Finite Group Databasebeta

← #34111689

Subgroups — #34111689

Order 20096.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097801C211yesmaximal, characteristic, complemented
2C2×#19097790C211yesmaximal, characteristic, complemented
3#23757929C211yesmaximal, characteristic, complemented
4#23757922C211yesmaximal, characteristic, complemented
5#23757664C211yesmaximal, characteristic, complemented
6#23757517C211yesmaximal, characteristic, complemented
7#23757513C211yesmaximal, characteristic, complemented
8#23757486C211yesmaximal, characteristic, complemented
9#23757482C211yesmaximal, characteristic, complemented
10#23757436C211yesmaximal, characteristic, complemented
11#23757432C211yesmaximal, characteristic, complemented
12#23757431C211yesmaximal, characteristic, complemented
13#23757370C211yesmaximal, characteristic, complemented
14#23757347C211yesmaximal, characteristic, complemented
15#449×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic
17C2×#14578953C2211yescharacteristic
18C22×Dic314C2211yescharacteristic
19#19097801—24no—
20#19097790—24no—
21C2×Dic628C2211yescharacteristic
22C2×Dic628C2211yescharacteristic
23C2×#14578944C2211yescharacteristic
24C2×#14578944C2211yescharacteristic
25C2×#14578942C2211yescharacteristic
26C2×#14578942C2211yescharacteristic
27C2×#14578942C2211yescharacteristic
28C2×#14578938C2211yescharacteristic
29C2×C8×D157C2211yescharacteristic
30#19097769C2211yescharacteristic
31#19097758C2211yescharacteristic
32#19097757—21no—
33#19097754—21no—
34#19097752—21no—
35#19097751C2211yescharacteristic
36#19097750C2211yescharacteristic
37#19097748—21no—
38#19097747C2211yescharacteristic
39#19097743—21no—
40#19097743—21no—
41#19097736C2211yescharacteristic
42D4×Dic157—21no—
43D4×Dic157C2211yescharacteristic
44C4×C22⋊D157C2211yescharacteristic
45C2×#14578933—21no—
46#19097704C2211yescharacteristic
47#19097700C2211yescharacteristic
48#19097696C2211yescharacteristic
49#19097696C2211yescharacteristic
50#19097694C2211yescharacteristic
51#19097692C2211yescharacteristic
52#19097683C2211yescharacteristic
53#19097681C2211yescharacteristic
54C2×#41×C157C2211yescharacteristic
55C2×SD16×C157C2211yescharacteristic
56C2×D8×C157C2211yescharacteristic
57#121×C157C2211yescharacteristic
58#103×C157C2211yescharacteristic
59#102×C157C2211yescharacteristic
60C22⋊C8×C157C2211yescharacteristic
61#14578953—22no—
62#14578953—22no—
63#14578953—41no—
64#14578953—41no—
65C2×Dic314—22no—
66C2×Dic314—21no—
67C2×Dic314—41no—
68C2×Dic314C2311yescharacteristic
69C2×Dic314C2311yescharacteristic
70C2×Dic314C2311yescharacteristic
71C2×C22⋊D157—21no—
72C2×C22⋊D157—21no—
73C2×C22⋊D157D411yescharacteristic
74C22×Dic157—21no—
75C22×Dic157—21no—
76C22×Dic157—21no—
77C22×Dic157—21no—
78C2×C4×D157C2311yescharacteristic
79Dic628—22no—
80Dic628—41no—
81#14578944—22no—
82#14578944—22no—
83#14578942—22no—
84#14578942—22no—
85#14578942D414yescomplemented
86#14578938—22no—
87C2×#11929434C2311yescharacteristic
88C2×#11929434C2311yescharacteristic
89C8×D157—22no—
90C4⋊Dic157—21no—
91C4⋊Dic157C2311yescharacteristic
92#14578933—21no—
93#14578933—21no—
94#14578933—21no—
95#14578933—21no—
96#14578933—21no—
97#14578933D411yescharacteristic
98C22⋊Dic157—21no—
99C22⋊Dic157—21no—
100C22⋊Dic157—21no—
101C22⋊Dic157—21no—
102C22⋊Dic157D411yescharacteristic
103#14578931—21no—
104#14578931—21no—
105#14578931D411yescharacteristic
106C4×Dic157—21no—
107C4×Dic157C2311yescharacteristic
108#41×C157—22no—
109#41×C157—41no—
110C2×Q8×C157C2311yescharacteristic
111C2×D4×C157—21no—
112C2×D4×C157—21no—
113C2×D4×C157C2311yescharacteristic
114C2×D4×C157C2311yescharacteristic
115C22×C4×C157—21no—
116C22×C4×C157C2311yescharacteristic
117SD16×C157—41no—
118D8×C157—22no—
119C2×C8×C157C2311yescharacteristic
120C2×C8×C157C2311yescharacteristic
121C4⋊C4×C157C2311yescharacteristic
122C22⋊C4×C157—21no—
123C22×D157C2×D411yescharacteristic
124Dic314—21no—
125Dic314—21no—
126Dic314—21no—
127Dic314—21no—
128Dic314—21no—
129Dic314—42no—
130Dic314C2×D412yes—
131C22⋊D157—41no—
132C22⋊D157—42no—
133C22⋊D157—42no—
134C2×Dic157—21no—
135C2×Dic157—21no—
136C2×Dic157—21no—
137C2×Dic157—21no—
138C2×Dic157—42no—
139C2×Dic157—42no—
140C2×Dic157—42no—
141C2×Dic157—42no—
142C2×Dic157C2×D411yescharacteristic
143C2×Dic157C2×D411yescharacteristic
144C2×Dic157C2×D411yescharacteristic
145C4×D157—21no—
146C4×D157—21no—
147#11929434—21no—
148#11929434C2×D412yes—
149C23×C157—21no—
150C23×C157—21no—
151C23×C157C2×D411yescharacteristic
152Q8×C157—21no—
153Q8×C157—21no—
154D4×C157—21no—
155D4×C157—21no—
156D4×C157—21no—
157D4×C157—42no—
158D4×C157—42no—
159D4×C157C2×D412yes—
160C2×C4×C157—21no—
161C2×C4×C157—21no—
162C2×C4×C157—21no—
163C2×C4×C157—21no—
164C2×C4×C157—42no—
165C2×C4×C157C2×D411yescharacteristic
166C2×C4×C157C2411yescharacteristic
167C8×C157—21no—
168C8×C157—21no—
169C22×C157—21no—
170C22×C157—22no—
171C22×C157—21no—
172C22×C157—21no—
173C22×C157—41no—
174C22×C157—41no—
175C22×C157—41no—
176C22×C157—41no—
177C22×C157—41no—
178C22×C157C22×D411yescharacteristic
179C2×D157—21no—
180C2×D157—21no—
181C2×D157—21no—
182C4×C157—21no—
183C4×C157—41no—
184C4×C157—41no—
185C4×C157C22×D411yescharacteristic
186C4×C157H(5,𝔽2)11yescharacteristic
187Dic157—21no—
188Dic157—22no—
189Dic157—21no—
190Dic157—21no—
191Dic157—41no—
192Dic157—41no—
193Dic157—41no—
194C2×C157—41no—
195C2×C157—42no—
196C2×C157—81no—
197C2×C157D4211yescharacteristic
198C2×C157#57511yescharacteristic
199C2×C157#57811yescharacteristic
200D157—41no—
201C157#326911yescharacteristic, complemented, Sylow
202#3269—1571nomaximal, Sylow
203C2×#137—1571no—
204C2×#135—1571no—
205#548—1571no—
206#547—1571no—
207#497—1571no—
208#468—1571no—
209#467—1571no—
210#463—1571no—
211#462—1571no—
212#452—1571no—
213#452—1571no—
214#451—1571no—
215#449D15711yescharacteristic, complemented
216#440—1571no—
217#435—1571no—
218C2×#41—1571no—
219C2×#41—1571no—
220C2×#41C2×D15711yescharacteristic
221C22×Q8—1571no—
222#137—3144no—
223#135—3144no—
224C2×Q16—1571no—
225C2×Q16—1571no—
226C2×Q16—1571no—
227C2×Q16—1571no—
228C2×SD16—1571no—
229C2×SD16—1571no—
230C2×SD16—1571no—
231C2×SD16C2×D15711yescharacteristic
232C2×D8C2×D15711yescharacteristic
233C2×M16—1571no—
234C22×C8—1571no—
235C4⋊Q8—1571no—
236#124—3141no—
237#124—1571no—
238#123—3141no—
239#123—3141no—
240C22⋊Q8—3141no—
241C22⋊Q8—1571no—
242C22⋊Q8—1571no—
243C22⋊Q8—1571no—
244#121—3141no—
245#121—3141no—
246#121C2×D15711yescharacteristic
247C4×Q8—1571no—
248C4×D4—3141no—
249C4×D4—1571no—
250C4×D4—1571no—
251C2×C4⋊C4—3141no—
252#107—1571no—
253C4⋊C8—1571no—
254#103—1571no—
255#103—1571no—
256#103—1571no—
257#103C2×D15711yescharacteristic
258#102—1571no—
259#102C2×D15711yescharacteristic
260C22⋊C8—1571no—
261C22⋊C8C2×D15711yescharacteristic
262#97—1571no—
263#41—22no—
264#41—41no—
265#41—3142no—
266#41—3142no—
267#41—6281no—
268#41—6281no—
269C2×Q8—3142no—
270C2×Q8—3141no—
271C2×Q8—6281no—
272C2×Q8—1571no—
273C2×Q8—1571no—
274C2×Q8—1571no—
275C2×Q8C22×D15711yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—3141no—
279C2×D4—3141no—
280C2×D4—1571no—
281C2×D4C22×D15711yescharacteristic
282C2×D4C22×D15711yescharacteristic
283C22×C4—21no—
284C22×C4—3141no—
285C22×C4—3141no—
286C22×C4—3141no—
287C22×C4—3141no—
288C22×C4—1571no—
289C22×C4C22×D15711yescharacteristic
290Q16—3142no—
291Q16—3142no—
292Q16—3142no—
293Q16—6281no—
294SD16—41no—
295SD16—3142no—
296SD16—3142no—
297SD16—1574no—
298D8—22no—
299M16—3142no—
300C2×C8—3142no—
301C2×C8—1571no—
302C2×C8—1571no—
303C2×C8C22×D15711yescharacteristic
304C2×C8C22×D15711yescharacteristic
305C4⋊C4—3141no—
306C4⋊C4—3141no—
307C4⋊C4—3141no—
308C4⋊C4—3141no—
309C4⋊C4—3141no—
310C4⋊C4—3141no—
311C4⋊C4—1571no—
312C4⋊C4—1571no—
313C4⋊C4C22×D15711yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—3141no—
316C22⋊C4—3141no—
317C22⋊C4—3141no—
318C22⋊C4—3141no—
319C22⋊C4—3141no—
320C22⋊C4—3141no—
321C22⋊C4—1571no—
322C22⋊C4—1571no—
323C42—3141no—
324C42—1571no—
325C23—21no—
326C23—21no—
327C23—1571no—
328C23D4×D15711yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—3141no—
332Q8—3141no—
333Q8—3141no—
334Q8—3141no—
335Q8—3141no—
336Q8—6282no—
337Q8—1572no—
338D4—21no—
339D4—21no—
340D4—21no—
341D4—42no—
342D4—42no—
343D4—6281no—
344D4—6282no—
345D4—6282no—
346D4D4×D15712yes—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—42no—
352C2×C4—3141no—
353C2×C4—3141no—
354C2×C4—3141no—
355C2×C4—3141no—
356C2×C4—3141no—
357C2×C4—3141no—
358C2×C4—6282no—
359C2×C4—6282no—
360C2×C4—6282no—
361C2×C4—6282no—
362C2×C4—1571no—
363C2×C4—1571no—
364C2×C4—1571no—
365C2×C4D4×D15711yescharacteristic
366C2×C4C23×D15711yescharacteristic
367C8—21no—
368C8—21no—
369C8—3141no—
370C8—1572no—
371C22—21no—
372C22—21no—
373C22—22no—
374C22—21no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—3141no—
381C22—3141no—
382C22—3141no—
383C22C2×D4×D15711yescharacteristic
384C4—21no—
385C4—41no—
386C4—41no—
387C4—3141no—
388C4—3141no—
389C4—3142no—
390C4—3141no—
391C4—6281no—
392C4—6281no—
393C4—6281no—
394C4C2×D4×D15711yescharacteristic
395C4#1909781611yescharacteristic
396C2—41no—
397C2—42no—
398C2—81no—
399C2—6281no—
400C2#2375790511yescharacteristic
401C2#2375806911yescharacteristic
402C2#2375808711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#449×C157C2
socleC22×C157C22×D4
hypercenter#449D157
hypocenterC157#3269
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#19097801C2semidirect
2. C2×#19097790C2semidirect
3. #23757929C2semidirect
4. #23757922C2semidirect
5. #23757664C2semidirect
6. #23757517C2semidirect
7. #23757513C2semidirect
8. #23757486C2semidirect
9. #23757482C2semidirect
10. #23757436C2semidirect
11. #23757432C2semidirect
12. #23757431C2semidirect
13. #23757370C2semidirect
14. #23757347C2semidirect
15. #449×C157C2semidirect
85. #14578942D4semidirect
201. C157#3269semidirect
215. #449D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978012
C2×#190977902
#237579292
#237579222
#237576642
#237575172
#237575132
#237574862
#237574822
#237574362
#237574322
#237574312
#237573702
#237573472
#449×C1572
#3269157