Finite Group Databasebeta

← #34111881

Subgroups — #34111881

Order 20096.

472 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14578944C211yesmaximal, characteristic, complemented
2C2×Q16×D157C211yesmaximal, characteristic, complemented
3Q8×C22⋊D157C211yesmaximal, characteristic
4#23757936C211yesmaximal, characteristic
5#23757651C211yesmaximal, characteristic, complemented
6#23757513C211yesmaximal, characteristic, complemented
7#23757512C211yesmaximal, characteristic, complemented
8#23757482C211yesmaximal, characteristic, complemented
9#23757481C211yesmaximal, characteristic, complemented
10#23757431C211yesmaximal, characteristic, complemented
11#23757430C211yesmaximal, characteristic, complemented
12#23757427C211yesmaximal, characteristic, complemented
13#23757365C211yesmaximal, characteristic, complemented
14#23757336C211yesmaximal, characteristic
15#451×C157C211yesmaximal, characteristic, complemented
16C22×Dic314C2211yescharacteristic
17C2×Q8×D157C2211yescharacteristic
18C2×Q8×D157C2211yescharacteristic
19C2×Dic628C2211yescharacteristic
20C2×Dic628C2211yescharacteristic
21C2×#14578944—24no—
22C2×#14578944C2211yescharacteristic
23C2×#14578944C2211yescharacteristic
24C2×#14578944C2211yescharacteristic
25C2×#14578944C2211yescharacteristic
26C2×#14578944C2211yescharacteristic
27Q16×D157—24no—
28C22×#11929434C2211yescharacteristic
29C2×C8×D157C2211yescharacteristic
30#19097770—21no—
31#19097769—21no—
32#19097769C2211yescharacteristic
33#19097769C2211yescharacteristic
34#19097751C2211yescharacteristic
35#19097750—21no—
36#19097750C2211yescharacteristic
37#19097749—21no—
38#19097748—21no—
39#19097747—21no—
40#19097747C2211yescharacteristic
41Q8×Dic157C2211yescharacteristic
42#19097736C2211yescharacteristic
43C4×C22⋊D157—21no—
44C4×C22⋊D157C2211yescharacteristic
45#19097731—21no—
46#19097704C2211yescharacteristic
47#19097700C2211yescharacteristic
48#19097696C2211yescharacteristic
49#19097696C2211yescharacteristic
50#19097695C2211yescharacteristic
51#19097694C2211yescharacteristic
52#19097683C2211yescharacteristic
53C8×Dic157C2211yescharacteristic
54C22×Q8×C157C2211yescharacteristic
55C2×Q16×C157C2211yescharacteristic
56C2×Q16×C157C2211yescharacteristic
57C22⋊Q8×C157C2211yescharacteristic
58#103×C157C2211yescharacteristic
59#103×C157C2211yescharacteristic
60C22⋊C8×C157C2211yescharacteristic
61C2×Dic314—21no—
62C2×Dic314—21no—
63C2×Dic314—22no—
64C2×Dic314—21no—
65C2×Dic314—41no—
66C2×Dic314C2311yescharacteristic
67C2×Dic314C2311yescharacteristic
68C2×Dic314C2311yescharacteristic
69Q8×D157—22no—
70Q8×D157—22no—
71Q8×D157—41no—
72Q8×D157—41no—
73C2×C22⋊D157D411yescharacteristic
74C22×Dic157—21no—
75C2×C4×D157—21no—
76C2×C4×D157—21no—
77C2×C4×D157C2311yescharacteristic
78Dic628—22no—
79Dic628—41no—
80#14578944—22no—
81#14578944—22no—
82#14578944—22no—
83#14578944—22no—
84#14578944D414yescomplemented
85C2×#11929434—22no—
86C2×#11929434C2311yescharacteristic
87C2×#11929434C2311yescharacteristic
88C8×D157—22no—
89C4⋊Dic157—21no—
90C4⋊Dic157—21no—
91C4⋊Dic157C2311yescharacteristic
92#14578933—21no—
93#14578933—21no—
94#14578933—21no—
95#14578933—21no—
96#14578933—21no—
97#14578933—21no—
98#14578933—21no—
99#14578933D411yescharacteristic
100C22⋊Dic157—21no—
101C22⋊Dic157D411yescharacteristic
102#14578931—21no—
103#14578931—21no—
104#14578931D411yescharacteristic
105C4×Dic157—21no—
106C4×Dic157—21no—
107C4×Dic157C2311yescharacteristic
108C2×Q8×C157—21no—
109C2×Q8×C157—22no—
110C2×Q8×C157—41no—
111C2×Q8×C157C2311yescharacteristic
112C2×Q8×C157C2311yescharacteristic
113C2×Q8×C157C2311yescharacteristic
114C22×C4×C157—21no—
115C22×C4×C157C2311yescharacteristic
116Q16×C157—22no—
117Q16×C157—41no—
118C2×C8×C157C2311yescharacteristic
119C2×C8×C157C2311yescharacteristic
120C4⋊C4×C157—21no—
121C4⋊C4×C157C2311yescharacteristic
122C22⋊C4×C157—21no—
123C22×D157C2×D411yescharacteristic
124Dic314—21no—
125Dic314—21no—
126Dic314—21no—
127Dic314—21no—
128Dic314—21no—
129Dic314—42no—
130Dic314—42no—
131Dic314—42no—
132Dic314C2×D412yes—
133C22⋊D157Q1614yescomplemented
134C2×Dic157—21no—
135C2×Dic157—21no—
136C2×Dic157—21no—
137C2×Dic157—21no—
138C2×Dic157—42no—
139C2×Dic157C2×D411yescharacteristic
140C2×Dic157C2×D411yescharacteristic
141C2×Dic157C2×D411yescharacteristic
142C4×D157—21no—
143C4×D157—21no—
144C4×D157—42no—
145C4×D157—42no—
146#11929434—21no—
147#11929434C2×D412yes—
148C23×C157C2×D411yescharacteristic
149Q8×C157—21no—
150Q8×C157—21no—
151Q8×C157—21no—
152Q8×C157—21no—
153Q8×C157—21no—
154Q8×C157—42no—
155Q8×C157C2×D412yes—
156C2×C4×C157—21no—
157C2×C4×C157—21no—
158C2×C4×C157—21no—
159C2×C4×C157—21no—
160C2×C4×C157—21no—
161C2×C4×C157—21no—
162C2×C4×C157—42no—
163C2×C4×C157C2×D411yescharacteristic
164C2×C4×C157C2411yescharacteristic
165C8×C157—21no—
166C8×C157—21no—
167C22×C157—21no—
168C22×C157—21no—
169C22×C157C2×Q1612yescomplemented
170C22×C157C22×D411yescharacteristic
171C2×D157—21no—
172C2×D157—21no—
173C2×D157C2×Q1612yescomplemented
174C4×C157—21no—
175C4×C157—22no—
176C4×C157—41no—
177C4×C157—41no—
178C4×C157—41no—
179C4×C157C22×D411yescharacteristic
180C4×C157H(5,𝔽2)11yescharacteristic
181Dic157—21no—
182Dic157—22no—
183Dic157—21no—
184Dic157—41no—
185Dic157—41no—
186Dic157—41no—
187Dic157C2×Q1612yescomplemented
188C2×C157—22no—
189C2×C157D4211yescharacteristic
190C2×C157C22×Q1611yescharacteristic
191C2×C157#57811yescharacteristic
192D157—22no—
193C157D4×Q1611yescharacteristic, complemented, Sylow
194D4×Q16—1571nomaximal, Sylow
195C22×Q16—1571no—
196C22×Q16—1571no—
197D4×Q8—1571no—
198D4×Q8—1571no—
199#494—1571no—
200#467—1571no—
201#467—1571no—
202#462—1571no—
203#462—1571no—
204#451—1571no—
205#451—1571no—
206#451—1571no—
207#451D15711yescharacteristic, complemented
208C4×Q16—1571no—
209C8×D4—1571no—
210C22×Q8—1571no—
211C22×Q8—1571no—
212C22×Q8—1571no—
213C22×Q8C2×D15711yescharacteristic
214C2×Q16—3144no—
215C2×Q16—3144no—
216C2×Q16—1571no—
217C2×Q16—1571no—
218C2×Q16—1571no—
219C2×Q16—1571no—
220C2×Q16—1571no—
221C2×Q16—1571no—
222C2×Q16—1571no—
223C2×Q16C2×D15711yescharacteristic
224C2×Q16C2×D15711yescharacteristic
225C22×C8—1571no—
226C22×C8—1571no—
227C4⋊Q8—3141no—
228C4⋊Q8—3141no—
229C4⋊Q8—1571no—
230C4⋊Q8—1571no—
231C22⋊Q8—3141no—
232C22⋊Q8—3141no—
233C22⋊Q8—3141no—
234C22⋊Q8—3141no—
235C22⋊Q8—1571no—
236C22⋊Q8—1571no—
237C22⋊Q8—1571no—
238C22⋊Q8C2×D15711yescharacteristic
239C4×Q8—1571no—
240C4×Q8—1571no—
241C4×D4—3141no—
242C4×D4—3141no—
243C4×D4—1571no—
244#107—1571no—
245C4⋊C8—1571no—
246#103—1571no—
247#103—1571no—
248#103—1571no—
249#103—1571no—
250#103C2×D15711yescharacteristic
251#103C2×D15711yescharacteristic
252C22⋊C8—1571no—
253C22⋊C8C2×D15711yescharacteristic
254C4×C8—1571no—
255C2×Q8—21no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—3141no—
259C2×Q8—3141no—
260C2×Q8—3142no—
261C2×Q8—3141no—
262C2×Q8—3142no—
263C2×Q8—3142no—
264C2×Q8—6281no—
265C2×Q8—6281no—
266C2×Q8—6281no—
267C2×Q8—1571no—
268C2×Q8—1571no—
269C2×Q8—1571no—
270C2×Q8C22×D15711yescharacteristic
271C2×Q8C22×D15711yescharacteristic
272C2×Q8C22×D15711yescharacteristic
273C2×D4—1571no—
274C22×C4—21no—
275C22×C4—3141no—
276C22×C4—3141no—
277C22×C4—3141no—
278C22×C4—1571no—
279C22×C4C22×D15711yescharacteristic
280Q16—22no—
281Q16—41no—
282Q16—3142no—
283Q16—3142no—
284Q16—3142no—
285Q16—3142no—
286Q16—3142no—
287Q16—6281no—
288Q16—1574no—
289C2×C8—3142no—
290C2×C8—3142no—
291C2×C8—1571no—
292C2×C8—1571no—
293C2×C8C22×D15711yescharacteristic
294C2×C8C22×D15711yescharacteristic
295C4⋊C4—21no—
296C4⋊C4—3141no—
297C4⋊C4—3141no—
298C4⋊C4—3141no—
299C4⋊C4—3141no—
300C4⋊C4—3141no—
301C4⋊C4—3141no—
302C4⋊C4—3141no—
303C4⋊C4—3141no—
304C4⋊C4—3141no—
305C4⋊C4—1571no—
306C4⋊C4—1571no—
307C4⋊C4C22×D15711yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—3141no—
310C22⋊C4—3141no—
311C22⋊C4—3141no—
312C22⋊C4—1571no—
313C22⋊C4—1571no—
314C42—3141no—
315C42—3141no—
316C42—1571no—
317C23—1571no—
318C23D4×D15711yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—42no—
325Q8—3141no—
326Q8—3141no—
327Q8—3141no—
328Q8—3141no—
329Q8—3141no—
330Q8—6282no—
331Q8—6282no—
332Q8—6282no—
333Q8—1572no—
334Q8D4×D15712yes—
335D4—1574no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—3141no—
344C2×C4—3141no—
345C2×C4—3141no—
346C2×C4—3141no—
347C2×C4—3141no—
348C2×C4—3141no—
349C2×C4—6282no—
350C2×C4—6282no—
351C2×C4—6282no—
352C2×C4—1571no—
353C2×C4—1571no—
354C2×C4—1571no—
355C2×C4D4×D15711yescharacteristic
356C2×C4C23×D15711yescharacteristic
357C8—21no—
358C8—21no—
359C8—3141no—
360C8—1572no—
361C22—21no—
362C22—21no—
363C22—3141no—
364C22—3141no—
365C22—1572no—
366C22Q16×D15712yescomplemented
367C22C2×D4×D15711yescharacteristic
368C4—21no—
369C4—22no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—3141no—
374C4—3141no—
375C4—3142no—
376C4—6281no—
377C4—6281no—
378C4—6281no—
379C4—1572no—
380C4C2×D4×D15711yescharacteristic
381C4#1909781611yescharacteristic
382C2—22no—
383C2—3142no—
384C2#2375790511yescharacteristic
385C2C2×Q16×D15711yescharacteristic
386C2#2375808911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#451×C157C2
socleC22×C157C22×D4
prime radicalC2×C22⋊D157D4
hypercenter#451D157
hypocenterC157D4×Q16
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C22×#14578944C2semidirect
2. C2×Q16×D157C2semidirect
5. #23757651C2semidirect
6. #23757513C2semidirect
7. #23757512C2semidirect
8. #23757482C2semidirect
9. #23757481C2semidirect
10. #23757431C2semidirect
11. #23757430C2semidirect
12. #23757427C2semidirect
13. #23757365C2semidirect
15. #451×C157C2semidirect
84. #14578944D4semidirect
133. C22⋊D157Q16semidirect
169. C22×C157C2×Q16semidirect
173. C2×D157C2×Q16semidirect
187. Dic157C2×Q16semidirect
193. C157D4×Q16semidirect
207. #451D157semidirect
366. C22Q16×D157semidirect

Maximal subgroups

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

subgroupindex
C22×#145789442
C2×Q16×D1572
Q8×C22⋊D1572
#237579362
#237576512
#237575132
#237575122
#237574822
#237574812
#237574312
#237574302
#237574272
#237573652
#237573362
#451×C1572
D4×Q16157