Finite Group Databasebeta

← #34101197

Subgroups — #34101197

Order 20096.

348 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 69 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#23758071C211yesmaximal, characteristic
2C2×#19097804C211yesmaximal, characteristic, complemented
3#23757844C211yesmaximal, characteristic, complemented
4#23757281C211yesmaximal, characteristic, complemented
5#23757246C211yesmaximal, characteristic, complemented
6#23757045C211yesmaximal, characteristic, complemented
7#421×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9C22×Dic314C2211yescharacteristic
10#19097804—22no—
11#19097804—22no—
12#19097802—21no—
13#19097801—21no—
14#19097797—21no—
15#19097796—21no—
16#19097791—22no—
17C2×Dic628—21no—
18C2×#14578943—21no—
19C2×#14578938C2211yescharacteristic
20#19097769—21no—
21#19097758—21no—
22#19097752—22no—
23#19097750—22no—
24#19097730C2211yescharacteristic
25#19097694—22no—
26#19097685—22no—
27C2×#41×C157C2211yescharacteristic
28C2×M16×C157C2211yescharacteristic
29#117×C157C2211yescharacteristic
30#104×C157—22no—
31#14578954—21no—
32#14578954—21no—
33#14578954—21no—
34#14578954—21no—
35C2×Dic314—21no—
36C2×Dic314—21no—
37C2×Dic314—21no—
38C2×Dic314—21no—
39C2×Dic314—21no—
40C2×Dic314D412yes—
41C2×Dic314D411yescharacteristic
42C2×D628D411yescharacteristic
43C2×C22⋊D157—21no—
44C22×Dic157—21no—
45C2×C4×D157—21no—
46Dic628—24no—
47#14578944—22no—
48#14578943—24no—
49#14578942—22no—
50#14578941—22no—
51#14578939—22no—
52#14578938—21no—
53#14578938—21no—
54C2×#11929434D412yes—
55C4⋊Dic157—21no—
56#14578933—24no—
57C22⋊Dic157—24no—
58C22⋊Dic157D412yes—
59C4×Dic157—21no—
60#41×C157—21no—
61#41×C157—21no—
62#41×C157—21no—
63#41×C157—21no—
64C2×Q8×C157D411yescharacteristic
65C2×D4×C157—21no—
66C2×D4×C157D411yescharacteristic
67C22×C4×C157—21no—
68C22×C4×C157C2311yescharacteristic
69M16×C157—21no—
70M16×C157D412yes—
71C2×C8×C157—21no—
72C4⋊C4×C157—21no—
73C22⋊C4×C157—21no—
74C42×C157—21no—
75C22×D157—21no—
76Dic314—21no—
77Dic314—21no—
78Dic314—22no—
79Dic314—22no—
80Dic314—41no—
81Dic314—41no—
82D628—21no—
83D628—21no—
84C22⋊D157—41no—
85C22⋊D157—41no—
86C2×Dic157—22no—
87C2×Dic157—22no—
88C2×Dic157—21no—
89C2×Dic157—41no—
90C2×Dic157—41no—
91C4×D157—41no—
92C4×D157—41no—
93#11929434—22no—
94C23×C157—21no—
95C23×C157C2×D411yescharacteristic
96Q8×C157—21no—
97Q8×C157—21no—
98D4×C157—21no—
99D4×C157—21no—
100D4×C157—41no—
101D4×C157—41no—
102C2×C4×C157—21no—
103C2×C4×C157—41no—
104C2×C4×C157—41no—
105C2×C4×C157—41no—
106C2×C4×C157C2×D411yescharacteristic
107C2×C4×C157C2×D411yescharacteristic
108C2×C4×C157C2×D411yescharacteristic
109C2×C4×C157C2×D411yescharacteristic
110C2×C4×C157C2×D411yescharacteristic
111C2×C4×C157#4111yescharacteristic
112C8×C157—22no—
113C22×C157—41no—
114C22×C157—42no—
115C22×C157—41no—
116C22×C157#12011yescharacteristic
117C22×C157#12111yescharacteristic
118C22×C157#12711yescharacteristic
119C2×D157—41no—
120C2×D157—81no—
121C4×C157—41no—
122C4×C157—81no—
123C4×C157#12011yescharacteristic
124C4×C157#12011yescharacteristic
125C4×C157#12111yescharacteristic
126C4×C157#12111yescharacteristic
127Dic157—42no—
128Dic157—41no—
129Dic157—42no—
130C2×C157—21no—
131C2×C157—21no—
132C2×C157—21no—
133C2×C157—81no—
134C2×C157#39211yescharacteristic
135D157—81no—
136C157#200911yescharacteristic, complemented, Sylow
137#2009—1571nomaximal, Sylow
138#575—1571no—
139C2×#137—1571no—
140#536—1571no—
141#421D15711yescharacteristic, complemented
142#419—1571no—
143#410—1571no—
144#344—1571no—
145C2×#41—1571no—
146C2×#41C2×D15711yescharacteristic
147C22×Q8—1571no—
148#137—3142no—
149#137—3141no—
150#137—3141no—
151#137—3142no—
152AGL(1,ℤ/8)—3141no—
153AGL(1,ℤ/8)—3141no—
154#135—3142no—
155C2×Q16—3141no—
156C2×SD16—3141no—
157C2×M16—1571no—
158C2×M16C2×D15711yescharacteristic
159C4⋊Q8—3141no—
160#124—3141no—
161#123—3142no—
162C22⋊Q8—3142no—
163#117—1571no—
164#117C2×D15711yescharacteristic
165#104—22no—
166#103—3142no—
167#99—3142no—
168#41—21no—
169#41—21no—
170#41—21no—
171#41—21no—
172#41—3141no—
173#41—3141no—
174#41—3141no—
175#41—3141no—
176C2×Q8—3141no—
177C2×Q8—3141no—
178C2×Q8—3141no—
179C2×Q8—3141no—
180C2×Q8—3141no—
181C2×Q8—1572no—
182C2×Q8—1571no—
183C2×Q8D62811yescharacteristic
184C2×D4—21no—
185C2×D4—3141no—
186C2×D4—1571no—
187C2×D4D62811yescharacteristic
188C22×C4—21no—
189C22×C4—3141no—
190C22×C4—3141no—
191C22×C4C22×D15711yescharacteristic
192Q16—3144no—
193Q16—3142no—
194SD16—3144no—
195SD16—3142no—
196SD16—3142no—
197D8—3142no—
198M16—21no—
199M16—3141no—
200M16—3141no—
201M16C22⋊D15712yes—
202C2×C8—21no—
203C2×C8—1572no—
204C4⋊C4—21no—
205C4⋊C4—3144no—
206C4⋊C4—3141no—
207C22⋊C4—21no—
208C22⋊C4—3144no—
209C22⋊C4—1572no—
210C42—21no—
211C42—3141no—
212C23—21no—
213C23—3141no—
214C23D4×D15711yescharacteristic
215Q8—21no—
216Q8—21no—
217Q8—3142no—
218Q8—3141no—
219Q8—3141no—
220Q8—3142no—
221Q8—6281no—
222Q8—6281no—
223D4—21no—
224D4—21no—
225D4—41no—
226D4—41no—
227D4—3141no—
228D4—3141no—
229D4—6281no—
230D4—6281no—
231C2×C4—21no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—3142no—
236C2×C4—3142no—
237C2×C4—3141no—
238C2×C4—6281no—
239C2×C4—6281no—
240C2×C4—6281no—
241C2×C4—6281no—
242C2×C4D4×D15711yescharacteristic
243C2×C4D4×D15711yescharacteristic
244C2×C4D4×D15711yescharacteristic
245C2×C4C2×C22⋊D15711yescharacteristic
246C2×C4C2×D62811yescharacteristic
247C2×C4#1457895411yescharacteristic
248C8—22no—
249C8—3142no—
250C22—41no—
251C22—42no—
252C22—41no—
253C22—6281no—
254C22—12561no—
255C22#1909774111yescharacteristic
256C22#1909774411yescharacteristic
257C22#1909776711yescharacteristic
258C4—41no—
259C4—81no—
260C4—6282no—
261C4—6281no—
262C4—6282no—
263C4#1909773911yescharacteristic
264C4#1909774111yescharacteristic
265C4#1909774411yescharacteristic
266C4#1909774511yescharacteristic
267C2—21no—
268C2—21no—
269C2—21no—
270C2—81no—
271C2—12561no—
272C2#2375718211yescharacteristic
subgroupG/H
centerC2#23757182
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#421×C157C2
socleC2×C157#392
prime radical#23758071C2
hypercenter#421D157
hypocenterC157#2009
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. C2×#19097804C2semidirect
3. #23757844C2semidirect
4. #23757281C2semidirect
5. #23757246C2semidirect
6. #23757045C2semidirect
7. #421×C157C2semidirect
136. C157#2009semidirect
141. #421D157semidirect

Maximal subgroups

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

subgroupindex
#237580712
C2×#190978042
#237578442
#237572812
#237572462
#237570452
#421×C1572
#2009157