Finite Group Databasebeta

← #28340349

Subgroups — #28340349

Order 13696.

718 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812080C211yesmaximal, characteristic, complemented
2C2×#16812071C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D107C211yesmaximal, characteristic, complemented
4#20292549C211yesmaximal, characteristic, complemented
5#20292536C211yesmaximal, characteristic, complemented
6#20292536C211yesmaximal, characteristic, complemented
7#20292535C211yesmaximal, characteristic, complemented
8#20292527C211yesmaximal, characteristic, complemented
9#20292515C211yesmaximal, characteristic, complemented
10#20292515C211yesmaximal, characteristic, complemented
11#20292513C211yesmaximal, characteristic, complemented
12#20292506C211yesmaximal, characteristic, complemented
13#20292503C211yesmaximal, characteristic, complemented
14#20292498C211yesmaximal, characteristic, complemented
15C2×#123×C107C211yesmaximal, characteristic, complemented
16C22×D428C2211yescharacteristic
17C22×C22⋊D107C2211yescharacteristic
18C22×C22⋊D107C2211yescharacteristic
19C22×C22⋊D107C2211yescharacteristic
20C22×C22⋊D107C2211yescharacteristic
21C22×C4×D107C2211yescharacteristic
22C22×C4×D107C2211yescharacteristic
23#16812080—24no—
24#16812071—24no—
25C4×C22⋊D107—24no—
26C2×C4⋊Dic107C2211yescharacteristic
27C2×Dic107⋊C4C2211yescharacteristic
28C2×Dic107⋊C4C2211yescharacteristic
29C2×C22⋊Dic107C2211yescharacteristic
30C2×C22⋊Dic107C2211yescharacteristic
31C2×C22⋊Dic107C2211yescharacteristic
32C2×C22⋊Dic107C2211yescharacteristic
33C2×(C2×D107)⋊C4C2211yescharacteristic
34C2×(C2×D107)⋊C4C2211yescharacteristic
35C2×(C2×D107)⋊C4C2211yescharacteristic
36C2×(C2×D107)⋊C4C2211yescharacteristic
37C2×(C2×D107)⋊C4C2211yescharacteristic
38C2×(C2×D107)⋊C4C2211yescharacteristic
39C2×C4×Dic107C2211yescharacteristic
40C2×C4×Dic107C2211yescharacteristic
41C107⋊#95C2211yescharacteristic
42C107⋊#95C2211yescharacteristic
43C107⋊#95C2211yescharacteristic
44C107⋊#95C2211yescharacteristic
45C107⋊#95C2211yescharacteristic
46C107⋊#95C2211yescharacteristic
47C22×D4×C107C2211yescharacteristic
48C23×C4×C107C2211yescharacteristic
49#123×C107—24no—
50C2×C4⋊C4×C107C2211yescharacteristic
51C2×C4⋊C4×C107C2211yescharacteristic
52C2×C22⋊C4×C107C2211yescharacteristic
53C2×C22⋊C4×C107C2211yescharacteristic
54C2×C22⋊C4×C107C2211yescharacteristic
55C23×D107C2311yescharacteristic
56C23×D107C2311yescharacteristic
57C2×D428—22no—
58C2×D428—22no—
59C2×D428—22no—
60C2×C22⋊D107—22no—
61C2×C22⋊D107—22no—
62C2×C22⋊D107—22no—
63C2×C22⋊D107—22no—
64C2×C22⋊D107—22no—
65C2×C22⋊D107—22no—
66C2×C22⋊D107—22no—
67C2×C22⋊D107—22no—
68C2×C22⋊D107—41no—
69C2×C22⋊D107—41no—
70C2×C22⋊D107—41no—
71C2×C22⋊D107—41no—
72C22×Dic107C2311yescharacteristic
73C22×Dic107C2311yescharacteristic
74C22×Dic107C2311yescharacteristic
75C22×Dic107C2311yescharacteristic
76C22×Dic107C2311yescharacteristic
77C22×Dic107C2311yescharacteristic
78C2×C4×D107—22no—
79C2×C4×D107—22no—
80C2×C4×D107—22no—
81C2×C4×D107—22no—
82C2×C4×D107—41no—
83C2×C4×D107—41no—
84C4⋊Dic107—22no—
85Dic107⋊C4—22no—
86Dic107⋊C4—41no—
87C22⋊Dic107—22no—
88C22⋊Dic107—22no—
89C22⋊Dic107—41no—
90C22⋊Dic107D414yescomplemented
91(C2×D107)⋊C4—22no—
92(C2×D107)⋊C4—22no—
93(C2×D107)⋊C4—22no—
94(C2×D107)⋊C4—22no—
95(C2×D107)⋊C4—41no—
96(C2×D107)⋊C4—41no—
97C4×Dic107—22no—
98C4×Dic107—22no—
99C24×C107C2311yescharacteristic
100C24×C107C2311yescharacteristic
101C2×D4×C107—22no—
102C2×D4×C107—22no—
103C2×D4×C107—41no—
104C22×C4×C107—22no—
105C22×C4×C107—22no—
106C22×C4×C107—41no—
107C22×C4×C107C2311yescharacteristic
108C22×C4×C107C2311yescharacteristic
109C22×C4×C107C2311yescharacteristic
110C22×C4×C107C2311yescharacteristic
111C22×C4×C107C2311yescharacteristic
112C4⋊C4×C107—22no—
113C4⋊C4×C107D414yescomplemented
114C22⋊C4×C107—22no—
115C22⋊C4×C107—22no—
116C22⋊C4×C107—22no—
117C23×C107—21no—
118C23×C107—21no—
119C23×C107—21no—
120C23×C107—21no—
121C23×C107—21no—
122C23×C107—21no—
123C23×C107—21no—
124C23×C107—21no—
125C23×C107—21no—
126C23×C107—21no—
127C23×C107—21no—
128C23×C107—21no—
129C23×C107—21no—
130C23×C107C2×D412yes—
131C23×C107C2411yescharacteristic
132C22×D107—21no—
133C22×D107—21no—
134C22×D107—21no—
135C22×D107—21no—
136C22×D107—21no—
137C22×D107—21no—
138C22×D107—21no—
139C22×D107—21no—
140C22×D107—21no—
141C22×D107—21no—
142C22×D107—21no—
143C22×D107—21no—
144C22×D107#4112yes—
145C22×D107#4112yes—
146D4×C107—44no—
147C2×C4×C107—21no—
148C2×C4×C107—21no—
149C2×C4×C107—21no—
150C2×C4×C107—21no—
151C2×C4×C107—21no—
152C2×C4×C107—21no—
153C2×C4×C107—21no—
154C2×C4×C107—21no—
155C2×C4×C107—21no—
156C2×C4×C107—21no—
157C2×C4×C107—21no—
158C2×C4×C107—21no—
159C2×C4×C107—44no—
160C2×C4×C107C2×D412yes—
161C2×C4×C107C2×D412yes—
162C2×C4×C107C2×D412yes—
163C22⋊D107—28no—
164C22⋊D107—44no—
165C22⋊D107—44no—
166C22⋊D107—44no—
167C2×Dic107—21no—
168C2×Dic107—21no—
169C2×Dic107—21no—
170C2×Dic107—21no—
171C2×Dic107—21no—
172C2×Dic107—21no—
173C2×Dic107—21no—
174C2×Dic107—21no—
175C2×Dic107—21no—
176C2×Dic107—21no—
177C2×Dic107—21no—
178C2×Dic107—21no—
179C2×Dic107—21no—
180C2×Dic107—21no—
181C2×Dic107—21no—
182C2×Dic107C2×D412yes—
183C2×Dic107C2×D412yes—
184C2×Dic107#4112yes—
185D428—28no—
186C4×D107—44no—
187C4×D107—44no—
188C22×C107—22no—
189C22×C107—22no—
190C22×C107—22no—
191C22×C107—41no—
192C22×C107—41no—
193C22×C107—41no—
194C22×C107—41no—
195C22×C107—41no—
196C22×C107—41no—
197C22×C107—41no—
198C22×C107—41no—
199C22×C107—41no—
200C22×C107—41no—
201C22×C107—41no—
202C22×C107C22×D411yescharacteristic
203C22×C107C22×D411yescharacteristic
204C22×C107C2×#4111yescharacteristic
205C22×C107C2×#4111yescharacteristic
206C22×C107C2×#4111yescharacteristic
207C22×C107H(5,𝔽2)11yescharacteristic
208C22×C107H(5,𝔽2)11yescharacteristic
209C2×D107—22no—
210C2×D107—22no—
211C2×D107—22no—
212C2×D107—22no—
213C2×D107—22no—
214C2×D107—22no—
215C2×D107—41no—
216C2×D107—41no—
217C2×D107—41no—
218C2×D107—41no—
219C2×D107—41no—
220C2×D107—41no—
221C2×D107—41no—
222C2×D107—41no—
223C4×C107—22no—
224C4×C107—22no—
225C4×C107—22no—
226C4×C107—41no—
227C4×C107—41no—
228Dic107—22no—
229Dic107—22no—
230Dic107—22no—
231Dic107—41no—
232Dic107—41no—
233Dic107—41no—
234C2×C107—42no—
235C2×C107—81no—
236C2×C107#52511yescharacteristic
237C2×C107#52511yescharacteristic
238C2×C107#53811yescharacteristic
239C2×C107D4211yescharacteristic
240C2×C107#54611yescharacteristic
241C2×C107#55011yescharacteristic
242C2×C107#55311yescharacteristic
243D107—42no—
244D107—42no—
245#2456—1071nomaximal, Sylow
246C107#245611yescharacteristic, complemented, Sylow
247C2×#123D10711yescharacteristic, complemented
248C2×#121—1071no—
249C2×C4×D4—1071no—
250C2×C4×D4—1071no—
251#396—1071no—
252#392—1071no—
253#392—1071no—
254#392—1071no—
255#390—1071no—
256#388—1071no—
257#388—1071no—
258#387—1071no—
259#386—1071no—
260#386—1071no—
261#385—1071no—
262C22×D4—1071no—
263C22×D4—1071no—
264C22×D4—1071no—
265C22×D4—1071no—
266C22×D4—1071no—
267C22×D4C2×D10711yescharacteristic
268C23×C4—1071no—
269C23×C4—1071no—
270C23×C4C2×D10711yescharacteristic
271#123—24no—
272#121—2144no—
273C4×D4—2144no—
274C4×D4—2144no—
275C2×C4⋊C4—1071no—
276C2×C4⋊C4—1071no—
277C2×C4⋊C4—1071no—
278C2×C4⋊C4C2×D10711yescharacteristic
279C2×C4⋊C4C2×D10711yescharacteristic
280C2×C22⋊C4—1071no—
281C2×C22⋊C4—1071no—
282C2×C22⋊C4—1071no—
283C2×C22⋊C4—1071no—
284C2×C22⋊C4—1071no—
285C2×C22⋊C4—1071no—
286C2×C22⋊C4—1071no—
287C2×C22⋊C4—1071no—
288C2×C22⋊C4—1071no—
289C2×C22⋊C4—1071no—
290C2×C22⋊C4C2×D10711yescharacteristic
291C2×C22⋊C4C2×D10711yescharacteristic
292C2×C22⋊C4C2×D10711yescharacteristic
293C2×C42—1071no—
294C2×C42—1071no—
295#95—1071no—
296#95—1071no—
297#95—1071no—
298#95—1071no—
299#95—1071no—
300#95—1071no—
301C24—1071no—
302C24—1071no—
303C24C22×D10711yescharacteristic
304C24C22×D10711yescharacteristic
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—41no—
308C2×D4—4281no—
309C2×D4—4281no—
310C2×D4—4281no—
311C2×D4—4281no—
312C2×D4—2142no—
313C2×D4—2142no—
314C2×D4—2142no—
315C2×D4—2142no—
316C2×D4—2142no—
317C2×D4—2142no—
318C2×D4—2142no—
319C2×D4—2142no—
320C2×D4—2142no—
321C2×D4—2142no—
322C2×D4—2142no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—41no—
326C22×C4—1071no—
327C22×C4—1071no—
328C22×C4—1071no—
329C22×C4—1071no—
330C22×C4—1071no—
331C22×C4—1071no—
332C22×C4—4281no—
333C22×C4—4281no—
334C22×C4—2142no—
335C22×C4—2142no—
336C22×C4—2142no—
337C22×C4—2142no—
338C22×C4C22×D10711yescharacteristic
339C22×C4C22×D10711yescharacteristic
340C22×C4C22×D10711yescharacteristic
341C22×C4C22×D10711yescharacteristic
342C22×C4C22×D10711yescharacteristic
343C4⋊C4—22no—
344C4⋊C4—4281no—
345C4⋊C4—2142no—
346C4⋊C4—2142no—
347C4⋊C4C22⋊D10714yescomplemented
348C22⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—1074no—
352C22⋊C4—4281no—
353C22⋊C4—4281no—
354C22⋊C4—4281no—
355C22⋊C4—2142no—
356C22⋊C4—2142no—
357C22⋊C4—2142no—
358C22⋊C4—2142no—
359C22⋊C4—2142no—
360C22⋊C4—2142no—
361C42—2142no—
362C42—2142no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—1072no—
377C23—1072no—
378C23—2141no—
379C23—2141no—
380C23—2141no—
381C23—2141no—
382C23—2141no—
383C23—2141no—
384C23—2141no—
385C23—2141no—
386C23—2141no—
387C23—2141no—
388C23—2141no—
389C23—2141no—
390C23D4×D10712yes—
391C23C23×D10711yescharacteristic
392D4—44no—
393D4—4284no—
394D4—4284no—
395D4—4284no—
396D4—2148no—
397D4—2148no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—44no—
411C2×C4—1072no—
412C2×C4—1072no—
413C2×C4—1072no—
414C2×C4—4284no—
415C2×C4—4284no—
416C2×C4—2141no—
417C2×C4—2141no—
418C2×C4—2141no—
419C2×C4—2141no—
420C2×C4—2141no—
421C2×C4—2141no—
422C2×C4—2141no—
423C2×C4—2141no—
424C2×C4—2141no—
425C2×C4—2141no—
426C2×C4—2141no—
427C2×C4—2141no—
428C2×C4—2141no—
429C2×C4—2141no—
430C2×C4—2141no—
431C2×C4C2×C22⋊D10712yes—
432C2×C4C2×C22⋊D10712yes—
433C2×C4C2×C22⋊D10712yes—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—4281no—
449C22—4281no—
450C22—4281no—
451C22—4281no—
452C22—4281no—
453C22—4281no—
454C22—4281no—
455C22—4281no—
456C22—2142no—
457C22—2142no—
458C22—2142no—
459C22—2142no—
460C22—2142no—
461C22—2142no—
462C22C2×D4×D10711yescharacteristic
463C22C22×C22⋊D10711yescharacteristic
464C22#41×D10711yescharacteristic
465C22#41×D10711yescharacteristic
466C22#41×D10711yescharacteristic
467C22#1681215411yescharacteristic
468C22#1681215511yescharacteristic
469C4—22no—
470C4—22no—
471C4—22no—
472C4—41no—
473C4—41no—
474C4—4281no—
475C4—4281no—
476C4—4281no—
477C4—2142no—
478C4—2142no—
479C4—2142no—
480C2—42no—
481C2—81no—
482C2—4282no—
483C2—4282no—
484C2#2029313411yescharacteristic
485C2#2029313511yescharacteristic
486C2#2029321311yescharacteristic
487C2D4×C22⋊D10711yescharacteristic
488C2#2029326911yescharacteristic
489C2#2029329611yescharacteristic
490C2#2029332211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×#123×C107C2
hypercenterC2×#123D107
hypocenterC107#2456
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#16812080C2semidirect
2. C2×#16812071C2semidirect
3. C2×C4×C22⋊D107C2semidirect
4. #20292549C2semidirect
5. #20292536C2semidirect
6. #20292536C2semidirect
7. #20292535C2semidirect
8. #20292527C2semidirect
9. #20292515C2semidirect
10. #20292515C2semidirect
11. #20292513C2semidirect
12. #20292506C2semidirect
13. #20292503C2semidirect
14. #20292498C2semidirect
15. C2×#123×C107C2semidirect
90. C22⋊Dic107D4semidirect
113. C4⋊C4×C107D4semidirect
246. C107#2456semidirect
247. C2×#123D107semidirect
347. C4⋊C4C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120802
C2×#168120712
C2×C4×C22⋊D1072
#202925492
#202925362
#202925352
#202925272
#202925152
#202925132
#202925062
#202925032
#202924982
C2×#123×C1072
#2456107