Finite Group Databasebeta

← #34110268

Subgroups — #34110268

Order 20096.

380 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 121 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#23757828C211yesmaximal, characteristic
2C4×Q8×D157C211yesmaximal, characteristic
3#23757605C212yesmaximal
4#23757555C211yesmaximal, characteristic, complemented
5#23757484C211yesmaximal, characteristic, complemented
6#23757483C211yesmaximal, characteristic, complemented
7#23757430C212yesmaximal
8#23757379C212yesmaximal, complemented
9#23757377C212yesmaximal, complemented
10#23757329C211yesmaximal, characteristic
11#475×C157C211yesmaximal, characteristic, complemented
12C2×Q8×D157C2211yescharacteristic
13C2×Dic628C2212yes—
14C2×#14578944C2212yes—
15#19097771C2211yescharacteristic
16#19097762—21no—
17#19097751C2212yes—
18#19097747—21no—
19Q8×Dic157—21no—
20Q8×Dic157C2211yescharacteristic
21C4×Dic314—21no—
22C4×Dic314C2211yescharacteristic
23#19097736—21no—
24#19097736C2212yes—
25#19097736C2211yescharacteristic
26#19097729—21no—
27C4⋊C4×D157—21no—
28C4⋊C4×D157C2211yescharacteristic
29C42×D157—21no—
30C42×D157C2211yescharacteristic
31#19097702C2212yes—
32C4⋊#11929434C2211yescharacteristic
33#19097696C2212yes—
34#19097694C2212yes—
35#19097694C2212yes—
36#19097683C2212yes—
37#19097681C2212yes—
38C4⋊Q8×C157C2211yescharacteristic
39C4×Q8×C157C2211yescharacteristic
40#106×C157C2212yes—
41C4⋊C8×C157C2211yescharacteristic
42#103×C157C2212yes—
43C2×Dic314—21no—
44C2×Dic314C2312yes—
45C2×Dic314C2311yescharacteristic
46Q8×D157—22no—
47Q8×D157—22no—
48C2×C4×D157—21no—
49C2×C4×D157—21no—
50C2×C4×D157D411yescharacteristic
51C2×C4×D157D411yescharacteristic
52C2×C4×D157C2311yescharacteristic
53Dic628—24no—
54#14578944—24no—
55C2×#11929434C2312yes—
56C4⋊Dic157—21no—
57C4⋊Dic157—22no—
58C4⋊Dic157—21no—
59C4⋊Dic157—21no—
60C4⋊Dic157C2311yescharacteristic
61#14578933—21no—
62#14578933—21no—
63#14578933—22no—
64#14578933—21no—
65#14578931—22no—
66C4×Dic157—21no—
67C4×Dic157—21no—
68C4×Dic157—22no—
69C4×Dic157—21no—
70C4×Dic157D411yescharacteristic
71C4×Dic157D411yescharacteristic
72C4×Dic157C2311yescharacteristic
73C2×Q8×C157—21no—
74C2×Q8×C157C2311yescharacteristic
75C2×C8×C157C2312yes—
76C4⋊C4×C157—21no—
77C4⋊C4×C157—21no—
78C4⋊C4×C157C2312yes—
79C4⋊C4×C157C2311yescharacteristic
80C42×C157—21no—
81C42×C157C2311yescharacteristic
82C22×D157C2×D411yescharacteristic
83Dic314—24no—
84Dic314—21no—
85Dic314—22no—
86Dic314—22no—
87Dic314#4112yes—
88C2×Dic157—21no—
89C2×Dic157—21no—
90C2×Dic157—22no—
91C2×Dic157C2×D411yescharacteristic
92C2×Dic157C2×D411yescharacteristic
93C2×Dic157C2×D411yescharacteristic
94C4×D157—24no—
95C4×D157—22no—
96C4×D157—22no—
97C4×D157—21no—
98C4×D157—21no—
99C4×D157—42no—
100#11929434—22no—
101Q8×C157—21no—
102Q8×C157—42no—
103Q8×C157#4112yes—
104C2×C4×C157—21no—
105C2×C4×C157—21no—
106C2×C4×C157—22no—
107C2×C4×C157C2×D411yescharacteristic
108C2×C4×C157C2×D411yescharacteristic
109C2×C4×C157C2411yescharacteristic
110C8×C157—22no—
111C22×C157C22×D411yescharacteristic
112C2×D157—21no—
113C2×D157—21no—
114C2×D157#13712yes—
115C4×C157—22no—
116C4×C157—21no—
117C4×C157—42no—
118C4×C157—41no—
119C4×C157#13712yes—
120C4×C157C2×#4111yescharacteristic
121C4×C157C2×#4111yescharacteristic
122Dic157—22no—
123Dic157—21no—
124Dic157—21no—
125Dic157—21no—
126Dic157—42no—
127Dic157—41no—
128C2×C157#52511yescharacteristic
129C2×C157C2×#13711yescharacteristic
130C2×C157C2×#13711yescharacteristic
131D157—22no—
132C157#311311yescharacteristic, complemented, Sylow
133#3113—1571nomaximal, Sylow
134#533—1571no—
135C2×C4×Q8—1571no—
136#483—1572no—
137#475—1571no—
138#475D15711yescharacteristic, complemented
139#462—1571no—
140#462—1571no—
141#451—1572no—
142#441—1572no—
143#441—1572no—
144#432—1571no—
145C22×Q8—1571no—
146C2×Q16—1572no—
147C2×Q16—1572no—
148C4⋊Q8—1571no—
149C4⋊Q8C2×D15711yescharacteristic
150#125—3141no—
151C22⋊Q8—3141no—
152C22⋊Q8—1572no—
153C4×Q8—3141no—
154C4×Q8—3141no—
155C4×Q8—3141no—
156C4×Q8—1571no—
157C4×Q8—1571no—
158C4×Q8—1572no—
159C4×Q8—1571no—
160C4×Q8C2×D15711yescharacteristic
161#117—3141no—
162C2×C4⋊C4—3141no—
163C2×C4⋊C4—1571no—
164C2×C42—3141no—
165C2×C42—1571no—
166#106—1572no—
167#106C2×D15712yes—
168C4⋊C8—1571no—
169C4⋊C8C2×D15711yescharacteristic
170#103—1572no—
171#103—1572no—
172#103—1572no—
173#103C2×D15712yes—
174C22⋊C8—1572no—
175#97—1572no—
176C2×Q8—21no—
177C2×Q8—3141no—
178C2×Q8—3142no—
179C2×Q8—3142no—
180C2×Q8—1572no—
181C2×Q8—1571no—
182C2×Q8C22×D15711yescharacteristic
183C22×C4—3141no—
184C22×C4—3141no—
185C22×C4—1571no—
186C22×C4—1571no—
187C22×C4—1571no—
188Q16—3144no—
189Q16—3144no—
190C2×C8—1572no—
191C2×C8C22×D15712yes—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—3141no—
195C4⋊C4—3141no—
196C4⋊C4—3141no—
197C4⋊C4—3142no—
198C4⋊C4—3142no—
199C4⋊C4—3141no—
200C4⋊C4—3141no—
201C4⋊C4—3141no—
202C4⋊C4—1571no—
203C4⋊C4C22×D15711yescharacteristic
204C4⋊C4C22×D15712yes—
205C22⋊C4—3142no—
206C42—21no—
207C42—3141no—
208C42—3141no—
209C42—3142no—
210C42—3141no—
211C42—1571no—
212C42—1571no—
213C42—1571no—
214C42C22×D15711yescharacteristic
215C23—1571no—
216Q8—21no—
217Q8—42no—
218Q8—3144no—
219Q8—3141no—
220Q8—3142no—
221Q8—3142no—
222Q8—1572no—
223Q8#1457895512yes—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—3144no—
228C2×C4—3142no—
229C2×C4—3141no—
230C2×C4—3142no—
231C2×C4—3141no—
232C2×C4—3141no—
233C2×C4—3141no—
234C2×C4—3142no—
235C2×C4—6282no—
236C2×C4—1571no—
237C2×C4—1571no—
238C2×C4—1571no—
239C2×C4D4×D15711yescharacteristic
240C2×C4D4×D15711yescharacteristic
241C2×C4C23×D15711yescharacteristic
242C8—22no—
243C8—3142no—
244C22—3141no—
245C22—3141no—
246C22—1572no—
247C22C2×D4×D15711yescharacteristic
248C4—22no—
249C4—21no—
250C4—41no—
251C4—42no—
252C4—3142no—
253C4—3141no—
254C4—3141no—
255C4—3141no—
256C4—6282no—
257C4—6281no—
258C4#1909780012yes—
259C4#41×D15711yescharacteristic
260C4C2×#1457895511yescharacteristic
261C2—3142no—
262C2#2375778611yescharacteristic
263C2#137×D15711yescharacteristic
264C2C2×#1909780011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#475×C157C2
socleC22×C157C22×D4
prime radicalC22×D157C2×D4
hypercenter#475D157
hypocenterC157#3113
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
4. #23757555C2semidirect
5. #23757484C2semidirect
6. #23757483C2semidirect
8. #23757379C2semidirect
9. #23757377C2semidirect
11. #475×C157C2semidirect
132. C157#3113semidirect
138. #475D157semidirect

Maximal subgroups

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

subgroupindex
#237578282
C4×Q8×D1572
#237576052
#237575552
#237574842
#237574832
#237574302
#237573792
#237573772
#237573292
#475×C1572
#3113157