Finite Group Databasebeta

← #28345007

Subgroups — #28345007

Order 13696.

738 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293463C211yesmaximal, characteristic
2#20293393C211yesmaximal, characteristic, complemented
3#20293095C211yesmaximal, characteristic, complemented
4#20292645C211yesmaximal, characteristic, complemented
5#20292642C214yesmaximal, complemented
6C2×#16812036C211yesmaximal, characteristic, complemented
7C2×#16812035C211yesmaximal, characteristic, complemented
8#104×D107C214yesmaximal, complemented
9#421×C107C211yesmaximal, characteristic, complemented
10#16812157C412yescomplemented
11#16812156—21no—
12#16812156C412yescomplemented
13#16812155—21no—
14#16812155C412yescomplemented
15#16812154—22no—
16#16812154C412yescomplemented
17C2×#13133818—22no—
18C2×#13133818C2211yescharacteristic
19C2×#13133818C2211yescharacteristic
20#41×D107—22no—
21#41×D107C2214yes—
22#16812117—22no—
23#16812115—22no—
24C2×#13133802C2211yescharacteristic, complemented
25M16×D107C2214yescomplemented
26#16812074—21no—
27#16812071—21no—
28(C22⋊D107)⋊C4—22no—
29#16812066C2212yescomplemented
30C2×C4×Dic107C2211yescharacteristic, complemented
31C42×D107C2212yescomplemented
32#16812037C2214yescomplemented
33#16812036C2212yescomplemented
34#16812036C2212yescomplemented
35#16812035C2212yescomplemented
36#16812035C2212yescomplemented
37C2×#41×C107C2211yescharacteristic
38C2×M16×C107C2211yescharacteristic, complemented
39#117×C107C2211yescharacteristic, complemented
40#104×C107C2214yescomplemented
41#13133819—22no—
42#13133819C2×C414yescomplemented
43#13133818—21no—
44#13133818—21no—
45#13133818—24no—
46#13133818—22no—
47#13133818—24no—
48#13133818—22no—
49#13133818C2×C412yes—
50#13133818C2×C412yes—
51#13133818C2×C412yescomplemented
52#13133818D412yes—
53#13133818D412yes—
54#13133818C2312yes—
55#13133817—24no—
56#13133817—24no—
57#13133817—22no—
58#13133817C2×C414yescomplemented
59C2×Dic214—22no—
60C2×Dic214C2×C411yescharacteristic, complemented
61C2×Dic214D411yescharacteristic
62Q8×D107—22no—
63Q8×D107C2×C414yescomplemented
64C2×D428—22no—
65C2×D428C2×C411yescharacteristic, complemented
66C2×D428D411yescharacteristic
67C2×C22⋊D107—21no—
68C2×C22⋊D107—24no—
69C2×C22⋊D107D412yes—
70D4×D107—24no—
71D4×D107—24no—
72D4×D107—22no—
73D4×D107C2×C414yescomplemented
74C22×Dic107—21no—
75C2×C4×D107—21no—
76C2×C4×D107—24no—
77C2×C4×D107—22no—
78C2×C4×D107C2312yes—
79#13133802—21no—
80#13133802C2312yes—
81#13133801—24no—
82C2×C107⋊C8—21no—
83C8×D107—24no—
84Dic107⋊C4—22no—
85(C2×D107)⋊C4—22no—
86C4×Dic107—21no—
87C4×Dic107—22no—
88C4×Dic107C2311yescharacteristic
89C4×Dic107C2311yescharacteristic
90#41×C107—21no—
91#41×C107—21no—
92#41×C107C2×C412yescomplemented
93#41×C107C2312yes—
94C2×Q8×C107C2×C411yescharacteristic, complemented
95C2×D4×C107—21no—
96C2×D4×C107C2×C411yescharacteristic, complemented
97C22×C4×C107—21no—
98C22×C4×C107C2311yescharacteristic
99M16×C107—21no—
100M16×C107C2312yes—
101C2×C8×C107—21no—
102C4⋊C4×C107—21no—
103C22⋊C4×C107—21no—
104C42×C107C2312yes—
105C23×C107—21no—
106C23×C107C2×D411yescharacteristic
107C22×D107—21no—
108C22×D107—24no—
109C22×D107C2×D412yes—
110Q8×C107—21no—
111Q8×C107C22×C412yes—
112D4×C107—22no—
113D4×C107—22no—
114D4×C107—21no—
115D4×C107C22×C412yes—
116C2×C4×C107—21no—
117C2×C4×C107—22no—
118C2×C4×C107—22no—
119C2×C4×C107—22no—
120C2×C4×C107C22×C411yescharacteristic
121C2×C4×C107C22×C411yescharacteristic
122C2×C4×C107C2×D411yescharacteristic
123C2×C4×C107C2×D411yescharacteristic
124C2×C4×C107C2×D411yescharacteristic
125C2×C4×C107C2411yescharacteristic
126C22⋊D107—22no—
127C22⋊D107—22no—
128C22⋊D107—24no—
129C22⋊D107—24no—
130C22⋊D107C22⋊C414yes—
131C22⋊D107C22⋊C414yes—
132C2×Dic107—21no—
133C2×Dic107—21no—
134C2×Dic107—24no—
135C2×Dic107—21no—
136C2×Dic107—42no—
137C2×Dic107C2×D412yes—
138D428—24no—
139D428—21no—
140D428C22⋊C412yes—
141D428C22⋊C412yes—
142D428C22×C412yes—
143C4×D107—24no—
144C4×D107—22no—
145C4×D107—22no—
146C4×D107—24no—
147C4×D107—44no—
148C4×D107C22×C414yes—
149C4×D107C2×D414yes—
150Dic214—21no—
151Dic214—24no—
152Dic214C22⋊C412yes—
153Dic214C22⋊C412yes—
154Dic214C22×C412yes—
155C8×C107—22no—
156C107⋊C8—22no—
157C22×C107—22no—
158C22×C107—41no—
159C22×C107—81no—
160C22×C107C2×C22⋊C411yescharacteristic
161C22×C107C2×C22⋊C411yescharacteristic
162C22×C107C22×D411yescharacteristic
163C2×D107—22no—
164C2×D107—42no—
165C2×D107—84no—
166C2×D107—81no—
167C2×D107C2×C22⋊C414yes—
168C4×C107—22no—
169C4×C107—42no—
170C4×C107C2×C22⋊C411yescharacteristic
171C4×C107C2×C22⋊C411yescharacteristic
172C4×C107C23×C411yescharacteristic
173C4×C107C22×D411yescharacteristic
174Dic107—22no—
175Dic107—22no—
176Dic107—41no—
177Dic107C2×C22⋊C414yes—
178C2×C107—21no—
179C2×C107—21no—
180C2×C107—21no—
181C2×C107—42no—
182C2×C107C22×C22⋊C411yescharacteristic
183D107—24no—
184D107—42no—
185#2901—1071nomaximal, Sylow
186C107#290111yescharacteristic, complemented, Sylow
187#585—1071no—
188#568—1071no—
189C4×#41—1071no—
190#421—1071no—
191#421—1074no—
192#421D10711yescharacteristic, complemented
193C2×#104—1071no—
194C2×#104—1071no—
195C2×#104—1074no—
1962−1+4—1072no—
1972−1+4—1072no—
1982−1+4—2141no—
199H(5,𝔽2)—1072no—
200H(5,𝔽2)—1072no—
201H(5,𝔽2)—2142no—
202H(5,𝔽2)—2141no—
203C2×#41—1071no—
204C2×#41—1074no—
205C2×#41—1071no—
206C2×#41—2142no—
207C2×#41—2142no—
208C2×#41C2×D10711yescharacteristic
209#131—2142no—
210#131—2142no—
211C2×M16—1071no—
212C2×M16—1074no—
213C2×M16C2×D10711yescharacteristic, complemented
214C4×Q8—2141no—
215C4×D4—2142no—
216C4×D4—2141no—
217#117—1072no—
218#117C2×D10711yescharacteristic, complemented
219C2×C42—1071no—
220C2×C42—1072no—
221#104—1072no—
222#104—1072no—
223#104—1072no—
224#104—1072no—
225#104—1074no—
226#104C2×D10714yescomplemented
227#41—21no—
228#41—21no—
229#41—1072no—
230#41—1072no—
231#41—1072no—
232#41—1072no—
233#41—1074no—
234#41—1074no—
235#41—1072no—
236#41—1072no—
237#41—2144no—
238#41—2144no—
239#41—2142no—
240#41—2142no—
241#41—2144no—
242#41—2144no—
243#41—2142no—
244#41—2142no—
245#41—2141no—
246#41—2141no—
247#41C4×D10712yescomplemented
248#41C22×D10712yes—
249C2×Q8—1071no—
250C2×Q8—1074no—
251C2×Q8—1071no—
252C2×Q8—2142no—
253C2×Q8—2142no—
254C2×Q8C4×D10711yescharacteristic, complemented
255C2×D4—21no—
256C2×D4—1072no—
257C2×D4—1074no—
258C2×D4—1071no—
259C2×D4—1071no—
260C2×D4—2142no—
261C2×D4—2144no—
262C2×D4—2142no—
263C2×D4—2144no—
264C2×D4—2144no—
265C2×D4—2141no—
266C2×D4C4×D10711yescharacteristic, complemented
267C22×C4—21no—
268C22×C4—1072no—
269C22×C4—2144no—
270C22×C4—2142no—
271C22×C4—2141no—
272C22×C4—2141no—
273C22×C4C22×D10711yescharacteristic
274M16—21no—
275M16—1072no—
276M16—2144no—
277M16—2141no—
278M16C22×D10712yes—
279C2×C8—21no—
280C2×C8—2144no—
281C2×C8—2141no—
282C4⋊C4—21no—
283C4⋊C4—2142no—
284C22⋊C4—21no—
285C22⋊C4—2142no—
286C42—1071no—
287C42—1071no—
288C42—2142no—
289C42—2141no—
290C42C22×D10712yes—
291C23—21no—
292C23—1072no—
293C23—2144no—
294C23—2141no—
295C23D4×D10711yescharacteristic
296Q8—21no—
297Q8—1072no—
298Q8—1072no—
299Q8—1072no—
300Q8—2144no—
301Q8—2141no—
302Q8C2×C4×D10712yes—
303D4—22no—
304D4—22no—
305D4—21no—
306D4—1074no—
307D4—1074no—
308D4—1072no—
309D4—1072no—
310D4—1072no—
311D4—2144no—
312D4—2144no—
313D4—2144no—
314D4—2142no—
315D4—2142no—
316D4—2141no—
317D4C2×C4×D10712yes—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—1072no—
323C2×C4—1074no—
324C2×C4—1074no—
325C2×C4—4284no—
326C2×C4—4282no—
327C2×C4—2144no—
328C2×C4—2144no—
329C2×C4—2144no—
330C2×C4—2141no—
331C2×C4—2141no—
332C2×C4—2141no—
333C2×C4—2142no—
334C2×C4—2142no—
335C2×C4C2×C4×D10711yescharacteristic
336C2×C4C2×C4×D10711yescharacteristic
337C2×C4D4×D10711yescharacteristic
338C2×C4D4×D10711yescharacteristic
339C2×C4D4×D10711yescharacteristic
340C2×C4C23×D10711yescharacteristic
341C8—22no—
342C8—2142no—
343C22—22no—
344C22—41no—
345C22—81no—
346C22—8561no—
347C22—8564no—
348C22—1074no—
349C22—4282no—
350C22—2142no—
351C22C22⋊C4×D10711yescharacteristic
352C22C22⋊C4×D10711yescharacteristic
353C22C2×D4×D10711yescharacteristic
354C4—22no—
355C4—42no—
356C4—1074no—
357C4—4281no—
358C4—2142no—
359C4—2142no—
360C4C22⋊C4×D10711yescharacteristic
361C4C22⋊C4×D10711yescharacteristic
362C4C22×C4×D10711yescharacteristic
363C4C2×D4×D10711yescharacteristic
364C2—21no—
365C2—21no—
366C2—21no—
367C2—42no—
368C2—4282no—
369C2—2144no—
370C2C2×C22⋊C4×D10711yescharacteristic
subgroupG/H
centerC4C2×D4×D107
commutatorC4×C107C23×C4
FrattiniC2×C4C23×D107
Fitting#421×C107C2
socleC2×C107C22×C22⋊C4
prime radical#20293463C2
cosocleC2×C4×C107C24
hypercenter#421D107
hypocenterC107#2901
LCSG ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #20293393C2semidirect
3. #20293095C2semidirect
4. #20292645C2semidirect
5. #20292642C2semidirect
6. C2×#16812036C2semidirect
7. C2×#16812035C2semidirect
8. #104×D107C2semidirect
9. #421×C107C2semidirect
10. #16812157C4semidirect
12. #16812156C4semidirect
14. #16812155C4semidirect
16. #16812154C4semidirect
24. C2×#13133802C22semidirect
25. M16×D107C22semidirect
29. #16812066C22semidirect
30. C2×C4×Dic107C22semidirect
31. C42×D107C22semidirect
32. #16812037C22semidirect
33. #16812036C22semidirect
34. #16812036C22semidirect
35. #16812035C22semidirect
36. #16812035C22semidirect
38. C2×M16×C107C22semidirect
39. #117×C107C22semidirect
40. #104×C107C22semidirect
42. #13133819C2×C4semidirect
51. #13133818C2×C4semidirect
58. #13133817C2×C4semidirect
60. C2×Dic214C2×C4semidirect
63. Q8×D107C2×C4semidirect
65. C2×D428C2×C4semidirect
73. D4×D107C2×C4semidirect
92. #41×C107C2×C4semidirect
94. C2×Q8×C107C2×C4semidirect
96. C2×D4×C107C2×C4semidirect
186. C107#2901semidirect
192. #421D107semidirect
213. C2×M16C2×D107semidirect
218. #117C2×D107semidirect
226. #104C2×D107semidirect
247. #41C4×D107semidirect
254. C2×Q8C4×D107semidirect
266. C2×D4C4×D107semidirect

Maximal subgroups

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

subgroupindex
#202934632
#202933932
#202930952
#202926452
#202926422
C2×#168120362
C2×#168120352
#104×D1072
#421×C1072
#2901107