Finite Group Databasebeta

← #28349049

Subgroups — #28349049

Order 13696.

386 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293341C211yesmaximal, characteristic, complemented
2#20293278C211yesmaximal, characteristic, complemented
3#20293024C211yesmaximal, characteristic, complemented
4#20292953C211yesmaximal, characteristic, complemented
5#20292952C211yesmaximal, characteristic, complemented
6#20292944C211yesmaximal, characteristic, complemented
7#20292916C211yesmaximal, characteristic, complemented
8#20292905C211yesmaximal, characteristic, complemented
9#20292879C211yesmaximal, characteristic, complemented
10#20292878C211yesmaximal, characteristic, complemented
11#20292707C211yesmaximal, characteristic, complemented
12#20292702C211yesmaximal, characteristic, complemented
13#20292667C211yesmaximal, characteristic, complemented
14#20292666C211yesmaximal, characteristic, complemented
15#481×C107C211yesmaximal, characteristic, complemented
16C2×#13133818—21no—
17C2×#13133803C2211yescharacteristic
18C2×#13133802C2211yescharacteristic
19C2×#13133801C2211yescharacteristic
20#16812108C2211yescharacteristic
21#16812108C2211yescharacteristic
22#16812096—21no—
23#16812095—21no—
24#16812094—21no—
25#16812092—21no—
26#16812087—21no—
27#16812084C2211yescharacteristic
28#16812083C2211yescharacteristic
29#16812081—21no—
30#16812080C2211yescharacteristic
31D4×Dic107C2211yescharacteristic
32#16812071C2211yescharacteristic
33C4×C22⋊D107C2211yescharacteristic
34#16812068C2211yescharacteristic
35#16812067C2211yescharacteristic
36#16812065—21no—
37C4⋊C4×D107C2211yescharacteristic
38#16812043C2211yescharacteristic
39#16812042C2211yescharacteristic
40#16812042C2211yescharacteristic
41#16812041C2211yescharacteristic
42#16812040C2211yescharacteristic
43#16812040C2211yescharacteristic
44#16812040C2211yescharacteristic
45Dic107⋊C8C2211yescharacteristic
46#16812031C2211yescharacteristic
47#16812031C2211yescharacteristic
48#16812030C2211yescharacteristic
49#16812029C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52#121×C107C2211yescharacteristic
53C2×C4⋊C4×C107C2211yescharacteristic
54#106×C107C2211yescharacteristic
55#106×C107C2211yescharacteristic
56#102×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58C22⋊C8×C107C2211yescharacteristic
59#13133818—44no—
60C2×Dic214—21no—
61C2×Dic214—21no—
62C2×D428—21no—
63C2×D428C2311yescharacteristic
64C2×C22⋊D107—21no—
65C2×C22⋊D107—21no—
66C2×C22⋊D107D411yescharacteristic
67C22×Dic107—21no—
68C2×C4×D107—21no—
69C2×C4×D107—21no—
70C2×C4×D107C2311yescharacteristic
71#13133803—22no—
72#13133802—22no—
73#13133801—22no—
74C2×C107⋊C8C2311yescharacteristic
75C2×C107⋊C8C2311yescharacteristic
76C4⋊Dic107C2311yescharacteristic
77C4⋊Dic107C2311yescharacteristic
78C4⋊Dic107C2311yescharacteristic
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4D411yescharacteristic
83C22⋊Dic107—21no—
84C22⋊Dic107—21no—
85C22⋊Dic107—21no—
86C22⋊Dic107D411yescharacteristic
87(C2×D107)⋊C4—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4D411yescharacteristic
92C4×Dic107—21no—
93C4×Dic107—21no—
94C4×Dic107C2311yescharacteristic
95C2×D4×C107—21no—
96C2×D4×C107C2311yescharacteristic
97C22×C4×C107—21no—
98C22×C4×C107C2311yescharacteristic
99C2×C8×C107C2311yescharacteristic
100C2×C8×C107C2311yescharacteristic
101C4⋊C4×C107—21no—
102C4⋊C4×C107C2311yescharacteristic
103C4⋊C4×C107C2311yescharacteristic
104C4⋊C4×C107C2311yescharacteristic
105C22⋊C4×C107—21no—
106C23×C107—21no—
107C23×C107C2×D411yescharacteristic
108C22×D107—21no—
109C22×D107C2×D411yescharacteristic
110D4×C107—21no—
111D4×C107—21no—
112D4×C107—81no—
113C2×C4×C107—21no—
114C2×C4×C107—21no—
115C2×C4×C107—21no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—42no—
119C2×C4×C107C2×D411yescharacteristic
120C2×C4×C107C2411yescharacteristic
121C22⋊D107—22no—
122C22⋊D107—42no—
123C22⋊D107—81no—
124C2×Dic107—21no—
125C2×Dic107—21no—
126C2×Dic107—21no—
127C2×Dic107—42no—
128C2×Dic107C2×D411yescharacteristic
129C2×Dic107C2×D411yescharacteristic
130C2×Dic107C2×D411yescharacteristic
131D428—21no—
132D428—21no—
133D428—42no—
134C4×D107—21no—
135C4×D107—21no—
136C4×D107—42no—
137C4×D107—42no—
138Dic214—42no—
139Dic214—81no—
140C8×C107—21no—
141C8×C107—21no—
142C107⋊C8—21no—
143C107⋊C8#4112yescomplemented
144C22×C107—21no—
145C22×C107—21no—
146C22×C107—21no—
147C22×C107—41no—
148C22×C107—41no—
149C22×C107—41no—
150C22×C107C22×D411yescharacteristic
151C2×D107—21no—
152C2×D107—21no—
153C2×D107—21no—
154C2×D107—41no—
155C2×D107—41no—
156C2×D107—41no—
157C4×C107—21no—
158C4×C107—41no—
159C4×C107—41no—
160C4×C107—41no—
161C4×C107C2×#4111yescharacteristic
162C4×C1072−1+411yescharacteristic
163Dic107—21no—
164Dic107—21no—
165Dic107—41no—
166Dic107—41no—
167Dic107—41no—
168Dic107AGL(1,ℤ/8)12yes—
169C2×C107—41no—
170C2×C107—81no—
171C2×C107#54711yescharacteristic
172C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
173C2×C107#57711yescharacteristic
174D107—41no—
175D107—81no—
176#3354—1071nomaximal, Sylow
177C107#335411yescharacteristic, complemented, Sylow
178#555—1071no—
179#547—1071no—
180#498—1071no—
181#482—1071no—
182#482—1071no—
183#481—1071no—
184#481D10711yescharacteristic, complemented
185#476—1071no—
186#474—1071no—
187#469—1071no—
188#469—1071no—
189C4×D8—1071no—
190#436—1071no—
191#428—1071no—
192#428—1071no—
193C2×#41—2141no—
194C2×D8—1071no—
195C2×M16—1071no—
196C2×M16—1071no—
197C4⋊Q8—1071no—
198C4⋊Q8—1071no—
199#124—2141no—
200#124—2141no—
201#123—2141no—
202#123—2141no—
203C22⋊Q8—2141no—
204#121—1071no—
205#121—1071no—
206#121—1071no—
207#121—2141no—
208#121C2×D10711yescharacteristic
209C4×D4—1071no—
210C4×D4—1071no—
211C4×D4—1071no—
212#117—1071no—
213#117—1071no—
214#117—2141no—
215C2×C4⋊C4—1071no—
216C2×C4⋊C4C2×D10711yescharacteristic
217#107—1071no—
218#107—1071no—
219#107—1071no—
220#106—1071no—
221#106—1071no—
222#106—1071no—
223#106—1071no—
224#106C2×D10711yescharacteristic
225#106C2×D10711yescharacteristic
226C4⋊C8—1071no—
227#102—1071no—
228#102—1071no—
229#102—1071no—
230#102—1071no—
231#102C2×D10711yescharacteristic
232#102C2×D10711yescharacteristic
233C22⋊C8—1071no—
234C22⋊C8C2×D10711yescharacteristic
235C4×C8—1071no—
236#41—4284no—
237C2×Q8—2141no—
238C2×Q8—2141no—
239C2×D4—21no—
240C2×D4—1071no—
241C2×D4—1071no—
242C2×D4—2141no—
243C2×D4—2141no—
244C2×D4—2141no—
245C2×D4C22×D10711yescharacteristic
246C22×C4—21no—
247C22×C4—1071no—
248C22×C4—2141no—
249C22×C4—2141no—
250C22×C4—2141no—
251C22×C4C22×D10711yescharacteristic
252D8—2142no—
253M16—2142no—
254M16—2142no—
255C2×C8—1071no—
256C2×C8—1071no—
257C2×C8C22×D10711yescharacteristic
258C2×C8C22×D10711yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—1071no—
261C4⋊C4—1071no—
262C4⋊C4—1071no—
263C4⋊C4—1071no—
264C4⋊C4—2141no—
265C4⋊C4—2141no—
266C4⋊C4—2141no—
267C4⋊C4C22×D10711yescharacteristic
268C4⋊C4C22×D10711yescharacteristic
269C4⋊C4C22×D10711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—1071no—
272C22⋊C4—1071no—
273C22⋊C4—2141no—
274C22⋊C4—2141no—
275C22⋊C4—2141no—
276C22⋊C4—2141no—
277C22⋊C4—2141no—
278C22⋊C4—2141no—
279C22⋊C4—2141no—
280C42—1071no—
281C42—2141no—
282C42—2141no—
283C23—21no—
284C23—1071no—
285C23—2141no—
286C23D4×D10711yescharacteristic
287Q8—8561no—
288Q8—4282no—
289D4—21no—
290D4—21no—
291D4—81no—
292D4—8561no—
293D4—4282no—
294D4—4282no—
295D4—2142no—
296D4—2141no—
297D4—2141no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—42no—
304C2×C4—1071no—
305C2×C4—1071no—
306C2×C4—1071no—
307C2×C4—4282no—
308C2×C4—4282no—
309C2×C4—4282no—
310C2×C4—2141no—
311C2×C4—2141no—
312C2×C4—2141no—
313C2×C4—2141no—
314C2×C4—2141no—
315C2×C4D4×D10711yescharacteristic
316C2×C4C23×D10711yescharacteristic
317C8—21no—
318C8—21no—
319C8—1072no—
320C8—2141no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—4281no—
328C22—4281no—
329C22—4281no—
330C22—2141no—
331C22—2141no—
332C22—2141no—
333C22C2×D4×D10711yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—1072no—
339C4—4281no—
340C4—4281no—
341C4—4281no—
342C4—2141no—
343C4—2141no—
344C4#41×D10711yescharacteristic
345C4#1681215611yescharacteristic
346C2—41no—
347C2—81no—
348C2—8561no—
349C2—4281no—
350C2#2029327511yescharacteristic
351C2AGL(1,ℤ/8)×D10711yescharacteristic
352C2#2029343311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#481×C107C2
socleC22×C107C22×D4
hypercenter#481D107
hypocenterC107#3354
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #20293341C2semidirect
2. #20293278C2semidirect
3. #20293024C2semidirect
4. #20292953C2semidirect
5. #20292952C2semidirect
6. #20292944C2semidirect
7. #20292916C2semidirect
8. #20292905C2semidirect
9. #20292879C2semidirect
10. #20292878C2semidirect
11. #20292707C2semidirect
12. #20292702C2semidirect
13. #20292667C2semidirect
14. #20292666C2semidirect
15. #481×C107C2semidirect
143. C107⋊C8#41semidirect
177. C107#3354semidirect
184. #481D107semidirect

Maximal subgroups

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

subgroupindex
#202933412
#202932782
#202930242
#202929532
#202929522
#202929442
#202929162
#202929052
#202928792
#202928782
#202927072
#202927022
#202926672
#202926662
#481×C1072
#3354107