Finite Group Databasebeta

← #34100105

Subgroups — #34100105

Order 20096.

588 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#19097817C211yesmaximal, characteristic
2#23757290C211yesmaximal, characteristic, complemented
3#23757279C211yesmaximal, characteristic, complemented
4#23757277C211yesmaximal, characteristic, complemented
5#23757255C211yesmaximal, characteristic, complemented
6#23757246C211yesmaximal, characteristic, complemented
7#421×C157C211yesmaximal, characteristic, complemented
8#19097817—22no—
9#19097817—22no—
10#19097817—22no—
11#19097817—22no—
12C2×#14578955C412yescomplemented
13C2×#14578954—21no—
14C2×#14578954C2211yescharacteristic
15C22×D628—21no—
16C22×D628C2211yescharacteristic
17C2×D4×D157—22no—
18C2×D4×D157C412yescomplemented
19C2×#14578938C2211yescharacteristic, complemented
20#19097730C2211yescharacteristic, complemented
21#19097698—22no—
22#19097693—22no—
23#19097691—22no—
24#19097689—21no—
25#19097687—21no—
26#19097685—22no—
27C2×#41×C157C2211yescharacteristic
28C2×M16×C157C2211yescharacteristic, complemented
29#117×C157C2211yescharacteristic, complemented
30#104×C157—22no—
31C23×D157—22no—
32C23×D157—21no—
33#14578955—24no—
34#14578955—24no—
35#14578954—24no—
36#14578954—24no—
37#14578954—21no—
38#14578954—21no—
39#14578954—21no—
40#14578954—21no—
41C2×Dic314D411yescharacteristic
42C2×D628—24no—
43C2×D628—24no—
44C2×D628—22no—
45C2×D628—21no—
46C2×D628—21no—
47C2×D628—21no—
48C2×D628C2×C412yes—
49C2×D628D411yescharacteristic
50C2×D628D412yes—
51C2×D628D412yes—
52C2×C22⋊D157—22no—
53C2×C22⋊D157D412yes—
54D4×D157—28no—
55D4×D157—28no—
56D4×D157—24no—
57D4×D157—24no—
58C2×C4×D157—22no—
59C2×C4×D157C2×C412yes—
60#14578938—21no—
61#14578938—21no—
62C2×#11929434—21no—
63C4⋊Dic157—21no—
64C22⋊Dic157—21no—
65C4×Dic157—21no—
66#41×C157—21no—
67#41×C157—21no—
68#41×C157D412yes—
69#41×C157D412yes—
70C2×Q8×C157C2×C411yescharacteristic
71C2×D4×C157—21no—
72C2×D4×C157C2×C411yescharacteristic
73C22×C4×C157—21no—
74C22×C4×C157C2311yescharacteristic
75M16×C157—21no—
76M16×C157—21no—
77C2×C8×C157—21no—
78C4⋊C4×C157—21no—
79C22⋊C4×C157—21no—
80C42×C157—21no—
81C22×D157—22no—
82C22×D157—28no—
83C22×D157—22no—
84C22×D157—22no—
85C22×D157—82no—
86C22×D157—84no—
87C22×D157C22⋊C412yes—
88Dic314—21no—
89Dic314—21no—
90D628—28no—
91D628—22no—
92D628—21no—
93D628—22no—
94D628—21no—
95D628—22no—
96D628C22⋊C414yes—
97C22⋊D157—28no—
98C22⋊D157—22no—
99C22⋊D157—22no—
100C2×Dic157—41no—
101C2×Dic157C22⋊C412yes—
102C4×D157—28no—
103C4×D157—22no—
104C4×D157—22no—
105#11929434—41no—
106C23×C157—21no—
107C23×C157C2×D411yescharacteristic
108Q8×C157—21no—
109Q8×C157C22⋊C412yes—
110D4×C157—22no—
111D4×C157—22no—
112D4×C157—21no—
113D4×C157C22⋊C412yes—
114C2×C4×C157—21no—
115C2×C4×C157—22no—
116C2×C4×C157—22no—
117C2×C4×C157—41no—
118C2×C4×C157C22×C411yescharacteristic
119C2×C4×C157C2×D411yescharacteristic
120C2×C4×C157C2×D411yescharacteristic
121C2×C4×C157C2×D411yescharacteristic
122C2×C4×C157C2×D411yescharacteristic
123C2×C4×C157C2×D411yescharacteristic
124C8×C157—41no—
125C22×C157—22no—
126C22×C157—41no—
127C22×C157—42no—
128C22×C157C2×C22⋊C411yescharacteristic
129C22×C157#12011yescharacteristic
130C22×C157#12011yescharacteristic
131C2×D157—22no—
132C2×D157—24no—
133C2×D157—44no—
134C2×D157—42no—
135C2×D157—82no—
136C2×D157—82no—
137C2×D157—88no—
138C4×C157—22no—
139C4×C157—81no—
140C4×C157C2×C22⋊C411yescharacteristic
141C4×C157C2×C22⋊C411yescharacteristic
142C4×C157#12011yescharacteristic
143C4×C157#12011yescharacteristic
144Dic157—22no—
145Dic157—81no—
146C2×C157—21no—
147C2×C157—21no—
148C2×C157—21no—
149C2×C157—42no—
150C2×C157#37911yescharacteristic
151D157—44no—
152D157—42no—
153C157#179511yescharacteristic, complemented, Sylow
154#1795—1571nomaximal, Sylow
155C2×H(5,𝔽2)—1571no—
156#421—1571no—
157#421D15711yescharacteristic, complemented
158#418—1571no—
159#418—1571no—
160#413—1571no—
161#410—1571no—
162H(5,𝔽2)—3142no—
163H(5,𝔽2)—3142no—
164H(5,𝔽2)—3142no—
165H(5,𝔽2)—3142no—
166C2×#41—3141no—
167C2×#41—1572no—
168C2×#41—1571no—
169C2×#41C2×D15711yescharacteristic
170C22×D4—3141no—
171C22×D4—3142no—
172C22×D4—1572no—
173C22×D4—1571no—
174C2×M16—1571no—
175C2×M16C2×D15711yescharacteristic, complemented
176#117—1571no—
177#117C2×D15711yescharacteristic, complemented
178#104—22no—
179#104—3142no—
180#102—3142no—
181#102—3142no—
182#101—3141no—
183#100—3141no—
184#99—3142no—
185C24—3142no—
186C24—3141no—
187#41—21no—
188#41—21no—
189#41—3144no—
190#41—3144no—
191#41—3144no—
192#41—3144no—
193#41—3141no—
194#41—3141no—
195#41—3141no—
196#41—3141no—
197#41C22⋊D15712yes—
198#41D62812yes—
199C2×Q8—1571no—
200C2×Q8C4×D15711yescharacteristic
201C2×D4—21no—
202C2×D4—3144no—
203C2×D4—3144no—
204C2×D4—3142no—
205C2×D4—3148no—
206C2×D4—3148no—
207C2×D4—3144no—
208C2×D4—3144no—
209C2×D4—3142no—
210C2×D4—3141no—
211C2×D4—3141no—
212C2×D4—3141no—
213C2×D4—1572no—
214C2×D4—1572no—
215C2×D4—1571no—
216C2×D4—1572no—
217C2×D4—1572no—
218C2×D4C4×D15711yescharacteristic
219C22×C4—21no—
220C22×C4—3142no—
221C22×C4—1572no—
222C22×C4C22×D15711yescharacteristic
223M16—21no—
224M16—21no—
225M16—3141no—
226M16—3141no—
227C2×C8—21no—
228C2×C8—3141no—
229C4⋊C4—21no—
230C4⋊C4—3141no—
231C22⋊C4—21no—
232C22⋊C4—3141no—
233C42—21no—
234C42—3141no—
235C23—21no—
236C23—3148no—
237C23—3142no—
238C23—3142no—
239C23—3142no—
240C23—1572no—
241C23—12562no—
242C23—12564no—
243C23D4×D15711yescharacteristic
244Q8—21no—
245Q8—3141no—
246Q8—3141no—
247Q8#1457893112yes—
248D4—22no—
249D4—22no—
250D4—21no—
251D4—3148no—
252D4—3148no—
253D4—3142no—
254D4—3142no—
255D4—3142no—
256D4—3141no—
257D4—3142no—
258D4—3141no—
259D4—3142no—
260D4—1574no—
261D4#1457893112yes—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—3148no—
267C2×C4—3142no—
268C2×C4—3142no—
269C2×C4—6281no—
270C2×C4—1572no—
271C2×C4C2×C4×D15711yescharacteristic
272C2×C4D4×D15711yescharacteristic
273C2×C4D4×D15711yescharacteristic
274C2×C4D4×D15711yescharacteristic
275C2×C4C2×C22⋊D15711yescharacteristic
276C2×C4C2×D62811yescharacteristic
277C8—41no—
278C8—6281no—
279C22—22no—
280C22—41no—
281C22—42no—
282C22—3142no—
283C22—3144no—
284C22—6282no—
285C22—6284no—
286C22—12562no—
287C22—12562no—
288C22—12568no—
289C22C22⋊C4×D15711yescharacteristic
290C22#1909773911yescharacteristic
291C22#1909774111yescharacteristic
292C4—22no—
293C4—81no—
294C4—3142no—
295C4—12561no—
296C4C22⋊C4×D15711yescharacteristic
297C4C2×#1457893111yescharacteristic
298C4#1909773911yescharacteristic
299C4#1909774111yescharacteristic
300C2—21no—
301C2—21no—
302C2—21no—
303C2—42no—
304C2—6284no—
305C2—6282no—
306C2#2375712811yescharacteristic
subgroupG/H
centerC2#23757128
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#421×C157C2
socleC2×C157#379
prime radicalC2×#19097817C2
cosocleC22×C4×C157C23
hypercenter#421D157
hypocenterC157#1795
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #23757290C2semidirect
3. #23757279C2semidirect
4. #23757277C2semidirect
5. #23757255C2semidirect
6. #23757246C2semidirect
7. #421×C157C2semidirect
12. C2×#14578955C4semidirect
18. C2×D4×D157C4semidirect
19. C2×#14578938C22semidirect
20. #19097730C22semidirect
28. C2×M16×C157C22semidirect
29. #117×C157C22semidirect
153. C157#1795semidirect
157. #421D157semidirect
175. C2×M16C2×D157semidirect
177. #117C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978172
#237572902
#237572792
#237572772
#237572552
#237572462
#421×C1572
#1795157