Finite Group Databasebeta

← #34116147

Subgroups — #34116147

Order 20096.

342 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 66 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#23758351C211yesmaximal, characteristic, complemented
2#23758260C211yesmaximal, characteristic, complemented
3#23758243C212yesmaximal, complemented
4#23758154C212yesmaximal, complemented
5#23757779C211yesmaximal, characteristic
6#19097858—24no—
7D157⋊M16—21no—
8C2×#14578963—21no—
9C2×#14578963C2212yes—
10C2×#14578963C2212yes—
11#19097832—21no—
12#19097831—21no—
13C22⋊#11929441—24no—
14C2×#14578953C2211yescharacteristic
15C23×Dic157C2211yescharacteristic
16#19097755C411yescharacteristic
17#19097752C411yescharacteristic
18#19097748—21no—
19#19097743—21no—
20#19097740C411yescharacteristic
21D4×Dic157—21no—
22#19097731—21no—
23#19097727C2211yescharacteristic
24#120×C157C411yescharacteristic
25#14578964—22no—
26#14578963—22no—
27#14578963—22no—
28#14578963—22no—
29#14578963—22no—
30#14578963—22no—
31D157⋊C8—22no—
32C2×#11929441—22no—
33C2×#11929441—22no—
34C2×#11929441D414yescomplemented
35#14578953—22no—
36#14578953—22no—
37C2×Dic314D411yescharacteristic
38C2×C22⋊D157—21no—
39C2×C22⋊D157C2×C411yescharacteristic
40C2×C22⋊D157D411yescharacteristic
41C22×Dic157—21no—
42C22×Dic157—21no—
43C22×Dic157—22no—
44C22×Dic157—22no—
45C22×Dic157C2×C411yescharacteristic
46C22×Dic157D411yescharacteristic
47C22×Dic157C2311yescharacteristic
48C2×C4×D157D411yescharacteristic
49C4⋊Dic157—21no—
50#14578933—21no—
51#14578933—21no—
52C22⋊Dic157—21no—
53C22⋊Dic157—21no—
54C22⋊Dic157C2×C411yescharacteristic
55#14578931—21no—
56C4×Dic157—21no—
57C24×C157C2×C411yescharacteristic
58C2×D4×C157—21no—
59C2×D4×C157C2×C411yescharacteristic
60C22⋊C4×C157—21no—
61C22⋊C4×C157C2×C411yescharacteristic
62#11929441—24no—
63#11929441—42no—
64C22×D157C22⋊C411yescharacteristic
65Dic314—22no—
66C22⋊D157—22no—
67C22⋊D157—22no—
68C22⋊D157—42no—
69C2×Dic157—21no—
70C2×Dic157—21no—
71C2×Dic157—21no—
72C2×Dic157—21no—
73C2×Dic157—22no—
74C2×Dic157—41no—
75C2×Dic157—42no—
76C2×Dic157—41no—
77C2×Dic157C22⋊C411yescharacteristic
78C2×Dic157C2×D412yes—
79C2×Dic157C2×D411yescharacteristic
80C2×Dic157C2×D411yescharacteristic
81C2×Dic157#4112yes—
82C4×D157—21no—
83C4×D157—21no—
84C23×C157—21no—
85C23×C157—22no—
86C23×C157—42no—
87C23×C157C22⋊C411yescharacteristic
88C23×C157C22×C411yescharacteristic
89D4×C157—22no—
90D4×C157—42no—
91C2×C4×C157—21no—
92C2×C4×C157C22⋊C411yescharacteristic
93C22×C157—22no—
94C22×C157—24no—
95C22×C157—21no—
96C22×C157—41no—
97C22×C157—41no—
98C22×C157—41no—
99C22×C157—42no—
100C22×C157—41no—
101C22×C157C2×C22⋊C411yescharacteristic
102C22×C157C4×D412yes—
103C2×D157—21no—
104C2×D157—41no—
105C4×C157—21no—
106C4×C157—41no—
107Dic157—21no—
108Dic157—41no—
109Dic157—41no—
110Dic157#12011yescharacteristic
111Dic157#12112yes—
112Dic157#12311yescharacteristic
113C2×C157—21no—
114C2×C157—22no—
115C2×C157—42no—
116C2×C157—41no—
117C2×C157#38611yescharacteristic
118D157—41no—
119C157#189111yescharacteristic, complemented, Sylow
120#1891—1571nomaximal, Sylow
121#568—1571no—
122#525—1571no—
123#413—1571no—
124#407—1572no—
125#344—1572no—
126C2×#41—1571no—
127C23×C4—1571no—
128#131—3144no—
129C2×M16—3141no—
130C2×M16—3141no—
131C2×M16—1572no—
132C2×M16—1572no—
133#123—1571no—
134#123—1571no—
135C22⋊Q8—3141no—
136#121—3141no—
137#120—1571no—
138#120#1059784511yescharacteristic
139C4×D4—3141no—
140C4×D4—3141no—
141#117—1571no—
142#101—3141no—
143#100—3141no—
144C22⋊C8—3144no—
145C24C2×#1059784511yescharacteristic
146#41—3142no—
147#41—3142no—
148C2×Q8—1571no—
149C2×D4—21no—
150C2×D4—3141no—
151C2×D4—1571no—
152C2×D4—1571no—
153C2×D4C2×#1059784511yescharacteristic
154C22×C4—3142no—
155C22×C4—3141no—
156C22×C4—3141no—
157C22×C4—3142no—
158C22×C4—1571no—
159C22×C4—1571no—
160C22×C4—1571no—
161C22×C4—1571no—
162M16—3142no—
163M16—3142no—
164M16—3142no—
165M16—3142no—
166M16—3142no—
167M16—3142no—
168C2×C8—3142no—
169C2×C8—3142no—
170C2×C8—3142no—
171C2×C8—1574no—
172C4⋊C4—3141no—
173C4⋊C4—3141no—
174C4⋊C4—3141no—
175C22⋊C4—21no—
176C22⋊C4—3141no—
177C22⋊C4—3141no—
178C22⋊C4—3141no—
179C22⋊C4—1571no—
180C22⋊C4C2×#1059784511yescharacteristic
181C42—3141no—
182C23—21no—
183C23—22no—
184C23—42no—
185C23—1571no—
186C23C22⋊#1059784511yescharacteristic
187C23C22×#1059784511yescharacteristic
188Q8—3142no—
189D4—22no—
190D4—42no—
191D4—3142no—
192D4—3142no—
193D4—6282no—
194C2×C4—21no—
195C2×C4—3141no—
196C2×C4—3141no—
197C2×C4—3141no—
198C2×C4—3141no—
199C2×C4—3141no—
200C2×C4—3141no—
201C2×C4—3142no—
202C2×C4—6281no—
203C2×C4—6282no—
204C2×C4—6281no—
205C2×C4—1571no—
206C2×C4—1571no—
207C2×C4—1571no—
208C2×C4—1572no—
209C2×C4—1572no—
210C2×C4C22⋊#1059784511yescharacteristic
211C8—3144no—
212C8—6282no—
213C22—22no—
214C22—24no—
215C22—21no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—42no—
220C22—41no—
221C22—3141no—
222C22—6281no—
223C22C2×C22⋊#1059784511yescharacteristic
224C22D4×#1059784512yes—
225C4—21no—
226C4—41no—
227C4—3141no—
228C4—6281no—
229C4—6281no—
230C4—1571no—
231C4—1572no—
232C4—1571no—
233C2—21no—
234C2—22no—
235C2—42no—
236C2—41no—
237C2—6281no—
238C2#2375822111yescharacteristic
subgroupG/H
centerC2#23758221
commutatorC23×C157C22×C4
FrattiniC23C22×#10597845
Fitting#120×C157C4
socleC2×C157#386
prime radical#23757779C2
cosocleC22×Dic157C23
hypercenter#120#10597845
hypocenterC157#1891
LCSG ▷ C23×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C23 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #23758351C2semidirect
2. #23758260C2semidirect
3. #23758243C2semidirect
4. #23758154C2semidirect
34. C2×#11929441D4semidirect
119. C157#1891semidirect

Maximal subgroups

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

subgroupindex
#237583512
#237582602
#237582432
#237581542
#237577792
#1891157