Finite Group Databasebeta

← #34108472

Subgroups — #34108472

Order 20096.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#19097748C211yesmaximal, characteristic
2C22×C4×Dic157C211yesmaximal, characteristic
3#23757177C212yesmaximal, complemented
4#23757168C212yesmaximal, complemented
5#23757159C212yesmaximal, complemented
6#23757155C212yesmaximal, complemented
7#23757146C211yesmaximal, characteristic
8#23757134C211yesmaximal, characteristic
9#23757134C211yesmaximal, characteristic
10#23757131C211yesmaximal, characteristic
11#385×C157C211yesmaximal, characteristic
12C22×Dic314C2211yescharacteristic
13C23×Dic157C2211yescharacteristic
14C23×Dic157C2211yescharacteristic
15#19097748—24no—
16C2×C4⋊Dic157C2211yescharacteristic
17C2×#14578933C2212yes—
18C2×#14578933C2212yes—
19C2×#14578933C2211yescharacteristic
20C2×#14578933C2211yescharacteristic
21C2×C22⋊Dic157C2211yescharacteristic
22C2×C22⋊Dic157C2211yescharacteristic
23C2×C4×Dic157—24no—
24C2×C4×Dic157C2212yes—
25C2×C4×Dic157C2212yes—
26#19097678C2212yes—
27#19097678C2212yes—
28#19097678C2212yes—
29#19097678C2212yes—
30#19097678C2212yes—
31#19097678C2212yes—
32C23×C4×C157C2211yescharacteristic
33C2×C4⋊C4×C157C2212yes—
34C2×C22⋊C4×C157C2211yescharacteristic
35C2×C22⋊C4×C157C2211yescharacteristic
36#95×C157C2212yes—
37C2×Dic314—22no—
38C2×Dic314—24no—
39C22×Dic157—24no—
40C22×Dic157—22no—
41C22×Dic157—22no—
42C22×Dic157D414yes—
43C22×Dic157Q814yescomplemented
44C22×Dic157C2312yes—
45C22×Dic157C2311yescharacteristic
46C22×Dic157C2311yescharacteristic
47C22×Dic157C2311yescharacteristic
48C22×Dic157C2312yes—
49C22×Dic157C2311yescharacteristic
50C4⋊Dic157—22no—
51#14578933—22no—
52#14578933—22no—
53#14578933—24no—
54#14578933—24no—
55C22⋊Dic157—22no—
56C22⋊Dic157—22no—
57C4×Dic157—24no—
58C4×Dic157—24no—
59C24×C157C2311yescharacteristic
60C22×C4×C157—22no—
61C22×C4×C157—24no—
62C22×C4×C157C2311yescharacteristic
63C22×C4×C157C2312yes—
64C22×C4×C157C2311yescharacteristic
65C22×C4×C157C2312yes—
66C4⋊C4×C157—24no—
67C22⋊C4×C157—22no—
68C22⋊C4×C157—22no—
69Dic314—44no—
70C2×Dic157—28no—
71C2×Dic157—22no—
72C2×Dic157—22no—
73C2×Dic157—22no—
74C2×Dic157—28no—
75C2×Dic157—21no—
76C2×Dic157—21no—
77C2×Dic157—22no—
78C2×Dic157—22no—
79C2×Dic157—22no—
80C2×Dic157—22no—
81C2×Dic157—22no—
82C2×Dic157C2×D412yes—
83C2×Dic157C2×D412yes—
84C2×Dic157C2×Q814yes—
85C2×Dic157#4114yes—
86C23×C157—22no—
87C23×C157—22no—
88C23×C157C2×D412yes—
89C23×C157C2×Q812yes—
90C23×C157#4112yes—
91C23×C157C2411yescharacteristic
92C2×C4×C157—28no—
93C2×C4×C157—22no—
94C2×C4×C157—22no—
95C2×C4×C157—21no—
96C2×C4×C157—21no—
97C2×C4×C157—22no—
98C2×C4×C157—22no—
99C2×C4×C157—22no—
100C2×C4×C157—22no—
101C2×C4×C157—22no—
102C2×C4×C157#4114yes—
103C22×C157—22no—
104C22×C157—22no—
105C22×C157—24no—
106C22×C157—24no—
107C22×C157C22⋊Q814yes—
108C22×C157C22×D411yescharacteristic
109C22×C157C22×Q811yescharacteristic
110C22×C157C2×#4112yes—
111C22×C157C2×#4112yes—
112C22×C157C2×#4111yescharacteristic
113C4×C157—24no—
114C4×C157—41no—
115C4×C157—41no—
116C4×C157—42no—
117Dic157—24no—
118Dic157—22no—
119Dic157—41no—
120Dic157—41no—
121Dic157—42no—
122Dic157#12314yes—
123C2×C157—24no—
124C2×C157C2×C22⋊Q811yescharacteristic
125C2×C157C2×#12311yescharacteristic
126C2×C157#52511yescharacteristic
127C2×C157#52912yes—
128C2×C157#53312yes—
129C157#285511yescharacteristic, complemented, Sylow
130#2855—1571nomaximal, Sylow
131C2×C22⋊Q8—1571no—
132C22×C42—1571no—
133#391—1572no—
134#389—1572no—
135#387—1572no—
136#387—1572no—
137#385—1571no—
138#385D15711yescharacteristic
139#381—1571no—
140#381—1571no—
141#380—1571no—
142C22×Q8—1571no—
143C23×C4—1571no—
144C23×C4—1571no—
145C23×C4C2×D15711yescharacteristic
146C22⋊Q8—3144no—
147C2×C4⋊C4—1571no—
148C2×C4⋊C4—1571no—
149C2×C4⋊C4—1571no—
150C2×C4⋊C4—1572no—
151C2×C4⋊C4—1572no—
152C2×C4⋊C4C2×D15712yes—
153C2×C22⋊C4—1571no—
154C2×C22⋊C4—1571no—
155C2×C22⋊C4C2×D15711yescharacteristic
156C2×C22⋊C4C2×D15711yescharacteristic
157C2×C42—3144no—
158C2×C42—1572no—
159C2×C42—1572no—
160#95—1572no—
161#95—1572no—
162#95—1572no—
163#95—1572no—
164#95—1572no—
165#95—1572no—
166#95C2×D15712yes—
167C24C22×D15711yescharacteristic
168C2×Q8—3142no—
169C2×Q8—3144no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—3142no—
173C22×C4—3142no—
174C22×C4—3144no—
175C22×C4—1572no—
176C22×C4—1571no—
177C22×C4—1571no—
178C22×C4—1574no—
179C22×C4—1574no—
180C22×C4—1572no—
181C22×C4—1571no—
182C22×C4—1571no—
183C22×C4C22×D15712yes—
184C22×C4C22×D15711yescharacteristic
185C22×C4C22×D15711yescharacteristic
186C22×C4C22×D15712yes—
187C4⋊C4—24no—
188C4⋊C4—3142no—
189C4⋊C4—3142no—
190C4⋊C4—3142no—
191C4⋊C4—3144no—
192C4⋊C4—3144no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—3142no—
196C22⋊C4—3142no—
197C42—3144no—
198C42—3144no—
199C23—22no—
200C23—22no—
201C23D4×D15712yes—
202C23Q8×D15712yes—
203C23#1457895412yes—
204C23C23×D15711yescharacteristic
205Q8—6284no—
206C2×C4—28no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—3148no—
217C2×C4—3142no—
218C2×C4—3142no—
219C2×C4—3142no—
220C2×C4—3148no—
221C2×C4—3141no—
222C2×C4—3141no—
223C2×C4—3142no—
224C2×C4—3142no—
225C2×C4—3142no—
226C2×C4—3142no—
227C2×C4—3142no—
228C2×C4—1572no—
229C2×C4—1574no—
230C2×C4—1572no—
231C2×C4—1574no—
232C2×C4#1457895314yes—
233C22—24no—
234C22—22no—
235C22—22no—
236C22—24no—
237C22#1909774914yes—
238C22C2×D4×D15711yescharacteristic
239C22C2×Q8×D15711yescharacteristic
240C22#41×D15712yes—
241C22C2×#1457895312yes—
242C22C2×#1457895411yescharacteristic
243C4—24no—
244C4—41no—
245C4—41no—
246C4—42no—
247C4—3144no—
248C4—3142no—
249C4—6281no—
250C4—6281no—
251C4—6282no—
252C4—1574no—
253C2—24no—
254C2C2×#1909774911yescharacteristic
255C2#123×D15711yescharacteristic
256C2#2375777911yescharacteristic
257C2#2375780312yes—
258C2#2375782512yes—
subgroupG/H
center, FrattiniC23C23×D157
commutator, socle, cosocleC23×C157C24
Fitting#385×C157C2
prime radicalC24×C157C23
hypercenter#385D157
hypocenterC157#2855
LCSG ▷ C23×C157 ▷ C157
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. #23757177C2semidirect
4. #23757168C2semidirect
5. #23757159C2semidirect
6. #23757155C2semidirect
43. C22×Dic157Q8semidirect
129. C157#2855semidirect

Maximal subgroups

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

subgroupindex
C2×#190977482
C22×C4×Dic1572
#237571772
#237571682
#237571592
#237571552
#237571462
#237571342
#237571312
#385×C1572
#2855157