Finite Group Databasebeta

← #34109940

Subgroups — #34109940

Order 20096.

376 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 135 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#23757826C211yesmaximal, characteristic
2C2×#19097736C211yesmaximal, characteristic
3#23757583C212yesmaximal, complemented
4#23757579C212yesmaximal, complemented
5#23757567C212yesmaximal, complemented
6#23757557C212yesmaximal, complemented
7#23757313C211yesmaximal, characteristic
8#23757296C211yesmaximal, characteristic
9#23757283C211yesmaximal, characteristic
10#23757281C211yesmaximal, characteristic
11#428×C157C211yesmaximal, characteristic
12C22×Dic314C2211yescharacteristic
13C2×#14578938C2211yescharacteristic
14#19097770C2211yescharacteristic
15#19097770C2211yescharacteristic
16#19097762C2211yescharacteristic
17#19097762C2211yescharacteristic
18#19097748—21no—
19#19097748—21no—
20C4×Dic314—21no—
21C4×Dic314—21no—
22#19097736—21no—
23#19097736—21no—
24#19097736C2212yes—
25#19097736C2212yes—
26#19097730C2211yescharacteristic
27C2×#14578933—21no—
28C2×C4×Dic157—21no—
29C2×C4×Dic157C2211yescharacteristic
30#19097704C2212yes—
31#19097702C2212yes—
32#19097700C2212yes—
33#19097700C2212yes—
34#19097696C2212yes—
35#19097696C2212yes—
36#19097694C2212yes—
37#19097694C2212yes—
38C2×M16×C157C2211yescharacteristic
39#117×C157C2211yescharacteristic
40C2×C4⋊C4×C157C2211yescharacteristic
41#107×C157C2212yes—
42#106×C157C2212yes—
43C2×Dic314—21no—
44C2×Dic314—21no—
45C2×Dic314—21no—
46C2×Dic314—22no—
47C2×Dic314Q814yescomplemented
48C2×Dic314C2312yes—
49C22×Dic157—21no—
50C22×Dic157D411yescharacteristic
51C22×Dic157D411yescharacteristic
52#14578938—22no—
53C2×#11929434C2312yes—
54C4⋊Dic157—21no—
55C4⋊Dic157—21no—
56C4⋊Dic157C2311yescharacteristic
57C4⋊Dic157C2311yescharacteristic
58#14578933—22no—
59#14578933—22no—
60#14578933—21no—
61#14578933—21no—
62#14578933—21no—
63#14578933—21no—
64C22⋊Dic157—21no—
65C4×Dic157—22no—
66C4×Dic157—22no—
67C4×Dic157—21no—
68C4×Dic157D411yescharacteristic
69C4×Dic157D411yescharacteristic
70C4×Dic157C2311yescharacteristic
71C4×Dic157C2311yescharacteristic
72C22×C4×C157—21no—
73C22×C4×C157C2311yescharacteristic
74M16×C157—22no—
75C2×C8×C157C2312yes—
76C4⋊C4×C157—21no—
77C4⋊C4×C157C2311yescharacteristic
78C4⋊C4×C157C2312yes—
79C4⋊C4×C157C2311yescharacteristic
80C22⋊C4×C157—21no—
81C42×C157—21no—
82Dic314—24no—
83Dic314—22no—
84Dic314—42no—
85Dic314—42no—
86Dic314C2×Q814yes—
87C2×Dic157—24no—
88C2×Dic157—22no—
89C2×Dic157—22no—
90C2×Dic157—21no—
91C2×Dic157—21no—
92C2×Dic157—41no—
93C2×Dic157C2×D411yescharacteristic
94C2×Dic157C2×D411yescharacteristic
95C2×Dic157C2×D411yescharacteristic
96C2×Dic157C2×D411yescharacteristic
97#11929434—22no—
98C23×C157C2×D411yescharacteristic
99C2×C4×C157—21no—
100C2×C4×C157—21no—
101C2×C4×C157—22no—
102C2×C4×C157—42no—
103C2×C4×C157C2×D411yescharacteristic
104C2×C4×C157C2×Q812yes—
105C2×C4×C157#4112yes—
106C2×C4×C157C2411yescharacteristic
107C8×C157—22no—
108C22×C157—21no—
109C22×C157—21no—
110C22×C157C22⋊Q812yes—
111C22×C157C22×D411yescharacteristic
112C4×C157—41no—
113C4×C157—42no—
114C4×C157—41no—
115C4×C157C22⋊Q812yes—
116C4×C157C22×Q811yescharacteristic
117C4×C157C2×#4111yescharacteristic
118Dic157—24no—
119Dic157—21no—
120Dic157—21no—
121Dic157—21no—
122Dic157—41no—
123Dic157—41no—
124Dic157#13712yes—
125C2×C157—22no—
126C2×C157C2×C22⋊Q811yescharacteristic
127C2×C157C2×#13711yescharacteristic
128C2×C157#57511yescharacteristic
129C157#307811yescharacteristic, complemented, Sylow
130#3078—1571nomaximal, Sylow
131#533—1571no—
132C2×C4×Q8—1571no—
133#479—1572no—
134#479—1572no—
135#477—1572no—
136#475—1572no—
137#428—1571no—
138#428D15711yescharacteristic
139#423—1571no—
140#419—1571no—
141#419—1571no—
142C22×Q8—1571no—
143C2×M16—1571no—
144C2×M16C2×D15711yescharacteristic
145C4⋊Q8—1571no—
146C4⋊Q8—1571no—
147#125—1571no—
148#125—1571no—
149C22⋊Q8—3141no—
150C22⋊Q8—3141no—
151C4×Q8—3141no—
152C4×Q8—3141no—
153C4×Q8—3141no—
154C4×Q8—3141no—
155C4×Q8—1572no—
156C4×Q8—1572no—
157#117—1571no—
158#117C2×D15711yescharacteristic
159C2×C4⋊C4—3141no—
160C2×C4⋊C4C2×D15711yescharacteristic
161C2×C42—3141no—
162C2×C42—1571no—
163#107—1572no—
164#107C2×D15712yes—
165#106—1572no—
166#106C2×D15712yes—
167C4⋊C8—1572no—
168C4⋊C8—1572no—
169#103—1572no—
170#103—1572no—
171#103—1572no—
172#103—1572no—
173C2×Q8—3141no—
174C2×Q8—3141no—
175C2×Q8—3141no—
176C2×Q8—3142no—
177C2×Q8—1574no—
178C2×Q8—1572no—
179C22×C4—21no—
180C22×C4—3141no—
181C22×C4—1571no—
182C22×C4—1571no—
183C22×C4C22×D15711yescharacteristic
184M16—22no—
185M16—3142no—
186C2×C8—1572no—
187C2×C8C22×D15712yes—
188C4⋊C4—21no—
189C4⋊C4—3142no—
190C4⋊C4—3142no—
191C4⋊C4—3141no—
192C4⋊C4—3141no—
193C4⋊C4—3141no—
194C4⋊C4—3141no—
195C4⋊C4—3141no—
196C4⋊C4—3141no—
197C4⋊C4—1571no—
198C4⋊C4—1571no—
199C4⋊C4C22×D15711yescharacteristic
200C4⋊C4C22×D15711yescharacteristic
201C4⋊C4C22×D15712yes—
202C22⋊C4—21no—
203C22⋊C4—3141no—
204C42—21no—
205C42—3142no—
206C42—3142no—
207C42—3141no—
208C42—1571no—
209C42—1571no—
210C42—1571no—
211C42—1571no—
212C23D4×D15711yescharacteristic
213Q8—3144no—
214Q8—3142no—
215Q8—6282no—
216Q8—6282no—
217Q8—1574no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—42no—
222C2×C4—3144no—
223C2×C4—3142no—
224C2×C4—3142no—
225C2×C4—3141no—
226C2×C4—3141no—
227C2×C4—6281no—
228C2×C4—1571no—
229C2×C4—1571no—
230C2×C4—1571no—
231C2×C4—1571no—
232C2×C4D4×D15711yescharacteristic
233C2×C4Q8×D15712yes—
234C2×C4#1457895412yes—
235C2×C4C23×D15711yescharacteristic
236C8—22no—
237C8—3142no—
238C22—21no—
239C22—21no—
240C22#1909774912yes—
241C22C2×D4×D15711yescharacteristic
242C4—41no—
243C4—42no—
244C4—41no—
245C4—3144no—
246C4—3141no—
247C4—3141no—
248C4—3141no—
249C4—6281no—
250C4—6281no—
251C4—1572no—
252C4#1909774912yes—
253C4C2×Q8×D15711yescharacteristic
254C4C2×#1457895411yescharacteristic
255C2—22no—
256C2C2×#1909774911yescharacteristic
257C2#137×D15711yescharacteristic
258C2#2375806911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#428×C157C2
socleC22×C157C22×D4
prime radicalC23×C157C2×D4
hypercenter#428D157
hypocenterC157#3078
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #23757583C2semidirect
4. #23757579C2semidirect
5. #23757567C2semidirect
6. #23757557C2semidirect
47. C2×Dic314Q8semidirect
129. C157#3078semidirect

Maximal subgroups

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

subgroupindex
#237578262
C2×#190977362
#237575832
#237575792
#237575672
#237575572
#237573132
#237572962
#237572832
#237572812
#428×C1572
#3078157