Finite Group Databasebeta

← #34110534

Subgroups — #34110534

Order 20096.

392 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 125 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#23757819C211yesmaximal, characteristic
2C2×#19097762C211yesmaximal, characteristic, complemented
3#23757640C212yesmaximal, complemented
4#23757636C212yesmaximal, complemented
5#23757629C212yesmaximal, complemented
6#23757626C212yesmaximal, complemented
7#23757275C211yesmaximal, characteristic, complemented
8#23757274C211yesmaximal, characteristic, complemented
9#23757272C211yesmaximal, characteristic, complemented
10M16×Dic157C211yesmaximal, characteristic, complemented
11#417×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13C2×#14578938C2211yescharacteristic
14#19097769C2211yescharacteristic
15#19097767C2211yescharacteristic
16#19097762—21no—
17#19097762—21no—
18#19097762C2212yes—
19#19097762C2212yes—
20#19097760C2211yescharacteristic
21#19097758C2211yescharacteristic
22#19097743—21no—
23#19097743—21no—
24C4×C22⋊D157—21no—
25C4×C22⋊D157—21no—
26C2×C4⋊Dic157—21no—
27C2×C4⋊Dic157C2211yescharacteristic
28C2×#14578933—21no—
29C2×C4×Dic157C2211yescharacteristic
30#19097696C2212yes—
31#19097695C2212yes—
32#19097694C2212yes—
33#19097693C2212yes—
34#19097692C2212yes—
35#19097691C2212yes—
36#19097681C2212yes—
37C8×Dic157C2212yes—
38C2×#41×C157C2211yescharacteristic
39C2×M16×C157C2211yescharacteristic
40C2×C4⋊C4×C157C2211yescharacteristic
41#103×C157C2212yes—
42#102×C157C2212yes—
43#14578954—22no—
44#14578954—22no—
45C2×Dic314C2311yescharacteristic
46C2×D628C2311yescharacteristic
47C2×C22⋊D157—21no—
48C2×C22⋊D157—21no—
49C2×C22⋊D157—21no—
50C22×Dic157—21no—
51C22×Dic157D411yescharacteristic
52C22×Dic157D411yescharacteristic
53C2×C4×D157—21no—
54#14578938—22no—
55C2×#11929434C2312yes—
56C4⋊Dic157—22no—
57C4⋊Dic157—22no—
58C4⋊Dic157—21no—
59C4⋊Dic157C2312yes—
60#14578933—22no—
61#14578933—22no—
62#14578933—21no—
63#14578933—21no—
64C22⋊Dic157—21no—
65C22⋊Dic157—21no—
66#14578931—21no—
67#14578931—21no—
68C4×Dic157D411yescharacteristic
69C4×Dic157D411yescharacteristic
70C4×Dic157C2311yescharacteristic
71C4×Dic157C2311yescharacteristic
72#41×C157—22no—
73#41×C157—22no—
74C2×Q8×C157C2311yescharacteristic
75C2×D4×C157—21no—
76C2×D4×C157C2311yescharacteristic
77C22×C4×C157—21no—
78C22×C4×C157—21no—
79C22×C4×C157C2311yescharacteristic
80M16×C157—22no—
81C2×C8×C157C2312yes—
82C4⋊C4×C157—21no—
83C4⋊C4×C157C2312yes—
84C22×D157—21no—
85Dic314—21no—
86Dic314—21no—
87D628—21no—
88D628—21no—
89C22⋊D157—42no—
90C22⋊D157—42no—
91C22⋊D157—42no—
92C2×Dic157—21no—
93C2×Dic157—22no—
94C2×Dic157—41no—
95C2×Dic157—41no—
96C2×Dic157—42no—
97C2×Dic157C2×D411yescharacteristic
98C2×Dic157C2×D411yescharacteristic
99C2×Dic157C2×D411yescharacteristic
100C2×Dic157C2×D411yescharacteristic
101C4×D157—42no—
102#11929434—22no—
103C23×C157—21no—
104C23×C157C2×D411yescharacteristic
105Q8×C157—21no—
106Q8×C157—21no—
107D4×C157—21no—
108D4×C157—21no—
109D4×C157—42no—
110C2×C4×C157—22no—
111C2×C4×C157—21no—
112C2×C4×C157—42no—
113C2×C4×C157—42no—
114C2×C4×C157C2×D411yescharacteristic
115C2×C4×C157#4112yes—
116C2×C4×C157#4112yes—
117C2×C4×C157C2411yescharacteristic
118C8×C157—22no—
119C22×C157—21no—
120C22×C157—21no—
121C22×C157—41no—
122C22×C157—41no—
123C22×C157—41no—
124C22×C157#12412yes—
125C22×C157C22×D411yescharacteristic
126C2×D157—41no—
127C2×D157—41no—
128C2×D157—41no—
129C4×C157—41no—
130C4×C157—42no—
131C4×C157#12412yes—
132C4×C157C2×#4111yescharacteristic
133C4×C157C2×#4111yescharacteristic
134Dic157—21no—
135Dic157—21no—
136Dic157—21no—
137Dic157—21no—
138Dic157—41no—
139Dic157—42no—
140C2×C157—22no—
141C2×C157—81no—
142C2×C157C2×#12411yescharacteristic
143C2×C157#57511yescharacteristic
144C2×C157#57511yescharacteristic
145D157—81no—
146C157#314211yescharacteristic, complemented, Sylow
147#3142—1571nomaximal, Sylow
148#532—1571no—
149C2×#125—1571no—
150#491—1572no—
151#490—1572no—
152#488—1572no—
153#488—1572no—
154#417—1571no—
155#417—1571no—
156#417—1571no—
157#417D15711yescharacteristic, complemented
158C4×M16—1571no—
159C2×#41—1571no—
160C2×#41C2×D15711yescharacteristic
161C2×M16—1571no—
162C2×M16C2×D15711yescharacteristic
163C4⋊Q8—1571no—
164#127—1571no—
165#125—3141no—
166#125—3141no—
167#125—1572no—
168#125—1572no—
169#124—1571no—
170#124—1571no—
171#121—3141no—
172#121—3141no—
173C4×D4—3141no—
174C4×D4—3141no—
175C2×C4⋊C4—3141no—
176C2×C4⋊C4—3141no—
177C2×C4⋊C4—1571no—
178C2×C4⋊C4C2×D15711yescharacteristic
179C2×C42—1571no—
180#103—1572no—
181#103—1572no—
182#103—1572no—
183#103C2×D15712yes—
184#102—1572no—
185#102—1572no—
186#102—1572no—
187#102C2×D15712yes—
188#97—1572no—
189C4×C8—1572no—
190#41—22no—
191#41—22no—
192#41—3142no—
193#41—3142no—
194C2×Q8—1571no—
195C2×Q8C22×D15711yescharacteristic
196C2×D4—21no—
197C2×D4—3141no—
198C2×D4—3141no—
199C2×D4—3141no—
200C2×D4—1571no—
201C2×D4C22×D15711yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—3141no—
205C22×C4—3141no—
206C22×C4—1571no—
207C22×C4—1571no—
208C22×C4C22×D15711yescharacteristic
209M16—22no—
210M16—3142no—
211C2×C8—1572no—
212C2×C8C22×D15712yes—
213C4⋊C4—21no—
214C4⋊C4—3142no—
215C4⋊C4—3142no—
216C4⋊C4—3142no—
217C4⋊C4—3142no—
218C4⋊C4—3141no—
219C4⋊C4—3141no—
220C4⋊C4—3141no—
221C4⋊C4—1572no—
222C4⋊C4C22×D15712yes—
223C22⋊C4—3141no—
224C22⋊C4—3141no—
225C22⋊C4—3141no—
226C22⋊C4—3141no—
227C42—1571no—
228C42—1571no—
229C42—1571no—
230C42—1571no—
231C23—21no—
232C23—3141no—
233C23D4×D15711yescharacteristic
234Q8—21no—
235Q8—21no—
236Q8—3141no—
237Q8—3141no—
238D4—21no—
239D4—21no—
240D4—42no—
241D4—3141no—
242D4—3141no—
243D4—6282no—
244D4—6282no—
245D4—6282no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—3141no—
251C2×C4—3142no—
252C2×C4—6281no—
253C2×C4—6281no—
254C2×C4—6282no—
255C2×C4—6282no—
256C2×C4—1571no—
257C2×C4—1571no—
258C2×C4—1571no—
259C2×C4—1571no—
260C2×C4D4×D15711yescharacteristic
261C2×C4#1457895312yes—
262C2×C4#1457895412yes—
263C2×C4C23×D15711yescharacteristic
264C8—22no—
265C8—3142no—
266C22—21no—
267C22—21no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—6281no—
272C22—6281no—
273C22—6281no—
274C22#1909775712yes—
275C22C2×D4×D15711yescharacteristic
276C4—41no—
277C4—42no—
278C4—3141no—
279C4—3141no—
280C4—3141no—
281C4—3141no—
282C4—6281no—
283C4—6282no—
284C4#1909775712yes—
285C4C2×#1457895311yescharacteristic
286C4C2×#1457895411yescharacteristic
287C2—22no—
288C2—81no—
289C2—12561no—
290C2C2×#1909775711yescharacteristic
291C2#2375806911yescharacteristic
292C2#2375807211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#417×C157C2
socleC22×C157C22×D4
prime radical#23757819C2
hypercenter#417D157
hypocenterC157#3142
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#19097762C2semidirect
3. #23757640C2semidirect
4. #23757636C2semidirect
5. #23757629C2semidirect
6. #23757626C2semidirect
7. #23757275C2semidirect
8. #23757274C2semidirect
9. #23757272C2semidirect
10. M16×Dic157C2semidirect
11. #417×C157C2semidirect
146. C157#3142semidirect
157. #417D157semidirect

Maximal subgroups

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

subgroupindex
#237578192
C2×#190977622
#237576402
#237576362
#237576292
#237576262
#237572752
#237572742
#237572722
M16×Dic1572
#417×C1572
#3142157