Finite Group Databasebeta

← #34101778

Subgroups — #34101778

Order 20096.

264 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 65 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#23757780C211yesmaximal, characteristic
2#23757247C211yesmaximal, characteristic, complemented
3#23757064C212yesmaximal, complemented
4#23757061C212yesmaximal, complemented
5#413×C157C211yesmaximal, characteristic, complemented
6C22×C4×D157C411yescharacteristic, complemented
7#19097752C411yescharacteristic, complemented
8#19097752C2211yescharacteristic
9#19097747—21no—
10#19097742—21no—
11#19097739C411yescharacteristic, complemented
12#19097739C2211yescharacteristic
13C4×C22⋊D157—21no—
14C4×C22⋊D157—21no—
15#19097730—21no—
16#19097730C411yescharacteristic, complemented
17#19097686—21no—
18#19097686C2212yes—
19C2×#41×C157C2211yescharacteristic
20C2×M16×C157—21no—
21#101×C157—21no—
22#100×C157C2212yes—
23C23×D157C2×C411yescharacteristic
24C2×C22⋊D157—21no—
25C2×C22⋊D157—21no—
26C22×Dic157C2×C411yescharacteristic
27C2×C4×D157—21no—
28C2×C4×D157—21no—
29C2×C4×D157—41no—
30C2×C4×D157—41no—
31C4⋊Dic157—21no—
32C4⋊Dic157—41no—
33#14578933—21no—
34#14578933—21no—
35C22⋊Dic157—21no—
36C22⋊Dic157—21no—
37C22⋊Dic157—22no—
38C22⋊Dic157C2×C411yescharacteristic
39C22⋊Dic157C2×C411yescharacteristic
40#14578931—21no—
41#14578931—21no—
42C4×Dic157—21no—
43C4×Dic157—41no—
44#41×C157—41no—
45#41×C157—41no—
46C2×Q8×C157D411yescharacteristic
47C2×D4×C157C2×C411yescharacteristic
48C2×D4×C157D411yescharacteristic
49C2×D4×C157C2311yescharacteristic
50C22×C4×C157C2×C411yescharacteristic
51C22×C4×C157D411yescharacteristic
52C22×C4×C157D411yescharacteristic
53M16×C157—22no—
54M16×C157—41no—
55C2×C8×C157—41no—
56C22×D157—21no—
57C22×D157—42no—
58C22×D157—41no—
59C22⋊D157—42no—
60C22⋊D157—42no—
61C2×Dic157—21no—
62C2×Dic157—21no—
63C2×Dic157—21no—
64C2×Dic157—41no—
65C2×Dic157—42no—
66C4×D157—42no—
67C4×D157—41no—
68C4×D157—41no—
69C23×C157C22⋊C411yescharacteristic
70C23×C157C22×C411yescharacteristic
71C23×C157C2×D411yescharacteristic
72Q8×C157—41no—
73D4×C157—41no—
74D4×C157—41no—
75D4×C157—41no—
76C2×C4×C157—21no—
77C2×C4×C157—21no—
78C2×C4×C157—21no—
79C2×C4×C157—21no—
80C2×C4×C157—21no—
81C2×C4×C157—21no—
82C2×C4×C157C22⋊C411yescharacteristic
83C2×C4×C157C22⋊C411yescharacteristic
84C2×C4×C157C22⋊C411yescharacteristic
85C2×C4×C157C2×D411yescharacteristic
86C8×C157—42no—
87C22×C157—21no—
88C22×C157—21no—
89C22×C157—21no—
90C22×C157—41no—
91C22×C157—41no—
92C22×C157—41no—
93C22×C157C2×C22⋊C411yescharacteristic
94C2×D157—22no—
95C2×D157—41no—
96C2×D157—42no—
97C2×D157—41no—
98C2×D157—42no—
99C4×C157—21no—
100C4×C157—21no—
101C4×C157—21no—
102C4×C157#9911yescharacteristic
103C4×C157#9911yescharacteristic
104Dic157—41no—
105Dic157—41no—
106Dic157—41no—
107Dic157—82no—
108C2×C157—21no—
109C2×C157—41no—
110C2×C157—41no—
111C2×C157—41no—
112C2×C157C2×#9911yescharacteristic
113D157—42no—
114C157#212011yescharacteristic, complemented, Sylow
115#2120—1571nomaximal, Sylow
116#525—1571no—
117#413D15711yescharacteristic, complemented
118#410—1571no—
119#352—1572no—
120AΣL(1,𝔽16)—1572no—
121C2×#41C2×D15711yescharacteristic
122C23×C4—1571no—
123C2×M16—21no—
124#123—1571no—
125#123—1571no—
126C22⋊Q8—3141no—
127#121—3141no—
128#120—1571no—
129#120—1571no—
130C4×D4—3141no—
131C4×D4—3141no—
132#117—3141no—
133#117—1571no—
134#101—21no—
135#100C2×D15712yes—
136#99—3141no—
137#99—1572no—
138C24—1571no—
139#41—41no—
140#41—41no—
141C2×Q8C22⋊D15711yescharacteristic
142C2×D4—3141no—
143C2×D4—3141no—
144C2×D4C4×D15711yescharacteristic
145C2×D4D62811yescharacteristic
146C2×D4C22×D15711yescharacteristic
147C22×C4—3141no—
148C22×C4—3141no—
149C22×C4—6281no—
150C22×C4—6281no—
151C22×C4—1571no—
152C22×C4C4×D15711yescharacteristic
153C22×C4C22⋊D15711yescharacteristic
154C22×C4D62811yescharacteristic
155M16—22no—
156M16—41no—
157C2×C8—41no—
158C4⋊C4—3141no—
159C4⋊C4—3141no—
160C4⋊C4—3141no—
161C4⋊C4—6281no—
162C22⋊C4—3141no—
163C22⋊C4—3141no—
164C22⋊C4—3141no—
165C22⋊C4—3141no—
166C22⋊C4—3142no—
167C22⋊C4—1571no—
168C22⋊C4—1571no—
169C42—3141no—
170C42—6281no—
171C23—3141no—
172C23—6281no—
173C23—6282no—
174C23#1457893111yescharacteristic
175C23C2×C4×D15711yescharacteristic
176C23C2×D62811yescharacteristic
177Q8—41no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—6282no—
182D4—6282no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—3141no—
190C2×C4—3141no—
191C2×C4—3141no—
192C2×C4—6282no—
193C2×C4—6281no—
194C2×C4—6281no—
195C2×C4—6281no—
196C2×C4—6282no—
197C2×C4#1457893111yescharacteristic
198C2×C4#1457893111yescharacteristic
199C2×C4#1457893111yescharacteristic
200C2×C4C2×C22⋊D15711yescharacteristic
201C8—42no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—3142no—
209C22—6281no—
210C22—6282no—
211C22—6282no—
212C22—6281no—
213C22C2×#1457893111yescharacteristic
214C4—21no—
215C4—21no—
216C4—21no—
217C4—6281no—
218C4—6281no—
219C4—6281no—
220C4—12562no—
221C4#1909768511yescharacteristic
222C4#1909768511yescharacteristic
223C2—21no—
224C2—41no—
225C2—41no—
226C2—41no—
227C2—6282no—
228C2C2×#1909768511yescharacteristic
subgroupG/H
centerC4#19097685
commutatorC23×C157C22×C4
FrattiniC2×D4C22×D157
Fitting#413×C157C2
socleC2×C157C2×#99
prime radical#23757780C2
cosocleC2×D4×C157C23
hypercenter#413D157
hypocenterC157#2120
LCSG ▷ C23×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #23757247C2semidirect
3. #23757064C2semidirect
4. #23757061C2semidirect
5. #413×C157C2semidirect
6. C22×C4×D157C4semidirect
7. #19097752C4semidirect
11. #19097739C4semidirect
16. #19097730C4semidirect
114. C157#2120semidirect
117. #413D157semidirect

Maximal subgroups

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

subgroupindex
#237577802
#237572472
#237570642
#237570612
#413×C1572
#2120157