Finite Group Databasebeta

← #28345303

Subgroups — #28345303

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×#16812141C211yesmaximal, characteristic
2#20293109C211yesmaximal, characteristic
3#20293103C211yesmaximal, characteristic, complemented
4#20292731C211yesmaximal, characteristic, complemented
5#20292730C211yesmaximal, characteristic, complemented
6#20292729C211yesmaximal, characteristic, complemented
7Q16×Dic107C211yesmaximal, characteristic, complemented
8C4×#13133808C211yesmaximal, characteristic, complemented
9C4×#13133805C211yesmaximal, characteristic, complemented
10#20292714C211yesmaximal, characteristic, complemented
11#20292666C211yesmaximal, characteristic, complemented
12#20292622C211yesmaximal, characteristic
13#103×D107C211yesmaximal, characteristic, complemented
14#20292584C211yesmaximal, characteristic, complemented
15#441×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×Q8×D107C2211yescharacteristic
18#16812141C418yescomplemented
19C2×#13133808C2211yescharacteristic
20C2×#13133807C2211yescharacteristic
21C2×#13133805C2211yescharacteristic
22C2×#13133801C2211yescharacteristic
23C2×C8×D107C2211yescharacteristic
24Q8×Dic107C2211yescharacteristic
25Q8×Dic107C2211yescharacteristic
26C4×Dic214—21no—
27C4×Dic214C2211yescharacteristic
28#16812074—21no—
29#16812074C2211yescharacteristic
30C4×D428—21no—
31C4×D428C2211yescharacteristic
32#16812071—21no—
33#16812071C2211yescharacteristic
34#16812067—21no—
35#16812067C2211yescharacteristic
36#16812066—21no—
37#16812066—21no—
38#16812066C2211yescharacteristic
39C4⋊C4×D107—21no—
40C4⋊C4×D107C2211yescharacteristic
41#16812043C2211yescharacteristic
42#16812042C2211yescharacteristic
43#16812040C2211yescharacteristic
44#16812034C2211yescharacteristic
45#16812033C2211yescharacteristic
46#16812033C2211yescharacteristic
47#16812032C2211yescharacteristic
48#16812031C2211yescharacteristic
49#16812029C2211yescharacteristic
50#16812019C2211yescharacteristic
51C4×C107⋊C8C2211yescharacteristic
52C8×Dic107C2211yescharacteristic
53C2×Q16×C107C2211yescharacteristic
54C4×Q8×C107C2211yescharacteristic
55C4×Q8×C107C2211yescharacteristic
56#106×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#103×C107C2211yescharacteristic
59#97×C107C2211yescharacteristic
60#13133819—22no—
61#13133819C2×C414yes—
62C2×Dic214—21no—
63C2×Dic214C2311yescharacteristic
64Q8×D107—22no—
65Q8×D107C2×C414yescomplemented
66C2×D428—21no—
67C2×D428C2311yescharacteristic
68C2×C4×D107—21no—
69C2×C4×D107—21no—
70C2×C4×D107—21no—
71C2×C4×D107—21no—
72C2×C4×D107—21no—
73C2×C4×D107C2311yescharacteristic
74#13133808C2×C414yescomplemented
75#13133807C2×C414yescomplemented
76#13133805C2×C414yescomplemented
77#13133801C2×C414yescomplemented
78C2×C107⋊C8C2311yescharacteristic
79C2×C107⋊C8C2311yescharacteristic
80C8×D107—22no—
81C4⋊Dic107—21no—
82C4⋊Dic107—21no—
83C4⋊Dic107C2311yescharacteristic
84C4⋊Dic107C2311yescharacteristic
85Dic107⋊C4—21no—
86Dic107⋊C4—21no—
87Dic107⋊C4—21no—
88Dic107⋊C4D411yescharacteristic
89Dic107⋊C4D411yescharacteristic
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4D411yescharacteristic
94(C2×D107)⋊C4D411yescharacteristic
95C4×Dic107—21no—
96C4×Dic107—21no—
97C4×Dic107—21no—
98C4×Dic107—21no—
99C4×Dic107—21no—
100C4×Dic107C2311yescharacteristic
101C2×Q8×C107C2311yescharacteristic
102C2×Q8×C107C2311yescharacteristic
103Q16×C107C2×C414yescomplemented
104C2×C8×C107C2311yescharacteristic
105C2×C8×C107C2311yescharacteristic
106C4⋊C4×C107—21no—
107C4⋊C4×C107—21no—
108C4⋊C4×C107C2311yescharacteristic
109C4⋊C4×C107C2311yescharacteristic
110C42×C107—21no—
111C42×C107—21no—
112C42×C107C2311yescharacteristic
113C22×D107—21no—
114C22×D107C2×D411yescharacteristic
115Q8×C107—21no—
116Q8×C107—21no—
117Q8×C107C22×C412yes—
118Q8×C107C22×C412yes—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107C2×D411yescharacteristic
124C2×C4×C107C2×D411yescharacteristic
125C2×C4×C107C2411yescharacteristic
126C2×Dic107—21no—
127C2×Dic107—21no—
128C2×Dic107—21no—
129C2×Dic107C2×D411yescharacteristic
130C2×Dic107C2×D411yescharacteristic
131C2×Dic107C2×D411yescharacteristic
132D428—24no—
133D428—21no—
134D428C22×C412yes—
135C4×D107—24no—
136C4×D107—24no—
137C4×D107—81no—
138C4×D107—81no—
139C4×D107—81no—
140C4×D107C22×C412yes—
141C4×D107#4112yes—
142Dic214—21no—
143Dic214—24no—
144Dic214C22×C412yes—
145C8×C107—21no—
146C8×C107C22×C412yes—
147C107⋊C8—21no—
148C107⋊C8C22×C412yes—
149C22×C107C22×D411yescharacteristic
150C2×D107—21no—
151C2×D107—21no—
152C2×D107—22no—
153C2×D107—41no—
154C2×D107—41no—
155C2×D107C4×D412yes—
156C4×C107—22no—
157C4×C107—22no—
158C4×C107—21no—
159C4×C107—21no—
160C4×C107—41no—
161C4×C107—41no—
162C4×C107C23×C411yescharacteristic
163C4×C107C2×#4111yescharacteristic
164Dic107—22no—
165Dic107—21no—
166Dic107—21no—
167Dic107—41no—
168Dic107—41no—
169Dic107C4×D412yes—
170C2×C107C2×C4×D411yescharacteristic
171C2×C107#57711yescharacteristic
172C2×C107#57811yescharacteristic
173D107—22no—
174D107—42no—
175#2947—1071nomaximal, Sylow
176C107#294711yescharacteristic, complemented, Sylow
177C2×#137—1071no—
178#520—1071no—
179#519—1071no—
180#441—1071no—
181#441D10711yescharacteristic, complemented
182#440—1071no—
183#440—1071no—
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×Q16C2×D10711yescharacteristic
197C2×SD16—1071no—
198C2×SD16—1071no—
199C2×M16—1071no—
200C22×C8—1071no—
201C4×Q8—1071no—
202C4×Q8—1071no—
203C4×Q8—1071no—
204C4×Q8—1071no—
205C4×Q8—2141no—
206C4×Q8—2141no—
207C4×Q8C2×D10711yescharacteristic
208C4×Q8C2×D10711yescharacteristic
209C4×D4—1071no—
210C4×D4—1071no—
211C4×D4—2141no—
212C4×D4—2141no—
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#107—1071no—
222#106—1071no—
223#106C2×D10711yescharacteristic
224#103—1071no—
225#103—1071no—
226#103—1071no—
227#103—1071no—
228#103C2×D10711yescharacteristic
229#103C2×D10711yescharacteristic
230#102—1071no—
231#102—1071no—
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—2141no—
241C2×Q8—2142no—
242C2×Q8C22×D10711yescharacteristic
243C2×Q8C22×D10711yescharacteristic
244C2×D4—1071no—
245C2×D4—2141no—
246C22×C4—1071no—
247C22×C4—2141no—
248C22×C4—2141no—
249C22×C4—2141no—
250C22×C4—2141no—
251C22×C4—2141no—
252Q16—1074no—
253Q16C4×D10714yescomplemented
254SD16—1074no—
255SD16—1074no—
256M16—1074no—
257C2×C8—1071no—
258C2×C8—1071no—
259C2×C8—2142no—
260C2×C8C22×D10711yescharacteristic
261C2×C8C22×D10711yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—1071no—
265C4⋊C4—1071no—
266C4⋊C4—1071no—
267C4⋊C4—1071no—
268C4⋊C4—2141no—
269C4⋊C4—2141no—
270C4⋊C4—2141no—
271C4⋊C4—2141no—
272C4⋊C4—2141no—
273C4⋊C4C22×D10711yescharacteristic
274C4⋊C4C22×D10711yescharacteristic
275C22⋊C4—1071no—
276C22⋊C4—1071no—
277C22⋊C4—2141no—
278C22⋊C4—2141no—
279C22⋊C4—2141no—
280C42—21no—
281C42—21no—
282C42—1071no—
283C42—2141no—
284C42—2141no—
285C42—2141no—
286C42—2141no—
287C42—2141no—
288C42C22×D10711yescharacteristic
289C23—1071no—
290C23—2141no—
291Q8—21no—
292Q8—21no—
293Q8—1072no—
294Q8—2144no—
295Q8—2141no—
296Q8C2×C4×D10712yes—
297Q8C2×C4×D10712yes—
298D4—1072no—
299D4—2144no—
300D4—2141no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—8561no—
306C2×C4—8561no—
307C2×C4—8561no—
308C2×C4—1071no—
309C2×C4—1072no—
310C2×C4—1072no—
311C2×C4—1071no—
312C2×C4—1071no—
313C2×C4—2144no—
314C2×C4—2144no—
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—1072no—
326C22—4281no—
327C22—4281no—
328C22—2142no—
329C22—2141no—
330C22—2141no—
331C22C2×D4×D10711yescharacteristic
332C4—21no—
333C4—21no—
334C4—22no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—1072no—
339C4—4281no—
340C4—4281no—
341C4—2141no—
342C4—2141no—
343C4—2142no—
344C4C22×C4×D10711yescharacteristic
345C4#41×D10711yescharacteristic
346C2—4282no—
347C2—2142no—
348C2C4×D4×D10711yescharacteristic
349C2#2029343611yescharacteristic
350C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutatorC4×C107C23×C4
FrattiniC2×C4C23×D107
Fitting#441×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133819C22
cosocleC2×C4×C107C24
hypercenter#441D107
hypocenterC107#2947
LCSG ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
3. #20293103C2semidirect
4. #20292731C2semidirect
5. #20292730C2semidirect
6. #20292729C2semidirect
7. Q16×Dic107C2semidirect
8. C4×#13133808C2semidirect
9. C4×#13133805C2semidirect
10. #20292714C2semidirect
11. #20292666C2semidirect
13. #103×D107C2semidirect
14. #20292584C2semidirect
15. #441×C107C2semidirect
18. #16812141C4semidirect
65. Q8×D107C2×C4semidirect
74. #13133808C2×C4semidirect
75. #13133807C2×C4semidirect
76. #13133805C2×C4semidirect
77. #13133801C2×C4semidirect
103. Q16×C107C2×C4semidirect
176. C107#2947semidirect
181. #441D107semidirect
253. Q16C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121412
#202931092
#202931032
#202927312
#202927302
#202927292
Q16×Dic1072
C4×#131338082
C4×#131338052
#202927142
#202926662
#202926222
#103×D1072
#202925842
#441×C1072
#2947107