Finite Group Databasebeta

← #34101172

Subgroups — #34101172

Order 20096.

362 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 75 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×#19097802C211yesmaximal, characteristic
2#23757847C211yesmaximal, characteristic
3C2×#19097699C211yesmaximal, characteristic
4#23757282C211yesmaximal, characteristic, complemented
5C2×#19097690C211yesmaximal, characteristic, complemented
6#23757033C211yesmaximal, characteristic, complemented
7C2×C4⋊Q8×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9#19097802—22no—
10#19097802—22no—
11C2×#14578944—21no—
12C2×#14578941—21no—
13C2×#14578938C2211yescharacteristic
14C2×#14578938C2211yescharacteristic
15#19097769—21no—
16#19097760—21no—
17#19097754—22no—
18#19097747—22no—
19#19097730C2211yescharacteristic
20#19097699—22no—
21#19097695—22no—
22#19097690—22no—
23C22×Q8×C157C2211yescharacteristic
24C22×Q8×C157C2211yescharacteristic
25C4⋊Q8×C157—22no—
26C4⋊Q8×C157—22no—
27C2×C4⋊C4×C157—22no—
28C2×C42×C157C2211yescharacteristic
29#14578954—21no—
30#14578954—21no—
31#14578954—21no—
32#14578954—21no—
33C2×Dic314D411yescharacteristic
34C2×D628D411yescharacteristic
35C2×C22⋊D157—21no—
36C2×C4×D157—21no—
37#14578944—24no—
38#14578941—24no—
39#14578938—21no—
40#14578938—21no—
41#14578938—21no—
42#14578938D412yes—
43C2×#11929434—21no—
44C2×#11929434—21no—
45C4⋊Dic157D412yes—
46#14578933—24no—
47C22⋊Dic157—21no—
48#14578931—24no—
49C4×Dic157—21no—
50C2×Q8×C157—21no—
51C2×Q8×C157—21no—
52C2×Q8×C157—21no—
53C2×Q8×C157—21no—
54C2×Q8×C157—21no—
55C2×Q8×C157—21no—
56C2×Q8×C157—21no—
57C2×Q8×C157—21no—
58C2×Q8×C157—21no—
59C2×Q8×C157—21no—
60C2×Q8×C157—21no—
61C2×Q8×C157D412yes—
62C22×C4×C157—21no—
63C22×C4×C157—21no—
64C22×C4×C157D412yes—
65C22×C4×C157C2311yescharacteristic
66C4⋊C4×C157—28no—
67C42×C157—21no—
68C42×C157D412yes—
69C22×D157—21no—
70Dic314—21no—
71Dic314—21no—
72D628—21no—
73D628—21no—
74C22⋊D157—41no—
75C22⋊D157—41no—
76C2×Dic157—21no—
77C2×Dic157—22no—
78C4×D157—41no—
79C4×D157—41no—
80#11929434—22no—
81#11929434—41no—
82C23×C157C2×D411yescharacteristic
83Q8×C157—22no—
84Q8×C157—22no—
85Q8×C157—41no—
86Q8×C157—41no—
87Q8×C157—41no—
88Q8×C157—41no—
89Q8×C157—41no—
90Q8×C157—41no—
91C2×C4×C157—24no—
92C2×C4×C157—22no—
93C2×C4×C157—22no—
94C2×C4×C157—22no—
95C2×C4×C157—22no—
96C2×C4×C157—22no—
97C2×C4×C157—41no—
98C2×C4×C157—41no—
99C2×C4×C157C2×D411yescharacteristic
100C2×C4×C157C2×D411yescharacteristic
101C2×C4×C157C2×D411yescharacteristic
102C2×C4×C157C2×D411yescharacteristic
103C2×C4×C157C2×D411yescharacteristic
104C2×C4×C157#4111yescharacteristic
105C22×C157—22no—
106C22×C157#12011yescharacteristic
107C22×C157#12011yescharacteristic
108C22×C157#12111yescharacteristic
109C2×D157—41no—
110C2×D157—42no—
111C4×C157—24no—
112C4×C157—42no—
113C4×C157—42no—
114C4×C157#12011yescharacteristic
115C4×C157#12111yescharacteristic
116C4×C157#12111yescharacteristic
117C4×C157#12711yescharacteristic
118Dic157—41no—
119Dic157—42no—
120C2×C157—21no—
121C2×C157—21no—
122C2×C157#39211yescharacteristic
123C2×C157#45612yes—
124D157—81no—
125C157#200511yescharacteristic, complemented, Sylow
126#2005—1571nomaximal, Sylow
127C2×#137—1571no—
128#536—1571no—
129C2×C4⋊Q8D15711yescharacteristic, complemented
130C2×#104—1571no—
131#419—1571no—
132C2×#101—1571no—
133#339—1571no—
134C2×#41—1571no—
135C22×Q8C2×D15711yescharacteristic
136C22×Q8C2×D15711yescharacteristic
137#137—3142no—
138#137—3142no—
139C2×Q16—3141no—
140C2×SD16—3141no—
141C2×M16—1571no—
142C2×M16—1571no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—3141no—
146#124—3141no—
147#123—3142no—
148C22⋊Q8—3142no—
149#117—1571no—
150C2×C4⋊C4—22no—
151C2×C42C2×D15711yescharacteristic
152#104—3142no—
153#103—3142no—
154#101—3142no—
155#41—3141no—
156#41—3141no—
157#41—3141no—
158#41—3141no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—1571no—
171C2×Q8C22⋊D15712yes—
172C2×D4—3141no—
173C2×D4—1571no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—3141no—
177C22×C4C22⋊D15712yes—
178C22×C4C22×D15711yescharacteristic
179Q16—3144no—
180SD16—3144no—
181M16—3141no—
182M16—3141no—
183M16—3141no—
184M16—1572no—
185C2×C8—3141no—
186C2×C8—3141no—
187C4⋊C4—28no—
188C4⋊C4—3144no—
189C4⋊C4—1572no—
190C22⋊C4—3144no—
191C22⋊C4—3141no—
192C42—21no—
193C42—3141no—
194C42C22⋊D15712yes—
195C23—3141no—
196C23C2×C22⋊D15711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201Q8—41no—
202Q8—41no—
203Q8—41no—
204Q8—41no—
205Q8—3141no—
206Q8—3141no—
207D4—3141no—
208D4—3141no—
209D4—6281no—
210D4—6281no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—3142no—
220C2×C4—3141no—
221C2×C4—6281no—
222C2×C4—6281no—
223C2×C4D4×D15711yescharacteristic
224C2×C4D4×D15711yescharacteristic
225C2×C4D4×D15711yescharacteristic
226C2×C4C2×C22⋊D15711yescharacteristic
227C2×C4C2×C22⋊D15711yescharacteristic
228C2×C4#1457895311yescharacteristic
229C8—3142no—
230C8—6281no—
231C22—22no—
232C22—6281no—
233C22—6282no—
234C22#1909773911yescharacteristic
235C22#1909774011yescharacteristic
236C22#1909774311yescharacteristic
237C4—24no—
238C4—42no—
239C4—42no—
240C4—6281no—
241C4—6282no—
242C4#1909773911yescharacteristic
243C4#1909774211yescharacteristic
244C4#1909774311yescharacteristic
245C4#1909776711yescharacteristic
246C2—21no—
247C2—21no—
248C2—12561no—
249C2#2375718111yescharacteristic
250C2#2375745512yes—
subgroupG/H
centerC22#19097739
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
FittingC2×C4⋊Q8×C157C2
socleC22×C157#120
prime radicalC2×#14578954C22
hypercenterC2×C4⋊Q8D157
hypocenterC157#2005
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. #23757282C2semidirect
5. C2×#19097690C2semidirect
6. #23757033C2semidirect
7. C2×C4⋊Q8×C157C2semidirect
125. C157#2005semidirect
129. C2×C4⋊Q8D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978022
#237578472
C2×#190976992
#237572822
C2×#190976902
#237570332
C2×C4⋊Q8×C1572
#2005157