Finite Group Databasebeta

← #28339777

Subgroups — #28339777

Order 13696.

1056 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
1C22×#13133817C211yesmaximal, characteristic, complemented
2C2×#16812078C211yesmaximal, characteristic, complemented
3C22×C22⋊Dic107C211yesmaximal, characteristic, complemented
4#20292539C212yesmaximal, complemented
5#20292535C212yesmaximal, complemented
6#20292507C212yesmaximal, complemented
7#20292507C211yesmaximal, characteristic, complemented
8#20292504C211yesmaximal, characteristic, complemented
9#20292503C212yesmaximal, complemented
10#20292488C211yesmaximal, characteristic, complemented
11C2×#120×C107C211yesmaximal, characteristic, complemented
12C2×#13133817—24no—
13C2×#13133817—24no—
14C2×#13133817—24no—
15C22×Dic214C2211yescharacteristic, complemented
16C22×C22⋊D107C2212yes—
17C22×C22⋊D107C2211yescharacteristic
18C22×C22⋊D107C2211yescharacteristic, complemented
19C23×Dic107C2211yescharacteristic
20C23×Dic107C2211yescharacteristic
21C23×Dic107C2211yescharacteristic, complemented
22C22×C4×D107C2211yescharacteristic, complemented
23#16812078—24no—
24C2×C22⋊Dic107—24no—
25C2×C22⋊Dic107C2211yescharacteristic
26C2×C22⋊Dic107C2212yes—
27C2×C22⋊Dic107C2212yes—
28C2×C22⋊Dic107C2212yes—
29C2×C22⋊Dic107C2212yescomplemented
30C2×(C2×D107)⋊C4C2212yescomplemented
31C107⋊#95C2212yescomplemented
32C107⋊#95C2212yescomplemented
33C107⋊#95C2212yescomplemented
34C107⋊#95C2212yescomplemented
35C25×C107C2211yescharacteristic
36C22×D4×C107C2212yes—
37C22×D4×C107C2211yescharacteristic
38#120×C107—24no—
39C2×C22⋊C4×C107C2211yescharacteristic
40C2×C22⋊C4×C107C2212yes—
41C23×D107C2311yescharacteristic
42#13133817—232no—
43C2×Dic214—22no—
44C2×Dic214—22no—
45C2×Dic214—22no—
46C2×C22⋊D107—22no—
47C2×C22⋊D107—22no—
48C2×C22⋊D107—22no—
49C2×C22⋊D107—22no—
50C2×C22⋊D107—24no—
51C2×C22⋊D107—24no—
52C2×C22⋊D107—22no—
53C2×C22⋊D107—42no—
54C2×C22⋊D107D414yescomplemented
55C22×Dic107—22no—
56C22×Dic107—22no—
57C22×Dic107—22no—
58C22×Dic107—22no—
59C22×Dic107—22no—
60C22×Dic107—22no—
61C22×Dic107—42no—
62C22×Dic107D414yescomplemented
63C22×Dic107C2312yes—
64C22×Dic107C2312yes—
65C22×Dic107C2311yescharacteristic
66C22×Dic107C2311yescharacteristic
67C22×Dic107C2311yescharacteristic
68C2×C4×D107—22no—
69C2×C4×D107—22no—
70C2×C4×D107—22no—
71C22⋊Dic107—22no—
72C22⋊Dic107—24no—
73C22⋊Dic107—24no—
74C22⋊Dic107—24no—
75C22⋊Dic107—42no—
76(C2×D107)⋊C4—42no—
77C24×C107—22no—
78C24×C107—24no—
79C24×C107—22no—
80C24×C107—22no—
81C24×C107—42no—
82C24×C107C2311yescharacteristic
83C24×C107C2312yes—
84C24×C107C2311yescharacteristic
85C2×D4×C107—22no—
86C2×D4×C107—22no—
87C2×D4×C107—24no—
88C2×D4×C107—24no—
89C2×D4×C107—42no—
90C2×D4×C107D414yescomplemented
91C22×C4×C107C2312yes—
92C22×C4×C107C2311yescharacteristic
93C22⋊C4×C107—22no—
94C22⋊C4×C107—24no—
95C23×C107—22no—
96C23×C107—22no—
97C23×C107—28no—
98C23×C107—22no—
99C23×C107—21no—
100C23×C107—21no—
101C23×C107—22no—
102C23×C107—22no—
103C23×C107—22no—
104C23×C107—22no—
105C23×C107—22no—
106C23×C107—22no—
107C23×C107—22no—
108C23×C107—48no—
109C23×C107—48no—
110C23×C107—44no—
111C23×C107—44no—
112C23×C107C2×D412yes—
113C23×C107C2×D412yes—
114C23×C107C2×D412yes—
115C23×C107C2×D412yes—
116C23×C107#4114yescomplemented
117C23×C107C2411yescharacteristic
118C22×D107—21no—
119C22×D107—21no—
120C22×D107—22no—
121C22×D107—22no—
122C22×D107C2×D412yes—
123D4×C107—28no—
124D4×C107—48no—
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107—22no—
128C2×C4×C107—22no—
129C2×C4×C107—22no—
130C2×C4×C107C2×D412yes—
131C22⋊D107—28no—
132C22⋊D107—28no—
133C22⋊D107—48no—
134C2×Dic107—28no—
135C2×Dic107—28no—
136C2×Dic107—21no—
137C2×Dic107—21no—
138C2×Dic107—21no—
139C2×Dic107—21no—
140C2×Dic107—21no—
141C2×Dic107—21no—
142C2×Dic107—22no—
143C2×Dic107—22no—
144C2×Dic107—22no—
145C2×Dic107—22no—
146C2×Dic107—22no—
147C2×Dic107—22no—
148C2×Dic107—44no—
149C2×Dic107C2×D412yes—
150C2×Dic107C2×D412yes—
151C2×Dic107C2×D412yes—
152C4×D107—28no—
153Dic214—28no—
154C22×C107—22no—
155C22×C107—22no—
156C22×C107—24no—
157C22×C107—24no—
158C22×C107—22no—
159C22×C107—24no—
160C22×C107—24no—
161C22×C107—24no—
162C22×C107—24no—
163C22×C107—416no—
164C22×C107—42no—
165C22×C107—41no—
166C22×C107—41no—
167C22×C107—42no—
168C22×C107—42no—
169C22×C107—42no—
170C22×C107—42no—
171C22×C107#12014yes—
172C22×C107C22×D411yescharacteristic
173C22×C107C22×D411yescharacteristic
174C22×C107C22×D411yescharacteristic
175C22×C107C2×#4112yes—
176C22×C107H(5,𝔽2)12yes—
177C2×D107—22no—
178C2×D107—24no—
179C2×D107—41no—
180C2×D107—41no—
181C2×D107—42no—
182C4×C107—22no—
183C4×C107—42no—
184Dic107—22no—
185Dic107—22no—
186Dic107—22no—
187Dic107—42no—
188Dic107—42no—
189C2×C107—24no—
190C2×C107—42no—
191C2×C107—44no—
192C2×C107C2×#12011yescharacteristic
193C2×C107#52511yescharacteristic
194C2×C107#53511yescharacteristic
195C2×C107#54612yes—
196C2×C107#54612yes—
197D107—42no—
198#2404—1071nomaximal, Sylow
199C107#240411yescharacteristic, complemented, Sylow
200C22×#41—1071no—
201C2×#120—1071no—
202C2×#120D10711yescharacteristic, complemented
203C22×C22⋊C4—1071no—
204#393—1072no—
205#392—1072no—
206#386—1072no—
207#386—1071no—
208#386—1072no—
209#386—1071no—
210#381—1071no—
211C25C2×D10711yescharacteristic
212C2×#41—2144no—
213C2×#41—2144no—
214C2×#41—2144no—
215C22×Q8—1071no—
216C22×D4—1071no—
217C22×D4—1072no—
218C22×D4—1071no—
219C22×D4C2×D10712yes—
220C22×D4C2×D10711yescharacteristic
221C23×C4—1071no—
222C23×C4—1071no—
223C23×C4—1071no—
224C23×C4—1071no—
225#120—24no—
226#120—2144no—
227C2×C22⋊C4—1072no—
228C2×C22⋊C4—1072no—
229C2×C22⋊C4—1072no—
230C2×C22⋊C4—1072no—
231C2×C22⋊C4—1071no—
232C2×C22⋊C4—1072no—
233C2×C22⋊C4—2144no—
234C2×C22⋊C4C2×D10711yescharacteristic
235C2×C22⋊C4C2×D10712yes—
236#95—1072no—
237#95—1072no—
238#95—1072no—
239#95—1072no—
240C24—22no—
241C24—24no—
242C24—22no—
243C24—22no—
244C24—42no—
245C24—1071no—
246C24C22×D10711yescharacteristic
247C24C22×D10712yes—
248C24C22×D10711yescharacteristic
249#41—21432no—
250C2×Q8—2142no—
251C2×Q8—2142no—
252C2×Q8—2142no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—42no—
258C2×D4—1074no—
259C2×D4—4282no—
260C2×D4—2142no—
261C2×D4—2144no—
262C2×D4—2144no—
263C2×D4—2142no—
264C2×D4—2142no—
265C2×D4—2142no—
266C2×D4—2142no—
267C2×D4C22⋊D10714yescomplemented
268C22×C4—1074no—
269C22×C4—1071no—
270C22×C4—1071no—
271C22×C4—1071no—
272C22×C4—1072no—
273C22×C4—1072no—
274C22×C4—4282no—
275C22×C4—2142no—
276C22×C4—2142no—
277C22×C4—2142no—
278C22×C4—2142no—
279C22×C4—2142no—
280C22×C4—2142no—
281C22×C4—2142no—
282C22×C4—2142no—
283C22×C4—2142no—
284C22×C4C22×D10712yes—
285C22×C4C22×D10711yescharacteristic
286C22⋊C4—22no—
287C22⋊C4—24no—
288C22⋊C4—4282no—
289C22⋊C4—4282no—
290C22⋊C4—2144no—
291C22⋊C4—2144no—
292C22⋊C4—2144no—
293C22⋊C4—2142no—
294C23—22no—
295C23—22no—
296C23—28no—
297C23—22no—
298C23—22no—
299C23—21no—
300C23—21no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—48no—
308C23—48no—
309C23—44no—
310C23—44no—
311C23—1072no—
312C23—2141no—
313C23—2141no—
314C23—2142no—
315C23—2142no—
316C23D4×D10712yes—
317C23D4×D10712yes—
318C23C2×C22⋊D10712yes—
319C23C2×C22⋊D10712yes—
320C23#1313381714yescomplemented
321C23C23×D10711yescharacteristic
322Q8—2148no—
323D4—28no—
324D4—48no—
325D4—4288no—
326D4—2148no—
327D4—2148no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—1072no—
334C2×C4—1072no—
335C2×C4—1072no—
336C2×C4—4284no—
337C2×C4—2148no—
338C2×C4—2142no—
339C2×C4—2142no—
340C2×C4—2142no—
341C2×C4—2142no—
342C2×C4—2142no—
343C2×C4—2142no—
344C2×C4—2148no—
345C2×C4—2148no—
346C2×C4—2141no—
347C2×C4—2141no—
348C2×C4—2141no—
349C2×C4—2141no—
350C2×C4—2141no—
351C2×C4—2141no—
352C2×C4C2×C22⋊D10712yes—
353C22—22no—
354C22—22no—
355C22—24no—
356C22—24no—
357C22—24no—
358C22—24no—
359C22—24no—
360C22—24no—
361C22—22no—
362C22—416no—
363C22—41no—
364C22—42no—
365C22—41no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—4281no—
371C22—4281no—
372C22—4282no—
373C22—2144no—
374C22—2142no—
375C22#1681207714yes—
376C22C2×D4×D10711yescharacteristic
377C22C2×D4×D10711yescharacteristic
378C22C22×C22⋊D10711yescharacteristic
379C22C2×#1313381712yes—
380C22#1681215412yes—
381C4—22no—
382C4—42no—
383C4—4282no—
384C4—4282no—
385C4—2142no—
386C4—2142no—
387C4—2142no—
388C2—24no—
389C2—44no—
390C2—42no—
391C2—4282no—
392C2C2×#1681207711yescharacteristic
393C2#2029313311yescharacteristic
394C2#2029319211yescharacteristic
395C2#2029326412yes—
396C2#2029327112yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×#120×C107C2
hypercenterC2×#120D107
hypocenterC107#2404
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C22×#13133817C2semidirect
2. C2×#16812078C2semidirect
3. C22×C22⋊Dic107C2semidirect
4. #20292539C2semidirect
5. #20292535C2semidirect
6. #20292507C2semidirect
7. #20292507C2semidirect
8. #20292504C2semidirect
9. #20292503C2semidirect
10. #20292488C2semidirect
11. C2×#120×C107C2semidirect
15. C22×Dic214C22semidirect
18. C22×C22⋊D107C22semidirect
21. C23×Dic107C22semidirect
22. C22×C4×D107C22semidirect
29. C2×C22⋊Dic107C22semidirect
30. C2×(C2×D107)⋊C4C22semidirect
31. C107⋊#95C22semidirect
32. C107⋊#95C22semidirect
33. C107⋊#95C22semidirect
34. C107⋊#95C22semidirect
54. C2×C22⋊D107D4semidirect
62. C22×Dic107D4semidirect
90. C2×D4×C107D4semidirect
116. C23×C107#41semidirect
199. C107#2404semidirect
202. C2×#120D107semidirect
267. C2×D4C22⋊D107semidirect
320. C23#13133817semidirect

Maximal subgroups

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

subgroupindex
C22×#131338172
C2×#168120782
C22×C22⋊Dic1072
#202925392
#202925352
#202925072
#202925042
#202925032
#202924882
C2×#120×C1072
#2404107