Finite Group Databasebeta

← #28339915

Subgroups — #28339915

Order 13696.

648 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812085C211yesmaximal, characteristic, complemented
2C2×#16812085C211yesmaximal, characteristic, complemented
3C2×C4×D428C211yesmaximal, characteristic
4C2×C4⋊C4×D107C211yesmaximal, characteristic, complemented
5#20292561C212yesmaximal, complemented
6#20292551C212yesmaximal, complemented
7#20292525C211yesmaximal, characteristic, complemented
8#20292505C212yesmaximal
9#20292486C212yesmaximal, complemented
10#20292484C211yesmaximal, characteristic, complemented
11#391×C107C211yesmaximal, characteristic, complemented
12C22×D428C2211yescharacteristic
13C22×C4×D107C2211yescharacteristic
14C22×C4×D107C2212yes—
15C22×C4×D107C2211yescharacteristic
16#16812085—24no—
17#16812085—24no—
18C4×D428—24no—
19C2×C4⋊Dic107C2211yescharacteristic
20C2×C4⋊Dic107C2211yescharacteristic
21C2×C4⋊Dic107C2211yescharacteristic
22C2×Dic107⋊C4C2212yes—
23C2×Dic107⋊C4C2212yes—
24C2×Dic107⋊C4C2212yes—
25C2×Dic107⋊C4C2212yes—
26C4⋊C4×D107—24no—
27C2×(C2×D107)⋊C4C2212yes—
28C2×(C2×D107)⋊C4C2212yes—
29C2×(C2×D107)⋊C4C2211yescharacteristic
30C2×(C2×D107)⋊C4C2211yescharacteristic
31C2×(C2×D107)⋊C4C2212yes—
32C107⋊#95C2212yes—
33C107⋊#95C2212yes—
34C22×Q8×C107C2211yescharacteristic
35C2×C4⋊C4×C107C2211yescharacteristic
36C2×C42×C107C2211yescharacteristic
37#95×C107C2212yes—
38#95×C107C2212yes—
39C23×D107C2311yescharacteristic
40C23×D107C2311yescharacteristic
41C2×D428—22no—
42C2×D428—22no—
43C2×D428Q814yescomplemented
44C22×Dic107C2312yes—
45C22×Dic107C2311yescharacteristic
46C22×Dic107C2311yescharacteristic
47C22×Dic107C2312yes—
48C2×C4×D107—22no—
49C2×C4×D107—22no—
50C2×C4×D107—24no—
51C2×C4×D107—22no—
52C2×C4×D107—22no—
53C2×C4×D107—22no—
54C2×C4×D107—24no—
55C2×C4×D107—42no—
56C2×C4×D107D414yes—
57C4⋊Dic107—22no—
58C4⋊Dic107—22no—
59C4⋊Dic107D414yescomplemented
60Dic107⋊C4—24no—
61Dic107⋊C4—24no—
62Dic107⋊C4—24no—
63Dic107⋊C4—42no—
64(C2×D107)⋊C4—22no—
65(C2×D107)⋊C4—24no—
66(C2×D107)⋊C4—22no—
67(C2×D107)⋊C4—24no—
68(C2×D107)⋊C4—42no—
69C2×Q8×C107—22no—
70C2×Q8×C107—22no—
71C2×Q8×C107—22no—
72C22×C4×C107C2311yescharacteristic
73C22×C4×C107C2312yes—
74C22×C4×C107C2312yes—
75C22×C4×C107C2311yescharacteristic
76C22×C4×C107C2311yescharacteristic
77C4⋊C4×C107—22no—
78C42×C107—22no—
79C23×C107C2411yescharacteristic
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×D107C2×Q812yes—
92C22×D107C2×Q812yes—
93C22×D107#4112yes—
94Q8×C107—44no—
95C2×C4×C107—21no—
96C2×C4×C107—21no—
97C2×C4×C107—21no—
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×C107—21no—
105C2×C4×C107—21no—
106C2×C4×C107C2×D412yes—
107C2×C4×C107C2×D412yes—
108C2×C4×C107C2×Q812yes—
109C2×C4×C107#4112yes—
110C2×Dic107—21no—
111C2×Dic107—22no—
112C2×Dic107—21no—
113C2×Dic107—22no—
114C2×Dic107—22no—
115C2×Dic107—21no—
116C2×Dic107—21no—
117C2×Dic107—21no—
118C2×Dic107—22no—
119C2×Dic107—22no—
120C2×Dic107C2×D412yes—
121C2×Dic107C2×D414yes—
122D428—28no—
123C4×D107—28no—
124C4×D107—48no—
125C4×D107—44no—
126C22×C107C22×D411yescharacteristic
127C22×C107C22×D411yescharacteristic
128C22×C107C22×Q811yescharacteristic
129C22×C107C2×#4111yescharacteristic
130C22×C107C2×#4111yescharacteristic
131C22×C107H(5,𝔽2)11yescharacteristic
132C22×C1072−1+411yescharacteristic
133C2×D107—22no—
134C2×D107—22no—
135C2×D107—22no—
136C2×D107—22no—
137C2×D107—22no—
138C2×D107—22no—
139C2×D107—22no—
140C2×D107—22no—
141C2×D107—22no—
142C2×D107—41no—
143C2×D107—41no—
144C2×D107—41no—
145C2×D107—41no—
146C2×D107C22⋊Q814yescomplemented
147C4×C107—22no—
148C4×C107—42no—
149C4×C107—41no—
150C4×C107—42no—
151C4×C107#12114yes—
152Dic107—22no—
153Dic107—24no—
154Dic107—41no—
155Dic107—42no—
156C2×C107C2×#12111yescharacteristic
157C2×C107C2×C22⋊Q811yescharacteristic
158C2×C107#53711yescharacteristic
159C2×C107#54611yescharacteristic
160C2×C107D4×Q811yescharacteristic
161C2×C107#55211yescharacteristic
162C2×C107#55411yescharacteristic
163D107—24no—
164D107—42no—
165#2417—1071nomaximal, Sylow
166C107#241711yescharacteristic, complemented, Sylow
167C2×C22⋊Q8—1071no—
168C2×C22⋊Q8—1071no—
169C2×C4×D4—1071no—
170C22×C4⋊C4—1071no—
171#398—1072no—
172#396—1072no—
173#391D10711yescharacteristic, complemented
174#389—1071no—
175#386—1072no—
176#380—1071no—
177#380—1072no—
178C22×Q8C2×D10711yescharacteristic
179C22×D4—1071no—
180C23×C4—1071no—
181C23×C4—1072no—
182C23×C4—1071no—
183C22⋊Q8—2144no—
184C22⋊Q8—2144no—
185C4×D4—2144no—
186C2×C4⋊C4—1071no—
187C2×C4⋊C4—1072no—
188C2×C4⋊C4—1072no—
189C2×C4⋊C4—1071no—
190C2×C4⋊C4—1072no—
191C2×C4⋊C4—1072no—
192C2×C4⋊C4—1071no—
193C2×C4⋊C4—2144no—
194C2×C4⋊C4C2×D10711yescharacteristic
195C2×C22⋊C4—1072no—
196C2×C22⋊C4—1071no—
197C2×C22⋊C4—1071no—
198C2×C22⋊C4—1072no—
199C2×C22⋊C4—1072no—
200C2×C42C2×D10711yescharacteristic
201#95—1072no—
202#95—1072no—
203#95C2×D10712yes—
204#95C2×D10712yes—
205C24—1071no—
206C24—1071no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×D4—1074no—
211C2×D4—2142no—
212C2×D4—2142no—
213C22×C4—1072no—
214C22×C4—1071no—
215C22×C4—1074no—
216C22×C4—1072no—
217C22×C4—1071no—
218C22×C4—4282no—
219C22×C4—2144no—
220C22×C4—2144no—
221C22×C4—2142no—
222C22×C4—2142no—
223C22×C4—2142no—
224C22×C4—2142no—
225C22×C4—2142no—
226C22×C4C22×D10712yes—
227C22×C4C22×D10711yescharacteristic
228C22×C4C22×D10712yes—
229C22×C4C22×D10711yescharacteristic
230C22×C4C22×D10711yescharacteristic
231C4⋊C4—22no—
232C4⋊C4—1074no—
233C4⋊C4—4282no—
234C4⋊C4—2144no—
235C4⋊C4—2142no—
236C4⋊C4—2144no—
237C4⋊C4—2144no—
238C4⋊C4—2142no—
239C22⋊C4—4282no—
240C22⋊C4—2142no—
241C22⋊C4—2144no—
242C22⋊C4—2142no—
243C22⋊C4—2144no—
244C42—22no—
245C23—1072no—
246C23—1072no—
247C23—1072no—
248C23—1072no—
249C23—2141no—
250C23—2141no—
251C23—2141no—
252C23—2141no—
253C23—2141no—
254C23—2141no—
255C23—2141no—
256C23—2141no—
257C23—2141no—
258C23—2141no—
259C23C23×D10711yescharacteristic
260Q8—44no—
261D4—2148no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—1074no—
274C2×C4—1072no—
275C2×C4—4288no—
276C2×C4—4284no—
277C2×C4—2142no—
278C2×C4—2141no—
279C2×C4—2141no—
280C2×C4—2141no—
281C2×C4—2142no—
282C2×C4—2142no—
283C2×C4—2141no—
284C2×C4—2142no—
285C2×C4—2141no—
286C2×C4—2142no—
287C2×C4—2148no—
288C2×C4D4×D10712yes—
289C2×C4D4×D10712yes—
290C2×C4Q8×D10712yes—
291C2×C4#1313381812yes—
292C22—1074no—
293C22—4281no—
294C22—4281no—
295C22—4281no—
296C22—4281no—
297C22—2142no—
298C22—2142no—
299C22—2142no—
300C22—2142no—
301C22—2142no—
302C22—2142no—
303C22—2142no—
304C22—2142no—
305C22—2142no—
306C22C2×D4×D10711yescharacteristic
307C22C2×D4×D10711yescharacteristic
308C22C2×Q8×D10711yescharacteristic
309C22#41×D10711yescharacteristic
310C22C2×#1313381811yescharacteristic
311C22#1681215511yescharacteristic
312C22#1681215711yescharacteristic
313C4—22no—
314C4—41no—
315C4—42no—
316C4—42no—
317C4—4282no—
318C4—4281no—
319C4—2144no—
320C4—2142no—
321C4#1681208214yes—
322C2—4282no—
323C2—2144no—
324C2C2×#1681208211yescharacteristic
325C2C22⋊Q8×D10711yescharacteristic
326C2#2029320711yescharacteristic
327C2#2029327011yescharacteristic
328C2#2029329211yescharacteristic
329C2#2029331811yescharacteristic
330C2#2029333711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#391×C107C2
prime radicalC22×D428C22
hypercenter#391D107
hypocenterC107#2417
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#16812085C2semidirect
2. C2×#16812085C2semidirect
4. C2×C4⋊C4×D107C2semidirect
5. #20292561C2semidirect
6. #20292551C2semidirect
7. #20292525C2semidirect
9. #20292486C2semidirect
10. #20292484C2semidirect
11. #391×C107C2semidirect
43. C2×D428Q8semidirect
59. C4⋊Dic107D4semidirect
146. C2×D107C22⋊Q8semidirect
166. C107#2417semidirect
173. #391D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120852
C2×C4×D4282
C2×C4⋊C4×D1072
#202925612
#202925512
#202925252
#202925052
#202924862
#202924842
#391×C1072
#2417107