Finite Group Databasebeta

← #28348363

Subgroups — #28348363

Order 13696.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293323C211yesmaximal, characteristic, complemented
2#20292981C211yesmaximal, characteristic, complemented
3#20292965C211yesmaximal, characteristic, complemented
4#20292943C211yesmaximal, characteristic, complemented
5#20292934C211yesmaximal, characteristic, complemented
6#20292861C211yesmaximal, characteristic, complemented
7#20292858C211yesmaximal, characteristic, complemented
8#20292839C211yesmaximal, characteristic, complemented
9#20292772C211yesmaximal, characteristic, complemented
10#20292766C211yesmaximal, characteristic, complemented
11C4×#13133805C211yesmaximal, characteristic, complemented
12D4×C107⋊C8C211yesmaximal, characteristic, complemented
13#20292623C211yesmaximal, characteristic, complemented
14C2×#16812033C211yesmaximal, characteristic, complemented
15#548×C107C211yesmaximal, characteristic, complemented
16C2×#13133805C2211yescharacteristic
17C2×#13133805C2211yescharacteristic
18C2×#13133803C2211yescharacteristic
19C22×C107⋊C8C2211yescharacteristic
20C22×C107⋊C8C2211yescharacteristic
21#16812104—21no—
22#16812101C2211yescharacteristic
23#16812091—21no—
24#16812083C2211yescharacteristic
25#16812083C2211yescharacteristic
26#16812081—21no—
27C4×D428C2211yescharacteristic
28(C22⋊D107)⋊C4—21no—
29#16812068C2211yescharacteristic
30C2×C4⋊Dic107C2211yescharacteristic
31#16812042C2211yescharacteristic
32#16812040C2211yescharacteristic
33#16812040C2211yescharacteristic
34C4⋊(C107⋊C8)C2211yescharacteristic
35#16812033C2211yescharacteristic
36#16812033C2211yescharacteristic
37#16812033C2211yescharacteristic
38#16812033C2211yescharacteristic
39#16812033C2211yescharacteristic
40#16812033C2211yescharacteristic
41#16812030C2211yescharacteristic
42#16812030C2211yescharacteristic
43#16812030C2211yescharacteristic
44#16812029C2211yescharacteristic
45C22⋊(C107⋊C8)C2211yescharacteristic
46C22⋊(C107⋊C8)C2211yescharacteristic
47C4×C107⋊C8C2211yescharacteristic
48C2×#41×C107C2211yescharacteristic
49C22×Q8×C107C2211yescharacteristic
50#124×C107—21no—
51#124×C107C2211yescharacteristic
52C22⋊Q8×C107—21no—
53C22⋊Q8×C107C2211yescharacteristic
54#121×C107—21no—
55#121×C107C2211yescharacteristic
56C4×Q8×C107C2211yescharacteristic
57C4×D4×C107—21no—
58C4×D4×C107C2211yescharacteristic
59C2×D428C2311yescharacteristic
60C2×C22⋊D107—21no—
61C2×C22⋊D107—21no—
62C22×Dic107—21no—
63C2×C4×D107—21no—
64#13133805—22no—
65#13133805—41no—
66#13133803—41no—
67C2×C107⋊C8—41no—
68C2×C107⋊C8—41no—
69C2×C107⋊C8C2311yescharacteristic
70C2×C107⋊C8C2311yescharacteristic
71C2×C107⋊C8C2311yescharacteristic
72C2×C107⋊C8C2311yescharacteristic
73C4⋊Dic107—21no—
74C4⋊Dic107C2311yescharacteristic
75C4⋊Dic107C2311yescharacteristic
76C4⋊Dic107C2311yescharacteristic
77Dic107⋊C4—21no—
78Dic107⋊C4—21no—
79C22⋊Dic107—21no—
80(C2×D107)⋊C4—21no—
81(C2×D107)⋊C4—21no—
82(C2×D107)⋊C4—21no—
83C4×Dic107—21no—
84#41×C107—22no—
85#41×C107—41no—
86C2×Q8×C107—21no—
87C2×Q8×C107—22no—
88C2×Q8×C107—41no—
89C2×Q8×C107C2311yescharacteristic
90C2×Q8×C107C2311yescharacteristic
91C2×D4×C107—21no—
92C2×D4×C107—21no—
93C2×D4×C107D411yescharacteristic
94C2×D4×C107C2311yescharacteristic
95C22×C4×C107—21no—
96C22×C4×C107—21no—
97C22×C4×C107C2311yescharacteristic
98C22×C4×C107C2311yescharacteristic
99C4⋊C4×C107—21no—
100C4⋊C4×C107—21no—
101C4⋊C4×C107D411yescharacteristic
102C4⋊C4×C107C2311yescharacteristic
103C22⋊C4×C107—21no—
104C22⋊C4×C107—21no—
105C22⋊C4×C107—21no—
106C22⋊C4×C107—21no—
107C22⋊C4×C107D411yescharacteristic
108C22⋊C4×C107D411yescharacteristic
109C42×C107—21no—
110C42×C107C2311yescharacteristic
111C23×C107—21no—
112C23×C107C2×D411yescharacteristic
113C23×C107C2×D411yescharacteristic
114C22×D107—21no—
115Q8×C107—21no—
116Q8×C107—21no—
117Q8×C107—21no—
118Q8×C107—42no—
119Q8×C107#4112yes—
120D4×C107—22no—
121D4×C107—21no—
122D4×C107—21no—
123D4×C107—42no—
124D4×C107—42no—
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107—21no—
128C2×C4×C107—21no—
129C2×C4×C107—21no—
130C2×C4×C107—21no—
131C2×C4×C107—21no—
132C2×C4×C107—42no—
133C2×C4×C107—42no—
134C2×C4×C107C2×D411yescharacteristic
135C2×C4×C107C2×D411yescharacteristic
136C2×C4×C107C2×D411yescharacteristic
137C2×C4×C107C2×D411yescharacteristic
138C2×C4×C107C2411yescharacteristic
139C22⋊D107—42no—
140C22⋊D107—81no—
141C2×Dic107—21no—
142C2×Dic107—21no—
143C2×Dic107—21no—
144C2×Dic107—81no—
145D428—21no—
146D428—21no—
147C4×D107—81no—
148C107⋊C8—21no—
149C107⋊C8—21no—
150C107⋊C8—21no—
151C107⋊C8—21no—
152C22×C107—21no—
153C22×C107—21no—
154C22×C107—21no—
155C22×C107—21no—
156C22×C107—21no—
157C22×C107—41no—
158C22×C107—41no—
159C22×C107—41no—
160C22×C107#13512yes—
161C22×C107C22×D411yescharacteristic
162C2×D107—41no—
163C2×D107—41no—
164C2×D107—41no—
165C4×C107—21no—
166C4×C107—21no—
167C4×C107—22no—
168C4×C107—21no—
169C4×C107—21no—
170C4×C107—41no—
171C4×C107—41no—
172C4×C107C2×#4111yescharacteristic
173C4×C107H(5,𝔽2)11yescharacteristic
174Dic107—41no—
175Dic107—41no—
176Dic107—41no—
177C2×C107—22no—
178C2×C107—41no—
179C2×C107—81no—
180C2×C107#54611yescharacteristic
181C2×C107C2×#13511yescharacteristic
182C2×C107#57711yescharacteristic
183D107—81no—
184#3301—1071nomaximal, Sylow
185C107#330111yescharacteristic, complemented, Sylow
186#553—1071no—
187#548D10711yescharacteristic, complemented
188#488—1071no—
189#485—1071no—
190#481—1071no—
191#479—1071no—
192#466—1071no—
193#465—1071no—
194#463—1071no—
195#449—1071no—
196#448—1071no—
197C4×SD16—1071no—
198C8×D4—1071no—
199#416—1071no—
200C2×#103—1071no—
201C2×#41C2×D10711yescharacteristic
202C22×Q8C2×D10711yescharacteristic
203C2×SD16—1071no—
204C2×SD16—1071no—
205C2×D8—1071no—
206C22×C8—1071no—
207C22×C8—1071no—
208#126—2141no—
209#125—1071no—
210#124—21no—
211#124C2×D10711yescharacteristic
212#123—2141no—
213C22⋊Q8—21no—
214C22⋊Q8C2×D10711yescharacteristic
215#121—21no—
216#121—1071no—
217#121—1071no—
218#121—2141no—
219#121C2×D10711yescharacteristic
220C4×Q8C2×D10711yescharacteristic
221C4×D4—21no—
222C4×D4—1071no—
223C4×D4—2141no—
224C4×D4C2×D10711yescharacteristic
225#117—1071no—
226C2×C4⋊C4—1071no—
227#107—1071no—
228#106—1071no—
229#106—1071no—
230C4⋊C8—1071no—
231#103—1071no—
232#103—1071no—
233#103—1071no—
234#103—1071no—
235#103—1071no—
236#103—1071no—
237#102—1071no—
238#102—1071no—
239#102—1071no—
240#102—1071no—
241C22⋊C8—1071no—
242C22⋊C8—1071no—
243C4×C8—1071no—
244#41—22no—
245#41—41no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C22×D10711yescharacteristic
250C2×Q8C22×D10711yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—1071no—
254C2×D4—2141no—
255C2×D4—2141no—
256C2×D4C22⋊D10711yescharacteristic
257C2×D4C22×D10711yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—2141no—
261C22×C4—2141no—
262C22×C4C22×D10711yescharacteristic
263C22×C4C22×D10711yescharacteristic
264SD16—4281no—
265SD16—2142no—
266D8—4281no—
267C2×C8—1071no—
268C2×C8—1071no—
269C2×C8—1071no—
270C2×C8—1071no—
271C2×C8—4281no—
272C2×C8—4281no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—1071no—
276C4⋊C4—1071no—
277C4⋊C4—1071no—
278C4⋊C4—2141no—
279C4⋊C4—2141no—
280C4⋊C4—2141no—
281C4⋊C4C22⋊D10711yescharacteristic
282C4⋊C4C22×D10711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—2141no—
288C22⋊C4—2141no—
289C22⋊C4—2141no—
290C22⋊C4—2141no—
291C22⋊C4C22⋊D10711yescharacteristic
292C22⋊C4C22⋊D10711yescharacteristic
293C42—21no—
294C42—2141no—
295C42C22×D10711yescharacteristic
296C23—21no—
297C23—2141no—
298C23C2×C22⋊D10711yescharacteristic
299C23C2×C22⋊D10711yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#1313381712yes—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—42no—
310D4—8561no—
311D4—4282no—
312D4—2141no—
313D4—2141no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—8561no—
324C2×C4—8561no—
325C2×C4—2141no—
326C2×C4—2141no—
327C2×C4—2141no—
328C2×C4C2×C22⋊D10711yescharacteristic
329C2×C4C2×C22⋊D10711yescharacteristic
330C2×C4C2×C22⋊D10711yescharacteristic
331C2×C4C2×C22⋊D10711yescharacteristic
332C2×C4C23×D10711yescharacteristic
333C8—2141no—
334C8—2141no—
335C8—2141no—
336C8—2141no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—4281no—
346C22—4281no—
347C22—4281no—
348C22#1681212912yes—
349C22C22×C22⋊D10711yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—4281no—
358C4—4281no—
359C4—4281no—
360C4C2×#1313381711yescharacteristic
361C4#1681215411yescharacteristic
362C2—22no—
363C2—41no—
364C2—81no—
365C2—8561no—
366C2#2029327111yescharacteristic
367C2C2×#1681212911yescharacteristic
368C2#2029343511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#548×C107C2
socleC22×C107C22×D4
hypercenter#548D107
hypocenterC107#3301
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. #20293323C2semidirect
2. #20292981C2semidirect
3. #20292965C2semidirect
4. #20292943C2semidirect
5. #20292934C2semidirect
6. #20292861C2semidirect
7. #20292858C2semidirect
8. #20292839C2semidirect
9. #20292772C2semidirect
10. #20292766C2semidirect
11. C4×#13133805C2semidirect
12. D4×C107⋊C8C2semidirect
13. #20292623C2semidirect
14. C2×#16812033C2semidirect
15. #548×C107C2semidirect
185. C107#3301semidirect
187. #548D107semidirect

Maximal subgroups

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

subgroupindex
#202933232
#202929812
#202929652
#202929432
#202929342
#202928612
#202928582
#202928392
#202927722
#202927662
C4×#131338052
D4×C107⋊C82
#202926232
C2×#168120332
#548×C1072
#3301107