Finite Group Databasebeta

← #34103348

Subgroups — #34103348

Order 20096.

944 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14578953C211yesmaximal, characteristic, complemented
2C2×D4×Dic157C212yesmaximal, complemented
3C2×#19097731C211yesmaximal, characteristic, complemented
4#23757194C211yesmaximal, characteristic, complemented
5#23757189C212yesmaximal, complemented
6#23757187C211yesmaximal, characteristic, complemented
7#23757185C211yesmaximal, characteristic, complemented
8#23757181C212yesmaximal, complemented
9C22⋊C4×Dic157C212yesmaximal, complemented
10#23757120C211yesmaximal, characteristic, complemented
11#392×C157C211yesmaximal, characteristic, complemented
12C2×#14578953—28no—
13C2×#14578953—24no—
14C22×Dic314C2211yescharacteristic
15C22×C22⋊D157C2212yes—
16C22×C22⋊D157C2211yescharacteristic
17C23×Dic157C2211yescharacteristic
18C23×Dic157C2212yes—
19C22×C4×D157C2211yescharacteristic
20D4×Dic157—28no—
21#19097731—24no—
22C2×C4⋊Dic157C2212yes—
23C2×#14578933C2211yescharacteristic
24C2×C22⋊Dic157C2212yes—
25C2×C22⋊Dic157C2212yes—
26C2×C22⋊Dic157C2212yes—
27C2×#14578931C2212yes—
28C2×#14578931C2211yescharacteristic
29C2×C4×Dic157C2212yes—
30C2×C4×Dic157C2211yescharacteristic
31#19097678C2211yescharacteristic
32#19097678C2212yes—
33#19097678C2212yes—
34C22×D4×C157C2211yescharacteristic
35C22×D4×C157C2212yes—
36C2×C22⋊C4×C157C2212yes—
37C2×C22⋊C4×C157C2211yescharacteristic
38#95×C157C2211yescharacteristic
39C23×D157C2311yescharacteristic
40#14578953—232no—
41C2×Dic314—24no—
42C2×Dic314—22no—
43C2×C22⋊D157—22no—
44C2×C22⋊D157—24no—
45C2×C22⋊D157—24no—
46C2×C22⋊D157—22no—
47C2×C22⋊D157—24no—
48C2×C22⋊D157—22no—
49C22×Dic157—24no—
50C22×Dic157—24no—
51C22×Dic157—22no—
52C22×Dic157—24no—
53C22×Dic157D418yescomplemented
54C22×Dic157C2312yes—
55C22×Dic157C2311yescharacteristic
56C22×Dic157C2312yes—
57C22×Dic157C2311yescharacteristic
58C22×Dic157C2311yescharacteristic
59C2×C4×D157—24no—
60C2×C4×D157D414yescomplemented
61C4⋊Dic157—24no—
62#14578933—22no—
63C22⋊Dic157—24no—
64C22⋊Dic157—24no—
65C22⋊Dic157—24no—
66#14578931—22no—
67#14578931—24no—
68C4×Dic157—22no—
69C4×Dic157—24no—
70C24×C157C2312yes—
71C24×C157C2311yescharacteristic
72C2×D4×C157—24no—
73C2×D4×C157—24no—
74C2×D4×C157—24no—
75C2×D4×C157—22no—
76C2×D4×C157—24no—
77C22×C4×C157C2311yescharacteristic
78C22×C4×C157C2311yescharacteristic
79C22×C4×C157C2312yes—
80C22⋊C4×C157—22no—
81C22⋊C4×C157—24no—
82C22×D157—22no—
83C22×D157—21no—
84C22×D157—21no—
85C22×D157—22no—
86C22×D157C2×D412yes—
87Dic314—28no—
88C22⋊D157—28no—
89C22⋊D157—216no—
90C2×Dic157—22no—
91C2×Dic157—22no—
92C2×Dic157—21no—
93C2×Dic157—21no—
94C2×Dic157—216no—
95C2×Dic157—21no—
96C2×Dic157—22no—
97C2×Dic157—22no—
98C2×Dic157—22no—
99C2×Dic157—22no—
100C2×Dic157—44no—
101C2×Dic157C2×D414yes—
102C2×Dic157C2×D414yes—
103C2×Dic157C2×D412yes—
104C2×Dic157#4112yescomplemented
105C4×D157—28no—
106C23×C157—21no—
107C23×C157—22no—
108C23×C157—22no—
109C23×C157—22no—
110C23×C157—21no—
111C23×C157—22no—
112C23×C157—22no—
113C23×C157—22no—
114C23×C157—22no—
115C23×C157—22no—
116C23×C157C2×D414yes—
117C23×C157#4112yescomplemented
118C23×C157C2411yescharacteristic
119D4×C157—28no—
120D4×C157—216no—
121C2×C4×C157—22no—
122C2×C4×C157—22no—
123C2×C4×C157—22no—
124C2×C4×C157—21no—
125C2×C4×C157—22no—
126C2×C4×C157C2×D412yes—
127C2×C4×C157#4114yescomplemented
128C22×C157—24no—
129C22×C157—24no—
130C22×C157—24no—
131C22×C157—22no—
132C22×C157—24no—
133C22×C157—41no—
134C22×C157—42no—
135C22×C157—42no—
136C22×C157—42no—
137C22×C157—42no—
138C22×C157—41no—
139C22×C157—42no—
140C22×C157C22×D412yes—
141C22×C157C22×D411yescharacteristic
142C22×C157C2×#4111yescharacteristic
143C22×C157C2×#4111yescharacteristic
144C22×C157C2×#4112yes—
145C2×D157—22no—
146C2×D157—22no—
147C2×D157—22no—
148C2×D157—42no—
149C2×D157—42no—
150C4×C157—22no—
151C4×C157—24no—
152C4×C157—41no—
153Dic157—24no—
154Dic157—22no—
155Dic157—41no—
156Dic157—42no—
157Dic157#12014yes—
158C2×C157—44no—
159C2×C157—42no—
160C2×C157C2×#12011yescharacteristic
161C2×C157#52512yes—
162C2×C157#52511yescharacteristic
163C2×C157#53212yes—
164C2×C157#53211yescharacteristic
165D157—42no—
166C157#238811yescharacteristic, complemented, Sylow
167#2388—1571nomaximal, Sylow
168C22×#41—1571no—
169C2×C4×D4—1572no—
170C2×C4×D4—1571no—
171#395—1571no—
172#394—1571no—
173#394—1572no—
174#393—1571no—
175#392—1572no—
176#392D15711yescharacteristic, complemented
177C4×C22⋊C4—1572no—
178C4×C22⋊C4—1571no—
179C2×#41—3148no—
180C2×#41—3144no—
181C22×Q8—1571no—
182C22×D4—1571no—
183C22×D4—1572no—
184C22×D4C2×D15712yes—
185C22×D4C2×D15711yescharacteristic
186C23×C4—1571no—
187C23×C4—1572no—
188C23×C4—1571no—
189C4×D4—3144no—
190C4×D4—3148no—
191C2×C4⋊C4—1571no—
192C2×C4⋊C4—1572no—
193C2×C22⋊C4—1572no—
194C2×C22⋊C4—1572no—
195C2×C22⋊C4—1572no—
196C2×C22⋊C4—1571no—
197C2×C22⋊C4—1572no—
198C2×C22⋊C4C2×D15711yescharacteristic
199C2×C22⋊C4C2×D15712yes—
200C2×C42—1572no—
201C2×C42—1571no—
202#95—1572no—
203#95—1571no—
204#95—1572no—
205#95C2×D15711yescharacteristic
206C24—1571no—
207C24C22×D15712yes—
208C24C22×D15711yescharacteristic
209#41—31432no—
210C2×Q8—3144no—
211C2×Q8—3142no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—22no—
216C2×D4—24no—
217C2×D4—3142no—
218C2×D4—3144no—
219C2×D4—3144no—
220C2×D4—3142no—
221C2×D4—3142no—
222C2×D4—3144no—
223C22×C4—3144no—
224C22×C4—3144no—
225C22×C4—3144no—
226C22×C4—3142no—
227C22×C4—3144no—
228C22×C4—1572no—
229C22×C4—1571no—
230C22×C4—1578no—
231C22×C4—1571no—
232C22×C4—1572no—
233C22×C4—1574no—
234C22×C4—1571no—
235C22×C4C22×D15711yescharacteristic
236C22×C4C22×D15711yescharacteristic
237C22×C4C22×D15712yes—
238C4⋊C4—3144no—
239C4⋊C4—3142no—
240C22⋊C4—22no—
241C22⋊C4—24no—
242C22⋊C4—3142no—
243C22⋊C4—3144no—
244C22⋊C4—3144no—
245C22⋊C4—3144no—
246C22⋊C4—3144no—
247C42—3144no—
248C42—3142no—
249C23—21no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—21no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—3142no—
260C23—3141no—
261C23—3142no—
262C23—3141no—
263C23—1572no—
264C23D4×D15714yes—
265C23#1457895312yescomplemented
266C23C23×D15711yescharacteristic
267Q8—3148no—
268D4—28no—
269D4—216no—
270D4—31416no—
271D4—3148no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—3148no—
278C2×C4—3141no—
279C2×C4—3142no—
280C2×C4—3142no—
281C2×C4—3141no—
282C2×C4—31416no—
283C2×C4—3142no—
284C2×C4—3142no—
285C2×C4—3141no—
286C2×C4—3142no—
287C2×C4—3142no—
288C2×C4—6284no—
289C2×C4—1574no—
290C2×C4—1572no—
291C2×C4—1574no—
292C2×C4—1572no—
293C2×C4D4×D15712yes—
294C2×C4#1457895314yescomplemented
295C22—24no—
296C22—24no—
297C22—24no—
298C22—22no—
299C22—24no—
300C22—41no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—41no—
306C22—42no—
307C22—3142no—
308C22—3142no—
309C22—3142no—
310C22—6282no—
311C22—6282no—
312C22C2×D4×D15712yes—
313C22C2×D4×D15711yescharacteristic
314C22#41×D15711yescharacteristic
315C22C2×#1457895311yescharacteristic
316C22C2×#1457895312yes—
317C4—22no—
318C4—24no—
319C4—41no—
320C4—3144no—
321C4—3142no—
322C4—6281no—
323C4—6282no—
324C4—1574no—
325C2—44no—
326C2—42no—
327C2—6282no—
328C2#120×D15711yescharacteristic
329C2#2375777912yes—
330C2#2375778311yescharacteristic
331C2#2375781812yes—
332C2#2375782111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutator, socle, cosocleC23×C157C24
Fitting#392×C157C2
hypercenter#392D157
hypocenterC157#2388
LCSG ▷ C23×C157 ▷ C157
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#14578953C2semidirect
2. C2×D4×Dic157C2semidirect
3. C2×#19097731C2semidirect
4. #23757194C2semidirect
5. #23757189C2semidirect
6. #23757187C2semidirect
7. #23757185C2semidirect
8. #23757181C2semidirect
9. C22⋊C4×Dic157C2semidirect
10. #23757120C2semidirect
11. #392×C157C2semidirect
53. C22×Dic157D4semidirect
60. C2×C4×D157D4semidirect
104. C2×Dic157#41semidirect
117. C23×C157#41semidirect
127. C2×C4×C157#41semidirect
166. C157#2388semidirect
176. #392D157semidirect
265. C23#14578953semidirect
294. C2×C4#14578953semidirect

Maximal subgroups

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

subgroupindex
C22×#145789532
C2×D4×Dic1572
C2×#190977312
#237571942
#237571892
#237571872
#237571852
#237571812
C22⋊C4×Dic1572
#237571202
#392×C1572
#2388157