Finite Group Databasebeta

← #34103477

Subgroups — #34103477

Order 20096.

908 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14578953C211yesmaximal, characteristic, complemented
2C2×#19097750C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D157C211yesmaximal, characteristic, complemented
4#23757188C212yesmaximal, complemented
5#23757181C212yesmaximal, complemented
6#23757178C211yesmaximal, characteristic, complemented
7#23757176C211yesmaximal, characteristic, complemented
8#23757153C212yesmaximal, complemented
9#23757145C212yesmaximal, complemented
10#23757130C211yesmaximal, characteristic, complemented
11C2×#121×C157C211yesmaximal, characteristic, complemented
12C2×#14578953—24no—
13C2×#14578953—24no—
14C2×#14578953—24no—
15C22×Dic314C2211yescharacteristic
16C22×C22⋊D157C2212yes—
17C22×C22⋊D157C2211yescharacteristic
18C23×Dic157C2212yes—
19C22×C4×D157C2211yescharacteristic
20#19097750—24no—
21C4×C22⋊D157—24no—
22C2×C4⋊Dic157C2211yescharacteristic
23C2×#14578933C2211yescharacteristic
24C2×#14578933C2212yes—
25C2×C22⋊Dic157C2212yes—
26C2×C22⋊Dic157C2211yescharacteristic
27C2×C22⋊Dic157C2212yes—
28C2×C22⋊Dic157C2212yes—
29C2×#14578931C2211yescharacteristic
30C2×#14578931C2212yes—
31C2×C4×Dic157C2211yescharacteristic
32#19097678C2212yes—
33#19097678C2212yes—
34#19097678C2212yes—
35C22×D4×C157C2212yes—
36C22×D4×C157C2211yescharacteristic
37C23×C4×C157C2211yescharacteristic
38#121×C157—24no—
39C2×C4⋊C4×C157C2211yescharacteristic
40C2×C22⋊C4×C157C2212yes—
41C23×D157C2311yescharacteristic
42#14578953—232no—
43C2×Dic314—22no—
44C2×Dic314—22no—
45C2×Dic314D414yescomplemented
46C2×C22⋊D157—22no—
47C2×C22⋊D157—24no—
48C2×C22⋊D157—24no—
49C2×C22⋊D157—22no—
50C2×C22⋊D157—24no—
51C2×C22⋊D157—22no—
52C22×Dic157—24no—
53C22×Dic157—24no—
54C22×Dic157—24no—
55C22×Dic157C2311yescharacteristic
56C22×Dic157C2312yes—
57C22×Dic157C2311yescharacteristic
58C22×Dic157C2312yes—
59C22×Dic157C2311yescharacteristic
60C2×C4×D157—22no—
61C2×C4×D157—22no—
62C2×C4×D157D414yescomplemented
63C4⋊Dic157—22no—
64#14578933—22no—
65#14578933—24no—
66C22⋊Dic157—22no—
67C22⋊Dic157—24no—
68C22⋊Dic157—24no—
69C22⋊Dic157—42no—
70#14578931—22no—
71#14578931—42no—
72C4×Dic157—22no—
73C24×C157C2312yes—
74C24×C157C2311yescharacteristic
75C2×D4×C157—24no—
76C2×D4×C157—24no—
77C2×D4×C157—22no—
78C2×D4×C157—22no—
79C2×D4×C157—42no—
80C2×D4×C157D414yescomplemented
81C22×C4×C157—22no—
82C22×C4×C157—22no—
83C22×C4×C157—22no—
84C22×C4×C157C2311yescharacteristic
85C22×C4×C157C2312yes—
86C22×C4×C157C2311yescharacteristic
87C4⋊C4×C157—22no—
88C22⋊C4×C157—24no—
89C22×D157—21no—
90C22×D157—21no—
91C22×D157—21no—
92C22×D157—21no—
93C22×D157—21no—
94C22×D157—21no—
95C22×D157C2×D412yes—
96Dic314—28no—
97C22⋊D157—28no—
98C22⋊D157—216no—
99C2×Dic157—216no—
100C2×Dic157—21no—
101C2×Dic157—21no—
102C2×Dic157—22no—
103C2×Dic157—22no—
104C2×Dic157—21no—
105C2×Dic157—21no—
106C2×Dic157—21no—
107C2×Dic157—22no—
108C2×Dic157—22no—
109C2×Dic157—22no—
110C2×Dic157—21no—
111C2×Dic157—21no—
112C2×Dic157C2×D412yes—
113C2×Dic157C2×D414yes—
114C2×Dic157#4112yescomplemented
115C4×D157—28no—
116C23×C157—22no—
117C23×C157—22no—
118C23×C157—21no—
119C23×C157—21no—
120C23×C157—21no—
121C23×C157—21no—
122C23×C157—22no—
123C23×C157—22no—
124C23×C157—21no—
125C23×C157—22no—
126C23×C157—22no—
127C23×C157—21no—
128C23×C157C2×D414yes—
129C23×C157#4112yescomplemented
130C23×C157C2411yescharacteristic
131D4×C157—28no—
132D4×C157—48no—
133C2×C4×C157—21no—
134C2×C4×C157—22no—
135C2×C4×C157—21no—
136C2×C4×C157—21no—
137C2×C4×C157—22no—
138C2×C4×C157—22no—
139C2×C4×C157—44no—
140C2×C4×C157C2×D412yes—
141C2×C4×C157C2×D412yes—
142C2×C4×C157C2×D412yes—
143C22×C157—24no—
144C22×C157—22no—
145C22×C157—22no—
146C22×C157—24no—
147C22×C157—24no—
148C22×C157—22no—
149C22×C157—41no—
150C22×C157—42no—
151C22×C157—42no—
152C22×C157—41no—
153C22×C157—41no—
154C22×C157—42no—
155C22×C157—42no—
156C22×C157—41no—
157C22×C157C22×D411yescharacteristic
158C22×C157C22×D411yescharacteristic
159C22×C157C22×D411yescharacteristic
160C22×C157C2×#4111yescharacteristic
161C22×C157C2×#4111yescharacteristic
162C22×C157H(5,𝔽2)11yescharacteristic
163C22×C1572−1+411yescharacteristic
164C2×D157—22no—
165C2×D157—22no—
166C2×D157—22no—
167C2×D157—41no—
168C2×D157—41no—
169C2×D157—41no—
170C2×D157—41no—
171C4×C157—41no—
172C4×C157—42no—
173C4×C157#12014yes—
174Dic157—22no—
175Dic157—24no—
176Dic157—22no—
177Dic157—41no—
178Dic157—42no—
179C2×C157—44no—
180C2×C157—42no—
181C2×C157C2×#12011yescharacteristic
182C2×C157#52511yescharacteristic
183C2×C157#53711yescharacteristic
184C2×C157#54611yescharacteristic
185C2×C157#54711yescharacteristic
186C2×C157#54811yescharacteristic
187C2×C157#55011yescharacteristic
188D157—42no—
189C157#240711yescharacteristic, complemented, Sylow
190#2407—1571nomaximal, Sylow
191C22×#41—1571no—
192C2×C22⋊Q8—1571no—
193C2×#121D15711yescharacteristic, complemented
194C2×C4×D4—1571no—
195#394—1572no—
196#392—1572no—
197#391—1571no—
198#390—1571no—
199#386—1572no—
200#385—1572no—
201#380—1571no—
202C2×#41—3144no—
203C2×#41—3144no—
204C2×#41—3144no—
205C22×Q8—1571no—
206C22×D4—1571no—
207C22×D4—1572no—
208C22×D4C2×D15712yes—
209C22×D4C2×D15711yescharacteristic
210C23×C4—1572no—
211C23×C4—1571no—
212C23×C4C2×D15711yescharacteristic
213C22⋊Q8—3144no—
214#121—24no—
215C4×D4—3144no—
216C2×C4⋊C4—1571no—
217C2×C4⋊C4—1571no—
218C2×C4⋊C4—1572no—
219C2×C4⋊C4C2×D15711yescharacteristic
220C2×C22⋊C4—1571no—
221C2×C22⋊C4—1572no—
222C2×C22⋊C4—1572no—
223C2×C22⋊C4—1571no—
224C2×C22⋊C4—1572no—
225C2×C22⋊C4—1572no—
226C2×C22⋊C4C2×D15712yes—
227C2×C42—1571no—
228#95—1572no—
229#95—1572no—
230#95—1572no—
231C24—1571no—
232C24C22×D15712yes—
233C24C22×D15711yescharacteristic
234#41—31432no—
235C2×Q8—3142no—
236C2×Q8—3142no—
237C2×Q8—1574no—
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—42no—
243C2×D4—3142no—
244C2×D4—3144no—
245C2×D4—3144no—
246C2×D4—3142no—
247C2×D4—3142no—
248C2×D4—3144no—
249C2×D4C22⋊D15714yescomplemented
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—3144no—
254C22×C4—3144no—
255C22×C4—3144no—
256C22×C4—3142no—
257C22×C4—3142no—
258C22×C4—1571no—
259C22×C4—1572no—
260C22×C4—1571no—
261C22×C4—1571no—
262C22×C4—1572no—
263C22×C4—1574no—
264C22×C4C22×D15712yes—
265C22×C4C22×D15711yescharacteristic
266C22×C4C22×D15711yescharacteristic
267C4⋊C4—22no—
268C4⋊C4—3142no—
269C4⋊C4—3144no—
270C4⋊C4—3142no—
271C22⋊C4—24no—
272C22⋊C4—3142no—
273C22⋊C4—3142no—
274C22⋊C4—3144no—
275C22⋊C4—3144no—
276C22⋊C4—6282no—
277C22⋊C4—6282no—
278C42—3142no—
279C23—21no—
280C23—21no—
281C23—22no—
282C23—22no—
283C23—21no—
284C23—22no—
285C23—21no—
286C23—22no—
287C23—22no—
288C23—21no—
289C23—22no—
290C23—21no—
291C23—3141no—
292C23—3141no—
293C23—3141no—
294C23—3141no—
295C23—3141no—
296C23—3141no—
297C23—1572no—
298C23C2×C22⋊D15714yes—
299C23#1457895312yescomplemented
300C23C23×D15711yescharacteristic
301Q8—3148no—
302D4—28no—
303D4—48no—
304D4—31416no—
305D4—3148no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—44no—
313C2×C4—31416no—
314C2×C4—3141no—
315C2×C4—3141no—
316C2×C4—3142no—
317C2×C4—3141no—
318C2×C4—3142no—
319C2×C4—3141no—
320C2×C4—3148no—
321C2×C4—3141no—
322C2×C4—3142no—
323C2×C4—3142no—
324C2×C4—3141no—
325C2×C4—3142no—
326C2×C4—3141no—
327C2×C4—1572no—
328C2×C4—1574no—
329C2×C4—1572no—
330C2×C4D4×D15712yes—
331C2×C4D4×D15712yes—
332C2×C4C2×C22⋊D15712yes—
333C22—22no—
334C22—24no—
335C22—24no—
336C22—22no—
337C22—24no—
338C22—22no—
339C22—41no—
340C22—42no—
341C22—41no—
342C22—42no—
343C22—41no—
344C22—42no—
345C22—41no—
346C22—42no—
347C22—3142no—
348C22—3142no—
349C22—3142no—
350C22—6281no—
351C22—6281no—
352C22—6281no—
353C22—6281no—
354C22C2×D4×D15711yescharacteristic
355C22C2×D4×D15711yescharacteristic
356C22C22×C22⋊D15711yescharacteristic
357C22#41×D15711yescharacteristic
358C22C2×#1457895311yescharacteristic
359C22#1909781611yescharacteristic
360C22#1909781811yescharacteristic
361C4—42no—
362C4—41no—
363C4—3142no—
364C4—3144no—
365C4—3142no—
366C4—6281no—
367C4—6282no—
368C4#1909773914yes—
369C2—42no—
370C2—44no—
371C2—6282no—
372C2C2×#1909773911yescharacteristic
373C2#2375778311yescharacteristic
374C2#2375785211yescharacteristic
375C2#2375791711yescharacteristic
376C2#2375792211yescharacteristic
377C2#2375792911yescharacteristic
378C2#2375794111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutator, socle, cosocleC23×C157C24
FittingC2×#121×C157C2
hypercenterC2×#121D157
hypocenterC157#2407
LCSG ▷ C23×C157 ▷ C157
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#14578953C2semidirect
2. C2×#19097750C2semidirect
3. C2×C4×C22⋊D157C2semidirect
4. #23757188C2semidirect
5. #23757181C2semidirect
6. #23757178C2semidirect
7. #23757176C2semidirect
8. #23757153C2semidirect
9. #23757145C2semidirect
10. #23757130C2semidirect
11. C2×#121×C157C2semidirect
45. C2×Dic314D4semidirect
62. C2×C4×D157D4semidirect
80. C2×D4×C157D4semidirect
114. C2×Dic157#41semidirect
129. C23×C157#41semidirect
189. C157#2407semidirect
193. C2×#121D157semidirect
249. C2×D4C22⋊D157semidirect
299. C23#14578953semidirect

Maximal subgroups

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

subgroupindex
C22×#145789532
C2×#190977502
C2×C4×C22⋊D1572
#237571882
#237571812
#237571782
#237571762
#237571532
#237571452
#237571302
C2×#121×C1572
#2407157