332C2×C4C23×D17313yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—3461no—
339C22—3461no—
340C22—1732no—
341C22#41×D17312yes—
342C22C24×D17311yescharacteristic
343C4—23no—
344C4—21no—
345C4—23no—
346C4—21no—
347C4—3463no—
348C4—3463no—
349C4—3463no—
350C4—3461no—
351C4—3463no—
352C4—1732no—
353C4—1732no—
354C4C2×#1458382516yes—
355C2—22no—
356C2—43no—
357C2—3462no—
358C2C2×#41×D17311yescharacteristic
359C2C22×#1458382511yescharacteristic
360C22−1+4×D17311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D173
commutator, socle, cosocleC22×C173C25
Fitting#548×C173C2
hypercenter#548D173
hypocenterC173#3482
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #26012217C2semidirect
2. #26012210C2semidirect
3. #26012181C2semidirect
4. #26012175C2semidirect
5. Q8×C22⋊D173C2semidirect
6. #26012148C2semidirect
7. #26012141C2semidirect
8. #26012043C2semidirect
9. #26012022C2semidirect
10. #26012020C2semidirect
11. #41×Dic173C2semidirect
12. C4×#14583825C2semidirect
13. C2×Q8×Dic173C2semidirect
14. C4×Q8×D173C2semidirect
15. #548×C173C2semidirect
124. C22⋊D173#41semidirect
142. Q8×C173#41semidirect
180. C173#3482semidirect
188. #548D173semidirect
307. Q8#14583825semidirect

Maximal subgroups

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

subgroupindex
#260122172
#260122102
#260121812
#260121752
Q8×C22⋊D1732
#260121482
#260121412
#260120432
#260120222
#260120202
#41×Dic1732
C4×#145838252
C2×Q8×Dic1732
C4×Q8×D1732
#548×C1732
#3482173