Finite Group Databasebeta

← #28349223

Subgroups — #28349223

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#20293355C211yesmaximal, characteristic, complemented
2#20292991C211yesmaximal, characteristic
3#20292986C211yesmaximal, characteristic, complemented
4#20292981C211yesmaximal, characteristic, complemented
5#20292934C211yesmaximal, characteristic, complemented
6#20292927C211yesmaximal, characteristic, complemented
7#20292917C211yesmaximal, characteristic, complemented
8#20292846C211yesmaximal, characteristic, complemented
9#20292846C211yesmaximal, characteristic, complemented
10#20292824C211yesmaximal, characteristic
11#20292758C211yesmaximal, characteristic, complemented
12#20292732C211yesmaximal, characteristic, complemented
13#20292727C211yesmaximal, characteristic, complemented
14C4×#13133806C211yesmaximal, characteristic, complemented
15#559×C107C211yesmaximal, characteristic
16C2×#13133808C2211yescharacteristic
17C2×#13133806C2211yescharacteristic
18C2×#13133806C2211yescharacteristic
19#16812109C2211yescharacteristic
20#16812107—21no—
21#16812101C2211yescharacteristic
22#16812101C2211yescharacteristic
23#16812099—21no—
24#16812099—21no—
25C4×Dic214C2211yescharacteristic
26C4×Dic214C2211yescharacteristic
27C4×Dic214C2211yescharacteristic
28#16812074—21no—
29#16812042C2211yescharacteristic
30#16812040C2211yescharacteristic
31#16812040C2211yescharacteristic
32C4⋊(C107⋊C8)C2211yescharacteristic
33C4⋊(C107⋊C8)C2211yescharacteristic
34C4⋊(C107⋊C8)C2211yescharacteristic
35#16812033C2211yescharacteristic
36#16812033C2211yescharacteristic
37#16812033C2211yescharacteristic
38#16812032C2211yescharacteristic
39#16812030C2211yescharacteristic
40#16812030C2211yescharacteristic
41#16812030C2211yescharacteristic
42#16812030C2211yescharacteristic
43#16812030C2211yescharacteristic
44#16812030C2211yescharacteristic
45#16812020C2211yescharacteristic
46#16812020C2211yescharacteristic
47C4×C107⋊C8C2211yescharacteristic
48#127×C107C2211yescharacteristic
49#124×C107—21no—
50#124×C107C2211yescharacteristic
51#124×C107C2211yescharacteristic
52#123×C107—21no—
53#123×C107—21no—
54C4×Q8×C107C2211yescharacteristic
55C4×Q8×C107C2211yescharacteristic
56C4×D4×C107—21no—
57C4×D4×C107C2211yescharacteristic
58C4×D4×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214C2311yescharacteristic
61#13133808—22no—
62#13133806—22no—
63#13133806—22no—
64C2×C107⋊C8C2311yescharacteristic
65C2×C107⋊C8C2311yescharacteristic
66C2×C107⋊C8C2311yescharacteristic
67C2×C107⋊C8C2311yescharacteristic
68C4⋊Dic107—21no—
69C4⋊Dic107C2311yescharacteristic
70C4⋊Dic107C2311yescharacteristic
71C4⋊Dic107C2311yescharacteristic
72Dic107⋊C4—21no—
73Dic107⋊C4—21no—
74Dic107⋊C4—21no—
75Dic107⋊C4—21no—
76Dic107⋊C4—21no—
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79C4×Dic107—21no—
80C4×Dic107—21no—
81C4×Dic107—21no—
82C2×Q8×C107D411yescharacteristic
83C2×Q8×C107C2311yescharacteristic
84C2×D4×C107—21no—
85C2×D4×C107—21no—
86C2×D4×C107C2311yescharacteristic
87C2×D4×C107C2311yescharacteristic
88C22×C4×C107—21no—
89C22×C4×C107—21no—
90C4⋊C4×C107—21no—
91C4⋊C4×C107D411yescharacteristic
92C4⋊C4×C107D411yescharacteristic
93C4⋊C4×C107D411yescharacteristic
94C4⋊C4×C107C2311yescharacteristic
95C22⋊C4×C107—21no—
96C22⋊C4×C107—21no—
97C22⋊C4×C107—21no—
98C22⋊C4×C107—21no—
99C22⋊C4×C107—21no—
100C22⋊C4×C107—21no—
101C42×C107—21no—
102C42×C107C2311yescharacteristic
103C42×C107C2311yescharacteristic
104C42×C107C2311yescharacteristic
105C23×C107—21no—
106C23×C107—21no—
107Q8×C107—21no—
108Q8×C107—21no—
109Q8×C107—41no—
110D4×C107—21no—
111D4×C107—21no—
112D4×C107—21no—
113D4×C107—21no—
114D4×C107—42no—
115D4×C107—81no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—81no—
119C2×C4×C107—81no—
120C2×C4×C107C2×D411yescharacteristic
121C2×C4×C107C2×D411yescharacteristic
122C2×C4×C107C2×D411yescharacteristic
123C2×C4×C107C2×D411yescharacteristic
124C2×C4×C107C2×D411yescharacteristic
125C2×C4×C107C2×D411yescharacteristic
126C2×C4×C107C2411yescharacteristic
127C2×Dic107—21no—
128C2×Dic107—21no—
129C2×Dic107—21no—
130C2×Dic107—21no—
131Dic214—21no—
132Dic214—42no—
133Dic214#4112yes—
134C107⋊C8—21no—
135C107⋊C8—21no—
136C107⋊C8—21no—
137C107⋊C8—21no—
138C22×C107—41no—
139C22×C107—41no—
140C22×C107—41no—
141C22×C107—41no—
142C22×C107—41no—
143C22×C107—41no—
144C22×C107C22×D411yescharacteristic
145C4×C107—21no—
146C4×C107—21no—
147C4×C107—21no—
148C4×C107—21no—
149C4×C107—21no—
150C4×C107—21no—
151C4×C107—41no—
152C4×C107—41no—
153C4×C107C2×#4111yescharacteristic
154C4×C1072−1+411yescharacteristic
155Dic107—22no—
156Dic107—41no—
157Dic107—41no—
158Dic107—41no—
159C2×C107—81no—
160C2×C107—81no—
161C2×C107#54811yescharacteristic
162C2×C107#57511yescharacteristic
163C2×C107#57711yescharacteristic
164#3371—1071nomaximal, Sylow
165C107#337111yescharacteristic, complemented, Sylow
166#559D10711yescharacteristic
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#446—1071no—
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#127C2×D10711yescharacteristic
187#125—1071no—
188#125—1071no—
189#125—2141no—
190#125—2141no—
191#124—21no—
192#124C2×D10711yescharacteristic
193#124C2×D10711yescharacteristic
194#123—21no—
195#123—21no—
196C4×Q8—1071no—
197C4×Q8—1071no—
198C4×Q8—1071no—
199C4×Q8—2141no—
200C4×Q8C2×D10711yescharacteristic
201C4×Q8C2×D10711yescharacteristic
202C4×D4—21no—
203C4×D4C2×D10711yescharacteristic
204C4×D4C2×D10711yescharacteristic
205#107—1071no—
206#106—1071no—
207#106—1071no—
208C4⋊C8—1071no—
209C4⋊C8—1071no—
210C4⋊C8—1071no—
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#97—1071no—
222#97—1071no—
223C4×C8—1071no—
224C2×Q8—1071no—
225C2×Q8—2141no—
226C2×Q8C22⋊D10711yescharacteristic
227C2×Q8C22×D10711yescharacteristic
228C2×D4—21no—
229C2×D4—21no—
230C2×D4C22×D10711yescharacteristic
231C2×D4C22×D10711yescharacteristic
232C22×C4—21no—
233C22×C4—21no—
234Q16—2142no—
235SD16—2142no—
236SD16—2142no—
237C2×C8—1071no—
238C2×C8—1071no—
239C2×C8—1071no—
240C2×C8—1071no—
241C4⋊C4—21no—
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⋊C4C22⋊D10711yescharacteristic
254C4⋊C4C22⋊D10711yescharacteristic
255C4⋊C4C22⋊D10711yescharacteristic
256C4⋊C4C22×D10711yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C42—21no—
264C42—2141no—
265C42—2141no—
266C42—2141no—
267C42C22×D10711yescharacteristic
268C42C22×D10711yescharacteristic
269C42C22×D10711yescharacteristic
270C23—21no—
271C23—21no—
272Q8—21no—
273Q8—21no—
274Q8—41no—
275Q8—1072no—
276Q8—4282no—
277Q8—2141no—
278D4—21no—
279D4—21no—
280D4—21no—
281D4—21no—
282D4—42no—
283D4—81no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—2141no—
289C2×C4—2141no—
290C2×C4—2141no—
291C2×C4—2141no—
292C2×C4C2×C22⋊D10711yescharacteristic
293C2×C4C2×C22⋊D10711yescharacteristic
294C2×C4C2×C22⋊D10711yescharacteristic
295C2×C4C2×C22⋊D10711yescharacteristic
296C2×C4C2×C22⋊D10711yescharacteristic
297C2×C4C2×C22⋊D10711yescharacteristic
298C2×C4C23×D10711yescharacteristic
299C8—2141no—
300C8—2141no—
301C8—2141no—
302C8—2141no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22C22×C22⋊D10711yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—4281no—
319C4—4281no—
320C4—4281no—
321C4—2142no—
322C4#41×D10711yescharacteristic
323C4#1681215611yescharacteristic
324C2—81no—
325C2—81no—
326C2#2029328411yescharacteristic
327C2#2029342511yescharacteristic
328C2#2029343811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#559×C107C2
socleC22×C107C22×D4
prime radical#127×C107C22
hypercenter#559D107
hypocenterC107#3371
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #20293355C2semidirect
3. #20292986C2semidirect
4. #20292981C2semidirect
5. #20292934C2semidirect
6. #20292927C2semidirect
7. #20292917C2semidirect
8. #20292846C2semidirect
9. #20292846C2semidirect
11. #20292758C2semidirect
12. #20292732C2semidirect
13. #20292727C2semidirect
14. C4×#13133806C2semidirect
165. C107#3371semidirect

Maximal subgroups

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

subgroupindex
#202933552
#202929912
#202929862
#202929812
#202929342
#202929272
#202929172
#202928462
#202928242
#202927582
#202927322
#202927272
C4×#131338062
#559×C1072
#3371107