Finite Group Databasebeta

← #28339357

Subgroups — #28339357

Order 13696.

776 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊D107C211yesmaximal, characteristic
2C2×(C22⋊D107)⋊C4C212yesmaximal
3C22×C4⋊Dic107C211yesmaximal, characteristic, complemented
4C2×C4⋊C4×D107C211yesmaximal, characteristic, complemented
5#20292524C212yesmaximal, complemented
6#20292501C212yesmaximal, complemented
7#20292499C212yesmaximal, complemented
8#20292496C211yesmaximal, characteristic, complemented
9#20292484C211yesmaximal, characteristic, complemented
10C4⋊C4×Dic107C211yesmaximal, characteristic, complemented
11#380×C107C211yesmaximal, characteristic, complemented
12C22×C22⋊D107C2211yescharacteristic
13C23×Dic107C2212yes—
14C22×C4×D107C2212yes—
15C22×C4×D107C2211yescharacteristic
16C4×C22⋊D107C418yescomplemented
17(C22⋊D107)⋊C4—28no—
18C2×C4⋊Dic107—24no—
19C2×C4⋊Dic107C2211yescharacteristic
20C2×C4⋊Dic107C2212yes—
21C2×C4⋊Dic107C2211yescharacteristic
22C2×Dic107⋊C4C2212yes—
23C2×Dic107⋊C4C2212yes—
24C2×Dic107⋊C4C2212yes—
25C2×Dic107⋊C4C2211yescharacteristic
26C4⋊C4×D107—24no—
27C2×C22⋊Dic107C2211yescharacteristic
28C2×(C2×D107)⋊C4C2212yes—
29C2×(C2×D107)⋊C4C2211yescharacteristic
30C2×C4×Dic107C2212yes—
31C2×C4×Dic107C2211yescharacteristic
32C107⋊#95C2212yes—
33C107⋊#95C2212yes—
34C23×C4×C107C2211yescharacteristic
35C2×C4⋊C4×C107C2211yescharacteristic
36C2×C4⋊C4×C107C2211yescharacteristic
37C2×C22⋊C4×C107C2212yes—
38#95×C107C2212yes—
39C23×D107C2311yescharacteristic
40C2×C22⋊D107C2×C414yes—
41C2×C22⋊D107D414yes—
42C2×C22⋊D107Q814yes—
43C22×Dic107—24no—
44C22×Dic107—24no—
45C22×Dic107—42no—
46C22×Dic107C2312yes—
47C22×Dic107C2311yescharacteristic
48C22×Dic107C2311yescharacteristic
49C22×Dic107C2312yes—
50C22×Dic107C2311yescharacteristic
51C2×C4×D107—22no—
52C2×C4×D107—22no—
53C2×C4×D107—24no—
54C2×C4×D107—24no—
55C2×C4×D107—42no—
56C2×C4×D107C2×C414yescomplemented
57C4⋊Dic107—22no—
58C4⋊Dic107—24no—
59C4⋊Dic107D414yescomplemented
60Dic107⋊C4—24no—
61Dic107⋊C4—24no—
62Dic107⋊C4—42no—
63Dic107⋊C4C2×C414yescomplemented
64C22⋊Dic107C2×C414yescomplemented
65(C2×D107)⋊C4—24no—
66(C2×D107)⋊C4C2×C414yescomplemented
67C4×Dic107—24no—
68C4×Dic107C2×C414yescomplemented
69C24×C107C2311yescharacteristic
70C22×C4×C107—22no—
71C22×C4×C107—22no—
72C22×C4×C107C2×C414yescomplemented
73C22×C4×C107C2312yes—
74C22×C4×C107C2312yes—
75C22×C4×C107C2311yescharacteristic
76C22×C4×C107C2311yescharacteristic
77C4⋊C4×C107—22no—
78C4⋊C4×C107—22no—
79C22⋊C4×C107—24no—
80C23×C107—21no—
81C23×C107—21no—
82C23×C107—21no—
83C23×C107—21no—
84C23×C107C22×C412yes—
85C23×C107C2×D412yes—
86C23×C107C2×Q812yes—
87C23×C107C2411yescharacteristic
88C22×D107—21no—
89C22×D107—21no—
90C22×D107—21no—
91C22×D107—21no—
92C22×D107C22×C412yes—
93C22×D107C2×D412yes—
94C22×D107C2×Q812yes—
95C2×C4×C107—21no—
96C2×C4×C107—21no—
97C2×C4×C107—28no—
98C2×C4×C107—22no—
99C2×C4×C107—22no—
100C2×C4×C107—22no—
101C2×C4×C107—22no—
102C2×C4×C107—22no—
103C2×C4×C107—22no—
104C2×C4×C107C22×C412yes—
105C2×C4×C107C22×C412yes—
106C2×C4×C107C2×D412yes—
107C2×C4×C107#4112yes—
108C22⋊D107C4⋊C4116yescomplemented
109C2×Dic107—22no—
110C2×Dic107—21no—
111C2×Dic107—21no—
112C2×Dic107—21no—
113C2×Dic107—22no—
114C2×Dic107—21no—
115C2×Dic107—22no—
116C2×Dic107—22no—
117C2×Dic107—22no—
118C2×Dic107—48no—
119C2×Dic107C22×C412yes—
120C2×Dic107C22×C412yes—
121C2×Dic107C22×C412yes—
122C2×Dic107C2×D414yes—
123C2×Dic107C2×D412yes—
124C2×Dic107C2×Q812yes—
125C4×D107—28no—
126C4×D107—48no—
127C22×C107—22no—
128C22×C107—22no—
129C22×C107—22no—
130C22×C107—22no—
131C22×C107—22no—
132C22×C107—22no—
133C22×C107C2×C4⋊C414yescomplemented
134C22×C107C23×C411yescharacteristic
135C22×C107C22×D411yescharacteristic
136C22×C107C22×D411yescharacteristic
137C22×C107C22×Q811yescharacteristic
138C22×C107C2×#4111yescharacteristic
139C22×C107H(5,𝔽2)11yescharacteristic
140C22×C1072−1+411yescharacteristic
141C2×D107—22no—
142C2×D107—22no—
143C2×D107—22no—
144C2×D107—22no—
145C2×D107—22no—
146C2×D107—22no—
147C2×D107C2×C4⋊C414yescomplemented
148C4×C107—22no—
149C4×C107—42no—
150C4×C107—42no—
151C4×C107C4×D414yescomplemented
152Dic107—22no—
153Dic107—22no—
154Dic107—24no—
155Dic107—42no—
156Dic107C2×C4⋊C414yescomplemented
157C2×C107—24no—
158C2×C107C22×C4⋊C411yescharacteristic
159C2×C107C2×C4×D411yescharacteristic
160C2×C107#52011yescharacteristic
161C2×C107D4211yescharacteristic
162C2×C107#54711yescharacteristic
163C2×C107D4×Q811yescharacteristic
164C2×C107#55411yescharacteristic
165D107—24no—
166D4×C4⋊C4—1071nomaximal, Sylow
167C107D4×C4⋊C411yescharacteristic, complemented, Sylow
168C2×C4×D4—1071no—
169C2×C4×D4—1072no—
170C22×C4⋊C4—1071no—
171C22×C4⋊C4—1071no—
172#389—1072no—
173#385—1072no—
174#385—1072no—
175#384—1071no—
176#380—1071no—
177#380D10711yescharacteristic, complemented
178C4×C4⋊C4—1071no—
179C22×D4—1071no—
180C23×C4—1071no—
181C23×C4—1072no—
182C23×C4—1072no—
183C23×C4C2×D10711yescharacteristic
184C4×D4—1078no—
185C4×D4—2148no—
186C2×C4⋊C4—1071no—
187C2×C4⋊C4—1071no—
188C2×C4⋊C4—1072no—
189C2×C4⋊C4—1072no—
190C2×C4⋊C4—1071no—
191C2×C4⋊C4—1072no—
192C2×C4⋊C4—1072no—
193C2×C4⋊C4—2144no—
194C2×C4⋊C4—2144no—
195C2×C4⋊C4C2×D10711yescharacteristic
196C2×C4⋊C4C2×D10711yescharacteristic
197C2×C22⋊C4—1071no—
198C2×C22⋊C4—1071no—
199C2×C22⋊C4—1072no—
200C2×C22⋊C4C2×D10712yes—
201C2×C42—1071no—
202C2×C42—1072no—
203#95—1072no—
204#95—1072no—
205#95C2×D10712yes—
206C24—1071no—
207C24C22×D10711yescharacteristic
208C2×D4—1074no—
209C2×D4—1074no—
210C2×D4—1074no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—1071no—
214C22×C4—1071no—
215C22×C4—1071no—
216C22×C4—1072no—
217C22×C4—1074no—
218C22×C4—1072no—
219C22×C4—4282no—
220C22×C4—4282no—
221C22×C4—2144no—
222C22×C4—2144no—
223C22×C4—2144no—
224C22×C4—2144no—
225C22×C4—2142no—
226C22×C4—2142no—
227C22×C4C4×D10714yescomplemented
228C22×C4C22×D10712yes—
229C22×C4C22×D10712yes—
230C22×C4C22×D10711yescharacteristic
231C22×C4C22×D10711yescharacteristic
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—1074no—
235C4⋊C4—1074no—
236C4⋊C4—4282no—
237C4⋊C4—2144no—
238C4⋊C4—2144no—
239C4⋊C4—2144no—
240C4⋊C4—2142no—
241C22⋊C4—24no—
242C22⋊C4—1074no—
243C22⋊C4—1074no—
244C22⋊C4—2144no—
245C42—1074no—
246C42—2144no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—1072no—
252C23—1072no—
253C23—1072no—
254C23—2141no—
255C23—2141no—
256C23—2141no—
257C23—2141no—
258C23C2×C4×D10712yes—
259C23D4×D10712yes—
260C23Q8×D10712yes—
261C23C23×D10711yescharacteristic
262D4—10716no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—28no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—1072no—
273C2×C4—1072no—
274C2×C4—1072no—
275C2×C4—1072no—
276C2×C4—1072no—
277C2×C4—1074no—
278C2×C4—4288no—
279C2×C4—4288no—
280C2×C4—2142no—
281C2×C4—2142no—
282C2×C4—2142no—
283C2×C4—2142no—
284C2×C4—2141no—
285C2×C4—2141no—
286C2×C4—2141no—
287C2×C4—2142no—
288C2×C4—2141no—
289C2×C4—2148no—
290C2×C4C2×C4×D10712yes—
291C2×C4C2×C4×D10712yes—
292C2×C4D4×D10712yes—
293C2×C4#1313381712yes—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—1074no—
301C22—2142no—
302C22—2142no—
303C22—2142no—
304C22—2142no—
305C22—2142no—
306C22—2142no—
307C22C4⋊C4×D10714yescomplemented
308C22C22×C4×D10711yescharacteristic
309C22C2×D4×D10711yescharacteristic
310C22C2×D4×D10711yescharacteristic
311C22C2×Q8×D10711yescharacteristic
312C22C2×#1313381711yescharacteristic
313C22#1681215511yescharacteristic
314C22#1681215611yescharacteristic
315C4—22no—
316C4—42no—
317C4—42no—
318C4—1074no—
319C4—4282no—
320C4—2144no—
321C4—2142no—
322C4—2142no—
323C4(C22⋊D107)⋊C414yescomplemented
324C2—24no—
325C2—2144no—
326C2C2×C4⋊C4×D10711yescharacteristic
327C2C2×(C22⋊D107)⋊C411yescharacteristic
328C2#2029310711yescharacteristic
329C2#2029326111yescharacteristic
330C2#2029327711yescharacteristic
331C2#2029329011yescharacteristic
332C2#2029333511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
Fitting#380×C107C2
socle, cosocleC23×C107C24
prime radicalC22×C22⋊D107C22
hypercenter#380D107
hypocenterC107D4×C4⋊C4
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
3. C22×C4⋊Dic107C2semidirect
4. C2×C4⋊C4×D107C2semidirect
5. #20292524C2semidirect
6. #20292501C2semidirect
7. #20292499C2semidirect
8. #20292496C2semidirect
9. #20292484C2semidirect
10. C4⋊C4×Dic107C2semidirect
11. #380×C107C2semidirect
16. C4×C22⋊D107C4semidirect
56. C2×C4×D107C2×C4semidirect
59. C4⋊Dic107D4semidirect
63. Dic107⋊C4C2×C4semidirect
64. C22⋊Dic107C2×C4semidirect
66. (C2×D107)⋊C4C2×C4semidirect
68. C4×Dic107C2×C4semidirect
72. C22×C4×C107C2×C4semidirect
108. C22⋊D107C4⋊C4semidirect
133. C22×C107C2×C4⋊C4semidirect
147. C2×D107C2×C4⋊C4semidirect
151. C4×C107C4×D4semidirect
156. Dic107C2×C4⋊C4semidirect
167. C107D4×C4⋊C4semidirect
177. #380D107semidirect
227. C22×C4C4×D107semidirect
307. C22C4⋊C4×D107semidirect
323. C4(C22⋊D107)⋊C4semidirect

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×(C22⋊D107)⋊C42
C22×C4⋊Dic1072
C2×C4⋊C4×D1072
#202925242
#202925012
#202924992
#202924962
#202924842
C4⋊C4×Dic1072
#380×C1072
D4×C4⋊C4107