Finite Group Databasebeta

← #28347971

Subgroups — #28347971

Order 13696.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812138C211yesmaximal, characteristic, complemented
2Q8×C22⋊D107C211yesmaximal, characteristic, complemented
3#20293271C211yesmaximal, characteristic
4#20293018C211yesmaximal, characteristic, complemented
5#20292870C211yesmaximal, characteristic, complemented
6#20292868C211yesmaximal, characteristic, complemented
7#20292847C211yesmaximal, characteristic, complemented
8#20292830C211yesmaximal, characteristic, complemented
9#20292786C211yesmaximal, characteristic, complemented
10#20292783C211yesmaximal, characteristic, complemented
11#20292778C211yesmaximal, characteristic, complemented
12#20292764C211yesmaximal, characteristic, complemented
13#20292728C211yesmaximal, characteristic, complemented
14#20292698C211yesmaximal, characteristic, complemented
15C22×SD16×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C2×Q8×D107C2211yescharacteristic, complemented
18#16812138—24no—
19C2×Dic428C2211yescharacteristic, complemented
20C2×#13133808C2211yescharacteristic, complemented
21C2×#13133808C2211yescharacteristic, complemented
22C2×#13133806C2211yescharacteristic
23C2×#13133806C2211yescharacteristic
24C2×#13133801C2211yescharacteristic, complemented
25#16812107—21no—
26#16812107C2211yescharacteristic, complemented
27#16812096C2211yescharacteristic
28#16812090—21no—
29#16812088—21no—
30#16812088C2211yescharacteristic
31#16812088C2211yescharacteristic
32#16812085—21no—
33#16812085C2211yescharacteristic, complemented
34#16812081—21no—
35#16812080C2211yescharacteristic
36#16812078—21no—
37Q8×Dic107C2211yescharacteristic, complemented
38D4×Dic107C2211yescharacteristic
39C4×C22⋊D107—21no—
40C4×C22⋊D107C2211yescharacteristic
41C2×C22⋊Dic107—21no—
42#16812043C2211yescharacteristic, complemented
43Dic107⋊C8C2211yescharacteristic, complemented
44#16812034C2211yescharacteristic, complemented
45#16812034C2211yescharacteristic, complemented
46#16812033C2211yescharacteristic, complemented
47#16812033C2211yescharacteristic, complemented
48#16812030C2211yescharacteristic
49#16812030C2211yescharacteristic
50C22⋊(C107⋊C8)C2211yescharacteristic
51(C2×D107)⋊C8C2211yescharacteristic, complemented
52#16812019C2211yescharacteristic, complemented
53C22×Q8×C107C2211yescharacteristic
54C22×D4×C107C2211yescharacteristic
55C2×SD16×C107—24no—
56C2×SD16×C107C2211yescharacteristic
57C2×SD16×C107C2211yescharacteristic
58C2×SD16×C107C2211yescharacteristic
59C2×SD16×C107C2211yescharacteristic
60C22×C8×C107C2211yescharacteristic
61#13133817—22no—
62#13133817—41no—
63C2×Dic214—21no—
64C2×Dic214C2311yescharacteristic
65C2×Dic214C2311yescharacteristic
66Q8×D107—22no—
67Q8×D107—41no—
68C2×C22⋊D107—21no—
69C2×C22⋊D107—21no—
70C2×C22⋊D107D411yescharacteristic
71C22×Dic107—21no—
72C22×Dic107—21no—
73C2×C4×D107—21no—
74C2×C4×D107C2311yescharacteristic
75Dic428—22no—
76#13133808—22no—
77#13133808—41no—
78#13133806—22no—
79#13133806—41no—
80#13133801—22no—
81C2×C107⋊C8C2311yescharacteristic
82C2×C107⋊C8C2311yescharacteristic
83C4⋊Dic107—21no—
84C4⋊Dic107C2311yescharacteristic
85C4⋊Dic107C2311yescharacteristic
86Dic107⋊C4—21no—
87Dic107⋊C4—21no—
88Dic107⋊C4—21no—
89Dic107⋊C4—21no—
90Dic107⋊C4—21no—
91Dic107⋊C4D411yescharacteristic
92C22⋊Dic107—21no—
93C22⋊Dic107—21no—
94C22⋊Dic107—21no—
95C22⋊Dic107—21no—
96C22⋊Dic107—21no—
97C22⋊Dic107—21no—
98C22⋊Dic107D411yescharacteristic
99(C2×D107)⋊C4—21no—
100(C2×D107)⋊C4D411yescharacteristic
101C4×Dic107—21no—
102C4×Dic107C2311yescharacteristic
103C24×C107—21no—
104C2×Q8×C107—21no—
105C2×Q8×C107—22no—
106C2×Q8×C107—41no—
107C2×Q8×C107C2311yescharacteristic
108C2×Q8×C107C2311yescharacteristic
109C2×D4×C107—22no—
110C2×D4×C107—21no—
111C2×D4×C107—41no—
112C2×D4×C107C2311yescharacteristic
113C2×D4×C107C2311yescharacteristic
114C22×C4×C107—21no—
115C22×C4×C107C2311yescharacteristic
116SD16×C107—22no—
117SD16×C107—22no—
118SD16×C107—22no—
119SD16×C107D414yescomplemented
120C2×C8×C107—22no—
121C2×C8×C107C2311yescharacteristic
122C2×C8×C107C2311yescharacteristic
123C23×C107—21no—
124C23×C107—21no—
125C23×C107—42no—
126C23×C107—42no—
127C23×C107—81no—
128C23×C107C2×D411yescharacteristic
129C22×D107C2×D411yescharacteristic
130Q8×C107—21no—
131Q8×C107—21no—
132Q8×C107—21no—
133Q8×C107—42no—
134Q8×C107C2×D412yes—
135D4×C107—21no—
136D4×C107—21no—
137D4×C107—21no—
138D4×C107—42no—
139D4×C107C2×D412yes—
140C2×C4×C107—21no—
141C2×C4×C107—21no—
142C2×C4×C107—21no—
143C2×C4×C107—21no—
144C2×C4×C107—42no—
145C2×C4×C107C2×D411yescharacteristic
146C2×C4×C107C2411yescharacteristic
147C22⋊D107—22no—
148C22⋊D107—42no—
149C22⋊D107—81no—
150C2×Dic107—21no—
151C2×Dic107—21no—
152C2×Dic107—21no—
153C2×Dic107—21no—
154C2×Dic107—42no—
155C2×Dic107—81no—
156C2×Dic107C2×D411yescharacteristic
157C2×Dic107C2×D411yescharacteristic
158C2×Dic107C2×D411yescharacteristic
159C4×D107—21no—
160C4×D107—21no—
161C4×D107—42no—
162Dic214—21no—
163Dic214—21no—
164Dic214—21no—
165Dic214—21no—
166Dic214—42no—
167C8×C107—21no—
168C8×C107C2×D412yes—
169C107⋊C8—21no—
170C107⋊C8—21no—
171C22×C107—21no—
172C22×C107—21no—
173C22×C107—22no—
174C22×C107—41no—
175C22×C107—41no—
176C22×C107—41no—
177C22×C107—41no—
178C22×C107—41no—
179C22×C107—84no—
180C22×C107#13712yescomplemented
181C22×C107C22×D411yescharacteristic
182C2×D107—21no—
183C2×D107—21no—
184C2×D107—21no—
185C4×C107—22no—
186C4×C107—21no—
187C4×C107—41no—
188C4×C107C22×D411yescharacteristic
189C4×C107H(5,𝔽2)11yescharacteristic
190Dic107—21no—
191Dic107—21no—
192Dic107—21no—
193Dic107—41no—
194Dic107—41no—
195Dic107—41no—
196Dic107—41no—
197C2×C107—22no—
198C2×C107—42no—
199C2×C107—81no—
200C2×C107D4211yescharacteristic
201C2×C107C2×#13711yescharacteristic
202C2×C107#57711yescharacteristic
203D107—41no—
204#3266—1071nomaximal, Sylow
205C107#326611yescharacteristic, complemented, Sylow
206C2×#137—1071no—
207C22×SD16D10711yescharacteristic, complemented
208D4×Q8—1071no—
209#546—1071no—
210#497—1071no—
211#468—1071no—
212#467—1071no—
213#464—1071no—
214#461—1071no—
215#452—1071no—
216#451—1071no—
217#450—1071no—
218#448—1071no—
219#440—1071no—
220#435—1071no—
221C2×#41—1071no—
222C22×Q8—1071no—
223C22×Q8C2×D10711yescharacteristic
224C22×D4C2×D10711yescharacteristic
225#137—2144no—
226C2×Q16—1071no—
227C2×Q16—1071no—
228C2×Q16—1071no—
229C2×SD16—24no—
230C2×SD16—1071no—
231C2×SD16—1071no—
232C2×SD16C2×D10711yescharacteristic
233C2×SD16C2×D10711yescharacteristic
234C2×SD16C2×D10711yescharacteristic
235C2×SD16C2×D10711yescharacteristic
236C2×M16—1071no—
237C22×C8C2×D10711yescharacteristic
238C4⋊Q8—1071no—
239C4⋊Q8—2141no—
240#124—1071no—
241#123—2141no—
242C22⋊Q8—1071no—
243C22⋊Q8—1071no—
244C22⋊Q8—1071no—
245C22⋊Q8—2141no—
246C22⋊Q8—2141no—
247#121—1071no—
248#121—2141no—
249#120—2141no—
250C4×Q8—1071no—
251C4×D4—1071no—
252C4×D4—1071no—
253C4×D4—2141no—
254C2×C22⋊C4—2141no—
255#107—1071no—
256C4⋊C8—1071no—
257#103—1071no—
258#103—1071no—
259#103—1071no—
260#103—1071no—
261#102—1071no—
262#102—1071no—
263C22⋊C8—1071no—
264C22⋊C8—1071no—
265#97—1071no—
266C24—21no—
267#41—4281no—
268#41—2142no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—1071no—
273C2×Q8—1071no—
274C2×Q8—4281no—
275C2×Q8—2141no—
276C2×Q8—2142no—
277C2×Q8C22×D10711yescharacteristic
278C2×Q8C22×D10711yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—1071no—
283C2×D4—2141no—
284C2×D4—2141no—
285C2×D4C22×D10711yescharacteristic
286C2×D4C22×D10711yescharacteristic
287C22×C4—21no—
288C22×C4—1071no—
289C22×C4—2141no—
290C22×C4—2141no—
291C22×C4—2141no—
292C22×C4C22×D10711yescharacteristic
293Q16—4281no—
294Q16—2142no—
295Q16—2142no—
296SD16—22no—
297SD16—22no—
298SD16—22no—
299SD16—4281no—
300SD16—2142no—
301SD16C22⋊D10714yescomplemented
302M16—2142no—
303C2×C8—22no—
304C2×C8—1071no—
305C2×C8—1071no—
306C2×C8C22×D10711yescharacteristic
307C2×C8C22×D10711yescharacteristic
308C4⋊C4—1071no—
309C4⋊C4—1071no—
310C4⋊C4—1071no—
311C4⋊C4—2141no—
312C4⋊C4—2141no—
313C4⋊C4—2141no—
314C4⋊C4—2141no—
315C4⋊C4—2141no—
316C4⋊C4—2141no—
317C22⋊C4—1071no—
318C22⋊C4—1071no—
319C22⋊C4—2141no—
320C22⋊C4—2141no—
321C22⋊C4—2141no—
322C22⋊C4—2141no—
323C22⋊C4—2141no—
324C22⋊C4—2141no—
325C22⋊C4—2141no—
326C42—1071no—
327C42—2141no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—1071no—
334C23D4×D10711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—42no—
339Q8—4282no—
340Q8—2141no—
341Q8—2141no—
342Q8—2141no—
343Q8—2141no—
344Q8C2×C22⋊D10712yes—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—8561no—
350D4—4282no—
351D4—2142no—
352D4C2×C22⋊D10712yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—8561no—
359C2×C4—1071no—
360C2×C4—1071no—
361C2×C4—1071no—
362C2×C4—4282no—
363C2×C4—4282no—
364C2×C4—2141no—
365C2×C4—2141no—
366C2×C4—2141no—
367C2×C4—2141no—
368C2×C4—2141no—
369C2×C4—2141no—
370C2×C4D4×D10711yescharacteristic
371C2×C4C23×D10711yescharacteristic
372C8—21no—
373C8—2141no—
374C8—2141no—
375C8C2×C22⋊D10712yes—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—2141no—
386C22—2141no—
387C22—2141no—
388C22#1681213812yescomplemented
389C22C2×D4×D10711yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—4281no—
394C4—4281no—
395C4—4281no—
396C4—4281no—
397C4—2141no—
398C4—2141no—
399C4—2141no—
400C4C22×C22⋊D10711yescharacteristic
401C4#1681215411yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—4281no—
406C2D4×C22⋊D10711yescharacteristic
407C2C2×#1681213811yescharacteristic
408C2#2029343311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC22×SD16×C107C2
socleC22×C107C22×D4
prime radical#20293271C2
hypercenterC22×SD16D107
hypocenterC107#3266
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16812138C2semidirect
2. Q8×C22⋊D107C2semidirect
4. #20293018C2semidirect
5. #20292870C2semidirect
6. #20292868C2semidirect
7. #20292847C2semidirect
8. #20292830C2semidirect
9. #20292786C2semidirect
10. #20292783C2semidirect
11. #20292778C2semidirect
12. #20292764C2semidirect
13. #20292728C2semidirect
14. #20292698C2semidirect
15. C22×SD16×C107C2semidirect
17. C2×Q8×D107C22semidirect
19. C2×Dic428C22semidirect
20. C2×#13133808C22semidirect
21. C2×#13133808C22semidirect
24. C2×#13133801C22semidirect
26. #16812107C22semidirect
33. #16812085C22semidirect
37. Q8×Dic107C22semidirect
42. #16812043C22semidirect
43. Dic107⋊C8C22semidirect
44. #16812034C22semidirect
45. #16812034C22semidirect
46. #16812033C22semidirect
47. #16812033C22semidirect
51. (C2×D107)⋊C8C22semidirect
52. #16812019C22semidirect
119. SD16×C107D4semidirect
180. C22×C107#137semidirect
205. C107#3266semidirect
207. C22×SD16D107semidirect
301. SD16C22⋊D107semidirect
388. C22#16812138semidirect

Maximal subgroups

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

subgroupindex
C2×#168121382
Q8×C22⋊D1072
#202932712
#202930182
#202928702
#202928682
#202928472
#202928302
#202927862
#202927832
#202927782
#202927642
#202927282
#202926982
C22×SD16×C1072
#3266107