Finite Group Databasebeta

← #28339405

Subgroups — #28339405

Order 13696.

632 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊D107C211yesmaximal, characteristic
2C2×#16812068C211yesmaximal, characteristic, complemented
3C2×#16812067C211yesmaximal, characteristic, complemented
4#20292527C212yesmaximal
5#20292521C212yesmaximal, complemented
6#20292516C212yesmaximal, complemented
7#20292496C211yesmaximal, characteristic, complemented
8#20292484C211yesmaximal, characteristic, complemented
9C4⋊C4×Dic107C211yesmaximal, characteristic, complemented
10#20292478C212yesmaximal, complemented
11#380×C107C211yesmaximal, characteristic, complemented
12C22×C22⋊D107C2211yescharacteristic
13C22×C4×D107C2211yescharacteristic
14C4×C22⋊D107C418yescomplemented
15#16812068—24no—
16#16812067—24no—
17C2×C4⋊Dic107C2211yescharacteristic
18C2×C4⋊Dic107C2211yescharacteristic
19C2×Dic107⋊C4C2212yes—
20C2×Dic107⋊C4C2212yes—
21C2×Dic107⋊C4C2211yescharacteristic
22C2×C22⋊Dic107C2212yes—
23C2×C22⋊Dic107C2211yescharacteristic
24C2×(C2×D107)⋊C4C2212yes—
25C2×(C2×D107)⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2211yescharacteristic
27C2×C4×Dic107C2212yes—
28C2×C4×Dic107C2212yes—
29C2×C4×Dic107C2212yes—
30C2×C4×Dic107C2211yescharacteristic
31C107⋊#95C2212yes—
32C107⋊#95C2212yes—
33C23×C4×C107C2211yescharacteristic
34C2×C4⋊C4×C107C2211yescharacteristic
35C2×C4⋊C4×C107C2211yescharacteristic
36C2×C22⋊C4×C107C2212yes—
37#95×C107C2212yes—
38C23×D107C2311yescharacteristic
39C2×C22⋊D107—22no—
40C2×C22⋊D107—22no—
41C2×C22⋊D107C2×C414yes—
42C22×Dic107C2312yes—
43C22×Dic107C2311yescharacteristic
44C22×Dic107C2311yescharacteristic
45C22×Dic107C2312yes—
46C22×Dic107C2311yescharacteristic
47C2×C4×D107—22no—
48C2×C4×D107—22no—
49C2×C4×D107C2×C414yes—
50C4⋊Dic107—22no—
51C4⋊Dic107D414yescomplemented
52Dic107⋊C4—24no—
53Dic107⋊C4—42no—
54Dic107⋊C4C2×C414yes—
55C22⋊Dic107—24no—
56C22⋊Dic107C2×C414yes—
57(C2×D107)⋊C4—24no—
58(C2×D107)⋊C4—42no—
59(C2×D107)⋊C4C2×C414yes—
60C4×Dic107—24no—
61C4×Dic107—24no—
62C4×Dic107—24no—
63C4×Dic107C2×C414yes—
64C24×C107C2311yescharacteristic
65C22×C4×C107—22no—
66C22×C4×C107—22no—
67C22×C4×C107C2×C414yes—
68C22×C4×C107C2312yes—
69C22×C4×C107C2312yes—
70C22×C4×C107C2311yescharacteristic
71C22×C4×C107C2311yescharacteristic
72C4⋊C4×C107—22no—
73C4⋊C4×C107—22no—
74C22⋊C4×C107—42no—
75C23×C107—21no—
76C23×C107—21no—
77C23×C107—21no—
78C23×C107—21no—
79C23×C107—21no—
80C23×C107—21no—
81C23×C107C22×C412yes—
82C23×C107C2411yescharacteristic
83C22×D107—21no—
84C22×D107—21no—
85C22×D107—21no—
86C22×D107—21no—
87C22×D107—21no—
88C22×D107—21no—
89C22×D107C22×C412yes—
90C2×C4×C107—21no—
91C2×C4×C107—21no—
92C2×C4×C107—28no—
93C2×C4×C107—22no—
94C2×C4×C107—22no—
95C2×C4×C107—22no—
96C2×C4×C107—22no—
97C2×C4×C107—22no—
98C2×C4×C107—22no—
99C2×C4×C107C22×C412yes—
100C2×C4×C107C22×C412yes—
101C2×C4×C107C2×D412yes—
102C2×C4×C107#4112yes—
103C22⋊D107—28no—
104C2×Dic107—21no—
105C2×Dic107—22no—
106C2×Dic107—21no—
107C2×Dic107—21no—
108C2×Dic107—21no—
109C2×Dic107—22no—
110C2×Dic107—22no—
111C2×Dic107—22no—
112C2×Dic107—22no—
113C2×Dic107C22×C412yes—
114C2×Dic107C22×C412yes—
115C2×Dic107C22×C412yes—
116C2×Dic107C2×D414yes—
117C2×Dic107#4112yes—
118C2×Dic107#4112yes—
119C4×D107—28no—
120C22×C107—22no—
121C22×C107—22no—
122C22×C107—22no—
123C22×C107—41no—
124C22×C107—41no—
125C22×C107—41no—
126C22×C107—41no—
127C22×C107C23×C411yescharacteristic
128C22×C107C22×D411yescharacteristic
129C22×C107C2×#4111yescharacteristic
130C22×C107C2×#4111yescharacteristic
131C22×C107C2×#4111yescharacteristic
132C22×C107H(5,𝔽2)11yescharacteristic
133C22×C1072−1+411yescharacteristic
134C2×D107—22no—
135C2×D107—22no—
136C2×D107—22no—
137C2×D107—41no—
138C2×D107—41no—
139C2×D107—41no—
140C2×D107—41no—
141C4×C107—22no—
142C4×C107—42no—
143C4×C107—42no—
144C4×C107C4×D414yes—
145Dic107—22no—
146Dic107—22no—
147Dic107—24no—
148Dic107—42no—
149Dic107#11714yescomplemented
150C2×C107—42no—
151C2×C107C2×#11711yescharacteristic
152C2×C107C2×C4×D411yescharacteristic
153C2×C107#52011yescharacteristic
154C2×C107#54711yescharacteristic
155C2×C107#55011yescharacteristic
156C2×C107#55511yescharacteristic
157C2×C107#55611yescharacteristic
158D107—42no—
159#2354—1071nomaximal, Sylow
160C107#235411yescharacteristic, complemented, Sylow
161C2×C4×D4—1071no—
162C2×#117—1071no—
163C2×#117—1071no—
164#390—1072no—
165#388—1072no—
166#388—1072no—
167#384—1071no—
168#380—1071no—
169#380D10711yescharacteristic, complemented
170C4×C4⋊C4—1071no—
171C4×C4⋊C4—1072no—
172C22×D4—1071no—
173C23×C4—1071no—
174C23×C4C2×D10711yescharacteristic
175C4×D4—1078no—
176#117—2144no—
177#117—2144no—
178C2×C4⋊C4—1071no—
179C2×C4⋊C4—1071no—
180C2×C4⋊C4—1072no—
181C2×C4⋊C4—1071no—
182C2×C4⋊C4—1072no—
183C2×C4⋊C4C2×D10711yescharacteristic
184C2×C4⋊C4C2×D10711yescharacteristic
185C2×C22⋊C4—1071no—
186C2×C22⋊C4—1071no—
187C2×C22⋊C4—1072no—
188C2×C22⋊C4—1072no—
189C2×C22⋊C4—1072no—
190C2×C22⋊C4C2×D10712yes—
191C2×C42—1071no—
192C2×C42—1072no—
193C2×C42—1072no—
194C2×C42—1072no—
195#95—1072no—
196#95—1072no—
197#95C2×D10712yes—
198C24—1071no—
199C24C22×D10711yescharacteristic
200C2×D4—1074no—
201C2×D4—2142no—
202C2×D4—2142no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—1071no—
206C22×C4—1071no—
207C22×C4—1071no—
208C22×C4—1072no—
209C22×C4—1074no—
210C22×C4—1072no—
211C22×C4—2142no—
212C22×C4—2142no—
213C22×C4C4×D10714yes—
214C22×C4C22×D10712yes—
215C22×C4C22×D10712yes—
216C22×C4C22×D10711yescharacteristic
217C22×C4C22×D10711yescharacteristic
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—1074no—
221C4⋊C4—1074no—
222C4⋊C4—4282no—
223C4⋊C4—2144no—
224C4⋊C4—2142no—
225C22⋊C4—42no—
226C22⋊C4—1074no—
227C22⋊C4—1074no—
228C22⋊C4—4282no—
229C22⋊C4—2144no—
230C22⋊C4—2144no—
231C42—1074no—
232C42—2144no—
233C42—2144no—
234C42—2144no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—1072no—
242C23—2141no—
243C23—2141no—
244C23—2141no—
245C23—2141no—
246C23—2141no—
247C23—2141no—
248C23C2×C4×D10712yes—
249C23C23×D10711yescharacteristic
250D4—2148no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—28no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—1072no—
261C2×C4—1072no—
262C2×C4—1072no—
263C2×C4—1072no—
264C2×C4—1072no—
265C2×C4—1074no—
266C2×C4—2142no—
267C2×C4—2142no—
268C2×C4—2142no—
269C2×C4—2141no—
270C2×C4—2142no—
271C2×C4—2141no—
272C2×C4—2141no—
273C2×C4—2142no—
274C2×C4—2141no—
275C2×C4—2148no—
276C2×C4C2×C4×D10712yes—
277C2×C4C2×C4×D10712yes—
278C2×C4D4×D10712yes—
279C2×C4#1313381712yes—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22—4281no—
288C22—4281no—
289C22—4281no—
290C22—4281no—
291C22—2142no—
292C22—2142no—
293C22—2142no—
294C22C22×C4×D10711yescharacteristic
295C22C2×D4×D10711yescharacteristic
296C22#41×D10711yescharacteristic
297C22#41×D10711yescharacteristic
298C22C2×#1313381711yescharacteristic
299C22#1681215411yescharacteristic
300C22#1681215711yescharacteristic
301C4—22no—
302C4—42no—
303C4—42no—
304C4—1074no—
305C4—4282no—
306C4—2144no—
307C4—2142no—
308C4—2142no—
309C4(C22⋊D107)⋊C414yes—
310C2—42no—
311C2—4282no—
312C2#117×D10711yescharacteristic
313C2C2×(C22⋊D107)⋊C411yescharacteristic
314C2#2029310811yescharacteristic
315C2#2029327911yescharacteristic
316C2#2029329511yescharacteristic
317C2#2029334111yescharacteristic
318C2#2029334811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
Fitting#380×C107C2
socle, cosocleC23×C107C24
prime radicalC22×C22⋊D107C22
hypercenter#380D107
hypocenterC107#2354
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#16812068C2semidirect
3. C2×#16812067C2semidirect
5. #20292521C2semidirect
6. #20292516C2semidirect
7. #20292496C2semidirect
8. #20292484C2semidirect
9. C4⋊C4×Dic107C2semidirect
10. #20292478C2semidirect
11. #380×C107C2semidirect
14. C4×C22⋊D107C4semidirect
51. C4⋊Dic107D4semidirect
149. Dic107#117semidirect
160. C107#2354semidirect
169. #380D107semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D1072
C2×#168120682
C2×#168120672
#202925272
#202925212
#202925162
#202924962
#202924842
C4⋊C4×Dic1072
#202924782
#380×C1072
#2354107