Finite Group Databasebeta

← #36367180

Subgroups — #36367180

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#26012213C213yesmaximal, complemented
3#26012182C213yesmaximal, complemented
4#26012176C213yesmaximal, complemented
5Q8×C22⋊D173C211yesmaximal, characteristic, complemented
6#26012153C211yesmaximal, characteristic, complemented
7#26012152C211yesmaximal, characteristic, complemented
8#26012142C213yesmaximal, complemented
9#26012045C213yesmaximal, complemented
10#26012028C213yesmaximal, complemented
11#26012025C213yesmaximal, complemented
12C4×#14583827C211yesmaximal, characteristic, complemented
13C4×#14583826C213yesmaximal, complemented
14C4×Q8×D173C211yesmaximal, characteristic, complemented
15C2×C4×Q8×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583826C2213yes—
18C2×Q8×D173C2211yescharacteristic
19#19149678C2213yes—
20#19149677C2213yes—
21#19149674C2216yes—
22#19149673C2216yes—
23#19149671C2213yes—
24#19149670C2213yes—
25#19149669C2216yes—
26#19149668C2213yes—
27#19149667C2213yes—
28#19149664C2216yes—
29#19149662C2216yes—
30#19149659C2213yes—
31#19149658C2213yes—
32#19149657C2216yes—
33#19149655C2213yes—
34#19149655C2213yes—
35#19149654C2213yes—
36#19149653C2213yes—
37Q8×Dic173C2211yescharacteristic
38Q8×Dic173C2211yescharacteristic
39C4×Dic346C2213yes—
40C4×Dic346C2213yes—
41C4×Dic346C2213yes—
42#19149644C2213yes—
43C4×D692C2213yes—
44C4×D692C2213yes—
45C4×D692C2213yes—
46#19149641C2213yes—
47C4×C22⋊D173C2213yes—
48C4×C22⋊D173C2211yescharacteristic
49C4×C22⋊D173C2213yes—
50C4×C22⋊D173C2213yes—
51#19149638C2213yes—
52#19149637C2213yes—
53#19149636C2213yes—
54#19149636C2213yes—
55#19149636C2216yes—
56C4⋊C4×D173C2213yes—
57C42×D173C2213yes—
58C42×D173C2213yes—
59C22×Q8×C173C2211yescharacteristic
60C4×Q8×C173C2213yes—
61C4×Q8×C173C2211yescharacteristic
62C4×Q8×C173C2211yescharacteristic
63C4×Q8×C173C2213yes—
64C2×C4⋊C4×C173C2213yes—
65C2×C42×C173C2213yes—
66#14583827—24no—
67#14583826—212no—
68C2×Dic346C2313yes—
69C2×Dic346C2313yes—
70Q8×D173—24no—
71C2×D692C2313yes—
72C2×D692C2313yes—
73C2×C22⋊D173C2313yes—
74C2×C22⋊D173C2311yescharacteristic
75C2×C4×D173C2311yescharacteristic
76C2×C4×D173C2313yes—
77C2×C4×D173C2313yes—
78C2×C4×D173C2313yes—
79C2×C4×D173C2313yes—
80C2×C4×D173C2313yes—
81C4⋊Dic173C2313yes—
82C4⋊Dic173C2313yes—
83C4⋊Dic173C2313yes—
84C4⋊Dic173C2313yes—
85#14583805C2316yes—
86#14583805C2311yescharacteristic
87#14583805C2313yes—
88#14583805C2313yes—
89#14583805C2313yes—
90#14583805C2316yes—
91#14583805C2316yes—
92C22⋊Dic173C2311yescharacteristic
93C22⋊Dic173C2313yes—
94#14583803C2311yescharacteristic
95#14583803C2313yes—
96#14583803C2313yes—
97#14583803C2313yes—
98#14583803C2316yes—
99#14583803C2316yes—
100#14583803C2316yes—
101C4×Dic173C2311yescharacteristic
102C4×Dic173C2313yes—
103C4×Dic173C2313yes—
104C4×Dic173C2313yes—
105C4×Dic173C2313yes—
106C4×Dic173C2313yes—
107C2×Q8×C173—24no—
108C2×Q8×C173C2313yes—
109C2×Q8×C173C2311yescharacteristic
110C22×C4×C173C2313yes—
111C22×C4×C173C2313yes—
112C22×C4×C173C2311yescharacteristic
113C4⋊C4×C173C2313yes—
114C4⋊C4×C173C2313yes—
115C4⋊C4×C173C2316yes—
116C42×C173C2313yes—
117C42×C173C2313yes—
118C42×C173C2313yes—
119C42×C173C2313yes—
120C22×D173C2413yes—
121C22×D173C2411yescharacteristic
122Dic346—26no—
123Dic346—26no—
124D692—26no—
125D692—26no—
126C22⋊D173—26no—
127C22⋊D173#4114yescomplemented
128C2×Dic173C2411yescharacteristic
129C2×Dic173C2413yes—
130C2×Dic173C2413yes—
131C2×Dic173C2411yescharacteristic
132C2×Dic173C2413yes—
133C2×Dic173C2411yescharacteristic
134C4×D173—26no—
135C4×D173—26no—
136C4×D173—26no—
137C4×D173—22no—
138C4×D173—26no—
139C4×D173—26no—
140C23×C173C2411yescharacteristic
141Q8×C173—26no—
142Q8×C173#4114yescomplemented
143C2×C4×C173—26no—
144C2×C4×C173—26no—
145C2×C4×C173—22no—
146C2×C4×C173C2413yes—
147C2×C4×C173C2413yes—
148C2×C4×C173C2411yescharacteristic
149C2×C4×C173C2413yes—
150C2×C4×C173C2413yes—
151C2×C4×C173C2411yescharacteristic
152C22×C173—21no—
153C22×C173—21no—
154C22×C173C2×#4112yes—
155C22×C173C2511yescharacteristic
156C2×D173—21no—
157C2×D173—21no—
158C2×D173—23no—
159C2×D173—23no—
160C2×D173—23no—
161C2×D173C2×#4112yes—
162C4×C173—23no—
163C4×C173—23no—
164C4×C173—23no—
165C4×C173—21no—
166C4×C173C2×#4116yes—
167C4×C1732−1+412yes—
168Dic173—21no—
169Dic173—23no—
170Dic173—23no—
171Dic173—21no—
172Dic173—23no—
173Dic173C2×#4112yes—
174C2×C173—22no—
175C2×C173C22×#4111yescharacteristic
176C2×C173C22×#4111yescharacteristic
177C2×C173C2×2−1+411yescharacteristic
178D173—22no—
179D173—43no—
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#548—1731no—
189#547—1733no—
190#533—1733no—
191#529—1733no—
192#529—1733no—
193C4×#41—1731no—
194C4×#41—1733no—
195C2×C4×Q8—1731no—
196C2×C4×Q8D17311yescharacteristic, complemented
197C2×#41—1731no—
198C2×#41—1733no—
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#124—1733no—
210#123—1736no—
211#123—1736no—
212C22⋊Q8—1733no—
213C22⋊Q8—1733no—
214C22⋊Q8—1733no—
215C22⋊Q8—1736no—
216C22⋊Q8—1733no—
217#121—1733no—
218#121—1733no—
219C4×Q8—1731no—
220C4×Q8—1731no—
221C4×Q8—1733no—
222C4×Q8—1733no—
223C4×Q8—1733no—
224C4×Q8—1733no—
225C4×Q8C2×D17311yescharacteristic
226C4×Q8C2×D17313yes—
227C4×Q8C2×D17311yescharacteristic
228C4×Q8C2×D17313yes—
229C4×D4—1731no—
230C4×D4—1733no—
231C4×D4—1733no—
232C4×D4—1733no—
233C4×D4—1733no—
234C4×D4—1733no—
235C4×D4—1733no—
236C4×D4—1733no—
237#117—1733no—
238#117—1733no—
239#117—1733no—
240#117—1733no—
241#117—1736no—
242C2×C4⋊C4—1733no—
243C2×C4⋊C4C2×D17313yes—
244C2×C42—1733no—
245C2×C42—1733no—
246C2×C42C2×D17313yes—
247#41—34612no—
248#41—3464no—
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×D4—1733no—
258C2×D4—1733no—
259C22×C4—1733no—
260C22×C4—1733no—
261C22×C4—1733no—
262C22×C4—1733no—
263C22×C4—1731no—
264C22×C4—1733no—
265C22×C4C22×D17313yes—
266C22×C4C22×D17313yes—
267C22×C4C22×D17311yescharacteristic
268C4⋊C4—1733no—
269C4⋊C4—1736no—
270C4⋊C4—1731no—
271C4⋊C4—1733no—
272C4⋊C4—1733no—
273C4⋊C4—1733no—
274C4⋊C4—1733no—
275C4⋊C4—1733no—
276C4⋊C4—1733no—
277C4⋊C4—1736no—
278C4⋊C4—1736no—
279C4⋊C4C22×D17313yes—
280C4⋊C4C22×D17313yes—
281C4⋊C4C22×D17316yes—
282C22⋊C4—1731no—
283C22⋊C4—1733no—
284C22⋊C4—1731no—
285C22⋊C4—1733no—
286C22⋊C4—1736no—
287C22⋊C4—1733no—
288C22⋊C4—1733no—
289C22⋊C4—1736no—
290C22⋊C4—1736no—
291C42—1733no—
292C42—1733no—
293C42—1733no—
294C42—1733no—
295C42—1731no—
296C42—1733no—
297C42C22×D17313yes—
298C42C22×D17313yes—
299C42C22×D17313yes—
300C42C22×D17313yes—
301C23—1733no—
302C23—1731no—
303C23C23×D17311yescharacteristic
304Q8—26no—
305Q8—3466no—
306Q8—3466no—
307Q8#1458382614yescomplemented
308D4—3466no—
309D4—3466no—
310D4—3466no—
311D4—1734no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—22no—
315C2×C4—3466no—
316C2×C4—3466no—
317C2×C4—3466no—
318C2×C4—3466no—
319C2×C4—3462no—
320C2×C4—3466no—
321C2×C4—1731no—
322C2×C4—1731no—
323C2×C4—1733no—
324C2×C4—1733no—
325C2×C4—1733no—
326C2×C4—1731no—
327C2×C4C23×D17313yes—
328C2×C4C23×D17311yescharacteristic
329C2×C4C23×D17313yes—
330C2×C4C23×D17313yes—
331C2×C4C23×D17313yes—
332C2×C4C23×D17311yescharacteristic
333C22—21no—
334C22—21no—
335C22—3461no—
336C22—3461no—
337C22—3463no—
338C22—3463no—
339C22—3463no—
340C22—1732no—
341C22#41×D17312yes—
342C22C24×D17311yescharacteristic
343C4—23no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—3463no—
348C4—3461no—
349C4—3463no—
350C4—3461no—
351C4—3463no—
352C4—1732no—
353C4C2×#1458382616yes—
354C4#1914972712yes—
355C2—22no—
356C2—3462no—
357C2—6923no—
358C2C2×#41×D17311yescharacteristic
359C2C22×#1458382611yescharacteristic
360C2C2×#1914972711yescharacteristic
subgroupG/H
centerC2×C4C23×D173
commutator, socle, cosocleC22×C173C25
FrattiniC22C24×D173
FittingC2×C4×Q8×C173C2
hypercenterC2×C4×Q8D173
hypocenterC173#3482
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #26012217C2semidirect
2. #26012213C2semidirect
3. #26012182C2semidirect
4. #26012176C2semidirect
5. Q8×C22⋊D173C2semidirect
6. #26012153C2semidirect
7. #26012152C2semidirect
8. #26012142C2semidirect
9. #26012045C2semidirect
10. #26012028C2semidirect
11. #26012025C2semidirect
12. C4×#14583827C2semidirect
13. C4×#14583826C2semidirect
14. C4×Q8×D173C2semidirect
15. C2×C4×Q8×C173C2semidirect
127. C22⋊D173#41semidirect
142. Q8×C173#41semidirect
180. C173#3482semidirect
196. C2×C4×Q8D173semidirect
307. Q8#14583826semidirect

Maximal subgroups

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

subgroupindex
#260122172
#260122132
#260121822
#260121762
Q8×C22⋊D1732
#260121532
#260121522
#260121422
#260120452
#260120282
#260120252
C4×#145838272
C4×#145838262
C4×Q8×D1732
C2×C4×Q8×C1732
#3482173