Finite Group Databasebeta

← #34101807

Subgroups — #34101807

Order 20096.

270 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 65 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#23757837C211yesmaximal, characteristic
2#23757247C211yesmaximal, characteristic, complemented
3#23757065C214yesmaximal, complemented
4#410×C157C211yesmaximal, characteristic, complemented
5C22×D628C411yescharacteristic, complemented
6#19097767C2212yes—
7#19097760C412yescomplemented
8#19097745—21no—
9#19097745—21no—
10#19097739—21no—
11#19097739—21no—
12#19097730—21no—
13#19097730C411yescharacteristic, complemented
14#19097686—21no—
15#19097686C2212yescomplemented
16C2×#41×C157C2211yescharacteristic
17#117×C157—21no—
18#99×C157—21no—
19#99×C157C2212yescomplemented
20C23×D157—21no—
21C2×D628—21no—
22C2×D628—41no—
23C2×D628—41no—
24C2×D628C2×C412yes—
25C2×C22⋊D157—22no—
26C2×C22⋊D157—22no—
27C4⋊Dic157—21no—
28C4⋊Dic157—41no—
29C22⋊Dic157—21no—
30C22⋊Dic157—22no—
31#14578931—22no—
32#14578931—22no—
33C4×Dic157—41no—
34C4×Dic157C2×C412yes—
35#41×C157—41no—
36#41×C157—41no—
37C2×Q8×C157C2×C411yescharacteristic
38C2×D4×C157D411yescharacteristic
39C2×D4×C157D411yescharacteristic
40C2×D4×C157C2311yescharacteristic
41C22×C4×C157C2×C411yescharacteristic
42C22×C4×C157D411yescharacteristic
43C22×C4×C157D411yescharacteristic
44C4⋊C4×C157—41no—
45C22⋊C4×C157—22no—
46C42×C157—41no—
47C22×D157—22no—
48C22×D157—42no—
49C22×D157—82no—
50D628—42no—
51D628—41no—
52D628—41no—
53C22⋊D157—82no—
54C22⋊D157—82no—
55C2×Dic157—22no—
56C2×Dic157—42no—
57C23×C157C22⋊C411yescharacteristic
58C23×C157C2×D411yescharacteristic
59C23×C157C2×D411yescharacteristic
60Q8×C157—41no—
61D4×C157—41no—
62D4×C157—41no—
63D4×C157—41no—
64C2×C4×C157—21no—
65C2×C4×C157—21no—
66C2×C4×C157—21no—
67C2×C4×C157—21no—
68C2×C4×C157—21no—
69C2×C4×C157—21no—
70C2×C4×C157—42no—
71C2×C4×C157C22⋊C411yescharacteristic
72C2×C4×C157C22⋊C411yescharacteristic
73C2×C4×C157C22⋊C411yescharacteristic
74C2×C4×C157C22×C411yescharacteristic
75C22×C157—21no—
76C22×C157—21no—
77C22×C157—21no—
78C22×C157—41no—
79C22×C157—41no—
80C22×C157—41no—
81C22×C157C2×C22⋊C411yescharacteristic
82C2×D157—42no—
83C2×D157—42no—
84C2×D157—42no—
85C2×D157—84no—
86C4×C157—21no—
87C4×C157—21no—
88C4×C157—21no—
89C4×C157—82no—
90C4×C157#9911yescharacteristic
91C4×C157#9911yescharacteristic
92Dic157—42no—
93Dic157—82no—
94C2×C157—21no—
95C2×C157—41no—
96C2×C157—41no—
97C2×C157—41no—
98C2×C157C2×#9911yescharacteristic
99D157—82no—
100C157#212811yescharacteristic, complemented, Sylow
101#2128—1571nomaximal, Sylow
102#535—1571no—
103#410—1571no—
104#410D15711yescharacteristic, complemented
105#353—1574no—
106C2×#41C2×D15711yescharacteristic
107C22×D4—1571no—
108#127—1572no—
109#124—1572no—
110#121—3141no—
111#121—3141no—
112#120—3141no—
113#120—3141no—
114#117—21no—
115#117—3141no—
116#117—1571no—
117#99—21no—
118#99—3141no—
119#99—1572no—
120#99C2×D15712yescomplemented
121C24—3141no—
122#41—41no—
123#41—41no—
124C2×Q8C4×D15711yescharacteristic
125C2×D4—3142no—
126C2×D4—3142no—
127C2×D4—3141no—
128C2×D4—6281no—
129C2×D4—6281no—
130C2×D4—1572no—
131C2×D4C22⋊D15711yescharacteristic
132C2×D4D62811yescharacteristic
133C2×D4C22×D15711yescharacteristic
134C22×C4C4×D15711yescharacteristic
135C22×C4C22⋊D15711yescharacteristic
136C22×C4D62811yescharacteristic
137C4⋊C4—41no—
138C4⋊C4—3141no—
139C4⋊C4—6281no—
140C22⋊C4—22no—
141C22⋊C4—3141no—
142C22⋊C4—3142no—
143C22⋊C4—3142no—
144C22⋊C4—3142no—
145C42—41no—
146C42—6281no—
147C42—1572no—
148C23—3142no—
149C23—6282no—
150C23—12562no—
151C23#1457893111yescharacteristic
152C23C2×C22⋊D15711yescharacteristic
153C23C2×D62811yescharacteristic
154Q8—41no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—6282no—
159D4—6281no—
160D4—6281no—
161D4—12562no—
162D4—12562no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—42no—
170C2×C4—3142no—
171C2×C4—6282no—
172C2×C4#1457893111yescharacteristic
173C2×C4#1457893111yescharacteristic
174C2×C4#1457893111yescharacteristic
175C2×C4C2×C4×D15711yescharacteristic
176C22—21no—
177C22—21no—
178C22—21no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—6282no—
183C22—6282no—
184C22—6282no—
185C22—12564no—
186C22C2×#1457893111yescharacteristic
187C4—21no—
188C4—21no—
189C4—21no—
190C4—82no—
191C4—6282no—
192C4—12562no—
193C4#1909768511yescharacteristic
194C4#1909768511yescharacteristic
195C2—21no—
196C2—41no—
197C2—41no—
198C2—41no—
199C2—12562no—
200C2C2×#1909768511yescharacteristic
subgroupG/H
centerC2C2×#19097685
commutatorC2×C4×C157C22×C4
FrattiniC2×D4C22×D157
Fitting#410×C157C2
socleC2×C157C2×#99
prime radical#23757837C2
cosocleC2×D4×C157C23
hypercenter#410D157
hypocenterC157#2128
LCSG ▷ C2×C4×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #23757247C2semidirect
3. #23757065C2semidirect
4. #410×C157C2semidirect
5. C22×D628C4semidirect
7. #19097760C4semidirect
13. #19097730C4semidirect
15. #19097686C22semidirect
19. #99×C157C22semidirect
100. C157#2128semidirect
104. #410D157semidirect
120. #99C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237578372
#237572472
#237570652
#410×C1572
#2128157