Finite Group Databasebeta

← #28339847

Subgroups — #28339847

Order 13696.

786 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812077C211yesmaximal, characteristic
2C2×#16812068C211yesmaximal, characteristic, complemented
3C22×(C2×D107)⋊C4C211yesmaximal, characteristic, complemented
4#20292549C212yesmaximal, complemented
5#20292543C212yesmaximal, complemented
6#20292527C212yesmaximal, complemented
7#20292517C212yesmaximal, complemented
8#20292507C211yesmaximal, characteristic, complemented
9#20292499C211yesmaximal, characteristic, complemented
10#20292482C211yesmaximal, characteristic, complemented
11C2×#123×C107C211yesmaximal, characteristic, complemented
12C24×D107C2211yescharacteristic
13C22×C22⋊D107C2212yes—
14C23×Dic107C2211yescharacteristic
15#16812077—24no—
16#16812068—24no—
17C2×C4⋊Dic107C2212yescomplemented
18C2×Dic107⋊C4C2212yescomplemented
19C2×C22⋊Dic107C2211yescharacteristic
20C2×C22⋊Dic107C2211yescharacteristic
21C2×C22⋊Dic107C2212yescomplemented
22C2×(C2×D107)⋊C4—24no—
23C2×(C2×D107)⋊C4C2212yes—
24C2×(C2×D107)⋊C4C2212yes—
25C2×(C2×D107)⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2212yes—
27C2×(C2×D107)⋊C4C2212yes—
28C2×C4×Dic107C2212yescomplemented
29C107⋊#95C2212yescomplemented
30C107⋊#95C2212yescomplemented
31C22×D4×C107C2211yescharacteristic
32C23×C4×C107C2211yescharacteristic
33#123×C107—24no—
34C2×C4⋊C4×C107C2211yescharacteristic, complemented
35C2×C4⋊C4×C107C2211yescharacteristic, complemented
36C2×C22⋊C4×C107C2211yescharacteristic
37C2×C22⋊C4×C107C2211yescharacteristic, complemented
38C2×C22⋊C4×C107C2211yescharacteristic, complemented
39C23×D107—22no—
40C23×D107—24no—
41C23×D107—22no—
42C23×D107—22no—
43C23×D107—42no—
44C23×D107C2312yes—
45C2×C22⋊D107—24no—
46C2×C22⋊D107—24no—
47C2×C22⋊D107—42no—
48C22×Dic107—22no—
49C22×Dic107—42no—
50C22×Dic107C2312yes—
51C22×Dic107C2312yes—
52C22×Dic107C2312yes—
53C4⋊Dic107—24no—
54Dic107⋊C4—42no—
55C22⋊Dic107—22no—
56C22⋊Dic107—42no—
57C22⋊Dic107D414yes—
58(C2×D107)⋊C4—24no—
59(C2×D107)⋊C4—24no—
60(C2×D107)⋊C4—24no—
61(C2×D107)⋊C4—24no—
62(C2×D107)⋊C4—42no—
63C4×Dic107—24no—
64C24×C107C2311yescharacteristic
65C24×C107C2311yescharacteristic
66C2×D4×C107—22no—
67C2×D4×C107—22no—
68C2×D4×C107—22no—
69C22×C4×C107—22no—
70C22×C4×C107—22no—
71C22×C4×C107D414yescomplemented
72C22×C4×C107C2311yescharacteristic
73C22×C4×C107C2311yescharacteristic
74C22×C4×C107C2311yescharacteristic
75C22×C4×C107C2311yescharacteristic
76C22×C4×C107C2311yescharacteristic
77C4⋊C4×C107—22no—
78C4⋊C4×C107—22no—
79C22⋊C4×C107—22no—
80C22⋊C4×C107—22no—
81C22⋊C4×C107—22no—
82C23×C107—22no—
83C23×C107—22no—
84C23×C107—21no—
85C23×C107—21no—
86C23×C107—21no—
87C23×C107—22no—
88C23×C107—22no—
89C23×C107C2×D412yes—
90C23×C107C2×D412yes—
91C23×C107#4112yes—
92C23×C107C2411yescharacteristic
93C22×D107—28no—
94C22×D107—22no—
95C22×D107—22no—
96C22×D107—22no—
97C22×D107—22no—
98C22×D107—22no—
99C22×D107—22no—
100C22×D107—48no—
101C22×D107—48no—
102C22×D107—44no—
103C22×D107—44no—
104C22×D107#4114yes—
105D4×C107—44no—
106C2×C4×C107—28no—
107C2×C4×C107—21no—
108C2×C4×C107—21no—
109C2×C4×C107—21no—
110C2×C4×C107—21no—
111C2×C4×C107—21no—
112C2×C4×C107—21no—
113C2×C4×C107—21no—
114C2×C4×C107—21no—
115C2×C4×C107—21no—
116C2×C4×C107—22no—
117C2×C4×C107—22no—
118C2×C4×C107C2×D412yes—
119C2×C4×C107C2×D412yes—
120C22⋊D107—48no—
121C2×Dic107—22no—
122C2×Dic107—22no—
123C2×Dic107—22no—
124C2×Dic107—22no—
125C2×Dic107—22no—
126C2×Dic107—22no—
127C2×Dic107—22no—
128C2×Dic107—22no—
129C2×Dic107—44no—
130C2×Dic107C2×D414yes—
131C22×C107—22no—
132C22×C107—22no—
133C22×C107—24no—
134C22×C107—24no—
135C22×C107—41no—
136C22×C107—41no—
137C22×C107—41no—
138C22×C107—42no—
139C22×C107—42no—
140C22×C107#12114yes—
141C22×C107C22×D411yescharacteristic
142C22×C107C22×D411yescharacteristic
143C22×C107C2×#4112yes—
144C22×C107C2×#4111yescharacteristic
145C22×C107H(5,𝔽2)12yes—
146C2×D107—24no—
147C2×D107—24no—
148C2×D107—24no—
149C2×D107—416no—
150C2×D107—42no—
151C2×D107—42no—
152C2×D107—42no—
153C2×D107—42no—
154C4×C107—22no—
155C4×C107—22no—
156C4×C107—41no—
157C4×C107—41no—
158C4×C107—41no—
159Dic107—24no—
160Dic107—42no—
161Dic107—42no—
162C2×C107—24no—
163C2×C107—81no—
164C2×C107C2×#12111yescharacteristic
165C2×C107#52511yescharacteristic
166C2×C107#53511yescharacteristic
167C2×C107#54612yes—
168C2×C107#55112yes—
169D107—44no—
170#2411—1071nomaximal, Sylow
171C107#241111yescharacteristic, complemented, Sylow
172C2×#123D10711yescharacteristic, complemented
173C2×#120—1071no—
174C2×#117—1071no—
175C22×C22⋊C4—1071no—
176#396—1072no—
177#394—1072no—
178#390—1072no—
179#388—1072no—
180#386—1071no—
181#385—1071no—
182#379—1071no—
183C25—1071no—
184C22×D4—1072no—
185C22×D4C2×D10711yescharacteristic
186C23×C4—1071no—
187C23×C4C2×D10711yescharacteristic
188#123—24no—
189#120—2144no—
190#117—2144no—
191C2×C4⋊C4—1072no—
192C2×C4⋊C4—1072no—
193C2×C4⋊C4C2×D10711yescharacteristic, complemented
194C2×C4⋊C4C2×D10711yescharacteristic, complemented
195C2×C22⋊C4—1071no—
196C2×C22⋊C4—1072no—
197C2×C22⋊C4—1072no—
198C2×C22⋊C4—1071no—
199C2×C22⋊C4—1072no—
200C2×C22⋊C4—1072no—
201C2×C22⋊C4—1072no—
202C2×C22⋊C4—1072no—
203C2×C22⋊C4—2144no—
204C2×C22⋊C4C2×D10711yescharacteristic
205C2×C22⋊C4C2×D10711yescharacteristic, complemented
206C2×C22⋊C4C2×D10711yescharacteristic, complemented
207C2×C42—1072no—
208#95—1072no—
209#95—1072no—
210C24—1072no—
211C24—4282no—
212C24—2144no—
213C24—2142no—
214C24—2142no—
215C24—2142no—
216C24C22×D10711yescharacteristic
217C24C22×D10711yescharacteristic
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—4282no—
222C2×D4—2144no—
223C2×D4—2144no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—1072no—
227C22×C4—1072no—
228C22×C4—1072no—
229C22×C4—4282no—
230C22×C4—2142no—
231C22×C4C22⋊D10714yescomplemented
232C22×C4C22×D10711yescharacteristic
233C22×C4C22×D10711yescharacteristic
234C22×C4C22×D10711yescharacteristic
235C22×C4C22×D10711yescharacteristic
236C22×C4C22×D10711yescharacteristic
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—4282no—
240C4⋊C4—2144no—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—22no—
244C22⋊C4—1074no—
245C22⋊C4—4282no—
246C22⋊C4—4282no—
247C22⋊C4—2144no—
248C22⋊C4—2144no—
249C22⋊C4—2142no—
250C22⋊C4—2144no—
251C22⋊C4—2144no—
252C42—2144no—
253C23—22no—
254C23—22no—
255C23—21no—
256C23—21no—
257C23—22no—
258C23—21no—
259C23—22no—
260C23—1074no—
261C23—4284no—
262C23—4288no—
263C23—4288no—
264C23—4284no—
265C23—2148no—
266C23—2142no—
267C23—2142no—
268C23—2142no—
269C23—2142no—
270C23—2142no—
271C23—2142no—
272C23D4×D10712yes—
273C23C2×C22⋊D10712yes—
274C23#1313381712yes—
275C23C23×D10711yescharacteristic
276D4—44no—
277D4—4288no—
278C2×C4—28no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—1074no—
291C2×C4—4284no—
292C2×C4—2142no—
293C2×C4—2142no—
294C2×C4—2142no—
295C2×C4—2142no—
296C2×C4—2142no—
297C2×C4—2142no—
298C2×C4—2142no—
299C2×C4—2142no—
300C2×C4C2×C22⋊D10712yes—
301C2×C4C2×C22⋊D10712yes—
302C22—22no—
303C22—22no—
304C22—24no—
305C22—24no—
306C22—41no—
307C22—41no—
308C22—42no—
309C22—42no—
310C22—41no—
311C22—42816no—
312C22—4282no—
313C22—4282no—
314C22—4282no—
315C22—4282no—
316C22—2144no—
317C22—2144no—
318C22—2144no—
319C22#1681208114yes—
320C22C2×D4×D10711yescharacteristic
321C22C22×C22⋊D10711yescharacteristic
322C22#41×D10712yes—
323C22C2×#1313381711yescharacteristic
324C22#1681215512yes—
325C4—22no—
326C4—22no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—4282no—
331C4—4282no—
332C4—2144no—
333C2—24no—
334C2—81no—
335C2—4284no—
336C2C2×#1681208111yescharacteristic
337C2#2029313411yescharacteristic
338C2#2029319111yescharacteristic
339C2#2029326912yes—
340C2#2029330012yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×#123×C107C2
prime radicalC2×#16812077C2
hypercenterC2×#123D107
hypocenterC107#2411
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#16812068C2semidirect
3. C22×(C2×D107)⋊C4C2semidirect
4. #20292549C2semidirect
5. #20292543C2semidirect
6. #20292527C2semidirect
7. #20292517C2semidirect
8. #20292507C2semidirect
9. #20292499C2semidirect
10. #20292482C2semidirect
11. C2×#123×C107C2semidirect
17. C2×C4⋊Dic107C22semidirect
18. C2×Dic107⋊C4C22semidirect
21. C2×C22⋊Dic107C22semidirect
28. C2×C4×Dic107C22semidirect
29. C107⋊#95C22semidirect
30. C107⋊#95C22semidirect
34. C2×C4⋊C4×C107C22semidirect
35. C2×C4⋊C4×C107C22semidirect
37. C2×C22⋊C4×C107C22semidirect
38. C2×C22⋊C4×C107C22semidirect
71. C22×C4×C107D4semidirect
171. C107#2411semidirect
172. C2×#123D107semidirect
193. C2×C4⋊C4C2×D107semidirect
194. C2×C4⋊C4C2×D107semidirect
205. C2×C22⋊C4C2×D107semidirect
206. C2×C22⋊C4C2×D107semidirect
231. C22×C4C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120772
C2×#168120682
C22×(C2×D107)⋊C42
#202925492
#202925432
#202925272
#202925172
#202925072
#202924992
#202924822
C2×#123×C1072
#2411107