Finite Group Databasebeta

← #28349052

Subgroups — #28349052

Order 13696.

386 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293340C211yesmaximal, characteristic, complemented
2#20293278C211yesmaximal, characteristic, complemented
3#20293024C211yesmaximal, characteristic, complemented
4#20292954C211yesmaximal, characteristic, complemented
5#20292951C211yesmaximal, characteristic, complemented
6#20292947C211yesmaximal, characteristic, complemented
7#20292945C211yesmaximal, characteristic, complemented
8#20292916C211yesmaximal, characteristic, complemented
9#20292906C211yesmaximal, characteristic, complemented
10#20292881C211yesmaximal, characteristic, complemented
11#20292880C211yesmaximal, characteristic, complemented
12#20292709C211yesmaximal, characteristic, complemented
13#20292703C211yesmaximal, characteristic, complemented
14#20292668C211yesmaximal, characteristic, complemented
15#428×C107C211yesmaximal, characteristic, complemented
16C2×#13133818—21no—
17C2×D856C2211yescharacteristic
18C2×#13133801C2211yescharacteristic
19#16812108C2211yescharacteristic
20#16812108C2211yescharacteristic
21#16812097—21no—
22#16812095—21no—
23#16812094—21no—
24#16812092—21no—
25#16812087—21no—
26#16812084C2211yescharacteristic
27#16812084C2211yescharacteristic
28#16812083C2211yescharacteristic
29#16812083C2211yescharacteristic
30#16812082—21no—
31#16812071C2211yescharacteristic
32#16812071C2211yescharacteristic
33C4×C22⋊D107C2211yescharacteristic
34#16812067C2211yescharacteristic
35#16812066—21no—
36C4⋊C4×D107C2211yescharacteristic
37#16812042C2211yescharacteristic
38#16812041C2211yescharacteristic
39#16812040C2211yescharacteristic
40#16812040C2211yescharacteristic
41#16812040C2211yescharacteristic
42Dic107⋊C8C2211yescharacteristic
43#16812031C2211yescharacteristic
44#16812031C2211yescharacteristic
45#16812029C2211yescharacteristic
46#16812029C2211yescharacteristic
47#16812029C2211yescharacteristic
48#16812029C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×M16×C107C2211yescharacteristic
53#117×C107C2211yescharacteristic
54C2×C4⋊C4×C107C2211yescharacteristic
55#107×C107C2211yescharacteristic
56#107×C107C2211yescharacteristic
57#106×C107C2211yescharacteristic
58#106×C107C2211yescharacteristic
59#13133818—44no—
60C2×Dic214—21no—
61C2×Dic214—21no—
62C2×D428—21no—
63C2×D428—21no—
64C2×D428C2311yescharacteristic
65C2×D428C2311yescharacteristic
66C2×C22⋊D107—21no—
67C2×C22⋊D107—21no—
68C2×C22⋊D107D411yescharacteristic
69C2×C4×D107—21no—
70C2×C4×D107—21no—
71C2×C4×D107—21no—
72C2×C4×D107C2311yescharacteristic
73D856—22no—
74#13133801—22no—
75C2×C107⋊C8C2311yescharacteristic
76C2×C107⋊C8C2311yescharacteristic
77C4⋊Dic107C2311yescharacteristic
78C4⋊Dic107C2311yescharacteristic
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4D411yescharacteristic
83C22⋊Dic107D411yescharacteristic
84(C2×D107)⋊C4—21no—
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4D411yescharacteristic
92C4×Dic107—21no—
93C4×Dic107C2311yescharacteristic
94C22×C4×C107—21no—
95C22×C4×C107C2311yescharacteristic
96M16×C107—22no—
97C2×C8×C107C2311yescharacteristic
98C2×C8×C107C2311yescharacteristic
99C4⋊C4×C107—21no—
100C4⋊C4×C107C2311yescharacteristic
101C4⋊C4×C107C2311yescharacteristic
102C4⋊C4×C107C2311yescharacteristic
103C4⋊C4×C107C2311yescharacteristic
104C22⋊C4×C107—21no—
105C42×C107—21no—
106C23×C107C2×D411yescharacteristic
107C22×D107—21no—
108C22×D107—21no—
109C22×D107C2×D411yescharacteristic
110C2×C4×C107—21no—
111C2×C4×C107—21no—
112C2×C4×C107—21no—
113C2×C4×C107—21no—
114C2×C4×C107—21no—
115C2×C4×C107—21no—
116C2×C4×C107—42no—
117C2×C4×C107C2×D411yescharacteristic
118C2×C4×C107C2411yescharacteristic
119C22⋊D107—22no—
120C22⋊D107—42no—
121C22⋊D107—81no—
122C2×Dic107—21no—
123C2×Dic107—21no—
124C2×Dic107C2×D411yescharacteristic
125C2×Dic107C2×D411yescharacteristic
126C2×Dic107C2×D411yescharacteristic
127D428—21no—
128D428—21no—
129D428—21no—
130D428—21no—
131D428—42no—
132D428—81no—
133C4×D107—21no—
134C4×D107—21no—
135C4×D107—42no—
136C4×D107—42no—
137C4×D107—42no—
138Dic214—42no—
139Dic214—81no—
140C8×C107—21no—
141C8×C107#4112yescomplemented
142C107⋊C8—21no—
143C107⋊C8—21no—
144C22×C107—21no—
145C22×C107—21no—
146C22×C107—21no—
147C22×C107C22×D411yescharacteristic
148C2×D107—21no—
149C2×D107—21no—
150C2×D107—21no—
151C2×D107—41no—
152C2×D107—41no—
153C2×D107—41no—
154C2×D107—41no—
155C2×D107—41no—
156C2×D107—41no—
157C4×C107—21no—
158C4×C107—41no—
159C4×C107—41no—
160C4×C107—41no—
161C4×C107—41no—
162C4×C107C2×#4111yescharacteristic
163C4×C1072−1+411yescharacteristic
164Dic107—21no—
165Dic107—21no—
166Dic107—41no—
167Dic107—41no—
168Dic107AGL(1,ℤ/8)12yes—
169C2×C107—41no—
170C2×C107#54711yescharacteristic
171C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
172C2×C107#57711yescharacteristic
173D107—41no—
174D107—81no—
175D107—81no—
176#3354—1071nomaximal, Sylow
177C107#335411yescharacteristic, complemented, Sylow
178#555—1071no—
179#547—1071no—
180#498—1071no—
181#482—1071no—
182#482—1071no—
183#481—1071no—
184#481—1071no—
185#476—1071no—
186#474—1071no—
187#469—1071no—
188#469—1071no—
189C4×D8—1071no—
190#436—1071no—
191#428—1071no—
192#428D10711yescharacteristic, complemented
193C2×#41—2141no—
194C2×D8—1071no—
195C2×M16—1071no—
196C2×M16C2×D10711yescharacteristic
197C4⋊Q8—1071no—
198C4⋊Q8—1071no—
199#124—2141no—
200#124—2141no—
201#123—2141no—
202#123—2141no—
203C22⋊Q8—2141no—
204#121—1071no—
205#121—1071no—
206#121—1071no—
207#121—1071no—
208#121—2141no—
209C4×D4—1071no—
210C4×D4—1071no—
211C4×D4—1071no—
212#117—1071no—
213#117—2141no—
214#117C2×D10711yescharacteristic
215C2×C4⋊C4—1071no—
216C2×C4⋊C4C2×D10711yescharacteristic
217#107—1071no—
218#107C2×D10711yescharacteristic
219#107C2×D10711yescharacteristic
220#106—1071no—
221#106—1071no—
222#106—1071no—
223#106—1071no—
224#106C2×D10711yescharacteristic
225#106C2×D10711yescharacteristic
226C4⋊C8—1071no—
227#102—1071no—
228#102—1071no—
229#102—1071no—
230#102—1071no—
231#102—1071no—
232#102—1071no—
233C22⋊C8—1071no—
234C22⋊C8—1071no—
235C4×C8—1071no—
236#41—4284no—
237C2×Q8—2141no—
238C2×Q8—2141no—
239C2×D4—1071no—
240C2×D4—1071no—
241C2×D4—1071no—
242C2×D4—2141no—
243C2×D4—2141no—
244C2×D4—2141no—
245C2×D4—2141no—
246C22×C4—21no—
247C22×C4—1071no—
248C22×C4—2141no—
249C22×C4—2141no—
250C22×C4—2141no—
251C22×C4C22×D10711yescharacteristic
252D8—2142no—
253M16—22no—
254M16—2142no—
255C2×C8—1071no—
256C2×C8—1071no—
257C2×C8C22×D10711yescharacteristic
258C2×C8C22×D10711yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—1071no—
261C4⋊C4—1071no—
262C4⋊C4—1071no—
263C4⋊C4—2141no—
264C4⋊C4—2141no—
265C4⋊C4—2141no—
266C4⋊C4C22×D10711yescharacteristic
267C4⋊C4C22×D10711yescharacteristic
268C4⋊C4C22×D10711yescharacteristic
269C4⋊C4C22×D10711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—1071no—
272C22⋊C4—1071no—
273C22⋊C4—2141no—
274C22⋊C4—2141no—
275C22⋊C4—2141no—
276C22⋊C4—2141no—
277C22⋊C4—2141no—
278C22⋊C4—2141no—
279C22⋊C4—2141no—
280C42—21no—
281C42—1071no—
282C42—2141no—
283C23—1071no—
284C23—2141no—
285C23—2141no—
286C23D4×D10711yescharacteristic
287Q8—8561no—
288Q8—4282no—
289D4—8561no—
290D4—8561no—
291D4—4282no—
292D4—4282no—
293D4—2142no—
294D4—2141no—
295D4—2141no—
296D4—2141no—
297D4—2141no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—42no—
305C2×C4—1071no—
306C2×C4—1071no—
307C2×C4—1071no—
308C2×C4—4282no—
309C2×C4—4282no—
310C2×C4—4282no—
311C2×C4—2141no—
312C2×C4—2141no—
313C2×C4—2141no—
314C2×C4—2141no—
315C2×C4D4×D10711yescharacteristic
316C2×C4C23×D10711yescharacteristic
317C8—21no—
318C8—2141no—
319C8—2141no—
320C8#1313381812yescomplemented
321C22—21no—
322C22—21no—
323C22—21no—
324C22—4281no—
325C22—4281no—
326C22—4281no—
327C22—4281no—
328C22—4281no—
329C22—4281no—
330C22—2141no—
331C22—2141no—
332C22—2141no—
333C22C2×D4×D10711yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—1072no—
340C4—4281no—
341C4—4281no—
342C4—2141no—
343C4—2141no—
344C4C2×#1313381811yescharacteristic
345C4#1681215711yescharacteristic
346C2—41no—
347C2—8561no—
348C2—8561no—
349C2—4281no—
350C2#2029327811yescharacteristic
351C2AGL(1,ℤ/8)×D10711yescharacteristic
352C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#428×C107C2
socleC22×C107C22×D4
hypercenter#428D107
hypocenterC107#3354
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #20293340C2semidirect
2. #20293278C2semidirect
3. #20293024C2semidirect
4. #20292954C2semidirect
5. #20292951C2semidirect
6. #20292947C2semidirect
7. #20292945C2semidirect
8. #20292916C2semidirect
9. #20292906C2semidirect
10. #20292881C2semidirect
11. #20292880C2semidirect
12. #20292709C2semidirect
13. #20292703C2semidirect
14. #20292668C2semidirect
15. #428×C107C2semidirect
141. C8×C107#41semidirect
177. C107#3354semidirect
192. #428D107semidirect
320. C8#13133818semidirect

Maximal subgroups

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

subgroupindex
#202933402
#202932782
#202930242
#202929542
#202929512
#202929472
#202929452
#202929162
#202929062
#202928812
#202928802
#202927092
#202927032
#202926682
#428×C1072
#3354107