Finite Group Databasebeta

← #34100109

Subgroups — #34100109

Order 20096.

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

normal subgroups only

This group has 536 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
1C2×#19097818C211yesmaximal, characteristic
2#23757291C212yesmaximal, complemented
3#23757281C212yesmaximal, complemented
4#23757257C211yesmaximal, characteristic, complemented
5#410×C157C211yesmaximal, characteristic, complemented
6#19097818—24no—
7#19097818—24no—
8C2×#14578954C412yescomplemented
9C2×#14578954C2211yescharacteristic
10C2×#14578953—22no—
11C2×#14578953—21no—
12C22×Dic314C412yescomplemented
13C22×Dic314C2211yescharacteristic
14C2×Q8×D157—21no—
15C2×#14578938C2212yes—
16#19097699—24no—
17#19097694—24no—
18#19097690—21no—
19#19097688—21no—
20C2×#41×C157C2211yescharacteristic
21#117×C157C2212yes—
22#99×C157—22no—
23#14578954—24no—
24#14578954—24no—
25#14578954—22no—
26#14578954D414yes—
27#14578953—216no—
28#14578953—28no—
29C2×Dic314—24no—
30C2×Dic314—24no—
31C2×Dic314—22no—
32C2×Dic314—21no—
33C2×Dic314—22no—
34C2×Dic314C2×C411yescharacteristic
35C2×Dic314C2×C412yes—
36C2×Dic314D414yes—
37Q8×D157—28no—
38C2×D628C2×C411yescharacteristic
39C2×C22⋊D157—22no—
40C2×C22⋊D157—21no—
41C22×Dic157—22no—
42C22×Dic157—21no—
43C2×C4×D157—22no—
44C2×C4×D157—21no—
45#14578938—22no—
46#14578938—22no—
47C2×#11929434—22no—
48#41×C157—22no—
49#41×C157—22no—
50C2×Q8×C157D411yescharacteristic
51C2×D4×C157D412yes—
52C2×D4×C157D411yescharacteristic
53C22×C4×C157C2×C412yes—
54C22×C4×C157C2311yescharacteristic
55C4⋊C4×C157—22no—
56C22⋊C4×C157—22no—
57C42×C157—22no—
58C22×D157—21no—
59Dic314—28no—
60Dic314—21no—
61Dic314—22no—
62Dic314—24no—
63Dic314C22⋊C412yes—
64Dic314C22⋊C414yes—
65D628—21no—
66D628C22⋊C412yes—
67C22⋊D157—28no—
68C22⋊D157—24no—
69C2×Dic157—28no—
70C2×Dic157—22no—
71C2×Dic157—21no—
72C2×Dic157—24no—
73C4×D157—28no—
74C4×D157—24no—
75#11929434—42no—
76C23×C157C22⋊C412yes—
77C23×C157C2×D411yescharacteristic
78Q8×C157—22no—
79D4×C157—24no—
80D4×C157—22no—
81C2×C4×C157—22no—
82C2×C4×C157—22no—
83C2×C4×C157—42no—
84C2×C4×C157C22⋊C412yes—
85C2×C4×C157C22×C411yescharacteristic
86C2×C4×C157C2×D412yes—
87C2×C4×C157C2×D412yes—
88C2×C4×C157C2×D411yescharacteristic
89C22×C157—22no—
90C22×C157—41no—
91C22×C157—42no—
92C22×C157C2×C22⋊C411yescharacteristic
93C22×C157#12012yes—
94C2×D157—22no—
95C2×D157—42no—
96C4×C157—22no—
97C4×C157—82no—
98C4×C157C2×C22⋊C411yescharacteristic
99C4×C157C2×C22⋊C411yescharacteristic
100C4×C157#12012yes—
101Dic157—24no—
102Dic157—22no—
103C2×C157—22no—
104C2×C157—21no—
105C2×C157—42no—
106C2×C157#37911yescharacteristic
107D157—42no—
108C157#179611yescharacteristic, complemented, Sylow
109#1796—1571nomaximal, Sylow
110C2×2−1+4—1571no—
111#421—1572no—
112#419—1572no—
113#413—1571no—
114#410D15711yescharacteristic, complemented
1152−1+4—3144no—
1162−1+4—3144no—
117C2×#41—3142no—
118C2×#41—3141no—
119C2×#41—1572no—
120C2×#41—1571no—
121C2×#41C2×D15711yescharacteristic
122C22×Q8—3141no—
123C22×Q8—1572no—
124C22×Q8—1571no—
125C2×M16—1572no—
126#117C2×D15712yes—
127#104—3144no—
128#103—3144no—
129#101—3141no—
130#100—3141no—
131#99—22no—
132#41—22no—
133#41—22no—
134#41—3144no—
135#41—3144no—
136#41—31416no—
137#41—3148no—
138#41—3142no—
139#41—1574no—
140C2×Q8—3144no—
141C2×Q8—3144no—
142C2×Q8—3148no—
143C2×Q8—3142no—
144C2×Q8—3141no—
145C2×Q8—3142no—
146C2×Q8—1571no—
147C2×Q8—1572no—
148C2×Q8—1574no—
149C2×Q8D62811yescharacteristic
150C2×D4—3141no—
151C2×D4—3142no—
152C2×D4—1571no—
153C2×D4C22⋊D15712yes—
154C2×D4D62811yescharacteristic
155C22×C4—3141no—
156C22×C4—3142no—
157C22×C4—3141no—
158C22×C4—3142no—
159C22×C4C4×D15712yes—
160C22×C4C22×D15711yescharacteristic
161M16—3142no—
162M16—3142no—
163C2×C8—3142no—
164C4⋊C4—22no—
165C22⋊C4—22no—
166C42—22no—
167C23—3141no—
168C23#1457893112yes—
169C23C2×C22⋊D15711yescharacteristic
170Q8—22no—
171Q8—3148no—
172Q8—3141no—
173Q8—3142no—
174Q8—3144no—
175Q8—1572no—
176Q8—1574no—
177D4—24no—
178D4—22no—
179D4—3148no—
180D4—3144no—
181D4—3141no—
182D4—1572no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—42no—
186C2×C4—3148no—
187C2×C4—3148no—
188C2×C4—3142no—
189C2×C4—3141no—
190C2×C4—3144no—
191C2×C4—3144no—
192C2×C4#1457893112yes—
193C2×C4C2×C4×D15711yescharacteristic
194C2×C4D4×D15712yes—
195C2×C4D4×D15712yes—
196C2×C4C2×D62811yescharacteristic
197C8—6282no—
198C22—22no—
199C22—41no—
200C22—42no—
201C22—3142no—
202C22—6282no—
203C22C2×#1457893111yescharacteristic
204C22#1909773912yes—
205C4—22no—
206C4—82no—
207C4—3144no—
208C4—3142no—
209C4C22⋊C4×D15711yescharacteristic
210C4C22⋊C4×D15711yescharacteristic
211C4#1909774112yes—
212C2—22no—
213C2—21no—
214C2—42no—
215C2—6282no—
216C2#2375712811yescharacteristic
subgroupG/H
centerC2#23757128
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#410×C157C2
socleC2×C157#379
prime radicalC2×#19097818C2
cosocleC22×C4×C157C23
hypercenter#410D157
hypocenterC157#1796
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #23757291C2semidirect
3. #23757281C2semidirect
4. #23757257C2semidirect
5. #410×C157C2semidirect
8. C2×#14578954C4semidirect
12. C22×Dic314C4semidirect
108. C157#1796semidirect
114. #410D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978182
#237572912
#237572812
#237572572
#410×C1572
#1796157