Finite Group Databasebeta

← #28346480

Subgroups — #28346480

Order 13696.

396 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293163C211yesmaximal, characteristic, complemented
2#41×Dic107C211yesmaximal, characteristic, complemented
3#20292965C211yesmaximal, characteristic, complemented
4#20292952C211yesmaximal, characteristic, complemented
5#20292932C211yesmaximal, characteristic, complemented
6#20292925C211yesmaximal, characteristic, complemented
7#20292851C211yesmaximal, characteristic, complemented
8#20292844C211yesmaximal, characteristic, complemented
9#20292772C211yesmaximal, characteristic
10Q16×Dic107C211yesmaximal, characteristic, complemented
11SD16×Dic107C211yesmaximal, characteristic, complemented
12#20292712C211yesmaximal, characteristic, complemented
13#20292707C211yesmaximal, characteristic, complemented
14#20292677C211yesmaximal, characteristic, complemented
15#452×C107C211yesmaximal, characteristic, complemented
16C2×#13133805C2211yescharacteristic
17C2×#13133803C2211yescharacteristic
18#16812104—21no—
19#16812100C2211yescharacteristic
20#16812098C2211yescharacteristic
21#16812093—21no—
22#16812083C2211yescharacteristic
23Q8×Dic107C2211yescharacteristic
24Q8×Dic107C2211yescharacteristic
25Q8×Dic107C2211yescharacteristic
26D4×Dic107—21no—
27D4×Dic107—21no—
28D4×Dic107C2211yescharacteristic
29D4×Dic107C2211yescharacteristic
30#16812071C2211yescharacteristic
31(C22⋊D107)⋊C4—21no—
32#16812068—21no—
33#16812068C2211yescharacteristic
34#16812067—21no—
35C2×C4×Dic107—21no—
36C2×C4×Dic107C2211yescharacteristic
37#16812043C2211yescharacteristic
38#16812042C2211yescharacteristic
39#16812041C2211yescharacteristic
40#16812040C2211yescharacteristic
41Dic107⋊C8C2211yescharacteristic
42Dic107⋊C8C2211yescharacteristic
43#16812033C2211yescharacteristic
44#16812033C2211yescharacteristic
45#16812033C2211yescharacteristic
46#16812031C2211yescharacteristic
47#16812031C2211yescharacteristic
48#16812030C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic
50C8×Dic107C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×#41×C107C2211yescharacteristic
53C2×Q16×C107C2211yescharacteristic
54C2×SD16×C107C2211yescharacteristic
55C22⋊Q8×C107C2211yescharacteristic
56#103×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58C22⋊C8×C107C2211yescharacteristic
59C2×D428C2311yescharacteristic
60C2×C22⋊D107—21no—
61C22×Dic107—21no—
62C22×Dic107—21no—
63C22×Dic107D411yescharacteristic
64C22×Dic107D411yescharacteristic
65C2×C4×D107—21no—
66#13133805—22no—
67#13133803—22no—
68C2×C107⋊C8C2311yescharacteristic
69C2×C107⋊C8C2311yescharacteristic
70C4⋊Dic107—21no—
71C4⋊Dic107—21no—
72C4⋊Dic107—21no—
73C4⋊Dic107—21no—
74C4⋊Dic107C2311yescharacteristic
75C4⋊Dic107C2311yescharacteristic
76C4⋊Dic107C2311yescharacteristic
77Dic107⋊C4—21no—
78C22⋊Dic107—21no—
79C22⋊Dic107—21no—
80C22⋊Dic107—21no—
81(C2×D107)⋊C4—21no—
82(C2×D107)⋊C4—21no—
83(C2×D107)⋊C4—21no—
84(C2×D107)⋊C4—21no—
85C4×Dic107—21no—
86C4×Dic107—21no—
87C4×Dic107—21no—
88C4×Dic107—21no—
89C4×Dic107D411yescharacteristic
90C4×Dic107D411yescharacteristic
91C4×Dic107C2311yescharacteristic
92C4×Dic107C2311yescharacteristic
93#41×C107—22no—
94#41×C107—22no—
95C2×Q8×C107C2311yescharacteristic
96C2×Q8×C107C2311yescharacteristic
97C2×D4×C107—21no—
98C2×D4×C107C2311yescharacteristic
99C22×C4×C107—21no—
100C22×C4×C107C2311yescharacteristic
101Q16×C107—22no—
102SD16×C107—22no—
103C2×C8×C107C2311yescharacteristic
104C2×C8×C107C2311yescharacteristic
105C4⋊C4×C107—21no—
106C4⋊C4×C107C2311yescharacteristic
107C22⋊C4×C107—21no—
108C23×C107—21no—
109C23×C107C2×D411yescharacteristic
110C22×D107—21no—
111Q8×C107—21no—
112Q8×C107—21no—
113Q8×C107—21no—
114Q8×C107#4112yes—
115D4×C107—24no—
116D4×C107—21no—
117D4×C107#4112yes—
118C2×C4×C107—24no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107C2×D411yescharacteristic
125C2×C4×C107C2411yescharacteristic
126C22⋊D107—42no—
127C2×Dic107—22no—
128C2×Dic107—21no—
129C2×Dic107—21no—
130C2×Dic107—21no—
131C2×Dic107—41no—
132C2×Dic107—42no—
133C2×Dic107—42no—
134C2×Dic107C2×D411yescharacteristic
135C2×Dic107C2×D411yescharacteristic
136C2×Dic107C2×D411yescharacteristic
137C2×Dic107C2×D411yescharacteristic
138D428—21no—
139D428—21no—
140C4×D107—42no—
141C8×C107—21no—
142C8×C107—21no—
143C107⋊C8—21no—
144C107⋊C8—21no—
145C22×C107—21no—
146C22×C107—22no—
147C22×C107—21no—
148C22×C107—21no—
149C22×C107—41no—
150C22×C107—41no—
151C22×C107C22×D411yescharacteristic
152C2×D107—41no—
153C2×D107—41no—
154C2×D107—41no—
155C4×C107—21no—
156C4×C107—22no—
157C4×C107—41no—
158C4×C107—41no—
159C4×C107C2×#4111yescharacteristic
160C4×C107C2×#4111yescharacteristic
161Dic107—21no—
162Dic107—21no—
163Dic107—21no—
164Dic107—41no—
165Dic107—41no—
166Dic107—41no—
167Dic107#13512yes—
168C2×C107—41no—
169C2×C107—42no—
170C2×C107#52511yescharacteristic
171C2×C107C2×#13511yescharacteristic
172C2×C107#57511yescharacteristic
173D107—81no—
174#3102—1071nomaximal, Sylow
175C107#310211yescharacteristic, complemented, Sylow
176#529—1071no—
177C4×#41—1071no—
178#485—1071no—
179#482—1071no—
180#479—1071no—
181#478—1071no—
182#464—1071no—
183#463—1071no—
184#452D10711yescharacteristic, complemented
185#449—1071no—
186C4×Q16—1071no—
187C4×SD16—1071no—
188C4×SD16—1071no—
189C4×D8—1071no—
190#431—1071no—
191C2×#41C2×D10711yescharacteristic
192C2×Q16C2×D10711yescharacteristic
193C2×SD16—1071no—
194C2×SD16C2×D10711yescharacteristic
195C2×D8—1071no—
196#126—2141no—
197#125—1071no—
198#124—1071no—
199#123—2141no—
200C22⋊Q8C2×D10711yescharacteristic
201#121—1071no—
202C4×Q8—1071no—
203C4×Q8—1071no—
204C4×Q8—1071no—
205C4×D4—1071no—
206C4×D4—1071no—
207C4×D4—1071no—
208C4×D4—2141no—
209C4×D4—2141no—
210C4×D4—2141no—
211#117—1071no—
212#117—2141no—
213#117—2141no—
214C2×C42—1071no—
215C2×C42—2141no—
216#107—1071no—
217#107—1071no—
218#106—1071no—
219#106—1071no—
220C4⋊C8—1071no—
221C4⋊C8—1071no—
222#103—1071no—
223#103—1071no—
224#103—1071no—
225#103C2×D10711yescharacteristic
226#102—1071no—
227#102—1071no—
228#102—1071no—
229#102C2×D10711yescharacteristic
230C22⋊C8—1071no—
231C22⋊C8C2×D10711yescharacteristic
232C4×C8—1071no—
233C4×C8—1071no—
234#41—22no—
235#41—22no—
236C2×Q8C22×D10711yescharacteristic
237C2×Q8C22×D10711yescharacteristic
238C2×D4—21no—
239C2×D4—1071no—
240C2×D4—2141no—
241C2×D4C22×D10711yescharacteristic
242C22×C4—21no—
243C22×C4—1071no—
244C22×C4—1071no—
245C22×C4—2141no—
246C22×C4—2141no—
247C22×C4—2141no—
248C22×C4C22×D10711yescharacteristic
249Q16—22no—
250SD16—22no—
251SD16—2142no—
252D8—2142no—
253C2×C8—1071no—
254C2×C8—1071no—
255C2×C8C22×D10711yescharacteristic
256C2×C8C22×D10711yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—1071no—
259C4⋊C4—1071no—
260C4⋊C4—1071no—
261C4⋊C4—2141no—
262C4⋊C4—2141no—
263C4⋊C4—2141no—
264C4⋊C4—2141no—
265C4⋊C4—2141no—
266C4⋊C4C22×D10711yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—2141no—
269C22⋊C4—2141no—
270C22⋊C4—2141no—
271C22⋊C4—2141no—
272C22⋊C4—2141no—
273C22⋊C4—2141no—
274C22⋊C4—2141no—
275C42—1071no—
276C42—1071no—
277C42—1071no—
278C42—1071no—
279C42—2141no—
280C42—2141no—
281C42—2141no—
282C42—2141no—
283C23—21no—
284C23—2141no—
285C23D4×D10711yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8#1313381712yes—
290D4—24no—
291D4—21no—
292D4—4282no—
293D4—2141no—
294D4—2141no—
295D4#1313381712yes—
296C2×C4—24no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—1071no—
303C2×C4—1071no—
304C2×C4—1071no—
305C2×C4—1071no—
306C2×C4—4282no—
307C2×C4—4282no—
308C2×C4—4282no—
309C2×C4—4281no—
310C2×C4—2142no—
311C2×C4—2141no—
312C2×C4—2141no—
313C2×C4—2141no—
314C2×C4D4×D10711yescharacteristic
315C2×C4C23×D10711yescharacteristic
316C8—21no—
317C8—21no—
318C8—2141no—
319C8—2141no—
320C22—21no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—4281no—
327C22—4281no—
328C22—4281no—
329C22C2×D4×D10711yescharacteristic
330C4—21no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—1072no—
335C4—4281no—
336C4—4281no—
337C4—4281no—
338C4—2141no—
339C4—2141no—
340C4—2141no—
341C4C2×#1313381711yescharacteristic
342C4C2×#1313381711yescharacteristic
343C2—41no—
344C2—42no—
345C2—8561no—
346C2#2029313311yescharacteristic
347C2#135×D10711yescharacteristic
348C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#452×C107C2
socleC22×C107C22×D4
prime radical#20292772C2
hypercenter#452D107
hypocenterC107#3102
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #20293163C2semidirect
2. #41×Dic107C2semidirect
3. #20292965C2semidirect
4. #20292952C2semidirect
5. #20292932C2semidirect
6. #20292925C2semidirect
7. #20292851C2semidirect
8. #20292844C2semidirect
10. Q16×Dic107C2semidirect
11. SD16×Dic107C2semidirect
12. #20292712C2semidirect
13. #20292707C2semidirect
14. #20292677C2semidirect
15. #452×C107C2semidirect
175. C107#3102semidirect
184. #452D107semidirect

Maximal subgroups

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

subgroupindex
#202931632
#41×Dic1072
#202929652
#202929522
#202929322
#202929252
#202928512
#202928442
#202927722
Q16×Dic1072
SD16×Dic1072
#202927122
#202927072
#202926772
#452×C1072
#3102107