Finite Group Databasebeta

← #34101314

Subgroups — #34101314

Order 20096.

278 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 63 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#23757899C211yesmaximal, characteristic, complemented
2C2×#19097685C211yesmaximal, characteristic
3C2×#19097685C211yesmaximal, characteristic
4C2×#19097685C211yesmaximal, characteristic
5#23757040C211yesmaximal, characteristic, complemented
6#23757039C211yesmaximal, characteristic, complemented
7#342×C157C211yesmaximal, characteristic, complemented
8C22×C22⋊D157C2211yescharacteristic
9#19097750—22no—
10#19097748—22no—
11#19097748—22no—
12C2×C22⋊Dic157C2211yescharacteristic
13C2×C22⋊Dic157C2211yescharacteristic
14C2×C22⋊Dic157C2211yescharacteristic
15#19097685—22no—
16#19097685—22no—
17#19097685—22no—
18C2×C22⋊C4×C157C2211yescharacteristic
19C2×C22⋊C4×C157C2211yescharacteristic
20C2×C22⋊C4×C157C2211yescharacteristic
21C23×D157Q811yescharacteristic, complemented
22C2×Dic314—22no—
23C2×C22⋊D157—21no—
24C2×C22⋊D157—21no—
25C2×C22⋊D157—21no—
26C2×C22⋊D157—21no—
27C2×C22⋊D157—21no—
28C2×C22⋊D157—21no—
29C22×Dic157—21no—
30C22×Dic157—21no—
31C22×Dic157—21no—
32C22×Dic157Q811yescharacteristic, complemented
33C4⋊Dic157—22no—
34#14578933—24no—
35C22⋊Dic157—21no—
36C22⋊Dic157—21no—
37C22⋊Dic157—21no—
38C22⋊Dic157—21no—
39C22⋊Dic157D412yes—
40C22⋊Dic157D412yes—
41C24×C157C2311yescharacteristic
42C22×C4×C157—21no—
43C22×C4×C157—21no—
44C22×C4×C157—21no—
45C22⋊C4×C157—21no—
46C22⋊C4×C157—21no—
47C22⋊C4×C157—21no—
48C22⋊C4×C157—21no—
49C22⋊C4×C157—21no—
50C22⋊C4×C157D412yes—
51C22×D157—21no—
52C22×D157—21no—
53C22×D157—21no—
54C22×D157—81no—
55Dic314—82no—
56C22⋊D157—41no—
57C22⋊D157—41no—
58C22⋊D157—41no—
59C22⋊D157—41no—
60C2×Dic157—22no—
61C2×Dic157—22no—
62C2×Dic157—21no—
63C2×Dic157—21no—
64C2×Dic157—21no—
65C2×Dic157—41no—
66C2×Dic157—81no—
67C2×Dic157—81no—
68C2×Dic157—81no—
69C23×C157—81no—
70C23×C157C2×D411yescharacteristic
71C23×C157C2×D411yescharacteristic
72C23×C157C2×D411yescharacteristic
73C23×C157C2×Q811yescharacteristic
74C23×C157#4111yescharacteristic
75C23×C157#4111yescharacteristic
76C23×C157#4111yescharacteristic
77C2×C4×C157—22no—
78C2×C4×C157—41no—
79C2×C4×C157—41no—
80C2×C4×C157—81no—
81C2×C4×C157—81no—
82C2×C4×C157—81no—
83C22×C157—41no—
84C22×C157—41no—
85C22×C157—41no—
86C22×C157—41no—
87C22×C157—41no—
88C22×C157—41no—
89C22×C157—41no—
90C22×C157#12011yescharacteristic
91C22×C157C22⋊Q811yescharacteristic
92C22×C157C22⋊Q811yescharacteristic
93C22×C157C22⋊Q811yescharacteristic
94C22×C157#12411yescharacteristic
95C22×C157#12411yescharacteristic
96C22×C157#12411yescharacteristic
97C2×D157—41no—
98C2×D157—81no—
99C2×D157—81no—
100C2×D157—81no—
101C4×C157—42no—
102C4×C157—81no—
103C4×C157—81no—
104Dic157—42no—
105Dic157—42no—
106Dic157—41no—
107Dic157—81no—
108C2×C157—21no—
109C2×C157—21no—
110C2×C157—21no—
111C2×C157—21no—
112C2×C157—21no—
113C2×C157—21no—
114C2×C157—21no—
115C2×C157#39311yescharacteristic
116D157—81no—
117C157#203211yescharacteristic, complemented, Sylow
118#2032—1571nomaximal, Sylow
119#543—1571no—
120C2×#99—1571no—
121C2×#99—1571no—
122C2×#99—1571no—
123#342—1571no—
124#342—1571no—
125#342D15711yescharacteristic, complemented
126C22×D4—1571no—
127C22⋊Q8—3142no—
128C22⋊Q8—3142no—
129C22⋊Q8—3142no—
130C2×C22⋊C4—1571no—
131C2×C22⋊C4—1571no—
132C2×C22⋊C4—1571no—
133C2×C22⋊C4C2×D15711yescharacteristic
134C2×C22⋊C4C2×D15711yescharacteristic
135C2×C22⋊C4C2×D15711yescharacteristic
136#99—3142no—
137#99—3142no—
138#99—3142no—
139C24—1571no—
140C24C22×D15711yescharacteristic
141C2×Q8—3142no—
142C2×D4—3141no—
143C2×D4—3141no—
144C2×D4—3141no—
145C2×D4—3141no—
146C2×D4—3141no—
147C2×D4—3141no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—3141no—
152C22×C4—3141no—
153C22×C4—3141no—
154C22×C4—1571no—
155C4⋊C4—3144no—
156C4⋊C4—3142no—
157C22⋊C4—21no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—3141no—
163C22⋊C4—3141no—
164C22⋊C4—3141no—
165C22⋊C4—3141no—
166C22⋊C4—1572no—
167C22⋊C4—1572no—
168C22⋊C4D62812yes—
169C23—81no—
170C23—3141no—
171C23—3141no—
172C23—3141no—
173C23—12561no—
174C23D4×D15711yescharacteristic
175C23D4×D15711yescharacteristic
176C23C2×D62811yescharacteristic
177C23Q8×D15711yescharacteristic
178C23#1457895311yescharacteristic
179C23#1457895411yescharacteristic
180C23#1457895411yescharacteristic
181Q8—12562no—
182D4—6281no—
183D4—6281no—
184D4—6281no—
185D4—6281no—
186C2×C4—22no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—3142no—
193C2×C4—3142no—
194C2×C4—3141no—
195C2×C4—3141no—
196C2×C4—3141no—
197C2×C4—6281no—
198C2×C4—12561no—
199C2×C4—12561no—
200C2×C4—12561no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—6281no—
209C22—12561no—
210C22—12561no—
211C22—12561no—
212C22#1909774111yescharacteristic
213C22#1909774911yescharacteristic
214C22#1909774911yescharacteristic
215C22#1909775111yescharacteristic
216C22#1909775711yescharacteristic
217C22#1909775711yescharacteristic
218C22#1909775911yescharacteristic
219C4—42no—
220C4—81no—
221C4—81no—
222C4—6282no—
223C4—6282no—
224C4—6281no—
225C4—12561no—
226C2—21no—
227C2—21no—
228C2—21no—
229C2—21no—
230C2—21no—
231C2—21no—
232C2—21no—
233C2—12561no—
234C2#2375718611yescharacteristic
subgroupG/H
centerC2#23757186
commutator, cosocleC24×C157C23
FrattiniC24C22×D157
Fitting#342×C157C2
socleC2×C157#393
prime radicalC22×C22⋊D157C22
hypercenter#342D157
hypocenterC157#2032
LCSG ▷ C24×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #23757899C2semidirect
5. #23757040C2semidirect
6. #23757039C2semidirect
7. #342×C157C2semidirect
21. C23×D157Q8semidirect
32. C22×Dic157Q8semidirect
117. C157#2032semidirect
125. #342D157semidirect

Maximal subgroups

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

subgroupindex
#237578992
C2×#190976852
#237570402
#237570392
#342×C1572
#2032157