Finite Group Databasebeta

← #28348332

Subgroups — #28348332

Order 13696.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293335C211yesmaximal, characteristic
2#20293290C211yesmaximal, characteristic
3#20292976C211yesmaximal, characteristic, complemented
4#20292964C211yesmaximal, characteristic, complemented
5#20292960C211yesmaximal, characteristic, complemented
6#20292921C211yesmaximal, characteristic, complemented
7#20292867C211yesmaximal, characteristic, complemented
8#20292835C211yesmaximal, characteristic, complemented
9#20292782C211yesmaximal, characteristic, complemented
10#20292781C211yesmaximal, characteristic, complemented
11#20292719C211yesmaximal, characteristic, complemented
12C8×C22⋊D107C211yesmaximal, characteristic
13C2×#16812034C211yesmaximal, characteristic, complemented
14#103×D107C211yesmaximal, characteristic, complemented
15#467×C107C211yesmaximal, characteristic, complemented
16C22×Dic214C2211yescharacteristic
17C2×Q8×D107C2211yescharacteristic
18C2×#13133808C2211yescharacteristic
19C2×#13133808C2211yescharacteristic
20C2×C8×D107C2211yescharacteristic
21#16812108—21no—
22#16812108C2211yescharacteristic
23#16812107C2211yescharacteristic
24#16812099—21no—
25#16812089—21no—
26#16812089C2211yescharacteristic
27#16812087—21no—
28#16812086—21no—
29#16812086—21no—
30#16812085C2211yescharacteristic
31Q8×Dic107C2211yescharacteristic
32#16812074C2211yescharacteristic
33C4×C22⋊D107C2211yescharacteristic
34(C22⋊D107)⋊C4—21no—
35(C22⋊D107)⋊C4—21no—
36C2×C4⋊Dic107C2211yescharacteristic
37C4⋊C4×D107C2211yescharacteristic
38#16812042C2211yescharacteristic
39#16812042C2211yescharacteristic
40Dic107⋊C8C2211yescharacteristic
41#16812034C2211yescharacteristic
42#16812034C2211yescharacteristic
43#16812034C2211yescharacteristic
44#16812034C2211yescharacteristic
45#16812033C2211yescharacteristic
46#16812033C2211yescharacteristic
47#16812032C2211yescharacteristic
48#16812032C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×Q16×C107C2211yescharacteristic
53C22×C8×C107C2211yescharacteristic
54C22⋊Q8×C107C2211yescharacteristic
55C22⋊Q8×C107C2211yescharacteristic
56#107×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#103×C107C2211yescharacteristic
59C2×Dic214—22no—
60C2×Dic214—21no—
61C2×Dic214—21no—
62C2×Dic214—21no—
63C2×Dic214—41no—
64C2×Dic214C2311yescharacteristic
65C2×Dic214C2311yescharacteristic
66Q8×D107—22no—
67Q8×D107—41no—
68C2×C22⋊D107D411yescharacteristic
69C22×Dic107—21no—
70C22×Dic107—21no—
71C2×C4×D107—21no—
72C2×C4×D107—21no—
73C2×C4×D107C2311yescharacteristic
74#13133808—22no—
75#13133808—41no—
76C2×C107⋊C8C2311yescharacteristic
77C2×C107⋊C8C2311yescharacteristic
78C8×D107—41no—
79C4⋊Dic107—21no—
80C4⋊Dic107—21no—
81C4⋊Dic107—21no—
82C4⋊Dic107C2311yescharacteristic
83C4⋊Dic107C2311yescharacteristic
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86Dic107⋊C4—21no—
87Dic107⋊C4—21no—
88Dic107⋊C4—21no—
89Dic107⋊C4—21no—
90Dic107⋊C4D411yescharacteristic
91C22⋊Dic107D411yescharacteristic
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4D411yescharacteristic
95C4×Dic107—21no—
96C4×Dic107—21no—
97C4×Dic107C2311yescharacteristic
98C2×Q8×C107C2311yescharacteristic
99C2×Q8×C107C2311yescharacteristic
100C22×C4×C107C2311yescharacteristic
101Q16×C107—41no—
102C2×C8×C107—41no—
103C2×C8×C107C2311yescharacteristic
104C2×C8×C107C2311yescharacteristic
105C4⋊C4×C107—21no—
106C4⋊C4×C107—21no—
107C4⋊C4×C107C2311yescharacteristic
108C4⋊C4×C107C2311yescharacteristic
109C22⋊C4×C107—21no—
110C22⋊C4×C107—21no—
111C23×C107C2×D411yescharacteristic
112C22×D107C2×D411yescharacteristic
113Q8×C107—21no—
114Q8×C107—21no—
115Q8×C107—21no—
116Q8×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107C2×D411yescharacteristic
124C2×C4×C107C2411yescharacteristic
125C22⋊D107Q1614yescomplemented
126C2×Dic107—21no—
127C2×Dic107—21no—
128C2×Dic107—21no—
129C2×Dic107—21no—
130C2×Dic107—42no—
131C2×Dic107—81no—
132C2×Dic107C2×D411yescharacteristic
133C2×Dic107C2×D411yescharacteristic
134C2×Dic107C2×D411yescharacteristic
135C4×D107—21no—
136C4×D107—21no—
137C4×D107—42no—
138C4×D107—81no—
139Dic214—21no—
140Dic214—21no—
141Dic214—21no—
142Dic214—42no—
143Dic214—42no—
144Dic214—81no—
145Dic214#4112yes—
146C8×C107—21no—
147C8×C107—21no—
148C107⋊C8—21no—
149C107⋊C8—21no—
150C22×C107—21no—
151C22×C107—21no—
152C22×C107C2×Q1612yes—
153C22×C107C22×D411yescharacteristic
154C2×D107—21no—
155C2×D107—21no—
156C2×D107C2×Q1612yes—
157C4×C107—21no—
158C4×C107—41no—
159C4×C107—41no—
160C4×C107—41no—
161C4×C107—41no—
162C4×C107C2×#4111yescharacteristic
163C4×C107H(5,𝔽2)11yescharacteristic
164Dic107—22no—
165Dic107—21no—
166Dic107—21no—
167Dic107—41no—
168Dic107—41no—
169Dic107—41no—
170Dic107C2×Q1612yes—
171C2×C107—22no—
172C2×C107#54611yescharacteristic
173C2×C107C22×Q1611yescharacteristic
174C2×C107#57711yescharacteristic
175D107—22no—
176#3299—1071nomaximal, Sylow
177C107#329911yescharacteristic, complemented, Sylow
178#554—1071no—
179D4×Q8—1071no—
180#487—1071no—
181#484—1071no—
182#484—1071no—
183#477—1071no—
184#467—1071no—
185#467D10711yescharacteristic, complemented
186#462—1071no—
187#451—1071no—
188#451—1071no—
189C4×Q16—1071no—
190C8×D4—1071no—
191C2×#103—1071no—
192C2×#103—1071no—
193C22×Q8—1071no—
194C22×Q8—1071no—
195C2×Q16—1071no—
196C2×Q16—1071no—
197C2×Q16C2×D10711yescharacteristic
198C22×C8—1071no—
199C22×C8C2×D10711yescharacteristic
200C4⋊Q8—1071no—
201C4⋊Q8—1071no—
202C4⋊Q8—2141no—
203#125—2141no—
204C22⋊Q8—1071no—
205C22⋊Q8—1071no—
206C22⋊Q8—2141no—
207C22⋊Q8—2141no—
208C22⋊Q8—2141no—
209C22⋊Q8—2141no—
210C22⋊Q8C2×D10711yescharacteristic
211C22⋊Q8C2×D10711yescharacteristic
212C4×Q8—1071no—
213C4×Q8—1071no—
214C4×D4—1071no—
215C4×D4—2141no—
216C4×D4—2141no—
217C2×C4⋊C4—1071no—
218C2×C4⋊C4—1071no—
219#107—1071no—
220#107—1071no—
221#107C2×D10711yescharacteristic
222C4⋊C8—1071no—
223#103—1071no—
224#103—1071no—
225#103—1071no—
226#103—1071no—
227#103—1071no—
228#103—1071no—
229#103—1071no—
230#103—1071no—
231#103C2×D10711yescharacteristic
232#103C2×D10711yescharacteristic
233C22⋊C8—1071no—
234C22⋊C8—1071no—
235C4×C8—1071no—
236C2×Q8—1071no—
237C2×Q8—1071no—
238C2×Q8—4281no—
239C2×Q8—4281no—
240C2×Q8—2141no—
241C2×Q8—2141no—
242C2×Q8—2142no—
243C2×Q8—2142no—
244C2×Q8—2141no—
245C2×Q8C22×D10711yescharacteristic
246C2×Q8C22×D10711yescharacteristic
247C2×D4—1071no—
248C22×C4—1071no—
249C22×C4—2141no—
250C22×C4—2141no—
251C22×C4—2141no—
252C22×C4—2141no—
253C22×C4C22×D10711yescharacteristic
254Q16—41no—
255Q16—4281no—
256Q16—2142no—
257C2×C8—41no—
258C2×C8—1071no—
259C2×C8—1071no—
260C2×C8—4281no—
261C2×C8C22×D10711yescharacteristic
262C2×C8C22×D10711yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—1071no—
266C4⋊C4—1071no—
267C4⋊C4—1071no—
268C4⋊C4—2141no—
269C4⋊C4—2141no—
270C4⋊C4—2141no—
271C4⋊C4—2141no—
272C4⋊C4—2141no—
273C4⋊C4—2141no—
274C4⋊C4—2141no—
275C4⋊C4—2141no—
276C4⋊C4—2141no—
277C4⋊C4C22×D10711yescharacteristic
278C4⋊C4C22×D10711yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—1071no—
282C22⋊C4—1071no—
283C22⋊C4—2141no—
284C22⋊C4—2141no—
285C42—1071no—
286C42—2141no—
287C42—2141no—
288C23—1071no—
289C23D4×D10711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—8561no—
295Q8—1072no—
296Q8—4282no—
297Q8—4282no—
298Q8—2141no—
299Q8—2141no—
300Q8—2141no—
301D4—1074no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—8561no—
309C2×C4—8561no—
310C2×C4—1071no—
311C2×C4—1071no—
312C2×C4—1071no—
313C2×C4—4282no—
314C2×C4—4282no—
315C2×C4—2141no—
316C2×C4—2141no—
317C2×C4—2141no—
318C2×C4—2141no—
319C2×C4—2141no—
320C2×C4—2141no—
321C2×C4D4×D10711yescharacteristic
322C2×C4C23×D10711yescharacteristic
323C8—21no—
324C8—21no—
325C8—2141no—
326C8—2141no—
327C22—21no—
328C22—21no—
329C22—1072no—
330C22—2141no—
331C22—2141no—
332C22Q16×D10712yes—
333C22C2×D4×D10711yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—1072no—
340C4—4281no—
341C4—4281no—
342C4—4281no—
343C4—2141no—
344C4—2141no—
345C4—2142no—
346C4#41×D10711yescharacteristic
347C4#1681215411yescharacteristic
348C2—22no—
349C2—2142no—
350C2#2029326611yescharacteristic
351C2C2×Q16×D10711yescharacteristic
352C2#2029343411yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#467×C107C2
socleC22×C107C22×D4
prime radicalC2×C22⋊D107D4
hypercenter#467D107
hypocenterC107#3299
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
3. #20292976C2semidirect
4. #20292964C2semidirect
5. #20292960C2semidirect
6. #20292921C2semidirect
7. #20292867C2semidirect
8. #20292835C2semidirect
9. #20292782C2semidirect
10. #20292781C2semidirect
11. #20292719C2semidirect
13. C2×#16812034C2semidirect
14. #103×D107C2semidirect
15. #467×C107C2semidirect
125. C22⋊D107Q16semidirect
177. C107#3299semidirect
185. #467D107semidirect

Maximal subgroups

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

subgroupindex
#202933352
#202932902
#202929762
#202929642
#202929602
#202929212
#202928672
#202928352
#202927822
#202927812
#202927192
C8×C22⋊D1072
C2×#168120342
#103×D1072
#467×C1072
#3299107