Finite Group Databasebeta

← #28346988

Subgroups — #28346988

Order 13696.

340 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293257C211yesmaximal, characteristic
2#20293254C211yesmaximal, characteristic
3#20293036C211yesmaximal, characteristic, complemented
4#20293035C211yesmaximal, characteristic, complemented
5#20293034C211yesmaximal, characteristic, complemented
6#20293028C211yesmaximal, characteristic, complemented
7#20293026C211yesmaximal, characteristic, complemented
8#20293024C211yesmaximal, characteristic, complemented
9#20293023C211yesmaximal, characteristic, complemented
10#20292668C211yesmaximal, characteristic
11#20292666C211yesmaximal, characteristic
12#20292664C211yesmaximal, characteristic
13#106×D107C211yesmaximal, characteristic
14#20292584C211yesmaximal, characteristic
15#501×C107C211yesmaximal, characteristic, complemented
16C2×#13133801C2211yescharacteristic
17C2×C8×D107C2211yescharacteristic
18#16812109C2211yescharacteristic
19#16812108—21no—
20#16812108C2211yescharacteristic
21#16812108C2211yescharacteristic
22#16812107—21no—
23#16812101C2211yescharacteristic
24#16812100C2211yescharacteristic
25#16812100C2211yescharacteristic
26#16812099—21no—
27#16812099—21no—
28#16812089—21no—
29#16812087—21no—
30#16812087—21no—
31#16812085—21no—
32#16812067C2211yescharacteristic
33#16812067C2211yescharacteristic
34#16812066C2211yescharacteristic
35C4⋊C4×D107C2211yescharacteristic
36C4⋊C4×D107C2211yescharacteristic
37#16812043C2211yescharacteristic
38#16812043C2211yescharacteristic
39#16812042C2211yescharacteristic
40#16812042C2211yescharacteristic
41#16812041C2211yescharacteristic
42#16812041C2211yescharacteristic
43#16812040C2211yescharacteristic
44#16812040C2211yescharacteristic
45#16812040C2211yescharacteristic
46#16812040C2211yescharacteristic
47#16812040C2211yescharacteristic
48#16812040C2211yescharacteristic
49#16812019C2211yescharacteristic
50C4×C107⋊C8C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C4⋊Q8×C107C2211yescharacteristic
53#125×C107C2211yescharacteristic
54#107×C107C2211yescharacteristic
55#107×C107C2211yescharacteristic
56#106×C107C2211yescharacteristic
57#106×C107C2211yescharacteristic
58#97×C107C2211yescharacteristic
59C2×Dic214—21no—
60C2×Dic214—21no—
61C2×Dic214—21no—
62C2×C4×D107—21no—
63C2×C4×D107—21no—
64C2×C4×D107C2311yescharacteristic
65#13133801Q814yescomplemented
66C2×C107⋊C8C2311yescharacteristic
67C2×C107⋊C8C2311yescharacteristic
68C8×D107Q814yescomplemented
69C4⋊Dic107—21no—
70C4⋊Dic107—21no—
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—
84Dic107⋊C4D411yescharacteristic
85Dic107⋊C4D411yescharacteristic
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4—21no—
88(C2×D107)⋊C4D411yescharacteristic
89(C2×D107)⋊C4D411yescharacteristic
90C4×Dic107—21no—
91C4×Dic107—21no—
92C4×Dic107C2311yescharacteristic
93C2×Q8×C107—21no—
94C2×C8×C107C2311yescharacteristic
95C2×C8×C107C2311yescharacteristic
96C4⋊C4×C107—21no—
97C4⋊C4×C107—21no—
98C4⋊C4×C107—21no—
99C4⋊C4×C107C2311yescharacteristic
100C4⋊C4×C107C2311yescharacteristic
101C4⋊C4×C107C2311yescharacteristic
102C4⋊C4×C107C2311yescharacteristic
103C42×C107C2311yescharacteristic
104C22×D107C2×D411yescharacteristic
105Q8×C107—81no—
106C2×C4×C107—21no—
107C2×C4×C107—21no—
108C2×C4×C107—21no—
109C2×C4×C107—21no—
110C2×C4×C107C2×D411yescharacteristic
111C2×C4×C107C2×D411yescharacteristic
112C2×C4×C107C2411yescharacteristic
113C2×Dic107—21no—
114C2×Dic107—21no—
115C2×Dic107—21no—
116C2×Dic107—21no—
117C2×Dic107C2×D411yescharacteristic
118C2×Dic107C2×D411yescharacteristic
119C2×Dic107C2×D411yescharacteristic
120C4×D107—81no—
121C4×D107—81no—
122C4×D107C2×Q812yes—
123C4×D107C2×Q812yes—
124Dic214—81no—
125Dic214—81no—
126Dic214—81no—
127C8×C107C2×Q812yes—
128C8×C107C2×Q812yes—
129C107⋊C8C2×Q812yes—
130C107⋊C8C2×Q812yes—
131C22×C107C22×D411yescharacteristic
132C2×D107—21no—
133C2×D107—21no—
134C2×D107C4⋊Q812yes—
135C4×C107—21no—
136C4×C107—21no—
137C4×C107—41no—
138C4×C107—41no—
139C4×C107—41no—
140C4×C107—41no—
141C4×C107C22×Q811yescharacteristic
142C4×C107C22×Q811yescharacteristic
143Dic107—21no—
144Dic107—21no—
145Dic107—41no—
146Dic107—41no—
147Dic107—41no—
148Dic107—41no—
149Dic107C4⋊Q812yes—
150C2×C107C2×C4⋊Q811yescharacteristic
151C2×C107#57711yescharacteristic
152C2×C107#57711yescharacteristic
153D107—22no—
154#3163—1071nomaximal, Sylow
155C107#316311yescharacteristic, complemented, Sylow
156#544—1071no—
157#544—1071no—
158#501—1071no—
159#501—1071no—
160#501—1071no—
161#501D10711yescharacteristic, complemented
162#499—1071no—
163#499—1071no—
164#498—1071no—
165#498—1071no—
166#428—1071no—
167#428—1071no—
168#427—1071no—
169C2×#106—1071no—
170#405—1071no—
171C2×M16—1071no—
172C22×C8—1071no—
173C4⋊Q8—1071no—
174C4⋊Q8—1071no—
175C4⋊Q8—1071no—
176C4⋊Q8—2141no—
177C4⋊Q8—2141no—
178C4⋊Q8C2×D10711yescharacteristic
179#125—1071no—
180#125—1071no—
181#125—1071no—
182#125—2141no—
183#125—2141no—
184#125C2×D10711yescharacteristic
185C22⋊Q8—2141no—
186C22⋊Q8—2141no—
187C22⋊Q8—2141no—
188C22⋊Q8—2141no—
189#117—1071no—
190#117—1071no—
191#117—1071no—
192C2×C4⋊C4—1071no—
193C2×C4⋊C4—1071no—
194#107—1071no—
195#107—1071no—
196#107—1071no—
197#107—1071no—
198#107C2×D10711yescharacteristic
199#107C2×D10711yescharacteristic
200#106—1071no—
201#106—1071no—
202#106—1071no—
203#106—1071no—
204#106—1071no—
205#106—1071no—
206#106—1071no—
207#106—1071no—
208#106C2×D10711yescharacteristic
209#106C2×D10711yescharacteristic
210#97—1071no—
211#97C2×D10711yescharacteristic
212C4×C8—1071no—
213C4×C8—1071no—
214C2×Q8—21no—
215C2×Q8—2141no—
216C2×Q8—2141no—
217C2×Q8—2141no—
218C22×C4—1071no—
219C22×C4—2141no—
220C22×C4—2141no—
221M16—1074no—
222C2×C8—1071no—
223C2×C8—1071no—
224C2×C8—1074no—
225C2×C8C22×D10711yescharacteristic
226C2×C8C22×D10711yescharacteristic
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—1071no—
231C4⋊C4—1071no—
232C4⋊C4—1071no—
233C4⋊C4—1071no—
234C4⋊C4—1071no—
235C4⋊C4—1071no—
236C4⋊C4—2141no—
237C4⋊C4—2141no—
238C4⋊C4—2141no—
239C4⋊C4—2141no—
240C4⋊C4—2141no—
241C4⋊C4—2141no—
242C4⋊C4—2141no—
243C4⋊C4—2141no—
244C4⋊C4—2141no—
245C4⋊C4—2141no—
246C4⋊C4—2141no—
247C4⋊C4C22×D10711yescharacteristic
248C4⋊C4C22×D10711yescharacteristic
249C4⋊C4C22×D10711yescharacteristic
250C4⋊C4C22×D10711yescharacteristic
251C22⋊C4—1071no—
252C22⋊C4—1071no—
253C22⋊C4—2141no—
254C22⋊C4—2141no—
255C42—1071no—
256C42—2141no—
257C42—2141no—
258C42C22×D10711yescharacteristic
259C23—1071no—
260Q8—81no—
261Q8—8561no—
262Q8—8561no—
263Q8—8561no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—8561no—
269C2×C4—8561no—
270C2×C4—1071no—
271C2×C4—1072no—
272C2×C4—1072no—
273C2×C4—1071no—
274C2×C4—1071no—
275C2×C4—2141no—
276C2×C4—2141no—
277C2×C4—2141no—
278C2×C4—2141no—
279C2×C4D4×D10711yescharacteristic
280C2×C4D4×D10711yescharacteristic
281C2×C4C23×D10711yescharacteristic
282C8—1072no—
283C8—1072no—
284C8Q8×D10712yes—
285C8Q8×D10712yes—
286C22—1072no—
287C22—2141no—
288C22—2141no—
289C22C2×D4×D10711yescharacteristic
290C4—21no—
291C4—21no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—1072no—
297C4—4281no—
298C4—4281no—
299C4—4281no—
300C4—4281no—
301C4—2141no—
302C4—2141no—
303C4C2×Q8×D10711yescharacteristic
304C4C2×Q8×D10711yescharacteristic
305C2—2142no—
306C2C4⋊Q8×D10711yescharacteristic
307C2#2029343611yescharacteristic
308C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#501×C107C2
socleC22×C107C22×D4
prime radicalC22×D107C2×D4
hypercenter#501D107
hypocenterC107#3163
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
3. #20293036C2semidirect
4. #20293035C2semidirect
5. #20293034C2semidirect
6. #20293028C2semidirect
7. #20293026C2semidirect
8. #20293024C2semidirect
9. #20293023C2semidirect
15. #501×C107C2semidirect
65. #13133801Q8semidirect
68. C8×D107Q8semidirect
155. C107#3163semidirect
161. #501D107semidirect

Maximal subgroups

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

subgroupindex
#202932572
#202932542
#202930362
#202930352
#202930342
#202930282
#202930262
#202930242
#202930232
#202926682
#202926662
#202926642
#106×D1072
#202925842
#501×C1072
#3163107