Finite Group Databasebeta

← #34100110

Subgroups — #34100110

Order 20096.

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

normal subgroups only

This group has 536 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×#19097818C211yesmaximal, characteristic
2#23757290C211yesmaximal, characteristic, complemented
3#23757283C211yesmaximal, characteristic, complemented
4#23757281C211yesmaximal, characteristic, complemented
5#23757255C211yesmaximal, characteristic, complemented
6#23757246C211yesmaximal, characteristic, complemented
7#421×C157C211yesmaximal, characteristic, complemented
8#19097818—22no—
9#19097818—22no—
10#19097818—22no—
11#19097818—22no—
12C2×#14578954—21no—
13C2×#14578954C2211yescharacteristic
14C2×#14578953—22no—
15C2×#14578953C412yescomplemented
16C22×Dic314—21no—
17C22×Dic314C2211yescharacteristic
18C2×Q8×D157C412yescomplemented
19C2×#14578938C2211yescharacteristic
20#19097730C2211yescharacteristic
21#19097698—22no—
22#19097696—22no—
23#19097694—22no—
24#19097689—21no—
25#19097687—21no—
26#19097685—22no—
27C2×#41×C157C2211yescharacteristic
28C2×M16×C157C2211yescharacteristic
29#117×C157C2211yescharacteristic
30#104×C157—22no—
31#14578954—24no—
32#14578954—24no—
33#14578954—21no—
34#14578954—21no—
35#14578954—21no—
36#14578954—21no—
37#14578953—28no—
38#14578953—28no—
39#14578953—24no—
40#14578953—24no—
41C2×Dic314—24no—
42C2×Dic314—24no—
43C2×Dic314—22no—
44C2×Dic314—21no—
45C2×Dic314—21no—
46C2×Dic314—21no—
47C2×Dic314C2×C412yes—
48C2×Dic314D412yes—
49C2×Dic314D412yes—
50C2×Dic314D411yescharacteristic
51Q8×D157—24no—
52Q8×D157—24no—
53C2×D628D411yescharacteristic
54C2×C22⋊D157—22no—
55C2×C22⋊D157D412yes—
56C22×Dic157—22no—
57C22×Dic157—21no—
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×D157C22⋊C412yes—
82Dic314—28no—
83Dic314—22no—
84Dic314—22no—
85Dic314—21no—
86Dic314—22no—
87Dic314—21no—
88Dic314C22⋊C414yes—
89D628—21no—
90D628—21no—
91C22⋊D157—28no—
92C22⋊D157—22no—
93C22⋊D157—22no—
94C2×Dic157—28no—
95C2×Dic157—22no—
96C2×Dic157—22no—
97C2×Dic157—22no—
98C2×Dic157—41no—
99C2×Dic157C22⋊C412yes—
100C4×D157—28no—
101C4×D157—22no—
102C4×D157—22no—
103#11929434—41no—
104C23×C157—21no—
105C23×C157C2×D411yescharacteristic
106Q8×C157—21no—
107Q8×C157C22⋊C412yes—
108D4×C157—22no—
109D4×C157—22no—
110D4×C157—21no—
111D4×C157C22⋊C412yes—
112C2×C4×C157—21no—
113C2×C4×C157—22no—
114C2×C4×C157—22no—
115C2×C4×C157—41no—
116C2×C4×C157C22×C411yescharacteristic
117C2×C4×C157C2×D411yescharacteristic
118C2×C4×C157C2×D411yescharacteristic
119C2×C4×C157C2×D411yescharacteristic
120C2×C4×C157C2×D411yescharacteristic
121C2×C4×C157C2×D411yescharacteristic
122C8×C157—41no—
123C22×C157—22no—
124C22×C157—41no—
125C22×C157—42no—
126C22×C157C2×C22⋊C411yescharacteristic
127C22×C157#12011yescharacteristic
128C22×C157#12011yescharacteristic
129C2×D157—22no—
130C2×D157—42no—
131C4×C157—22no—
132C4×C157—81no—
133C4×C157C2×C22⋊C411yescharacteristic
134C4×C157C2×C22⋊C411yescharacteristic
135C4×C157#12011yescharacteristic
136C4×C157#12011yescharacteristic
137Dic157—24no—
138Dic157—22no—
139Dic157—81no—
140C2×C157—21no—
141C2×C157—21no—
142C2×C157—21no—
143C2×C157—42no—
144C2×C157#37911yescharacteristic
145D157—42no—
146C157#179611yescharacteristic, complemented, Sylow
147#1796—1571nomaximal, Sylow
148C2×2−1+4—1571no—
149#421—1571no—
150#421D15711yescharacteristic, complemented
151#419—1571no—
152#419—1571no—
153#413—1571no—
154#410—1571no—
1552−1+4—3142no—
1562−1+4—3142no—
1572−1+4—3142no—
1582−1+4—3142no—
159C2×#41—3141no—
160C2×#41—3142no—
161C2×#41—1572no—
162C2×#41—1571no—
163C2×#41C2×D15711yescharacteristic
164C22×Q8—3141no—
165C22×Q8—1572no—
166C22×Q8—1571no—
167C2×M16—1571no—
168C2×M16C2×D15711yescharacteristic
169#117—1571no—
170#117C2×D15711yescharacteristic
171#104—22no—
172#104—3142no—
173#103—3142no—
174#103—3142no—
175#101—3141no—
176#100—3141no—
177#99—3142no—
178#41—21no—
179#41—21no—
180#41—3144no—
181#41—3144no—
182#41—3148no—
183#41—3148no—
184#41—3144no—
185#41—3144no—
186#41—3141no—
187#41—3141no—
188#41—3141no—
189#41—3141no—
190#41C22⋊D15712yes—
191#41D62812yes—
192C2×Q8—3144no—
193C2×Q8—3144no—
194C2×Q8—3142no—
195C2×Q8—3144no—
196C2×Q8—3144no—
197C2×Q8—3141no—
198C2×Q8—3141no—
199C2×Q8—3141no—
200C2×Q8—1572no—
201C2×Q8—1572no—
202C2×Q8—1572no—
203C2×Q8—1571no—
204C2×Q8C4×D15711yescharacteristic
205C2×D4—21no—
206C2×D4—3142no—
207C2×D4—1572no—
208C2×D4—1571no—
209C2×D4C4×D15711yescharacteristic
210C22×C4—21no—
211C22×C4—3142no—
212C22×C4—3141no—
213C22×C4—3142no—
214C22×C4—1572no—
215C22×C4C22×D15711yescharacteristic
216M16—21no—
217M16—21no—
218M16—3141no—
219M16—3141no—
220C2×C8—21no—
221C2×C8—3141no—
222C4⋊C4—21no—
223C4⋊C4—3141no—
224C22⋊C4—21no—
225C22⋊C4—3141no—
226C42—21no—
227C42—3141no—
228C23—21no—
229C23—1572no—
230C23D4×D15711yescharacteristic
231Q8—21no—
232Q8—3148no—
233Q8—3142no—
234Q8—3142no—
235Q8—3141no—
236Q8—3142no—
237Q8—3141no—
238Q8—1574no—
239Q8#1457893112yes—
240D4—22no—
241D4—22no—
242D4—21no—
243D4—3148no—
244D4—3142no—
245D4—3142no—
246D4—3141no—
247D4—3141no—
248D4#1457893112yes—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—3148no—
254C2×C4—3148no—
255C2×C4—3142no—
256C2×C4—3142no—
257C2×C4—3142no—
258C2×C4—3142no—
259C2×C4—3142no—
260C2×C4—6281no—
261C2×C4—1572no—
262C2×C4C2×C4×D15711yescharacteristic
263C2×C4D4×D15711yescharacteristic
264C2×C4D4×D15711yescharacteristic
265C2×C4D4×D15711yescharacteristic
266C2×C4C2×C22⋊D15711yescharacteristic
267C2×C4C2×D62811yescharacteristic
268C8—41no—
269C8—6281no—
270C22—22no—
271C22—41no—
272C22—42no—
273C22—3142no—
274C22—6282no—
275C22C22⋊C4×D15711yescharacteristic
276C22#1909773911yescharacteristic
277C22#1909774111yescharacteristic
278C4—22no—
279C4—81no—
280C4—3144no—
281C4—3142no—
282C4—12561no—
283C4C22⋊C4×D15711yescharacteristic
284C4C2×#1457893111yescharacteristic
285C4#1909773911yescharacteristic
286C4#1909774111yescharacteristic
287C2—21no—
288C2—21no—
289C2—21no—
290C2—42no—
291C2—6282no—
292C2#2375712811yescharacteristic
subgroupG/H
centerC2#23757128
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#421×C157C2
socleC2×C157#379
prime radicalC2×#19097818C2
cosocleC22×C4×C157C23
hypercenter#421D157
hypocenterC157#1796
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. #23757283C2semidirect
4. #23757281C2semidirect
5. #23757255C2semidirect
6. #23757246C2semidirect
7. #421×C157C2semidirect
15. C2×#14578953C4semidirect
18. C2×Q8×D157C4semidirect
146. C157#1796semidirect
150. #421D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978182
#237572902
#237572832
#237572812
#237572552
#237572462
#421×C1572
#1796157