Finite Group Databasebeta

← #34101356

Subgroups — #34101356

Order 20096.

278 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 67 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#23757849C211yesmaximal, characteristic
2#23757316C211yesmaximal, characteristic, complemented
3C2×#19097699C211yesmaximal, characteristic
4#23757275C211yesmaximal, characteristic
5C2×#19097689C211yesmaximal, characteristic, complemented
6#23757179C211yesmaximal, characteristic, complemented
7#339×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9C22×Dic314C2211yescharacteristic
10C2×#14578938C2211yescharacteristic
11#19097771—21no—
12#19097759—21no—
13#19097755—22no—
14#19097751—21no—
15#19097751—21no—
16C2×C4⋊Dic157C2211yescharacteristic
17#19097704—21no—
18#19097702—21no—
19#19097699—22no—
20#19097696—21no—
21#19097693—21no—
22#19097689—22no—
23#19097678—22no—
24C2×M16×C157C2211yescharacteristic
25#117×C157C2211yescharacteristic
26C2×C42×C157C2211yescharacteristic
27#14578954—21no—
28#14578954—21no—
29#14578954—21no—
30#14578954—21no—
31C2×Dic314—21no—
32C2×Dic314—21no—
33C2×Dic314—21no—
34C2×Dic314—21no—
35C2×Dic314—21no—
36C2×Dic314—21no—
37C2×Dic314D411yescharacteristic
38C2×D628D411yescharacteristic
39C2×C22⋊D157—21no—
40C22×Dic157—21no—
41C22×Dic157D412yes—
42C2×C4×D157—21no—
43#14578938—21no—
44#14578938D412yes—
45C2×#11929434—21no—
46C4⋊Dic157—22no—
47C4⋊Dic157—22no—
48C4⋊Dic157—21no—
49C4⋊Dic157—21no—
50#14578931—22no—
51#14578931—22no—
52C22×C4×C157—21no—
53C22×C4×C157C2311yescharacteristic
54M16×C157—21no—
55M16×C157—21no—
56C2×C8×C157—21no—
57C4⋊C4×C157D411yescharacteristic
58C4⋊C4×C157D411yescharacteristic
59C22⋊C4×C157—21no—
60C42×C157—21no—
61C42×C157—21no—
62C42×C157—21no—
63C22×D157—21no—
64Dic314—21no—
65Dic314—21no—
66Dic314—41no—
67Dic314—41no—
68Dic314—41no—
69Dic314—41no—
70D628—21no—
71D628—21no—
72C22⋊D157—41no—
73C22⋊D157—41no—
74C2×Dic157—21no—
75C2×Dic157—22no—
76C2×Dic157—22no—
77C2×Dic157—22no—
78C2×Dic157—42no—
79C2×Dic157—41no—
80C4×D157—41no—
81C4×D157—41no—
82#11929434—22no—
83C23×C157C2×D411yescharacteristic
84C2×C4×C157—22no—
85C2×C4×C157—21no—
86C2×C4×C157—21no—
87C2×C4×C157—41no—
88C2×C4×C157—41no—
89C2×C4×C157C2×D411yescharacteristic
90C2×C4×C157C2×D411yescharacteristic
91C2×C4×C157C2×D411yescharacteristic
92C2×C4×C157#4111yescharacteristic
93C2×C4×C157#4111yescharacteristic
94C2×C4×C157#4111yescharacteristic
95C8×C157—41no—
96C22×C157—21no—
97C22×C157—21no—
98C22×C157#12011yescharacteristic
99C22×C157#12111yescharacteristic
100C22×C157#12311yescharacteristic
101C2×D157—41no—
102C2×D157—41no—
103C2×D157—41no—
104C4×C157—42no—
105C4×C157—41no—
106C4×C157—41no—
107C4×C157#12111yescharacteristic
108C4×C157#12111yescharacteristic
109C4×C157#12311yescharacteristic
110C4×C157#12411yescharacteristic
111Dic157—41no—
112Dic157—42no—
113Dic157—81no—
114C2×C157—21no—
115C2×C157—21no—
116C2×C157#39411yescharacteristic
117C2×C157#45411yescharacteristic
118C2×C157#45611yescharacteristic
119D157—81no—
120C157#203811yescharacteristic, complemented, Sylow
121#2038—1571nomaximal, Sylow
122#536—1571no—
123#428—1571no—
124C2×#104—1571no—
125#417—1571no—
126C2×#101—1571no—
127#391—1571no—
128#339D15711yescharacteristic, complemented
129C2×#41—1571no—
130C22×Q8—1571no—
131C2×M16—1571no—
132C2×M16C2×D15711yescharacteristic
133C4⋊Q8—3141no—
134#124—3141no—
135#123—3142no—
136C22⋊Q8—3141no—
137C22⋊Q8—3141no—
138#117C2×D15711yescharacteristic
139C2×C4⋊C4—1571no—
140C2×C42C2×D15711yescharacteristic
141#107—3141no—
142#106—3141no—
143#104—3142no—
144#103—3141no—
145#102—3141no—
146#101—3142no—
147#95—3142no—
148#41—3141no—
149#41—3141no—
150#41—3141no—
151#41—3141no—
152C2×Q8—3141no—
153C2×Q8—3141no—
154C2×Q8—3141no—
155C2×Q8—3141no—
156C2×Q8—3141no—
157C2×Q8—3141no—
158C2×Q8—1571no—
159C2×D4—3141no—
160C2×D4—1571no—
161C22×C4—21no—
162C22×C4—3141no—
163C22×C4—3141no—
164C22×C4—1572no—
165C22×C4C22×D15711yescharacteristic
166M16—21no—
167M16—21no—
168M16—3141no—
169M16—1572no—
170C2×C8—21no—
171C2×C8—3141no—
172C4⋊C4—3142no—
173C4⋊C4—3142no—
174C4⋊C4—3141no—
175C4⋊C4—3141no—
176C4⋊C4D62811yescharacteristic
177C4⋊C4D62811yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—3142no—
180C22⋊C4—3142no—
181C42—21no—
182C42—21no—
183C42—21no—
184C23—3141no—
185C23D4×D15711yescharacteristic
186Q8—3141no—
187Q8—3141no—
188Q8—6281no—
189Q8—6281no—
190Q8—6281no—
191Q8—6281no—
192D4—3141no—
193D4—3141no—
194D4—6281no—
195D4—6281no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—3142no—
202C2×C4—3141no—
203C2×C4—3142no—
204C2×C4—3142no—
205C2×C4—6281no—
206C2×C4—6281no—
207C2×C4—6282no—
208C2×C4—6281no—
209C2×C4D4×D15711yescharacteristic
210C2×C4D4×D15711yescharacteristic
211C2×C4C2×D62811yescharacteristic
212C2×C4#1457895411yescharacteristic
213C2×C4#1457895411yescharacteristic
214C2×C4#1457895511yescharacteristic
215C8—41no—
216C8—3142no—
217C22—21no—
218C22—21no—
219C22—6281no—
220C22—6281no—
221C22—6281no—
222C22#1909774111yescharacteristic
223C22#1909774411yescharacteristic
224C22#1909775611yescharacteristic
225C4—42no—
226C4—41no—
227C4—41no—
228C4—6282no—
229C4—6281no—
230C4—12561no—
231C4#1909774411yescharacteristic
232C4#1909774611yescharacteristic
233C4#1909775611yescharacteristic
234C4#1909775911yescharacteristic
235C2—21no—
236C2—21no—
237C2—12561no—
238C2#2375719111yescharacteristic
239C2#2375744711yescharacteristic
240C2#2375745711yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#339×C157C2
socleC22×C157#120
prime radicalC2×#14578954C22
hypercenter#339D157
hypocenterC157#2038
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #23757316C2semidirect
5. C2×#19097689C2semidirect
6. #23757179C2semidirect
7. #339×C157C2semidirect
120. C157#2038semidirect
128. #339D157semidirect

Maximal subgroups

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

subgroupindex
#237578492
#237573162
C2×#190976992
#237572752
C2×#190976892
#237571792
#339×C1572
#2038157