Finite Group Databasebeta

← #28340093

Subgroups — #28340093

Order 13696.

1348 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 181 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×#16812105C212yesmaximal, complemented
3C2×#16812082C211yesmaximal, characteristic, complemented
4C2×#16812079C212yesmaximal, complemented
5C2×#16812077C212yesmaximal, complemented
6#20292537C211yesmaximal, characteristic, complemented
7#20292536C211yesmaximal, characteristic, complemented
8#20292535C211yesmaximal, characteristic, complemented
9#20292527C212yesmaximal, complemented
10#20292474C211yesmaximal, characteristic, complemented
11#392×C107C211yesmaximal, characteristic, complemented
12C24×D107C2212yes—
13C22×D428C2211yescharacteristic
14C22×D428C2211yescharacteristic
15C22×D428C2212yescomplemented
16C22×C22⋊D107C2211yescharacteristic
17C22×C22⋊D107C2212yes—
18C22×C22⋊D107C2211yescharacteristic, complemented
19C22×C22⋊D107C2212yescomplemented
20C22×C22⋊D107C2212yescomplemented
21C2×D4×D107—28no—
22C2×D4×D107—24no—
23C22×C4×D107C2211yescharacteristic
24#16812105—28no—
25#16812082—24no—
26#16812079—28no—
27#16812077—28no—
28C2×Dic107⋊C4C2211yescharacteristic, complemented
29C2×C22⋊Dic107C2212yescomplemented
30C2×(C2×D107)⋊C4C2212yes—
31C2×(C2×D107)⋊C4C2212yescomplemented
32C2×(C2×D107)⋊C4C2212yescomplemented
33C2×(C2×D107)⋊C4C2211yescharacteristic, complemented
34C2×C4×Dic107C2212yescomplemented
35C107⋊#95C2211yescharacteristic, complemented
36C22×D4×C107C2211yescharacteristic
37C22×D4×C107C2212yescomplemented
38C2×C22⋊C4×C107C2212yescomplemented
39C2×C22⋊C4×C107C2211yescharacteristic, complemented
40#95×C107C2211yescharacteristic, complemented
41C23×D107—24no—
42C23×D107—24no—
43C23×D107—24no—
44C23×D107—24no—
45C23×D107—24no—
46C23×D107—42no—
47C23×D107—42no—
48C23×D107C2311yescharacteristic
49C23×D107C2312yes—
50C23×D107C2311yescharacteristic
51C2×D428—24no—
52C2×D428—22no—
53C2×D428—24no—
54C2×D428—22no—
55C2×D428—24no—
56C2×D428—24no—
57C2×D428—42no—
58C2×C22⋊D107—24no—
59C2×C22⋊D107—24no—
60C2×C22⋊D107—22no—
61C2×C22⋊D107—24no—
62C2×C22⋊D107—24no—
63C2×C22⋊D107—24no—
64C2×C22⋊D107—22no—
65C2×C22⋊D107—24no—
66C2×C22⋊D107—24no—
67C2×C22⋊D107—24no—
68C2×C22⋊D107—42no—
69C2×C22⋊D107—42no—
70C2×C22⋊D107D418yescomplemented
71D4×D107—232no—
72C22×Dic107C2311yescharacteristic
73C22×Dic107C2312yes—
74C22×Dic107C2311yescharacteristic
75C2×C4×D107—24no—
76C2×C4×D107D414yescomplemented
77Dic107⋊C4—22no—
78C22⋊Dic107—24no—
79(C2×D107)⋊C4—22no—
80(C2×D107)⋊C4—24no—
81(C2×D107)⋊C4—24no—
82(C2×D107)⋊C4D418yescomplemented
83C4×Dic107—24no—
84C24×C107C2311yescharacteristic
85C24×C107C2312yes—
86C2×D4×C107—24no—
87C2×D4×C107—24no—
88C2×D4×C107—22no—
89C2×D4×C107—24no—
90C2×D4×C107—42no—
91C22×C4×C107C2311yescharacteristic
92C22×C4×C107C2311yescharacteristic
93C22×C4×C107C2312yes—
94C22⋊C4×C107—22no—
95C22⋊C4×C107—24no—
96C23×C107—22no—
97C23×C107—22no—
98C23×C107—22no—
99C23×C107—22no—
100C23×C107—22no—
101C23×C107—22no—
102C23×C107—21no—
103C23×C107—22no—
104C23×C107—22no—
105C23×C107—21no—
106C23×C107—21no—
107C23×C107C2×D414yes—
108C23×C107C2411yescharacteristic
109C22×D107—22no—
110C22×D107—21no—
111C22×D107—21no—
112C22×D107—216no—
113C22×D107—22no—
114C22×D107—22no—
115C22×D107—22no—
116C22×D107—22no—
117C22×D107—22no—
118C22×D107—22no—
119C22×D107—21no—
120C22×D107—22no—
121C22×D107—22no—
122C22×D107—21no—
123C22×D107—21no—
124C22×D107—48no—
125C22×D107—48no—
126C22×D107—48no—
127C22×D107—48no—
128C22×D107—48no—
129C22×D107—48no—
130C22×D107C2×D414yes—
131C22×D107C2×D414yes—
132C22×D107C2×D412yes—
133D4×C107—28no—
134D4×C107—48no—
135C2×C4×C107—22no—
136C2×C4×C107—22no—
137C2×C4×C107—22no—
138C2×C4×C107—22no—
139C2×C4×C107—21no—
140C2×C4×C107C2×D412yes—
141C2×C4×C107C2×D414yescomplemented
142C22⋊D107—216no—
143C22⋊D107—48no—
144C22⋊D107—48no—
145C22⋊D107—44no—
146C22⋊D107—44no—
147C2×Dic107—22no—
148C2×Dic107—22no—
149C2×Dic107—22no—
150C2×Dic107—21no—
151C2×Dic107C2×D414yes—
152C2×Dic107C2×D412yes—
153C2×Dic107C2×D414yescomplemented
154D428—28no—
155D428—48no—
156D428—44no—
157C4×D107—28no—
158C22×C107—24no—
159C22×C107—24no—
160C22×C107—24no—
161C22×C107—42no—
162C22×C107—42no—
163C22×C107—42no—
164C22×C107—42no—
165C22×C107—41no—
166C22×C107—41no—
167C22×C107—41no—
168C22×C107—42no—
169C22×C107—42no—
170C22×C107C22×D411yescharacteristic
171C22×C107C22×D412yes—
172C22×C107C22×D412yes—
173C22×C107H(5,𝔽2)11yescharacteristic
174C22×C107H(5,𝔽2)11yescharacteristic
175C2×D107—22no—
176C2×D107—22no—
177C2×D107—24no—
178C2×D107—24no—
179C2×D107—24no—
180C2×D107—24no—
181C2×D107—24no—
182C2×D107—432no—
183C2×D107—42no—
184C2×D107—42no—
185C2×D107—42no—
186C2×D107—42no—
187C2×D107—41no—
188C2×D107—41no—
189C2×D107—41no—
190C2×D107—42no—
191C2×D107—42no—
192C2×D107#12714yescomplemented
193C4×C107—22no—
194C4×C107—24no—
195C4×C107—41no—
196Dic107—24no—
197Dic107—41no—
198Dic107#12014yescomplemented
199C2×C107—44no—
200C2×C107—81no—
201C2×C107C2×#12011yescharacteristic
202C2×C107C2×#12711yescharacteristic
203C2×C107#53511yescharacteristic
204C2×C107D4212yes—
205C2×C107D4212yes—
206D107—24no—
207D107—44no—
208D107—81no—
209#2433—1071nomaximal, Sylow
210C107#243311yescharacteristic, complemented, Sylow
211C23×D4—1071no—
212C2×#127—1072no—
213C2×#121—1071no—
214C2×#120—1072no—
215C2×#120—1072no—
216#392—1071no—
217#392—1071no—
218#392—1071no—
219#392D10711yescharacteristic, complemented
220#390—1072no—
221C4×C22⋊C4—1071no—
222C25—1072no—
223C22×D4—1072no—
224C22×D4—1072no—
225C22×D4—1072no—
226C22×D4—1071no—
227C22×D4—1071no—
228C22×D4—1072no—
229C22×D4—1071no—
230C22×D4—1071no—
231C22×D4—2144no—
232C22×D4—2148no—
233C22×D4C2×D10711yescharacteristic
234C22×D4C2×D10712yescomplemented
235C23×C4—1071no—
236#127—2148no—
237#121—2144no—
238#120—2148no—
239#120—2148no—
240C2×C4⋊C4—1071no—
241C2×C22⋊C4—1072no—
242C2×C22⋊C4—1072no—
243C2×C22⋊C4—1072no—
244C2×C22⋊C4—1072no—
245C2×C22⋊C4—1071no—
246C2×C22⋊C4C2×D10712yescomplemented
247C2×C22⋊C4C2×D10711yescharacteristic, complemented
248C2×C42—1072no—
249#95—1071no—
250#95C2×D10711yescharacteristic, complemented
251C24—1071no—
252C24—1072no—
253C24—1071no—
254C24—4282no—
255C24—4282no—
256C24—2144no—
257C24—2144no—
258C24—2144no—
259C24—2144no—
260C24—2144no—
261C24C22×D10711yescharacteristic
262C24C22×D10712yes—
263C2×D4—24no—
264C2×D4—24no—
265C2×D4—22no—
266C2×D4—24no—
267C2×D4—42no—
268C2×D4—1078no—
269C2×D4—4282no—
270C2×D4—4282no—
271C2×D4—4282no—
272C2×D4—2144no—
273C2×D4—2142no—
274C2×D4—2144no—
275C2×D4—2144no—
276C2×D4—2144no—
277C2×D4—2144no—
278C2×D4—2142no—
279C2×D4—2144no—
280C2×D4—2144no—
281C2×D4—2144no—
282C2×D4—21432no—
283C2×D4—2144no—
284C2×D4—2142no—
285C2×D4—2144no—
286C2×D4—2144no—
287C2×D4—2142no—
288C2×D4—2144no—
289C22×C4—1071no—
290C22×C4—1072no—
291C22×C4—1074no—
292C22×C4—1071no—
293C22×C4—2144no—
294C22×C4C22×D10711yescharacteristic
295C22×C4C22×D10711yescharacteristic
296C22×C4C22×D10712yes—
297C4⋊C4—2142no—
298C22⋊C4—22no—
299C22⋊C4—24no—
300C22⋊C4—1078no—
301C22⋊C4—2144no—
302C22⋊C4—2144no—
303C22⋊C4—2142no—
304C22⋊C4—2144no—
305C42—2144no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—21no—
313C23—22no—
314C23—22no—
315C23—21no—
316C23—21no—
317C23—1074no—
318C23—1072no—
319C23—1074no—
320C23—4288no—
321C23—4288no—
322C23—4288no—
323C23—4288no—
324C23—4288no—
325C23—4288no—
326C23—21416no—
327C23—2141no—
328C23—2142no—
329C23—2142no—
330C23—2141no—
331C23—2141no—
332C23—2142no—
333C23—2142no—
334C23—2142no—
335C23—2142no—
336C23—2142no—
337C23—2142no—
338C23—2142no—
339C23—2141no—
340C23—2141no—
341C23D4×D10714yes—
342C23C23×D10711yescharacteristic
343D4—28no—
344D4—48no—
345D4—4288no—
346D4—4284no—
347D4—4284no—
348D4—4284no—
349D4—4288no—
350D4—4288no—
351D4—2148no—
352D4—21416no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—1074no—
359C2×C4—1072no—
360C2×C4—1074no—
361C2×C4—2148no—
362C2×C4—2142no—
363C2×C4—2141no—
364C2×C4—2142no—
365C2×C4—2142no—
366C2×C4D4×D10712yes—
367C2×C4D4×D10714yescomplemented
368C22—24no—
369C22—24no—
370C22—24no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—42no—
378C22—42no—
379C22—41no—
380C22—1074no—
381C22—42832no—
382C22—4281no—
383C22—4281no—
384C22—4282no—
385C22—4282no—
386C22—4281no—
387C22—4282no—
388C22—4282no—
389C22—4282no—
390C22—4282no—
391C22—2144no—
392C22—2144no—
393C22—2144no—
394C22—2142no—
395C22—2144no—
396C22—2144no—
397C22—2142no—
398C22C2×D4×D10711yescharacteristic
399C22C2×D4×D10712yes—
400C22C2×D4×D10712yes—
401C22#1681215411yescharacteristic
402C22#1681215511yescharacteristic
403C4—24no—
404C4—22no—
405C4—41no—
406C4—1074no—
407C4—4281no—
408C4—2144no—
409C2—44no—
410C2—81no—
411C2—8561no—
412C2—4284no—
413C2—2144no—
414C2#120×D10711yescharacteristic
415C2#127×D10711yescharacteristic
416C2#2029319311yescharacteristic
417C2#2029325912yes—
418C2#2029326112yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#392×C107C2
hypercenter#392D107
hypocenterC107#2433
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×#16812105C2semidirect
3. C2×#16812082C2semidirect
4. C2×#16812079C2semidirect
5. C2×#16812077C2semidirect
6. #20292537C2semidirect
7. #20292536C2semidirect
8. #20292535C2semidirect
9. #20292527C2semidirect
10. #20292474C2semidirect
11. #392×C107C2semidirect
15. C22×D428C22semidirect
18. C22×C22⋊D107C22semidirect
19. C22×C22⋊D107C22semidirect
20. C22×C22⋊D107C22semidirect
28. C2×Dic107⋊C4C22semidirect
29. C2×C22⋊Dic107C22semidirect
31. C2×(C2×D107)⋊C4C22semidirect
32. C2×(C2×D107)⋊C4C22semidirect
33. C2×(C2×D107)⋊C4C22semidirect
34. C2×C4×Dic107C22semidirect
35. C107⋊#95C22semidirect
37. C22×D4×C107C22semidirect
38. C2×C22⋊C4×C107C22semidirect
39. C2×C22⋊C4×C107C22semidirect
40. #95×C107C22semidirect
70. C2×C22⋊D107D4semidirect
76. C2×C4×D107D4semidirect
82. (C2×D107)⋊C4D4semidirect
141. C2×C4×C107C2×D4semidirect
153. C2×Dic107C2×D4semidirect
192. C2×D107#127semidirect
198. Dic107#120semidirect
210. C107#2433semidirect
219. #392D107semidirect
234. C22×D4C2×D107semidirect
246. C2×C22⋊C4C2×D107semidirect
247. C2×C22⋊C4C2×D107semidirect
250. #95C2×D107semidirect
367. C2×C4D4×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×#168121052
C2×#168120822
C2×#168120792
C2×#168120772
#202925372
#202925362
#202925352
#202925272
#202924742
#392×C1072
#2433107