Finite Group Databasebeta

← #28350551

Subgroups — #28350551

Order 13696.

992 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293355C213yesmaximal, complemented
2#20293353C211yesmaximal, characteristic, complemented
3#20293345C213yesmaximal, complemented
4#20293318C213yesmaximal, complemented
5#20293316C213yesmaximal, complemented
6Q8×D428C211yesmaximal, characteristic, complemented
7Q8×C22⋊D107C211yesmaximal, characteristic, complemented
8Q8×C22⋊D107C211yesmaximal, characteristic, complemented
9#20293239C213yesmaximal, complemented
10#20293235C213yesmaximal, complemented
11#20293181C213yesmaximal, complemented
12#20293155C213yesmaximal
13#20293103C211yesmaximal, characteristic, complemented
14C4×Q8×D107C211yesmaximal, characteristic, complemented
15#519×C107C211yesmaximal, characteristic, complemented
16C2×Q8×D107C2211yescharacteristic
17C2×Q8×D107C2211yescharacteristic
18#16812109C2213yes—
19#16812108C2213yes—
20#16812107C2213yes—
21#16812107C2213yes—
22#16812104C2213yes—
23#16812103C2216yes—
24#16812102C2213yes—
25#16812102C2216yes—
26#16812101C2216yes—
27#16812101C2213yes—
28#16812100C2213yes—
29#16812099C2213yes—
30#16812099C2216yes—
31#16812099C2216yes—
32#16812095C2213yes—
33#16812094C2213yes—
34#16812091C2213yes—
35#16812089C2213yes—
36#16812089C2213yes—
37#16812088C2213yes—
38#16812088C2213yes—
39#16812087C2213yes—
40#16812085C2213yes—
41#16812085C2213yes—
42#16812082C2211yescharacteristic
43Q8×Dic107C2211yescharacteristic
44Q8×Dic107C2211yescharacteristic
45C4×Dic214C2213yes—
46C4×Dic214C2213yes—
47C4×Dic214C2216yes—
48C4×Dic214C2213yes—
49#16812074C2213yes—
50#16812074C2213yes—
51C4×D428C2213yes—
52C4×C22⋊D107C2213yes—
53C4×C22⋊D107C2213yes—
54#16812067C2213yes—
55#16812066C2213yes—
56#16812066C2216yes—
57C4⋊C4×D107C2213yes—
58C42×D107C2213yes—
59C22×Q8×C107C2211yescharacteristic
60C4×Q8×C107C2213yes—
61C4×Q8×C107C2211yescharacteristic
62C4×Q8×C107C2213yes—
63C4×Q8×C107C2211yescharacteristic
64#117×C107C2213yes—
65#117×C107C2213yes—
66C2×Dic214C2313yes—
67C2×Dic214C2313yes—
68C2×Dic214C2313yes—
69Q8×D107—24no—
70Q8×D107—24no—
71C2×D428C2311yescharacteristic
72C2×C22⋊D107C2311yescharacteristic
73C2×C22⋊D107C2311yescharacteristic
74C2×C4×D107C2311yescharacteristic
75C2×C4×D107C2313yes—
76C2×C4×D107C2313yes—
77C2×C4×D107C2313yes—
78C4⋊Dic107C2313yes—
79C4⋊Dic107C2313yes—
80C4⋊Dic107C2313yes—
81C4⋊Dic107C2316yes—
82C4⋊Dic107C2313yes—
83Dic107⋊C4C2316yes—
84Dic107⋊C4C2313yes—
85Dic107⋊C4C2313yes—
86Dic107⋊C4C2316yes—
87Dic107⋊C4C2316yes—
88Dic107⋊C4C2313yes—
89Dic107⋊C4C2313yes—
90Dic107⋊C4C2311yescharacteristic
91Dic107⋊C4C2316yes—
92Dic107⋊C4C2313yes—
93C22⋊Dic107C2313yes—
94C22⋊Dic107C2313yes—
95(C2×D107)⋊C4C2311yescharacteristic
96(C2×D107)⋊C4C2313yes—
97(C2×D107)⋊C4C2313yes—
98(C2×D107)⋊C4C2313yes—
99(C2×D107)⋊C4C2313yes—
100(C2×D107)⋊C4C2313yes—
101C4×Dic107C2313yes—
102C4×Dic107C2313yes—
103C4×Dic107C2313yes—
104C4×Dic107C2316yes—
105C4×Dic107C2313yes—
106C2×Q8×C107—24no—
107C2×Q8×C107C2313yes—
108C2×Q8×C107C2311yescharacteristic
109C22×C4×C107C2313yes—
110C4⋊C4×C107C2313yes—
111C4⋊C4×C107C2313yes—
112C4⋊C4×C107C2313yes—
113C4⋊C4×C107C2313yes—
114C22⋊C4×C107C2311yescharacteristic
115C22⋊C4×C107C2313yes—
116C42×C107C2313yes—
117C42×C107C2316yes—
118C42×C107C2313yes—
119C23×C107C2411yescharacteristic
120C22×D107C2411yescharacteristic
121C22×D107C2411yescharacteristic
122Q8×C107—26no—
123Q8×C107#4114yes—
124C2×C4×C107—26no—
125C2×C4×C107C2411yescharacteristic
126C2×C4×C107C2413yes—
127C2×C4×C107C2413yes—
128C2×C4×C107C2411yescharacteristic
129C2×C4×C107C2413yes—
130C2×C4×C107C2413yes—
131C22⋊D107—22no—
132C22⋊D107—22no—
133C2×Dic107C2413yes—
134C2×Dic107C2411yescharacteristic
135C2×Dic107C2413yes—
136C2×Dic107C2413yes—
137C2×Dic107C2413yes—
138C2×Dic107C2411yescharacteristic
139D428—22no—
140C4×D107—26no—
141C4×D107—26no—
142C4×D107—26no—
143C4×D107—22no—
144Dic214—26no—
145Dic214—26no—
146Dic214—26no—
147C22×C107—21no—
148C22×C107—21no—
149C22×C107—21no—
150C22×C107C2511yescharacteristic
151C2×D107—21no—
152C2×D107—21no—
153C2×D107—21no—
154C2×D107—21no—
155C2×D107—21no—
156C2×D1072−1+412yes—
157C4×C107—23no—
158C4×C107—23no—
159C4×C107—23no—
160C4×C107—21no—
161C4×C107—21no—
162C4×C107C2×#4116yes—
163Dic107—23no—
164Dic107—23no—
165Dic107—23no—
166Dic107—23no—
167Dic107—21no—
168Dic1072−1+412yes—
169C2×C107—41no—
170C2×C107C22×#4111yescharacteristic
171C2×C107C2×2−1+411yescharacteristic
172C2×C107C2×2−1+411yescharacteristic
173D107—22no—
174D107—41no—
175#3523—1071nomaximal, Sylow
176C107#352311yescharacteristic, complemented, Sylow
177#557—1071no—
178#557—1073no—
179#556—1073no—
180#552—1073no—
181#552—1073no—
182D4×Q8—1071no—
183D4×Q8—1071no—
184D4×Q8—1071no—
185#541—1073no—
186#541—1073no—
187#533—1073no—
188#529—1073no—
189#519—1071no—
190#519D10711yescharacteristic, complemented
191C2×C4×Q8—1071no—
192C22×Q8—1071no—
193C22×Q8—1071no—
194C22×Q8C2×D10711yescharacteristic
195C4⋊Q8—1073no—
196C4⋊Q8—1073no—
197C4⋊Q8—1073no—
198C4⋊Q8—1073no—
199#126—1073no—
200#126—1073no—
201#126—1076no—
202#126—1076no—
203#125—1073no—
204#125—1076no—
205#125—1076no—
206#125—1073no—
207#125—1076no—
208#125—1073no—
209#124—1073no—
210#123—1073no—
211#123—1073no—
212C22⋊Q8—1073no—
213C22⋊Q8—1073no—
214C22⋊Q8—1073no—
215C22⋊Q8—1073no—
216C22⋊Q8—1073no—
217C22⋊Q8—1073no—
218C22⋊Q8—1073no—
219#121—1071no—
220C4×Q8—1071no—
221C4×Q8—1071no—
222C4×Q8—1073no—
223C4×Q8—1073no—
224C4×Q8—1073no—
225C4×Q8—1073no—
226C4×Q8—1073no—
227C4×Q8—1076no—
228C4×Q8C2×D10711yescharacteristic
229C4×Q8C2×D10713yes—
230C4×Q8C2×D10711yescharacteristic
231C4×Q8C2×D10713yes—
232C4×D4—1073no—
233C4×D4—1073no—
234C4×D4—1073no—
235#117—1073no—
236#117—1073no—
237#117—1076no—
238#117C2×D10713yes—
239#117C2×D10713yes—
240C2×C4⋊C4—1073no—
241C2×C42—1073no—
242C2×Q8—24no—
243C2×Q8—1073no—
244C2×Q8—1073no—
245C2×Q8—1073no—
246C2×Q8—2144no—
247C2×Q8—2144no—
248C2×Q8C22×D10713yes—
249C2×Q8C22×D10711yescharacteristic
250C2×D4—1071no—
251C2×D4—1071no—
252C2×D4—1071no—
253C22×C4—1071no—
254C22×C4—1073no—
255C22×C4—1073no—
256C22×C4—1073no—
257C22×C4C22×D10713yes—
258C4⋊C4—1073no—
259C4⋊C4—1076no—
260C4⋊C4—1073no—
261C4⋊C4—1073no—
262C4⋊C4—1073no—
263C4⋊C4—1076no—
264C4⋊C4—1076no—
265C4⋊C4—1073no—
266C4⋊C4—1073no—
267C4⋊C4—1073no—
268C4⋊C4—1071no—
269C4⋊C4—1076no—
270C4⋊C4—1076no—
271C4⋊C4—1073no—
272C4⋊C4—1073no—
273C4⋊C4C22×D10713yes—
274C4⋊C4C22×D10713yes—
275C4⋊C4C22×D10713yes—
276C4⋊C4C22×D10713yes—
277C22⋊C4—1073no—
278C22⋊C4—1073no—
279C22⋊C4—1071no—
280C22⋊C4—1073no—
281C22⋊C4—1073no—
282C22⋊C4—1073no—
283C22⋊C4—1073no—
284C22⋊C4—1073no—
285C22⋊C4C22×D10711yescharacteristic
286C22⋊C4C22×D10713yes—
287C42—1073no—
288C42—1073no—
289C42—1073no—
290C42—1076no—
291C42—1073no—
292C42C22×D10713yes—
293C42C22×D10716yes—
294C42C22×D10713yes—
295C23—1071no—
296C23—1071no—
297C23C23×D10711yescharacteristic
298Q8—26no—
299Q8—2146no—
300Q8—2146no—
301Q8—2146no—
302Q8#1313381814yes—
303D4—2142no—
304D4—2142no—
305D4—2142no—
306C2×C4—26no—
307C2×C4—1071no—
308C2×C4—1073no—
309C2×C4—1071no—
310C2×C4—1073no—
311C2×C4—1073no—
312C2×C4—1073no—
313C2×C4—2142no—
314C2×C4—2146no—
315C2×C4—2146no—
316C2×C4—2146no—
317C2×C4C23×D10713yes—
318C2×C4C23×D10711yescharacteristic
319C2×C4C23×D10713yes—
320C2×C4C23×D10711yescharacteristic
321C2×C4C23×D10713yes—
322C2×C4C23×D10713yes—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—1072no—
327C22—2141no—
328C22—2141no—
329C22—2141no—
330C22—2141no—
331C22—2141no—
332C22C24×D10711yescharacteristic
333C4—23no—
334C4—21no—
335C4—23no—
336C4—21no—
337C4—23no—
338C4—1072no—
339C4—2143no—
340C4—2143no—
341C4—2143no—
342C4—2141no—
343C4—2143no—
344C4C2×#1313381816yes—
345C2—41no—
346C2—4281no—
347C2—2142no—
348C2C22×#1313381811yescharacteristic
349C22−1+4×D10711yescharacteristic
350C22−1+4×D10711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D107
commutator, socle, cosocleC22×C107C25
Fitting#519×C107C2
prime radical#16812082C22
hypercenter#519D107
hypocenterC107#3523
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
1. #20293355C2semidirect
2. #20293353C2semidirect
3. #20293345C2semidirect
4. #20293318C2semidirect
5. #20293316C2semidirect
6. Q8×D428C2semidirect
7. Q8×C22⋊D107C2semidirect
8. Q8×C22⋊D107C2semidirect
9. #20293239C2semidirect
10. #20293235C2semidirect
11. #20293181C2semidirect
13. #20293103C2semidirect
14. C4×Q8×D107C2semidirect
15. #519×C107C2semidirect
176. C107#3523semidirect
190. #519D107semidirect

Maximal subgroups

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

subgroupindex
#202933552
#202933532
#202933452
#202933182
#202933162
Q8×D4282
Q8×C22⋊D1072
#202932392
#202932352
#202931812
#202931552
#202931032
C4×Q8×D1072
#519×C1072
#3523107