Finite Group Databasebeta

← #34101787

Subgroups — #34101787

Order 20096.

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

normal subgroups only

This group has 266 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#23757061C214yesmaximal, complemented
4#413×C157C211yesmaximal, characteristic, complemented
5C22×D628C411yescharacteristic, complemented
6#19097767—21no—
7#19097760—21no—
8#19097745—21no—
9#19097745—21no—
10#19097739C412yescomplemented
11#19097739C2212yes—
12#19097730—21no—
13#19097730C411yescharacteristic, complemented
14#19097686—21no—
15#19097686C2212yes—
16C2×#41×C157C2211yescharacteristic
17C2×M16×C157—21no—
18#101×C157—21no—
19#100×C157C2212yes—
20C23×D157C2×C412yes—
21C2×D628—21no—
22C2×D628—21no—
23C2×D628—41no—
24C2×D628—41no—
25C2×C22⋊D157—22no—
26C2×C22⋊D157—22no—
27C4⋊Dic157—21no—
28C4⋊Dic157—41no—
29C22⋊Dic157—22no—
30C22⋊Dic157C2×C412yes—
31#14578931—22no—
32#14578931—22no—
33C4×Dic157—21no—
34C4×Dic157—41no—
35#41×C157—41no—
36#41×C157—41no—
37C2×Q8×C157D411yescharacteristic
38C2×D4×C157C2×C411yescharacteristic
39C2×D4×C157D411yescharacteristic
40C2×D4×C157C2311yescharacteristic
41C22×C4×C157C2×C411yescharacteristic
42C22×C4×C157D411yescharacteristic
43C22×C4×C157D411yescharacteristic
44M16×C157—22no—
45M16×C157—41no—
46C2×C8×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×C157C22×C411yescharacteristic
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×C157C22⋊C411yescharacteristic
71C2×C4×C157C22⋊C411yescharacteristic
72C2×C4×C157C22⋊C411yescharacteristic
73C2×C4×C157C2×D411yescharacteristic
74C8×C157—42no—
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#9911yescharacteristic
90C4×C157#9911yescharacteristic
91Dic157—42no—
92Dic157—82no—
93C2×C157—21no—
94C2×C157—41no—
95C2×C157—41no—
96C2×C157—41no—
97C2×C157C2×#9911yescharacteristic
98D157—82no—
99C157#212211yescharacteristic, complemented, Sylow
100#2122—1571nomaximal, Sylow
101#535—1571no—
102#413D15711yescharacteristic, complemented
103#410—1571no—
104AΣL(1,𝔽16)—1574no—
105C2×#41C2×D15711yescharacteristic
106C22×D4—1571no—
107C2×M16—21no—
108#127—3141no—
109#124—3141no—
110#121—3141no—
111#121—3141no—
112#120—1572no—
113#120—1572no—
114#117—3141no—
115#117—1571no—
116#101—21no—
117#100C2×D15712yes—
118#99—3141no—
119#99—1572no—
120C24—1572no—
121#41—41no—
122#41—41no—
123C2×Q8C22⋊D15711yescharacteristic
124C2×D4—3142no—
125C2×D4—3142no—
126C2×D4—3141no—
127C2×D4—3141no—
128C2×D4—6281no—
129C2×D4—6281no—
130C2×D4C4×D15711yescharacteristic
131C2×D4D62811yescharacteristic
132C2×D4C22×D15711yescharacteristic
133C22×C4C4×D15711yescharacteristic
134C22×C4C22⋊D15711yescharacteristic
135C22×C4D62811yescharacteristic
136M16—22no—
137M16—41no—
138C2×C8—41no—
139C4⋊C4—3141no—
140C4⋊C4—6281no—
141C22⋊C4—3142no—
142C22⋊C4—3142no—
143C22⋊C4—3142no—
144C22⋊C4—1572no—
145C42—3141no—
146C42—6281no—
147C23—3142no—
148C23—6282no—
149C23—12562no—
150C23#1457893111yescharacteristic
151C23C2×C4×D15711yescharacteristic
152C23C2×D62811yescharacteristic
153Q8—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—6282no—
158D4—6281no—
159D4—6281no—
160D4—12562no—
161D4—12562no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—3142no—
169C2×C4—6282no—
170C2×C4#1457893111yescharacteristic
171C2×C4#1457893111yescharacteristic
172C2×C4#1457893111yescharacteristic
173C2×C4C2×C22⋊D15711yescharacteristic
174C8—42no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—6282no—
182C22—6282no—
183C22—6282no—
184C22—12564no—
185C22C2×#1457893111yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—6282no—
190C4—12562no—
191C4#1909768511yescharacteristic
192C4#1909768511yescharacteristic
193C2—21no—
194C2—41no—
195C2—41no—
196C2—41no—
197C2—12562no—
198C2C2×#1909768511yescharacteristic
subgroupG/H
centerC2C2×#19097685
commutatorC23×C157C22×C4
FrattiniC2×D4C22×D157
Fitting#413×C157C2
socleC2×C157C2×#99
prime radical#23757837C2
cosocleC2×D4×C157C23
hypercenter#413D157
hypocenterC157#2122
LCSG ▷ C23×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #23757247C2semidirect
3. #23757061C2semidirect
4. #413×C157C2semidirect
5. C22×D628C4semidirect
10. #19097739C4semidirect
13. #19097730C4semidirect
99. C157#2122semidirect
102. #413D157semidirect

Maximal subgroups

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

subgroupindex
#237578372
#237572472
#237570612
#413×C1572
#2122157