Finite Group Databasebeta

← #28345301

Subgroups — #28345301

Order 13696.

470 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812138C211yesmaximal, characteristic
2#20293106C211yesmaximal, characteristic
3#20293103C211yesmaximal, characteristic, complemented
4#20292735C211yesmaximal, characteristic, complemented
5#20292731C211yesmaximal, characteristic, complemented
6#20292724C211yesmaximal, characteristic, complemented
7#20292721C211yesmaximal, characteristic, complemented
8C4×#13133808C211yesmaximal, characteristic, complemented
9SD16×Dic107C211yesmaximal, characteristic, complemented
10C4×#13133806C211yesmaximal, characteristic, complemented
11#20292668C211yesmaximal, characteristic, complemented
12#20292620C211yesmaximal, characteristic
13#103×D107C211yesmaximal, characteristic, complemented
14#20292584C211yesmaximal, characteristic, complemented
15#440×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C2×Q8×D107C2211yescharacteristic
18#16812138C418yescomplemented
19C2×Dic428C2211yescharacteristic
20C2×#13133808C2211yescharacteristic
21C2×#13133806C2211yescharacteristic
22C2×#13133801C2211yescharacteristic
23C2×C8×D107C2211yescharacteristic
24Q8×Dic107C2211yescharacteristic
25C4×Dic214—21no—
26C4×Dic214C2211yescharacteristic
27C4×Dic214C2211yescharacteristic
28#16812074—21no—
29#16812074C2211yescharacteristic
30#16812074C2211yescharacteristic
31D4×Dic107C2211yescharacteristic
32C4×C22⋊D107—21no—
33(C22⋊D107)⋊C4—21no—
34#16812067—21no—
35#16812067C2211yescharacteristic
36#16812066—21no—
37#16812066C2211yescharacteristic
38#16812065—21no—
39C2×Dic107⋊C4—21no—
40C4⋊C4×D107C2211yescharacteristic
41#16812042C2211yescharacteristic
42#16812041C2211yescharacteristic
43#16812040C2211yescharacteristic
44#16812034C2211yescharacteristic
45#16812034C2211yescharacteristic
46#16812033C2211yescharacteristic
47#16812032C2211yescharacteristic
48#16812032C2211yescharacteristic
49#16812030C2211yescharacteristic
50#16812019C2211yescharacteristic
51C4×C107⋊C8C2211yescharacteristic
52C8×Dic107C2211yescharacteristic
53C2×SD16×C107C2211yescharacteristic
54C4×Q8×C107C2211yescharacteristic
55C4×D4×C107C2211yescharacteristic
56#107×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59#97×C107C2211yescharacteristic
60#13133817—22no—
61#13133817C2×C414yes—
62C2×Dic214—21no—
63C2×Dic214C2311yescharacteristic
64C2×Dic214C2311yescharacteristic
65Q8×D107—22no—
66Q8×D107C2×C414yescomplemented
67C2×C22⋊D107—21no—
68C22×Dic107—21no—
69C22×Dic107—21no—
70C2×C4×D107—21no—
71C2×C4×D107—21no—
72C2×C4×D107C2311yescharacteristic
73Dic428C2×C414yescomplemented
74#13133808C2×C414yescomplemented
75#13133806C2×C414yescomplemented
76#13133801C2×C414yescomplemented
77C2×C107⋊C8C2311yescharacteristic
78C2×C107⋊C8C2311yescharacteristic
79C8×D107—22no—
80C4⋊Dic107—21no—
81C4⋊Dic107C2311yescharacteristic
82C4⋊Dic107C2311yescharacteristic
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86Dic107⋊C4—21no—
87Dic107⋊C4—21no—
88Dic107⋊C4D411yescharacteristic
89Dic107⋊C4D411yescharacteristic
90C22⋊Dic107—21no—
91(C2×D107)⋊C4—21no—
92(C2×D107)⋊C4D411yescharacteristic
93(C2×D107)⋊C4D411yescharacteristic
94C4×Dic107—21no—
95C4×Dic107—21no—
96C4×Dic107—21no—
97C4×Dic107—21no—
98C4×Dic107—21no—
99C4×Dic107—21no—
100C4×Dic107C2311yescharacteristic
101C2×Q8×C107C2311yescharacteristic
102C2×D4×C107C2311yescharacteristic
103C22×C4×C107—21no—
104SD16×C107C2×C414yescomplemented
105C2×C8×C107C2311yescharacteristic
106C2×C8×C107C2311yescharacteristic
107C4⋊C4×C107—21no—
108C4⋊C4×C107C2311yescharacteristic
109C4⋊C4×C107C2311yescharacteristic
110C22⋊C4×C107—21no—
111C42×C107—21no—
112C42×C107C2311yescharacteristic
113C23×C107—21no—
114C22×D107C2×D411yescharacteristic
115Q8×C107—21no—
116Q8×C107C22×C412yes—
117D4×C107—21no—
118D4×C107C22×C412yes—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—81no—
123C2×C4×C107C2×D411yescharacteristic
124C2×C4×C107C2×D411yescharacteristic
125C2×C4×C107C2411yescharacteristic
126C22⋊D107—24no—
127C2×Dic107—21no—
128C2×Dic107—21no—
129C2×Dic107—24no—
130C2×Dic107—21no—
131C2×Dic107—21no—
132C2×Dic107—81no—
133C2×Dic107C2×D411yescharacteristic
134C2×Dic107C2×D411yescharacteristic
135C2×Dic107C2×D411yescharacteristic
136C4×D107—24no—
137C4×D107—81no—
138C4×D107C22×C412yes—
139C4×D107#4112yes—
140Dic214—21no—
141Dic214—21no—
142Dic214—24no—
143Dic214C22×C412yes—
144Dic214C22×C412yes—
145C8×C107—21no—
146C8×C107C22×C412yes—
147C107⋊C8—21no—
148C107⋊C8C22×C412yes—
149C22×C107—22no—
150C22×C107—41no—
151C22×C107—41no—
152C22×C107C22×D411yescharacteristic
153C2×D107—21no—
154C2×D107—21no—
155C2×D107C4×D412yes—
156C4×C107—22no—
157C4×C107—21no—
158C4×C107—21no—
159C4×C107—41no—
160C4×C107—41no—
161C4×C107C23×C411yescharacteristic
162C4×C107C2×#4111yescharacteristic
163Dic107—22no—
164Dic107—22no—
165Dic107—21no—
166Dic107—21no—
167Dic107—41no—
168Dic107—41no—
169Dic107C4×D412yes—
170C2×C107—42no—
171C2×C107C2×C4×D411yescharacteristic
172C2×C107#57711yescharacteristic
173C2×C107#57811yescharacteristic
174D107—22no—
175#2947—1071nomaximal, Sylow
176C107#294711yescharacteristic, complemented, Sylow
177C2×#137—1071no—
178#520—1071no—
179#519—1071no—
180#441—1071no—
181#441—1071no—
182#440—1071no—
183#440D10711yescharacteristic, complemented
184C4×Q16—1071no—
185C4×Q16—1071no—
186C4×SD16—1071no—
187C4×SD16—1071no—
188#428—1071no—
189#416—1071no—
190C2×#103—1071no—
191#405—1071no—
192C2×#41—1071no—
193C22×Q8—1071no—
194#137—1078no—
195C2×Q16—1071no—
196C2×Q16—1071no—
197C2×SD16—1071no—
198C2×SD16C2×D10711yescharacteristic
199C2×M16—1071no—
200C22×C8—1071no—
201C4×Q8—1071no—
202C4×Q8—1071no—
203C4×Q8—1071no—
204C4×Q8—1071no—
205C4×Q8—1071no—
206C4×Q8—2141no—
207C4×Q8—2141no—
208C4×Q8C2×D10711yescharacteristic
209C4×D4—1071no—
210C4×D4—2141no—
211C4×D4—2141no—
212C4×D4C2×D10711yescharacteristic
213#117—1071no—
214#117—1071no—
215#117—2141no—
216#117—2141no—
217#117—2141no—
218C2×C4⋊C4—1071no—
219C2×C4⋊C4—2141no—
220#107—1071no—
221#107C2×D10711yescharacteristic
222#106—1071no—
223#106—1071no—
224#103—1071no—
225#103—1071no—
226#103—1071no—
227#103—1071no—
228#103—1071no—
229#103C2×D10711yescharacteristic
230#102—1071no—
231#102C2×D10711yescharacteristic
232#97—1071no—
233#97C2×D10711yescharacteristic
234C4×C8—1071no—
235C4×C8—1071no—
236#41—1074no—
237#41—2142no—
238C2×Q8—1074no—
239C2×Q8—1071no—
240C2×Q8—1071no—
241C2×Q8—2141no—
242C2×Q8—2142no—
243C2×Q8C22×D10711yescharacteristic
244C2×D4—2141no—
245C2×D4C22×D10711yescharacteristic
246C22×C4—21no—
247C22×C4—1071no—
248C22×C4—2141no—
249C22×C4—2141no—
250C22×C4—2141no—
251C22×C4—2141no—
252Q16—1074no—
253Q16—1074no—
254SD16—1074no—
255SD16C4×D10714yescomplemented
256M16—1074no—
257C2×C8—1071no—
258C2×C8—1071no—
259C2×C8—2142no—
260C2×C8C22×D10711yescharacteristic
261C2×C8C22×D10711yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—1071no—
264C4⋊C4—1071no—
265C4⋊C4—1071no—
266C4⋊C4—1071no—
267C4⋊C4—2141no—
268C4⋊C4—2141no—
269C4⋊C4—2141no—
270C4⋊C4—2141no—
271C4⋊C4—2141no—
272C4⋊C4—2141no—
273C4⋊C4C22×D10711yescharacteristic
274C4⋊C4C22×D10711yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—1071no—
277C22⋊C4—1071no—
278C22⋊C4—2141no—
279C22⋊C4—2141no—
280C42—21no—
281C42—1071no—
282C42—2141no—
283C42—2141no—
284C42—2141no—
285C42—2141no—
286C42—2141no—
287C42—2141no—
288C42C22×D10711yescharacteristic
289C23—21no—
290C23—1071no—
291Q8—21no—
292Q8—1072no—
293Q8—1072no—
294Q8—2144no—
295Q8—2141no—
296Q8—2141no—
297Q8C2×C4×D10712yes—
298D4—21no—
299D4—2144no—
300D4C2×C4×D10712yes—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—81no—
305C2×C4—8561no—
306C2×C4—8561no—
307C2×C4—1071no—
308C2×C4—1072no—
309C2×C4—1072no—
310C2×C4—1071no—
311C2×C4—1071no—
312C2×C4—2144no—
313C2×C4—2144no—
314C2×C4—2141no—
315C2×C4—2141no—
316C2×C4—2141no—
317C2×C4—2141no—
318C2×C4D4×D10711yescharacteristic
319C2×C4D4×D10711yescharacteristic
320C2×C4C23×D10711yescharacteristic
321C8—21no—
322C8—1072no—
323C8—2141no—
324C8C2×C4×D10712yes—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—1072no—
329C22—2141no—
330C22—2141no—
331C22C2×D4×D10711yescharacteristic
332C4—21no—
333C4—21no—
334C4—22no—
335C4—41no—
336C4—41no—
337C4—1072no—
338C4—4281no—
339C4—4281no—
340C4—2141no—
341C4—2141no—
342C4—2142no—
343C4—2142no—
344C4C22×C4×D10711yescharacteristic
345C4#41×D10711yescharacteristic
346C2—42no—
347C2—2142no—
348C2C4×D4×D10711yescharacteristic
349C2#2029343711yescharacteristic
350C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutatorC4×C107C23×C4
FrattiniC2×C4C23×D107
Fitting#440×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133817C22
cosocleC2×C4×C107C24
hypercenter#440D107
hypocenterC107#2947
LCSG ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
3. #20293103C2semidirect
4. #20292735C2semidirect
5. #20292731C2semidirect
6. #20292724C2semidirect
7. #20292721C2semidirect
8. C4×#13133808C2semidirect
9. SD16×Dic107C2semidirect
10. C4×#13133806C2semidirect
11. #20292668C2semidirect
13. #103×D107C2semidirect
14. #20292584C2semidirect
15. #440×C107C2semidirect
18. #16812138C4semidirect
66. Q8×D107C2×C4semidirect
73. Dic428C2×C4semidirect
74. #13133808C2×C4semidirect
75. #13133806C2×C4semidirect
76. #13133801C2×C4semidirect
104. SD16×C107C2×C4semidirect
176. C107#2947semidirect
183. #440D107semidirect
255. SD16C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121382
#202931062
#202931032
#202927352
#202927312
#202927242
#202927212
C4×#131338082
SD16×Dic1072
C4×#131338062
#202926682
#202926202
#103×D1072
#202925842
#440×C1072
#2947107