Finite Group Databasebeta

← #28346408

Subgroups — #28346408

Order 13696.

410 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×D107C211yesmaximal, characteristic
2#117×D107C211yesmaximal, characteristic
3#20292969C211yesmaximal, characteristic, complemented
4#20292968C211yesmaximal, characteristic, complemented
5#20292965C211yesmaximal, characteristic, complemented
6#20292964C211yesmaximal, characteristic, complemented
7#20292961C211yesmaximal, characteristic, complemented
8#20292959C211yesmaximal, characteristic, complemented
9#20292956C211yesmaximal, characteristic, complemented
10#20292668C211yesmaximal, characteristic, complemented
11#20292666C211yesmaximal, characteristic, complemented
12#20292634C211yesmaximal, characteristic, complemented
13#20292634C211yesmaximal, characteristic, complemented
14#20292589C211yesmaximal, characteristic
15#485×C107C211yesmaximal, characteristic, complemented
16C2×Q8×D107C2211yescharacteristic
17C22×C4×D107C2211yescharacteristic
18C2×#13133801C2211yescharacteristic
19C2×#13133801C2211yescharacteristic
20#16812089C2211yescharacteristic
21#16812088C2211yescharacteristic
22#16812087—21no—
23#16812086—21no—
24#16812085C2211yescharacteristic
25#16812068C2211yescharacteristic
26#16812067C2211yescharacteristic
27#16812067C2211yescharacteristic
28#16812066—21no—
29#16812065—21no—
30C4⋊C4×D107—21no—
31C4⋊C4×D107C2211yescharacteristic
32C4⋊C4×D107C2211yescharacteristic
33C4⋊C4×D107C2211yescharacteristic
34C22⋊C4×D107—21no—
35C22⋊C4×D107—21no—
36C42×D107—21no—
37#16812043C2211yescharacteristic, complemented
38#16812042C2211yescharacteristic, complemented
39#16812042C2211yescharacteristic, complemented
40#16812041C2211yescharacteristic, complemented
41#16812040C2211yescharacteristic, complemented
42#16812040C2211yescharacteristic, complemented
43#16812034C2211yescharacteristic, complemented
44#16812034C2211yescharacteristic, complemented
45#16812033C2211yescharacteristic, complemented
46#16812033C2211yescharacteristic, complemented
47#16812032C2211yescharacteristic, complemented
48#16812032C2211yescharacteristic, complemented
49C22⋊(C107⋊C8)C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51(C2×D107)⋊C8C2211yescharacteristic
52C22⋊Q8×C107C2211yescharacteristic
53#117×C107C2211yescharacteristic
54#107×C107C2211yescharacteristic, complemented
55#106×C107C2211yescharacteristic, complemented
56#103×C107C2211yescharacteristic, complemented
57#103×C107C2211yescharacteristic, complemented
58C22⋊C8×C107C2211yescharacteristic
59C23×D107D411yescharacteristic
60C2×Dic214—21no—
61C2×Dic214C2311yescharacteristic
62Q8×D107—22no—
63Q8×D107—22no—
64C22×Dic107D411yescharacteristic
65C2×C4×D107—22no—
66C2×C4×D107—22no—
67C2×C4×D107—21no—
68C2×C4×D107—21no—
69C2×C4×D107—21no—
70C2×C4×D107—21no—
71C2×C4×D107D411yescharacteristic
72C2×C4×D107D411yescharacteristic
73C2×C4×D107C2311yescharacteristic
74C2×C4×D107C2311yescharacteristic
75#13133801—22no—
76#13133801—22no—
77C2×C107⋊C8C2311yescharacteristic
78C2×C107⋊C8C2311yescharacteristic
79C4⋊Dic107—21no—
80C4⋊Dic107C2311yescharacteristic
81C4⋊Dic107C2311yescharacteristic
82C4⋊Dic107C2311yescharacteristic
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86Dic107⋊C4—21no—
87Dic107⋊C4—21no—
88C22⋊Dic107—21no—
89C22⋊Dic107—21no—
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94C4×Dic107—21no—
95C4×Dic107—21no—
96C4×Dic107—21no—
97C2×Q8×C107C2311yescharacteristic
98C22×C4×C107C2311yescharacteristic
99C2×C8×C107C2311yescharacteristic
100C2×C8×C107C2311yescharacteristic
101C4⋊C4×C107—21no—
102C4⋊C4×C107C2311yescharacteristic
103C4⋊C4×C107C2311yescharacteristic
104C4⋊C4×C107C2311yescharacteristic
105C22⋊C4×C107—21no—
106C22⋊C4×C107—21no—
107C42×C107—21no—
108C23×C107C2×D411yescharacteristic
109C22×D107—22no—
110C22×D107—22no—
111C22×D107—41no—
112C22×D107C2×D411yescharacteristic
113C22×D107C2×D411yescharacteristic
114Q8×C107—21no—
115Q8×C107—21no—
116C2×C4×C107—21no—
117C2×C4×C107—21no—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107C2×D411yescharacteristic
123C2×C4×C107C2411yescharacteristic
124C2×Dic107—22no—
125C2×Dic107—21no—
126C2×Dic107—21no—
127C2×Dic107—21no—
128C2×Dic107—21no—
129C2×Dic107C2×D411yescharacteristic
130C2×Dic107C2×D411yescharacteristic
131C4×D107—22no—
132C4×D107—21no—
133C4×D107—21no—
134C4×D107—22no—
135C4×D107—42no—
136C4×D107—42no—
137C4×D107—42no—
138C4×D107—42no—
139C4×D107#4112yes—
140C4×D107#4112yes—
141Dic214—21no—
142Dic214—21no—
143Dic214—42no—
144C8×C107—21no—
145C8×C107—21no—
146C107⋊C8—21no—
147C107⋊C8—21no—
148C22×C107—21no—
149C22×C107—21no—
150C22×C107—21no—
151C22×C107C22×D411yescharacteristic
152C2×D107—21no—
153C2×D107—21no—
154C2×D107—21no—
155C2×D107—21no—
156C2×D107—44no—
157C2×D107#12312yes—
158C2×D107#13712yes—
159C4×C107—21no—
160C4×C107—41no—
161C4×C107—41no—
162C4×C107—41no—
163C4×C107—41no—
164C4×C107C2×#4111yescharacteristic
165C4×C107C2×#4111yescharacteristic
166Dic107—21no—
167Dic107—41no—
168Dic107—41no—
169Dic107—41no—
170Dic107—41no—
171Dic107#12312yes—
172C2×C107—41no—
173C2×C107C2×#12311yescharacteristic
174C2×C107C2×#13711yescharacteristic
175C2×C107#57511yescharacteristic
176D107—22no—
177D107—22no—
178#3095—1071nomaximal, Sylow
179C107#309511yescharacteristic, complemented, Sylow
180C2×C22⋊Q8—1071no—
181C2×#117—1071no—
182#485—1071no—
183#485—1071no—
184#485—1071no—
185#485D10711yescharacteristic, complemented
186#484—1071no—
187#484—1071no—
188#483—1071no—
189#483—1071no—
190#428—1071no—
191#428—1071no—
192#419—1071no—
193#419—1071no—
194#407—1071no—
195C22×Q8—1071no—
196C23×C4—1071no—
197C2×M16—1071no—
198C2×M16—1071no—
199C22⋊Q8—1071no—
200C22⋊Q8—1071no—
201C22⋊Q8—1071no—
202C22⋊Q8—2141no—
203C22⋊Q8—2141no—
204C22⋊Q8C2×D10711yescharacteristic
205#117—1071no—
206#117—1071no—
207#117—1071no—
208#117—2141no—
209#117—2141no—
210#117C2×D10711yescharacteristic
211C2×C4⋊C4—1071no—
212C2×C4⋊C4—1071no—
213C2×C4⋊C4—1071no—
214C2×C4⋊C4—2141no—
215C2×C22⋊C4—2141no—
216C2×C22⋊C4—2141no—
217C2×C42—2141no—
218#107—1071no—
219#107—1071no—
220#107—1071no—
221#107C2×D10711yescharacteristic, complemented
222#106—1071no—
223#106—1071no—
224#106—1071no—
225#106C2×D10711yescharacteristic, complemented
226#103—1071no—
227#103—1071no—
228#103—1071no—
229#103—1071no—
230#103—1071no—
231#103—1071no—
232#103C2×D10711yescharacteristic, complemented
233#103C2×D10711yescharacteristic, complemented
234C22⋊C8—1071no—
235C22⋊C8—1071no—
236C22⋊C8—1071no—
237C22⋊C8C2×D10711yescharacteristic
238C24—1071no—
239C2×Q8—1071no—
240C2×Q8—2141no—
241C2×Q8—2142no—
242C2×Q8—2142no—
243C2×Q8C22×D10711yescharacteristic
244C22×C4—1071no—
245C22×C4—1071no—
246C22×C4—1071no—
247C22×C4—1071no—
248C22×C4—1071no—
249C22×C4—2142no—
250C22×C4—2142no—
251C22×C4—2141no—
252C22×C4—2141no—
253C22×C4—2141no—
254C22×C4—2141no—
255C22×C4C22×D10711yescharacteristic
256M16—2142no—
257M16—2142no—
258C2×C8—1071no—
259C2×C8—1071no—
260C2×C8C22×D10711yescharacteristic
261C2×C8C22×D10711yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—1071no—
264C4⋊C4—1071no—
265C4⋊C4—1071no—
266C4⋊C4—2141no—
267C4⋊C4—2141no—
268C4⋊C4—2141no—
269C4⋊C4—2141no—
270C4⋊C4—2141no—
271C4⋊C4—2141no—
272C4⋊C4C22×D10711yescharacteristic
273C4⋊C4C22×D10711yescharacteristic
274C4⋊C4C22×D10711yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—2141no—
278C22⋊C4—2141no—
279C22⋊C4—2141no—
280C22⋊C4—2141no—
281C22⋊C4—2141no—
282C22⋊C4—2141no—
283C42—21no—
284C42—2141no—
285C42—2141no—
286C42—2141no—
287C23—1071no—
288C23—1071no—
289C23—4281no—
290C23—2142no—
291C23—2142no—
292C23D4×D10711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—4282no—
296Q8—2141no—
297Q8—2141no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—1071no—
305C2×C4—1072no—
306C2×C4—1072no—
307C2×C4—1071no—
308C2×C4—4282no—
309C2×C4—4282no—
310C2×C4—4282no—
311C2×C4—4282no—
312C2×C4—2142no—
313C2×C4—2142no—
314C2×C4—2142no—
315C2×C4—2141no—
316C2×C4—2141no—
317C2×C4—2141no—
318C2×C4—2141no—
319C2×C4—2141no—
320C2×C4—2141no—
321C2×C4D4×D10711yescharacteristic
322C2×C4C23×D10711yescharacteristic
323C8—21no—
324C8—21no—
325C8—2141no—
326C8—2141no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—1072no—
331C22—1072no—
332C22—4284no—
333C22—2141no—
334C22—2141no—
335C22—2141no—
336C22—2141no—
337C22C2×D4×D10711yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—1072no—
344C4—4281no—
345C4—4281no—
346C4—4281no—
347C4—4281no—
348C4—2141no—
349C4#41×D10711yescharacteristic
350C4#41×D10711yescharacteristic
351C2—41no—
352C2—2142no—
353C2—2142no—
354C2#123×D10711yescharacteristic
355C2#137×D10711yescharacteristic
356C2#2029342411yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#485×C107C2
socleC22×C107C22×D4
prime radicalC23×D107D4
hypercenter#485D107
hypocenterC107#3095
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
3. #20292969C2semidirect
4. #20292968C2semidirect
5. #20292965C2semidirect
6. #20292964C2semidirect
7. #20292961C2semidirect
8. #20292959C2semidirect
9. #20292956C2semidirect
10. #20292668C2semidirect
11. #20292666C2semidirect
12. #20292634C2semidirect
13. #20292634C2semidirect
15. #485×C107C2semidirect
37. #16812043C22semidirect
38. #16812042C22semidirect
39. #16812042C22semidirect
40. #16812041C22semidirect
41. #16812040C22semidirect
42. #16812040C22semidirect
43. #16812034C22semidirect
44. #16812034C22semidirect
45. #16812033C22semidirect
46. #16812033C22semidirect
47. #16812032C22semidirect
48. #16812032C22semidirect
54. #107×C107C22semidirect
55. #106×C107C22semidirect
56. #103×C107C22semidirect
57. #103×C107C22semidirect
179. C107#3095semidirect
185. #485D107semidirect
221. #107C2×D107semidirect
225. #106C2×D107semidirect
232. #103C2×D107semidirect
233. #103C2×D107semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1072
#117×D1072
#202929692
#202929682
#202929652
#202929642
#202929612
#202929592
#202929562
#202926682
#202926662
#202926342
#202925892
#485×C1072
#3095107