Finite Group Databasebeta

← #28346702

Subgroups — #28346702

Order 13696.

418 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293231C211yesmaximal, characteristic, complemented
2#20293099C211yesmaximal, characteristic, complemented
3#20292950C211yesmaximal, characteristic, complemented
4#20292949C211yesmaximal, characteristic, complemented
5#20292928C211yesmaximal, characteristic, complemented
6#20292927C211yesmaximal, characteristic, complemented
7#20292819C211yesmaximal, characteristic, complemented
8#20292759C211yesmaximal, characteristic, complemented
9#20292759C211yesmaximal, characteristic, complemented
10#20292739C211yesmaximal, characteristic, complemented
11#20292736C211yesmaximal, characteristic, complemented
12D8×Dic107C211yesmaximal, characteristic, complemented
13#20292707C211yesmaximal, characteristic, complemented
14#20292687C211yesmaximal, characteristic, complemented
15#459×C107C211yesmaximal, characteristic, complemented
16C2×D4×D107C2211yescharacteristic
17C2×#13133803C2211yescharacteristic
18C2×#13133803C2211yescharacteristic
19#16812101C2211yescharacteristic, complemented
20#16812090—21no—
21#16812090—21no—
22#16812081—21no—
23#16812081—21no—
24#16812080C2211yescharacteristic
25#16812080C2211yescharacteristic
26C4×D428C2211yescharacteristic
27D4×Dic107C2211yescharacteristic
28D4×Dic107C2211yescharacteristic, complemented
29D4×Dic107C2211yescharacteristic, complemented
30#16812071C2211yescharacteristic
31C4×C22⋊D107—21no—
32(C22⋊D107)⋊C4—21no—
33#16812067C2211yescharacteristic
34#16812066C2211yescharacteristic
35C2×(C2×D107)⋊C4—21no—
36C22⋊C4×D107—21no—
37#16812043C2211yescharacteristic, complemented
38#16812042C2211yescharacteristic, complemented
39#16812041C2211yescharacteristic, complemented
40#16812040C2211yescharacteristic, complemented
41C4⋊(C107⋊C8)C2211yescharacteristic, complemented
42#16812031C2211yescharacteristic
43#16812031C2211yescharacteristic
44#16812030C2211yescharacteristic
45#16812030C2211yescharacteristic
46#16812030C2211yescharacteristic, complemented
47#16812030C2211yescharacteristic, complemented
48(C2×D107)⋊C8C2211yescharacteristic
49(C2×D107)⋊C8C2211yescharacteristic
50#16812019C2211yescharacteristic, complemented
51C8×Dic107C2211yescharacteristic, complemented
52C2×D8×C107C2211yescharacteristic
53C2×D8×C107C2211yescharacteristic
54#127×C107C2211yescharacteristic, complemented
55C4×D4×C107C2211yescharacteristic
56C4⋊C8×C107C2211yescharacteristic, complemented
57#102×C107C2211yescharacteristic, complemented
58#102×C107C2211yescharacteristic, complemented
59C23×D107—21no—
60C2×D428C2311yescharacteristic
61C2×C22⋊D107—21no—
62C2×C22⋊D107—21no—
63C2×C22⋊D107—21no—
64D4×D107—22no—
65D4×D107—22no—
66C22×Dic107—21no—
67C22×Dic107—21no—
68C22×Dic107—21no—
69C2×C4×D107—21no—
70C2×C4×D107—21no—
71C2×C4×D107C2311yescharacteristic
72#13133803—22no—
73#13133803—22no—
74C2×C107⋊C8C2311yescharacteristic
75C2×C107⋊C8C2311yescharacteristic
76C4⋊Dic107C2311yescharacteristic
77C4⋊Dic107C2311yescharacteristic
78C4⋊Dic107C2311yescharacteristic
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4D411yescharacteristic
82Dic107⋊C4D411yescharacteristic
83C22⋊Dic107—21no—
84C22⋊Dic107—21no—
85C22⋊Dic107—21no—
86C22⋊Dic107—21no—
87C22⋊Dic107—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4D411yescharacteristic
92(C2×D107)⋊C4D411yescharacteristic
93C4×Dic107—21no—
94C4×Dic107C2311yescharacteristic
95C2×D4×C107—21no—
96C2×D4×C107—21no—
97C2×D4×C107C2311yescharacteristic
98C2×D4×C107C2311yescharacteristic
99C2×D4×C107C2311yescharacteristic
100C22×C4×C107—21no—
101D8×C107—22no—
102D8×C107—22no—
103C2×C8×C107C2311yescharacteristic
104C2×C8×C107C2311yescharacteristic
105C4⋊C4×C107C2311yescharacteristic
106C22⋊C4×C107—21no—
107C42×C107C2311yescharacteristic
108C23×C107—21no—
109C23×C107—21no—
110C23×C107—21no—
111C22×D107—21no—
112C22×D107—24no—
113C22×D107—81no—
114C22×D107—81no—
115C22×D107C2×D411yescharacteristic
116D4×C107—21no—
117D4×C107—21no—
118D4×C107—21no—
119D4×C107—21no—
120D4×C107—21no—
121D4×C107—81no—
122D4×C107—81no—
123D4×C107#4112yes—
124C2×C4×C107—21no—
125C2×C4×C107—81no—
126C2×C4×C107C2×D411yescharacteristic
127C2×C4×C107C2×D411yescharacteristic
128C2×C4×C107C2411yescharacteristic
129C22⋊D107—24no—
130C22⋊D107—81no—
131C22⋊D107—81no—
132C2×Dic107—21no—
133C2×Dic107—21no—
134C2×Dic107—21no—
135C2×Dic107—81no—
136C2×Dic107—81no—
137C2×Dic107—81no—
138C2×Dic107C2×D411yescharacteristic
139C2×Dic107C2×D411yescharacteristic
140C2×Dic107C2×D411yescharacteristic
141D428—21no—
142D428#4112yes—
143C4×D107—21no—
144C4×D107—21no—
145C4×D107—81no—
146C4×D107—81no—
147C8×C107—21no—
148C8×C107—21no—
149C107⋊C8—21no—
150C107⋊C8—21no—
151C22×C107—22no—
152C22×C107—41no—
153C22×C107—41no—
154C22×C107—41no—
155C22×C107—41no—
156C22×C107—41no—
157C22×C107—41no—
158C22×C107—41no—
159C22×C107—41no—
160C22×C107C22×D411yescharacteristic
161C2×D107—21no—
162C2×D107—22no—
163C2×D107—21no—
164C2×D107—21no—
165C2×D107—41no—
166C2×D107—41no—
167C2×D107—84no—
168C4×C107—21no—
169C4×C107—21no—
170C4×C107—41no—
171C4×C107C2×#4111yescharacteristic
172C4×C107C2×#4111yescharacteristic
173Dic107—21no—
174Dic107—21no—
175Dic107—21no—
176Dic107—41no—
177Dic107—41no—
178Dic107—41no—
179C2×C107—42no—
180C2×C107—81no—
181C2×C107—81no—
182C2×C107#52511yescharacteristic
183C2×C107#57611yescharacteristic
184C2×C107#57611yescharacteristic
185D107—41no—
186D107—42no—
187#3122—1071nomaximal, Sylow
188C107#312211yescharacteristic, complemented, Sylow
189#540—1071no—
190#518—1071no—
191#482—1071no—
192#482—1071no—
193#478—1071no—
194#478—1071no—
195#459—1071no—
196#459D10711yescharacteristic, complemented
197#447—1071no—
198#447—1071no—
199#442—1071no—
200#442—1071no—
201C4×D8—1071no—
202C4×D8—1071no—
203#433—1071no—
204C22×D4—1071no—
205C2×D8—1071no—
206C2×D8—1071no—
207C2×D8C2×D10711yescharacteristic
208C2×D8C2×D10711yescharacteristic
209#127C2×D10711yescharacteristic, complemented
210#125—1071no—
211#123—2141no—
212#123—2141no—
213#121—1071no—
214#121—1071no—
215#121—2141no—
216#121—2141no—
217C4×D4—1071no—
218C4×D4—1071no—
219C4×D4—1071no—
220C4×D4—1071no—
221C4×D4—1071no—
222C4×D4—2141no—
223C4×D4—2141no—
224C4×D4C2×D10711yescharacteristic
225#117—1071no—
226#117—1071no—
227C2×C22⋊C4—2141no—
228C2×C22⋊C4—2141no—
229#107—1071no—
230#107—1071no—
231#106—1071no—
232#106—1071no—
233C4⋊C8—1071no—
234C4⋊C8C2×D10711yescharacteristic, complemented
235#102—1071no—
236#102—1071no—
237#102—1071no—
238#102—1071no—
239#102—1071no—
240#102—1071no—
241#102C2×D10711yescharacteristic, complemented
242#102C2×D10711yescharacteristic, complemented
243C22⋊C8—1071no—
244C22⋊C8—1071no—
245#97—1071no—
246C4×C8—1071no—
247C24—2141no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—1071no—
251C2×D4—2141no—
252C2×D4—2142no—
253C2×D4—2141no—
254C2×D4—2141no—
255C2×D4—2142no—
256C2×D4C22×D10711yescharacteristic
257C2×D4C22×D10711yescharacteristic
258C2×D4C22×D10711yescharacteristic
259C22×C4—21no—
260C22×C4—1071no—
261C22×C4—2141no—
262C22×C4—2141no—
263C22×C4—2141no—
264C22×C4—2141no—
265C22×C4—2141no—
266D8—22no—
267D8—22no—
268D8—2142no—
269D8—2142no—
270C2×C8—1071no—
271C2×C8—1071no—
272C2×C8C22×D10711yescharacteristic
273C2×C8C22×D10711yescharacteristic
274C4⋊C4—1071no—
275C4⋊C4—1071no—
276C4⋊C4—1071no—
277C4⋊C4—1071no—
278C4⋊C4—1071no—
279C4⋊C4—2141no—
280C4⋊C4—2141no—
281C4⋊C4C22×D10711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—1071no—
284C22⋊C4—1071no—
285C22⋊C4—2141no—
286C22⋊C4—2141no—
287C22⋊C4—2141no—
288C22⋊C4—2141no—
289C22⋊C4—2141no—
290C22⋊C4—2141no—
291C22⋊C4—2141no—
292C22⋊C4—2141no—
293C42—1071no—
294C42—2141no—
295C42C22×D10711yescharacteristic
296C23—21no—
297C23—21no—
298C23—21no—
299C23—8561no—
300C23—8561no—
301C23—1071no—
302C23—2144no—
303C23—2141no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—81no—
310D4—81no—
311D4—8561no—
312D4—8561no—
313D4—1072no—
314D4—2144no—
315D4—2141no—
316D4#1313381712yes—
317C2×C4—21no—
318C2×C4—81no—
319C2×C4—8561no—
320C2×C4—8561no—
321C2×C4—8561no—
322C2×C4—8561no—
323C2×C4—8561no—
324C2×C4—1071no—
325C2×C4—1071no—
326C2×C4—1071no—
327C2×C4—2141no—
328C2×C4—2141no—
329C2×C4—2141no—
330C2×C4—2141no—
331C2×C4—2141no—
332C2×C4D4×D10711yescharacteristic
333C2×C4D4×D10711yescharacteristic
334C2×C4C23×D10711yescharacteristic
335C8—21no—
336C8—21no—
337C8—2141no—
338C8—2141no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—8564no—
349C22—4281no—
350C22—4281no—
351C22—2141no—
352C22—2142no—
353C22—2141no—
354C22—2141no—
355C22C2×D4×D10711yescharacteristic
356C4—21no—
357C4—21no—
358C4—41no—
359C4—4281no—
360C4—4281no—
361C4—4281no—
362C4—2141no—
363C4—2141no—
364C4—2141no—
365C4#41×D10711yescharacteristic
366C4C2×#1313381711yescharacteristic
367C2—42no—
368C2—81no—
369C2—81no—
370C2—4281no—
371C2—4282no—
372C2#2029313711yescharacteristic
373C2#2029342811yescharacteristic
374C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#459×C107C2
socleC22×C107C22×D4
hypercenter#459D107
hypocenterC107#3122
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #20293231C2semidirect
2. #20293099C2semidirect
3. #20292950C2semidirect
4. #20292949C2semidirect
5. #20292928C2semidirect
6. #20292927C2semidirect
7. #20292819C2semidirect
8. #20292759C2semidirect
9. #20292759C2semidirect
10. #20292739C2semidirect
11. #20292736C2semidirect
12. D8×Dic107C2semidirect
13. #20292707C2semidirect
14. #20292687C2semidirect
15. #459×C107C2semidirect
19. #16812101C22semidirect
28. D4×Dic107C22semidirect
29. D4×Dic107C22semidirect
37. #16812043C22semidirect
38. #16812042C22semidirect
39. #16812041C22semidirect
40. #16812040C22semidirect
41. C4⋊(C107⋊C8)C22semidirect
46. #16812030C22semidirect
47. #16812030C22semidirect
50. #16812019C22semidirect
51. C8×Dic107C22semidirect
54. #127×C107C22semidirect
56. C4⋊C8×C107C22semidirect
57. #102×C107C22semidirect
58. #102×C107C22semidirect
188. C107#3122semidirect
196. #459D107semidirect
209. #127C2×D107semidirect
234. C4⋊C8C2×D107semidirect
241. #102C2×D107semidirect
242. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202932312
#202930992
#202929502
#202929492
#202929282
#202929272
#202928192
#202927592
#202927392
#202927362
D8×Dic1072
#202927072
#202926872
#459×C1072
#3122107