Finite Group Databasebeta

← #28339978

Subgroups — #28339978

Order 13696.

652 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812094C212yesmaximal, complemented
2C2×#16812084C211yesmaximal, characteristic
3C2×#16812071C211yesmaximal, characteristic, complemented
4#20292553C212yesmaximal, complemented
5#20292544C212yesmaximal, complemented
6#20292529C211yesmaximal, characteristic, complemented
7#20292523C211yesmaximal, characteristic, complemented
8#20292498C212yesmaximal, complemented
9#20292484C211yesmaximal, characteristic, complemented
10C4×(C2×D107)⋊C4C211yesmaximal, characteristic, complemented
11#384×C107C211yesmaximal, characteristic, complemented
12C22×D428C2212yes—
13C22×D428C2211yescharacteristic
14C22×C4×D107C2211yescharacteristic
15C22×C4×D107C2212yes—
16#16812094—28no—
17#16812084—24no—
18#16812071—24no—
19C2×C4⋊Dic107C2211yescharacteristic
20C2×Dic107⋊C4C2212yes—
21C2×(C2×D107)⋊C4C2212yes—
22C2×(C2×D107)⋊C4C2212yes—
23C2×(C2×D107)⋊C4C2212yes—
24C2×(C2×D107)⋊C4C2211yescharacteristic
25C2×(C2×D107)⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2212yes—
27C2×(C2×D107)⋊C4C2211yescharacteristic
28C2×C4×Dic107C2211yescharacteristic
29C107⋊#95C2211yescharacteristic
30C107⋊#95C2211yescharacteristic
31C107⋊#95C2212yes—
32C107⋊#95C2212yes—
33C2×C4⋊C4×C107C2211yescharacteristic
34C2×C4⋊C4×C107C2212yes—
35C2×C4⋊C4×C107C2211yescharacteristic
36C2×C4⋊C4×C107C2212yes—
37C2×C42×C107C2211yescharacteristic
38C23×D107C2312yes—
39C23×D107C2311yescharacteristic
40C2×D428—22no—
41C2×D428—22no—
42C2×D428—24no—
43C2×D428—24no—
44C2×D428—42no—
45C2×D428D414yes—
46C22×Dic107C2311yescharacteristic
47C22×Dic107C2312yes—
48C22×Dic107C2311yescharacteristic
49C22×Dic107C2311yescharacteristic
50C2×C4×D107—24no—
51C2×C4×D107—22no—
52C2×C4×D107—22no—
53C2×C4×D107—22no—
54C2×C4×D107—24no—
55C2×C4×D107—42no—
56C4⋊Dic107—22no—
57Dic107⋊C4—24no—
58(C2×D107)⋊C4—24no—
59(C2×D107)⋊C4—24no—
60(C2×D107)⋊C4—24no—
61(C2×D107)⋊C4—24no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4—24no—
64(C2×D107)⋊C4D414yescomplemented
65C4×Dic107—22no—
66C22×C4×C107C2312yes—
67C22×C4×C107C2312yes—
68C22×C4×C107C2311yescharacteristic
69C22×C4×C107C2311yescharacteristic
70C22×C4×C107C2311yescharacteristic
71C4⋊C4×C107—22no—
72C4⋊C4×C107—22no—
73C4⋊C4×C107—24no—
74C4⋊C4×C107—42no—
75C42×C107—22no—
76C23×C107C2411yescharacteristic
77C22×D107—22no—
78C22×D107—21no—
79C22×D107—21no—
80C22×D107—22no—
81C22×D107—21no—
82C22×D107—21no—
83C22×D107—22no—
84C22×D107—22no—
85C22×D107—22no—
86C22×D107—21no—
87C22×D107—22no—
88C22×D107—21no—
89C22×D107C2×D412yes—
90C22×D107C2×D414yes—
91C2×C4×C107—21no—
92C2×C4×C107—21no—
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×C107—22no—
101C2×C4×C107C2×D412yes—
102C2×C4×C107C2×D412yes—
103C2×C4×C107#4112yes—
104C2×C4×C107#4112yes—
105C2×C4×C107#4112yes—
106C2×Dic107—21no—
107C2×Dic107—21no—
108C2×Dic107—21no—
109C2×Dic107—22no—
110C2×Dic107—21no—
111C2×Dic107—22no—
112C2×Dic107—21no—
113C2×Dic107—22no—
114C2×Dic107—21no—
115C2×Dic107—21no—
116C2×Dic107—21no—
117C2×Dic107C2×D412yes—
118D428—28no—
119D428—48no—
120C4×D107—48no—
121C4×D107—44no—
122C22×C107C22×D411yescharacteristic
123C22×C107C22×D411yescharacteristic
124C22×C107C2×#4111yescharacteristic
125C22×C107C2×#4111yescharacteristic
126C22×C107C2×#4111yescharacteristic
127C22×C107H(5,𝔽2)11yescharacteristic
128C22×C1072−1+411yescharacteristic
129C2×D107—24no—
130C2×D107—22no—
131C2×D107—24no—
132C2×D107—22no—
133C2×D107—22no—
134C2×D107—24no—
135C2×D107—41no—
136C2×D107—42no—
137C2×D107—41no—
138C2×D107—42no—
139C2×D107—42no—
140C2×D107—41no—
141C2×D107—41no—
142C2×D107—42no—
143C4×C107—22no—
144C4×C107—22no—
145C4×C107—42no—
146C4×C107—42no—
147C4×C107#12314yescomplemented
148Dic107—22no—
149Dic107—41no—
150Dic107—42no—
151Dic107—41no—
152C2×C107C2×#12311yescharacteristic
153C2×C107#53211yescharacteristic
154C2×C107#54211yescharacteristic
155C2×C107D4211yescharacteristic
156C2×C107#54711yescharacteristic
157C2×C107#54711yescharacteristic
158C2×C107#55311yescharacteristic
159D107—44no—
160D107—42no—
161#2422—1071nomaximal, Sylow
162C107#242211yescharacteristic, complemented, Sylow
163C2×#123—1072no—
164C2×#121—1071no—
165C2×C4×D4—1071no—
166#397—1072no—
167#394—1072no—
168#390—1071no—
169#389—1071no—
170#385—1072no—
171#384D10711yescharacteristic, complemented
172#380—1071no—
173C4×C22⋊C4—1071no—
174C22×D4—1071no—
175C22×D4—1072no—
176C23×C4—1072no—
177C23×C4—1071no—
178#123—2148no—
179#121—2144no—
180C4×D4—2144no—
181C2×C4⋊C4—1072no—
182C2×C4⋊C4—1071no—
183C2×C4⋊C4C2×D10711yescharacteristic
184C2×C4⋊C4C2×D10711yescharacteristic
185C2×C4⋊C4C2×D10712yes—
186C2×C4⋊C4C2×D10712yes—
187C2×C22⋊C4—1071no—
188C2×C22⋊C4—1072no—
189C2×C22⋊C4—1072no—
190C2×C22⋊C4—1072no—
191C2×C22⋊C4—1072no—
192C2×C22⋊C4—1071no—
193C2×C22⋊C4—1072no—
194C2×C42—1071no—
195C2×C42C2×D10711yescharacteristic
196#95—1071no—
197#95—1071no—
198#95—1072no—
199#95—1072no—
200C24—1071no—
201C24—1072no—
202C2×D4—1074no—
203C2×D4—4282no—
204C2×D4—2144no—
205C2×D4—2144no—
206C2×D4—2142no—
207C2×D4—2142no—
208C22×C4—1071no—
209C22×C4—1071no—
210C22×C4—1071no—
211C22×C4—1072no—
212C22×C4—4282no—
213C22×C4—2144no—
214C22×C4—2144no—
215C22×C4—2142no—
216C22×C4—2142no—
217C22×C4—2142no—
218C22×C4C22×D10712yes—
219C22×C4C22×D10712yes—
220C22×C4C22×D10711yescharacteristic
221C22×C4C22×D10711yescharacteristic
222C22×C4C22×D10711yescharacteristic
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—24no—
226C4⋊C4—42no—
227C4⋊C4—2144no—
228C4⋊C4—2142no—
229C22⋊C4—1074no—
230C22⋊C4—2144no—
231C22⋊C4—2144no—
232C22⋊C4—2144no—
233C22⋊C4—2144no—
234C22⋊C4—2142no—
235C22⋊C4—2144no—
236C42—22no—
237C42—2142no—
238C23—1072no—
239C23—1074no—
240C23—2142no—
241C23—2141no—
242C23—2141no—
243C23—2142no—
244C23—2142no—
245C23—2141no—
246C23—2141no—
247C23—2142no—
248C23—2142no—
249C23—2141no—
250C23—2142no—
251C23—2141no—
252C23C23×D10711yescharacteristic
253D4—4288no—
254D4—2148no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—1072no—
266C2×C4—4288no—
267C2×C4—4284no—
268C2×C4—2141no—
269C2×C4—2142no—
270C2×C4—2141no—
271C2×C4—2141no—
272C2×C4—2142no—
273C2×C4—2141no—
274C2×C4—2142no—
275C2×C4—2141no—
276C2×C4—2141no—
277C2×C4—2141no—
278C2×C4—2141no—
279C2×C4D4×D10712yes—
280C2×C4D4×D10712yes—
281C2×C4#1313381812yes—
282C2×C4#1313381912yes—
283C2×C4#1313381912yes—
284C22—4281no—
285C22—4282no—
286C22—4281no—
287C22—4282no—
288C22—4282no—
289C22—4281no—
290C22—4281no—
291C22—4282no—
292C22—2144no—
293C22—2142no—
294C22—2144no—
295C22—2142no—
296C22—2142no—
297C22—2144no—
298C22C2×D4×D10711yescharacteristic
299C22C2×D4×D10711yescharacteristic
300C22C2×#1313381811yescharacteristic
301C22C2×#1313381911yescharacteristic
302C22C2×#1313381911yescharacteristic
303C22#1681215411yescharacteristic
304C22#1681215711yescharacteristic
305C4—22no—
306C4—22no—
307C4—42no—
308C4—42no—
309C4—4281no—
310C4—4282no—
311C4—4281no—
312C4—2142no—
313C4#1681209414yescomplemented
314C2—4282no—
315C2—4284no—
316C2C2×#1681209411yescharacteristic
317C2#2029317811yescharacteristic
318C2#2029325111yescharacteristic
319C2#2029326211yescharacteristic
320C2#2029327811yescharacteristic
321C2#2029328011yescharacteristic
322C2#2029332511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#384×C107C2
prime radicalC2×#16812084C2
hypercenter#384D107
hypocenterC107#2422
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
1. C2×#16812094C2semidirect
3. C2×#16812071C2semidirect
4. #20292553C2semidirect
5. #20292544C2semidirect
6. #20292529C2semidirect
7. #20292523C2semidirect
8. #20292498C2semidirect
9. #20292484C2semidirect
10. C4×(C2×D107)⋊C4C2semidirect
11. #384×C107C2semidirect
64. (C2×D107)⋊C4D4semidirect
147. C4×C107#123semidirect
162. C107#2422semidirect
171. #384D107semidirect
313. C4#16812094semidirect

Maximal subgroups

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

subgroupindex
C2×#168120942
C2×#168120842
C2×#168120712
#202925532
#202925442
#202925292
#202925232
#202924982
#202924842
C4×(C2×D107)⋊C42
#384×C1072
#2422107