Finite Group Databasebeta

← #28339825

Subgroups — #28339825

Order 13696.

840 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×Dic214C211yesmaximal, characteristic, complemented
2C2×#16812087C212yesmaximal, complemented
3C2×(C22⋊D107)⋊C4C211yesmaximal, characteristic
4#20292547C212yesmaximal, complemented
5#20292540C212yesmaximal, complemented
6#20292531C212yesmaximal, complemented
7#20292506C211yesmaximal, characteristic
8#20292504C211yesmaximal, characteristic
9#20292499C211yesmaximal, characteristic, complemented
10#20292485C211yesmaximal, characteristic, complemented
11#385×C107C211yesmaximal, characteristic, complemented
12C22×Dic214—28no—
13C22×Dic214—24no—
14C22×Dic214C2212yes—
15C22×Dic214C2212yes—
16C22×C22⋊D107C2211yescharacteristic
17C23×Dic107C2211yescharacteristic
18C23×Dic107C2211yescharacteristic
19C22×C4×D107C2211yescharacteristic
20#16812087—28no—
21(C22⋊D107)⋊C4—24no—
22C2×Dic107⋊C4C2212yes—
23C2×Dic107⋊C4C2212yes—
24C2×Dic107⋊C4C2211yescharacteristic
25C2×C22⋊Dic107C2211yescharacteristic
26C2×(C2×D107)⋊C4C2212yes—
27C2×(C2×D107)⋊C4C2212yes—
28C2×(C2×D107)⋊C4C2211yescharacteristic
29C2×C4×Dic107C2211yescharacteristic
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C107⋊#95C2212yes—
33C107⋊#95C2212yes—
34C23×C4×C107C2211yescharacteristic
35C2×C4⋊C4×C107C2212yes—
36C2×C22⋊C4×C107C2211yescharacteristic
37C2×C22⋊C4×C107C2211yescharacteristic
38#95×C107C2212yes—
39C23×D107C2311yescharacteristic
40C2×Dic214—232no—
41C2×Dic214—24no—
42C2×Dic214—24no—
43C2×Dic214—24no—
44C2×Dic214—24no—
45C2×Dic214—24no—
46C2×Dic214D418yescomplemented
47C2×C22⋊D107—22no—
48C2×C22⋊D107—22no—
49C2×C22⋊D107Q814yescomplemented
50C22×Dic107—22no—
51C22×Dic107—24no—
52C22×Dic107—24no—
53C22×Dic107D414yes—
54C22×Dic107C2312yes—
55C22×Dic107C2312yes—
56C22×Dic107C2311yescharacteristic
57C22×Dic107C2311yescharacteristic
58C22×Dic107C2311yescharacteristic
59C2×C4×D107—22no—
60C2×C4×D107—24no—
61Dic107⋊C4—22no—
62Dic107⋊C4—24no—
63Dic107⋊C4—24no—
64C22⋊Dic107—22no—
65(C2×D107)⋊C4—24no—
66(C2×D107)⋊C4—22no—
67(C2×D107)⋊C4—24no—
68C4×Dic107—22no—
69C24×C107C2311yescharacteristic
70C22×C4×C107—22no—
71C22×C4×C107—24no—
72C22×C4×C107C2311yescharacteristic
73C22×C4×C107C2312yes—
74C22×C4×C107C2311yescharacteristic
75C22×C4×C107C2312yes—
76C4⋊C4×C107—24no—
77C22⋊C4×C107—22no—
78C22⋊C4×C107—22no—
79C23×C107—22no—
80C23×C107—22no—
81C23×C107C2×D412yes—
82C23×C107C2×Q812yes—
83C23×C107#4112yes—
84C23×C107C2411yescharacteristic
85C22×D107—22no—
86C22×D107—21no—
87C22×D107—21no—
88C22×D107—22no—
89C22×D107C2×Q812yescomplemented
90C2×C4×C107—28no—
91C2×C4×C107—22no—
92C2×C4×C107—22no—
93C2×C4×C107—21no—
94C2×C4×C107—21no—
95C2×C4×C107—22no—
96C2×C4×C107—22no—
97C2×C4×C107—22no—
98C2×C4×C107—22no—
99C2×C4×C107—22no—
100C2×C4×C107C2×D414yes—
101C22⋊D107—28no—
102C2×Dic107—28no—
103C2×Dic107—21no—
104C2×Dic107—22no—
105C2×Dic107—22no—
106C2×Dic107—22no—
107C2×Dic107—21no—
108C2×Dic107—28no—
109C2×Dic107—22no—
110C2×Dic107—22no—
111C2×Dic107—22no—
112C2×Dic107C2×D414yes—
113C2×Dic107C2×D412yes—
114C2×Dic107C2×D412yes—
115C2×Dic107C2×D414yes—
116C2×Dic107C2×Q812yescomplemented
117C4×D107—44no—
118Dic214—216no—
119Dic214—216no—
120C22×C107—22no—
121C22×C107—22no—
122C22×C107—24no—
123C22×C107—24no—
124C22×C107C22⋊Q814yescomplemented
125C22×C107C22×D411yescharacteristic
126C22×C107C22×D412yes—
127C22×C107C22×Q811yescharacteristic
128C22×C107C2×#4111yescharacteristic
129C22×C1072−1+412yes—
130C2×D107—22no—
131C2×D107—24no—
132C2×D107—41no—
133C2×D107—41no—
134C2×D107—42no—
135C4×C107—24no—
136C4×C107—41no—
137C4×C107—41no—
138C4×C107—42no—
139Dic107—22no—
140Dic107—24no—
141Dic107—22no—
142Dic107—42no—
143Dic107#12014yes—
144C2×C107—24no—
145C2×C107C2×#12011yescharacteristic
146C2×C107C2×C22⋊Q811yescharacteristic
147C2×C107#53611yescharacteristic
148C2×C107#54812yes—
149C2×C107D4×Q812yes—
150D107—42no—
151#2409—1071nomaximal, Sylow
152C107#240911yescharacteristic, complemented, Sylow
153C23×Q8—1071no—
154C2×C22⋊Q8—1072no—
155C2×C4×D4—1071no—
156#395—1072no—
157#393—1072no—
158#391—1072no—
159#386—1071no—
160#386—1071no—
161#385—1071no—
162#385D10711yescharacteristic, complemented
163#380—1071no—
164C22×Q8—1072no—
165C22×Q8—1072no—
166C22×Q8—2144no—
167C22×Q8—2148no—
168C22×D4—1071no—
169C23×C4—1071no—
170C23×C4—1071no—
171C23×C4—1071no—
172C23×C4C2×D10711yescharacteristic
173C22⋊Q8—2148no—
174C4×D4—2144no—
175C2×C4⋊C4—1071no—
176C2×C4⋊C4—1072no—
177C2×C4⋊C4—1072no—
178C2×C4⋊C4C2×D10712yes—
179C2×C22⋊C4—1071no—
180C2×C22⋊C4—1071no—
181C2×C22⋊C4—1072no—
182C2×C22⋊C4—1072no—
183C2×C22⋊C4C2×D10711yescharacteristic
184C2×C22⋊C4C2×D10711yescharacteristic
185C2×C42—1071no—
186#95—1072no—
187#95—1072no—
188#95—1072no—
189#95—1072no—
190#95C2×D10712yes—
191C24—1071no—
192C24C22×D10711yescharacteristic
193C2×Q8—1078no—
194C2×Q8—2144no—
195C2×Q8—2144no—
196C2×Q8—2144no—
197C2×Q8—2144no—
198C2×Q8—21432no—
199C2×Q8—2144no—
200C2×D4—1074no—
201C2×D4—2142no—
202C2×D4—2142no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—1071no—
206C22×C4—1074no—
207C22×C4—1071no—
208C22×C4—1071no—
209C22×C4—1072no—
210C22×C4—1072no—
211C22×C4—2142no—
212C22×C4—2144no—
213C22×C4—2144no—
214C22×C4—2144no—
215C22×C4—2142no—
216C22×C4C22×D10711yescharacteristic
217C22×C4C22×D10712yes—
218C22×C4C22×D10711yescharacteristic
219C22×C4C22×D10712yes—
220C4⋊C4—24no—
221C4⋊C4—2142no—
222C4⋊C4—2144no—
223C4⋊C4—2144no—
224C22⋊C4—22no—
225C22⋊C4—22no—
226C22⋊C4—2142no—
227C22⋊C4—2144no—
228C22⋊C4—2142no—
229C22⋊C4—2144no—
230C42—2142no—
231C23—22no—
232C23—22no—
233C23—1072no—
234C23—2142no—
235C23—2141no—
236C23—2141no—
237C23—2142no—
238C23D4×D10712yes—
239C23Q8×D10712yes—
240C23#1313381812yes—
241C23C23×D10711yescharacteristic
242Q8—21416no—
243Q8—21416no—
244D4—2148no—
245C2×C4—28no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—1072no—
256C2×C4—1074no—
257C2×C4—1072no—
258C2×C4—1074no—
259C2×C4—1072no—
260C2×C4—4284no—
261C2×C4—2148no—
262C2×C4—2142no—
263C2×C4—2142no—
264C2×C4—2142no—
265C2×C4—2148no—
266C2×C4—2141no—
267C2×C4—2142no—
268C2×C4—2142no—
269C2×C4—2142no—
270C2×C4—2141no—
271C2×C4D4×D10714yes—
272C22—22no—
273C22—24no—
274C22—24no—
275C22—22no—
276C22—4281no—
277C22—4282no—
278C22—4281no—
279C22—2142no—
280C22—2144no—
281C22#1681208714yescomplemented
282C22C2×D4×D10711yescharacteristic
283C22C2×D4×D10712yes—
284C22C2×Q8×D10711yescharacteristic
285C22C2×#1313381811yescharacteristic
286C22#1681215612yes—
287C4—24no—
288C4—41no—
289C4—42no—
290C4—41no—
291C4—1074no—
292C4—4282no—
293C4—2142no—
294C4—2142no—
295C4—2144no—
296C2—24no—
297C2—4282no—
298C2#120×D10711yescharacteristic
299C2C2×#1681208711yescharacteristic
300C2#2029319911yescharacteristic
301C2#2029328612yes—
302C2#2029329012yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#385×C107C2
prime radicalC22×C22⋊D107C22
hypercenter#385D107
hypocenterC107#2409
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C23×Dic214C2semidirect
2. C2×#16812087C2semidirect
4. #20292547C2semidirect
5. #20292540C2semidirect
6. #20292531C2semidirect
9. #20292499C2semidirect
10. #20292485C2semidirect
11. #385×C107C2semidirect
46. C2×Dic214D4semidirect
49. C2×C22⋊D107Q8semidirect
89. C22×D107C2×Q8semidirect
116. C2×Dic107C2×Q8semidirect
124. C22×C107C22⋊Q8semidirect
152. C107#2409semidirect
162. #385D107semidirect
281. C22#16812087semidirect

Maximal subgroups

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

subgroupindex
C23×Dic2142
C2×#168120872
C2×(C22⋊D107)⋊C42
#202925472
#202925402
#202925312
#202925062
#202925042
#202924992
#202924852
#385×C1072
#2409107