Finite Group Databasebeta

← #28344752

Subgroups — #28344752

Order 13696.

1102 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
1C23×C4×D107C211yesmaximal, characteristic, complemented
2C2×#16812091C212yesmaximal, complemented
3C2×#16812077C211yesmaximal, characteristic
4#20292507C211yesmaximal, characteristic, complemented
5#20292503C212yesmaximal, complemented
6#20292502C211yesmaximal, characteristic, complemented
7#20292501C212yesmaximal, complemented
8#20292500C211yesmaximal, characteristic, complemented
9#20292489C212yesmaximal, complemented
10#20292482C211yesmaximal, characteristic, complemented
11#386×C107C211yesmaximal, characteristic, complemented
12C24×D107C2211yescharacteristic
13C22×C22⋊D107C2211yescharacteristic
14C22×C22⋊D107C2211yescharacteristic
15C23×Dic107C2211yescharacteristic
16C22×C4×D107—24no—
17C22×C4×D107—28no—
18C22×C4×D107C2212yes—
19C22×C4×D107C2212yes—
20#16812091—28no—
21#16812077—24no—
22C2×C4⋊Dic107C2212yescomplemented
23C2×Dic107⋊C4C2212yescomplemented
24C2×C22⋊Dic107C2211yescharacteristic
25C2×C22⋊Dic107C2211yescharacteristic
26C2×C22⋊Dic107C2212yescomplemented
27C2×(C2×D107)⋊C4C2212yes—
28C2×(C2×D107)⋊C4C2212yes—
29C2×(C2×D107)⋊C4C2211yescharacteristic
30C2×(C2×D107)⋊C4C2211yescharacteristic
31C107⋊#95C2212yescomplemented
32C107⋊#95C2212yescomplemented
33C107⋊#95C2212yescomplemented
34C22×D4×C107C2211yescharacteristic
35C23×C4×C107C2211yescharacteristic
36C2×C22⋊C4×C107C2211yescharacteristic
37C2×C22⋊C4×C107C2212yescomplemented
38#95×C107C2212yescomplemented
39C23×D107—22no—
40C23×D107—22no—
41C23×D107—24no—
42C23×D107—24no—
43C23×D107D414yes—
44C23×D107C2311yescharacteristic
45C23×D107C2311yescharacteristic
46C2×C22⋊D107—22no—
47C2×C22⋊D107—22no—
48C2×C22⋊D107—24no—
49C2×C22⋊D107—24no—
50C22×Dic107—22no—
51C22×Dic107—24no—
52C22×Dic107C2312yes—
53C22×Dic107C2311yescharacteristic
54C22×Dic107C2311yescharacteristic
55C22×Dic107C2312yes—
56C2×C4×D107—232no—
57C2×C4×D107—24no—
58C2×C4×D107—24no—
59C2×C4×D107—24no—
60C2×C4×D107—24no—
61C2×C4×D107—24no—
62C2×C4×D107D418yescomplemented
63C4⋊Dic107—24no—
64Dic107⋊C4—24no—
65C22⋊Dic107—22no—
66C22⋊Dic107—22no—
67C22⋊Dic107—24no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4—22no—
70(C2×D107)⋊C4—24no—
71(C2×D107)⋊C4—24no—
72C24×C107C2311yescharacteristic
73C24×C107C2311yescharacteristic
74C2×D4×C107—22no—
75C2×D4×C107—22no—
76C2×D4×C107—22no—
77C22×C4×C107—22no—
78C22×C4×C107—24no—
79C22×C4×C107C2312yes—
80C22×C4×C107C2311yescharacteristic
81C22×C4×C107C2312yes—
82C22⋊C4×C107—22no—
83C22⋊C4×C107—24no—
84C23×C107—22no—
85C23×C107—22no—
86C23×C107—22no—
87C23×C107—21no—
88C23×C107—21no—
89C23×C107—21no—
90C23×C107—22no—
91C23×C107C2×D412yes—
92C23×C107#4112yes—
93C23×C107#4112yes—
94C23×C107C2411yescharacteristic
95C22×D107—216no—
96C22×D107—216no—
97C22×D107—28no—
98C22×D107—22no—
99C22×D107—22no—
100C22×D107—21no—
101C22×D107—21no—
102C22×D107—21no—
103C22×D107—21no—
104C22×D107—28no—
105C22×D107—28no—
106C22×D107C2×D412yes—
107C22×D107C2×D414yes—
108C22×D107C2×D412yes—
109C22×D107#4114yes—
110D4×C107—44no—
111C2×C4×C107—28no—
112C2×C4×C107—22no—
113C2×C4×C107—22no—
114C2×C4×C107—22no—
115C2×C4×C107—21no—
116C2×C4×C107—22no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—22no—
120C2×C4×C107C2×D414yes—
121C22⋊D107—44no—
122C22⋊D107—44no—
123C2×Dic107—28no—
124C2×Dic107—22no—
125C2×Dic107—22no—
126C2×Dic107—22no—
127C2×Dic107—21no—
128C2×Dic107—21no—
129C2×Dic107—22no—
130C2×Dic107—22no—
131C2×Dic107—22no—
132C2×Dic107—22no—
133C2×Dic107C2×D414yes—
134C4×D107—216no—
135C4×D107—216no—
136C22×C107—22no—
137C22×C107—22no—
138C22×C107—24no—
139C22×C107—24no—
140C22×C107—41no—
141C22×C107—41no—
142C22×C107—41no—
143C22×C107—42no—
144C22×C107—42no—
145C22×C107#12314yescomplemented
146C22×C107C22×D411yescharacteristic
147C22×C107C22×D412yes—
148C22×C107C2×#4112yes—
149C22×C107C2×#4111yescharacteristic
150C22×C107C2×#4111yescharacteristic
151C2×D107—232no—
152C2×D107—22no—
153C2×D107—22no—
154C2×D107—22no—
155C2×D107—22no—
156C2×D107—24no—
157C2×D107—24no—
158C2×D107—24no—
159C2×D107—24no—
160C2×D107#12014yes—
161C2×D107#12314yescomplemented
162C4×C107—24no—
163C4×C107—41no—
164C4×C107—42no—
165Dic107—24no—
166Dic107—41no—
167Dic107—41no—
168Dic107—42no—
169C2×C107—24no—
170C2×C107—81no—
171C2×C107C2×#12011yescharacteristic
172C2×C107C2×#12311yescharacteristic
173C2×C107C2×#12311yescharacteristic
174C2×C107#52512yes—
175C2×C107#52512yes—
176D107—24no—
177D107—24no—
178#2849—1071nomaximal, Sylow
179C107#284911yescharacteristic, complemented, Sylow
180C24×C4—1071no—
181C2×#123—1072no—
182C2×#120—1071no—
183#386—1071no—
184#386—1072no—
185#386D10711yescharacteristic, complemented
186#385—1071no—
187#385—1071no—
188#385—1072no—
189#381—1072no—
190#379—1071no—
191C25—1071no—
192C22×D4—1071no—
193C22×D4—1071no—
194C22×D4C2×D10711yescharacteristic
195C23×C4—1072no—
196C23×C4—1071no—
197C23×C4—1072no—
198C23×C4—2148no—
199C23×C4—2144no—
200C23×C4C2×D10711yescharacteristic
201#123—2148no—
202#120—2144no—
203C2×C4⋊C4—1072no—
204C2×C4⋊C4—1072no—
205C2×C22⋊C4—1071no—
206C2×C22⋊C4—1072no—
207C2×C22⋊C4—1071no—
208C2×C22⋊C4—1071no—
209C2×C22⋊C4—1071no—
210C2×C22⋊C4—1072no—
211C2×C22⋊C4—1072no—
212C2×C22⋊C4C2×D10711yescharacteristic
213C2×C22⋊C4C2×D10712yescomplemented
214#95—1072no—
215#95—1072no—
216#95—1072no—
217#95C2×D10712yescomplemented
218C24—1071no—
219C24—1071no—
220C24—1074no—
221C24—2144no—
222C24—2144no—
223C24—2142no—
224C24—2142no—
225C24C22×D10711yescharacteristic
226C24C22×D10711yescharacteristic
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—2142no—
231C2×D4—2142no—
232C2×D4—2144no—
233C2×D4—2144no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—1072no—
237C22×C4—1078no—
238C22×C4—1071no—
239C22×C4—1072no—
240C22×C4—1071no—
241C22×C4—2144no—
242C22×C4—2144no—
243C22×C4—2144no—
244C22×C4—2144no—
245C22×C4—21432no—
246C22×C4—2144no—
247C22×C4—2142no—
248C22×C4—2144no—
249C22×C4C22×D10711yescharacteristic
250C22×C4C22×D10712yes—
251C22×C4C22×D10712yes—
252C4⋊C4—2144no—
253C4⋊C4—2144no—
254C22⋊C4—22no—
255C22⋊C4—24no—
256C22⋊C4—2142no—
257C22⋊C4—2142no—
258C22⋊C4—2144no—
259C22⋊C4—2144no—
260C22⋊C4—2144no—
261C22⋊C4—2142no—
262C22⋊C4—2142no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—21no—
267C23—22no—
268C23—21no—
269C23—21no—
270C23—1074no—
271C23—1072no—
272C23—1072no—
273C23—1074no—
274C23—2148no—
275C23—2148no—
276C23—21416no—
277C23—21416no—
278C23—2148no—
279C23—2142no—
280C23—2142no—
281C23—2141no—
282C23—2141no—
283C23—2141no—
284C23—2141no—
285C23D4×D10712yes—
286C23#1313381712yes—
287C23#1313381812yes—
288C23C23×D10711yescharacteristic
289D4—44no—
290D4—4284no—
291D4—4284no—
292C2×C4—28no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—1074no—
302C2×C4—21416no—
303C2×C4—21416no—
304C2×C4—2142no—
305C2×C4—2141no—
306C2×C4—2141no—
307C2×C4—2142no—
308C2×C4—2142no—
309C2×C4—2142no—
310C2×C4—2142no—
311C2×C4—2148no—
312C2×C4—2142no—
313C2×C4—2142no—
314C2×C4D4×D10714yes—
315C22—22no—
316C22—24no—
317C22—24no—
318C22—22no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22—42no—
323C22—41no—
324C22—1074no—
325C22—1074no—
326C22—21432no—
327C22—2142no—
328C22—2142no—
329C22—2142no—
330C22—2144no—
331C22—2144no—
332C22—2144no—
333C22—2144no—
334C22—2142no—
335C22#1681209114yescomplemented
336C22C2×D4×D10711yescharacteristic
337C22C2×D4×D10712yes—
338C22#41×D10712yes—
339C22C2×#1313381711yescharacteristic
340C22C2×#1313381811yescharacteristic
341C4—24no—
342C4—41no—
343C4—42no—
344C4—4281no—
345C4—4282no—
346C4—4281no—
347C4—2144no—
348C2—24no—
349C2—81no—
350C2—2144no—
351C2—2144no—
352C2#120×D10711yescharacteristic
353C2#123×D10711yescharacteristic
354C2C2×#1681209111yescharacteristic
355C2#2029313612yes—
356C2#2029313712yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#386×C107C2
prime radicalC2×#16812077C2
hypercenter#386D107
hypocenterC107#2849
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C23×C4×D107C2semidirect
2. C2×#16812091C2semidirect
4. #20292507C2semidirect
5. #20292503C2semidirect
6. #20292502C2semidirect
7. #20292501C2semidirect
8. #20292500C2semidirect
9. #20292489C2semidirect
10. #20292482C2semidirect
11. #386×C107C2semidirect
22. C2×C4⋊Dic107C22semidirect
23. C2×Dic107⋊C4C22semidirect
26. C2×C22⋊Dic107C22semidirect
31. C107⋊#95C22semidirect
32. C107⋊#95C22semidirect
33. C107⋊#95C22semidirect
37. C2×C22⋊C4×C107C22semidirect
38. #95×C107C22semidirect
62. C2×C4×D107D4semidirect
145. C22×C107#123semidirect
161. C2×D107#123semidirect
179. C107#2849semidirect
185. #386D107semidirect
213. C2×C22⋊C4C2×D107semidirect
217. #95C2×D107semidirect
335. C22#16812091semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D1072
C2×#168120912
C2×#168120772
#202925072
#202925032
#202925022
#202925012
#202925002
#202924892
#202924822
#386×C1072
#2849107