Finite Group Databasebeta

← #34101788

Subgroups — #34101788

Order 20096.

234 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 55 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#23757845C211yesmaximal, characteristic
2#23757254C211yesmaximal, characteristic
3#23757245C211yesmaximal, characteristic
4#23757064C211yesmaximal, characteristic, complemented
5#23757063C211yesmaximal, characteristic, complemented
6#23757062C211yesmaximal, characteristic, complemented
7#352×C157C211yesmaximal, characteristic, complemented
8C2×#14578953C2211yescharacteristic
9C22×Dic314C411yescharacteristic, complemented
10M16×D157—21no—
11#19097770—21no—
12#19097757—21no—
13#19097755C411yescharacteristic, complemented
14#19097754C411yescharacteristic, complemented
15#19097752C2211yescharacteristic
16#19097749—21no—
17#19097748—21no—
18#19097727—21no—
19#19097727C411yescharacteristic, complemented
20#19097689—21no—
21#19097688C2211yescharacteristic
22#19097686C2211yescharacteristic
23#19097685—21no—
24#123×C157C2211yescharacteristic
25#100×C157C2211yescharacteristic
26#99×C157C2211yescharacteristic
27#14578953—41no—
28#14578953—41no—
29C2×Dic314—21no—
30C2×Dic314—21no—
31C2×Dic314—41no—
32C2×Dic314—41no—
33C2×Dic314D411yescharacteristic
34C2×C22⋊D157C2×C411yescharacteristic
35C2×C22⋊D157D411yescharacteristic
36C22×Dic157C2×C411yescharacteristic
37C22×Dic157C2×C411yescharacteristic
38C22×Dic157D411yescharacteristic
39C2×C4×D157D411yescharacteristic
40#14578938—21no—
41#14578937—41no—
42C8×D157—41no—
43C4⋊Dic157—21no—
44#14578933—21no—
45#14578933—21no—
46#14578933—21no—
47#14578933—41no—
48C22⋊Dic157—21no—
49C22⋊Dic157—21no—
50C22⋊Dic157C2×C411yescharacteristic
51#14578931—21no—
52#14578931—21no—
53C4×Dic157—21no—
54C4×Dic157—41no—
55C2×D4×C157C2311yescharacteristic
56C22×C4×C157C2×C411yescharacteristic
57M16×C157—21no—
58C4⋊C4×C157—21no—
59C22⋊C4×C157—21no—
60C22⋊C4×C157—21no—
61C22⋊C4×C157C2×C411yescharacteristic
62C22×D157C22⋊C411yescharacteristic
63Dic314—41no—
64Dic314—42no—
65Dic314—41no—
66Dic314—41no—
67C22⋊D157—41no—
68C22⋊D157—41no—
69C2×Dic157—21no—
70C2×Dic157—21no—
71C2×Dic157—21no—
72C2×Dic157—21no—
73C2×Dic157—21no—
74C2×Dic157—21no—
75C2×Dic157—41no—
76C2×Dic157—41no—
77C2×Dic157C22⋊C411yescharacteristic
78C2×Dic157C22⋊C411yescharacteristic
79C2×Dic157C22⋊C411yescharacteristic
80C4×D157—21no—
81C4×D157—21no—
82#11929434—41no—
83C23×C157C22×C411yescharacteristic
84C23×C157C2×D411yescharacteristic
85D4×C157—41no—
86C2×C4×C157—21no—
87C2×C4×C157—21no—
88C2×C4×C157—41no—
89C2×C4×C157—41no—
90C2×C4×C157C2×D411yescharacteristic
91C8×C157—41no—
92C22×C157—21no—
93C22×C157—21no—
94C22×C157—41no—
95C22×C157—41no—
96C22×C157C2×C22⋊C411yescharacteristic
97C2×D157—21no—
98C2×D157—41no—
99C4×C157—21no—
100C4×C157—41no—
101C4×C157—41no—
102C4×C157—81no—
103Dic157—21no—
104Dic157—21no—
105Dic157—41no—
106Dic157—41no—
107Dic157—81no—
108Dic157#9911yescharacteristic
109Dic157#9911yescharacteristic
110C2×C157—21no—
111C2×C157—41no—
112C2×C157—41no—
113C2×C157C2×#9911yescharacteristic
114D157—41no—
115C157#212311yescharacteristic, complemented, Sylow
116#2123—1571nomaximal, Sylow
117#536—1571no—
118#413—1571no—
119#410—1571no—
120#352—1571no—
121#352—1571no—
122#352—1571no—
123#352D15711yescharacteristic, complemented
124C2×#41—1571no—
125C22×Q8—1571no—
126C2×M16—3141no—
127C4⋊Q8—3141no—
128#124—3141no—
129#123—1571no—
130#123—1571no—
131#123—1571no—
132#123C2×D15711yescharacteristic
133C22⋊Q8—3141no—
134C22⋊Q8—3141no—
135#117—3141no—
136#117—1571no—
137#101—3141no—
138#100—1571no—
139#100C2×D15711yescharacteristic
140#99—3141no—
141#99—1571no—
142#99C2×D15711yescharacteristic
143#41—6281no—
144#41—6281no—
145C2×Q8—3141no—
146C2×Q8—3141no—
147C2×Q8—6281no—
148C2×Q8—6281no—
149C2×Q8—1571no—
150C2×D4—1571no—
151C2×D4—1571no—
152C2×D4C22×D15711yescharacteristic
153C22×C4—1571no—
154C22×C4—1571no—
155C22×C4—1571no—
156C22×C4—1571no—
157C22×C4C4×D15711yescharacteristic
158M16—21no—
159M16—3141no—
160M16—6281no—
161C2×C8—6281no—
162C4⋊C4—21no—
163C4⋊C4—3141no—
164C4⋊C4—3141no—
165C4⋊C4—3141no—
166C4⋊C4—3141no—
167C4⋊C4—6281no—
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—3141no—
171C22⋊C4—3141no—
172C22⋊C4—3141no—
173C22⋊C4—3141no—
174C22⋊C4—1571no—
175C22⋊C4C4×D15711yescharacteristic
176C42—3141no—
177C42—6281no—
178C23—1571no—
179C23C2×C4×D15711yescharacteristic
180C23D4×D15711yescharacteristic
181Q8—6281no—
182Q8—6282no—
183Q8—6281no—
184Q8—6281no—
185D4—41no—
186D4—6281no—
187D4—6281no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—3141no—
193C2×C4—3141no—
194C2×C4—3141no—
195C2×C4—3141no—
196C2×C4—3141no—
197C2×C4—3141no—
198C2×C4—3141no—
199C2×C4—3141no—
200C2×C4—6281no—
201C2×C4—6281no—
202C2×C4—1571no—
203C2×C4—1571no—
204C2×C4—1571no—
205C2×C4D4×D15711yescharacteristic
206C8—41no—
207C8—6281no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—3141no—
213C22—6281no—
214C22C22⋊C4×D15711yescharacteristic
215C4—21no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—3141no—
220C4—3141no—
221C4—6281no—
222C4—6281no—
223C4—1571no—
224C4—1571no—
225C4—12561no—
226C2—21no—
227C2—41no—
228C2—41no—
229C2—6281no—
230C2#99×D15711yescharacteristic
subgroupG/H
centerC2#99×D157
commutatorC23×C157C22×C4
FrattiniC2×D4C22×D157
Fitting#352×C157C2
socleC2×C157C2×#99
prime radicalC2×#14578953C22
cosocleC2×D4×C157C23
hypercenter#352D157
hypocenterC157#2123
LCSG ▷ C23×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
4. #23757064C2semidirect
5. #23757063C2semidirect
6. #23757062C2semidirect
7. #352×C157C2semidirect
9. C22×Dic314C4semidirect
13. #19097755C4semidirect
14. #19097754C4semidirect
19. #19097727C4semidirect
115. C157#2123semidirect
123. #352D157semidirect

Maximal subgroups

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

subgroupindex
#237578452
#237572542
#237572452
#237570642
#237570632
#237570622
#352×C1572
#2123157