Finite Group Databasebeta

← #28339910

Subgroups — #28339910

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×#16812089C211yesmaximal, characteristic, complemented
2C2×#16812089C211yesmaximal, characteristic, complemented
3C2×C4×D428C211yesmaximal, characteristic
4C2×C4⋊C4×D107C211yesmaximal, characteristic, complemented
5#20292559C212yesmaximal, complemented
6#20292549C212yesmaximal, complemented
7#20292533C211yesmaximal, characteristic, complemented
8#20292505C212yesmaximal
9#20292484C212yesmaximal, complemented
10#20292484C211yesmaximal, characteristic, complemented
11#389×C107C211yesmaximal, characteristic, complemented
12C22×Dic214C2211yescharacteristic
13C22×D428C2211yescharacteristic
14C22×C4×D107C2211yescharacteristic
15C22×C4×D107C2212yes—
16C22×C4×D107C2211yescharacteristic
17#16812089—24no—
18#16812089—24no—
19C4×D428—24no—
20C2×C4⋊Dic107C2212yes—
21C2×C4⋊Dic107C2212yes—
22C2×C4⋊Dic107C2211yescharacteristic
23C2×C4⋊Dic107C2211yescharacteristic
24C2×Dic107⋊C4C2212yes—
25C4⋊C4×D107—24no—
26C2×(C2×D107)⋊C4C2212yes—
27C2×(C2×D107)⋊C4C2212yes—
28C2×(C2×D107)⋊C4C2211yescharacteristic
29C2×(C2×D107)⋊C4C2211yescharacteristic
30C2×(C2×D107)⋊C4C2212yes—
31C107⋊#95C2212yes—
32C107⋊#95C2212yes—
33C107⋊#95C2212yes—
34C2×C4⋊C4×C107C2211yescharacteristic
35C2×C4⋊C4×C107C2212yes—
36C2×C4⋊C4×C107C2211yescharacteristic
37C2×C42×C107C2211yescharacteristic
38#95×C107C2212yes—
39C23×D107C2311yescharacteristic
40C23×D107C2311yescharacteristic
41C2×Dic214—22no—
42C2×Dic214—22no—
43C2×Dic214—22no—
44C2×D428—22no—
45C2×D428—22no—
46C2×D428Q814yescomplemented
47C22×Dic107C2312yes—
48C22×Dic107C2312yes—
49C22×Dic107C2311yescharacteristic
50C22×Dic107C2311yescharacteristic
51C2×C4×D107—22no—
52C2×C4×D107—22no—
53C2×C4×D107—24no—
54C2×C4×D107—24no—
55C2×C4×D107—22no—
56C2×C4×D107—22no—
57C2×C4×D107—22no—
58C2×C4×D107—42no—
59C2×C4×D107D414yes—
60C4⋊Dic107—22no—
61C4⋊Dic107—24no—
62C4⋊Dic107—22no—
63C4⋊Dic107—24no—
64Dic107⋊C4—24no—
65(C2×D107)⋊C4—22no—
66(C2×D107)⋊C4—24no—
67(C2×D107)⋊C4—24no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4—42no—
70C22×C4×C107C2312yes—
71C22×C4×C107C2311yescharacteristic
72C22×C4×C107C2312yes—
73C22×C4×C107C2311yescharacteristic
74C22×C4×C107C2311yescharacteristic
75C4⋊C4×C107—22no—
76C4⋊C4×C107—42no—
77C4⋊C4×C107D414yescomplemented
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—
94C2×C4×C107—21no—
95C2×C4×C107—22no—
96C2×C4×C107—21no—
97C2×C4×C107—22no—
98C2×C4×C107—21no—
99C2×C4×C107—22no—
100C2×C4×C107—21no—
101C2×C4×C107—22no—
102C2×C4×C107—22no—
103C2×C4×C107—21no—
104C2×C4×C107C2×D412yes—
105C2×C4×C107C2×D414yes—
106C2×C4×C107C2×D412yes—
107C2×C4×C107C2×Q812yes—
108C2×C4×C107#4112yes—
109C2×Dic107—21no—
110C2×Dic107—21no—
111C2×Dic107—21no—
112C2×Dic107—22no—
113C2×Dic107—22no—
114C2×Dic107—22no—
115C2×Dic107—21no—
116C2×Dic107—22no—
117C2×Dic107—21no—
118C2×Dic107—22no—
119C2×Dic107—22no—
120C2×Dic107C2×D412yes—
121D428—28no—
122C4×D107—28no—
123C4×D107—48no—
124C4×D107—44no—
125Dic214—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—24no—
149C4×C107—41no—
150C4×C107—42no—
151C4×C107#12114yes—
152Dic107—22no—
153Dic107—42no—
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#391—1071no—
174#389D10711yescharacteristic, complemented
175#386—1072no—
176#380—1071no—
177#380—1072no—
178C22×Q8—1071no—
179C22×D4—1071no—
180C23×C4—1071no—
181C23×C4—1072no—
182C23×C4—1071no—
183C22⋊Q8—2144no—
184C22⋊Q8—2144no—
185C4×D4—2144no—
186C2×C4⋊C4—1072no—
187C2×C4⋊C4—1071no—
188C2×C4⋊C4—1071no—
189C2×C4⋊C4—1072no—
190C2×C4⋊C4—1072no—
191C2×C4⋊C4—2144no—
192C2×C4⋊C4C2×D10711yescharacteristic
193C2×C4⋊C4C2×D10712yes—
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#95—1072no—
204#95C2×D10712yes—
205C24—1071no—
206C24—1071no—
207C2×Q8—2142no—
208C2×Q8—2142no—
209C2×Q8—2142no—
210C2×D4—1074no—
211C2×D4—2142no—
212C2×D4—2142no—
213C22×C4—1071no—
214C22×C4—1074no—
215C22×C4—1072no—
216C22×C4—1072no—
217C22×C4—1071no—
218C22×C4—4282no—
219C22×C4—2144no—
220C22×C4—2142no—
221C22×C4—2144no—
222C22×C4—2142no—
223C22×C4—2142no—
224C22×C4—2142no—
225C22×C4—2142no—
226C22×C4C22×D10711yescharacteristic
227C22×C4C22×D10712yes—
228C22×C4C22×D10712yes—
229C22×C4C22×D10711yescharacteristic
230C22×C4C22×D10711yescharacteristic
231C4⋊C4—22no—
232C4⋊C4—42no—
233C4⋊C4—2142no—
234C4⋊C4—2144no—
235C4⋊C4—2144no—
236C4⋊C4—2142no—
237C4⋊C4—2144no—
238C4⋊C4D42814yescomplemented
239C22⋊C4—4282no—
240C22⋊C4—2142no—
241C22⋊C4—2142no—
242C22⋊C4—2144no—
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—4284no—
261D4—2148no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—1072no—
273C2×C4—4288no—
274C2×C4—4284no—
275C2×C4—2141no—
276C2×C4—2142no—
277C2×C4—2142no—
278C2×C4—2141no—
279C2×C4—2142no—
280C2×C4—2142no—
281C2×C4—2141no—
282C2×C4—2142no—
283C2×C4—2142no—
284C2×C4—2141no—
285C2×C4—2141no—
286C2×C4—2148no—
287C2×C4D4×D10712yes—
288C2×C4C2×D42814yes—
289C2×C4C2×D42812yes—
290C2×C4Q8×D10712yes—
291C2×C4#1313381912yes—
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
307C22C22×D42811yescharacteristic
308C22C2×Q8×D10711yescharacteristic
309C22#41×D10711yescharacteristic
310C22C2×#1313381911yescharacteristic
311C22#1681215411yescharacteristic
312C22#1681215611yescharacteristic
313C4—22no—
314C4—24no—
315C4—41no—
316C4—42no—
317C4—4281no—
318C4—4282no—
319C4—4282no—
320C4—2142no—
321C4#1681208414yes—
322C2—4282no—
323C2—2144no—
324C2C2×#1681208411yescharacteristic
325C2C22⋊Q8×D10711yescharacteristic
326C2#2029320611yescharacteristic
327C2#2029327211yescharacteristic
328C2Q8×D42811yescharacteristic
329C2#2029331511yescharacteristic
330C2#2029333811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#389×C107C2
prime radicalC22×D428C22
hypercenter#389D107
hypocenterC107#2417
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C2×#16812089C2semidirect
2. C2×#16812089C2semidirect
4. C2×C4⋊C4×D107C2semidirect
5. #20292559C2semidirect
6. #20292549C2semidirect
7. #20292533C2semidirect
9. #20292484C2semidirect
10. #20292484C2semidirect
11. #389×C107C2semidirect
46. C2×D428Q8semidirect
77. C4⋊C4×C107D4semidirect
146. C2×D107C22⋊Q8semidirect
166. C107#2417semidirect
174. #389D107semidirect
238. C4⋊C4D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168120892
C2×C4×D4282
C2×C4⋊C4×D1072
#202925592
#202925492
#202925332
#202925052
#202924842
#389×C1072
#2417107