Finite Group Databasebeta

← #34114212

Subgroups — #34114212

Order 20096.

1104 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#23758005C213yesmaximal, complemented
2Q8×Dic314C211yesmaximal, characteristic, complemented
3#23757981C213yesmaximal, complemented
4#23757942C211yesmaximal, characteristic, complemented
5#23757934C211yesmaximal, characteristic, complemented
6#23757927C213yesmaximal, complemented
7#23757923C213yesmaximal, complemented
8#23757896C213yesmaximal, complemented
9#23757892C213yesmaximal, complemented
10#23757818C213yesmaximal, complemented
11#23757803C213yesmaximal, complemented
12#23757756C211yesmaximal, characteristic, complemented
13#23757755C211yesmaximal, characteristic, complemented
14C2×Q8×Dic157C211yesmaximal, characteristic, complemented
15#548×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578953C2216yes—
18#19097771C2213yes—
19#19097770C2213yes—
20#19097770C2213yes—
21#19097767C2213yes—
22#19097766C2213yes—
23#19097764C2213yes—
24#19097761C2213yes—
25#19097760C2213yes—
26#19097757C2216yes—
27#19097757C2213yes—
28#19097755C2216yes—
29#19097755C2213yes—
30#19097753C2216yes—
31#19097752C2213yes—
32#19097751C2216yes—
33#19097748C2216yes—
34#19097748C2211yescharacteristic
35#19097747C2213yes—
36#19097745C2213yes—
37#19097745C2213yes—
38#19097743C2213yes—
39#19097742C2213yes—
40Q8×Dic157C2213yes—
41Q8×Dic157C2211yescharacteristic
42Q8×Dic157C2211yescharacteristic
43Q8×Dic157C2213yes—
44Q8×Dic157C2211yescharacteristic
45C4×Dic314C2213yes—
46C4×D628C2213yes—
47D4×Dic157C2213yes—
48D4×Dic157C2213yes—
49#19097733C2213yes—
50C4×C22⋊D157C2213yes—
51C4×C22⋊D157C2213yes—
52#19097731C2216yes—
53#19097730C2213yes—
54#19097728C2213yes—
55C2×C4⋊Dic157C2213yes—
56C2×C4⋊Dic157C2213yes—
57C4⋊C4×D157C2213yes—
58C2×C4×Dic157C2213yes—
59C2×#41×C157C2211yescharacteristic
60C22×Q8×C157C2211yescharacteristic
61#124×C157C2213yes—
62C22⋊Q8×C157C2213yes—
63#121×C157C2213yes—
64C4×Q8×C157C2211yescharacteristic
65C4×D4×C157C2213yes—
66#14578955—24no—
67#14578953—412no—
68C2×Dic314C2311yescharacteristic
69C2×Dic314C2316yes—
70C2×D628C2313yes—
71C2×C22⋊D157C2313yes—
72C2×C22⋊D157C2313yes—
73C2×C22⋊D157C2316yes—
74C22×Dic157C2311yescharacteristic
75C22×Dic157C2313yes—
76C22×Dic157C2316yes—
77C2×C4×D157C2313yes—
78C2×C4×D157C2316yes—
79C4⋊Dic157C2313yes—
80C4⋊Dic157C2313yes—
81C4⋊Dic157C2313yes—
82C4⋊Dic157C2313yes—
83C4⋊Dic157C2316yes—
84C4⋊Dic157C2311yescharacteristic
85C4⋊Dic157C2313yes—
86#14578933C2313yes—
87#14578933C2311yescharacteristic
88#14578933C2313yes—
89#14578933C2311yescharacteristic
90#14578933C2313yes—
91#14578933C2313yes—
92C22⋊Dic157C2313yes—
93C22⋊Dic157C2313yes—
94C22⋊Dic157C2311yescharacteristic
95C22⋊Dic157C2313yes—
96#14578931C2313yes—
97#14578931C2313yes—
98#14578931C2316yes—
99#14578931C2313yes—
100#14578931C2313yes—
101C4×Dic157C2313yes—
102C4×Dic157C2313yes—
103C4×Dic157C2313yes—
104C4×Dic157C2313yes—
105C4×Dic157C2313yes—
106#41×C157—24no—
107C2×Q8×C157—24no—
108C2×Q8×C157C2313yes—
109C2×Q8×C157C2311yescharacteristic
110C2×D4×C157C2313yes—
111C2×D4×C157C2313yes—
112C22×C4×C157C2313yes—
113C22×C4×C157C2313yes—
114C4⋊C4×C157C2313yes—
115C4⋊C4×C157C2313yes—
116C22⋊C4×C157C2311yescharacteristic
117C22⋊C4×C157C2313yes—
118C22⋊C4×C157C2316yes—
119C42×C157C2313yes—
120C22×D157C2413yes—
121Dic314—22no—
122Dic314—46no—
123D628—26no—
124C22⋊D157—26no—
125C22⋊D157—26no—
126C22⋊D157—46no—
127C2×Dic157—26no—
128C2×Dic157—22no—
129C2×Dic157—46no—
130C2×Dic157C2411yescharacteristic
131C2×Dic157C2413yes—
132C2×Dic157C2413yes—
133C2×Dic157C2413yes—
134C2×Dic157C2411yescharacteristic
135C2×Dic157C2411yescharacteristic
136C2×Dic157C2411yescharacteristic
137C4×D157—26no—
138C4×D157—46no—
139C23×C157C2413yes—
140C23×C157C2411yescharacteristic
141Q8×C157—26no—
142Q8×C157#4114yescomplemented
143D4×C157—26no—
144D4×C157—26no—
145C2×C4×C157—26no—
146C2×C4×C157—26no—
147C2×C4×C157C2413yes—
148C2×C4×C157C2411yescharacteristic
149C2×C4×C157C2413yes—
150C2×C4×C157C2411yescharacteristic
151C2×C4×C157C2413yes—
152C22×C157—21no—
153C22×C157—21no—
154C22×C157—23no—
155C22×C157—23no—
156C22×C157—23no—
157C22×C157H(5,𝔽2)12yes—
158C22×C157C2511yescharacteristic
159C2×D157—23no—
160C2×D157—23no—
161C2×D157—23no—
162C4×C157—21no—
163C4×C157—23no—
164C4×C157—23no—
165C4×C157—21no—
166C4×C157C2×#4116yes—
167Dic157—21no—
168Dic157—23no—
169Dic157—23no—
170Dic157—23no—
171Dic157—21no—
172Dic157—21no—
173Dic1572−1+412yes—
174C2×C157—22no—
175C2×C157—43no—
176C2×C157C22×#4111yescharacteristic
177C2×C157C2×H(5,𝔽2)11yescharacteristic
178C2×C157C2×2−1+411yescharacteristic
179D157—43no—
180C157#352211yescharacteristic, complemented, Sylow
181#3522—1571nomaximal, Sylow
182#559—1573no—
183Q82—1571no—
184#554—1573no—
185#550—1571no—
186#548—1571no—
187#548D15711yescharacteristic, complemented
188#547—1573no—
189#547—1573no—
190#542—1573no—
191#542—1573no—
192#532—1573no—
193#529—1573no—
194#520—1571no—
195#520—1571no—
196C2×C4×Q8—1571no—
197C2×#41—1571no—
198C2×#41—1576no—
199C2×#41C2×D15711yescharacteristic
200C22×Q8C2×D15711yescharacteristic
201C4⋊Q8—1573no—
202C4⋊Q8—1573no—
203C4⋊Q8—1573no—
204#127—1573no—
205#126—1573no—
206#126—1573no—
207#125—1573no—
208#124—1573no—
209#124—1573no—
210#124—1576no—
211#124C2×D15713yes—
212#123—1573no—
213#123—1573no—
214#123—1576no—
215#123—1576no—
216C22⋊Q8—1573no—
217C22⋊Q8—1576no—
218C22⋊Q8—1571no—
219C22⋊Q8—1576no—
220C22⋊Q8C2×D15713yes—
221#121—1573no—
222#121—1573no—
223#121—1573no—
224#121—1573no—
225#121C2×D15713yes—
226C4×Q8—1571no—
227C4×Q8—1573no—
228C4×Q8—1571no—
229C4×Q8—1573no—
230C4×Q8—1571no—
231C4×Q8—1573no—
232C4×Q8C2×D15711yescharacteristic
233C4×D4—1573no—
234C4×D4—1573no—
235C4×D4—1573no—
236C4×D4—1573no—
237C4×D4—1573no—
238C4×D4—1576no—
239C4×D4—1573no—
240C4×D4C2×D15713yes—
241#117—1573no—
242#117—1573no—
243C2×C4⋊C4—1573no—
244C2×C4⋊C4—1573no—
245C2×C4⋊C4—1573no—
246C2×C42—1573no—
247#41—24no—
248#41—3144no—
249#41—62812no—
250C2×Q8—24no—
251C2×Q8—1571no—
252C2×Q8—1576no—
253C2×Q8C22×D15713yes—
254C2×Q8C22×D15711yescharacteristic
255C2×D4—1573no—
256C2×D4—1573no—
257C2×D4—1576no—
258C2×D4—1573no—
259C2×D4C22×D15713yes—
260C2×D4C22×D15713yes—
261C22×C4—1571no—
262C22×C4—1573no—
263C22×C4—1573no—
264C22×C4—1576no—
265C22×C4—1576no—
266C22×C4C22×D15713yes—
267C22×C4C22×D15713yes—
268C4⋊C4—1573no—
269C4⋊C4—1573no—
270C4⋊C4—1573no—
271C4⋊C4—1573no—
272C4⋊C4—1573no—
273C4⋊C4—1576no—
274C4⋊C4—1571no—
275C4⋊C4—1571no—
276C4⋊C4—1571no—
277C4⋊C4—1573no—
278C4⋊C4—1573no—
279C4⋊C4—1573no—
280C4⋊C4—1573no—
281C4⋊C4C22×D15713yes—
282C4⋊C4C22×D15713yes—
283C22⋊C4—1573no—
284C22⋊C4—1573no—
285C22⋊C4—1571no—
286C22⋊C4—1573no—
287C22⋊C4—1573no—
288C22⋊C4—1573no—
289C22⋊C4—1573no—
290C22⋊C4—1576no—
291C22⋊C4—1573no—
292C22⋊C4C22×D15716yes—
293C22⋊C4C22×D15711yescharacteristic
294C22⋊C4C22×D15713yes—
295C42—1573no—
296C42—1573no—
297C42—1573no—
298C42—1573no—
299C42—1573no—
300C42C22×D15713yes—
301C23—1573no—
302C23C23×D15713yes—
303C23C23×D15711yescharacteristic
304Q8—26no—
305Q8—3142no—
306Q8—6286no—
307Q8#1457895314yescomplemented
308D4—26no—
309D4—26no—
310D4—3146no—
311D4—3146no—
312D4—3146no—
313D4—6286no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—3146no—
317C2×C4—3142no—
318C2×C4—3146no—
319C2×C4—6286no—
320C2×C4—6286no—
321C2×C4—1571no—
322C2×C4—1571no—
323C2×C4—1573no—
324C2×C4—1573no—
325C2×C4—1573no—
326C2×C4—1571no—
327C2×C4—1571no—
328C2×C4C23×D15713yes—
329C2×C4C23×D15711yescharacteristic
330C2×C4C23×D15713yes—
331C2×C4C23×D15711yescharacteristic
332C2×C4C23×D15713yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—3143no—
339C22—3143no—
340C22—3143no—
341C22#1909781712yes—
342C22C24×D15711yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—3141no—
348C4—3143no—
349C4—3143no—
350C4—3143no—
351C4—3141no—
352C4—3141no—
353C4—1572no—
354C4C2×#1457895316yes—
355C2—22no—
356C2—43no—
357C2—6283no—
358C2C22×#1457895311yescharacteristic
359C2C2×#1909781711yescharacteristic
360C22−1+4×D15711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D157
commutator, socle, cosocleC22×C157C25
Fitting#548×C157C2
hypercenter#548D157
hypocenterC157#3522
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #23758005C2semidirect
2. Q8×Dic314C2semidirect
3. #23757981C2semidirect
4. #23757942C2semidirect
5. #23757934C2semidirect
6. #23757927C2semidirect
7. #23757923C2semidirect
8. #23757896C2semidirect
9. #23757892C2semidirect
10. #23757818C2semidirect
11. #23757803C2semidirect
12. #23757756C2semidirect
13. #23757755C2semidirect
14. C2×Q8×Dic157C2semidirect
15. #548×C157C2semidirect
142. Q8×C157#41semidirect
180. C157#3522semidirect
187. #548D157semidirect
307. Q8#14578953semidirect

Maximal subgroups

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

subgroupindex
#237580052
Q8×Dic3142
#237579812
#237579422
#237579342
#237579272
#237579232
#237578962
#237578922
#237578182
#237578032
#237577562
#237577552
C2×Q8×Dic1572
#548×C1572
#3522157