Finite Group Databasebeta

← #34100364

Subgroups — #34100364

Order 20096.

302 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 97 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#23757756C211yesmaximal, characteristic
2#23757315C211yesmaximal, characteristic, complemented
3#23757274C211yesmaximal, characteristic
4#23757143C211yesmaximal, characteristic, complemented
5#23757041C211yesmaximal, characteristic, complemented
6#23757033C211yesmaximal, characteristic, complemented
7C2×#104×C157C211yesmaximal, characteristic
8C2×#14578938C2211yescharacteristic
9Q8×Dic157—21no—
10D4×Dic157—21no—
11D4×Dic157—22no—
12#19097730—21no—
13#19097730C2211yescharacteristic
14C2×C4⋊Dic157—21no—
15C2×C4⋊Dic157—22no—
16C2×C4⋊Dic157C2211yescharacteristic
17C2×C4⋊Dic157C2211yescharacteristic
18#19097705—21no—
19#19097703—21no—
20#19097695—21no—
21#19097692—21no—
22C2×#41×C157C2211yescharacteristic
23C2×M16×C157C2211yescharacteristic
24C2×C42×C157C2211yescharacteristic
25#104×C157C414yescomplemented
26C22×Dic157—22no—
27C22×Dic157—21no—
28C22×Dic157D412yes—
29#14578938—21no—
30#14578938—21no—
31C2×#11929434—21no—
32C4⋊Dic157—22no—
33C4⋊Dic157—21no—
34C4⋊Dic157—21no—
35C4⋊Dic157—21no—
36C4⋊Dic157—21no—
37C4⋊Dic157—44no—
38C4⋊Dic157D411yescharacteristic
39C4⋊Dic157D411yescharacteristic
40C22⋊Dic157—22no—
41C22⋊Dic157—21no—
42C4×Dic157—22no—
43C4×Dic157—21no—
44#41×C157—21no—
45#41×C157—21no—
46#41×C157C2×C412yes—
47#41×C157Q812yes—
48C2×Q8×C157D411yescharacteristic
49C2×D4×C157—21no—
50C2×D4×C157D411yescharacteristic
51C22×C4×C157—21no—
52C22×C4×C157—21no—
53C22×C4×C157C2311yescharacteristic
54M16×C157—21no—
55M16×C157C2×C412yescomplemented
56C2×C8×C157—21no—
57C42×C157—21no—
58C42×C157C2×C412yescomplemented
59C2×Dic157—21no—
60C2×Dic157—22no—
61C2×Dic157—21no—
62C2×Dic157—42no—
63C2×Dic157—42no—
64C2×Dic157—41no—
65C2×Dic157—82no—
66#11929434—41no—
67C23×C157—21no—
68C23×C157C2×D411yescharacteristic
69Q8×C157—21no—
70Q8×C157C4⋊C412yescomplemented
71D4×C157—22no—
72D4×C157—22no—
73D4×C157—21no—
74D4×C157C4⋊C412yescomplemented
75C2×C4×C157—22no—
76C2×C4×C157—22no—
77C2×C4×C157—22no—
78C2×C4×C157—21no—
79C2×C4×C157—22no—
80C2×C4×C157—22no—
81C2×C4×C157C22×C411yescharacteristic
82C2×C4×C157C2×D411yescharacteristic
83C2×C4×C157C2×D411yescharacteristic
84C2×C4×C157C2×Q811yescharacteristic
85C2×C4×C157#4111yescharacteristic
86C2×C4×C157#4111yescharacteristic
87C8×C157—22no—
88C22×C157—21no—
89C22×C157—21no—
90C22×C157—22no—
91C22×C157—41no—
92C22×C157—41no—
93C22×C157C4×D411yescharacteristic
94C22×C157#12011yescharacteristic
95C22×C157C22⋊Q811yescharacteristic
96C4×C157—24no—
97C4×C157—22no—
98C4×C157C2×C4⋊C411yescharacteristic
99C4×C157C4×D411yescharacteristic
100C4×C157C22⋊Q811yescharacteristic
101C4×C157#12311yescharacteristic
102Dic157—42no—
103Dic157—41no—
104Dic157—41no—
105Dic157—81no—
106C2×C157—21no—
107C2×C157—21no—
108C2×C157—42no—
109C2×C157#38511yescharacteristic
110C2×C157#45311yescharacteristic
111C2×C157#45611yescharacteristic
112C157#185911yescharacteristic, complemented, Sylow
113#1859—1571nomaximal, Sylow
114#520—1571no—
115#428—1571no—
116C2×#104D15711yescharacteristic
117#417—1571no—
118#384—1571no—
119#343—1571no—
120#339—1571no—
121C2×#41C2×D15711yescharacteristic
122C2×M16—1571no—
123C2×M16C2×D15711yescharacteristic
124C4×Q8—3141no—
125C4×D4—3141no—
126C4×D4—3142no—
127#117—3141no—
128#117—1571no—
129C2×C4⋊C4—3141no—
130C2×C4⋊C4—3142no—
131C2×C4⋊C4—1571no—
132C2×C4⋊C4—1571no—
133C2×C42C2×D15711yescharacteristic
134#107—3141no—
135#106—3141no—
136#104Dic15714yescomplemented
137#103—3141no—
138#102—3141no—
139#41—21no—
140#41—21no—
141#41C2×Dic15712yes—
142#41Dic31412yes—
143C2×Q8D62811yescharacteristic
144C2×D4—21no—
145C2×D4D62811yescharacteristic
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—3142no—
149C22×C4—3141no—
150C22×C4—1572no—
151C22×C4C22×D15711yescharacteristic
152M16—21no—
153M16—3141no—
154M16—3141no—
155M16C2×Dic15712yescomplemented
156C2×C8—21no—
157C2×C8—3141no—
158C4⋊C4—3142no—
159C4⋊C4—3141no—
160C4⋊C4—3141no—
161C4⋊C4—3141no—
162C4⋊C4—3141no—
163C4⋊C4—6284no—
164C4⋊C4—1571no—
165C4⋊C4—1571no—
166C22⋊C4—3142no—
167C22⋊C4—3141no—
168C42—21no—
169C42—3142no—
170C42—3141no—
171C42C2×Dic15712yescomplemented
172C23—21no—
173C23D4×D15711yescharacteristic
174Q8—21no—
175Q8C4⋊Dic15712yescomplemented
176D4—22no—
177D4—22no—
178D4—21no—
179D4C4⋊Dic15712yescomplemented
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—3141no—
187C2×C4—3142no—
188C2×C4—3141no—
189C2×C4—6282no—
190C2×C4—6282no—
191C2×C4—6281no—
192C2×C4—12562no—
193C2×C4C22×Dic15711yescharacteristic
194C2×C4D4×D15711yescharacteristic
195C2×C4C2×D62811yescharacteristic
196C2×C4C2×Dic31411yescharacteristic
197C2×C4#1457895311yescharacteristic
198C2×C4#1457895311yescharacteristic
199C8—22no—
200C8—6281no—
201C22—21no—
202C22—21no—
203C22—22no—
204C22—41no—
205C22—41no—
206C22D4×Dic15711yescharacteristic
207C22#1909774111yescharacteristic
208C22#1909774811yescharacteristic
209C4—24no—
210C4—22no—
211C4—6282no—
212C4—6281no—
213C4—6281no—
214C4—12561no—
215C4C2×C4⋊Dic15711yescharacteristic
216C4D4×Dic15711yescharacteristic
217C4#1909774811yescharacteristic
218C4#1909775511yescharacteristic
219C2—21no—
220C2—21no—
221C2—42no—
222C2#2375714811yescharacteristic
223C2#2375744211yescharacteristic
224C2#2375745711yescharacteristic
subgroupG/H
centerC22#19097741
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
FittingC2×#104×C157C2
socleC22×C157#120
prime radicalC2×#41×C157C22
cosocleC22×C4×C157C23
hypercenterC2×#104D157
hypocenterC157#1859
LCSG ▷ C2×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. #23757143C2semidirect
5. #23757041C2semidirect
6. #23757033C2semidirect
25. #104×C157C4semidirect
55. M16×C157C2×C4semidirect
58. C42×C157C2×C4semidirect
70. Q8×C157C4⋊C4semidirect
74. D4×C157C4⋊C4semidirect
112. C157#1859semidirect
136. #104Dic157semidirect
155. M16C2×Dic157semidirect
171. C42C2×Dic157semidirect
175. Q8C4⋊Dic157semidirect
179. D4C4⋊Dic157semidirect

Maximal subgroups

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

subgroupindex
#237577562
#237573152
#237572742
#237571432
#237570412
#237570332
C2×#104×C1572
#1859157