Finite Group Databasebeta

← #28348350

Subgroups — #28348350

Order 13696.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293317C211yesmaximal, characteristic, complemented
2#20293286C211yesmaximal, characteristic
3#20292982C211yesmaximal, characteristic, complemented
4#20292969C211yesmaximal, characteristic, complemented
5#20292941C211yesmaximal, characteristic, complemented
6#20292935C211yesmaximal, characteristic, complemented
7#20292868C211yesmaximal, characteristic, complemented
8#20292856C211yesmaximal, characteristic, complemented
9#20292850C211yesmaximal, characteristic, complemented
10#20292791C211yesmaximal, characteristic, complemented
11#20292785C211yesmaximal, characteristic, complemented
12C4×Dic428C211yesmaximal, characteristic, complemented
13#20292624C211yesmaximal, characteristic, complemented
14C2×#16812034C211yesmaximal, characteristic, complemented
15C8×D4×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic
17C22×Dic214C2211yescharacteristic
18C2×Dic428C2211yescharacteristic
19C2×Dic428C2211yescharacteristic
20C2×#13133807C2211yescharacteristic
21#16812102—21no—
22#16812101C2211yescharacteristic
23#16812099—21no—
24#16812097C2211yescharacteristic
25#16812095—21no—
26#16812090—21no—
27#16812088C2211yescharacteristic
28#16812088C2211yescharacteristic
29#16812088C2211yescharacteristic
30#16812087—21no—
31#16812083C2211yescharacteristic
32#16812081—21no—
33C4×Dic214C2211yescharacteristic
34C4×Dic214C2211yescharacteristic
35(C22⋊D107)⋊C4—21no—
36#16812068C2211yescharacteristic
37#16812065—21no—
38C2×C4⋊Dic107C2211yescharacteristic
39#16812043C2211yescharacteristic
40#16812043C2211yescharacteristic
41#16812041C2211yescharacteristic
42#16812034C2211yescharacteristic
43#16812034C2211yescharacteristic
44#16812034C2211yescharacteristic
45#16812034C2211yescharacteristic
46#16812034C2211yescharacteristic
47#16812034C2211yescharacteristic
48#16812034C2211yescharacteristic
49#16812031C2211yescharacteristic
50#16812031C2211yescharacteristic
51#16812031C2211yescharacteristic
52C22×C8×C107C2211yescharacteristic
53C22×C8×C107C2211yescharacteristic
54C4×D4×C107C2211yescharacteristic
55C4⋊C8×C107C2211yescharacteristic
56C22⋊C8×C107C2211yescharacteristic
57C22⋊C8×C107C2211yescharacteristic
58C4×C8×C107C2211yescharacteristic
59#13133818—22no—
60#13133818—41no—
61C2×Dic214—22no—
62C2×Dic214—21no—
63C2×Dic214—41no—
64C2×Dic214C2311yescharacteristic
65C2×Dic214C2311yescharacteristic
66C2×Dic214C2311yescharacteristic
67C2×D428C2311yescharacteristic
68C2×C22⋊D107—21no—
69C2×C22⋊D107—21no—
70C22×Dic107—21no—
71C22×Dic107—21no—
72C2×C4×D107—21no—
73Dic428—22no—
74Dic428—41no—
75#13133807—41no—
76C4⋊Dic107—21no—
77C4⋊Dic107C2311yescharacteristic
78C4⋊Dic107C2311yescharacteristic
79C4⋊Dic107C2311yescharacteristic
80C4⋊Dic107C2311yescharacteristic
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86Dic107⋊C4—21no—
87Dic107⋊C4—21no—
88C22⋊Dic107—21no—
89C22⋊Dic107—21no—
90C22⋊Dic107—21no—
91C22⋊Dic107—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95C4×Dic107—21no—
96C4×Dic107—21no—
97C4×Dic107—21no—
98C2×D4×C107D411yescharacteristic
99C22×C4×C107C2311yescharacteristic
100C22×C4×C107C2311yescharacteristic
101C2×C8×C107—41no—
102C2×C8×C107—41no—
103C2×C8×C107C2311yescharacteristic
104C2×C8×C107C2311yescharacteristic
105C2×C8×C107C2311yescharacteristic
106C2×C8×C107C2311yescharacteristic
107C4⋊C4×C107D411yescharacteristic
108C22⋊C4×C107D411yescharacteristic
109C22⋊C4×C107D411yescharacteristic
110C42×C107C2311yescharacteristic
111C23×C107C2×D411yescharacteristic
112C23×C107C2×D411yescharacteristic
113C22×D107—21no—
114D4×C107—22no—
115C2×C4×C107—21no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107C2×D411yescharacteristic
120C2×C4×C107C2×D411yescharacteristic
121C2×C4×C107C2×D411yescharacteristic
122C2×C4×C107C2×D411yescharacteristic
123C2×C4×C107C2411yescharacteristic
124C22⋊D107—42no—
125C22⋊D107—42no—
126C2×Dic107—21no—
127C2×Dic107—21no—
128C2×Dic107—21no—
129C2×Dic107—21no—
130C2×Dic107—21no—
131C2×Dic107—21no—
132C2×Dic107—21no—
133C2×Dic107—42no—
134C2×Dic107—42no—
135D428—21no—
136D428—21no—
137C4×D107—42no—
138Dic214—21no—
139Dic214—21no—
140Dic214—21no—
141Dic214—21no—
142Dic214—21no—
143Dic214—42no—
144Dic214#4112yes—
145C8×C107—21no—
146C8×C107—21no—
147C8×C107—21no—
148C8×C107—21no—
149C22×C107—21no—
150C22×C107—21no—
151C22×C107—21no—
152C22×C107—21no—
153C22×C107—21no—
154C22×C107#13512yes—
155C22×C107C22×D411yescharacteristic
156C2×D107—41no—
157C2×D107—41no—
158C2×D107—41no—
159C4×C107—21no—
160C4×C107—21no—
161C4×C107—21no—
162C4×C107—21no—
163C4×C107C2×#4111yescharacteristic
164C4×C107H(5,𝔽2)11yescharacteristic
165Dic107—22no—
166Dic107—41no—
167Dic107—41no—
168Dic107—41no—
169Dic107—41no—
170Dic107—41no—
171Dic107—41no—
172C2×C107—22no—
173C2×C107—41no—
174C2×C107#54611yescharacteristic
175C2×C107C2×#13511yescharacteristic
176C2×C107#57811yescharacteristic
177D107—81no—
178#3300—1071nomaximal, Sylow
179C107#330011yescharacteristic, complemented, Sylow
180#552—1071no—
181#548—1071no—
182#488—1071no—
183#485—1071no—
184#480—1071no—
185#479—1071no—
186#467—1071no—
187#465—1071no—
188#464—1071no—
189#452—1071no—
190#451—1071no—
191C4×Q16—1071no—
192C8×D4D10711yescharacteristic, complemented
193#416—1071no—
194C2×#103—1071no—
195C2×#41—1071no—
196C22×Q8—1071no—
197C2×Q16—1071no—
198C2×Q16—1071no—
199C2×SD16—1071no—
200C22×C8C2×D10711yescharacteristic
201C22×C8C2×D10711yescharacteristic
202#126—2141no—
203#125—1071no—
204#125—2141no—
205#124—1071no—
206#124—2141no—
207#123—2141no—
208C22⋊Q8—1071no—
209C22⋊Q8—1071no—
210C22⋊Q8—1071no—
211C22⋊Q8—2141no—
212#121—1071no—
213#121—2141no—
214C4×Q8—1071no—
215C4×Q8—1071no—
216C4×D4—2141no—
217C4×D4C2×D10711yescharacteristic
218#117—1071no—
219#117—2141no—
220C2×C4⋊C4—1071no—
221#107—1071no—
222#107—1071no—
223#106—1071no—
224C4⋊C8C2×D10711yescharacteristic
225#103—1071no—
226#103—1071no—
227#103—1071no—
228#103—1071no—
229#103—1071no—
230#103—1071no—
231#103—1071no—
232#102—1071no—
233#102—1071no—
234#102—1071no—
235C22⋊C8C2×D10711yescharacteristic
236C22⋊C8C2×D10711yescharacteristic
237C4×C8C2×D10711yescharacteristic
238#41—4281no—
239#41—2142no—
240C2×Q8—1071no—
241C2×Q8—1071no—
242C2×Q8—1071no—
243C2×Q8—4281no—
244C2×Q8—2142no—
245C2×Q8—2141no—
246C2×D4—1071no—
247C2×D4—2141no—
248C2×D4—2141no—
249C2×D4D42811yescharacteristic
250C22×C4—2141no—
251C22×C4—2141no—
252C22×C4—2141no—
253C22×C4C22×D10711yescharacteristic
254C22×C4C22×D10711yescharacteristic
255Q16—4281no—
256Q16—2142no—
257SD16—4281no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8C22×D10711yescharacteristic
261C2×C8C22×D10711yescharacteristic
262C2×C8C22×D10711yescharacteristic
263C2×C8C22×D10711yescharacteristic
264C4⋊C4—1071no—
265C4⋊C4—1071no—
266C4⋊C4—1071no—
267C4⋊C4—1071no—
268C4⋊C4—2141no—
269C4⋊C4—2141no—
270C4⋊C4—2141no—
271C4⋊C4—2141no—
272C4⋊C4—2141no—
273C4⋊C4—2141no—
274C4⋊C4—2141no—
275C4⋊C4—2141no—
276C4⋊C4D42811yescharacteristic
277C22⋊C4—2141no—
278C22⋊C4—2141no—
279C22⋊C4—2141no—
280C22⋊C4—2141no—
281C22⋊C4—2141no—
282C22⋊C4—2141no—
283C22⋊C4—2141no—
284C22⋊C4D42811yescharacteristic
285C22⋊C4D42811yescharacteristic
286C42—2141no—
287C42—2141no—
288C42—2141no—
289C42C22×D10711yescharacteristic
290C23—2141no—
291C23C2×D42811yescharacteristic
292C23C2×D42811yescharacteristic
293Q8—1072no—
294Q8—4282no—
295Q8—2141no—
296Q8—2141no—
297Q8—2141no—
298Q8—2141no—
299Q8—2141no—
300D4—22no—
301D4—4282no—
302D4—4282no—
303D4—2141no—
304D4—2141no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—4282no—
310C2×C4—4282no—
311C2×C4—4282no—
312C2×C4—2141no—
313C2×C4—2141no—
314C2×C4—2141no—
315C2×C4—2141no—
316C2×C4—2141no—
317C2×C4—2141no—
318C2×C4—2141no—
319C2×C4C2×D42811yescharacteristic
320C2×C4C2×D42811yescharacteristic
321C2×C4C2×D42811yescharacteristic
322C2×C4C2×D42811yescharacteristic
323C2×C4C23×D10711yescharacteristic
324C8—21no—
325C8—21no—
326C8—21no—
327C8—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—4281no—
334C22—4281no—
335C22—4281no—
336C22#1681213112yes—
337C22C22×D42811yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—4281no—
343C4—4281no—
344C4—4281no—
345C4—4281no—
346C4—4281no—
347C4—4281no—
348C4—2142no—
349C4#41×D10711yescharacteristic
350C4#1681215411yescharacteristic
351C2—22no—
352C2—41no—
353C2—8561no—
354C2#2029327211yescharacteristic
355C2C2×#1681213111yescharacteristic
356C2#2029344411yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC8×D4×C107C2
socleC22×C107C22×D4
prime radical#20293286C2
hypercenterC8×D4D107
hypocenterC107#3300
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #20293317C2semidirect
3. #20292982C2semidirect
4. #20292969C2semidirect
5. #20292941C2semidirect
6. #20292935C2semidirect
7. #20292868C2semidirect
8. #20292856C2semidirect
9. #20292850C2semidirect
10. #20292791C2semidirect
11. #20292785C2semidirect
12. C4×Dic428C2semidirect
13. #20292624C2semidirect
14. C2×#16812034C2semidirect
15. C8×D4×C107C2semidirect
179. C107#3300semidirect
192. C8×D4D107semidirect

Maximal subgroups

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

subgroupindex
#202933172
#202932862
#202929822
#202929692
#202929412
#202929352
#202928682
#202928562
#202928502
#202927912
#202927852
C4×Dic4282
#202926242
C2×#168120342
C8×D4×C1072
#3300107