Finite Group Databasebeta

← #28348010

Subgroups — #28348010

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×#16812130C211yesmaximal, characteristic, complemented
2#20293288C211yesmaximal, characteristic, complemented
3#20293276C211yesmaximal, characteristic, complemented
4#20293020C211yesmaximal, characteristic, complemented
5#20292873C211yesmaximal, characteristic, complemented
6#20292865C211yesmaximal, characteristic, complemented
7#20292844C211yesmaximal, characteristic, complemented
8#20292835C211yesmaximal, characteristic, complemented
9#20292788C211yesmaximal, characteristic, complemented
10#20292786C211yesmaximal, characteristic, complemented
11#20292783C211yesmaximal, characteristic, complemented
12#20292772C211yesmaximal, characteristic, complemented
13#20292728C211yesmaximal, characteristic, complemented
14#20292696C211yesmaximal, characteristic, complemented
15C2×#137×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133817C2211yescharacteristic
18#16812130—24no—
19C2×#13133808C2211yescharacteristic
20C2×#13133808C2211yescharacteristic
21C2×#13133807C2211yescharacteristic
22C2×#13133805C2211yescharacteristic
23C2×#13133803C2211yescharacteristic
24C2×C8×D107C2211yescharacteristic
25#16812107C2211yescharacteristic
26#16812098—21no—
27#16812098C2211yescharacteristic
28#16812092—21no—
29#16812090—21no—
30#16812088C2211yescharacteristic
31#16812085—21no—
32#16812085C2211yescharacteristic
33#16812085C2211yescharacteristic
34#16812083—21no—
35#16812083C2211yescharacteristic
36#16812081—21no—
37Q8×Dic107C2211yescharacteristic
38D4×Dic107C2211yescharacteristic
39C4×C22⋊D107—21no—
40C4×C22⋊D107C2211yescharacteristic
41C2×Dic107⋊C4—21no—
42#16812043C2211yescharacteristic
43Dic107⋊C8C2211yescharacteristic
44#16812034C2211yescharacteristic
45#16812033C2211yescharacteristic
46#16812033C2211yescharacteristic
47#16812033C2211yescharacteristic
48#16812031C2211yescharacteristic
49#16812030C2211yescharacteristic
50C22⋊(C107⋊C8)C2211yescharacteristic
51(C2×D107)⋊C8C2211yescharacteristic
52#16812019C2211yescharacteristic
53C2×#41×C107C2211yescharacteristic
54C22×Q8×C107C2211yescharacteristic
55#137×C107—24no—
56C2×Q16×C107C2211yescharacteristic
57C2×Q16×C107C2211yescharacteristic
58C2×SD16×C107C2211yescharacteristic
59C2×SD16×C107C2211yescharacteristic
60C2×M16×C107C2211yescharacteristic
61#13133819—22no—
62#13133819—41no—
63#13133817—22no—
64#13133817—41no—
65C2×Dic214C2311yescharacteristic
66C2×D428—21no—
67C2×D428C2311yescharacteristic
68C2×C22⋊D107—21no—
69C2×C22⋊D107—21no—
70C2×C22⋊D107D411yescharacteristic
71C22×Dic107—21no—
72C22×Dic107—21no—
73C2×C4×D107—21no—
74C2×C4×D107C2311yescharacteristic
75#13133808—22no—
76#13133808—41no—
77#13133807—22no—
78#13133805—41no—
79#13133803—22no—
80C2×C107⋊C8C2311yescharacteristic
81C2×C107⋊C8C2311yescharacteristic
82C8×D107—22no—
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⋊Dic107D411yescharacteristic
95(C2×D107)⋊C4—21no—
96(C2×D107)⋊C4—21no—
97(C2×D107)⋊C4—21no—
98(C2×D107)⋊C4—21no—
99(C2×D107)⋊C4—21no—
100(C2×D107)⋊C4D411yescharacteristic
101C4×Dic107—21no—
102C4×Dic107C2311yescharacteristic
103#41×C107—22no—
104#41×C107—41no—
105C2×Q8×C107—21no—
106C2×Q8×C107—22no—
107C2×Q8×C107—41no—
108C2×Q8×C107C2311yescharacteristic
109C2×Q8×C107C2311yescharacteristic
110C2×Q8×C107C2311yescharacteristic
111C2×D4×C107—21no—
112C2×D4×C107C2311yescharacteristic
113C22×C4×C107—21no—
114C22×C4×C107—21no—
115C22×C4×C107C2311yescharacteristic
116Q16×C107—22no—
117Q16×C107—22no—
118SD16×C107—22no—
119SD16×C107D414yescomplemented
120M16×C107—22no—
121C2×C8×C107C2311yescharacteristic
122C2×C8×C107C2311yescharacteristic
123C23×C107—21no—
124C23×C107C2×D411yescharacteristic
125C22×D107—21no—
126C22×D107C2×D411yescharacteristic
127Q8×C107—21no—
128Q8×C107—21no—
129Q8×C107—21no—
130Q8×C107—21no—
131Q8×C107—21no—
132Q8×C107—42no—
133Q8×C107C2×D412yes—
134D4×C107—21no—
135D4×C107—42no—
136D4×C107C2×D412yes—
137C2×C4×C107—21no—
138C2×C4×C107—21no—
139C2×C4×C107—21no—
140C2×C4×C107—21no—
141C2×C4×C107—21no—
142C2×C4×C107—42no—
143C2×C4×C107—42no—
144C2×C4×C107C2×D411yescharacteristic
145C2×C4×C107C2411yescharacteristic
146C22⋊D107—41no—
147C22⋊D107—42no—
148C22⋊D107—42no—
149C2×Dic107—21no—
150C2×Dic107—21no—
151C2×Dic107—21no—
152C2×Dic107—42no—
153C2×Dic107—42no—
154C2×Dic107C2×D411yescharacteristic
155C2×Dic107C2×D411yescharacteristic
156C2×Dic107C2×D411yescharacteristic
157D428—21no—
158D428—21no—
159D428—42no—
160C4×D107—21no—
161C4×D107—21no—
162C4×D107—42no—
163Dic214—21no—
164Dic214—21no—
165C8×C107—21no—
166C8×C107C2×D412yes—
167C107⋊C8—21no—
168C107⋊C8—21no—
169C22×C107—22no—
170C22×C107—21no—
171C22×C107—21no—
172C22×C107—21no—
173C22×C107—41no—
174C22×C107—41no—
175C22×C107C22×D411yescharacteristic
176C2×D107—21no—
177C2×D107—21no—
178C2×D107—21no—
179C2×D107—41no—
180C2×D107—41no—
181C2×D107—41no—
182C4×C107—22no—
183C4×C107—21no—
184C4×C107—41no—
185C4×C107—41no—
186C4×C107C22×D411yescharacteristic
187C4×C107H(5,𝔽2)11yescharacteristic
188Dic107—21no—
189Dic107—21no—
190Dic107—21no—
191Dic107—41no—
192Dic107—41no—
193Dic107—41no—
194C2×C107—41no—
195C2×C107—42no—
196C2×C107D4211yescharacteristic
197C2×C107#57511yescharacteristic
198C2×C107#57811yescharacteristic
199D107—41no—
200D107—81no—
201#3269—1071nomaximal, Sylow
202C107#326911yescharacteristic, complemented, Sylow
203C2×#137D10711yescharacteristic, complemented
204C2×#135—1071no—
205#548—1071no—
206#547—1071no—
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×#41—1071no—
220C2×#41C2×D10711yescharacteristic
221C22×Q8C2×D10711yescharacteristic
222#137—24no—
223#135—2144no—
224C2×Q16—1071no—
225C2×Q16—1071no—
226C2×Q16C2×D10711yescharacteristic
227C2×Q16C2×D10711yescharacteristic
228C2×SD16—1071no—
229C2×SD16—1071no—
230C2×SD16C2×D10711yescharacteristic
231C2×SD16C2×D10711yescharacteristic
232C2×D8—1071no—
233C2×M16C2×D10711yescharacteristic
234C22×C8—1071no—
235C4⋊Q8—1071no—
236#124—1071no—
237#124—2141no—
238#123—2141no—
239#123—2141no—
240C22⋊Q8—1071no—
241C22⋊Q8—1071no—
242C22⋊Q8—1071no—
243C22⋊Q8—2141no—
244#121—1071no—
245#121—2141no—
246#121—2141no—
247C4×Q8—1071no—
248C4×D4—1071no—
249C4×D4—1071no—
250C4×D4—2141no—
251C2×C4⋊C4—2141no—
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—41no—
265#41—4281no—
266#41—4281no—
267#41—2142no—
268#41—2142no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—1071no—
273C2×Q8C22×D10711yescharacteristic
274C2×Q8C22×D10711yescharacteristic
275C2×Q8C22×D10711yescharacteristic
276C2×D4—21no—
277C2×D4—1071no—
278C2×D4—1071no—
279C2×D4—2141no—
280C2×D4—2141no—
281C2×D4—2141no—
282C2×D4C22×D10711yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—1071no—
286C22×C4—2141no—
287C22×C4—2141no—
288C22×C4—2141no—
289C22×C4C22×D10711yescharacteristic
290Q16—22no—
291Q16—22no—
292Q16—4281no—
293Q16—2142no—
294SD16—22no—
295SD16—4281no—
296SD16—2142no—
297SD16C22⋊D10714yescomplemented
298D8—2142no—
299M16—22no—
300C2×C8—1071no—
301C2×C8—1071no—
302C2×C8—2142no—
303C2×C8C22×D10711yescharacteristic
304C2×C8C22×D10711yescharacteristic
305C4⋊C4—1071no—
306C4⋊C4—1071no—
307C4⋊C4—1071no—
308C4⋊C4—2141no—
309C4⋊C4—2141no—
310C4⋊C4—2141no—
311C4⋊C4—2141no—
312C4⋊C4—2141no—
313C4⋊C4—2141no—
314C22⋊C4—1071no—
315C22⋊C4—1071no—
316C22⋊C4—2141no—
317C22⋊C4—2141no—
318C22⋊C4—2141no—
319C22⋊C4—2141no—
320C22⋊C4—2141no—
321C22⋊C4—2141no—
322C22⋊C4—2141no—
323C42—1071no—
324C42—2141no—
325C23—21no—
326C23—1071no—
327C23—2141no—
328C23D4×D10711yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—21no—
333Q8—21no—
334Q8—42no—
335Q8—2141no—
336Q8—2141no—
337Q8C2×C22⋊D10712yes—
338D4—21no—
339D4—42no—
340D4—4282no—
341D4—4282no—
342D4—4282no—
343D4—4281no—
344D4—2141no—
345D4—2141no—
346D4C2×C22⋊D10712yes—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—1071no—
355C2×C4—1071no—
356C2×C4—1071no—
357C2×C4—4282no—
358C2×C4—4282no—
359C2×C4—4282no—
360C2×C4—2141no—
361C2×C4—2141no—
362C2×C4—2141no—
363C2×C4—2141no—
364C2×C4—2141no—
365C2×C4D4×D10711yescharacteristic
366C2×C4C23×D10711yescharacteristic
367C8—21no—
368C8—2141no—
369C8—2141no—
370C8C2×C22⋊D10712yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—22no—
375C22—41no—
376C22—41no—
377C22—4281no—
378C22—4281no—
379C22—4281no—
380C22—2141no—
381C22—2141no—
382C22—2141no—
383C22C2×D4×D10711yescharacteristic
384C4—21no—
385C4—22no—
386C4—41no—
387C4—41no—
388C4—4281no—
389C4—4281no—
390C4—4281no—
391C4—2141no—
392C4—2141no—
393C4—2141no—
394C4C22×C22⋊D10711yescharacteristic
395C4#1681215411yescharacteristic
396C2—41no—
397C2—42no—
398C2—8561no—
399C2—4281no—
400C2D4×C22⋊D10711yescharacteristic
401C2#2029342611yescharacteristic
402C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#137×C107C2
socleC22×C107C22×D4
hypercenterC2×#137D107
hypocenterC107#3269
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#16812130C2semidirect
2. #20293288C2semidirect
3. #20293276C2semidirect
4. #20293020C2semidirect
5. #20292873C2semidirect
6. #20292865C2semidirect
7. #20292844C2semidirect
8. #20292835C2semidirect
9. #20292788C2semidirect
10. #20292786C2semidirect
11. #20292783C2semidirect
12. #20292772C2semidirect
13. #20292728C2semidirect
14. #20292696C2semidirect
15. C2×#137×C107C2semidirect
119. SD16×C107D4semidirect
202. C107#3269semidirect
203. C2×#137D107semidirect
297. SD16C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121302
#202932882
#202932762
#202930202
#202928732
#202928652
#202928442
#202928352
#202927882
#202927862
#202927832
#202927722
#202927282
#202926962
C2×#137×C1072
#3269107