Finite Group Databasebeta

← #28339607

Subgroups — #28339607

Order 13696.

512 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812093C211yesmaximal, characteristic
2#20292527C211yesmaximal, characteristic
3#20292525C211yesmaximal, characteristic, complemented
4#20292523C211yesmaximal, characteristic, complemented
5#20292520C211yesmaximal, characteristic, complemented
6#20292518C211yesmaximal, characteristic, complemented
7#20292516C211yesmaximal, characteristic, complemented
8#20292512C211yesmaximal, characteristic, complemented
9#20292510C211yesmaximal, characteristic, complemented
10#20292503C211yesmaximal, characteristic
11#20292501C211yesmaximal, characteristic, complemented
12#20292498C211yesmaximal, characteristic, complemented
13#20292485C211yesmaximal, characteristic, complemented
14#20292485C211yesmaximal, characteristic, complemented
15#388×C107C211yesmaximal, characteristic, complemented
16C22×C22⋊D107C2211yescharacteristic
17C23×Dic107C2211yescharacteristic
18C22×C4×D107C2211yescharacteristic
19#16812093C418yescomplemented
20C2×C4⋊Dic107C2211yescharacteristic
21C2×C4⋊Dic107C2211yescharacteristic
22C2×C4⋊Dic107C2211yescharacteristic
23C2×Dic107⋊C4C2211yescharacteristic
24C2×Dic107⋊C4C2211yescharacteristic
25C2×Dic107⋊C4C2211yescharacteristic
26C2×Dic107⋊C4C2211yescharacteristic
27C2×Dic107⋊C4C2211yescharacteristic
28C2×C22⋊Dic107C2211yescharacteristic
29C2×C22⋊Dic107C2211yescharacteristic
30C2×(C2×D107)⋊C4C2211yescharacteristic
31C2×(C2×D107)⋊C4C2211yescharacteristic
32C2×(C2×D107)⋊C4C2211yescharacteristic
33C2×(C2×D107)⋊C4C2211yescharacteristic
34C2×(C2×D107)⋊C4C2211yescharacteristic
35C2×C4×Dic107C2211yescharacteristic
36C2×C4×Dic107C2211yescharacteristic
37C107⋊#95C2211yescharacteristic
38C107⋊#95C2211yescharacteristic
39C107⋊#95C2211yescharacteristic
40C107⋊#95C2211yescharacteristic
41C107⋊#95C2211yescharacteristic
42C107⋊#95C2211yescharacteristic
43C107⋊#95C2211yescharacteristic
44C107⋊#95C2211yescharacteristic
45C2×C4⋊C4×C107C2211yescharacteristic
46C2×C22⋊C4×C107C2211yescharacteristic
47C2×C22⋊C4×C107C2211yescharacteristic
48C2×C22⋊C4×C107C2211yescharacteristic
49C2×C42×C107C2211yescharacteristic
50#95×C107C2211yescharacteristic
51#95×C107C2211yescharacteristic
52C23×D107C2311yescharacteristic
53C2×C22⋊D107—41no—
54C2×C22⋊D107—41no—
55C2×C22⋊D107C2×C414yes—
56C22×Dic107—41no—
57C22×Dic107—41no—
58C22×Dic107C2×C414yescomplemented
59C22×Dic107C2311yescharacteristic
60C22×Dic107C2311yescharacteristic
61C22×Dic107C2311yescharacteristic
62C22×Dic107C2311yescharacteristic
63C22×Dic107C2311yescharacteristic
64C22×Dic107C2311yescharacteristic
65C22×Dic107C2311yescharacteristic
66C2×C4×D107—41no—
67C2×C4×D107—41no—
68C2×C4×D107—41no—
69C4⋊Dic107—41no—
70C4⋊Dic107C2×C414yescomplemented
71C4⋊Dic107C2×C414yescomplemented
72Dic107⋊C4—41no—
73Dic107⋊C4—41no—
74Dic107⋊C4—41no—
75Dic107⋊C4—41no—
76Dic107⋊C4—41no—
77C22⋊Dic107—41no—
78C22⋊Dic107—41no—
79(C2×D107)⋊C4—41no—
80(C2×D107)⋊C4—41no—
81(C2×D107)⋊C4—41no—
82(C2×D107)⋊C4C2×C414yes—
83(C2×D107)⋊C4C2×C414yes—
84C4×Dic107—41no—
85C4×Dic107—41no—
86C24×C107C2311yescharacteristic
87C22×C4×C107C2311yescharacteristic
88C22×C4×C107C2311yescharacteristic
89C22×C4×C107C2311yescharacteristic
90C22×C4×C107C2311yescharacteristic
91C22×C4×C107C2311yescharacteristic
92C22×C4×C107C2311yescharacteristic
93C4⋊C4×C107—41no—
94C22⋊C4×C107—41no—
95C22⋊C4×C107—41no—
96C22⋊C4×C107C2×C414yescomplemented
97C42×C107—41no—
98C23×C107—21no—
99C23×C107—21no—
100C23×C107—21no—
101C23×C107—21no—
102C23×C107—21no—
103C23×C107—21no—
104C23×C107C22×C412yes—
105C23×C107C2411yescharacteristic
106C22×D107—21no—
107C22×D107—21no—
108C22×D107—21no—
109C22×D107—21no—
110C22×D107—21no—
111C22×D107—21no—
112C22×D107C22×C412yes—
113C2×C4×C107—21no—
114C2×C4×C107—21no—
115C2×C4×C107—21no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107—21no—
128C2×C4×C107—21no—
129C2×C4×C107C22×C412yes—
130C2×C4×C107C22×C412yes—
131C22⋊D107—44no—
132C2×Dic107—21no—
133C2×Dic107—21no—
134C2×Dic107—21no—
135C2×Dic107—21no—
136C2×Dic107—21no—
137C2×Dic107—21no—
138C2×Dic107—21no—
139C2×Dic107—21no—
140C2×Dic107—21no—
141C2×Dic107—21no—
142C2×Dic107—21no—
143C2×Dic107—21no—
144C2×Dic107—21no—
145C2×Dic107—21no—
146C2×Dic107—21no—
147C2×Dic107—21no—
148C2×Dic107—21no—
149C2×Dic107—21no—
150C2×Dic107—44no—
151C2×Dic107C22×C412yes—
152C2×Dic107C22×C412yes—
153C2×Dic107C22×C412yes—
154C4×D107—44no—
155C22×C107—22no—
156C22×C107—22no—
157C22×C107—22no—
158C22×C107—41no—
159C22×C107—41no—
160C22×C107—41no—
161C22×C107—41no—
162C22×C107C23×C411yescharacteristic
163C22×C107H(5,𝔽2)11yescharacteristic
164C22×C107H(5,𝔽2)11yescharacteristic
165C22×C107H(5,𝔽2)11yescharacteristic
166C22×C107H(5,𝔽2)11yescharacteristic
167C22×C1072−1+411yescharacteristic
168C22×C1072−1+411yescharacteristic
169C2×D107—22no—
170C2×D107—22no—
171C2×D107—22no—
172C2×D107—41no—
173C2×D107—41no—
174C2×D107—41no—
175C2×D107—41no—
176C4×C107—22no—
177C4×C107—22no—
178C4×C107—41no—
179C4×C107—41no—
180C4×C107—41no—
181C4×C107—41no—
182Dic107—22no—
183Dic107—22no—
184Dic107—22no—
185Dic107—41no—
186Dic107—41no—
187Dic107—41no—
188Dic107—41no—
189C2×C107—42no—
190C2×C107#51811yescharacteristic
191C2×C107#52011yescharacteristic
192C2×C107#52011yescharacteristic
193C2×C107#56011yescharacteristic
194C2×C107#56211yescharacteristic
195C2×C107#56211yescharacteristic
196C2×C107#56311yescharacteristic
197D107—42no—
198#2376—1071nomaximal, Sylow
199C107#237611yescharacteristic, complemented, Sylow
200C2×#123—1071no—
201#390—1071no—
202#389—1071no—
203#389—1071no—
204#388—1071no—
205#388—1071no—
206#388—1071no—
207#388D10711yescharacteristic, complemented
208#387—1071no—
209#387—1071no—
210#386—1071no—
211#385—1071no—
212#385—1071no—
213#380—1071no—
214#380—1071no—
215C22×D4—1071no—
216C23×C4—1071no—
217C23×C4—1071no—
218#123—1078no—
219C2×C4⋊C4—1071no—
220C2×C4⋊C4—1071no—
221C2×C4⋊C4—1071no—
222C2×C4⋊C4—1071no—
223C2×C4⋊C4—1071no—
224C2×C4⋊C4—1071no—
225C2×C4⋊C4—1071no—
226C2×C4⋊C4—1071no—
227C2×C4⋊C4C2×D10711yescharacteristic
228C2×C22⋊C4—1071no—
229C2×C22⋊C4—1071no—
230C2×C22⋊C4—1071no—
231C2×C22⋊C4—1071no—
232C2×C22⋊C4—1071no—
233C2×C22⋊C4—1071no—
234C2×C22⋊C4—1071no—
235C2×C22⋊C4C2×D10711yescharacteristic
236C2×C22⋊C4C2×D10711yescharacteristic
237C2×C22⋊C4C2×D10711yescharacteristic
238C2×C42—1071no—
239C2×C42—1071no—
240C2×C42C2×D10711yescharacteristic
241#95—1071no—
242#95—1071no—
243#95—1071no—
244#95—1071no—
245#95—1071no—
246#95—1071no—
247#95—1071no—
248#95—1071no—
249#95C2×D10711yescharacteristic
250#95C2×D10711yescharacteristic
251C24—1071no—
252C24C22×D10711yescharacteristic
253C2×D4—1074no—
254C2×D4—4281no—
255C2×D4—4281no—
256C22×C4—1074no—
257C22×C4—1071no—
258C22×C4—1071no—
259C22×C4—1071no—
260C22×C4—1071no—
261C22×C4—1071no—
262C22×C4—1071no—
263C22×C4—1071no—
264C22×C4—4281no—
265C22×C4—4281no—
266C22×C4—4281no—
267C22×C4—4281no—
268C22×C4—4281no—
269C22×C4C22×D10711yescharacteristic
270C22×C4C22×D10711yescharacteristic
271C22×C4C22×D10711yescharacteristic
272C22×C4C22×D10711yescharacteristic
273C22×C4C22×D10711yescharacteristic
274C22×C4C22×D10711yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—1074no—
277C4⋊C4—1074no—
278C4⋊C4—4281no—
279C4⋊C4—4281no—
280C4⋊C4—4281no—
281C4⋊C4—4281no—
282C4⋊C4—4281no—
283C4⋊C4—4281no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—1074no—
287C22⋊C4—1074no—
288C22⋊C4—4281no—
289C22⋊C4—4281no—
290C22⋊C4—4281no—
291C22⋊C4—4281no—
292C22⋊C4—4281no—
293C22⋊C4C4×D10714yescomplemented
294C42—41no—
295C42—4281no—
296C42—4281no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—1072no—
304C23—2141no—
305C23—2141no—
306C23—2141no—
307C23—2141no—
308C23—2141no—
309C23—2141no—
310C23C2×C4×D10712yes—
311C23C23×D10711yescharacteristic
312D4—4284no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—1072no—
330C2×C4—1072no—
331C2×C4—1072no—
332C2×C4—4284no—
333C2×C4—4284no—
334C2×C4—2141no—
335C2×C4—2141no—
336C2×C4—2141no—
337C2×C4—2141no—
338C2×C4—2141no—
339C2×C4—2141no—
340C2×C4—2141no—
341C2×C4—2141no—
342C2×C4—2141no—
343C2×C4—2141no—
344C2×C4—2141no—
345C2×C4—2141no—
346C2×C4—2141no—
347C2×C4—2141no—
348C2×C4—2141no—
349C2×C4—2141no—
350C2×C4—2141no—
351C2×C4—2141no—
352C2×C4C2×C4×D10712yes—
353C2×C4C2×C4×D10712yes—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—4281no—
362C22—4281no—
363C22—4281no—
364C22—4281no—
365C22—2142no—
366C22—2142no—
367C22—2142no—
368C22C22×C4×D10711yescharacteristic
369C22#1681215411yescharacteristic
370C22#1681215411yescharacteristic
371C22#1681215511yescharacteristic
372C22#1681215511yescharacteristic
373C22#1681215611yescharacteristic
374C22#1681215611yescharacteristic
375C4—22no—
376C4—22no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—4281no—
382C4—4281no—
383C4—4281no—
384C4—4281no—
385C4—2142no—
386C4—2142no—
387C4—2142no—
388C2—42no—
389C2—4282no—
390C2#2029310111yescharacteristic
391C2#2029310611yescharacteristic
392C2#2029310611yescharacteristic
393C2#2029336511yescharacteristic
394C2#2029337111yescharacteristic
395C2#2029337411yescharacteristic
396C2#2029337911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
Fitting#388×C107C2
socle, cosocleC23×C107C24
prime radicalC22×C22⋊D107C22
hypercenter#388D107
hypocenterC107#2376
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
3. #20292525C2semidirect
4. #20292523C2semidirect
5. #20292520C2semidirect
6. #20292518C2semidirect
7. #20292516C2semidirect
8. #20292512C2semidirect
9. #20292510C2semidirect
11. #20292501C2semidirect
12. #20292498C2semidirect
13. #20292485C2semidirect
14. #20292485C2semidirect
15. #388×C107C2semidirect
19. #16812093C4semidirect
58. C22×Dic107C2×C4semidirect
70. C4⋊Dic107C2×C4semidirect
71. C4⋊Dic107C2×C4semidirect
96. C22⋊C4×C107C2×C4semidirect
199. C107#2376semidirect
207. #388D107semidirect
293. C22⋊C4C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120932
#202925272
#202925252
#202925232
#202925202
#202925182
#202925162
#202925122
#202925102
#202925032
#202925012
#202924982
#202924852
#388×C1072
#2376107