Finite Group Databasebeta

← #34101184

Subgroups — #34101184

Order 20096.

370 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C2×#19097804C211yesmaximal, characteristic, complemented
2C2×#19097801C211yesmaximal, characteristic, complemented
3#23757851C211yesmaximal, characteristic
4C2×#19097699C211yesmaximal, characteristic, complemented
5#23757179C211yesmaximal, characteristic, complemented
6#23757041C211yesmaximal, characteristic, complemented
7C2×#104×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9C22×Dic314C2211yescharacteristic
10#19097804—22no—
11#19097804—22no—
12#19097801—22no—
13#19097801—22no—
14C2×Dic628—21no—
15C2×#14578944—21no—
16C2×#14578943—21no—
17C2×#14578942—21no—
18C2×#14578938C2211yescharacteristic
19#19097750—21no—
20#19097747—21no—
21#19097745—21no—
22#19097743—22no—
23#19097742—21no—
24C2×C4⋊Dic157C2211yescharacteristic
25#19097699—22no—
26#19097678—22no—
27C2×#41×C157C2211yescharacteristic
28C2×M16×C157C2211yescharacteristic
29C2×C42×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×Dic314D411yescharacteristic
41C2×Dic314D412yescomplemented
42C2×D628D411yescharacteristic
43C2×C22⋊D157—22no—
44C2×C22⋊D157—21no—
45C22×Dic157—21no—
46C22×Dic157D412yes—
47C2×C4×D157—21no—
48Dic628—24no—
49#14578944—24no—
50#14578943—24no—
51#14578942—24no—
52#14578938—21no—
53#14578938D412yescomplemented
54C2×#11929434—21no—
55C4⋊Dic157—21no—
56C4⋊Dic157—21no—
57#14578933—22no—
58C22⋊Dic157—22no—
59#14578931—22no—
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×C157—21no—
69C22×C4×C157C2311yescharacteristic
70M16×C157—21no—
71M16×C157D412yescomplemented
72C2×C8×C157—21no—
73C42×C157—21no—
74C42×C157—21no—
75C22×D157—21no—
76Dic314—22no—
77Dic314—21no—
78Dic314—22no—
79Dic314—21no—
80Dic314—41no—
81Dic314—41no—
82D628—21no—
83D628—21no—
84C22⋊D157—41no—
85C22⋊D157—41no—
86C22⋊D157—82no—
87C2×Dic157—21no—
88C2×Dic157—22no—
89C2×Dic157—22no—
90C2×Dic157—41no—
91C2×Dic157—41no—
92C2×Dic157—42no—
93C4×D157—41no—
94C4×D157—41no—
95#11929434—22no—
96C23×C157—21no—
97C23×C157C2×D411yescharacteristic
98Q8×C157—21no—
99Q8×C157—21no—
100D4×C157—21no—
101D4×C157—21no—
102D4×C157—41no—
103D4×C157—41no—
104C2×C4×C157—22no—
105C2×C4×C157—21no—
106C2×C4×C157—41no—
107C2×C4×C157—41no—
108C2×C4×C157—41no—
109C2×C4×C157—41no—
110C2×C4×C157C2×D411yescharacteristic
111C2×C4×C157C2×D411yescharacteristic
112C2×C4×C157C2×D411yescharacteristic
113C2×C4×C157C2×D411yescharacteristic
114C2×C4×C157C2×D411yescharacteristic
115C2×C4×C157#4111yescharacteristic
116C8×C157—22no—
117C22×C157—21no—
118C22×C157—21no—
119C22×C157—41no—
120C22×C157—41no—
121C22×C157—41no—
122C22×C157#12011yescharacteristic
123C22×C157#12111yescharacteristic
124C22×C157#12711yescharacteristic
125C2×D157—41no—
126C2×D157—41no—
127C2×D157—41no—
128C4×C157—42no—
129C4×C157—41no—
130C4×C157#12011yescharacteristic
131C4×C157#12011yescharacteristic
132C4×C157#12111yescharacteristic
133C4×C157#12111yescharacteristic
134Dic157—41no—
135Dic157—42no—
136Dic157—42no—
137C2×C157—21no—
138C2×C157—21no—
139C2×C157—81no—
140C2×C157#39211yescharacteristic
141C2×C157#45511yescharacteristic
142C2×C157#45611yescharacteristic
143D157—81no—
144C157#200711yescharacteristic, complemented, Sylow
145#2007—1571nomaximal, Sylow
146C2×#137—1571no—
147C2×#137—1571no—
148#537—1571no—
149C2×#104—1571no—
150C2×#104D15711yescharacteristic, complemented
151#391—1571no—
152#343—1571no—
153C2×#41—1571no—
154C2×#41C2×D15711yescharacteristic
155C22×Q8—1571no—
156#137—3142no—
157#137—3142no—
158#137—3142no—
159#137—3142no—
160C2×Q16—3141no—
161C2×Q16—3141no—
162C2×SD16—3141no—
163C2×SD16—3141no—
164C2×M16—1571no—
165C2×M16C2×D15711yescharacteristic
166C22⋊Q8—3141no—
167C22⋊Q8—3141no—
168#121—3141no—
169#121—3141no—
170#121—3142no—
171C2×C4⋊C4—1571no—
172C2×C42C2×D15711yescharacteristic
173#104—22no—
174#104—3142no—
175#95—3142no—
176#41—21no—
177#41—21no—
178#41—21no—
179#41—21no—
180#41—3141no—
181#41—3141no—
182#41—3141no—
183#41—3141no—
184C2×Q8—3141no—
185C2×Q8—3141no—
186C2×Q8—3141no—
187C2×Q8—3141no—
188C2×Q8—3141no—
189C2×Q8—1571no—
190C2×Q8—1572no—
191C2×Q8D62811yescharacteristic
192C2×D4—21no—
193C2×D4—3141no—
194C2×D4—3142no—
195C2×D4—1571no—
196C2×D4D62811yescharacteristic
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—3141no—
200C22×C4—3141no—
201C22×C4—1572no—
202C22×C4C22×D15711yescharacteristic
203Q16—3144no—
204Q16—3144no—
205SD16—3144no—
206SD16—3144no—
207M16—21no—
208M16—3141no—
209M16—1572no—
210M16C22⋊D15712yescomplemented
211C2×C8—21no—
212C2×C8—3141no—
213C4⋊C4—3142no—
214C4⋊C4—3141no—
215C4⋊C4—3141no—
216C22⋊C4—3142no—
217C22⋊C4—3142no—
218C42—21no—
219C42—21no—
220C23—21no—
221C23—3141no—
222C23D4×D15711yescharacteristic
223Q8—21no—
224Q8—21no—
225Q8—3142no—
226Q8—3141no—
227Q8—3142no—
228Q8—3141no—
229Q8—6281no—
230Q8—6281no—
231D4—21no—
232D4—21no—
233D4—41no—
234D4—41no—
235D4—3141no—
236D4—3141no—
237D4—6281no—
238D4—6281no—
239D4—12562no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—3142no—
247C2×C4—3141no—
248C2×C4—3142no—
249C2×C4—6281no—
250C2×C4—6281no—
251C2×C4—6281no—
252C2×C4—6281no—
253C2×C4—6282no—
254C2×C4D4×D15711yescharacteristic
255C2×C4D4×D15711yescharacteristic
256C2×C4D4×D15711yescharacteristic
257C2×C4C2×C22⋊D15711yescharacteristic
258C2×C4C2×D62811yescharacteristic
259C2×C4#1457895411yescharacteristic
260C8—22no—
261C8—3142no—
262C22—21no—
263C22—21no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—6281no—
268C22—6281no—
269C22—6281no—
270C22#1909774111yescharacteristic
271C22#1909774411yescharacteristic
272C22#1909776711yescharacteristic
273C4—42no—
274C4—41no—
275C4—6281no—
276C4—6282no—
277C4—6282no—
278C4#1909773911yescharacteristic
279C4#1909774111yescharacteristic
280C4#1909774411yescharacteristic
281C4#1909774511yescharacteristic
282C2—21no—
283C2—21no—
284C2—81no—
285C2—12561no—
286C2#2375718211yescharacteristic
287C2#2375745211yescharacteristic
288C2#2375745711yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
FittingC2×#104×C157C2
socleC22×C157#120
prime radical#23757851C2
hypercenterC2×#104D157
hypocenterC157#2007
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
1. C2×#19097804C2semidirect
2. C2×#19097801C2semidirect
4. C2×#19097699C2semidirect
5. #23757179C2semidirect
6. #23757041C2semidirect
7. C2×#104×C157C2semidirect
41. C2×Dic314D4semidirect
53. #14578938D4semidirect
71. M16×C157D4semidirect
144. C157#2007semidirect
150. C2×#104D157semidirect
210. M16C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978042
C2×#190978012
#237578512
C2×#190976992
#237571792
#237570412
C2×#104×C1572
#2007157