Finite Group Databasebeta

← #28339808

Subgroups — #28339808

Order 13696.

908 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133819C211yesmaximal, characteristic, complemented
2C2×#16812083C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D107C211yesmaximal, characteristic, complemented
4#20292544C212yesmaximal, complemented
5#20292536C212yesmaximal, complemented
6#20292534C211yesmaximal, characteristic, complemented
7#20292528C211yesmaximal, characteristic, complemented
8#20292503C212yesmaximal, complemented
9#20292498C212yesmaximal, complemented
10#20292484C211yesmaximal, characteristic, complemented
11C2×C22⋊Q8×C107C211yesmaximal, characteristic, complemented
12C2×#13133819—24no—
13C2×#13133819—24no—
14C2×#13133819—24no—
15C22×D428C2212yes—
16C22×D428C2211yescharacteristic
17C22×C22⋊D107C2211yescharacteristic
18C22×C22⋊D107C2212yes—
19C22×C4×D107C2212yes—
20C22×C4×D107C2211yescharacteristic
21#16812083—24no—
22C4×C22⋊D107—24no—
23C2×C4⋊Dic107C2211yescharacteristic
24C2×Dic107⋊C4C2211yescharacteristic
25C2×C22⋊Dic107C2211yescharacteristic
26C2×(C2×D107)⋊C4C2212yes—
27C2×(C2×D107)⋊C4C2212yes—
28C2×(C2×D107)⋊C4C2211yescharacteristic
29C2×(C2×D107)⋊C4C2212yes—
30C2×(C2×D107)⋊C4C2212yes—
31C2×C4×Dic107C2211yescharacteristic
32C107⋊#95C2212yes—
33C107⋊#95C2212yes—
34C107⋊#95C2212yes—
35C22×Q8×C107C2211yescharacteristic
36C23×C4×C107C2211yescharacteristic
37C22⋊Q8×C107—24no—
38C2×C4⋊C4×C107C2211yescharacteristic
39C2×C4⋊C4×C107C2212yes—
40C2×C22⋊C4×C107C2212yes—
41C23×D107C2312yes—
42C23×D107C2311yescharacteristic
43#13133819—232no—
44C2×D428—24no—
45C2×D428—24no—
46C2×D428—22no—
47C2×D428—24no—
48C2×D428—22no—
49C2×D428D414yescomplemented
50C2×C22⋊D107—22no—
51C2×C22⋊D107—22no—
52C2×C22⋊D107—24no—
53C2×C22⋊D107—24no—
54C2×C22⋊D107—22no—
55C2×C22⋊D107—42no—
56C22×Dic107C2311yescharacteristic
57C22×Dic107C2312yes—
58C22×Dic107C2311yescharacteristic
59C22×Dic107C2311yescharacteristic
60C2×C4×D107—24no—
61C2×C4×D107—22no—
62C2×C4×D107—24no—
63C2×C4×D107—22no—
64C2×C4×D107—24no—
65C2×C4×D107D414yescomplemented
66C4⋊Dic107—22no—
67Dic107⋊C4—22no—
68C22⋊Dic107—22no—
69(C2×D107)⋊C4—22no—
70(C2×D107)⋊C4—24no—
71(C2×D107)⋊C4—24no—
72(C2×D107)⋊C4—42no—
73(C2×D107)⋊C4—42no—
74C4×Dic107—22no—
75C24×C107C2311yescharacteristic
76C2×Q8×C107—22no—
77C2×Q8×C107—22no—
78C2×Q8×C107D414yescomplemented
79C22×C4×C107—22no—
80C22×C4×C107—22no—
81C22×C4×C107—22no—
82C22×C4×C107C2311yescharacteristic
83C22×C4×C107C2312yes—
84C22×C4×C107C2312yes—
85C22×C4×C107C2311yescharacteristic
86C4⋊C4×C107—22no—
87C4⋊C4×C107—24no—
88C22⋊C4×C107—24no—
89C23×C107—21no—
90C23×C107—21no—
91C23×C107—21no—
92C23×C107—21no—
93C23×C107—21no—
94C23×C107—21no—
95C23×C107#4112yescomplemented
96C23×C107C2411yescharacteristic
97C22×D107—22no—
98C22×D107—21no—
99C22×D107—22no—
100C22×D107—21no—
101C22×D107—22no—
102C22×D107—21no—
103C22×D107—21no—
104C22×D107—22no—
105C22×D107—21no—
106C22×D107—22no—
107C22×D107—21no—
108C22×D107—22no—
109C22×D107C2×D414yes—
110C22×D107C2×D412yes—
111Q8×C107—28no—
112C2×C4×C107—22no—
113C2×C4×C107—22no—
114C2×C4×C107—22no—
115C2×C4×C107—21no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—22no—
119C2×C4×C107—22no—
120C2×C4×C107—44no—
121C2×C4×C107C2×D412yes—
122C2×C4×C107C2×D412yes—
123C2×C4×C107C2×D414yes—
124C2×C4×C107C2×D412yes—
125C22⋊D107—28no—
126C22⋊D107—48no—
127C2×Dic107—21no—
128C2×Dic107—21no—
129C2×Dic107—21no—
130C2×Dic107—21no—
131C2×Dic107—21no—
132C2×Dic107—22no—
133C2×Dic107—22no—
134C2×Dic107—21no—
135C2×Dic107—22no—
136C2×Dic107—21no—
137C2×Dic107C2×D412yes—
138C2×Dic107#4112yescomplemented
139D428—216no—
140D428—28no—
141C4×D107—216no—
142C4×D107—28no—
143C22×C107—22no—
144C22×C107—22no—
145C22×C107—22no—
146C22×C107—41no—
147C22×C107—41no—
148C22×C107—41no—
149C22×C107—41no—
150C22×C107C22×D411yescharacteristic
151C22×C107C22×D411yescharacteristic
152C22×C107C22×D411yescharacteristic
153C22×C107C2×#4111yescharacteristic
154C22×C107C2×#4111yescharacteristic
155C22×C107H(5,𝔽2)11yescharacteristic
156C22×C1072−1+411yescharacteristic
157C2×D107—24no—
158C2×D107—22no—
159C2×D107—22no—
160C2×D107—22no—
161C2×D107—24no—
162C2×D107—24no—
163C2×D107—42no—
164C2×D107—42no—
165C2×D107—41no—
166C2×D107—41no—
167C2×D107—41no—
168C2×D107—42no—
169C2×D107—42no—
170C2×D107—41no—
171C4×C107—24no—
172C4×C107—41no—
173C4×C107—42no—
174C4×C107#12014yes—
175Dic107—22no—
176Dic107—22no—
177Dic107—41no—
178Dic107—42no—
179C2×C107—42no—
180C2×C107C2×#12011yescharacteristic
181C2×C107#52511yescharacteristic
182C2×C107#53711yescharacteristic
183C2×C107#54611yescharacteristic
184C2×C107#54711yescharacteristic
185C2×C107#54811yescharacteristic
186C2×C107#55011yescharacteristic
187D107—42no—
188D107—44no—
189#2407—1071nomaximal, Sylow
190C107#240711yescharacteristic, complemented, Sylow
191C22×#41—1071no—
192C2×C22⋊Q8D10711yescharacteristic, complemented
193C2×#121—1071no—
194C2×C4×D4—1071no—
195#394—1072no—
196#392—1072no—
197#391—1071no—
198#390—1071no—
199#386—1072no—
200#385—1072no—
201#380—1071no—
202C2×#41—2144no—
203C2×#41—2144no—
204C2×#41—2144no—
205C22×Q8C2×D10711yescharacteristic
206C22×D4—1072no—
207C22×D4—1071no—
208C22×D4—1071no—
209C22×D4—1072no—
210C23×C4—1071no—
211C23×C4—1072no—
212C23×C4C2×D10711yescharacteristic
213C22⋊Q8—24no—
214#121—2144no—
215C4×D4—2144no—
216C2×C4⋊C4—1071no—
217C2×C4⋊C4—1071no—
218C2×C4⋊C4C2×D10711yescharacteristic
219C2×C4⋊C4C2×D10712yes—
220C2×C22⋊C4—1072no—
221C2×C22⋊C4—1072no—
222C2×C22⋊C4—1071no—
223C2×C22⋊C4—1071no—
224C2×C22⋊C4—1072no—
225C2×C22⋊C4—1072no—
226C2×C22⋊C4C2×D10712yes—
227C2×C42—1071no—
228#95—1072no—
229#95—1072no—
230#95—1072no—
231C24—1071no—
232C24—1072no—
233C24C22×D10711yescharacteristic
234#41—21432no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8C22⋊D10714yescomplemented
238C2×D4—1074no—
239C2×D4—4282no—
240C2×D4—2144no—
241C2×D4—2144no—
242C2×D4—2142no—
243C2×D4—2144no—
244C2×D4—2144no—
245C2×D4—2142no—
246C2×D4—2142no—
247C2×D4—2142no—
248C2×D4—2144no—
249C2×D4—2142no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—1071no—
254C22×C4—1071no—
255C22×C4—1074no—
256C22×C4—1072no—
257C22×C4—1071no—
258C22×C4—2144no—
259C22×C4—2144no—
260C22×C4—2142no—
261C22×C4—2144no—
262C22×C4—2142no—
263C22×C4C22×D10712yes—
264C22×C4C22×D10711yescharacteristic
265C22×C4C22×D10712yes—
266C22×C4C22×D10711yescharacteristic
267C4⋊C4—22no—
268C4⋊C4—24no—
269C4⋊C4—2142no—
270C4⋊C4—2142no—
271C22⋊C4—24no—
272C22⋊C4—4282no—
273C22⋊C4—4282no—
274C22⋊C4—2144no—
275C22⋊C4—2142no—
276C22⋊C4—2144no—
277C22⋊C4—2142no—
278C42—2142no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—1072no—
286C23—1074no—
287C23—2142no—
288C23—2141no—
289C23—2142no—
290C23—2141no—
291C23—2142no—
292C23—2141no—
293C23—2142no—
294C23—2141no—
295C23—2141no—
296C23—2142no—
297C23—2142no—
298C23—2141no—
299C23#1313381912yescomplemented
300C23C23×D10711yescharacteristic
301Q8—28no—
302D4—4288no—
303D4—21416no—
304D4—2148no—
305D4—2148no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—44no—
315C2×C4—1072no—
316C2×C4—1072no—
317C2×C4—21416no—
318C2×C4—2141no—
319C2×C4—2142no—
320C2×C4—2142no—
321C2×C4—2141no—
322C2×C4—2142no—
323C2×C4—2141no—
324C2×C4—2141no—
325C2×C4—2141no—
326C2×C4—2141no—
327C2×C4—2141no—
328C2×C4—2148no—
329C2×C4D4×D10712yes—
330C2×C4D4×D10712yes—
331C2×C4C2×C22⋊D10714yes—
332C2×C4C2×C22⋊D10712yes—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—4282no—
341C22—4282no—
342C22—4281no—
343C22—4281no—
344C22—4282no—
345C22—4281no—
346C22—4282no—
347C22—4281no—
348C22—2142no—
349C22—2142no—
350C22—2144no—
351C22—2144no—
352C22—2144no—
353C22—2142no—
354C22C2×D4×D10711yescharacteristic
355C22C2×D4×D10711yescharacteristic
356C22C22×C22⋊D10711yescharacteristic
357C22#41×D10711yescharacteristic
358C22C2×#1313381911yescharacteristic
359C22#1681215511yescharacteristic
360C22#1681215711yescharacteristic
361C4—24no—
362C4—42no—
363C4—41no—
364C4—4281no—
365C4—4282no—
366C4—2142no—
367C4—2142no—
368C4#1681207714yes—
369C2—42no—
370C2—4284no—
371C2—4282no—
372C2C2×#1681207711yescharacteristic
373C2#2029314011yescharacteristic
374C2#2029320711yescharacteristic
375C2#2029326811yescharacteristic
376C2#2029327911yescharacteristic
377C2#2029328811yescharacteristic
378C2#2029329611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×C22⋊Q8×C107C2
hypercenterC2×C22⋊Q8D107
hypocenterC107#2407
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#13133819C2semidirect
2. C2×#16812083C2semidirect
3. C2×C4×C22⋊D107C2semidirect
4. #20292544C2semidirect
5. #20292536C2semidirect
6. #20292534C2semidirect
7. #20292528C2semidirect
8. #20292503C2semidirect
9. #20292498C2semidirect
10. #20292484C2semidirect
11. C2×C22⋊Q8×C107C2semidirect
49. C2×D428D4semidirect
65. C2×C4×D107D4semidirect
78. C2×Q8×C107D4semidirect
95. C23×C107#41semidirect
138. C2×Dic107#41semidirect
190. C107#2407semidirect
192. C2×C22⋊Q8D107semidirect
237. C2×Q8C22⋊D107semidirect
299. C23#13133819semidirect

Maximal subgroups

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

subgroupindex
C22×#131338192
C2×#168120832
C2×C4×C22⋊D1072
#202925442
#202925362
#202925342
#202925282
#202925032
#202924982
#202924842
C2×C22⋊Q8×C1072
#2407107