Finite Group Databasebeta

← #34100086

Subgroups — #34100086

Order 20096.

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

normal subgroups only

This group has 548 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#23758109C211yesmaximal, characteristic
2#23757257C211yesmaximal, characteristic, complemented
3#23757255C212yesmaximal, complemented
4#23757239C212yesmaximal, complemented
5#408×C157C211yesmaximal, characteristic, complemented
6#19097819—21no—
7#19097818—22no—
8#19097817—22no—
9#19097816—21no—
10#19097816C414yes—
11C2×#14578954—22no—
12C2×#14578954C2212yes—
13#41×D157—24no—
14C2×#14578938C2212yescomplemented
15#19097690—21no—
16#19097689—22no—
17#19097688—21no—
18#19097687—22no—
19#19097683—24no—
20C2×#41×C157C2211yescharacteristic
21C2×M16×C157C2211yescharacteristic, complemented
22C22×C8×C157C2211yescharacteristic, complemented
23C22⋊C8×C157—22no—
24#14578955—24no—
25#14578954—24no—
26#14578954—24no—
27#14578954—24no—
28#14578954—22no—
29#14578954—22no—
30#14578954—22no—
31#14578954—22no—
32#14578953—24no—
33#14578953—28no—
34C2×Dic314—22no—
35C2×Dic314D412yes—
36Q8×D157—24no—
37C2×D628—22no—
38C2×D628D412yes—
39C2×C22⋊D157—24no—
40C2×C22⋊D157C2×C414yes—
41D4×D157—24no—
42D4×D157—28no—
43C2×C4×D157—24no—
44C2×C4×D157D414yes—
45#14578938—22no—
46#14578938—22no—
47C2×#11929434—22no—
48#41×C157—22no—
49#41×C157—22no—
50C2×Q8×C157D411yescharacteristic
51C2×D4×C157C2×C412yes—
52C2×D4×C157D411yescharacteristic
53C22×C4×C157D412yes—
54C22×C4×C157C2311yescharacteristic
55M16×C157—22no—
56C2×C8×C157—21no—
57C2×C8×C157—21no—
58C2×C8×C157—22no—
59C22×D157—24no—
60C22×D157C22⋊C414yes—
61Dic314—24no—
62Dic314—22no—
63Dic314—22no—
64D628—24no—
65D628—22no—
66D628—22no—
67C22⋊D157—28no—
68C22⋊D157—24no—
69C22⋊D157—24no—
70C2×Dic157—24no—
71C2×Dic157C22⋊C414yes—
72C4×D157—28no—
73C4×D157—24no—
74C4×D157—24no—
75#11929434—42no—
76C23×C157C22⋊C412yes—
77C23×C157C22×C411yescharacteristic
78Q8×C157—22no—
79D4×C157—24no—
80D4×C157—22no—
81C2×C4×C157—22no—
82C2×C4×C157—22no—
83C2×C4×C157C22⋊C412yes—
84C2×C4×C157C2×D412yes—
85C2×C4×C157C2×D412yes—
86C2×C4×C157C2×D411yescharacteristic
87C2×C4×C157C2×D411yescharacteristic
88C8×C157—41no—
89C8×C157—41no—
90C22×C157—22no—
91C22×C157—41no—
92C22×C157—42no—
93C22×C157C2×C22⋊C412yes—
94C22×C157C2×C22⋊C411yescharacteristic
95C2×D157—24no—
96C2×D157—44no—
97C2×D157—84no—
98C4×C157—22no—
99C4×C157#12011yescharacteristic
100C4×C157#12011yescharacteristic
101C4×C157#12012yes—
102Dic157—24no—
103C2×C157—22no—
104C2×C157—21no—
105C2×C157—42no—
106C2×C157#37911yescharacteristic
107D157—44no—
108C157#179111yescharacteristic, complemented, Sylow
109#1791—1571nomaximal, Sylow
110#585—1571no—
111#413—1571no—
112#413—1572no—
113#408—1572no—
114#408D15711yescharacteristic, complemented
1152−1+4—3142no—
1162−1+4—3141no—
117H(5,𝔽2)—3142no—
118H(5,𝔽2)—3141no—
119H(5,𝔽2)—1574no—
120C2×#41—3144no—
121C2×#41—3142no—
122C2×#41—1572no—
123C2×#41C2×D15711yescharacteristic
124C2×M16—1572no—
125C2×M16C2×D15711yescharacteristic, complemented
126C22×C8C2×D15711yescharacteristic, complemented
127#101—3142no—
128#101—3141no—
129#100—3142no—
130#100—3141no—
131C22⋊C8—22no—
132C22⋊C8—3144no—
133#41—22no—
134#41—22no—
135#41—3144no—
136#41—3144no—
137#41—3144no—
138#41—3144no—
139#41—3148no—
140#41—3142no—
141#41—3142no—
142#41—3142no—
143#41—3142no—
144#41—3144no—
145C2×Q8—3142no—
146C2×Q8—3144no—
147C2×Q8—1572no—
148C2×Q8D62811yescharacteristic
149C2×D4—3142no—
150C2×D4—3144no—
151C2×D4—3148no—
152C2×D4—3144no—
153C2×D4—1574no—
154C2×D4—1572no—
155C2×D4C4×D15712yes—
156C2×D4D62811yescharacteristic
157C22×C4—3144no—
158C22×C4—1574no—
159C22×C4C22⋊D15712yes—
160C22×C4C22×D15711yescharacteristic
161M16—22no—
162M16—3142no—
163M16—3142no—
164C2×C8—22no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—3142no—
168C23—3144no—
169C23—1574no—
170C23#1457893112yes—
171C23C2×C4×D15711yescharacteristic
172Q8—22no—
173Q8—3144no—
174Q8—3142no—
175Q8—3142no—
176D4—24no—
177D4—22no—
178D4—3144no—
179D4—3148no—
180D4—3144no—
181D4—3144no—
182D4—3142no—
183D4—3142no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—3148no—
187C2×C4—3144no—
188C2×C4—3144no—
189C2×C4—3144no—
190C2×C4—1574no—
191C2×C4#1457893112yes—
192C2×C4D4×D15712yes—
193C2×C4D4×D15712yes—
194C2×C4C2×C22⋊D15711yescharacteristic
195C2×C4C2×D62811yescharacteristic
196C8—41no—
197C8—41no—
198C8—6282no—
199C22—22no—
200C22—41no—
201C22—42no—
202C22—3144no—
203C22—6284no—
204C22—12564no—
205C22C22⋊C4×D15712yes—
206C22C2×#1457893111yescharacteristic
207C4—22no—
208C4—3144no—
209C4#1909773911yescharacteristic
210C4#1909773911yescharacteristic
211C4#1909774112yes—
212C2—21no—
213C2—22no—
214C2—42no—
215C2—6284no—
216C2#2375712811yescharacteristic
subgroupG/H
centerC4#19097739
commutatorC23×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#408×C157C2
socleC2×C157#379
prime radical#23758109C2
cosocleC22×C4×C157C23
hypercenter#408D157
hypocenterC157#1791
LCSG ▷ C23×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #23757257C2semidirect
3. #23757255C2semidirect
4. #23757239C2semidirect
5. #408×C157C2semidirect
14. C2×#14578938C22semidirect
21. C2×M16×C157C22semidirect
22. C22×C8×C157C22semidirect
108. C157#1791semidirect
114. #408D157semidirect
125. C2×M16C2×D157semidirect
126. C22×C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237581092
#237572572
#237572552
#237572392
#408×C1572
#1791157