Finite Group Databasebeta

← #34103177

Subgroups — #34103177

Order 20096.

688 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097743C211yesmaximal, characteristic
2C2×C4×C22⋊D157C211yesmaximal, characteristic, complemented
3C2×#19097727C211yesmaximal, characteristic, complemented
4#23757176C211yesmaximal, characteristic, complemented
5#23757173C211yesmaximal, characteristic, complemented
6#23757173C211yesmaximal, characteristic, complemented
7#23757170C211yesmaximal, characteristic, complemented
8#23757167C211yesmaximal, characteristic, complemented
9#23757165C211yesmaximal, characteristic, complemented
10#23757149C211yesmaximal, characteristic, complemented
11#23757124C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic157C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic157C211yesmaximal, characteristic, complemented
14#23757120C211yesmaximal, characteristic, complemented
15#386×C157C211yesmaximal, characteristic, complemented
16C22×C22⋊D157C2211yescharacteristic
17C22×C22⋊D157C2211yescharacteristic
18C23×Dic157C2211yescharacteristic
19C22×C4×D157C2211yescharacteristic
20#19097743C418yescomplemented
21C4×C22⋊D157—24no—
22#19097727—24no—
23C2×C4⋊Dic157C2211yescharacteristic
24C2×#14578933C2211yescharacteristic
25C2×#14578933C2211yescharacteristic
26C2×#14578933C2211yescharacteristic
27C2×#14578933C2211yescharacteristic
28C2×C22⋊Dic157C2211yescharacteristic
29C2×C22⋊Dic157C2211yescharacteristic
30C2×C22⋊Dic157C2211yescharacteristic
31C2×C22⋊Dic157C2211yescharacteristic
32C2×C22⋊Dic157C2211yescharacteristic
33C2×#14578931C2211yescharacteristic
34C2×#14578931C2211yescharacteristic
35C2×#14578931C2211yescharacteristic
36C2×#14578931C2211yescharacteristic
37C2×C4×Dic157C2211yescharacteristic
38C2×C4×Dic157C2211yescharacteristic
39C2×C4×Dic157C2211yescharacteristic
40C2×C4×Dic157C2211yescharacteristic
41C2×C4×Dic157C2211yescharacteristic
42C2×C4×Dic157C2211yescharacteristic
43#19097678C2211yescharacteristic
44#19097678C2211yescharacteristic
45#19097678C2211yescharacteristic
46#19097678C2211yescharacteristic
47C22×D4×C157C2211yescharacteristic
48C23×C4×C157C2211yescharacteristic
49C2×C22⋊C4×C157C2211yescharacteristic
50C2×C22⋊C4×C157C2211yescharacteristic
51C2×C22⋊C4×C157C2211yescharacteristic
52#95×C157C2211yescharacteristic
53#95×C157C2211yescharacteristic
54C23×D157C2311yescharacteristic
55C2×C22⋊D157—22no—
56C2×C22⋊D157—22no—
57C2×C22⋊D157—41no—
58C2×C22⋊D157—41no—
59C2×C22⋊D157C2×C414yes—
60C2×C22⋊D157C2×C414yescomplemented
61C22×Dic157—22no—
62C22×Dic157—22no—
63C22×Dic157C2×C414yescomplemented
64C22×Dic157C2311yescharacteristic
65C22×Dic157C2311yescharacteristic
66C22×Dic157C2311yescharacteristic
67C22×Dic157C2311yescharacteristic
68C22×Dic157C2311yescharacteristic
69C22×Dic157C2311yescharacteristic
70C22×Dic157C2311yescharacteristic
71C2×C4×D157—22no—
72C2×C4×D157—22no—
73C2×C4×D157—41no—
74C4⋊Dic157—41no—
75#14578933—22no—
76#14578933—41no—
77#14578933C2×C414yescomplemented
78#14578933D414yescomplemented
79C22⋊Dic157—22no—
80C22⋊Dic157—22no—
81C22⋊Dic157—22no—
82C22⋊Dic157—22no—
83C22⋊Dic157C2×C414yescomplemented
84#14578931—22no—
85#14578931—22no—
86#14578931—41no—
87#14578931C2×C414yescomplemented
88C4×Dic157—22no—
89C4×Dic157—22no—
90C4×Dic157—22no—
91C4×Dic157—22no—
92C4×Dic157—41no—
93C4×Dic157—41no—
94C24×C157C2311yescharacteristic
95C24×C157C2311yescharacteristic
96C2×D4×C157—41no—
97C2×D4×C157—41no—
98C2×D4×C157C2×C414yescomplemented
99C22×C4×C157—22no—
100C22×C4×C157—22no—
101C22×C4×C157—41no—
102C22×C4×C157C2311yescharacteristic
103C22×C4×C157C2311yescharacteristic
104C22×C4×C157C2311yescharacteristic
105C22×C4×C157C2311yescharacteristic
106C22×C4×C157C2311yescharacteristic
107C22⋊C4×C157—22no—
108C22⋊C4×C157—22no—
109C22⋊C4×C157—41no—
110C22×D157—21no—
111C22×D157—21no—
112C22×D157—21no—
113C22×D157—21no—
114C22×D157—21no—
115C22×D157—21no—
116C22×D157C22×C412yes—
117C22⋊D157—28no—
118C22⋊D157—44no—
119C2×Dic157—21no—
120C2×Dic157—21no—
121C2×Dic157—21no—
122C2×Dic157—21no—
123C2×Dic157—21no—
124C2×Dic157—21no—
125C2×Dic157—21no—
126C2×Dic157—21no—
127C2×Dic157—28no—
128C2×Dic157—21no—
129C2×Dic157—21no—
130C2×Dic157—21no—
131C2×Dic157—21no—
132C2×Dic157—21no—
133C2×Dic157—21no—
134C2×Dic157C22×C412yes—
135C2×Dic157C22×C412yes—
136C2×Dic157C22×C412yes—
137C2×Dic157C2×D412yes—
138C2×Dic157C2×D412yes—
139C2×Dic157#4112yes—
140C2×Dic157#4112yes—
141C4×D157—44no—
142C23×C157—21no—
143C23×C157—21no—
144C23×C157—21no—
145C23×C157—21no—
146C23×C157—21no—
147C23×C157—21no—
148C23×C157—21no—
149C23×C157—21no—
150C23×C157—21no—
151C23×C157—21no—
152C23×C157—21no—
153C23×C157—21no—
154C23×C157C22×C412yes—
155C23×C157C22×C412yes—
156C23×C157C2411yescharacteristic
157D4×C157—44no—
158C2×C4×C157—21no—
159C2×C4×C157—21no—
160C2×C4×C157—21no—
161C2×C4×C157—21no—
162C2×C4×C157—21no—
163C2×C4×C157—21no—
164C2×C4×C157—21no—
165C2×C4×C157—21no—
166C2×C4×C157—21no—
167C2×C4×C157—21no—
168C2×C4×C157—21no—
169C2×C4×C157—21no—
170C2×C4×C157—44no—
171C2×C4×C157C22×C412yes—
172C2×C4×C157C2×D412yes—
173C2×C4×C157#4112yes—
174C22×C157—22no—
175C22×C157—22no—
176C22×C157—22no—
177C22×C157—22no—
178C22×C157—22no—
179C22×C157—22no—
180C22×C157—41no—
181C22×C157—41no—
182C22×C157—41no—
183C22×C157—41no—
184C22×C157—41no—
185C22×C157—41no—
186C22×C157—41no—
187C22×C157—41no—
188C22×C157C23×C411yescharacteristic
189C22×C157C22×D411yescharacteristic
190C22×C157C2×#4111yescharacteristic
191C22×C157C2×#4111yescharacteristic
192C22×C157C2×#4111yescharacteristic
193C22×C157H(5,𝔽2)11yescharacteristic
194C22×C157H(5,𝔽2)11yescharacteristic
195C2×D157—22no—
196C2×D157—22no—
197C2×D157—22no—
198C2×D157—41no—
199C2×D157—41no—
200C2×D157—41no—
201C2×D157—41no—
202C4×C157—22no—
203C4×C157—22no—
204C4×C157—22no—
205C4×C157—41no—
206C4×C157—41no—
207Dic157—22no—
208Dic157—22no—
209Dic157—22no—
210Dic157—22no—
211Dic157—41no—
212Dic157—41no—
213Dic157C4×D414yes—
214C2×C157—42no—
215C2×C157—42no—
216C2×C157C2×C4×D411yescharacteristic
217C2×C157C4×#4111yescharacteristic
218C2×C157#51811yescharacteristic
219C2×C157#54611yescharacteristic
220C2×C157#55011yescharacteristic
221C2×C157#55311yescharacteristic
222C2×C157#55511yescharacteristic
223D157—42no—
224C157#236311yescharacteristic, complemented, Sylow
225#2363—1571nomaximal, Sylow
226C2×#121—1571no—
227C2×C4×D4—1571no—
228C2×#117—1571no—
229#390—1571no—
230#390—1571no—
231#390—1571no—
232#389—1571no—
233#388—1571no—
234#388—1571no—
235#386—1571no—
236#386D15711yescharacteristic, complemented
237C4×C4⋊C4—1571no—
238C4×C22⋊C4—1571no—
239C4×C22⋊C4—1571no—
240C4×C22⋊C4—1571no—
241C22×D4—1571no—
242C22×D4—1571no—
243C22×D4C2×D15711yescharacteristic
244C23×C4—1571no—
245C23×C4—1571no—
246C23×C4C2×D15711yescharacteristic
247#121—1578no—
248C4×D4—3144no—
249#117—3144no—
250C2×C4⋊C4—1571no—
251C2×C4⋊C4—1571no—
252C2×C4⋊C4—1571no—
253C2×C4⋊C4—1571no—
254C2×C4⋊C4—1571no—
255C2×C22⋊C4—1571no—
256C2×C22⋊C4—1571no—
257C2×C22⋊C4—1571no—
258C2×C22⋊C4—1571no—
259C2×C22⋊C4—1571no—
260C2×C22⋊C4—1571no—
261C2×C22⋊C4—1571no—
262C2×C22⋊C4—1571no—
263C2×C22⋊C4—1571no—
264C2×C22⋊C4C2×D15711yescharacteristic
265C2×C22⋊C4C2×D15711yescharacteristic
266C2×C22⋊C4C2×D15711yescharacteristic
267C2×C42—1571no—
268C2×C42—1571no—
269C2×C42—1571no—
270C2×C42—1571no—
271C2×C42—1571no—
272C2×C42—1571no—
273#95—1571no—
274#95—1571no—
275#95—1571no—
276#95—1571no—
277#95C2×D15711yescharacteristic
278#95C2×D15711yescharacteristic
279C24—1571no—
280C24C22×D15711yescharacteristic
281C24C22×D15711yescharacteristic
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—3142no—
285C2×D4—3142no—
286C2×D4—6281no—
287C2×D4—6281no—
288C2×D4—1574no—
289C2×D4—1574no—
290C2×D4C4×D15714yescomplemented
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—41no—
294C22×C4—3142no—
295C22×C4—3142no—
296C22×C4—3142no—
297C22×C4—3142no—
298C22×C4—6281no—
299C22×C4—1571no—
300C22×C4—1571no—
301C22×C4—1574no—
302C22×C4—1571no—
303C22×C4—1571no—
304C22×C4—1571no—
305C22×C4—1571no—
306C22×C4—1571no—
307C22×C4C22×D15711yescharacteristic
308C22×C4C22×D15711yescharacteristic
309C22×C4C22×D15711yescharacteristic
310C22×C4C22×D15711yescharacteristic
311C22×C4C22×D15711yescharacteristic
312C4⋊C4—3142no—
313C4⋊C4—6281no—
314C4⋊C4—6281no—
315C4⋊C4—1574no—
316C4⋊C4—1574no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—41no—
320C22⋊C4—3142no—
321C22⋊C4—3142no—
322C22⋊C4—3142no—
323C22⋊C4—3142no—
324C22⋊C4—3142no—
325C22⋊C4—3142no—
326C22⋊C4—6281no—
327C22⋊C4—1574no—
328C22⋊C4—1574no—
329C42—3142no—
330C42—3142no—
331C42—3142no—
332C42—3142no—
333C42—6281no—
334C42—6281no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—3141no—
348C23—3141no—
349C23—3141no—
350C23—3141no—
351C23—3141no—
352C23—3141no—
353C23—1572no—
354C23C2×C4×D15712yes—
355C23C2×C4×D15712yes—
356C23C23×D15711yescharacteristic
357D4—44no—
358D4—3148no—
359D4—6284no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—44no—
373C2×C4—3141no—
374C2×C4—3141no—
375C2×C4—3141no—
376C2×C4—3141no—
377C2×C4—3141no—
378C2×C4—3141no—
379C2×C4—3141no—
380C2×C4—3141no—
381C2×C4—3148no—
382C2×C4—3141no—
383C2×C4—3141no—
384C2×C4—3141no—
385C2×C4—3141no—
386C2×C4—3141no—
387C2×C4—3141no—
388C2×C4—6284no—
389C2×C4—1572no—
390C2×C4—1572no—
391C2×C4—1572no—
392C2×C4—1572no—
393C2×C4—1572no—
394C2×C4—1572no—
395C2×C4—1572no—
396C2×C4C2×C4×D15712yes—
397C2×C4D4×D15712yes—
398C2×C4#1457895312yes—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—3142no—
414C22—3142no—
415C22—3142no—
416C22—6281no—
417C22—6281no—
418C22—6281no—
419C22—6281no—
420C22C22×C4×D15711yescharacteristic
421C22C2×D4×D15711yescharacteristic
422C22#41×D15711yescharacteristic
423C22#41×D15711yescharacteristic
424C22C2×#1457895311yescharacteristic
425C22#1909781611yescharacteristic
426C22#1909781611yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—41no—
431C4—41no—
432C4—3142no—
433C4—3142no—
434C4—3142no—
435C4—3142no—
436C4—6281no—
437C4—6281no—
438C4—1574no—
439C2—42no—
440C2—42no—
441C2—6282no—
442C2C4×D4×D15711yescharacteristic
443C2C4×#1457895311yescharacteristic
444C2#2375774611yescharacteristic
445C2#2375791011yescharacteristic
446C2#2375794111yescharacteristic
447C2#2375796811yescharacteristic
448C2#2375798711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutatorC22×C157C23×C4
Fitting#386×C157C2
socle, cosocleC23×C157C24
prime radicalC2×#19097743C2
hypercenter#386D157
hypocenterC157#2363
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D157C2semidirect
3. C2×#19097727C2semidirect
4. #23757176C2semidirect
5. #23757173C2semidirect
6. #23757173C2semidirect
7. #23757170C2semidirect
8. #23757167C2semidirect
9. #23757165C2semidirect
10. #23757149C2semidirect
11. #23757124C2semidirect
12. C22⋊C4×Dic157C2semidirect
13. C4×C22⋊Dic157C2semidirect
14. #23757120C2semidirect
15. #386×C157C2semidirect
20. #19097743C4semidirect
60. C2×C22⋊D157C2×C4semidirect
63. C22×Dic157C2×C4semidirect
77. #14578933C2×C4semidirect
78. #14578933D4semidirect
83. C22⋊Dic157C2×C4semidirect
87. #14578931C2×C4semidirect
98. C2×D4×C157C2×C4semidirect
224. C157#2363semidirect
236. #386D157semidirect
290. C2×D4C4×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977432
C2×C4×C22⋊D1572
C2×#190977272
#237571762
#237571732
#237571702
#237571672
#237571652
#237571492
#237571242
C22⋊C4×Dic1572
C4×C22⋊Dic1572
#237571202
#386×C1572
#2363157