Finite Group Databasebeta

← #28348267

Subgroups — #28348267

Order 13696.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293277C211yesmaximal, characteristic, complemented
2D4×C22⋊D107C211yesmaximal, characteristic, complemented
3#20292972C211yesmaximal, characteristic, complemented
4#20292941C211yesmaximal, characteristic, complemented
5#20292939C211yesmaximal, characteristic, complemented
6#20292905C211yesmaximal, characteristic, complemented
7#20292861C211yesmaximal, characteristic, complemented
8#20292860C211yesmaximal, characteristic, complemented
9#20292820C211yesmaximal, characteristic, complemented
10#20292759C211yesmaximal, characteristic, complemented
11D8×Dic107C211yesmaximal, characteristic, complemented
12(C22⋊D107)⋊C8C211yesmaximal, characteristic, complemented
13C2×#16812030C211yesmaximal, characteristic, complemented
14#102×D107C211yesmaximal, characteristic, complemented
15#447×C107C211yesmaximal, characteristic, complemented
16C2×#13133817—21no—
17C22×C22⋊D107—21no—
18C2×D4×D107C2211yescharacteristic
19C2×#13133803C2211yescharacteristic, complemented
20C22×C107⋊C8C2211yescharacteristic, complemented
21C2×C8×D107C2211yescharacteristic, complemented
22#16812108C2211yescharacteristic, complemented
23#16812105C2211yescharacteristic
24#16812091—21no—
25#16812090—21no—
26#16812086—21no—
27#16812083C2211yescharacteristic
28#16812081—21no—
29#16812080C2211yescharacteristic
30#16812080C2211yescharacteristic, complemented
31#16812078—21no—
32#16812077—21no—
33D4×Dic107C2211yescharacteristic
34D4×Dic107C2211yescharacteristic, complemented
35C4×C22⋊D107C2211yescharacteristic
36C2×C4⋊Dic107C2211yescharacteristic, complemented
37C4⋊C4×D107C2211yescharacteristic, complemented
38#16812043C2211yescharacteristic, complemented
39#16812043C2211yescharacteristic, complemented
40#16812042C2211yescharacteristic, complemented
41Dic107⋊C8C2211yescharacteristic, complemented
42#16812031C2211yescharacteristic, complemented
43#16812031C2211yescharacteristic, complemented
44#16812030C2211yescharacteristic
45#16812030C2211yescharacteristic
46#16812030C2211yescharacteristic, complemented
47#16812030C2211yescharacteristic, complemented
48#16812030C2211yescharacteristic, complemented
49#16812029C2211yescharacteristic, complemented
50(C2×D107)⋊C8C2211yescharacteristic, complemented
51C8×Dic107C2211yescharacteristic, complemented
52C22×D4×C107C2211yescharacteristic
53C2×D8×C107C2211yescharacteristic
54C2×D8×C107C2211yescharacteristic, complemented
55#121×C107C2211yescharacteristic, complemented
56#102×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic, complemented
58C22⋊C8×C107C2211yescharacteristic, complemented
59C23×D107—21no—
60#13133817—44no—
61C2×Dic214—21no—
62C2×D428C2311yescharacteristic
63C2×C22⋊D107—21no—
64C2×C22⋊D107—21no—
65C2×C22⋊D107—21no—
66C2×C22⋊D107—21no—
67C2×C22⋊D107—21no—
68C2×C22⋊D107—21no—
69C2×C22⋊D107—44no—
70C2×C22⋊D107D411yescharacteristic
71D4×D107—22no—
72D4×D107—41no—
73C22×Dic107—21no—
74C22×Dic107—21no—
75C22×Dic107—21no—
76C2×C4×D107—21no—
77C2×C4×D107C2311yescharacteristic
78#13133803—41no—
79C2×C107⋊C8—41no—
80C2×C107⋊C8C2311yescharacteristic
81C2×C107⋊C8C2311yescharacteristic
82C8×D107—41no—
83C4⋊Dic107—21no—
84C4⋊Dic107C2311yescharacteristic
85C4⋊Dic107C2311yescharacteristic
86C4⋊Dic107C2311yescharacteristic
87Dic107⋊C4—21no—
88Dic107⋊C4—21no—
89Dic107⋊C4D411yescharacteristic
90C22⋊Dic107—21no—
91C22⋊Dic107—21no—
92C22⋊Dic107—21no—
93C22⋊Dic107—21no—
94C22⋊Dic107D411yescharacteristic
95(C2×D107)⋊C4—21no—
96(C2×D107)⋊C4D411yescharacteristic
97C4×Dic107C2311yescharacteristic
98C24×C107—21no—
99C2×D4×C107—22no—
100C2×D4×C107—21no—
101C2×D4×C107—21no—
102C2×D4×C107—41no—
103C2×D4×C107C2311yescharacteristic
104C2×D4×C107C2311yescharacteristic
105C2×D4×C107C2311yescharacteristic
106C22×C4×C107C2311yescharacteristic
107D8×C107—22no—
108D8×C107—41no—
109C2×C8×C107C2311yescharacteristic
110C2×C8×C107C2311yescharacteristic
111C4⋊C4×C107C2311yescharacteristic
112C22⋊C4×C107—21no—
113C23×C107—21no—
114C23×C107—21no—
115C23×C107—21no—
116C23×C107—42no—
117C23×C107—42no—
118C23×C107—81no—
119C23×C107C2×D411yescharacteristic
120C22×D107—21no—
121C22×D107—42no—
122C22×D107—42no—
123C22×D107—81no—
124C22×D107C2×D411yescharacteristic
125D4×C107—21no—
126D4×C107—21no—
127D4×C107—21no—
128D4×C107—21no—
129D4×C107—21no—
130D4×C107—42no—
131D4×C107—42no—
132D4×C107#4112yescomplemented
133C2×C4×C107—21no—
134C2×C4×C107—21no—
135C2×C4×C107—21no—
136C2×C4×C107C2×D411yescharacteristic
137C2×C4×C107C2411yescharacteristic
138C22⋊D107—42no—
139C22⋊D107—42no—
140C22⋊D107—42no—
141C22⋊D107—42no—
142C22⋊D107—42no—
143C22⋊D107—81no—
144C22⋊D107D814yescomplemented
145C2×Dic107—21no—
146C2×Dic107—21no—
147C2×Dic107—21no—
148C2×Dic107—42no—
149C2×Dic107—42no—
150C2×Dic107—42no—
151C2×Dic107C2×D411yescharacteristic
152C2×Dic107C2×D411yescharacteristic
153C2×Dic107C2×D411yescharacteristic
154D428—21no—
155D428—21no—
156C4×D107—21no—
157C4×D107—21no—
158C4×D107—42no—
159Dic214—42no—
160C8×C107—21no—
161C8×C107—21no—
162C107⋊C8—21no—
163C107⋊C8—21no—
164C22×C107—21no—
165C22×C107—21no—
166C22×C107—22no—
167C22×C107—41no—
168C22×C107—41no—
169C22×C107—41no—
170C22×C107—41no—
171C22×C107—41no—
172C22×C107—41no—
173C22×C107—41no—
174C22×C107—41no—
175C22×C107—84no—
176C22×C107C2×D812yes—
177C22×C107C22×D411yescharacteristic
178C2×D107—21no—
179C2×D107—21no—
180C2×D107—41no—
181C2×D107—41no—
182C2×D107—41no—
183C2×D107—84no—
184C2×D107C2×D812yes—
185C4×C107—21no—
186C4×C107—41no—
187C4×C107C2×#4111yescharacteristic
188C4×C107H(5,𝔽2)11yescharacteristic
189Dic107—21no—
190Dic107—21no—
191Dic107—41no—
192Dic107—41no—
193Dic107—41no—
194Dic107C2×D812yes—
195C2×C107—22no—
196C2×C107—42no—
197C2×C107—81no—
198C2×C107—81no—
199C2×C107#54611yescharacteristic
200C2×C107C22×D811yescharacteristic
201C2×C107#57711yescharacteristic
202D107—22no—
203D107—81no—
204#3294—1071nomaximal, Sylow
205C107#329411yescharacteristic, complemented, Sylow
206#547—1071no—
207D42—1071no—
208#486—1071no—
209#480—1071no—
210#480—1071no—
211#474—1071no—
212#466—1071no—
213#466—1071no—
214#459—1071no—
215#447—1071no—
216#447D10711yescharacteristic, complemented
217C4×D8—1071no—
218C8×D4—1071no—
219C2×#102—1071no—
220C2×#102—1071no—
221C2×#41—2141no—
222C22×D4—1071no—
223C22×D4—2141no—
224C22×D4C2×D10711yescharacteristic
225C2×D8—1071no—
226C2×D8C2×D10711yescharacteristic
227C2×D8C2×D10711yescharacteristic, complemented
228C22×C8—1071no—
229C22×C8—1071no—
230C4⋊Q8—1071no—
231#127—1071no—
232#123—2141no—
233#123—2141no—
234C22⋊Q8—2141no—
235#121—1071no—
236#121—1071no—
237#121—1071no—
238#121—2141no—
239#121C2×D10711yescharacteristic, complemented
240#120—2141no—
241#120—2141no—
242C4×D4—1071no—
243C4×D4—1071no—
244C4×D4—1071no—
245C2×C4⋊C4—1071no—
246C2×C4⋊C4—1071no—
247#107—1071no—
248#107—1071no—
249#107—1071no—
250C4⋊C8—1071no—
251#102—1071no—
252#102—1071no—
253#102—1071no—
254#102—1071no—
255#102—1071no—
256#102—1071no—
257#102—1071no—
258#102—1071no—
259#102C2×D10711yescharacteristic
260#102C2×D10711yescharacteristic, complemented
261C22⋊C8—1071no—
262C22⋊C8C2×D10711yescharacteristic, complemented
263C4×C8—1071no—
264C24—21no—
265C24—2141no—
266#41—4284no—
267C2×Q8—2141no—
268C2×D4—22no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—41no—
272C2×D4—1071no—
273C2×D4—1071no—
274C2×D4—4284no—
275C2×D4—4281no—
276C2×D4—2141no—
277C2×D4—2141no—
278C2×D4—2141no—
279C2×D4—2141no—
280C2×D4—2141no—
281C2×D4—2141no—
282C2×D4—2142no—
283C2×D4C22×D10711yescharacteristic
284C2×D4C22×D10711yescharacteristic
285C2×D4C22×D10711yescharacteristic
286C22×C4—1071no—
287C22×C4—2141no—
288C22×C4—2141no—
289C22×C4—2141no—
290C22×C4—2141no—
291C22×C4C22×D10711yescharacteristic
292D8—22no—
293D8—41no—
294D8—4281no—
295C2×C8—1071no—
296C2×C8—1071no—
297C2×C8—4281no—
298C2×C8—4281no—
299C2×C8C22×D10711yescharacteristic
300C2×C8C22×D10711yescharacteristic
301C4⋊C4—1071no—
302C4⋊C4—1071no—
303C4⋊C4—1071no—
304C4⋊C4—1071no—
305C4⋊C4—2141no—
306C4⋊C4—2141no—
307C4⋊C4—2141no—
308C4⋊C4C22×D10711yescharacteristic
309C22⋊C4—21no—
310C22⋊C4—1071no—
311C22⋊C4—1071no—
312C22⋊C4—2141no—
313C22⋊C4—2141no—
314C22⋊C4—2141no—
315C22⋊C4—2141no—
316C22⋊C4—2141no—
317C42—1071no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—42no—
322C23—42no—
323C23—81no—
324C23—8561no—
325C23—1071no—
326C23—4282no—
327C23—4282no—
328C23—2141no—
329C23D4×D10711yescharacteristic
330Q8—4282no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—42no—
338D4—8561no—
339D4—1074no—
340D4—4282no—
341D4—4282no—
342D4—4282no—
343D4—4282no—
344D4—4282no—
345D4—2141no—
346D4—2141no—
347D4#1313381712yescomplemented
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—1071no—
352C2×C4—1071no—
353C2×C4—1071no—
354C2×C4—4282no—
355C2×C4—4282no—
356C2×C4—4282no—
357C2×C4—4282no—
358C2×C4—2141no—
359C2×C4—2141no—
360C2×C4—2141no—
361C2×C4—2141no—
362C2×C4—2141no—
363C2×C4D4×D10711yescharacteristic
364C2×C4C23×D10711yescharacteristic
365C8—21no—
366C8—21no—
367C8—2141no—
368C8—2141no—
369C22—21no—
370C22—21no—
371C22—22no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—84no—
381C22—8564no—
382C22—1072no—
383C22—4281no—
384C22—4281no—
385C22—4281no—
386C22—2141no—
387C22—2141no—
388C22D8×D10712yes—
389C22C2×D4×D10711yescharacteristic
390C4—21no—
391C4—41no—
392C4—1072no—
393C4—4281no—
394C4—4281no—
395C4—4281no—
396C4—2141no—
397C4—2141no—
398C4C2×#1313381711yescharacteristic
399C4#1681215411yescharacteristic
400C2—22no—
401C2—42no—
402C2—81no—
403C2—81no—
404C2—8561no—
405C2—2142no—
406C2#2029326411yescharacteristic
407C2C2×D8×D10711yescharacteristic
408C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#447×C107C2
socleC22×C107C22×D4
hypercenter#447D107
hypocenterC107#3294
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #20293277C2semidirect
2. D4×C22⋊D107C2semidirect
3. #20292972C2semidirect
4. #20292941C2semidirect
5. #20292939C2semidirect
6. #20292905C2semidirect
7. #20292861C2semidirect
8. #20292860C2semidirect
9. #20292820C2semidirect
10. #20292759C2semidirect
11. D8×Dic107C2semidirect
12. (C22⋊D107)⋊C8C2semidirect
13. C2×#16812030C2semidirect
14. #102×D107C2semidirect
15. #447×C107C2semidirect
19. C2×#13133803C22semidirect
20. C22×C107⋊C8C22semidirect
21. C2×C8×D107C22semidirect
22. #16812108C22semidirect
30. #16812080C22semidirect
34. D4×Dic107C22semidirect
36. C2×C4⋊Dic107C22semidirect
37. C4⋊C4×D107C22semidirect
38. #16812043C22semidirect
39. #16812043C22semidirect
40. #16812042C22semidirect
41. Dic107⋊C8C22semidirect
42. #16812031C22semidirect
43. #16812031C22semidirect
46. #16812030C22semidirect
47. #16812030C22semidirect
48. #16812030C22semidirect
49. #16812029C22semidirect
50. (C2×D107)⋊C8C22semidirect
51. C8×Dic107C22semidirect
54. C2×D8×C107C22semidirect
55. #121×C107C22semidirect
57. #102×C107C22semidirect
58. C22⋊C8×C107C22semidirect
132. D4×C107#41semidirect
144. C22⋊D107D8semidirect
205. C107#3294semidirect
216. #447D107semidirect
227. C2×D8C2×D107semidirect
239. #121C2×D107semidirect
260. #102C2×D107semidirect
262. C22⋊C8C2×D107semidirect
347. D4#13133817semidirect

Maximal subgroups

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

subgroupindex
#202932772
D4×C22⋊D1072
#202929722
#202929412
#202929392
#202929052
#202928612
#202928602
#202928202
#202927592
D8×Dic1072
(C22⋊D107)⋊C82
C2×#168120302
#102×D1072
#447×C1072
#3294107