Finite Group Databasebeta

← #34101385

Subgroups — #34101385

Order 20096.

382 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 67 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
1#23757832C211yesmaximal, characteristic
2C2×#19097686C211yesmaximal, characteristic, complemented
3C2×#19097685C211yesmaximal, characteristic, complemented
4#23757152C211yesmaximal, characteristic, complemented
5#23757146C211yesmaximal, characteristic, complemented
6#23757039C211yesmaximal, characteristic, complemented
7#386×C157C211yesmaximal, characteristic, complemented
8C22×C22⋊D157C2211yescharacteristic
9#19097754—21no—
10#19097752—21no—
11#19097743—21no—
12#19097743—21no—
13#19097740—21no—
14#19097739—21no—
15C2×#14578933—21no—
16C2×C22⋊Dic157C2211yescharacteristic
17C2×C22⋊Dic157C2211yescharacteristic
18C2×C22⋊Dic157C2211yescharacteristic
19C2×#14578931—21no—
20#19097686—22no—
21#19097685—22no—
22#19097678—21no—
23#19097678—21no—
24C22×D4×C157C2211yescharacteristic
25C23×C4×C157C2211yescharacteristic
26C2×C22⋊C4×C157—21no—
27C2×C22⋊C4×C157C2211yescharacteristic
28#95×C157—21no—
29C23×D157D411yescharacteristic
30C2×C22⋊D157—21no—
31C2×C22⋊D157—21no—
32C2×C22⋊D157—21no—
33C2×C22⋊D157—21no—
34C2×C22⋊D157—21no—
35C2×C22⋊D157—21no—
36C2×C22⋊D157—21no—
37C2×C22⋊D157—21no—
38C22×Dic157—21no—
39C22×Dic157—21no—
40C22×Dic157D411yescharacteristic
41C22×Dic157D411yescharacteristic
42C22×Dic157D411yescharacteristic
43#14578933—21no—
44#14578933—21no—
45#14578933—42no—
46C22⋊Dic157—21no—
47C22⋊Dic157—21no—
48C22⋊Dic157—21no—
49C22⋊Dic157—21no—
50C22⋊Dic157—21no—
51C22⋊Dic157—21no—
52C22⋊Dic157—21no—
53C22⋊Dic157D412yescomplemented
54#14578931—21no—
55#14578931—21no—
56#14578931—42no—
57C24×C157D411yescharacteristic
58C24×C157C2311yescharacteristic
59C2×D4×C157—21no—
60C2×D4×C157—21no—
61C2×D4×C157—21no—
62C2×D4×C157—21no—
63C2×D4×C157—21no—
64C2×D4×C157—21no—
65C22×C4×C157—21no—
66C22×C4×C157—21no—
67C22×C4×C157—21no—
68C22×C4×C157—21no—
69C22×C4×C157—42no—
70C22×C4×C157D411yescharacteristic
71C22⋊C4×C157—21no—
72C22⋊C4×C157—21no—
73C22⋊C4×C157—42no—
74C22×D157—21no—
75C22×D157—21no—
76C22×D157—21no—
77C22×D157—41no—
78C22×D157—41no—
79C22⋊D157—41no—
80C22⋊D157—41no—
81C22⋊D157—41no—
82C22⋊D157—41no—
83C22⋊D157—81no—
84C22⋊D157—81no—
85C2×Dic157—21no—
86C2×Dic157—21no—
87C2×Dic157—21no—
88C2×Dic157—22no—
89C2×Dic157—21no—
90C2×Dic157—21no—
91C2×Dic157—41no—
92C2×Dic157—41no—
93C2×Dic157—41no—
94C2×Dic157—41no—
95C2×Dic157—41no—
96C2×Dic157—41no—
97C2×Dic157—41no—
98C23×C157—21no—
99C23×C157—21no—
100C23×C157—21no—
101C23×C157—41no—
102C23×C157—41no—
103C23×C157—41no—
104C23×C157—41no—
105C23×C157C2×D411yescharacteristic
106C23×C157C2×D411yescharacteristic
107C23×C157C2×D411yescharacteristic
108C23×C157C2×D411yescharacteristic
109C23×C157#4111yescharacteristic
110C23×C157#4111yescharacteristic
111C23×C157#4111yescharacteristic
112D4×C157—41no—
113D4×C157—41no—
114D4×C157—41no—
115D4×C157—41no—
116C2×C4×C157—22no—
117C2×C4×C157—22no—
118C2×C4×C157—22no—
119C2×C4×C157—21no—
120C2×C4×C157—21no—
121C2×C4×C157—21no—
122C2×C4×C157—41no—
123C2×C4×C157—41no—
124C2×C4×C157—41no—
125C2×C4×C157—41no—
126C2×C4×C157—41no—
127C2×C4×C157—41no—
128C2×C4×C157—41no—
129C22×C157—21no—
130C22×C157—21no—
131C22×C157—21no—
132C22×C157—21no—
133C22×C157—21no—
134C22×C157—21no—
135C22×C157—41no—
136C22×C157—41no—
137C22×C157—41no—
138C22×C157—41no—
139C22×C157—42no—
140C22×C157—41no—
141C22×C157—41no—
142C22×C157—41no—
143C22×C157—81no—
144C22×C157#12011yescharacteristic
145C22×C157#12111yescharacteristic
146C22×C157#12111yescharacteristic
147C22×C157#12111yescharacteristic
148C22×C157#12311yescharacteristic
149C22×C157#12311yescharacteristic
150C22×C157#12411yescharacteristic
151C2×D157—42no—
152C2×D157—41no—
153C2×D157—41no—
154C2×D157—41no—
155C2×D157—81no—
156C4×C157—41no—
157C4×C157—42no—
158C4×C157—81no—
159Dic157—42no—
160Dic157—41no—
161Dic157—41no—
162Dic157—41no—
163Dic157—81no—
164C2×C157—21no—
165C2×C157—21no—
166C2×C157—21no—
167C2×C157—21no—
168C2×C157—21no—
169C2×C157—21no—
170C2×C157—81no—
171C2×C157#39411yescharacteristic
172C2×C157B(4,𝔽2)11yescharacteristic
173C2×C157#45811yescharacteristic
174D157—81no—
175C157#204411yescharacteristic, complemented, Sylow
176#2044—1571nomaximal, Sylow
177#534—1571no—
178C2×#99—1571no—
179C2×#99—1571no—
180#386—1571no—
181#386D15711yescharacteristic, complemented
182#385—1571no—
183#342—1571no—
184C22×D4—1571no—
185C22×D4C2×D15711yescharacteristic
186C23×C4C2×D15711yescharacteristic
187#123—3141no—
188#123—3141no—
189#121—3141no—
190#121—3141no—
191#120—3141no—
192#120—3141no—
193C2×C4⋊C4—3141no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—3141no—
196C2×C22⋊C4—1571no—
197C2×C22⋊C4—1571no—
198C2×C22⋊C4—1571no—
199C2×C22⋊C4C2×D15711yescharacteristic
200#99—3142no—
201#99—3142no—
202#95—21no—
203#95—3141no—
204#95—3141no—
205C24—1571no—
206C24C22⋊D15711yescharacteristic
207C24C22×D15711yescharacteristic
208C2×D4—21no—
209C2×D4—21no—
210C2×D4—21no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—21no—
214C2×D4—3141no—
215C2×D4—3141no—
216C2×D4—3141no—
217C2×D4—3141no—
218C2×D4—3141no—
219C2×D4—3141no—
220C2×D4—3141no—
221C2×D4—3141no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—3141no—
228C22×C4—3141no—
229C22×C4—1571no—
230C22×C4—1571no—
231C22×C4—1571no—
232C22×C4C22⋊D15711yescharacteristic
233C4⋊C4—3141no—
234C4⋊C4—3141no—
235C4⋊C4—6282no—
236C22⋊C4—21no—
237C22⋊C4—21no—
238C22⋊C4—42no—
239C22⋊C4—3141no—
240C22⋊C4—3141no—
241C22⋊C4—3141no—
242C22⋊C4—3141no—
243C22⋊C4—3141no—
244C22⋊C4—3141no—
245C22⋊C4—3141no—
246C22⋊C4—3141no—
247C22⋊C4—3141no—
248C22⋊C4—6282no—
249C22⋊C4—1572no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—3141no—
258C23—3141no—
259C23—3141no—
260C23—6281no—
261C23—6281no—
262C23D4×D15711yescharacteristic
263C23D4×D15711yescharacteristic
264C23D4×D15711yescharacteristic
265C23C2×C22⋊D15711yescharacteristic
266C23#1457895311yescharacteristic
267C23#1457895411yescharacteristic
268C23#1457895411yescharacteristic
269D4—41no—
270D4—41no—
271D4—41no—
272D4—41no—
273D4—6281no—
274D4—6281no—
275D4—6281no—
276D4—6281no—
277D4—12561no—
278D4—12561no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—3141no—
293C2×C4—3141no—
294C2×C4—3141no—
295C2×C4—3142no—
296C2×C4—3141no—
297C2×C4—3141no—
298C2×C4—6281no—
299C2×C4—6281no—
300C2×C4—6281no—
301C2×C4—6281no—
302C2×C4—6281no—
303C2×C4—6281no—
304C2×C4—6281no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—81no—
320C22—6282no—
321C22—6281no—
322C22—6281no—
323C22—6281no—
324C22—12561no—
325C22#1909773911yescharacteristic
326C22#1909774311yescharacteristic
327C22#1909774411yescharacteristic
328C22#1909774411yescharacteristic
329C22#1909775311yescharacteristic
330C22#1909775411yescharacteristic
331C22#1909775711yescharacteristic
332C4—41no—
333C4—42no—
334C4—81no—
335C4—6282no—
336C4—6281no—
337C4—6281no—
338C4—6281no—
339C4—12561no—
340C2—21no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2—21no—
345C2—21no—
346C2—81no—
347C2—12561no—
348C2#2375718811yescharacteristic
349C2#2375745911yescharacteristic
350C2#2375746211yescharacteristic
subgroupG/H
centerC22#19097739
commutator, cosocleC24×C157C23
FrattiniC24C22×D157
Fitting#386×C157C2
socleC22×C157#120
prime radical#23757832C2
hypercenter#386D157
hypocenterC157#2044
LCSG ▷ C24×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C24 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#19097686C2semidirect
3. C2×#19097685C2semidirect
4. #23757152C2semidirect
5. #23757146C2semidirect
6. #23757039C2semidirect
7. #386×C157C2semidirect
53. C22⋊Dic157D4semidirect
175. C157#2044semidirect
181. #386D157semidirect

Maximal subgroups

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

subgroupindex
#237578322
C2×#190976862
C2×#190976852
#237571522
#237571462
#237570392
#386×C1572
#2044157