Finite Group Databasebeta

← #28350548

Subgroups — #28350548

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#20293353C211yesmaximal, characteristic, complemented
2#20293352C213yesmaximal, complemented
3#20293348C213yesmaximal, complemented
4#20293319C213yesmaximal, complemented
5#20293317C213yesmaximal, complemented
6Q8×C22⋊D107C211yesmaximal, characteristic, complemented
7Q8×C22⋊D107C211yesmaximal, characteristic, complemented
8#20293238C213yesmaximal, complemented
9#20293237C213yesmaximal, complemented
10#20293179C213yesmaximal, complemented
11#20293157C213yesmaximal
12#20293105C211yesmaximal, characteristic, complemented
13#20293103C211yesmaximal, characteristic, complemented
14C2×Q8×Dic107C211yesmaximal, characteristic, complemented
15D4×Q8×C107C211yesmaximal, characteristic, complemented
16C2×Q8×D107C2211yescharacteristic
17#16812108C2213yes—
18#16812107C2213yes—
19#16812107C2213yes—
20#16812104C2213yes—
21#16812104C2216yes—
22#16812102C2213yes—
23#16812102C2216yes—
24#16812101C2213yes—
25#16812100C2216yes—
26#16812100C2213yes—
27#16812099C2213yes—
28#16812099C2216yes—
29#16812099C2216yes—
30#16812095C2213yes—
31#16812093C2213yes—
32#16812090C2213yes—
33#16812088C2213yes—
34#16812088C2213yes—
35#16812086C2213yes—
36#16812085C2213yes—
37#16812085C2213yes—
38#16812081C2211yescharacteristic
39Q8×Dic107C2213yes—
40Q8×Dic107C2213yes—
41Q8×Dic107C2211yescharacteristic
42Q8×Dic107C2211yescharacteristic
43Q8×Dic107C2213yes—
44Q8×Dic107C2213yes—
45Q8×Dic107C2211yescharacteristic
46C4×Dic214C2213yes—
47C4×Dic214C2213yes—
48#16812074C2213yes—
49#16812074C2216yes—
50C4×C22⋊D107C2213yes—
51C4×C22⋊D107C2213yes—
52#16812068C2213yes—
53#16812068C2213yes—
54#16812067C2213yes—
55#16812066C2213yes—
56#16812065C2216yes—
57C2×C4⋊Dic107C2213yes—
58C2×C4×Dic107C2213yes—
59C22×Q8×C107C2211yescharacteristic
60C22×Q8×C107C2211yescharacteristic
61C4⋊Q8×C107C2213yes—
62C22⋊Q8×C107C2213yes—
63C22⋊Q8×C107C2213yes—
64C4×Q8×C107C2211yescharacteristic
65C4×D4×C107C2213yes—
66C2×Dic214C2313yes—
67C2×Dic214C2313yes—
68Q8×D107—24no—
69C2×C22⋊D107C2311yescharacteristic
70C2×C22⋊D107C2311yescharacteristic
71C22×Dic107C2311yescharacteristic
72C22×Dic107C2313yes—
73C2×C4×D107C2313yes—
74C4⋊Dic107C2313yes—
75C4⋊Dic107C2313yes—
76C4⋊Dic107C2316yes—
77C4⋊Dic107C2313yes—
78C4⋊Dic107C2313yes—
79C4⋊Dic107C2313yes—
80C4⋊Dic107C2313yes—
81Dic107⋊C4C2316yes—
82Dic107⋊C4C2316yes—
83Dic107⋊C4C2313yes—
84Dic107⋊C4C2313yes—
85Dic107⋊C4C2313yes—
86Dic107⋊C4C2313yes—
87Dic107⋊C4C2316yes—
88Dic107⋊C4C2311yescharacteristic
89Dic107⋊C4C2313yes—
90C22⋊Dic107C2313yes—
91C22⋊Dic107C2311yescharacteristic
92C22⋊Dic107C2313yes—
93C22⋊Dic107C2313yes—
94(C2×D107)⋊C4C2311yescharacteristic
95(C2×D107)⋊C4C2313yes—
96(C2×D107)⋊C4C2313yes—
97(C2×D107)⋊C4C2313yes—
98C4×Dic107C2313yes—
99C4×Dic107C2313yes—
100C4×Dic107C2313yes—
101C4×Dic107C2313yes—
102C4×Dic107C2316yes—
103C4×Dic107C2316yes—
104C4×Dic107C2313yes—
105C2×Q8×C107—24no—
106C2×Q8×C107—24no—
107C2×Q8×C107C2313yes—
108C2×Q8×C107C2313yes—
109C2×Q8×C107C2311yescharacteristic
110C2×D4×C107C2311yescharacteristic
111C22×C4×C107C2313yes—
112C22×C4×C107C2313yes—
113C4⋊C4×C107C2313yes—
114C4⋊C4×C107C2316yes—
115C4⋊C4×C107C2313yes—
116C22⋊C4×C107C2313yes—
117C22⋊C4×C107C2313yes—
118C42×C107C2313yes—
119C23×C107C2411yescharacteristic
120C23×C107C2411yescharacteristic
121C22×D107C2411yescharacteristic
122Q8×C107—26no—
123Q8×C107—26no—
124Q8×C107#4114yes—
125D4×C107—22no—
126C2×C4×C107—26no—
127C2×C4×C107—26no—
128C2×C4×C107C2411yescharacteristic
129C2×C4×C107C2413yes—
130C2×C4×C107C2413yes—
131C2×C4×C107C2413yes—
132C2×C4×C107C2413yes—
133C22⋊D107—22no—
134C22⋊D107—22no—
135C2×Dic107—26no—
136C2×Dic107—22no—
137C2×Dic107C2411yescharacteristic
138C2×Dic107C2413yes—
139C2×Dic107C2413yes—
140C2×Dic107C2411yescharacteristic
141C2×Dic107C2413yes—
142C2×Dic107C2413yes—
143C2×Dic107C2411yescharacteristic
144C4×D107—26no—
145Dic214—26no—
146Dic214—26no—
147C22×C107—21no—
148C22×C107—21no—
149C22×C107—21no—
150C22×C107—21no—
151C22×C107—21no—
152C22×C1072−1+412yes—
153C22×C107C2511yescharacteristic
154C2×D107—21no—
155C2×D107—21no—
156C2×D107—21no—
157C4×C107—23no—
158C4×C107—23no—
159C4×C107—23no—
160C4×C107—21no—
161C4×C107C2×#4116yes—
162Dic107—23no—
163Dic107—23no—
164Dic107—21no—
165Dic107—23no—
166Dic107—23no—
167Dic107—21no—
168Dic1072−1+412yes—
169C2×C107—22no—
170C2×C107—41no—
171C2×C107C22×#4111yescharacteristic
172C2×C107C2×2−1+411yescharacteristic
173C2×C107C2×2−1+411yescharacteristic
174D107—41no—
175#3523—1071nomaximal, Sylow
176C107#352311yescharacteristic, complemented, Sylow
177#557—1073no—
178#557—1071no—
179#556—1073no—
180#552—1073no—
181#552—1073no—
182D4×Q8—1071no—
183D4×Q8—1071no—
184D4×Q8D10711yescharacteristic, complemented
185#541—1073no—
186#541—1073no—
187#533—1073no—
188#529—1073no—
189#519—1071no—
190#519—1071no—
191C2×C4×Q8—1071no—
192C22×Q8—1071no—
193C22×Q8C2×D10711yescharacteristic
194C22×Q8C2×D10711yescharacteristic
195C4⋊Q8—1073no—
196C4⋊Q8—1073no—
197C4⋊Q8—1073no—
198C4⋊Q8C2×D10713yes—
199#126—1073no—
200#126—1076no—
201#126—1073no—
202#126—1076no—
203#125—1073no—
204#125—1076no—
205#125—1076no—
206#125—1076no—
207#125—1073no—
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⋊Q8C2×D10713yes—
218C22⋊Q8C2×D10713yes—
219#121—1071no—
220C4×Q8—1071no—
221C4×Q8—1073no—
222C4×Q8—1071no—
223C4×Q8—1073no—
224C4×Q8—1073no—
225C4×Q8—1071no—
226C4×Q8—1073no—
227C4×Q8—1073no—
228C4×Q8—1076no—
229C4×Q8—1073no—
230C4×Q8—1073no—
231C4×Q8C2×D10711yescharacteristic
232C4×D4—1073no—
233C4×D4—1073no—
234C4×D4C2×D10713yes—
235#117—1073no—
236#117—1073no—
237#117—1073no—
238#117—1076no—
239#117—1073no—
240C2×C4⋊C4—1073no—
241C2×C42—1073no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—1073no—
245C2×Q8—1073no—
246C2×Q8—2144no—
247C2×Q8C22×D10713yes—
248C2×Q8C22×D10713yes—
249C2×Q8C22×D10711yescharacteristic
250C2×D4—1071no—
251C2×D4—1071no—
252C2×D4C22×D10711yescharacteristic
253C22×C4—1071no—
254C22×C4—1073no—
255C22×C4—1073no—
256C22×C4C22×D10713yes—
257C22×C4C22×D10713yes—
258C4⋊C4—1073no—
259C4⋊C4—1073no—
260C4⋊C4—1076no—
261C4⋊C4—1073no—
262C4⋊C4—1073no—
263C4⋊C4—1073no—
264C4⋊C4—1073no—
265C4⋊C4—1073no—
266C4⋊C4—1076no—
267C4⋊C4—1076no—
268C4⋊C4—1073no—
269C4⋊C4—1073no—
270C4⋊C4—1071no—
271C4⋊C4—1076no—
272C4⋊C4—1073no—
273C4⋊C4—1073no—
274C4⋊C4C22×D10713yes—
275C4⋊C4C22×D10713yes—
276C4⋊C4C22×D10716yes—
277C22⋊C4—1071no—
278C22⋊C4—1073no—
279C22⋊C4—1073no—
280C22⋊C4—1073no—
281C22⋊C4—1071no—
282C22⋊C4—1073no—
283C22⋊C4—1073no—
284C22⋊C4—1073no—
285C22⋊C4C22×D10713yes—
286C22⋊C4C22×D10713yes—
287C42—1073no—
288C42—1073no—
289C42—1073no—
290C42—1073no—
291C42—1076no—
292C42—1076no—
293C42—1073no—
294C42C22×D10713yes—
295C23—1071no—
296C23C23×D10711yescharacteristic
297C23C23×D10711yescharacteristic
298Q8—26no—
299Q8—26no—
300Q8—2146no—
301Q8—2146no—
302Q8#1313381714yes—
303D4—22no—
304D4—2142no—
305D4—2142no—
306C2×C4—26no—
307C2×C4—26no—
308C2×C4—1071no—
309C2×C4—1071no—
310C2×C4—1073no—
311C2×C4—1073no—
312C2×C4—1071no—
313C2×C4—1073no—
314C2×C4—1073no—
315C2×C4—2142no—
316C2×C4—2146no—
317C2×C4—2146no—
318C2×C4C23×D10713yes—
319C2×C4C23×D10713yes—
320C2×C4C23×D10713yes—
321C2×C4C23×D10711yescharacteristic
322C2×C4C23×D10713yes—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—2141no—
329C22—2141no—
330C22—2141no—
331C22#1681215712yes—
332C22C24×D10711yescharacteristic
333C4—23no—
334C4—23no—
335C4—23no—
336C4—21no—
337C4—1072no—
338C4—2143no—
339C4—2141no—
340C4—2143no—
341C4—2143no—
342C4—2143no—
343C4—2141no—
344C4C2×#1313381716yes—
345C2—22no—
346C2—41no—
347C2—4281no—
348C2C22×#1313381711yescharacteristic
349C22−1+4×D10711yescharacteristic
350C2C2×#1681215711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D107
commutator, socle, cosocleC22×C107C25
FittingD4×Q8×C107C2
prime radical#16812081C22
hypercenterD4×Q8D107
hypocenterC107#3523
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. #20293353C2semidirect
2. #20293352C2semidirect
3. #20293348C2semidirect
4. #20293319C2semidirect
5. #20293317C2semidirect
6. Q8×C22⋊D107C2semidirect
7. Q8×C22⋊D107C2semidirect
8. #20293238C2semidirect
9. #20293237C2semidirect
10. #20293179C2semidirect
12. #20293105C2semidirect
13. #20293103C2semidirect
14. C2×Q8×Dic107C2semidirect
15. D4×Q8×C107C2semidirect
176. C107#3523semidirect
184. D4×Q8D107semidirect

Maximal subgroups

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

subgroupindex
#202933532
#202933522
#202933482
#202933192
#202933172
Q8×C22⋊D1072
#202932382
#202932372
#202931792
#202931572
#202931052
#202931032
C2×Q8×Dic1072
D4×Q8×C1072
#3523107