Finite Group Databasebeta

← #34099175

Subgroups — #34099175

Order 20096.

246 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 53 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#23757854C211yesmaximal, characteristic
2#23757592C212yesmaximal, complemented
3#23757212C211yesmaximal, characteristic, complemented
4#23757008C212yesmaximal, complemented
5#324×C157C211yesmaximal, characteristic, complemented
6C2×#14578954—21no—
7#19097751—22no—
8#19097746—22no—
9#19097745—21no—
10#19097745C2212yes—
11C2×C4⋊Dic157C2211yescharacteristic
12#19097703—22no—
13#19097693—22no—
14#19097678—21no—
15#19097678—21no—
16#19097678C2211yescharacteristic
17C2×C4⋊C4×C157C2211yescharacteristic
18C22⋊C8×C157C2212yes—
19#14578954—42no—
20#14578954—42no—
21C2×Dic314—21no—
22C2×D628—21no—
23C2×D628—21no—
24C2×D628D412yes—
25C2×C22⋊D157—22no—
26C22×Dic157—21no—
27C22×Dic157—21no—
28C2×C4×D157—22no—
29C4⋊Dic157—21no—
30C4⋊Dic157—22no—
31C4⋊Dic157D412yes—
32#14578931—22no—
33#14578931—22no—
34C22×C4×C157D411yescharacteristic
35C22×C4×C157D411yescharacteristic
36C22×C4×C157C2311yescharacteristic
37C2×C8×C157—22no—
38C4⋊C4×C157—22no—
39C22×D157—22no—
40Dic314—42no—
41D628—42no—
42D628—42no—
43D628—81no—
44C22⋊D157—42no—
45C22⋊D157—42no—
46C2×Dic157—22no—
47C2×Dic157—41no—
48C2×Dic157—41no—
49C2×Dic157—42no—
50C2×Dic157—41no—
51C2×Dic157—41no—
52C4×D157—44no—
53C23×C157C2×D411yescharacteristic
54C2×C4×C157—21no—
55C2×C4×C157—21no—
56C2×C4×C157—21no—
57C2×C4×C157—41no—
58C2×C4×C157—41no—
59C2×C4×C157SD1612yes—
60C2×C4×C157C2×D412yes—
61C8×C157—42no—
62C22×C157—21no—
63C22×C157—21no—
64C22×C157#12011yescharacteristic
65C22×C157C2×SD1611yescharacteristic
66C22×C157AGL(1,ℤ/8)11yescharacteristic
67C2×D157—42no—
68C2×D157—42no—
69C2×D157—42no—
70C4×C157—22no—
71C4×C157—22no—
72C4×C157—41no—
73Dic157—42no—
74Dic157—81no—
75Dic157—81no—
76C2×C157—21no—
77C2×C157—21no—
78C2×C157#45011yescharacteristic
79C2×C157#45411yescharacteristic
80C2×C157#45811yescharacteristic
81D157—82no—
82C157#161611yescharacteristic, complemented, Sylow
83#1616—1571nomaximal, Sylow
84#537—1571no—
85#481—1572no—
86#400—1571no—
87#327—1572no—
88#324D15711yescharacteristic, complemented
89C2×#41—3141no—
90C22⋊Q8—3142no—
91#121—3142no—
92#121—3141no—
93#121—1572no—
94C2×C4⋊C4—1571no—
95C2×C4⋊C4C2×D15711yescharacteristic
96#106—3142no—
97#102—3142no—
98C22⋊C8C2×D15712yes—
99#95—3141no—
100#95—3141no—
101#95—1571no—
102#41—6282no—
103#41—6282no—
104C2×Q8—3141no—
105C2×D4—3141no—
106C2×D4—3141no—
107C2×D4—3142no—
108C2×D4—1572no—
109C22×C4—3142no—
110C22×C4—3141no—
111C22×C4—3141no—
112C22×C4D62811yescharacteristic
113C22×C4D62811yescharacteristic
114C22×C4C22×D15711yescharacteristic
115C2×C8—22no—
116C4⋊C4—22no—
117C4⋊C4—3141no—
118C4⋊C4—3142no—
119C4⋊C4—1572no—
120C22⋊C4—3142no—
121C22⋊C4—3142no—
122C23—3142no—
123C23C2×D62811yescharacteristic
124Q8—6282no—
125D4—6282no—
126D4—6282no—
127D4—6282no—
128D4—6282no—
129D4—12561no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—3142no—
136C2×C4—6284no—
137C2×C4—6281no—
138C2×C4—6281no—
139C2×C4—6282no—
140C2×C4—6281no—
141C2×C4—6281no—
142C2×C4#1457894312yes—
143C2×C4D4×D15712yes—
144C8—42no—
145C22—21no—
146C22—21no—
147C22—6282no—
148C22—6282no—
149C22—6282no—
150C22#1909774111yescharacteristic
151C22C2×#1457894311yescharacteristic
152C22#1909779911yescharacteristic
153C4—22no—
154C4—22no—
155C4—41no—
156C4—6282no—
157C4—12561no—
158C4—12561no—
159C2—21no—
160C2—21no—
161C2—12562no—
162C2#2375742611yescharacteristic
163C2#2375744811yescharacteristic
164C2#2375746411yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#324×C157C2
socleC22×C157#120
prime radical#23757854C2
hypercenter#324D157
hypocenterC157#1616
LCSG ▷ C22×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #23757592C2semidirect
3. #23757212C2semidirect
4. #23757008C2semidirect
5. #324×C157C2semidirect
82. C157#1616semidirect
88. #324D157semidirect

Maximal subgroups

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

subgroupindex
#237578542
#237575922
#237572122
#237570082
#324×C1572
#1616157