Finite Group Databasebeta

← #36367153

Subgroups — #36367153

Order 22144.

1112 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012223C213yesmaximal, complemented
2#26012217C211yesmaximal, characteristic, complemented
3#26012212C213yesmaximal, complemented
4#26012193C213yesmaximal, complemented
5#26012187C213yesmaximal, complemented
6#26012160C211yesmaximal, characteristic, complemented
7#26012152C211yesmaximal, characteristic, complemented
8#26012036C213yesmaximal, complemented
9#26012027C213yesmaximal, complemented
10#26012020C213yesmaximal, complemented
11#41×Dic173C211yesmaximal, characteristic, complemented
12C4×#14583827C211yesmaximal, characteristic, complemented
13C4×#14583825C213yesmaximal, complemented
14C2×Q8×Dic173C211yesmaximal, characteristic, complemented
15#548×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583825C2213yes—
18#19149678C2213yes—
19#19149675C2213yes—
20#19149674C2216yes—
21#19149672C2216yes—
22#19149671C2213yes—
23#19149670C2213yes—
24#19149668C2213yes—
25#19149665C2216yes—
26#19149663C2216yes—
27#19149661C2216yes—
28#19149655C2213yes—
29#19149653C2213yes—
30#19149653C2213yes—
31#19149651C2216yes—
32#19149650C2213yes—
33Q8×Dic173C2211yescharacteristic
34Q8×Dic173C2213yes—
35Q8×Dic173C2211yescharacteristic
36Q8×Dic173C2213yes—
37Q8×Dic173C2211yescharacteristic
38C4×Dic346C2213yes—
39#19149644C2213yes—
40C4×D692C2213yes—
41D4×Dic173C2213yes—
42D4×Dic173C2213yes—
43D4×Dic173C2213yes—
44#19149641C2213yes—
45C4×C22⋊D173C2213yes—
46C4×C22⋊D173C2213yes—
47#19149639C2213yes—
48#19149639C2213yes—
49#19149639C2216yes—
50#19149638C2213yes—
51#19149637C2213yes—
52#19149637C2213yes—
53#19149635C2211yescharacteristic
54#19149635C2213yes—
55C2×C4⋊Dic173C2213yes—
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#14583827—24no—
67#14583825—212no—
68C2×Dic346C2313yes—
69C2×D692C2313yes—
70C2×C22⋊D173C2313yes—
71C2×C22⋊D173C2313yes—
72C2×C22⋊D173C2316yes—
73C22×Dic173C2313yes—
74C22×Dic173C2313yes—
75C22×Dic173C2313yes—
76C22×Dic173C2311yescharacteristic
77C2×C4×D173C2313yes—
78C2×C4×D173C2313yes—
79C2×C4×D173C2313yes—
80C4⋊Dic173C2313yes—
81C4⋊Dic173C2313yes—
82C4⋊Dic173C2316yes—
83C4⋊Dic173C2313yes—
84C4⋊Dic173C2313yes—
85#14583805C2311yescharacteristic
86#14583805C2313yes—
87#14583805C2311yescharacteristic
88#14583805C2313yes—
89#14583805C2316yes—
90C22⋊Dic173C2311yescharacteristic
91C22⋊Dic173C2313yes—
92C22⋊Dic173C2316yes—
93#14583803C2313yes—
94#14583803C2313yes—
95#14583803C2316yes—
96#14583803C2316yes—
97C4×Dic173C2311yescharacteristic
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×D173C2413yes—
121Dic346—26no—
122D692—26no—
123C22⋊D173—26no—
124C22⋊D173—26no—
125C22⋊D173—46no—
126C2×Dic173—26no—
127C2×Dic173—43no—
128C2×Dic173—43no—
129C2×Dic173#4114yescomplemented
130C2×Dic173C2411yescharacteristic
131C2×Dic173C2413yes—
132C2×Dic173C2411yescharacteristic
133C2×Dic173C2413yes—
134C2×Dic173C2413yes—
135C2×Dic173C2411yescharacteristic
136C2×Dic173C2411yescharacteristic
137C4×D173—26no—
138C4×D173—26no—
139C4×D173—43no—
140C23×C173C2413yes—
141C23×C173C2411yescharacteristic
142Q8×C173—26no—
143Q8×C173#4114yescomplemented
144D4×C173—26no—
145D4×C173—26no—
146C2×C4×C173—26no—
147C2×C4×C173—26no—
148C2×C4×C173C2411yescharacteristic
149C2×C4×C173C2413yes—
150C2×C4×C173C2413yes—
151C2×C4×C173C2411yescharacteristic
152C2×C4×C173C2413yes—
153C22×C173—21no—
154C22×C173—21no—
155C22×C173—23no—
156C22×C173—23no—
157C22×C173—23no—
158C22×C173C2×#4112yes—
159C22×C173C2511yescharacteristic
160C2×D173—23no—
161C2×D173—23no—
162C2×D173—23no—
163C4×C173—21no—
164C4×C173—23no—
165C4×C173—23no—
166C4×C173—21no—
167C4×C173C2×#4116yes—
168Dic173—21no—
169Dic173—23no—
170Dic173—23no—
171Dic173—21no—
172Dic173—23no—
173Dic173C2×#4112yes—
174Dic173C2×#4112yes—
175C2×C173—22no—
176C2×C173—43no—
177C2×C173C22×#4111yescharacteristic
178C2×C173C22×#4111yescharacteristic
179C2×C173#58511yescharacteristic
180D173—43no—
181C173#348011yescharacteristic, complemented, Sylow
182#3480—1731nomaximal, Sylow
183#559—1733no—
184#557—1731no—
185#556—1733no—
186#553—1733no—
187#553—1733no—
188#550—1731no—
189#548—1731no—
190#548D17311yescharacteristic, complemented
191#532—1733no—
192#529—1733no—
193#529—1733no—
194C4×#41—1731no—
195C4×#41—1731no—
196C4×#41—1733no—
197C2×C4×Q8—1731no—
198C2×#41—1731no—
199C2×#41—1733no—
200C2×#41C2×D17311yescharacteristic
201C22×Q8C2×D17311yescharacteristic
202C4⋊Q8—1733no—
203#127—1733no—
204#126—1736no—
205#126—1736no—
206#125—1733no—
207#125—1733no—
208#124—1733no—
209#124—1736no—
210#124C2×D17313yes—
211#123—1736no—
212#123—1736no—
213C22⋊Q8—1733no—
214C22⋊Q8C2×D17313yes—
215#121—1733no—
216#121—1736no—
217#121—1733no—
218#121—1733no—
219#121C2×D17313yes—
220C4×Q8—1731no—
221C4×Q8—1733no—
222C4×Q8—1731no—
223C4×Q8—1733no—
224C4×Q8—1731no—
225C4×Q8—1733no—
226C4×Q8—1733no—
227C4×Q8C2×D17311yescharacteristic
228C4×D4—1733no—
229C4×D4—1733no—
230C4×D4—1733no—
231C4×D4—1733no—
232C4×D4—1733no—
233C4×D4—1733no—
234C4×D4—1733no—
235C4×D4—1733no—
236C4×D4—1736no—
237C4×D4—1733no—
238C4×D4C2×D17313yes—
239#117—1733no—
240#117—1733no—
241#117—1733no—
242#117—1731no—
243#117—1733no—
244C2×C4⋊C4—1733no—
245C2×C42—1733no—
246C2×C42—1733no—
247C2×C42—1733no—
248#41—24no—
249#41—3464no—
250#41—34612no—
251C2×Q8—24no—
252C2×Q8—1733no—
253C2×Q8C22×D17313yes—
254C2×Q8C22×D17311yescharacteristic
255C2×D4—1733no—
256C2×D4—1733no—
257C2×D4—1736no—
258C2×D4—1733no—
259C2×D4C22×D17313yes—
260C2×D4C22×D17313yes—
261C22×C4—1733no—
262C22×C4—1733no—
263C22×C4—1733no—
264C22×C4—1731no—
265C22×C4—1733no—
266C22×C4—1733no—
267C22×C4—1733no—
268C22×C4C22×D17313yes—
269C22×C4C22×D17313yes—
270C4⋊C4—1733no—
271C4⋊C4—1733no—
272C4⋊C4—1736no—
273C4⋊C4—1731no—
274C4⋊C4—1731no—
275C4⋊C4—1733no—
276C4⋊C4—1733no—
277C4⋊C4—1733no—
278C4⋊C4—1733no—
279C4⋊C4—1736no—
280C4⋊C4C22×D17313yes—
281C4⋊C4C22×D17313yes—
282C22⋊C4—1731no—
283C22⋊C4—1733no—
284C22⋊C4—1736no—
285C22⋊C4—1733no—
286C22⋊C4—1736no—
287C22⋊C4—1733no—
288C22⋊C4—1736no—
289C22⋊C4C22×D17311yescharacteristic
290C22⋊C4C22×D17313yes—
291C22⋊C4C22×D17316yes—
292C42—1731no—
293C42—1731no—
294C42—1733no—
295C42—1733no—
296C42—1733no—
297C42—1733no—
298C42—1733no—
299C42—1733no—
300C42—1733no—
301C42C22×D17313yes—
302C23—1733no—
303C23C23×D17313yes—
304C23C23×D17311yescharacteristic
305Q8—26no—
306Q8—3466no—
307Q8#1458382514yescomplemented
308D4—26no—
309D4—26no—
310D4—3466no—
311D4—3466no—
312D4—3466no—
313D4—6926no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—3466no—
317C2×C4—3466no—
318C2×C4—3466no—
319C2×C4—1731no—
320C2×C4—1733no—
321C2×C4—1731no—
322C2×C4—1733no—
323C2×C4—1733no—
324C2×C4—1734no—
325C2×C4—1731no—
326C2×C4—1731no—
327C2×C4—6923no—
328C2×C4—6923no—
329C2×C4—6923no—
330C2×C4C23×D17311yescharacteristic
331C2×C4C23×D17313yes—
332C2×C4C23×D17311yescharacteristic
333C2×C4C23×D17313yes—
334C2×C4C23×D17313yes—
335C22—21no—
336C22—21no—
337C22—23no—
338C22—23no—
339C22—23no—
340C22—3463no—
341C22—3463no—
342C22—3463no—
343C22#41×D17312yes—
344C22C24×D17311yescharacteristic
345C4—23no—
346C4—21no—
347C4—23no—
348C4—21no—
349C4—3463no—
350C4—3461no—
351C4—3463no—
352C4—3461no—
353C4—3463no—
354C4—1732no—
355C4—1732no—
356C4C2×#1458382516yes—
357C2—22no—
358C2—43no—
359C2—6923no—
360C2C2×#41×D17311yescharacteristic
361C2C22×#1458382511yescharacteristic
362C2#2601232811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D173
commutator, socle, cosocleC22×C173C25
Fitting#548×C173C2
hypercenter#548D173
hypocenterC173#3480
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #26012223C2semidirect
2. #26012217C2semidirect
3. #26012212C2semidirect
4. #26012193C2semidirect
5. #26012187C2semidirect
6. #26012160C2semidirect
7. #26012152C2semidirect
8. #26012036C2semidirect
9. #26012027C2semidirect
10. #26012020C2semidirect
11. #41×Dic173C2semidirect
12. C4×#14583827C2semidirect
13. C4×#14583825C2semidirect
14. C2×Q8×Dic173C2semidirect
15. #548×C173C2semidirect
129. C2×Dic173#41semidirect
143. Q8×C173#41semidirect
181. C173#3480semidirect
190. #548D173semidirect
307. Q8#14583825semidirect

Maximal subgroups

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

subgroupindex
#260122232
#260122172
#260122122
#260121932
#260121872
#260121602
#260121522
#260120362
#260120272
#260120202
#41×Dic1732
C4×#145838272
C4×#145838252
C2×Q8×Dic1732
#548×C1732
#3480173