Finite Group Databasebeta

← #28348047

Subgroups — #28348047

Order 13696.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812140C211yesmaximal, characteristic, complemented
2C2×#16812129C211yesmaximal, characteristic, complemented
3#20293298C211yesmaximal, characteristic, complemented
4#20293017C211yesmaximal, characteristic, complemented
5#20292878C211yesmaximal, characteristic, complemented
6#20292858C211yesmaximal, characteristic, complemented
7#20292846C211yesmaximal, characteristic, complemented
8#20292827C211yesmaximal, characteristic, complemented
9#20292787C211yesmaximal, characteristic, complemented
10#20292772C211yesmaximal, characteristic, complemented
11#20292770C211yesmaximal, characteristic, complemented
12#20292766C211yesmaximal, characteristic, complemented
13#20292732C211yesmaximal, characteristic, complemented
14#20292699C211yesmaximal, characteristic, complemented
15#548×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic
17C2×#13133818C2211yescharacteristic
18#16812140—24no—
19#16812129—24no—
20C2×#13133808C2211yescharacteristic
21C2×#13133808C2211yescharacteristic
22C2×#13133806C2211yescharacteristic
23C2×#13133805C2211yescharacteristic
24C2×#13133805C2211yescharacteristic
25C2×#13133805C2211yescharacteristic
26C2×#13133805C2211yescharacteristic
27C2×#13133803C2211yescharacteristic
28C2×#13133803C2211yescharacteristic
29C2×#13133802C2211yescharacteristic
30C22×C107⋊C8C2211yescharacteristic
31#16812106C2211yescharacteristic
32#16812105—21no—
33#16812083C2211yescharacteristic
34#16812083C2211yescharacteristic
35#16812082—21no—
36#16812082—21no—
37C4×Dic214C2211yescharacteristic
38#16812071—21no—
39#16812040C2211yescharacteristic
40C4⋊(C107⋊C8)C2211yescharacteristic
41#16812033C2211yescharacteristic
42#16812033C2211yescharacteristic
43#16812032C2211yescharacteristic
44#16812030C2211yescharacteristic
45#16812029C2211yescharacteristic
46#16812029C2211yescharacteristic
47C22⋊(C107⋊C8)C2211yescharacteristic
48C22⋊(C107⋊C8)C2211yescharacteristic
49#16812020C2211yescharacteristic
50C2×#41×C107C2211yescharacteristic
51C22×Q8×C107C2211yescharacteristic
52#124×C107—21no—
53#124×C107C2211yescharacteristic
54C22⋊Q8×C107—21no—
55C22⋊Q8×C107C2211yescharacteristic
56#121×C107—21no—
57#121×C107C2211yescharacteristic
58C4×Q8×C107C2211yescharacteristic
59C4×D4×C107—21no—
60C4×D4×C107C2211yescharacteristic
61#13133818—22no—
62#13133818—22no—
63#13133818—41no—
64#13133818—41no—
65C2×Dic214C2311yescharacteristic
66C2×D428—21no—
67C2×D428—21no—
68C2×D428C2311yescharacteristic
69C2×D428C2311yescharacteristic
70C2×C22⋊D107—21no—
71C2×C22⋊D107—21no—
72C2×C22⋊D107—21no—
73C2×C22⋊D107—21no—
74C2×C4×D107—21no—
75C2×C4×D107—21no—
76#13133808—22no—
77#13133808D414yescomplemented
78#13133806—22no—
79#13133805—22no—
80#13133805—22no—
81#13133805—22no—
82#13133805—41no—
83#13133803—22no—
84#13133803—41no—
85#13133802—22no—
86C2×C107⋊C8—22no—
87C2×C107⋊C8C2311yescharacteristic
88C2×C107⋊C8C2311yescharacteristic
89C2×C107⋊C8C2311yescharacteristic
90C2×C107⋊C8C2311yescharacteristic
91C4⋊Dic107C2311yescharacteristic
92Dic107⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95C4×Dic107—21no—
96#41×C107—22no—
97#41×C107—41no—
98C2×Q8×C107—21no—
99C2×Q8×C107—22no—
100C2×Q8×C107—41no—
101C2×Q8×C107C2311yescharacteristic
102C2×Q8×C107C2311yescharacteristic
103C2×D4×C107—21no—
104C2×D4×C107—21no—
105C2×D4×C107D411yescharacteristic
106C2×D4×C107C2311yescharacteristic
107C22×C4×C107—21no—
108C22×C4×C107—21no—
109C22×C4×C107C2311yescharacteristic
110C22×C4×C107C2311yescharacteristic
111C4⋊C4×C107—21no—
112C4⋊C4×C107—21no—
113C4⋊C4×C107D411yescharacteristic
114C4⋊C4×C107C2311yescharacteristic
115C22⋊C4×C107—21no—
116C22⋊C4×C107—21no—
117C22⋊C4×C107—21no—
118C22⋊C4×C107—21no—
119C22⋊C4×C107D411yescharacteristic
120C22⋊C4×C107D411yescharacteristic
121C42×C107—21no—
122C42×C107C2311yescharacteristic
123C23×C107—21no—
124C23×C107C2×D411yescharacteristic
125C23×C107C2×D411yescharacteristic
126C22×D107—21no—
127C22×D107—21no—
128Q8×C107—21no—
129Q8×C107—21no—
130Q8×C107—21no—
131Q8×C107—42no—
132Q8×C107C2×D412yes—
133D4×C107—21no—
134D4×C107—21no—
135D4×C107—41no—
136D4×C107—42no—
137D4×C107—42no—
138C2×C4×C107—21no—
139C2×C4×C107—21no—
140C2×C4×C107—21no—
141C2×C4×C107—21no—
142C2×C4×C107—21no—
143C2×C4×C107—21no—
144C2×C4×C107—21no—
145C2×C4×C107—42no—
146C2×C4×C107—42no—
147C2×C4×C107C2×D411yescharacteristic
148C2×C4×C107C2×D411yescharacteristic
149C2×C4×C107C2×D411yescharacteristic
150C2×C4×C107C2×D411yescharacteristic
151C2×C4×C107C2411yescharacteristic
152C22⋊D107—42no—
153C22⋊D107—42no—
154C22⋊D107—81no—
155C22⋊D107—81no—
156C2×Dic107—21no—
157C2×Dic107—21no—
158D428—21no—
159D428—21no—
160D428—21no—
161D428—21no—
162D428—42no—
163D428—81no—
164C4×D107—42no—
165C4×D107—42no—
166Dic214—21no—
167Dic214C2×D412yes—
168C107⋊C8—21no—
169C107⋊C8—21no—
170C107⋊C8—21no—
171C107⋊C8C2×D412yes—
172C22×C107—21no—
173C22×C107—21no—
174C22×C107—21no—
175C22×C107—21no—
176C22×C107—21no—
177C22×C107—21no—
178C22×C107—41no—
179C22×C107—41no—
180C22×C107—41no—
181C22×C107C22×D411yescharacteristic
182C2×D107—41no—
183C2×D107—41no—
184C2×D107—41no—
185C2×D107—41no—
186C2×D107—41no—
187C2×D107—41no—
188C4×C107—21no—
189C4×C107—21no—
190C4×C107—22no—
191C4×C107—21no—
192C4×C107—21no—
193C4×C107—41no—
194C4×C107—41no—
195C4×C107C22×D411yescharacteristic
196C4×C107H(5,𝔽2)11yescharacteristic
197Dic107—22no—
198Dic107—41no—
199C2×C107—41no—
200C2×C107—41no—
201C2×C107—81no—
202C2×C107D4211yescharacteristic
203C2×C107#57511yescharacteristic
204C2×C107#57711yescharacteristic
205D107—81no—
206D107—81no—
207#3271—1071nomaximal, Sylow
208C107#327111yescharacteristic, complemented, Sylow
209C2×#137—1071no—
210C2×#135—1071no—
211#550—1071no—
212#548D10711yescharacteristic, complemented
213#496—1071no—
214#469—1071no—
215#465—1071no—
216#464—1071no—
217#460—1071no—
218#452—1071no—
219#449—1071no—
220#449—1071no—
221#448—1071no—
222#441—1071no—
223#435—1071no—
224C2×#41—1071no—
225C2×#41—1071no—
226C2×#41C2×D10711yescharacteristic
227C22×Q8C2×D10711yescharacteristic
228#137—2144no—
229#135—2144no—
230C2×Q16—1071no—
231C2×Q16—1071no—
232C2×SD16—1071no—
233C2×SD16—1071no—
234C2×SD16—1071no—
235C2×SD16—1071no—
236C2×SD16—1071no—
237C2×D8—1071no—
238C2×D8—1071no—
239C2×M16—1071no—
240C22×C8—1071no—
241#127—1071no—
242#127—2141no—
243#124—21no—
244#124C2×D10711yescharacteristic
245C22⋊Q8—21no—
246C22⋊Q8C2×D10711yescharacteristic
247#121—21no—
248#121—1071no—
249#121—1071no—
250#121—2141no—
251#121—2141no—
252#121C2×D10711yescharacteristic
253C4×Q8—1071no—
254C4×Q8C2×D10711yescharacteristic
255C4×D4—21no—
256C4×D4—2141no—
257C4×D4C2×D10711yescharacteristic
258#106—1071no—
259C4⋊C8—1071no—
260#103—1071no—
261#103—1071no—
262#103—1071no—
263#102—1071no—
264#102—1071no—
265#102—1071no—
266C22⋊C8—1071no—
267C22⋊C8—1071no—
268#97—1071no—
269#41—22no—
270#41—41no—
271#41—4281no—
272#41—4281no—
273#41—2142no—
274#41—2142no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—1071no—
279C2×Q8C22×D10711yescharacteristic
280C2×Q8C22×D10711yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—1071no—
284C2×D4—1071no—
285C2×D4—2141no—
286C2×D4—2141no—
287C2×D4—2141no—
288C2×D4—2141no—
289C2×D4—2141no—
290C2×D4—2141no—
291C2×D4C22⋊D10711yescharacteristic
292C2×D4C22×D10711yescharacteristic
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—2141no—
296C22×C4—2141no—
297C22×C4C22×D10711yescharacteristic
298C22×C4C22×D10711yescharacteristic
299Q16—1074no—
300Q16—2142no—
301SD16—4281no—
302SD16—2142no—
303SD16—2142no—
304SD16—2142no—
305SD16—2142no—
306D8—4281no—
307D8—2142no—
308M16—2142no—
309C2×C8—1071no—
310C2×C8—1071no—
311C2×C8—1071no—
312C2×C8—1071no—
313C2×C8—2142no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—1071no—
317C4⋊C4—2141no—
318C4⋊C4C22⋊D10711yescharacteristic
319C4⋊C4C22×D10711yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—2141no—
325C22⋊C4—2141no—
326C22⋊C4C22⋊D10711yescharacteristic
327C22⋊C4C22⋊D10711yescharacteristic
328C42—21no—
329C42—2141no—
330C42C22×D10711yescharacteristic
331C23—21no—
332C23—2141no—
333C23—2141no—
334C23C2×C22⋊D10711yescharacteristic
335C23C2×C22⋊D10711yescharacteristic
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—42no—
340Q8—1072no—
341Q8—2141no—
342Q8D4×D10712yes—
343D4—21no—
344D4—21no—
345D4—41no—
346D4—42no—
347D4—42no—
348D4—8561no—
349D4—8561no—
350D4—8561no—
351D4—4282no—
352D4—4282no—
353D4—4282no—
354D4—2141no—
355D4—2141no—
356D4—2141no—
357D4—2141no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—4282no—
368C2×C4—4282no—
369C2×C4—2141no—
370C2×C4—2141no—
371C2×C4C2×C22⋊D10711yescharacteristic
372C2×C4C2×C22⋊D10711yescharacteristic
373C2×C4C2×C22⋊D10711yescharacteristic
374C2×C4C2×C22⋊D10711yescharacteristic
375C2×C4C23×D10711yescharacteristic
376C8—1072no—
377C8—2141no—
378C8—2141no—
379C8—2141no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—4281no—
390C22—4281no—
391C22—4281no—
392C22—4281no—
393C22—4281no—
394C22—4281no—
395C22C22×C22⋊D10711yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—22no—
401C4—41no—
402C4—41no—
403C4—4281no—
404C4—2142no—
405C4C2×D4×D10711yescharacteristic
406C4#1681215411yescharacteristic
407C2—41no—
408C2—41no—
409C2—81no—
410C2—8561no—
411C2—8561no—
412C2D4×C22⋊D10711yescharacteristic
413C2#2029342511yescharacteristic
414C2#2029343511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#548×C107C2
socleC22×C107C22×D4
hypercenter#548D107
hypocenterC107#3271
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#16812140C2semidirect
2. C2×#16812129C2semidirect
3. #20293298C2semidirect
4. #20293017C2semidirect
5. #20292878C2semidirect
6. #20292858C2semidirect
7. #20292846C2semidirect
8. #20292827C2semidirect
9. #20292787C2semidirect
10. #20292772C2semidirect
11. #20292770C2semidirect
12. #20292766C2semidirect
13. #20292732C2semidirect
14. #20292699C2semidirect
15. #548×C107C2semidirect
77. #13133808D4semidirect
208. C107#3271semidirect
212. #548D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121402
C2×#168121292
#202932982
#202930172
#202928782
#202928582
#202928462
#202928272
#202927872
#202927722
#202927702
#202927662
#202927322
#202926992
#548×C1072
#3271107