Finite Group Databasebeta

← #34111994

Subgroups — #34111994

Order 20096.

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⋊D157C211yesmaximal, characteristic, complemented
2#23757923C211yesmaximal, characteristic
3#23757622C211yesmaximal, characteristic, complemented
4#23757604C211yesmaximal, characteristic, complemented
5#23757602C211yesmaximal, characteristic, complemented
6#23757556C211yesmaximal, characteristic, complemented
7#23757500C211yesmaximal, characteristic, complemented
8#23757499C211yesmaximal, characteristic, complemented
9#23757476C211yesmaximal, characteristic, complemented
10#23757410C211yesmaximal, characteristic, complemented
11SD16×Dic157C211yesmaximal, characteristic, complemented
12#23757336C211yesmaximal, characteristic, complemented
13C2×#19097695C211yesmaximal, characteristic, complemented
14#103×D157C211yesmaximal, characteristic, complemented
15#448×C157C211yesmaximal, characteristic, complemented
16C2×#14578953—21no—
17C2×Q8×D157C2211yescharacteristic
18C2×#14578942C2211yescharacteristic
19C22×#11929434C2211yescharacteristic
20C2×C8×D157C2211yescharacteristic
21#19097770C2211yescharacteristic
22#19097769—21no—
23#19097769C2211yescharacteristic
24#19097753—21no—
25#19097752—21no—
26#19097750—21no—
27#19097750C2211yescharacteristic
28#19097748—21no—
29#19097747—21no—
30#19097747C2211yescharacteristic
31#19097742C2211yescharacteristic
32Q8×Dic157C2211yescharacteristic
33D4×Dic157C2211yescharacteristic
34C4×C22⋊D157—21no—
35C4×C22⋊D157C2211yescharacteristic
36C2×C4⋊Dic157C2211yescharacteristic
37C4⋊C4×D157C2211yescharacteristic
38#19097703C2211yescharacteristic
39#19097703C2211yescharacteristic
40#19097702C2211yescharacteristic
41#19097700C2211yescharacteristic
42#19097696C2211yescharacteristic
43#19097696C2211yescharacteristic
44#19097695C2211yescharacteristic
45#19097695C2211yescharacteristic
46#19097695C2211yescharacteristic
47#19097695C2211yescharacteristic
48#19097694C2211yescharacteristic
49#19097692C2211yescharacteristic
50#19097683C2211yescharacteristic
51C8×Dic157C2211yescharacteristic
52C22×Q8×C157C2211yescharacteristic
53C2×SD16×C157C2211yescharacteristic
54C2×SD16×C157C2211yescharacteristic
55#121×C157C2211yescharacteristic
56#103×C157C2211yescharacteristic
57#103×C157C2211yescharacteristic
58C22⋊C8×C157C2211yescharacteristic
59#14578953—44no—
60C2×Dic314—21no—
61C2×Dic314—21no—
62C2×Dic314C2311yescharacteristic
63Q8×D157—22no—
64Q8×D157—41no—
65C2×C22⋊D157—21no—
66C2×C22⋊D157D411yescharacteristic
67C22×Dic157—21no—
68C22×Dic157—21no—
69C2×C4×D157—21no—
70C2×C4×D157—21no—
71C2×C4×D157C2311yescharacteristic
72#14578942—41no—
73C2×#11929434—41no—
74C2×#11929434C2311yescharacteristic
75C2×#11929434C2311yescharacteristic
76C8×D157—41no—
77C4⋊Dic157—21no—
78C4⋊Dic157—21no—
79C4⋊Dic157C2311yescharacteristic
80C4⋊Dic157C2311yescharacteristic
81C4⋊Dic157C2311yescharacteristic
82#14578933—21no—
83#14578933—21no—
84#14578933—21no—
85#14578933—21no—
86#14578933—21no—
87#14578933—21no—
88#14578933D411yescharacteristic
89C22⋊Dic157—21no—
90C22⋊Dic157—21no—
91C22⋊Dic157—21no—
92C22⋊Dic157D411yescharacteristic
93#14578931—21no—
94#14578931D411yescharacteristic
95C4×Dic157—21no—
96C4×Dic157C2311yescharacteristic
97C2×Q8×C157—21no—
98C2×Q8×C157—22no—
99C2×Q8×C157—41no—
100C2×Q8×C157C2311yescharacteristic
101C2×Q8×C157C2311yescharacteristic
102C2×D4×C157—21no—
103C2×D4×C157C2311yescharacteristic
104C22×C4×C157—21no—
105C22×C4×C157C2311yescharacteristic
106SD16×C157—22no—
107SD16×C157—41no—
108C2×C8×C157C2311yescharacteristic
109C2×C8×C157C2311yescharacteristic
110C4⋊C4×C157C2311yescharacteristic
111C22⋊C4×C157—21no—
112C22×D157C2×D411yescharacteristic
113Dic314—21no—
114Dic314—21no—
115Dic314—42no—
116Dic314—42no—
117C22⋊D157—42no—
118C22⋊D157SD1614yescomplemented
119C2×Dic157—21no—
120C2×Dic157—21no—
121C2×Dic157—21no—
122C2×Dic157—21no—
123C2×Dic157—42no—
124C2×Dic157—42no—
125C2×Dic157C2×D411yescharacteristic
126C2×Dic157C2×D411yescharacteristic
127C2×Dic157C2×D411yescharacteristic
128C4×D157—21no—
129C4×D157—21no—
130C4×D157—42no—
131C4×D157—42no—
132#11929434—21no—
133#11929434—21no—
134C23×C157—21no—
135C23×C157C2×D411yescharacteristic
136Q8×C157—21no—
137Q8×C157—21no—
138Q8×C157—21no—
139Q8×C157—42no—
140Q8×C157#4112yescomplemented
141D4×C157—21no—
142D4×C157—21no—
143D4×C157—42no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157—21no—
147C2×C4×C157—21no—
148C2×C4×C157—21no—
149C2×C4×C157—42no—
150C2×C4×C157C2×D411yescharacteristic
151C2×C4×C157C2411yescharacteristic
152C8×C157—21no—
153C8×C157—21no—
154C22×C157—21no—
155C22×C157—21no—
156C22×C157—41no—
157C22×C157—41no—
158C22×C157—41no—
159C22×C157C2×SD1612yes—
160C22×C157C22×D411yescharacteristic
161C2×D157—21no—
162C2×D157—21no—
163C2×D157C2×SD1612yes—
164C4×C157—21no—
165C4×C157—22no—
166C4×C157—41no—
167C4×C157—41no—
168C4×C157C2×#4111yescharacteristic
169C4×C157H(5,𝔽2)11yescharacteristic
170Dic157—21no—
171Dic157—21no—
172Dic157—41no—
173Dic157—41no—
174Dic157—41no—
175Dic157—41no—
176Dic157C2×SD1612yes—
177C2×C157—22no—
178C2×C157—81no—
179C2×C157#54611yescharacteristic
180C2×C157C22×SD1611yescharacteristic
181C2×C157#57811yescharacteristic
182D157—22no—
183C157#329811yescharacteristic, complemented, Sylow
184#3298—1571nomaximal, Sylow
185D4×Q8—1571no—
186#547—1571no—
187#487—1571no—
188#483—1571no—
189#483—1571no—
190#475—1571no—
191#465—1571no—
192#465—1571no—
193#461—1571no—
194#448—1571no—
195#448D15711yescharacteristic, complemented
196C4×SD16—1571no—
197C8×D4—1571no—
198C2×#103—1571no—
199C2×#103—1571no—
200C2×#41—3141no—
201C22×Q8—1571no—
202C22×Q8C2×D15711yescharacteristic
203C2×SD16—1571no—
204C2×SD16C2×D15711yescharacteristic
205C2×SD16C2×D15711yescharacteristic
206C22×C8—1571no—
207C22×C8—1571no—
208C4⋊Q8—3141no—
209C4⋊Q8—1571no—
210C4⋊Q8—1571no—
211#123—3141no—
212#123—3141no—
213C22⋊Q8—3141no—
214C22⋊Q8—3141no—
215C22⋊Q8—3141no—
216C22⋊Q8—1571no—
217C22⋊Q8—1571no—
218#121—1571no—
219#121C2×D15711yescharacteristic
220C4×Q8—1571no—
221C4×D4—3141no—
222C4×D4—1571no—
223C4×D4—1571no—
224C2×C4⋊C4—1571no—
225C2×C4⋊C4—1571no—
226#106—1571no—
227#106—1571no—
228#106—1571no—
229C4⋊C8—1571no—
230#103—1571no—
231#103—1571no—
232#103—1571no—
233#103—1571no—
234#103—1571no—
235#103—1571no—
236#103—1571no—
237#103C2×D15711yescharacteristic
238#103C2×D15711yescharacteristic
239#102—1571no—
240C22⋊C8—1571no—
241C22⋊C8C2×D15711yescharacteristic
242C4×C8—1571no—
243#41—6284no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—3141no—
248C2×Q8—3141no—
249C2×Q8—3142no—
250C2×Q8—6281no—
251C2×Q8—1571no—
252C2×Q8C22×D15711yescharacteristic
253C2×Q8C22×D15711yescharacteristic
254C2×D4—21no—
255C2×D4—3141no—
256C2×D4—1571no—
257C2×D4C22×D15711yescharacteristic
258C22×C4—21no—
259C22×C4—3141no—
260C22×C4—3141no—
261C22×C4—3141no—
262C22×C4—3141no—
263C22×C4—1571no—
264C22×C4C22×D15711yescharacteristic
265SD16—22no—
266SD16—41no—
267SD16—6281no—
268C2×C8—6281no—
269C2×C8—6281no—
270C2×C8—1571no—
271C2×C8—1571no—
272C2×C8C22×D15711yescharacteristic
273C2×C8C22×D15711yescharacteristic
274C4⋊C4—3141no—
275C4⋊C4—3141no—
276C4⋊C4—3141no—
277C4⋊C4—3141no—
278C4⋊C4—3141no—
279C4⋊C4—3141no—
280C4⋊C4—3141no—
281C4⋊C4—3141no—
282C4⋊C4—1571no—
283C4⋊C4—1571no—
284C4⋊C4—1571no—
285C4⋊C4—1571no—
286C4⋊C4C22×D15711yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—3141no—
289C22⋊C4—3141no—
290C22⋊C4—3141no—
291C22⋊C4—3141no—
292C22⋊C4—1571no—
293C22⋊C4—1571no—
294C42—3141no—
295C42—1571no—
296C23—21no—
297C23—1571no—
298C23D4×D15711yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—21no—
302Q8—42no—
303Q8—3141no—
304Q8—3141no—
305Q8—6282no—
306Q8—6282no—
307Q8#1457895312yescomplemented
308D4—21no—
309D4—21no—
310D4—42no—
311D4—6282no—
312D4—1574no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—3141no—
320C2×C4—3141no—
321C2×C4—3141no—
322C2×C4—3141no—
323C2×C4—3141no—
324C2×C4—3141no—
325C2×C4—6282no—
326C2×C4—6282no—
327C2×C4—6282no—
328C2×C4—6282no—
329C2×C4—1571no—
330C2×C4—1571no—
331C2×C4—1571no—
332C2×C4D4×D15711yescharacteristic
333C2×C4C23×D15711yescharacteristic
334C8—21no—
335C8—21no—
336C8—3141no—
337C8—3141no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—3141no—
344C22—3141no—
345C22—1572no—
346C22SD16×D15712yes—
347C22C2×D4×D15711yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—41no—
352C4—3141no—
353C4—3141no—
354C4—6281no—
355C4—6281no—
356C4—6281no—
357C4—6281no—
358C4—1572no—
359C4C2×#1457895311yescharacteristic
360C4#1909781611yescharacteristic
361C2—22no—
362C2—81no—
363C2—3142no—
364C2#2375791011yescharacteristic
365C2C2×SD16×D15711yescharacteristic
366C2#2375808911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#448×C157C2
socleC22×C157C22×D4
prime radical#23757923C2
hypercenter#448D157
hypocenterC157#3298
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. Q8×C22⋊D157C2semidirect
3. #23757622C2semidirect
4. #23757604C2semidirect
5. #23757602C2semidirect
6. #23757556C2semidirect
7. #23757500C2semidirect
8. #23757499C2semidirect
9. #23757476C2semidirect
10. #23757410C2semidirect
11. SD16×Dic157C2semidirect
12. #23757336C2semidirect
13. C2×#19097695C2semidirect
14. #103×D157C2semidirect
15. #448×C157C2semidirect
118. C22⋊D157SD16semidirect
140. Q8×C157#41semidirect
183. C157#3298semidirect
195. #448D157semidirect
307. Q8#14578953semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊D1572
#237579232
#237576222
#237576042
#237576022
#237575562
#237575002
#237574992
#237574762
#237574102
SD16×Dic1572
#237573362
C2×#190976952
#103×D1572
#448×C1572
#3298157