Finite Group Databasebeta

← #28339888

Subgroups — #28339888

Order 13696.

604 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812089C211yesmaximal, characteristic, complemented
2C2×#16812089C211yesmaximal, characteristic, complemented
3C2×C4×D428C211yesmaximal, characteristic
4C2×#16812067C211yesmaximal, characteristic, complemented
5#20292554C212yesmaximal, complemented
6#20292544C212yesmaximal
7#20292533C211yesmaximal, characteristic, complemented
8#20292526C212yesmaximal, complemented
9#20292519C212yesmaximal, complemented
10#20292484C211yesmaximal, characteristic, complemented
11#389×C107C211yesmaximal, characteristic, complemented
12C22×Dic214C2211yescharacteristic
13C22×D428C2211yescharacteristic
14C22×C4×D107C2211yescharacteristic
15C22×C4×D107C2211yescharacteristic
16#16812089—24no—
17#16812089—24no—
18C4×D428—24no—
19#16812067—24no—
20C2×C4⋊Dic107C2212yes—
21C2×C4⋊Dic107C2212yes—
22C2×C4⋊Dic107C2211yescharacteristic
23C2×C4⋊Dic107C2211yescharacteristic
24C2×(C2×D107)⋊C4C2212yes—
25C2×(C2×D107)⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2212yes—
27C2×(C2×D107)⋊C4C2211yescharacteristic
28C2×(C2×D107)⋊C4C2212yes—
29C2×(C2×D107)⋊C4C2211yescharacteristic
30C2×C4×Dic107C2212yes—
31C107⋊#95C2212yes—
32C107⋊#95C2212yes—
33C107⋊#95C2212yes—
34C2×C4⋊C4×C107C2211yescharacteristic
35C2×C4⋊C4×C107C2212yes—
36C2×C4⋊C4×C107C2211yescharacteristic
37C2×C42×C107C2211yescharacteristic
38#95×C107C2212yes—
39C23×D107C2311yescharacteristic
40C23×D107C2311yescharacteristic
41C2×Dic214—22no—
42C2×Dic214—22no—
43C2×Dic214—22no—
44C2×D428—22no—
45C2×D428—22no—
46C2×D428—22no—
47C22×Dic107C2312yes—
48C22×Dic107C2312yes—
49C22×Dic107C2311yescharacteristic
50C22×Dic107C2311yescharacteristic
51C2×C4×D107—22no—
52C2×C4×D107—22no—
53C2×C4×D107—22no—
54C2×C4×D107—22no—
55C2×C4×D107—22no—
56C2×C4×D107D414yes—
57C4⋊Dic107—22no—
58C4⋊Dic107—24no—
59C4⋊Dic107—22no—
60C4⋊Dic107—24no—
61(C2×D107)⋊C4—22no—
62(C2×D107)⋊C4—24no—
63(C2×D107)⋊C4—24no—
64(C2×D107)⋊C4—24no—
65(C2×D107)⋊C4—24no—
66(C2×D107)⋊C4—22no—
67C4×Dic107—24no—
68C22×C4×C107C2312yes—
69C22×C4×C107C2311yescharacteristic
70C22×C4×C107C2312yes—
71C22×C4×C107C2311yescharacteristic
72C22×C4×C107C2311yescharacteristic
73C4⋊C4×C107—22no—
74C4⋊C4×C107—24no—
75C4⋊C4×C107D414yescomplemented
76C42×C107—22no—
77C23×C107C2411yescharacteristic
78C22×D107—21no—
79C22×D107—21no—
80C22×D107—21no—
81C22×D107—21no—
82C22×D107—21no—
83C22×D107—21no—
84C22×D107—21no—
85C22×D107—21no—
86C22×D107—21no—
87C22×D107—21no—
88C22×D107—21no—
89C22×D107—21no—
90C22×D107C2×D412yes—
91C22×D107#4112yes—
92C2×C4×C107—21no—
93C2×C4×C107—22no—
94C2×C4×C107—21no—
95C2×C4×C107—22no—
96C2×C4×C107—21no—
97C2×C4×C107—22no—
98C2×C4×C107—22no—
99C2×C4×C107—22no—
100C2×C4×C107—21no—
101C2×C4×C107—21no—
102C2×C4×C107C2×D412yes—
103C2×C4×C107C2×D414yes—
104C2×C4×C107C2×D412yes—
105C2×C4×C107#4112yes—
106C2×C4×C107#4112yes—
107C2×Dic107—21no—
108C2×Dic107—21no—
109C2×Dic107—22no—
110C2×Dic107—21no—
111C2×Dic107—22no—
112C2×Dic107—22no—
113C2×Dic107—21no—
114C2×Dic107—22no—
115C2×Dic107—21no—
116C2×Dic107—22no—
117C2×Dic107—22no—
118C2×Dic107C2×D412yes—
119D428—28no—
120C4×D107—28no—
121C4×D107—44no—
122Dic214—44no—
123C22×C107C22×D411yescharacteristic
124C22×C107C22×D411yescharacteristic
125C22×C107C2×#4111yescharacteristic
126C22×C107C2×#4111yescharacteristic
127C22×C107C2×#4111yescharacteristic
128C22×C1072−1+411yescharacteristic
129C22×C1072−1+411yescharacteristic
130C2×D107—22no—
131C2×D107—22no—
132C2×D107—22no—
133C2×D107—22no—
134C2×D107—22no—
135C2×D107—22no—
136C2×D107—41no—
137C2×D107—41no—
138C2×D107—41no—
139C2×D107—41no—
140C2×D107—41no—
141C2×D107—41no—
142C2×D107—41no—
143C2×D107—41no—
144C4×C107—22no—
145C4×C107—24no—
146C4×C107—42no—
147C4×C107—41no—
148C4×C107#12114yes—
149Dic107—22no—
150Dic107—42no—
151Dic107—41no—
152Dic107—42no—
153C2×C107C2×#12111yescharacteristic
154C2×C107#52511yescharacteristic
155C2×C107#53611yescharacteristic
156C2×C107#54711yescharacteristic
157C2×C107#54811yescharacteristic
158C2×C107#55211yescharacteristic
159C2×C107#55611yescharacteristic
160D107—42no—
161D107—42no—
162#2415—1071nomaximal, Sylow
163C107#241511yescharacteristic, complemented, Sylow
164C2×C22⋊Q8—1071no—
165C2×C22⋊Q8—1071no—
166C2×C4×D4—1071no—
167C2×#117—1071no—
168#397—1072no—
169#394—1072no—
170#391—1071no—
171#389—1072no—
172#389D10711yescharacteristic, complemented
173#388—1072no—
174#380—1071no—
175C22×Q8—1071no—
176C22×D4—1071no—
177C23×C4—1071no—
178C23×C4—1071no—
179C22⋊Q8—2144no—
180C22⋊Q8—2144no—
181C4×D4—2144no—
182#117—2144no—
183C2×C4⋊C4—1071no—
184C2×C4⋊C4—1071no—
185C2×C4⋊C4—1072no—
186C2×C4⋊C4—1072no—
187C2×C4⋊C4C2×D10711yescharacteristic
188C2×C4⋊C4C2×D10712yes—
189C2×C4⋊C4C2×D10711yescharacteristic
190C2×C22⋊C4—1072no—
191C2×C22⋊C4—1071no—
192C2×C22⋊C4—1071no—
193C2×C22⋊C4—1072no—
194C2×C22⋊C4—1072no—
195C2×C22⋊C4—1072no—
196C2×C42—1072no—
197C2×C42C2×D10711yescharacteristic
198#95—1072no—
199#95—1072no—
200#95—1072no—
201#95C2×D10712yes—
202C24—1071no—
203C24—1071no—
204C2×Q8—2142no—
205C2×Q8—2142no—
206C2×Q8—2142no—
207C2×D4—2142no—
208C2×D4—2142no—
209C2×D4—2142no—
210C22×C4—1071no—
211C22×C4—1074no—
212C22×C4—1072no—
213C22×C4—1072no—
214C22×C4—1071no—
215C22×C4—2142no—
216C22×C4—2142no—
217C22×C4—2142no—
218C22×C4—2142no—
219C22×C4—2142no—
220C22×C4C22×D10711yescharacteristic
221C22×C4C22×D10712yes—
222C22×C4C22×D10712yes—
223C22×C4C22×D10711yescharacteristic
224C22×C4C22×D10711yescharacteristic
225C4⋊C4—22no—
226C4⋊C4—24no—
227C4⋊C4—2142no—
228C4⋊C4—2144no—
229C4⋊C4—2142no—
230C4⋊C4—2144no—
231C4⋊C4D42814yescomplemented
232C22⋊C4—2142no—
233C22⋊C4—2142no—
234C22⋊C4—2144no—
235C22⋊C4—2144no—
236C22⋊C4—2144no—
237C22⋊C4—2144no—
238C42—22no—
239C42—2144no—
240C23—1072no—
241C23—1072no—
242C23—2141no—
243C23—2141no—
244C23—2141no—
245C23—2141no—
246C23—2141no—
247C23—2141no—
248C23—2141no—
249C23—2141no—
250C23—2141no—
251C23—2141no—
252C23—2141no—
253C23—2141no—
254C23C23×D10711yescharacteristic
255Q8—4284no—
256D4—2148no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—1072no—
268C2×C4—4284no—
269C2×C4—2141no—
270C2×C4—2142no—
271C2×C4—2142no—
272C2×C4—2142no—
273C2×C4—2141no—
274C2×C4—2142no—
275C2×C4—2141no—
276C2×C4—2142no—
277C2×C4—2142no—
278C2×C4—2141no—
279C2×C4—2141no—
280C2×C4—2148no—
281C2×C4D4×D10712yes—
282C2×C4C2×D42814yes—
283C2×C4C2×D42812yes—
284C2×C4#1313381712yes—
285C2×C4#1313381912yes—
286C22—4281no—
287C22—4281no—
288C22—4281no—
289C22—4281no—
290C22—4281no—
291C22—4281no—
292C22—4281no—
293C22—4281no—
294C22—2142no—
295C22—2142no—
296C22—2142no—
297C22—2142no—
298C22—2142no—
299C22—2142no—
300C22C2×D4×D10711yescharacteristic
301C22C22×D42811yescharacteristic
302C22#41×D10711yescharacteristic
303C22C2×#1313381711yescharacteristic
304C22C2×#1313381911yescharacteristic
305C22#1681215611yescharacteristic
306C22#1681215711yescharacteristic
307C4—22no—
308C4—24no—
309C4—42no—
310C4—41no—
311C4—4281no—
312C4—4282no—
313C4—4282no—
314C4—2142no—
315C4#1681208414yes—
316C2—4282no—
317C2—4282no—
318C2C2×#1681208411yescharacteristic
319C2#2029313711yescharacteristic
320C2#2029320011yescharacteristic
321C2#2029328111yescharacteristic
322C2#2029328911yescharacteristic
323C2#2029331511yescharacteristic
324C2#2029335111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#389×C107C2
prime radicalC22×D428C22
hypercenter#389D107
hypocenterC107#2415
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C2×#16812089C2semidirect
2. C2×#16812089C2semidirect
4. C2×#16812067C2semidirect
5. #20292554C2semidirect
7. #20292533C2semidirect
8. #20292526C2semidirect
9. #20292519C2semidirect
10. #20292484C2semidirect
11. #389×C107C2semidirect
75. C4⋊C4×C107D4semidirect
163. C107#2415semidirect
172. #389D107semidirect
231. C4⋊C4D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168120892
C2×C4×D4282
C2×#168120672
#202925542
#202925442
#202925332
#202925262
#202925192
#202924842
#389×C1072
#2415107