Finite Group Databasebeta

← #34101799

Subgroups — #34101799

Order 20096.

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

normal subgroups only

This group has 258 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#23757819C211yesmaximal, characteristic
2#23757247C211yesmaximal, characteristic, complemented
3#23757067C212yesmaximal, complemented
4#23757065C212yesmaximal, complemented
5#410×C157C211yesmaximal, characteristic, complemented
6C2×#14578954C411yescharacteristic, complemented
7#19097769C411yescharacteristic, complemented
8#19097767C2211yescharacteristic
9#19097760C411yescharacteristic, complemented
10#19097758C2211yescharacteristic
11#19097743—21no—
12#19097743—21no—
13C4×C22⋊D157—21no—
14C4×C22⋊D157—21no—
15#19097730—21no—
16C2×C4×Dic157C411yescharacteristic, complemented
17#19097686—21no—
18#19097686C2212yes—
19C2×#41×C157C2211yescharacteristic
20#117×C157—21no—
21#99×C157—21no—
22#99×C157C2212yes—
23#14578954—41no—
24#14578954—41no—
25C2×Dic314C2×C411yescharacteristic
26C2×D628C2×C411yescharacteristic
27C2×C22⋊D157—21no—
28C2×C22⋊D157—21no—
29C2×C22⋊D157—21no—
30C22×Dic157—21no—
31C2×C4×D157—21no—
32C4⋊Dic157—41no—
33#14578933—21no—
34#14578933—21no—
35C22⋊Dic157—21no—
36C22⋊Dic157—21no—
37C22⋊Dic157—22no—
38#14578931—21no—
39#14578931—21no—
40C4×Dic157—21no—
41C4×Dic157—41no—
42C4×Dic157C2×C411yescharacteristic
43C4×Dic157C2×C411yescharacteristic
44#41×C157—41no—
45#41×C157—41no—
46C2×Q8×C157C2×C411yescharacteristic
47C2×D4×C157D411yescharacteristic
48C2×D4×C157D411yescharacteristic
49C2×D4×C157C2311yescharacteristic
50C22×C4×C157C2×C411yescharacteristic
51C22×C4×C157D411yescharacteristic
52C22×C4×C157D411yescharacteristic
53C4⋊C4×C157—41no—
54C22⋊C4×C157—22no—
55C42×C157—41no—
56C22×D157—21no—
57Dic314—41no—
58D628—41no—
59C22⋊D157—42no—
60C22⋊D157—42no—
61C22⋊D157—42no—
62C2×Dic157—21no—
63C2×Dic157—21no—
64C2×Dic157—21no—
65C2×Dic157—42no—
66C2×Dic157—42no—
67C4×D157—41no—
68C4×D157—41no—
69C23×C157C22⋊C411yescharacteristic
70C23×C157C2×D411yescharacteristic
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×C157—42no—
83C2×C4×C157C22⋊C411yescharacteristic
84C2×C4×C157C22⋊C411yescharacteristic
85C2×C4×C157C22⋊C411yescharacteristic
86C2×C4×C157C22×C411yescharacteristic
87C22×C157—21no—
88C22×C157—21no—
89C22×C157—21no—
90C22×C157—41no—
91C22×C157—41no—
92C22×C157—41no—
93C22×C157C2×C22⋊C411yescharacteristic
94C2×D157—41no—
95C2×D157—41no—
96C2×D157—41no—
97C4×C157—21no—
98C4×C157—21no—
99C4×C157—21no—
100C4×C157—82no—
101C4×C157#9911yescharacteristic
102C4×C157#9911yescharacteristic
103Dic157—22no—
104Dic157—41no—
105Dic157—41no—
106Dic157—82no—
107C2×C157—21no—
108C2×C157—41no—
109C2×C157—41no—
110C2×C157—41no—
111C2×C157C2×#9911yescharacteristic
112D157—81no—
113C157#212611yescharacteristic, complemented, Sylow
114#2126—1571nomaximal, Sylow
115#532—1571no—
116#410—1571no—
117#410D15711yescharacteristic, complemented
118#354—1572no—
119#353—1572no—
120C2×#41—1571no—
121C2×#41C2×D15711yescharacteristic
122C4⋊Q8—1571no—
123#127—1571no—
124#124—1571no—
125#124—1571no—
126#121—3141no—
127#121—3141no—
128C4×D4—3141no—
129C4×D4—3141no—
130#117—21no—
131#117—3141no—
132C2×C42—1571no—
133#99—21no—
134#99—3141no—
135#99—1572no—
136#99C2×D15712yes—
137#41—41no—
138#41—41no—
139#41—6281no—
140#41—6281no—
141C2×Q8—1571no—
142C2×Q8C4×D15711yescharacteristic
143C2×D4—3141no—
144C2×D4—3141no—
145C2×D4—3141no—
146C2×D4—1571no—
147C2×D4C22⋊D15711yescharacteristic
148C2×D4D62811yescharacteristic
149C2×D4C22×D15711yescharacteristic
150C22×C4—3141no—
151C22×C4—3141no—
152C22×C4C4×D15711yescharacteristic
153C22×C4C22⋊D15711yescharacteristic
154C22×C4D62811yescharacteristic
155C4⋊C4—41no—
156C4⋊C4—3141no—
157C4⋊C4—3141no—
158C4⋊C4—6281no—
159C22⋊C4—22no—
160C22⋊C4—3141no—
161C22⋊C4—3141no—
162C22⋊C4—3141no—
163C22⋊C4—3141no—
164C22⋊C4—3142no—
165C42—41no—
166C42—3141no—
167C42—6281no—
168C42—1571no—
169C42—1571no—
170C23—3141no—
171C23#1457893111yescharacteristic
172C23C2×C22⋊D15711yescharacteristic
173C23C2×D62811yescharacteristic
174Q8—41no—
175Q8—6281no—
176D4—41no—
177D4—41no—
178D4—41no—
179D4—6281no—
180D4—6282no—
181D4—6282no—
182D4—6282no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—3141no—
191C2×C4—3141no—
192C2×C4—3141no—
193C2×C4—6282no—
194C2×C4—6281no—
195C2×C4—6281no—
196C2×C4—6282no—
197C2×C4#1457893111yescharacteristic
198C2×C4#1457893111yescharacteristic
199C2×C4#1457893111yescharacteristic
200C2×C4C2×C4×D15711yescharacteristic
201C22—21no—
202C22—21no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—6281no—
208C22—6281no—
209C22—6281no—
210C22C2×#1457893111yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—82no—
215C4—3142no—
216C4—6281no—
217C4—6281no—
218C4—12562no—
219C4#1909768511yescharacteristic
220C4#1909768511yescharacteristic
221C2—21no—
222C2—41no—
223C2—41no—
224C2—41no—
225C2—12561no—
226C2C2×#1909768511yescharacteristic
subgroupG/H
centerC4#19097685
commutatorC2×C4×C157C22×C4
FrattiniC2×D4C22×D157
Fitting#410×C157C2
socleC2×C157C2×#99
prime radical#23757819C2
cosocleC2×D4×C157C23
hypercenter#410D157
hypocenterC157#2126
LCSG ▷ C2×C4×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #23757247C2semidirect
3. #23757067C2semidirect
4. #23757065C2semidirect
5. #410×C157C2semidirect
6. C2×#14578954C4semidirect
7. #19097769C4semidirect
9. #19097760C4semidirect
16. C2×C4×Dic157C4semidirect
113. C157#2126semidirect
117. #410D157semidirect

Maximal subgroups

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

subgroupindex
#237578192
#237572472
#237570672
#237570652
#410×C1572
#2126157