Finite Group Databasebeta

← #34103409

Subgroups — #34103409

Order 20096.

528 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 151 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×#19097736C211yesmaximal, characteristic, complemented
2C2×#19097729C211yesmaximal, characteristic
3#23757208C211yesmaximal, characteristic
4#23757207C212yesmaximal, complemented
5#23757197C212yesmaximal
6#23757193C211yesmaximal, characteristic, complemented
7#23757186C211yesmaximal, characteristic
8#23757124C211yesmaximal, characteristic, complemented
9#23757124C212yesmaximal, complemented
10#23757120C212yesmaximal
11#400×C157C211yesmaximal, characteristic, complemented
12C22×Dic314C2211yescharacteristic
13C22×C4×D157C2211yescharacteristic
14#19097736—24no—
15#19097729—24no—
16C2×C4⋊Dic157C2211yescharacteristic
17C2×#14578933C2212yes—
18C2×#14578933C2212yes—
19C2×#14578933C2212yes—
20C2×#14578933C2212yes—
21C2×#14578931C2212yes—
22C2×#14578931C2212yes—
23C2×#14578931C2211yescharacteristic
24C2×#14578931C2211yescharacteristic
25C2×C4×Dic157C2212yes—
26C2×C4×Dic157C2212yes—
27C2×C4×Dic157C2211yescharacteristic
28C2×C4×Dic157C2211yescharacteristic
29#19097678C2212yes—
30#19097678C2212yes—
31#19097678C2211yescharacteristic
32C2×C4⋊C4×C157C2211yescharacteristic
33C2×C4⋊C4×C157C2211yescharacteristic
34#95×C157C2212yes—
35#95×C157C2211yescharacteristic
36#95×C157C2212yes—
37C23×D157C2311yescharacteristic
38C2×Dic314—24no—
39C2×Dic314—22no—
40C22×Dic157C2311yescharacteristic
41C22×Dic157C2311yescharacteristic
42C22×Dic157C2312yes—
43C22×Dic157C2312yes—
44C22×Dic157C2311yescharacteristic
45C2×C4×D157—22no—
46C2×C4×D157—22no—
47C2×C4×D157Q814yescomplemented
48C4⋊Dic157—22no—
49#14578933—24no—
50#14578933—24no—
51#14578933—24no—
52#14578933—24no—
53#14578931—22no—
54#14578931—24no—
55#14578931—24no—
56#14578931—22no—
57C4×Dic157—22no—
58C4×Dic157—24no—
59C4×Dic157—24no—
60C4×Dic157D414yes—
61C22×C4×C157C2312yes—
62C22×C4×C157C2311yescharacteristic
63C22×C4×C157C2312yes—
64C22×C4×C157C2311yescharacteristic
65C22×C4×C157C2311yescharacteristic
66C4⋊C4×C157—22no—
67C4⋊C4×C157—22no—
68C22×D157—21no—
69C22×D157—21no—
70C22×D157—22no—
71C22×D157—22no—
72C22×D157C2×Q812yes—
73Dic314—28no—
74C2×Dic157—22no—
75C2×Dic157—22no—
76C2×Dic157—22no—
77C2×Dic157—22no—
78C2×Dic157—22no—
79C2×Dic157—21no—
80C2×Dic157—22no—
81C2×Dic157C2×D412yes—
82C2×Dic157C2×D412yes—
83C2×Dic157C2×Q812yes—
84C2×Dic157#4114yes—
85C2×Dic157#4114yes—
86C2×Dic157#4112yes—
87C4×D157—28no—
88C23×C157C2411yescharacteristic
89C2×C4×C157—21no—
90C2×C4×C157—21no—
91C2×C4×C157—22no—
92C2×C4×C157—21no—
93C2×C4×C157—22no—
94C2×C4×C157—22no—
95C2×C4×C157—22no—
96C2×C4×C157—22no—
97C2×C4×C157—22no—
98C2×C4×C157—22no—
99C2×C4×C157—22no—
100C2×C4×C157C2×D412yes—
101C2×C4×C157C2×Q812yes—
102C22×C157C22×D411yescharacteristic
103C22×C157C22×Q811yescharacteristic
104C22×C157C2×#4111yescharacteristic
105C22×C157C2×#4112yes—
106C22×C157C2×#4112yes—
107C2×D157—22no—
108C2×D157—24no—
109C2×D157—41no—
110C2×D157—42no—
111C2×D157—41no—
112C4×C157—22no—
113C4×C157—22no—
114C4×C157—41no—
115C4×C157—42no—
116C4×C157—42no—
117Dic157—24no—
118Dic157—24no—
119Dic157—22no—
120Dic157—41no—
121Dic157C22⋊Q814yes—
122C2×C157C2×C22⋊Q811yescharacteristic
123C2×C157#52511yescharacteristic
124C2×C157#52912yes—
125C2×C157#53211yescharacteristic
126C2×C157#53312yes—
127D157—42no—
128C157#239611yescharacteristic, complemented, Sylow
129#2396—1571nomaximal, Sylow
130C2×C4×Q8—1571no—
131C2×#117—1571no—
132#400D15711yescharacteristic, complemented
133#399—1571no—
134#398—1572no—
135#396—1572no—
136#395—1571no—
137#393—1571no—
138C4×C4⋊C4—1572no—
139C4×C4⋊C4—1571no—
140C4×C22⋊C4—1572no—
141C22×Q8—1571no—
142C23×C4—1571no—
143C4×Q8—3144no—
144#117—3144no—
145C2×C4⋊C4—1571no—
146C2×C4⋊C4—1572no—
147C2×C4⋊C4—1572no—
148C2×C4⋊C4—1572no—
149C2×C4⋊C4—1572no—
150C2×C4⋊C4C2×D15711yescharacteristic
151C2×C4⋊C4C2×D15711yescharacteristic
152C2×C22⋊C4—1571no—
153C2×C22⋊C4—1571no—
154C2×C22⋊C4—1572no—
155C2×C22⋊C4—1572no—
156C2×C42—1571no—
157C2×C42—1572no—
158C2×C42—1572no—
159C2×C42—1571no—
160#95—1572no—
161#95—1572no—
162#95—1571no—
163#95C2×D15712yes—
164#95C2×D15711yescharacteristic
165#95C2×D15712yes—
166C24—1571no—
167C2×Q8—3144no—
168C2×Q8—3142no—
169C22×C4—3142no—
170C22×C4—3142no—
171C22×C4—1571no—
172C22×C4—1572no—
173C22×C4—1572no—
174C22×C4—1571no—
175C22×C4—1574no—
176C22×C4—1571no—
177C22×C4C22×D15712yes—
178C22×C4C22×D15711yescharacteristic
179C22×C4C22×D15712yes—
180C22×C4C22×D15711yescharacteristic
181C22×C4C22×D15711yescharacteristic
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—3142no—
185C4⋊C4—3144no—
186C4⋊C4—3144no—
187C4⋊C4—3144no—
188C4⋊C4—3144no—
189C22⋊C4—3142no—
190C22⋊C4—3144no—
191C22⋊C4—3144no—
192C22⋊C4—3142no—
193C42—3142no—
194C42—3144no—
195C42—3144no—
196C42—1574no—
197C23—3141no—
198C23—3142no—
199C23—3142no—
200C23—3141no—
201C23—1572no—
202C23C23×D15711yescharacteristic
203Q8—3148no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—3148no—
216C2×C4—3142no—
217C2×C4—3142no—
218C2×C4—3142no—
219C2×C4—3142no—
220C2×C4—3142no—
221C2×C4—3141no—
222C2×C4—3142no—
223C2×C4—1572no—
224C2×C4—1574no—
225C2×C4—1574no—
226C2×C4—1572no—
227C2×C4—1572no—
228C2×C4—1572no—
229C2×C4D4×D15712yes—
230C2×C4Q8×D15712yes—
231C22—3144no—
232C22—3142no—
233C22—6281no—
234C22—6282no—
235C22—6281no—
236C22C2×D4×D15711yescharacteristic
237C22C2×Q8×D15711yescharacteristic
238C22#41×D15712yes—
239C22#41×D15712yes—
240C22#41×D15711yescharacteristic
241C4—22no—
242C4—22no—
243C4—41no—
244C4—42no—
245C4—42no—
246C4—3144no—
247C4—3144no—
248C4—3142no—
249C4—6281no—
250C4—1574no—
251C2—6282no—
252C2C22⋊Q8×D15711yescharacteristic
253C2#2375778111yescharacteristic
254C2#2375780612yes—
255C2#2375782211yescharacteristic
256C2#2375782912yes—
subgroupG/H
center, FrattiniC23C23×D157
commutator, socle, cosocleC23×C157C24
Fitting#400×C157C2
prime radicalC23×D157C23
hypercenter#400D157
hypocenterC157#2396
LCSG ▷ C23×C157 ▷ C157
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#19097736C2semidirect
4. #23757207C2semidirect
6. #23757193C2semidirect
8. #23757124C2semidirect
9. #23757124C2semidirect
11. #400×C157C2semidirect
47. C2×C4×D157Q8semidirect
128. C157#2396semidirect
132. #400D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977362
C2×#190977292
#237572082
#237572072
#237571972
#237571932
#237571862
#237571242
#237571202
#400×C1572
#2396157