Finite Group Databasebeta

← #28348349

Subgroups — #28348349

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#20293316C211yesmaximal, characteristic, complemented
2#20293284C211yesmaximal, characteristic
3#20292980C211yesmaximal, characteristic, complemented
4#20292968C211yesmaximal, characteristic, complemented
5#20292939C211yesmaximal, characteristic, complemented
6#20292936C211yesmaximal, characteristic, complemented
7#20292868C211yesmaximal, characteristic, complemented
8#20292855C211yesmaximal, characteristic, complemented
9#20292848C211yesmaximal, characteristic, complemented
10#20292789C211yesmaximal, characteristic, complemented
11#20292784C211yesmaximal, characteristic, complemented
12#20292719C211yesmaximal, characteristic, complemented
13C8×C22⋊D107C211yesmaximal, characteristic, complemented
14#103×D107C211yesmaximal, characteristic, complemented
15#416×C107C211yesmaximal, characteristic, complemented
16C2×Q8×D107C2211yescharacteristic
17C2×Dic428C2211yescharacteristic
18C2×#13133808C2211yescharacteristic
19C2×#13133806C2211yescharacteristic
20C2×C8×D107C2211yescharacteristic
21#16812102—21no—
22#16812101—21no—
23#16812100C2211yescharacteristic
24#16812096—21no—
25#16812096C2211yescharacteristic
26#16812091—21no—
27#16812089C2211yescharacteristic
28#16812088—21no—
29#16812088C2211yescharacteristic
30#16812088C2211yescharacteristic
31#16812080—21no—
32#16812080C2211yescharacteristic
33Q8×Dic107C2211yescharacteristic
34#16812074C2211yescharacteristic
35C4×C22⋊D107—21no—
36C4×C22⋊D107C2211yescharacteristic
37#16812066—21no—
38C4⋊C4×D107C2211yescharacteristic
39#16812043C2211yescharacteristic
40#16812042C2211yescharacteristic
41#16812040C2211yescharacteristic
42Dic107⋊C8C2211yescharacteristic
43#16812034C2211yescharacteristic
44#16812034C2211yescharacteristic
45#16812033C2211yescharacteristic
46#16812032C2211yescharacteristic
47#16812032C2211yescharacteristic
48#16812030C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×#41×C107C2211yescharacteristic
53C22×C8×C107C2211yescharacteristic
54#117×C107C2211yescharacteristic
55#103×C107C2211yescharacteristic
56#103×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214C2311yescharacteristic
61C2×Dic214C2311yescharacteristic
62Q8×D107—22no—
63Q8×D107—41no—
64C2×C22⋊D107—21no—
65C2×C22⋊D107D411yescharacteristic
66C2×C4×D107—21no—
67C2×C4×D107—21no—
68C2×C4×D107C2311yescharacteristic
69Dic428—41no—
70#13133808—22no—
71#13133806—41no—
72C2×C107⋊C8C2311yescharacteristic
73C2×C107⋊C8C2311yescharacteristic
74C8×D107—41no—
75C4⋊Dic107—21no—
76C4⋊Dic107—21no—
77C4⋊Dic107—21no—
78C4⋊Dic107C2311yescharacteristic
79C4⋊Dic107C2311yescharacteristic
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85Dic107⋊C4D411yescharacteristic
86C22⋊Dic107—21no—
87C22⋊Dic107—21no—
88C22⋊Dic107—21no—
89C22⋊Dic107—21no—
90C22⋊Dic107—21no—
91C22⋊Dic107D411yescharacteristic
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4D411yescharacteristic
94C4×Dic107—21no—
95C4×Dic107—21no—
96C4×Dic107C2311yescharacteristic
97#41×C107—22no—
98#41×C107—41no—
99C2×Q8×C107C2311yescharacteristic
100C2×D4×C107—21no—
101C2×D4×C107C2311yescharacteristic
102C22×C4×C107—21no—
103C22×C4×C107C2311yescharacteristic
104C2×C8×C107—41no—
105C2×C8×C107C2311yescharacteristic
106C2×C8×C107C2311yescharacteristic
107C4⋊C4×C107C2311yescharacteristic
108C4⋊C4×C107C2311yescharacteristic
109C22⋊C4×C107—21no—
110C42×C107—21no—
111C23×C107—21no—
112C23×C107C2×D411yescharacteristic
113C22×D107C2×D411yescharacteristic
114Q8×C107—21no—
115Q8×C107#4112yes—
116D4×C107—21no—
117D4×C107—21no—
118D4×C107—42no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—42no—
125C2×C4×C107C2×D411yescharacteristic
126C2×C4×C107C2411yescharacteristic
127C22⋊D107—22no—
128C22⋊D107—42no—
129C2×Dic107—21no—
130C2×Dic107—21no—
131C2×Dic107—21no—
132C2×Dic107—21no—
133C2×Dic107C2×D411yescharacteristic
134C2×Dic107C2×D411yescharacteristic
135C2×Dic107C2×D411yescharacteristic
136C4×D107—21no—
137C4×D107—21no—
138C4×D107—42no—
139C4×D107—42no—
140Dic214—21no—
141Dic214—21no—
142Dic214—21no—
143Dic214—21no—
144Dic214—42no—
145C8×C107—21no—
146C8×C107—21no—
147C107⋊C8—21no—
148C107⋊C8—21no—
149C22×C107—21no—
150C22×C107—21no—
151C22×C107—21no—
152C22×C107—41no—
153C22×C107—41no—
154C22×C107—41no—
155C22×C107C22×D411yescharacteristic
156C2×D107—21no—
157C2×D107—21no—
158C2×D107#13512yes—
159C4×C107—22no—
160C4×C107—21no—
161C4×C107—41no—
162C4×C107—41no—
163C4×C107C2×#4111yescharacteristic
164C4×C107H(5,𝔽2)11yescharacteristic
165Dic107—21no—
166Dic107—21no—
167Dic107—21no—
168Dic107—41no—
169Dic107—41no—
170Dic107—41no—
171Dic107—41no—
172C2×C107—41no—
173C2×C107—81no—
174C2×C107#54611yescharacteristic
175C2×C107C2×#13511yescharacteristic
176C2×C107#57811yescharacteristic
177D107—22no—
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×D4—1071no—
193#416D10711yescharacteristic, complemented
194C2×#103—1071no—
195C2×#41C2×D10711yescharacteristic
196C22×Q8—1071no—
197C2×Q16—1071no—
198C2×Q16—1071no—
199C2×SD16—1071no—
200C22×C8—1071no—
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—1071no—
217C4×D4—2141no—
218#117—2141no—
219#117C2×D10711yescharacteristic
220C2×C4⋊C4—1071no—
221#107—1071no—
222#107—1071no—
223#106—1071no—
224C4⋊C8—1071no—
225#103—1071no—
226#103—1071no—
227#103—1071no—
228#103—1071no—
229#103—1071no—
230#103C2×D10711yescharacteristic
231#103C2×D10711yescharacteristic
232#102—1071no—
233#102C2×D10711yescharacteristic
234#102C2×D10711yescharacteristic
235C22⋊C8—1071no—
236C22⋊C8—1071no—
237C4×C8—1071no—
238#41—22no—
239#41—41no—
240C2×Q8—1071no—
241C2×Q8—1071no—
242C2×Q8—4281no—
243C2×Q8—2141no—
244C2×Q8—2142no—
245C2×Q8C22×D10711yescharacteristic
246C2×D4—21no—
247C2×D4—1071no—
248C2×D4—2141no—
249C2×D4C22×D10711yescharacteristic
250C22×C4—21no—
251C22×C4—1071no—
252C22×C4—2141no—
253C22×C4—2141no—
254C22×C4C22×D10711yescharacteristic
255Q16—4281no—
256Q16—2142no—
257SD16—4281no—
258C2×C8—41no—
259C2×C8—1071no—
260C2×C8—1071no—
261C2×C8—4281no—
262C2×C8C22×D10711yescharacteristic
263C2×C8C22×D10711yescharacteristic
264C4⋊C4—1071no—
265C4⋊C4—1071no—
266C4⋊C4—1071no—
267C4⋊C4—2141no—
268C4⋊C4—2141no—
269C4⋊C4—2141no—
270C4⋊C4—2141no—
271C4⋊C4—2141no—
272C4⋊C4—2141no—
273C4⋊C4—2141no—
274C4⋊C4—2141no—
275C4⋊C4C22×D10711yescharacteristic
276C4⋊C4C22×D10711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—1071no—
279C22⋊C4—1071no—
280C22⋊C4—2141no—
281C22⋊C4—2141no—
282C22⋊C4—2141no—
283C22⋊C4—2141no—
284C22⋊C4—2141no—
285C22⋊C4—2141no—
286C42—21no—
287C42—1071no—
288C42—2141no—
289C42—2141no—
290C23—21no—
291C23—1071no—
292C23D4×D10711yescharacteristic
293Q8—21no—
294Q8—4282no—
295Q8—2141no—
296Q8—2141no—
297Q8—2141no—
298Q8—2141no—
299Q8#1313381812yes—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—4282no—
304D4—2142no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—1071no—
312C2×C4—1071no—
313C2×C4—1071no—
314C2×C4—4282no—
315C2×C4—4282no—
316C2×C4—2141no—
317C2×C4—2141no—
318C2×C4—2141no—
319C2×C4—2141no—
320C2×C4—2141no—
321C2×C4—2141no—
322C2×C4D4×D10711yescharacteristic
323C2×C4C23×D10711yescharacteristic
324C8—21no—
325C8—21no—
326C8—2141no—
327C8—2141no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—1072no—
335C22—2141no—
336C22—2141no—
337C22C2×D4×D10711yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—4281no—
343C4—4281no—
344C4—4281no—
345C4—4281no—
346C4—2141no—
347C4—2141no—
348C4—2141no—
349C4C2×#1313381811yescharacteristic
350C4#1681215411yescharacteristic
351C2—41no—
352C2—81no—
353C2—2142no—
354C2#2029326711yescharacteristic
355C2#135×D10711yescharacteristic
356C2#2029344111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#416×C107C2
socleC22×C107C22×D4
prime radical#20293284C2
hypercenter#416D107
hypocenterC107#3300
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #20293316C2semidirect
3. #20292980C2semidirect
4. #20292968C2semidirect
5. #20292939C2semidirect
6. #20292936C2semidirect
7. #20292868C2semidirect
8. #20292855C2semidirect
9. #20292848C2semidirect
10. #20292789C2semidirect
11. #20292784C2semidirect
12. #20292719C2semidirect
13. C8×C22⋊D107C2semidirect
14. #103×D107C2semidirect
15. #416×C107C2semidirect
179. C107#3300semidirect
193. #416D107semidirect

Maximal subgroups

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

subgroupindex
#202933162
#202932842
#202929802
#202929682
#202929392
#202929362
#202928682
#202928552
#202928482
#202927892
#202927842
#202927192
C8×C22⋊D1072
#103×D1072
#416×C1072
#3300107