Finite Group Databasebeta

← #34110520

Subgroups — #34110520

Order 20096.

400 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 125 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#23757826C211yesmaximal, characteristic, complemented
2C2×#19097758C211yesmaximal, characteristic
3#23757631C214yesmaximal, complemented
4#23757626C214yesmaximal, complemented
5#23757283C211yesmaximal, characteristic, complemented
6#23757281C211yesmaximal, characteristic, complemented
7#23757278C211yesmaximal, characteristic
8M16×Dic157C211yesmaximal, characteristic, complemented
9#418×C157C211yesmaximal, characteristic
10C22×Dic314C2211yescharacteristic
11C2×#14578938C2211yescharacteristic
12#19097770C2212yescomplemented
13#19097762C2212yescomplemented
14#19097758—22no—
15#19097758C2212yes—
16#19097758C2212yes—
17#19097748—22no—
18C4×Dic314—22no—
19#19097730C2211yescharacteristic
20C2×C22⋊Dic157—22no—
21C2×C4×Dic157C2211yescharacteristic
22#19097696C2214yescomplemented
23#19097694C2214yescomplemented
24#19097692C2214yes—
25#19097681C2212yescomplemented
26C8×Dic157C2212yescomplemented
27C22×D4×C157C2211yescharacteristic
28C2×M16×C157C2211yescharacteristic
29#117×C157C2211yescharacteristic
30#102×C157C2214yes—
31C2×Dic314—22no—
32C2×Dic314—21no—
33C2×Dic314—22no—
34C2×Dic314—22no—
35C2×Dic314C2312yes—
36C22×Dic157—21no—
37C22×Dic157D412yes—
38#14578938—22no—
39C2×#11929434C2312yes—
40C4⋊Dic157—22no—
41C4⋊Dic157C2312yes—
42#14578933—22no—
43#14578933—22no—
44C22⋊Dic157—24no—
45C22⋊Dic157—24no—
46C22⋊Dic157—21no—
47C4×Dic157—21no—
48C4×Dic157D412yes—
49C4×Dic157C2311yescharacteristic
50C4×Dic157C2311yescharacteristic
51C24×C157—21no—
52C2×D4×C157—21no—
53C2×D4×C157—22no—
54C2×D4×C157—22no—
55C2×D4×C157C2312yes—
56C22×C4×C157C2311yescharacteristic
57M16×C157—22no—
58C2×C8×C157C2312yes—
59C4⋊C4×C157C2312yes—
60C22⋊C4×C157—21no—
61C42×C157—21no—
62Dic314—22no—
63Dic314—22no—
64Dic314—44no—
65Dic314—42no—
66C2×Dic157—22no—
67C2×Dic157—22no—
68C2×Dic157—42no—
69C2×Dic157—42no—
70C2×Dic157C2×D412yes—
71C2×Dic157C2×D412yes—
72#11929434—22no—
73C23×C157—22no—
74C23×C157—44no—
75C23×C157—42no—
76C23×C157C2×D411yescharacteristic
77D4×C157—22no—
78D4×C157—22no—
79D4×C157—42no—
80C2×C4×C157—22no—
81C2×C4×C157C2×D411yescharacteristic
82C2×C4×C157#4112yes—
83C2×C4×C157#4112yes—
84C2×C4×C157C2411yescharacteristic
85C8×C157—22no—
86C22×C157—22no—
87C22×C157—42no—
88C22×C157—44no—
89C22×C157—84no—
90C22×C157#12412yes—
91C22×C157C22×D411yescharacteristic
92C4×C157—42no—
93C4×C157#12412yes—
94C4×C157C2×#4111yescharacteristic
95C4×C157C2×#4111yescharacteristic
96Dic157—22no—
97Dic157—22no—
98Dic157—42no—
99Dic157—42no—
100C2×C157—22no—
101C2×C157—82no—
102C2×C157C2×#12411yescharacteristic
103C2×C157#57512yes—
104C157#314011yescharacteristic, complemented, Sylow
105#3140—1571nomaximal, Sylow
106#533—1571no—
107C2×#124—1571no—
108#489—1574no—
109#488—1574no—
110#419—1571no—
111#419—1571no—
112#418—1571no—
113#418D15711yescharacteristic
114C4×M16—1571no—
115C22×Q8—1571no—
116C22×D4C2×D15711yescharacteristic
117C2×M16—1571no—
118C2×M16C2×D15711yescharacteristic
119C4⋊Q8—1572no—
120#125—1572no—
121#124—3142no—
122#124—1572no—
123#124—1572no—
124C22⋊Q8—3142no—
125C4×Q8—3142no—
126#117—1571no—
127#117C2×D15711yescharacteristic
128C2×C22⋊C4—3142no—
129C2×C42—1571no—
130#103—1574no—
131#103—1574no—
132#102—1574no—
133#102C2×D15714yes—
134#97—1572no—
135C4×C8—1572no—
136C24—21no—
137C2×Q8—3142no—
138C2×Q8—3141no—
139C2×Q8—3142no—
140C2×Q8—3142no—
141C2×Q8—1572no—
142C2×D4—21no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4C22×D15712yes—
146C22×C4—3141no—
147C22×C4—1572no—
148C22×C4C22×D15711yescharacteristic
149M16—22no—
150M16—3142no—
151C2×C8—1572no—
152C2×C8C22×D15712yes—
153C4⋊C4—3142no—
154C4⋊C4—3142no—
155C4⋊C4—3142no—
156C4⋊C4—1572no—
157C4⋊C4C22×D15712yes—
158C22⋊C4—21no—
159C22⋊C4—3141no—
160C22⋊C4—3144no—
161C22⋊C4—3144no—
162C42—21no—
163C42—3141no—
164C42—1572no—
165C42—1571no—
166C42—1571no—
167C23—22no—
168C23—44no—
169C23—42no—
170C23D4×D15711yescharacteristic
171Q8—3142no—
172Q8—3142no—
173Q8—6284no—
174Q8—6282no—
175D4—22no—
176D4—22no—
177D4—42no—
178C2×C4—22no—
179C2×C4—3142no—
180C2×C4—3142no—
181C2×C4—6282no—
182C2×C4—6282no—
183C2×C4—1572no—
184C2×C4—1572no—
185C2×C4D4×D15711yescharacteristic
186C2×C4#1457895312yes—
187C2×C4#1457895412yes—
188C2×C4C23×D15711yescharacteristic
189C8—22no—
190C8—3142no—
191C22—22no—
192C22—42no—
193C22—44no—
194C22—84no—
195C22#1909775712yes—
196C22C2×D4×D15711yescharacteristic
197C4—42no—
198C4—3142no—
199C4—3142no—
200C4—6282no—
201C4—6282no—
202C4#1909775712yes—
203C4C2×#1457895311yescharacteristic
204C4C2×#1457895411yescharacteristic
205C2—22no—
206C2—82no—
207C2C2×#1909775711yescharacteristic
208C2#2375806912yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#418×C157C2
socleC22×C157C22×D4
prime radicalC22×D4×C157C22
hypercenter#418D157
hypocenterC157#3140
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #23757826C2semidirect
3. #23757631C2semidirect
4. #23757626C2semidirect
5. #23757283C2semidirect
6. #23757281C2semidirect
8. M16×Dic157C2semidirect
12. #19097770C22semidirect
13. #19097762C22semidirect
22. #19097696C22semidirect
23. #19097694C22semidirect
25. #19097681C22semidirect
26. C8×Dic157C22semidirect
104. C157#3140semidirect

Maximal subgroups

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

subgroupindex
#237578262
C2×#190977582
#237576312
#237576262
#237572832
#237572812
#237572782
M16×Dic1572
#418×C1572
#3140157