Finite Group Databasebeta

← #28339810

Subgroups — #28339810

Order 13696.

1232 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 171 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D107C211yesmaximal, characteristic, complemented
2C2×#16812081C212yesmaximal, complemented
3C2×#16812077C212yesmaximal, complemented
4C2×(C22⋊D107)⋊C4C211yesmaximal, characteristic, complemented
5#20292545C211yesmaximal, characteristic, complemented
6#20292541C211yesmaximal, characteristic, complemented
7#20292536C211yesmaximal, characteristic, complemented
8#20292527C212yesmaximal, complemented
9#20292486C211yesmaximal, characteristic, complemented
10#20292482C212yesmaximal, complemented
11#392×C107C211yesmaximal, characteristic, complemented
12C24×D107C2212yes—
13C22×D428C2211yescharacteristic
14C22×C22⋊D107C2212yes—
15C22×C22⋊D107C2211yescharacteristic
16C22×C22⋊D107C2212yescomplemented
17C2×D4×D107—28no—
18C2×D4×D107—24no—
19C23×Dic107C2211yescharacteristic, complemented
20C22×C4×D107C2211yescharacteristic
21#16812081—28no—
22#16812077—28no—
23(C22⋊D107)⋊C4—24no—
24C2×Dic107⋊C4C2212yescomplemented
25C2×Dic107⋊C4C2211yescharacteristic, complemented
26C2×C22⋊Dic107C2212yescomplemented
27C2×C22⋊Dic107C2212yescomplemented
28C2×(C2×D107)⋊C4C2211yescharacteristic
29C2×(C2×D107)⋊C4C2212yes—
30C2×(C2×D107)⋊C4C2212yescomplemented
31C2×(C2×D107)⋊C4C2212yescomplemented
32C2×(C2×D107)⋊C4C2212yescomplemented
33C2×C4×Dic107C2211yescharacteristic, complemented
34C107⋊#95C2211yescharacteristic, complemented
35C22×D4×C107C2211yescharacteristic
36C22×D4×C107C2212yescomplemented
37C2×C22⋊C4×C107C2211yescharacteristic, complemented
38C2×C22⋊C4×C107C2212yescomplemented
39#95×C107C2211yescharacteristic, complemented
40C23×D107—24no—
41C23×D107—24no—
42C23×D107—24no—
43C23×D107—24no—
44C23×D107—24no—
45C23×D107—42no—
46C23×D107—42no—
47C23×D107C2312yes—
48C23×D107C2311yescharacteristic
49C2×D428—24no—
50C2×D428—22no—
51C2×C22⋊D107—22no—
52C2×C22⋊D107—22no—
53C2×C22⋊D107—24no—
54C2×C22⋊D107—24no—
55C2×C22⋊D107—24no—
56C2×C22⋊D107—24no—
57C2×C22⋊D107—42no—
58C2×C22⋊D107D418yescomplemented
59C2×C22⋊D107D414yescomplemented
60D4×D107—232no—
61C22×Dic107—22no—
62C22×Dic107—24no—
63C22×Dic107C2311yescharacteristic
64C22×Dic107C2312yes—
65C22×Dic107C2311yescharacteristic
66C22×Dic107C2311yescharacteristic
67C2×C4×D107—24no—
68C2×C4×D107D414yescomplemented
69Dic107⋊C4—22no—
70Dic107⋊C4—24no—
71C22⋊Dic107—24no—
72C22⋊Dic107—24no—
73(C2×D107)⋊C4—22no—
74(C2×D107)⋊C4—24no—
75(C2×D107)⋊C4—24no—
76(C2×D107)⋊C4—24no—
77(C2×D107)⋊C4—42no—
78C4×Dic107—22no—
79C24×C107C2312yes—
80C24×C107C2311yescharacteristic
81C2×D4×C107—24no—
82C2×D4×C107—24no—
83C2×D4×C107—24no—
84C2×D4×C107—22no—
85C2×D4×C107—42no—
86C22×C4×C107C2311yescharacteristic
87C22×C4×C107C2312yes—
88C22×C4×C107C2311yescharacteristic
89C22⋊C4×C107—22no—
90C22⋊C4×C107—42no—
91C23×C107—21no—
92C23×C107—22no—
93C23×C107—22no—
94C23×C107—22no—
95C23×C107—22no—
96C23×C107—21no—
97C23×C107—22no—
98C23×C107—22no—
99C23×C107—22no—
100C23×C107—22no—
101C23×C107C2×D414yes—
102C23×C107C2×D412yescomplemented
103C23×C107C2411yescharacteristic
104C22×D107—22no—
105C22×D107—22no—
106C22×D107—216no—
107C22×D107—22no—
108C22×D107—22no—
109C22×D107—22no—
110C22×D107—22no—
111C22×D107—22no—
112C22×D107—22no—
113C22×D107—48no—
114C22×D107—48no—
115C22×D107—48no—
116C22×D107—48no—
117C22×D107—48no—
118C22×D107—48no—
119C22×D107C2×D414yes—
120C22×D107C2×D412yes—
121C22×D107C2×D412yes—
122C22×D107#4112yes—
123D4×C107—28no—
124D4×C107—48no—
125C2×C4×C107—22no—
126C2×C4×C107—21no—
127C2×C4×C107—22no—
128C2×C4×C107—22no—
129C2×C4×C107—22no—
130C2×C4×C107—22no—
131C2×C4×C107C2×D412yes—
132C22⋊D107—28no—
133C22⋊D107—216no—
134C22⋊D107—48no—
135C2×Dic107—21no—
136C2×Dic107—21no—
137C2×Dic107—22no—
138C2×Dic107—22no—
139C2×Dic107—21no—
140C2×Dic107—22no—
141C2×Dic107—22no—
142C2×Dic107—44no—
143C2×Dic107C2×D412yes—
144C2×Dic107C2×D414yes—
145C2×Dic107C2×D412yescomplemented
146D428—28no—
147C4×D107—28no—
148C22×C107—24no—
149C22×C107—22no—
150C22×C107—24no—
151C22×C107—24no—
152C22×C107—24no—
153C22×C107—41no—
154C22×C107—42no—
155C22×C107—42no—
156C22×C107—41no—
157C22×C107—42no—
158C22×C107—42no—
159C22×C107—42no—
160C22×C107C22×D411yescharacteristic
161C22×C107C22×D412yes—
162C22×C107C22×D411yescharacteristic
163C22×C107C2×#4111yescharacteristic
164C22×C107H(5,𝔽2)12yes—
165C2×D107—22no—
166C2×D107—22no—
167C2×D107—24no—
168C2×D107—24no—
169C2×D107—24no—
170C2×D107—24no—
171C2×D107—24no—
172C2×D107—432no—
173C2×D107—42no—
174C2×D107—42no—
175C2×D107—42no—
176C2×D107—42no—
177C2×D107#12114yescomplemented
178C4×C107—22no—
179C4×C107—41no—
180C4×C107—42no—
181Dic107—22no—
182Dic107—41no—
183Dic107—42no—
184Dic107#12014yescomplemented
185C2×C107—44no—
186C2×C107—42no—
187C2×C107C2×#12011yescharacteristic
188C2×C107C2×#12111yescharacteristic
189C2×C107#53511yescharacteristic
190C2×C107D4212yes—
191C2×C107#54612yes—
192D107—24no—
193D107—44no—
194#2408—1071nomaximal, Sylow
195C107#240811yescharacteristic, complemented, Sylow
196C23×D4—1071no—
197C2×#121—1072no—
198C2×#120—1072no—
199C2×C4×D4—1071no—
200#394—1071no—
201#394—1071no—
202#392—1071no—
203#392D10711yescharacteristic, complemented
204#390—1072no—
205#380—1071no—
206#379—1072no—
207C25—1072no—
208C22×D4—1071no—
209C22×D4—1072no—
210C22×D4—1072no—
211C22×D4—1071no—
212C22×D4—2144no—
213C22×D4—2148no—
214C22×D4C2×D10711yescharacteristic
215C22×D4C2×D10712yescomplemented
216C23×C4—1071no—
217C23×C4—1071no—
218#121—2148no—
219#120—2148no—
220C4×D4—2144no—
221C2×C4⋊C4—1071no—
222C2×C4⋊C4—1072no—
223C2×C22⋊C4—1072no—
224C2×C22⋊C4—1072no—
225C2×C22⋊C4—1072no—
226C2×C22⋊C4—1071no—
227C2×C22⋊C4—1072no—
228C2×C22⋊C4—1072no—
229C2×C22⋊C4—1072no—
230C2×C22⋊C4C2×D10711yescharacteristic, complemented
231C2×C22⋊C4C2×D10712yescomplemented
232C2×C42—1071no—
233#95—1071no—
234#95C2×D10711yescharacteristic, complemented
235C24—1071no—
236C24—1072no—
237C24—4282no—
238C24—4282no—
239C24—2144no—
240C24—2144no—
241C24—2144no—
242C24—2144no—
243C24—2144no—
244C24C22×D10712yes—
245C24C22×D10711yescharacteristic
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—22no—
250C2×D4—42no—
251C2×D4—1078no—
252C2×D4—1074no—
253C2×D4—4282no—
254C2×D4—2144no—
255C2×D4—2144no—
256C2×D4—2144no—
257C2×D4—21432no—
258C2×D4—2144no—
259C2×D4—2144no—
260C2×D4—2142no—
261C2×D4—2142no—
262C2×D4—2142no—
263C22×C4—1071no—
264C22×C4—1071no—
265C22×C4—1074no—
266C22×C4—1072no—
267C22×C4—1071no—
268C22×C4—2144no—
269C22×C4—2142no—
270C22×C4—2144no—
271C22×C4C22×D10712yes—
272C22×C4C22×D10711yescharacteristic
273C22×C4C22×D10711yescharacteristic
274C4⋊C4—2144no—
275C4⋊C4—2142no—
276C22⋊C4—22no—
277C22⋊C4—42no—
278C22⋊C4—4282no—
279C22⋊C4—2144no—
280C22⋊C4—2142no—
281C22⋊C4—2144no—
282C22⋊C4—2144no—
283C22⋊C4—2144no—
284C22⋊C4—2144no—
285C42—2142no—
286C23—21no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—21no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—1072no—
297C23—1072no—
298C23—1072no—
299C23—1074no—
300C23—4288no—
301C23—4288no—
302C23—4288no—
303C23—4288no—
304C23—4288no—
305C23—4288no—
306C23—21416no—
307C23—2142no—
308C23—2142no—
309C23—2142no—
310C23—2142no—
311C23—2142no—
312C23—2142no—
313C23—2142no—
314C23—2142no—
315C23D4×D10714yes—
316C23D4×D10712yescomplemented
317C23C23×D10711yescharacteristic
318D4—28no—
319D4—48no—
320D4—4288no—
321D4—2148no—
322D4—21416no—
323D4—2148no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—1072no—
331C2×C4—1074no—
332C2×C4—1072no—
333C2×C4—4284no—
334C2×C4—2148no—
335C2×C4—2142no—
336C2×C4—2142no—
337C2×C4—2141no—
338C2×C4—2142no—
339C2×C4—2142no—
340C2×C4—2141no—
341C2×C4—2141no—
342C2×C4D4×D10712yes—
343C22—24no—
344C22—24no—
345C22—24no—
346C22—24no—
347C22—22no—
348C22—42no—
349C22—41no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—41no—
354C22—42no—
355C22—1074no—
356C22—42832no—
357C22—4282no—
358C22—4282no—
359C22—4282no—
360C22—4282no—
361C22—2144no—
362C22—2144no—
363C22—2144no—
364C22—2142no—
365C22—2144no—
366C22—2144no—
367C22—2142no—
368C22C2×D4×D10711yescharacteristic
369C22C2×D4×D10711yescharacteristic
370C22C2×D4×D10712yes—
371C22#41×D10711yescharacteristic
372C22#1681215412yes—
373C4—22no—
374C4—41no—
375C4—42no—
376C4—1074no—
377C4—4282no—
378C4—4281no—
379C4—2142no—
380C2—44no—
381C2—42no—
382C2—4284no—
383C2—2144no—
384C2#120×D10711yescharacteristic
385C2#121×D10711yescharacteristic
386C2#2029319011yescharacteristic
387C2#2029325912yes—
388C2#2029326612yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#392×C107C2
hypercenter#392D107
hypocenterC107#2408
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D4×D107C2semidirect
2. C2×#16812081C2semidirect
3. C2×#16812077C2semidirect
4. C2×(C22⋊D107)⋊C4C2semidirect
5. #20292545C2semidirect
6. #20292541C2semidirect
7. #20292536C2semidirect
8. #20292527C2semidirect
9. #20292486C2semidirect
10. #20292482C2semidirect
11. #392×C107C2semidirect
16. C22×C22⋊D107C22semidirect
19. C23×Dic107C22semidirect
24. C2×Dic107⋊C4C22semidirect
25. C2×Dic107⋊C4C22semidirect
26. C2×C22⋊Dic107C22semidirect
27. C2×C22⋊Dic107C22semidirect
30. C2×(C2×D107)⋊C4C22semidirect
31. C2×(C2×D107)⋊C4C22semidirect
32. C2×(C2×D107)⋊C4C22semidirect
33. C2×C4×Dic107C22semidirect
34. C107⋊#95C22semidirect
36. C22×D4×C107C22semidirect
37. C2×C22⋊C4×C107C22semidirect
38. C2×C22⋊C4×C107C22semidirect
39. #95×C107C22semidirect
58. C2×C22⋊D107D4semidirect
59. C2×C22⋊D107D4semidirect
68. C2×C4×D107D4semidirect
102. C23×C107C2×D4semidirect
145. C2×Dic107C2×D4semidirect
177. C2×D107#121semidirect
184. Dic107#120semidirect
195. C107#2408semidirect
203. #392D107semidirect
215. C22×D4C2×D107semidirect
230. C2×C22⋊C4C2×D107semidirect
231. C2×C22⋊C4C2×D107semidirect
234. #95C2×D107semidirect
316. C23D4×D107semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1072
C2×#168120812
C2×#168120772
C2×(C22⋊D107)⋊C42
#202925452
#202925412
#202925362
#202925272
#202924862
#202924822
#392×C1072
#2408107