Finite Group Databasebeta

← #34101352

Subgroups — #34101352

Order 20096.

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

normal subgroups only

This group has 278 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#23757844C211yesmaximal, characteristic
2#23757315C211yesmaximal, characteristic, complemented
3#23757274C211yesmaximal, characteristic
4C2×#19097689C211yesmaximal, characteristic, complemented
5#23757179C211yesmaximal, characteristic, complemented
6#23757033C211yesmaximal, characteristic, complemented
7C2×#104×C157C211yesmaximal, characteristic
8C22×Dic314C2211yescharacteristic
9C2×#14578938C2211yescharacteristic
10#19097769—21no—
11#19097758—21no—
12#19097752—22no—
13#19097750—21no—
14#19097750—21no—
15#19097730C2211yescharacteristic
16C2×C4⋊Dic157C2211yescharacteristic
17#19097705—21no—
18#19097703—21no—
19#19097695—21no—
20#19097692—21no—
21#19097689—22no—
22#19097678—22no—
23C2×#41×C157C2211yescharacteristic
24C2×M16×C157C2211yescharacteristic
25C2×C42×C157C2211yescharacteristic
26#104×C157—22no—
27C2×Dic314—21no—
28C2×Dic314—21no—
29C2×Dic314—21no—
30C2×Dic314—21no—
31C2×Dic314—21no—
32C2×Dic314—21no—
33C22×Dic157—21no—
34C22×Dic157D412yes—
35#14578938—21no—
36#14578938—21no—
37C2×#11929434—21no—
38C4⋊Dic157—21no—
39C4⋊Dic157—21no—
40C4⋊Dic157D411yescharacteristic
41C4⋊Dic157D411yescharacteristic
42#14578933—22no—
43#14578933—22no—
44C22⋊Dic157—22no—
45C22⋊Dic157—22no—
46C22⋊Dic157—21no—
47C4×Dic157—21no—
48#41×C157—21no—
49#41×C157—21no—
50#41×C157—21no—
51#41×C157—21no—
52C2×Q8×C157D411yescharacteristic
53C2×D4×C157—21no—
54C2×D4×C157D411yescharacteristic
55C22×C4×C157—21no—
56C22×C4×C157—21no—
57C22×C4×C157C2311yescharacteristic
58M16×C157—21no—
59M16×C157D412yes—
60C2×C8×C157—21no—
61C42×C157—21no—
62C42×C157—21no—
63Dic314—41no—
64Dic314—41no—
65Dic314—41no—
66Dic314—41no—
67C2×Dic157—22no—
68C2×Dic157—21no—
69C2×Dic157—21no—
70C2×Dic157—22no—
71C2×Dic157—22no—
72C2×Dic157—42no—
73C2×Dic157—41no—
74#11929434—41no—
75C23×C157—21no—
76C23×C157C2×D411yescharacteristic
77Q8×C157—21no—
78Q8×C157—21no—
79D4×C157—21no—
80D4×C157—21no—
81D4×C157—41no—
82D4×C157—41no—
83C2×C4×C157—22no—
84C2×C4×C157—21no—
85C2×C4×C157—41no—
86C2×C4×C157—41no—
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—22no—
96C22×C157—21no—
97C22×C157—21no—
98C22×C157—41no—
99C22×C157—41no—
100C22×C157—41no—
101C22×C157#12011yescharacteristic
102C22×C157#12111yescharacteristic
103C22×C157#12311yescharacteristic
104C4×C157—42no—
105C4×C157—41no—
106C4×C157#12111yescharacteristic
107C4×C157#12111yescharacteristic
108C4×C157#12311yescharacteristic
109C4×C157#12411yescharacteristic
110Dic157—42no—
111Dic157—41no—
112Dic157—41no—
113Dic157—81no—
114C2×C157—21no—
115C2×C157—21no—
116C2×C157—81no—
117C2×C157#39411yescharacteristic
118C2×C157#45411yescharacteristic
119C2×C157#45611yescharacteristic
120C157#203811yescharacteristic, complemented, Sylow
121#2038—1571nomaximal, Sylow
122#536—1571no—
123#428—1571no—
124C2×#104D15711yescharacteristic
125#417—1571no—
126C2×#101—1571no—
127#391—1571no—
128#339—1571no—
129C2×#41C2×D15711yescharacteristic
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#117—1571no—
139C2×C4⋊C4—1571no—
140C2×C42C2×D15711yescharacteristic
141#107—3141no—
142#106—3141no—
143#104—22no—
144#103—3141no—
145#102—3141no—
146#101—3142no—
147#95—3142no—
148#41—21no—
149#41—21no—
150#41—21no—
151#41—21no—
152C2×Q8—3141no—
153C2×Q8—3141no—
154C2×Q8—3141no—
155C2×Q8—3141no—
156C2×Q8—3141no—
157C2×Q8—3141no—
158C2×Q8D62811yescharacteristic
159C2×D4—21no—
160C2×D4D62811yescharacteristic
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—3141no—
164C22×C4—1572no—
165C22×C4C22×D15711yescharacteristic
166M16—21no—
167M16—3141no—
168M16—3141no—
169M16C22⋊D15712yes—
170C2×C8—21no—
171C2×C8—3141no—
172C4⋊C4—3142no—
173C4⋊C4—3142no—
174C4⋊C4—3141no—
175C4⋊C4—3141no—
176C4⋊C4—1571no—
177C4⋊C4—1571no—
178C22⋊C4—3142no—
179C22⋊C4—3142no—
180C22⋊C4—3141no—
181C42—21no—
182C42—21no—
183C42—3141no—
184C23—21no—
185C23D4×D15711yescharacteristic
186Q8—21no—
187Q8—21no—
188Q8—6281no—
189Q8—6281no—
190Q8—6281no—
191Q8—6281no—
192D4—21no—
193D4—21no—
194D4—41no—
195D4—41no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—3141no—
203C2×C4—3141no—
204C2×C4—3142no—
205C2×C4—3142no—
206C2×C4—3142no—
207C2×C4—6282no—
208C2×C4—6281no—
209C2×C4D4×D15711yescharacteristic
210C2×C4C2×C22⋊D15711yescharacteristic
211C2×C4C2×D62811yescharacteristic
212C2×C4#1457895311yescharacteristic
213C2×C4#1457895311yescharacteristic
214C2×C4#1457895411yescharacteristic
215C8—22no—
216C8—6281no—
217C22—21no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22#1909774111yescharacteristic
223C22#1909774211yescharacteristic
224C22#1909775311yescharacteristic
225C4—42no—
226C4—41no—
227C4—6282no—
228C4—6281no—
229C4—6281no—
230C4—12561no—
231C4#1909774311yescharacteristic
232C4#1909774511yescharacteristic
233C4#1909775511yescharacteristic
234C4#1909775711yescharacteristic
235C2—21no—
236C2—21no—
237C2—81no—
238C2#2375718911yescharacteristic
239C2#2375744711yescharacteristic
240C2#2375745711yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
FittingC2×#104×C157C2
socleC22×C157#120
prime radicalC2×#41×C157C22
hypercenterC2×#104D157
hypocenterC157#2038
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
2. #23757315C2semidirect
4. C2×#19097689C2semidirect
5. #23757179C2semidirect
6. #23757033C2semidirect
120. C157#2038semidirect

Maximal subgroups

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

subgroupindex
#237578442
#237573152
#237572742
C2×#190976892
#237571792
#237570332
C2×#104×C1572
#2038157