Finite Group Databasebeta

← #28348012

Subgroups — #28348012

Order 13696.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812139C211yesmaximal, characteristic, complemented
2C2×#16812129C211yesmaximal, characteristic, complemented
3#20293286C211yesmaximal, characteristic, complemented
4#20293019C211yesmaximal, characteristic, complemented
5#20292875C211yesmaximal, characteristic, complemented
6#20292866C211yesmaximal, characteristic, complemented
7#20292842C211yesmaximal, characteristic, complemented
8#20292834C211yesmaximal, characteristic, complemented
9#20292789C211yesmaximal, characteristic, complemented
10#20292787C211yesmaximal, characteristic, complemented
11#20292782C211yesmaximal, characteristic, complemented
12#20292772C211yesmaximal, characteristic, complemented
13#20292727C211yesmaximal, characteristic, complemented
14#20292699C211yesmaximal, characteristic, complemented
15#547×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic
17C22×Dic214C2211yescharacteristic
18#16812139—24no—
19#16812129—24no—
20C2×#13133808C2211yescharacteristic
21C2×#13133808C2211yescharacteristic
22C2×#13133808C2211yescharacteristic
23C2×#13133808C2211yescharacteristic
24C2×#13133806C2211yescharacteristic
25C2×#13133806C2211yescharacteristic
26C2×#13133805C2211yescharacteristic
27C2×#13133805C2211yescharacteristic
28C2×#13133803C2211yescharacteristic
29C2×#13133802C2211yescharacteristic
30C22×C107⋊C8C2211yescharacteristic
31#16812097C2211yescharacteristic
32#16812095—21no—
33#16812088C2211yescharacteristic
34#16812087—21no—
35#16812083C2211yescharacteristic
36#16812081—21no—
37C4×Dic214C2211yescharacteristic
38(C22⋊D107)⋊C4—21no—
39#16812042C2211yescharacteristic
40C4⋊(C107⋊C8)C2211yescharacteristic
41#16812033C2211yescharacteristic
42#16812033C2211yescharacteristic
43#16812032C2211yescharacteristic
44#16812032C2211yescharacteristic
45#16812030C2211yescharacteristic
46#16812029C2211yescharacteristic
47C22⋊(C107⋊C8)C2211yescharacteristic
48C22⋊(C107⋊C8)C2211yescharacteristic
49#16812020C2211yescharacteristic
50C2×#41×C107C2211yescharacteristic
51C2×#41×C107C2211yescharacteristic
52C4⋊Q8×C107C2211yescharacteristic
53#123×C107—21no—
54#123×C107—21no—
55C22⋊Q8×C107C2211yescharacteristic
56C22⋊Q8×C107C2211yescharacteristic
57#121×C107—21no—
58C4×D4×C107C2211yescharacteristic
59C4×D4×C107C2211yescharacteristic
60C2×C4⋊C4×C107—21no—
61#13133818—22no—
62#13133818—41no—
63C2×Dic214—22no—
64C2×Dic214—21no—
65C2×Dic214—41no—
66C2×Dic214C2311yescharacteristic
67C2×Dic214C2311yescharacteristic
68C2×D428C2311yescharacteristic
69C2×C22⋊D107—21no—
70C2×C22⋊D107—21no—
71C22×Dic107—21no—
72C2×C4×D107—21no—
73#13133808—22no—
74#13133808—22no—
75#13133808—22no—
76#13133808—41no—
77#13133806—22no—
78#13133806D414yescomplemented
79#13133805—22no—
80#13133805—41no—
81#13133803—22no—
82#13133802—22no—
83C2×C107⋊C8—22no—
84C2×C107⋊C8C2311yescharacteristic
85C2×C107⋊C8C2311yescharacteristic
86C2×C107⋊C8C2311yescharacteristic
87C2×C107⋊C8C2311yescharacteristic
88C4⋊Dic107C2311yescharacteristic
89Dic107⋊C4—21no—
90Dic107⋊C4—21no—
91C22⋊Dic107—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95C4×Dic107—21no—
96#41×C107—22no—
97#41×C107—22no—
98#41×C107—41no—
99#41×C107—41no—
100C2×Q8×C107C2311yescharacteristic
101C2×Q8×C107C2311yescharacteristic
102C2×D4×C107—21no—
103C2×D4×C107—21no—
104C2×D4×C107D411yescharacteristic
105C2×D4×C107C2311yescharacteristic
106C22×C4×C107—21no—
107C22×C4×C107—21no—
108C22×C4×C107—21no—
109C22×C4×C107C2311yescharacteristic
110C22×C4×C107C2311yescharacteristic
111C4⋊C4×C107—21no—
112C4⋊C4×C107—21no—
113C4⋊C4×C107—21no—
114C4⋊C4×C107—21no—
115C4⋊C4×C107D411yescharacteristic
116C4⋊C4×C107C2311yescharacteristic
117C22⋊C4×C107—21no—
118C22⋊C4×C107—21no—
119C22⋊C4×C107—21no—
120C22⋊C4×C107D411yescharacteristic
121C22⋊C4×C107D411yescharacteristic
122C42×C107C2311yescharacteristic
123C23×C107—21no—
124C23×C107C2×D411yescharacteristic
125C23×C107C2×D411yescharacteristic
126C22×D107—21no—
127Q8×C107—21no—
128Q8×C107—21no—
129Q8×C107—21no—
130Q8×C107—21no—
131D4×C107—21no—
132D4×C107—41no—
133D4×C107—42no—
134D4×C107—42no—
135D4×C107C2×D412yes—
136C2×C4×C107—21no—
137C2×C4×C107—21no—
138C2×C4×C107—21no—
139C2×C4×C107—21no—
140C2×C4×C107—21no—
141C2×C4×C107—21no—
142C2×C4×C107—21no—
143C2×C4×C107—42no—
144C2×C4×C107—42no—
145C2×C4×C107—42no—
146C2×C4×C107C2×D411yescharacteristic
147C2×C4×C107C2×D411yescharacteristic
148C2×C4×C107C2×D411yescharacteristic
149C2×C4×C107C2×D411yescharacteristic
150C2×C4×C107C2411yescharacteristic
151C22⋊D107—42no—
152C22⋊D107—42no—
153C2×Dic107—21no—
154C2×Dic107—21no—
155C2×Dic107—21no—
156C2×Dic107—42no—
157D428—21no—
158D428—21no—
159C4×D107—42no—
160Dic214—21no—
161Dic214—21no—
162Dic214—21no—
163Dic214—42no—
164Dic214C2×D412yes—
165C107⋊C8—21no—
166C107⋊C8—21no—
167C107⋊C8—21no—
168C107⋊C8C2×D412yes—
169C22×C107—21no—
170C22×C107—21no—
171C22×C107—22no—
172C22×C107—21no—
173C22×C107—21no—
174C22×C107—21no—
175C22×C107—21no—
176C22×C107—41no—
177C22×C107—41no—
178C22×C107C22×D411yescharacteristic
179C2×D107—41no—
180C2×D107—41no—
181C2×D107—41no—
182C4×C107—21no—
183C4×C107—21no—
184C4×C107—21no—
185C4×C107—21no—
186C4×C107—41no—
187C4×C107—41no—
188C4×C107—41no—
189C4×C107C22×D411yescharacteristic
190C4×C107H(5,𝔽2)11yescharacteristic
191Dic107—22no—
192Dic107—41no—
193Dic107—41no—
194C2×C107—41no—
195C2×C107—41no—
196C2×C107—42no—
197C2×C107D4211yescharacteristic
198C2×C107#57511yescharacteristic
199C2×C107#57811yescharacteristic
200D107—81no—
201#3269—1071nomaximal, Sylow
202C107#326911yescharacteristic, complemented, Sylow
203C2×#137—1071no—
204C2×#135—1071no—
205#548—1071no—
206#547D10711yescharacteristic, complemented
207#497—1071no—
208#468—1071no—
209#467—1071no—
210#463—1071no—
211#462—1071no—
212#452—1071no—
213#452—1071no—
214#451—1071no—
215#449—1071no—
216#440—1071no—
217#435—1071no—
218C2×#41—1071no—
219C2×#41C2×D10711yescharacteristic
220C2×#41C2×D10711yescharacteristic
221C22×Q8—1071no—
222#137—2144no—
223#135—2144no—
224C2×Q16—1071no—
225C2×Q16—1071no—
226C2×Q16—1071no—
227C2×Q16—1071no—
228C2×SD16—1071no—
229C2×SD16—1071no—
230C2×SD16—1071no—
231C2×SD16—1071no—
232C2×D8—1071no—
233C2×M16—1071no—
234C22×C8—1071no—
235C4⋊Q8C2×D10711yescharacteristic
236#124—1071no—
237#124—2141no—
238#123—21no—
239#123—21no—
240C22⋊Q8—1071no—
241C22⋊Q8—2141no—
242C22⋊Q8C2×D10711yescharacteristic
243C22⋊Q8C2×D10711yescharacteristic
244#121—21no—
245#121—1071no—
246#121—2141no—
247C4×Q8—1071no—
248C4×D4—2141no—
249C4×D4C2×D10711yescharacteristic
250C4×D4C2×D10711yescharacteristic
251C2×C4⋊C4—21no—
252#107—1071no—
253C4⋊C8—1071no—
254#103—1071no—
255#103—1071no—
256#103—1071no—
257#103—1071no—
258#102—1071no—
259#102—1071no—
260C22⋊C8—1071no—
261C22⋊C8—1071no—
262#97—1071no—
263#41—22no—
264#41—22no—
265#41—41no—
266#41—41no—
267#41—4281no—
268#41—2142no—
269C2×Q8—1071no—
270C2×Q8—1071no—
271C2×Q8—4281no—
272C2×Q8—2142no—
273C2×Q8—2141no—
274C2×Q8C22×D10711yescharacteristic
275C2×Q8C22×D10711yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—1071no—
279C2×D4—2141no—
280C2×D4—2141no—
281C2×D4C22⋊D10711yescharacteristic
282C2×D4C22×D10711yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—2141no—
287C22×C4—2141no—
288C22×C4C22×D10711yescharacteristic
289C22×C4C22×D10711yescharacteristic
290Q16—4281no—
291Q16—2142no—
292Q16—2142no—
293Q16—2142no—
294SD16—1074no—
295SD16—4281no—
296SD16—2142no—
297SD16—2142no—
298D8—2142no—
299M16—2142no—
300C2×C8—1071no—
301C2×C8—1071no—
302C2×C8—1071no—
303C2×C8—1071no—
304C2×C8—2142no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—1071no—
310C4⋊C4—2141no—
311C4⋊C4—2141no—
312C4⋊C4C22⋊D10711yescharacteristic
313C4⋊C4C22×D10711yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—2141no—
318C22⋊C4—2141no—
319C22⋊C4—2141no—
320C22⋊C4—2141no—
321C22⋊C4C22⋊D10711yescharacteristic
322C22⋊C4C22⋊D10711yescharacteristic
323C42—2141no—
324C42C22×D10711yescharacteristic
325C23—21no—
326C23—2141no—
327C23C2×C22⋊D10711yescharacteristic
328C23C2×C22⋊D10711yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—21no—
333Q8—1072no—
334Q8—4282no—
335Q8—2141no—
336Q8—2141no—
337Q8—2141no—
338D4—21no—
339D4—41no—
340D4—42no—
341D4—42no—
342D4—4282no—
343D4—4282no—
344D4—2141no—
345D4—2141no—
346D4D4×D10712yes—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—4282no—
358C2×C4—4282no—
359C2×C4—2141no—
360C2×C4—2141no—
361C2×C4—2141no—
362C2×C4C2×C22⋊D10711yescharacteristic
363C2×C4C2×C22⋊D10711yescharacteristic
364C2×C4C2×C22⋊D10711yescharacteristic
365C2×C4C2×C22⋊D10711yescharacteristic
366C2×C4C23×D10711yescharacteristic
367C8—1072no—
368C8—2141no—
369C8—2141no—
370C8—2141no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—22no—
378C22—41no—
379C22—41no—
380C22—4281no—
381C22—4281no—
382C22—4281no—
383C22C22×C22⋊D10711yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—4281no—
392C4—4281no—
393C4—2142no—
394C4C2×D4×D10711yescharacteristic
395C4#1681215411yescharacteristic
396C2—41no—
397C2—41no—
398C2—42no—
399C2—8561no—
400C2D4×C22⋊D10711yescharacteristic
401C2#2029342511yescharacteristic
402C2#2029344211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#547×C107C2
socleC22×C107C22×D4
hypercenter#547D107
hypocenterC107#3269
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#16812139C2semidirect
2. C2×#16812129C2semidirect
3. #20293286C2semidirect
4. #20293019C2semidirect
5. #20292875C2semidirect
6. #20292866C2semidirect
7. #20292842C2semidirect
8. #20292834C2semidirect
9. #20292789C2semidirect
10. #20292787C2semidirect
11. #20292782C2semidirect
12. #20292772C2semidirect
13. #20292727C2semidirect
14. #20292699C2semidirect
15. #547×C107C2semidirect
78. #13133806D4semidirect
202. C107#3269semidirect
206. #547D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121392
C2×#168121292
#202932862
#202930192
#202928752
#202928662
#202928422
#202928342
#202927892
#202927872
#202927822
#202927722
#202927272
#202926992
#547×C1072
#3269107