Finite Group Databasebeta

← #28346720

Subgroups — #28346720

Order 13696.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293246C211yesmaximal, characteristic, complemented
2#20293106C211yesmaximal, characteristic, complemented
3#20292961C211yesmaximal, characteristic, complemented
4#20292939C211yesmaximal, characteristic, complemented
5#20292923C211yesmaximal, characteristic, complemented
6#20292907C211yesmaximal, characteristic, complemented
7#20292846C211yesmaximal, characteristic, complemented
8#20292786C211yesmaximal, characteristic, complemented
9#20292769C211yesmaximal, characteristic
10#20292728C211yesmaximal, characteristic, complemented
11#20292724C211yesmaximal, characteristic, complemented
12#20292719C211yesmaximal, characteristic, complemented
13D8×Dic107C211yesmaximal, characteristic, complemented
14#20292687C211yesmaximal, characteristic, complemented
15#463×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C2×#13133808C2211yescharacteristic
18C2×#13133806C2211yescharacteristic
19#16812109C2211yescharacteristic
20#16812102—21no—
21#16812095—21no—
22#16812093—21no—
23#16812086—21no—
24#16812085C2211yescharacteristic
25#16812080C2211yescharacteristic
26Q8×Dic107C2211yescharacteristic
27C4×Dic214C2211yescharacteristic
28#16812074C2211yescharacteristic
29D4×Dic107C2211yescharacteristic
30D4×Dic107C2211yescharacteristic
31C4×C22⋊D107—21no—
32(C22⋊D107)⋊C4—21no—
33#16812066C2211yescharacteristic
34#16812065—21no—
35C2×Dic107⋊C4—21no—
36C4⋊C4×D107C2211yescharacteristic
37#16812043C2211yescharacteristic
38#16812043C2211yescharacteristic
39#16812042C2211yescharacteristic
40#16812042C2211yescharacteristic
41C4⋊(C107⋊C8)C2211yescharacteristic
42#16812034C2211yescharacteristic
43#16812034C2211yescharacteristic
44#16812033C2211yescharacteristic
45#16812030C2211yescharacteristic
46#16812030C2211yescharacteristic
47#16812030C2211yescharacteristic
48(C2×D107)⋊C8C2211yescharacteristic
49(C2×D107)⋊C8C2211yescharacteristic
50#16812019C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C2×SD16×C107C2211yescharacteristic
53C2×D8×C107C2211yescharacteristic
54#124×C107C2211yescharacteristic
55C4×D4×C107C2211yescharacteristic
56C4⋊C8×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59#13133817—22no—
60#13133817—22no—
61C2×Dic214—21no—
62C2×Dic214C2311yescharacteristic
63C2×C22⋊D107—21no—
64C2×C22⋊D107—21no—
65C22×Dic107—21no—
66C22×Dic107—21no—
67C22×Dic107—21no—
68C2×C4×D107—21no—
69C2×C4×D107C2311yescharacteristic
70#13133808—22no—
71#13133806—22no—
72C2×C107⋊C8C2311yescharacteristic
73C2×C107⋊C8C2311yescharacteristic
74C4⋊Dic107—21no—
75C4⋊Dic107—21no—
76C4⋊Dic107C2311yescharacteristic
77C4⋊Dic107C2311yescharacteristic
78C4⋊Dic107C2311yescharacteristic
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4D411yescharacteristic
84Dic107⋊C4D411yescharacteristic
85C22⋊Dic107—21no—
86C22⋊Dic107—21no—
87C22⋊Dic107—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4D411yescharacteristic
90(C2×D107)⋊C4D411yescharacteristic
91C4×Dic107—21no—
92C4×Dic107—21no—
93C4×Dic107—21no—
94C4×Dic107C2311yescharacteristic
95C2×Q8×C107C2311yescharacteristic
96C2×D4×C107C2311yescharacteristic
97C2×D4×C107C2311yescharacteristic
98C22×C4×C107—21no—
99SD16×C107—22no—
100D8×C107—22no—
101C2×C8×C107C2311yescharacteristic
102C2×C8×C107C2311yescharacteristic
103C4⋊C4×C107C2311yescharacteristic
104C22⋊C4×C107—21no—
105C22⋊C4×C107—21no—
106C22⋊C4×C107—21no—
107C42×C107C2311yescharacteristic
108C23×C107—21no—
109C23×C107—21no—
110C22×D107C2×D411yescharacteristic
111Q8×C107—21no—
112Q8×C107—21no—
113D4×C107—21no—
114D4×C107—21no—
115D4×C107—21no—
116D4×C107#4112yes—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—81no—
120C2×C4×C107C2×D411yescharacteristic
121C2×C4×C107C2×D411yescharacteristic
122C2×C4×C107C2411yescharacteristic
123C22⋊D107—24no—
124C22⋊D107—81no—
125C2×Dic107—24no—
126C2×Dic107—21no—
127C2×Dic107—21no—
128C2×Dic107—21no—
129C2×Dic107—21no—
130C2×Dic107—81no—
131C2×Dic107—81no—
132C2×Dic107C2×D411yescharacteristic
133C2×Dic107C2×D411yescharacteristic
134C2×Dic107C2×D411yescharacteristic
135C4×D107—21no—
136C4×D107—21no—
137C4×D107—81no—
138Dic214—21no—
139Dic214—81no—
140Dic214#4112yes—
141C8×C107—21no—
142C8×C107—21no—
143C107⋊C8—21no—
144C107⋊C8—21no—
145C22×C107—22no—
146C22×C107—41no—
147C22×C107—41no—
148C22×C107—41no—
149C22×C107—41no—
150C22×C107—41no—
151C22×C107C22×D411yescharacteristic
152C2×D107—21no—
153C2×D107—21no—
154C2×D107—21no—
155C4×C107—21no—
156C4×C107—21no—
157C4×C107—41no—
158C4×C107—41no—
159C4×C107C2×#4111yescharacteristic
160C4×C107C2×#4111yescharacteristic
161Dic107—22no—
162Dic107—21no—
163Dic107—21no—
164Dic107—21no—
165Dic107—41no—
166Dic107—41no—
167Dic107—41no—
168C2×C107—42no—
169C2×C107—81no—
170C2×C107#52511yescharacteristic
171C2×C107#57611yescharacteristic
172C2×C107#57811yescharacteristic
173D107—41no—
174#3124—1071nomaximal, Sylow
175C107#312411yescharacteristic, complemented, Sylow
176#542—1071no—
177#520—1071no—
178#484—1071no—
179#480—1071no—
180#477—1071no—
181#474—1071no—
182#464—1071no—
183#463D10711yescharacteristic, complemented
184#452—1071no—
185#449—1071no—
186#440—1071no—
187#440—1071no—
188C4×Q16—1071no—
189C4×D8—1071no—
190#433—1071no—
191C2×#41—1071no—
192C2×Q16—1071no—
193C2×SD16—1071no—
194C2×SD16C2×D10711yescharacteristic
195C2×D8C2×D10711yescharacteristic
196C4⋊Q8—1071no—
197#126—2141no—
198#124—2141no—
199#124C2×D10711yescharacteristic
200#123—2141no—
201C22⋊Q8—1071no—
202C22⋊Q8—2141no—
203#121—1071no—
204C4×Q8—1071no—
205C4×Q8—1071no—
206C4×Q8—1071no—
207C4×D4—1071no—
208C4×D4—1071no—
209C4×D4—2141no—
210C4×D4—2141no—
211C4×D4C2×D10711yescharacteristic
212#117—1071no—
213#117—2141no—
214C2×C4⋊C4—1071no—
215C2×C4⋊C4—2141no—
216#107—1071no—
217#107—1071no—
218#107—1071no—
219#107—1071no—
220C4⋊C8—1071no—
221C4⋊C8C2×D10711yescharacteristic
222#103—1071no—
223#103—1071no—
224#103—1071no—
225#103C2×D10711yescharacteristic
226#102—1071no—
227#102—1071no—
228#102—1071no—
229#102C2×D10711yescharacteristic
230C22⋊C8—1071no—
231C22⋊C8—1071no—
232#97—1071no—
233C4×C8—1071no—
234#41—2142no—
235#41—2142no—
236C2×Q8—1071no—
237C2×Q8—2141no—
238C2×Q8C22×D10711yescharacteristic
239C2×D4—2141no—
240C2×D4—2141no—
241C2×D4C22×D10711yescharacteristic
242C2×D4C22×D10711yescharacteristic
243C22×C4—21no—
244C22×C4—1071no—
245C22×C4—2141no—
246C22×C4—2141no—
247C22×C4—2141no—
248C22×C4—2141no—
249Q16—2142no—
250SD16—22no—
251SD16—2142no—
252D8—22no—
253C2×C8—1071no—
254C2×C8—1071no—
255C2×C8C22×D10711yescharacteristic
256C2×C8C22×D10711yescharacteristic
257C4⋊C4—1071no—
258C4⋊C4—1071no—
259C4⋊C4—1071no—
260C4⋊C4—1071no—
261C4⋊C4—1071no—
262C4⋊C4—2141no—
263C4⋊C4—2141no—
264C4⋊C4—2141no—
265C4⋊C4—2141no—
266C4⋊C4—2141no—
267C4⋊C4—2141no—
268C4⋊C4C22×D10711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—1071no—
273C22⋊C4—1071no—
274C22⋊C4—2141no—
275C22⋊C4—2141no—
276C22⋊C4—2141no—
277C22⋊C4—2141no—
278C42—1071no—
279C42—2141no—
280C42—2141no—
281C42—2141no—
282C42C22×D10711yescharacteristic
283C23—21no—
284C23—21no—
285C23—1071no—
286Q8—21no—
287Q8—21no—
288Q8—8561no—
289Q8—1072no—
290Q8—2141no—
291D4—21no—
292D4—21no—
293D4—21no—
294D4—8561no—
295D4—2144no—
296D4#1313381712yes—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—81no—
300C2×C4—8561no—
301C2×C4—8561no—
302C2×C4—8561no—
303C2×C4—1071no—
304C2×C4—1071no—
305C2×C4—1071no—
306C2×C4—2141no—
307C2×C4—2141no—
308C2×C4—2141no—
309C2×C4—2144no—
310C2×C4—2141no—
311C2×C4—2141no—
312C2×C4—2141no—
313C2×C4D4×D10711yescharacteristic
314C2×C4D4×D10711yescharacteristic
315C2×C4C23×D10711yescharacteristic
316C8—21no—
317C8—21no—
318C8—2141no—
319C8—2141no—
320C22—22no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—2141no—
327C22—2141no—
328C22—2141no—
329C22C2×D4×D10711yescharacteristic
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—4281no—
335C4—4281no—
336C4—4281no—
337C4—2141no—
338C4—2142no—
339C4—2141no—
340C4—2141no—
341C4#41×D10711yescharacteristic
342C4C2×#1313381711yescharacteristic
343C2—42no—
344C2—81no—
345C2—4281no—
346C2#2029313711yescharacteristic
347C2#2029343111yescharacteristic
348C2#2029344111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#463×C107C2
socleC22×C107C22×D4
prime radical#20292769C2
hypercenter#463D107
hypocenterC107#3124
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #20293246C2semidirect
2. #20293106C2semidirect
3. #20292961C2semidirect
4. #20292939C2semidirect
5. #20292923C2semidirect
6. #20292907C2semidirect
7. #20292846C2semidirect
8. #20292786C2semidirect
10. #20292728C2semidirect
11. #20292724C2semidirect
12. #20292719C2semidirect
13. D8×Dic107C2semidirect
14. #20292687C2semidirect
15. #463×C107C2semidirect
175. C107#3124semidirect
183. #463D107semidirect

Maximal subgroups

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

subgroupindex
#202932462
#202931062
#202929612
#202929392
#202929232
#202929072
#202928462
#202927862
#202927692
#202927282
#202927242
#202927192
D8×Dic1072
#202926872
#463×C1072
#3124107