Finite Group Databasebeta

← #34103217

Subgroups — #34103217

Order 20096.

548 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097748C211yesmaximal, characteristic
2#23757178C211yesmaximal, characteristic, complemented
3#23757177C211yesmaximal, characteristic, complemented
4#23757158C211yesmaximal, characteristic, complemented
5#23757155C211yesmaximal, characteristic, complemented
6#23757154C211yesmaximal, characteristic, complemented
7#23757154C211yesmaximal, characteristic, complemented
8#23757145C211yesmaximal, characteristic
9C4⋊C4×Dic157C211yesmaximal, characteristic, complemented
10#23757124C211yesmaximal, characteristic, complemented
11C22⋊C4×Dic157C211yesmaximal, characteristic
12C22⋊C4×Dic157C211yesmaximal, characteristic
13C4×C22⋊Dic157C211yesmaximal, characteristic
14C2×#19097678C211yesmaximal, characteristic
15#385×C157C211yesmaximal, characteristic
16C22×Dic314C2211yescharacteristic
17C23×Dic157C2211yescharacteristic
18C23×Dic157C2211yescharacteristic
19#19097748C418yescomplemented
20C2×C4⋊Dic157C2211yescharacteristic
21C2×#14578933C2211yescharacteristic
22C2×#14578933C2211yescharacteristic
23C2×#14578933C2211yescharacteristic
24C2×#14578933C2211yescharacteristic
25C2×C22⋊Dic157C2211yescharacteristic
26C2×C22⋊Dic157C2211yescharacteristic
27C2×C4×Dic157C2211yescharacteristic
28C2×C4×Dic157C2211yescharacteristic
29C2×C4×Dic157C2211yescharacteristic
30C2×C4×Dic157C2211yescharacteristic
31C2×C4×Dic157C2211yescharacteristic
32C2×C4×Dic157C2211yescharacteristic
33#19097678C2211yescharacteristic
34#19097678C2211yescharacteristic
35#19097678C2211yescharacteristic
36#19097678C2211yescharacteristic
37#19097678C2211yescharacteristic
38#19097678C2211yescharacteristic
39#19097678C2211yescharacteristic
40#19097678C2211yescharacteristic
41#19097678C2211yescharacteristic
42#19097678C2211yescharacteristic
43#19097678C2211yescharacteristic
44#19097678C2211yescharacteristic
45C23×C4×C157C2211yescharacteristic
46C2×C4⋊C4×C157C2211yescharacteristic
47C2×C4⋊C4×C157C2211yescharacteristic
48C2×C22⋊C4×C157C2211yescharacteristic
49C2×C22⋊C4×C157C2211yescharacteristic
50#95×C157C2211yescharacteristic
51#95×C157C2211yescharacteristic
52C2×Dic314—41no—
53C2×Dic314—41no—
54C2×Dic314C2×C414yescomplemented
55C22×Dic157—22no—
56C22×Dic157—22no—
57C22×Dic157—41no—
58C22×Dic157—41no—
59C22×Dic157—41no—
60C22×Dic157C2×C414yes—
61C22×Dic157C2311yescharacteristic
62C22×Dic157C2311yescharacteristic
63C22×Dic157C2311yescharacteristic
64C22×Dic157C2311yescharacteristic
65C22×Dic157C2311yescharacteristic
66C22×Dic157C2311yescharacteristic
67C22×Dic157C2311yescharacteristic
68C22×Dic157C2311yescharacteristic
69C4⋊Dic157C2×C414yescomplemented
70#14578933—22no—
71#14578933—41no—
72#14578933C2×C414yescomplemented
73#14578933C2×C414yescomplemented
74C22⋊Dic157C2×C414yes—
75C22⋊Dic157Q814yescomplemented
76C4×Dic157—22no—
77C4×Dic157—22no—
78C4×Dic157—41no—
79C4×Dic157—41no—
80C4×Dic157—41no—
81C4×Dic157—41no—
82C24×C157C2311yescharacteristic
83C22×C4×C157—22no—
84C22×C4×C157—22no—
85C22×C4×C157—41no—
86C22×C4×C157C2311yescharacteristic
87C22×C4×C157C2311yescharacteristic
88C22×C4×C157C2311yescharacteristic
89C22×C4×C157C2311yescharacteristic
90C22×C4×C157C2311yescharacteristic
91C22×C4×C157C2311yescharacteristic
92C4⋊C4×C157—22no—
93C4⋊C4×C157—41no—
94C22⋊C4×C157—22no—
95C22⋊C4×C157C2×C414yes—
96Dic314—44no—
97C2×Dic157—21no—
98C2×Dic157—21no—
99C2×Dic157—21no—
100C2×Dic157—21no—
101C2×Dic157—21no—
102C2×Dic157—21no—
103C2×Dic157—21no—
104C2×Dic157—21no—
105C2×Dic157—21no—
106C2×Dic157—21no—
107C2×Dic157—21no—
108C2×Dic157—21no—
109C2×Dic157—21no—
110C2×Dic157—21no—
111C2×Dic157—21no—
112C2×Dic157—21no—
113C2×Dic157—21no—
114C2×Dic157—44no—
115C2×Dic157—44no—
116C2×Dic157C22×C412yes—
117C2×Dic157C22×C412yes—
118C2×Dic157C22×C412yes—
119C2×Dic157C22×C412yes—
120C2×Dic157C2×Q812yes—
121C2×Dic157C2×Q812yes—
122C2×Dic157#4112yes—
123C23×C157—21no—
124C23×C157—21no—
125C23×C157—21no—
126C23×C157—21no—
127C23×C157C22×C412yes—
128C23×C157C2×Q812yes—
129C23×C157#4112yes—
130C23×C157C2411yescharacteristic
131C2×C4×C157—21no—
132C2×C4×C157—21no—
133C2×C4×C157—21no—
134C2×C4×C157—21no—
135C2×C4×C157—21no—
136C2×C4×C157—21no—
137C2×C4×C157—21no—
138C2×C4×C157—21no—
139C2×C4×C157—21no—
140C2×C4×C157—21no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157—44no—
147C2×C4×C157C22×C412yes—
148C2×C4×C157C22×C412yes—
149C2×C4×C157#4112yes—
150C22×C157—22no—
151C22×C157—22no—
152C22×C157—22no—
153C22×C157—22no—
154C22×C157—22no—
155C22×C157—22no—
156C22×C157C4×Q814yes—
157C22×C157C23×C411yescharacteristic
158C22×C157C22×Q811yescharacteristic
159C22×C157C2×#4111yescharacteristic
160C22×C157C2×#4111yescharacteristic
161C22×C157C2×#4111yescharacteristic
162C22×C157H(5,𝔽2)11yescharacteristic
163C22×C157H(5,𝔽2)11yescharacteristic
164C4×C157—22no—
165C4×C157—22no—
166C4×C157—22no—
167C4×C157—41no—
168C4×C157—41no—
169C4×C157—41no—
170Dic157—22no—
171Dic157—22no—
172Dic157—22no—
173Dic157—22no—
174Dic157—22no—
175Dic157—22no—
176Dic157—22no—
177Dic157—41no—
178C2×C157—24no—
179C2×C157C2×C4×Q811yescharacteristic
180C2×C157C4×#4111yescharacteristic
181C2×C157#51811yescharacteristic
182C2×C157#55111yescharacteristic
183C2×C157#55111yescharacteristic
184C2×C157#55411yescharacteristic
185C2×C157#55411yescharacteristic
186C157#236811yescharacteristic, complemented, Sylow
187#2368—1571nomaximal, Sylow
188C2×C22⋊Q8—1571no—
189#391—1571no—
190#391—1571no—
191#387—1571no—
192#387—1571no—
193#387—1571no—
194#387—1571no—
195#385—1571no—
196#385D15711yescharacteristic
197C4×C4⋊C4—1571no—
198C4×C4⋊C4—1571no—
199C4×C22⋊C4—1571no—
200C4×C22⋊C4—1571no—
201C4×C22⋊C4—1571no—
202C2×#95—1571no—
203C22×Q8—1571no—
204C23×C4—1571no—
205C23×C4—1571no—
206C23×C4C2×D15711yescharacteristic
207C22⋊Q8—1578no—
208C2×C4⋊C4—1571no—
209C2×C4⋊C4—1571no—
210C2×C4⋊C4—1571no—
211C2×C4⋊C4—1571no—
212C2×C4⋊C4—1571no—
213C2×C4⋊C4C2×D15711yescharacteristic
214C2×C4⋊C4C2×D15711yescharacteristic
215C2×C22⋊C4—1571no—
216C2×C22⋊C4—1571no—
217C2×C22⋊C4C2×D15711yescharacteristic
218C2×C22⋊C4C2×D15711yescharacteristic
219C2×C42—1571no—
220C2×C42—1571no—
221C2×C42—1571no—
222C2×C42—1571no—
223C2×C42—1571no—
224C2×C42—1571no—
225#95—1571no—
226#95—1571no—
227#95—1571no—
228#95—1571no—
229#95—1571no—
230#95—1571no—
231#95—1571no—
232#95—1571no—
233#95—1571no—
234#95—1571no—
235#95—1571no—
236#95—1571no—
237#95C2×D15711yescharacteristic
238#95C2×D15711yescharacteristic
239C24C22×D15711yescharacteristic
240C2×Q8—6281no—
241C2×Q8—6281no—
242C2×Q8—1574no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4—3142no—
247C22×C4—3142no—
248C22×C4—6281no—
249C22×C4—6281no—
250C22×C4—6281no—
251C22×C4—1571no—
252C22×C4—1571no—
253C22×C4—1571no—
254C22×C4—1571no—
255C22×C4—1571no—
256C22×C4—1571no—
257C22×C4—1571no—
258C22×C4—1571no—
259C22×C4—1574no—
260C22×C4C22×D15711yescharacteristic
261C22×C4C22×D15711yescharacteristic
262C22×C4C22×D15711yescharacteristic
263C22×C4C22×D15711yescharacteristic
264C22×C4C22×D15711yescharacteristic
265C22×C4C22×D15711yescharacteristic
266C4⋊C4—22no—
267C4⋊C4—41no—
268C4⋊C4—3142no—
269C4⋊C4—6281no—
270C4⋊C4—1574no—
271C4⋊C4—1574no—
272C4⋊C4—1574no—
273C22⋊C4—22no—
274C22⋊C4—1574no—
275C22⋊C4—1574no—
276C22⋊C4C4×D15714yes—
277C42—3142no—
278C42—3142no—
279C42—6281no—
280C42—6281no—
281C42—6281no—
282C42—6281no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23C2×C4×D15712yes—
288C23Q8×D15712yes—
289C23#1457895312yes—
290C23C23×D15711yescharacteristic
291Q8—6284no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—44no—
308C2×C4—3141no—
309C2×C4—3141no—
310C2×C4—3141no—
311C2×C4—3141no—
312C2×C4—3141no—
313C2×C4—3141no—
314C2×C4—3141no—
315C2×C4—3141no—
316C2×C4—3141no—
317C2×C4—3141no—
318C2×C4—3141no—
319C2×C4—3141no—
320C2×C4—3141no—
321C2×C4—3141no—
322C2×C4—3141no—
323C2×C4—3141no—
324C2×C4—3141no—
325C2×C4—6284no—
326C2×C4—6284no—
327C2×C4—1572no—
328C2×C4—1572no—
329C2×C4—1572no—
330C2×C4—1572no—
331C2×C4—1572no—
332C2×C4—1572no—
333C2×C4—1572no—
334C2×C4C2×C4×D15712yes—
335C2×C4C2×C4×D15712yes—
336C2×C4#1457895312yes—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22#1909773614yes—
344C22C22×C4×D15711yescharacteristic
345C22C2×Q8×D15711yescharacteristic
346C22#41×D15711yescharacteristic
347C22C2×#1457895311yescharacteristic
348C22C2×#1457895311yescharacteristic
349C22#1909781611yescharacteristic
350C22#1909781711yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—3142no—
358C4—3142no—
359C4—3142no—
360C4—3142no—
361C4—3142no—
362C4—3142no—
363C4—3142no—
364C4—6281no—
365C2—24no—
366C2C2×#1909773611yescharacteristic
367C2C4×#1457895311yescharacteristic
368C2#2375774511yescharacteristic
369C2#2375794611yescharacteristic
370C2#2375795411yescharacteristic
371C2#2375797711yescharacteristic
372C2#2375798111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutatorC22×C157C23×C4
Fitting#385×C157C2
socle, cosocleC23×C157C24
prime radicalC24×C157C23
hypercenter#385D157
hypocenterC157#2368
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. #23757178C2semidirect
3. #23757177C2semidirect
4. #23757158C2semidirect
5. #23757155C2semidirect
6. #23757154C2semidirect
7. #23757154C2semidirect
9. C4⋊C4×Dic157C2semidirect
10. #23757124C2semidirect
19. #19097748C4semidirect
54. C2×Dic314C2×C4semidirect
69. C4⋊Dic157C2×C4semidirect
72. #14578933C2×C4semidirect
73. #14578933C2×C4semidirect
75. C22⋊Dic157Q8semidirect
186. C157#2368semidirect

Maximal subgroups

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

subgroupindex
C2×#190977482
#237571782
#237571772
#237571582
#237571552
#237571542
#237571452
C4⋊C4×Dic1572
#237571242
C22⋊C4×Dic1572
C4×C22⋊Dic1572
C2×#190976782
#385×C1572
#2368157