Finite Group Databasebeta

← #28349231

Subgroups — #28349231

Order 13696.

344 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293362C211yesmaximal, characteristic
2#20293355C211yesmaximal, characteristic, complemented
3#20292992C211yesmaximal, characteristic
4#20292987C211yesmaximal, characteristic, complemented
5#20292982C211yesmaximal, characteristic, complemented
6#20292935C211yesmaximal, characteristic, complemented
7#20292928C211yesmaximal, characteristic, complemented
8#20292918C211yesmaximal, characteristic, complemented
9#20292850C211yesmaximal, characteristic, complemented
10#20292850C211yesmaximal, characteristic, complemented
11#20292828C211yesmaximal, characteristic
12#20292735C211yesmaximal, characteristic, complemented
13#20292729C211yesmaximal, characteristic, complemented
14C4×#13133807C211yesmaximal, characteristic, complemented
15#446×C107C211yesmaximal, characteristic, complemented
16C2×Dic428C2211yescharacteristic
17C2×#13133807C2211yescharacteristic
18C2×#13133807C2211yescharacteristic
19#16812109C2211yescharacteristic
20#16812107—21no—
21#16812106C2211yescharacteristic
22#16812101C2211yescharacteristic
23#16812101C2211yescharacteristic
24#16812099—21no—
25#16812099—21no—
26#16812098—21no—
27#16812097C2211yescharacteristic
28#16812097C2211yescharacteristic
29#16812094—21no—
30#16812094—21no—
31C4×Dic214C2211yescharacteristic
32C4×Dic214C2211yescharacteristic
33C4×Dic214C2211yescharacteristic
34C4×Dic214C2211yescharacteristic
35#16812074—21no—
36C4×D428C2211yescharacteristic
37C4×D428C2211yescharacteristic
38#16812071—21no—
39#16812043C2211yescharacteristic
40#16812041C2211yescharacteristic
41#16812041C2211yescharacteristic
42#16812034C2211yescharacteristic
43#16812034C2211yescharacteristic
44#16812034C2211yescharacteristic
45#16812034C2211yescharacteristic
46#16812031C2211yescharacteristic
47#16812031C2211yescharacteristic
48#16812031C2211yescharacteristic
49#16812031C2211yescharacteristic
50#16812031C2211yescharacteristic
51#16812031C2211yescharacteristic
52C4×Q8×C107C2211yescharacteristic
53C4⋊C8×C107C2211yescharacteristic
54C4⋊C8×C107C2211yescharacteristic
55C4⋊C8×C107C2211yescharacteristic
56#97×C107C2211yescharacteristic
57#97×C107C2211yescharacteristic
58C4×C8×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214C2311yescharacteristic
61C2×Dic214C2311yescharacteristic
62C2×D428—21no—
63C2×D428—21no—
64C2×D428C2311yescharacteristic
65C2×D428C2311yescharacteristic
66C2×C4×D107—21no—
67C2×C4×D107—21no—
68Dic428—22no—
69#13133807—22no—
70#13133807—22no—
71C4⋊Dic107—21no—
72C4⋊Dic107C2311yescharacteristic
73C4⋊Dic107C2311yescharacteristic
74C4⋊Dic107C2311yescharacteristic
75C4⋊Dic107C2311yescharacteristic
76Dic107⋊C4—21no—
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
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—
90C4×Dic107—21no—
91C4×Dic107—21no—
92C4×Dic107—21no—
93C4×Dic107—21no—
94C2×Q8×C107D411yescharacteristic
95C2×C8×C107C2311yescharacteristic
96C2×C8×C107C2311yescharacteristic
97C2×C8×C107C2311yescharacteristic
98C2×C8×C107C2311yescharacteristic
99C4⋊C4×C107D411yescharacteristic
100C4⋊C4×C107D411yescharacteristic
101C4⋊C4×C107D411yescharacteristic
102C42×C107C2311yescharacteristic
103C42×C107C2311yescharacteristic
104C42×C107C2311yescharacteristic
105C22×D107—21no—
106C22×D107—21no—
107Q8×C107—41no—
108C2×C4×C107C2×D411yescharacteristic
109C2×C4×C107C2×D411yescharacteristic
110C2×C4×C107C2×D411yescharacteristic
111C2×C4×C107C2×D411yescharacteristic
112C2×C4×C107C2×D411yescharacteristic
113C2×C4×C107C2×D411yescharacteristic
114C2×C4×C107C2411yescharacteristic
115C2×Dic107—21no—
116C2×Dic107—21no—
117C2×Dic107—21no—
118C2×Dic107—21no—
119C2×Dic107—21no—
120C2×Dic107—21no—
121D428—21no—
122D428—21no—
123D428—21no—
124D428—21no—
125D428—42no—
126D428—81no—
127C4×D107—81no—
128C4×D107—81no—
129Dic214—21no—
130Dic214—21no—
131Dic214—21no—
132Dic214—42no—
133Dic214#4112yes—
134C8×C107—21no—
135C8×C107—21no—
136C8×C107—21no—
137C8×C107—21no—
138C22×C107C22×D411yescharacteristic
139C2×D107—41no—
140C2×D107—41no—
141C2×D107—41no—
142C2×D107—41no—
143C2×D107—41no—
144C2×D107—41no—
145C4×C107—21no—
146C4×C107—21no—
147C4×C107—21no—
148C4×C107—21no—
149C4×C107—21no—
150C4×C107—21no—
151C4×C107C2×#4111yescharacteristic
152C4×C1072−1+411yescharacteristic
153Dic107—22no—
154Dic107—41no—
155Dic107—41no—
156Dic107—41no—
157Dic107—41no—
158Dic107—41no—
159C2×C107#54811yescharacteristic
160C2×C107#57511yescharacteristic
161C2×C107#57711yescharacteristic
162D107—81no—
163D107—81no—
164#3371—1071nomaximal, Sylow
165C107#337111yescharacteristic, complemented, Sylow
166#559—1071no—
167#557—1071no—
168#490—1071no—
169#489—1071no—
170#488—1071no—
171#479—1071no—
172#478—1071no—
173#476—1071no—
174#464—1071no—
175#464—1071no—
176#460—1071no—
177#446D10711yescharacteristic, complemented
178#441—1071no—
179#440—1071no—
180C4×SD16—1071no—
181C2×Q16—1071no—
182C2×SD16—1071no—
183C2×SD16—1071no—
184C4⋊Q8—1071no—
185C4⋊Q8—2141no—
186#127—1071no—
187#125—1071no—
188#125—1071no—
189#125—2141no—
190#125—2141no—
191#124—1071no—
192#124—1071no—
193#124—2141no—
194#123—2141no—
195#123—2141no—
196C4×Q8—1071no—
197C4×Q8—1071no—
198C4×Q8—1071no—
199C4×Q8—1071no—
200C4×Q8—2141no—
201C4×Q8C2×D10711yescharacteristic
202C4×D4—1071no—
203C4×D4—1071no—
204C4×D4—2141no—
205#107—1071no—
206#106—1071no—
207#106—1071no—
208C4⋊C8C2×D10711yescharacteristic
209C4⋊C8C2×D10711yescharacteristic
210C4⋊C8C2×D10711yescharacteristic
211#103—1071no—
212#103—1071no—
213#103—1071no—
214#103—1071no—
215#102—1071no—
216#102—1071no—
217#102—1071no—
218#102—1071no—
219#102—1071no—
220#102—1071no—
221#97C2×D10711yescharacteristic
222#97C2×D10711yescharacteristic
223C4×C8C2×D10711yescharacteristic
224C2×Q8—1071no—
225C2×Q8—1071no—
226C2×Q8—2141no—
227C2×Q8D42811yescharacteristic
228C2×D4—1071no—
229C2×D4—1071no—
230C2×D4—2141no—
231C2×D4—2141no—
232C22×C4—2141no—
233C22×C4—2141no—
234Q16—2142no—
235SD16—2142no—
236SD16—2142no—
237C2×C8C22×D10711yescharacteristic
238C2×C8C22×D10711yescharacteristic
239C2×C8C22×D10711yescharacteristic
240C2×C8C22×D10711yescharacteristic
241C4⋊C4—1071no—
242C4⋊C4—1071no—
243C4⋊C4—1071no—
244C4⋊C4—1071no—
245C4⋊C4—2141no—
246C4⋊C4—2141no—
247C4⋊C4—2141no—
248C4⋊C4—2141no—
249C4⋊C4—2141no—
250C4⋊C4—2141no—
251C4⋊C4—2141no—
252C4⋊C4—2141no—
253C4⋊C4—2141no—
254C4⋊C4D42811yescharacteristic
255C4⋊C4D42811yescharacteristic
256C4⋊C4D42811yescharacteristic
257C22⋊C4—2141no—
258C22⋊C4—2141no—
259C22⋊C4—2141no—
260C22⋊C4—2141no—
261C22⋊C4—2141no—
262C22⋊C4—2141no—
263C42—2141no—
264C42—2141no—
265C42—2141no—
266C42—2141no—
267C42C22×D10711yescharacteristic
268C42C22×D10711yescharacteristic
269C42C22×D10711yescharacteristic
270C23—2141no—
271C23—2141no—
272Q8—41no—
273Q8—1072no—
274Q8—4282no—
275Q8—2141no—
276Q8—2141no—
277Q8—2141no—
278D4—8561no—
279D4—4282no—
280D4—2141no—
281D4—2141no—
282D4—2141no—
283D4—2141no—
284C2×C4—8561no—
285C2×C4—8561no—
286C2×C4—2141no—
287C2×C4—2141no—
288C2×C4—2141no—
289C2×C4—2141no—
290C2×C4—2141no—
291C2×C4—2141no—
292C2×C4C2×D42811yescharacteristic
293C2×C4C2×D42811yescharacteristic
294C2×C4C2×D42811yescharacteristic
295C2×C4C2×D42811yescharacteristic
296C2×C4C2×D42811yescharacteristic
297C2×C4C2×D42811yescharacteristic
298C2×C4C23×D10711yescharacteristic
299C8—21no—
300C8—21no—
301C8—21no—
302C8—21no—
303C22—4281no—
304C22—4281no—
305C22—4281no—
306C22—4281no—
307C22—4281no—
308C22—4281no—
309C22C22×D42811yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—4281no—
317C4—4281no—
318C4—4281no—
319C4—4281no—
320C4—4281no—
321C4—2142no—
322C4#41×D10711yescharacteristic
323C4#1681215711yescharacteristic
324C2—8561no—
325C2—8561no—
326C2#2029328911yescharacteristic
327C2#2029342711yescharacteristic
328C2#2029343911yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#446×C107C2
socleC22×C107C22×D4
prime radical#16812106C22
hypercenter#446D107
hypocenterC107#3371
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #446

Complements

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

NH ≅ G/Ntype
2. #20293355C2semidirect
4. #20292987C2semidirect
5. #20292982C2semidirect
6. #20292935C2semidirect
7. #20292928C2semidirect
8. #20292918C2semidirect
9. #20292850C2semidirect
10. #20292850C2semidirect
12. #20292735C2semidirect
13. #20292729C2semidirect
14. C4×#13133807C2semidirect
15. #446×C107C2semidirect
165. C107#3371semidirect
177. #446D107semidirect

Maximal subgroups

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

subgroupindex
#202933622
#202933552
#202929922
#202929872
#202929822
#202929352
#202929282
#202929182
#202928502
#202928282
#202927352
#202927292
C4×#131338072
#446×C1072
#3371107