Finite Group Databasebeta

← #28348319

Subgroups — #28348319

Order 13696.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C22⋊D107C211yesmaximal, characteristic, complemented
2#20293277C211yesmaximal, characteristic
3#20292976C211yesmaximal, characteristic, complemented
4#20292958C211yesmaximal, characteristic, complemented
5#20292956C211yesmaximal, characteristic, complemented
6#20292910C211yesmaximal, characteristic, complemented
7#20292854C211yesmaximal, characteristic, complemented
8#20292853C211yesmaximal, characteristic, complemented
9#20292830C211yesmaximal, characteristic, complemented
10#20292764C211yesmaximal, characteristic, complemented
11SD16×Dic107C211yesmaximal, characteristic, complemented
12(C22⋊D107)⋊C8C211yesmaximal, characteristic, complemented
13C2×#16812033C211yesmaximal, characteristic, complemented
14#103×D107C211yesmaximal, characteristic, complemented
15#448×C107C211yesmaximal, characteristic, complemented
16C2×#13133817—21no—
17C2×Q8×D107C2211yescharacteristic
18C2×#13133806C2211yescharacteristic
19C22×C107⋊C8C2211yescharacteristic
20C2×C8×D107C2211yescharacteristic
21#16812108C2211yescharacteristic
22#16812107—21no—
23#16812107C2211yescharacteristic
24#16812091—21no—
25#16812090—21no—
26#16812088—21no—
27#16812088C2211yescharacteristic
28#16812086—21no—
29#16812085—21no—
30#16812085C2211yescharacteristic
31#16812080C2211yescharacteristic
32Q8×Dic107C2211yescharacteristic
33D4×Dic107C2211yescharacteristic
34C4×C22⋊D107—21no—
35C4×C22⋊D107C2211yescharacteristic
36C2×C4⋊Dic107C2211yescharacteristic
37C4⋊C4×D107C2211yescharacteristic
38#16812041C2211yescharacteristic
39#16812041C2211yescharacteristic
40#16812040C2211yescharacteristic
41Dic107⋊C8C2211yescharacteristic
42#16812034C2211yescharacteristic
43#16812034C2211yescharacteristic
44#16812033C2211yescharacteristic
45#16812033C2211yescharacteristic
46#16812033C2211yescharacteristic
47#16812033C2211yescharacteristic
48#16812032C2211yescharacteristic
49#16812030C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51C8×Dic107C2211yescharacteristic
52C22×Q8×C107C2211yescharacteristic
53C2×SD16×C107C2211yescharacteristic
54C2×SD16×C107C2211yescharacteristic
55#121×C107C2211yescharacteristic
56#103×C107C2211yescharacteristic
57#103×C107C2211yescharacteristic
58C22⋊C8×C107C2211yescharacteristic
59#13133817—44no—
60C2×Dic214—21no—
61C2×Dic214—21no—
62C2×Dic214C2311yescharacteristic
63Q8×D107—22no—
64Q8×D107—41no—
65C2×C22⋊D107—21no—
66C2×C22⋊D107D411yescharacteristic
67C22×Dic107—21no—
68C22×Dic107—21no—
69C2×C4×D107—21no—
70C2×C4×D107—21no—
71C2×C4×D107C2311yescharacteristic
72#13133806—41no—
73C2×C107⋊C8—41no—
74C2×C107⋊C8C2311yescharacteristic
75C2×C107⋊C8C2311yescharacteristic
76C8×D107—41no—
77C4⋊Dic107—21no—
78C4⋊Dic107—21no—
79C4⋊Dic107C2311yescharacteristic
80C4⋊Dic107C2311yescharacteristic
81C4⋊Dic107C2311yescharacteristic
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86Dic107⋊C4—21no—
87Dic107⋊C4—21no—
88Dic107⋊C4D411yescharacteristic
89C22⋊Dic107—21no—
90C22⋊Dic107—21no—
91C22⋊Dic107—21no—
92C22⋊Dic107D411yescharacteristic
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4D411yescharacteristic
95C4×Dic107—21no—
96C4×Dic107C2311yescharacteristic
97C2×Q8×C107—21no—
98C2×Q8×C107—22no—
99C2×Q8×C107—41no—
100C2×Q8×C107C2311yescharacteristic
101C2×Q8×C107C2311yescharacteristic
102C2×D4×C107—21no—
103C2×D4×C107C2311yescharacteristic
104C22×C4×C107—21no—
105C22×C4×C107C2311yescharacteristic
106SD16×C107—22no—
107SD16×C107—41no—
108C2×C8×C107C2311yescharacteristic
109C2×C8×C107C2311yescharacteristic
110C4⋊C4×C107C2311yescharacteristic
111C22⋊C4×C107—21no—
112C23×C107—21no—
113C23×C107C2×D411yescharacteristic
114C22×D107C2×D411yescharacteristic
115Q8×C107—21no—
116Q8×C107—21no—
117Q8×C107—21no—
118Q8×C107—42no—
119Q8×C107#4112yescomplemented
120D4×C107—21no—
121D4×C107—21no—
122D4×C107—42no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107—21no—
128C2×C4×C107—42no—
129C2×C4×C107C2×D411yescharacteristic
130C2×C4×C107C2411yescharacteristic
131C22⋊D107—42no—
132C22⋊D107SD1614yescomplemented
133C2×Dic107—21no—
134C2×Dic107—21no—
135C2×Dic107—21no—
136C2×Dic107—21no—
137C2×Dic107—42no—
138C2×Dic107—42no—
139C2×Dic107C2×D411yescharacteristic
140C2×Dic107C2×D411yescharacteristic
141C2×Dic107C2×D411yescharacteristic
142C4×D107—21no—
143C4×D107—21no—
144C4×D107—42no—
145C4×D107—42no—
146Dic214—21no—
147Dic214—21no—
148Dic214—42no—
149Dic214—42no—
150C8×C107—21no—
151C8×C107—21no—
152C107⋊C8—21no—
153C107⋊C8—21no—
154C22×C107—21no—
155C22×C107—21no—
156C22×C107—41no—
157C22×C107—41no—
158C22×C107—41no—
159C22×C107C2×SD1612yes—
160C22×C107C22×D411yescharacteristic
161C2×D107—21no—
162C2×D107—21no—
163C2×D107C2×SD1612yes—
164C4×C107—22no—
165C4×C107—21no—
166C4×C107—41no—
167C4×C107—41no—
168C4×C107C2×#4111yescharacteristic
169C4×C107H(5,𝔽2)11yescharacteristic
170Dic107—21no—
171Dic107—21no—
172Dic107—41no—
173Dic107—41no—
174Dic107—41no—
175Dic107—41no—
176Dic107C2×SD1612yes—
177C2×C107—22no—
178C2×C107—81no—
179C2×C107#54611yescharacteristic
180C2×C107C22×SD1611yescharacteristic
181C2×C107#57811yescharacteristic
182D107—22no—
183#3298—1071nomaximal, Sylow
184C107#329811yescharacteristic, complemented, Sylow
185D4×Q8—1071no—
186#547—1071no—
187#487—1071no—
188#483—1071no—
189#483—1071no—
190#475—1071no—
191#465—1071no—
192#465—1071no—
193#461—1071no—
194#448—1071no—
195#448D10711yescharacteristic, complemented
196C4×SD16—1071no—
197C8×D4—1071no—
198C2×#103—1071no—
199C2×#103—1071no—
200C2×#41—2141no—
201C22×Q8—1071no—
202C22×Q8C2×D10711yescharacteristic
203C2×SD16—1071no—
204C2×SD16C2×D10711yescharacteristic
205C2×SD16C2×D10711yescharacteristic
206C22×C8—1071no—
207C22×C8—1071no—
208C4⋊Q8—1071no—
209C4⋊Q8—1071no—
210C4⋊Q8—2141no—
211#123—2141no—
212#123—2141no—
213C22⋊Q8—1071no—
214C22⋊Q8—1071no—
215C22⋊Q8—2141no—
216C22⋊Q8—2141no—
217C22⋊Q8—2141no—
218#121—1071no—
219#121C2×D10711yescharacteristic
220C4×Q8—1071no—
221C4×D4—1071no—
222C4×D4—1071no—
223C4×D4—2141no—
224C2×C4⋊C4—1071no—
225C2×C4⋊C4—1071no—
226#106—1071no—
227#106—1071no—
228#106—1071no—
229C4⋊C8—1071no—
230#103—1071no—
231#103—1071no—
232#103—1071no—
233#103—1071no—
234#103—1071no—
235#103—1071no—
236#103—1071no—
237#103C2×D10711yescharacteristic
238#103C2×D10711yescharacteristic
239#102—1071no—
240C22⋊C8—1071no—
241C22⋊C8C2×D10711yescharacteristic
242C4×C8—1071no—
243#41—4284no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—1071no—
248C2×Q8—4281no—
249C2×Q8—2141no—
250C2×Q8—2141no—
251C2×Q8—2142no—
252C2×Q8C22×D10711yescharacteristic
253C2×Q8C22×D10711yescharacteristic
254C2×D4—21no—
255C2×D4—1071no—
256C2×D4—2141no—
257C2×D4C22×D10711yescharacteristic
258C22×C4—21no—
259C22×C4—1071no—
260C22×C4—2141no—
261C22×C4—2141no—
262C22×C4—2141no—
263C22×C4—2141no—
264C22×C4C22×D10711yescharacteristic
265SD16—22no—
266SD16—41no—
267SD16—4281no—
268C2×C8—1071no—
269C2×C8—1071no—
270C2×C8—4281no—
271C2×C8—4281no—
272C2×C8C22×D10711yescharacteristic
273C2×C8C22×D10711yescharacteristic
274C4⋊C4—1071no—
275C4⋊C4—1071no—
276C4⋊C4—1071no—
277C4⋊C4—1071no—
278C4⋊C4—2141no—
279C4⋊C4—2141no—
280C4⋊C4—2141no—
281C4⋊C4—2141no—
282C4⋊C4—2141no—
283C4⋊C4—2141no—
284C4⋊C4—2141no—
285C4⋊C4—2141no—
286C4⋊C4C22×D10711yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—1071no—
289C22⋊C4—1071no—
290C22⋊C4—2141no—
291C22⋊C4—2141no—
292C22⋊C4—2141no—
293C22⋊C4—2141no—
294C42—1071no—
295C42—2141no—
296C23—21no—
297C23—1071no—
298C23D4×D10711yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—21no—
302Q8—42no—
303Q8—4282no—
304Q8—4282no—
305Q8—2141no—
306Q8—2141no—
307Q8#1313381712yescomplemented
308D4—21no—
309D4—21no—
310D4—42no—
311D4—1074no—
312D4—4282no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—1071no—
320C2×C4—1071no—
321C2×C4—1071no—
322C2×C4—4282no—
323C2×C4—4282no—
324C2×C4—4282no—
325C2×C4—4282no—
326C2×C4—2141no—
327C2×C4—2141no—
328C2×C4—2141no—
329C2×C4—2141no—
330C2×C4—2141no—
331C2×C4—2141no—
332C2×C4D4×D10711yescharacteristic
333C2×C4C23×D10711yescharacteristic
334C8—21no—
335C8—21no—
336C8—2141no—
337C8—2141no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—1072no—
344C22—2141no—
345C22—2141no—
346C22SD16×D10712yes—
347C22C2×D4×D10711yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—41no—
352C4—1072no—
353C4—4281no—
354C4—4281no—
355C4—4281no—
356C4—4281no—
357C4—2141no—
358C4—2141no—
359C4C2×#1313381711yescharacteristic
360C4#1681215411yescharacteristic
361C2—22no—
362C2—81no—
363C2—2142no—
364C2#2029326411yescharacteristic
365C2C2×SD16×D10711yescharacteristic
366C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#448×C107C2
socleC22×C107C22×D4
prime radical#20293277C2
hypercenter#448D107
hypocenterC107#3298
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. Q8×C22⋊D107C2semidirect
3. #20292976C2semidirect
4. #20292958C2semidirect
5. #20292956C2semidirect
6. #20292910C2semidirect
7. #20292854C2semidirect
8. #20292853C2semidirect
9. #20292830C2semidirect
10. #20292764C2semidirect
11. SD16×Dic107C2semidirect
12. (C22⋊D107)⋊C8C2semidirect
13. C2×#16812033C2semidirect
14. #103×D107C2semidirect
15. #448×C107C2semidirect
119. Q8×C107#41semidirect
132. C22⋊D107SD16semidirect
184. C107#3298semidirect
195. #448D107semidirect
307. Q8#13133817semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊D1072
#202932772
#202929762
#202929582
#202929562
#202929102
#202928542
#202928532
#202928302
#202927642
SD16×Dic1072
(C22⋊D107)⋊C82
C2×#168120332
#103×D1072
#448×C1072
#3298107