Finite Group Databasebeta

← #34102151

Subgroups — #34102151

Order 20096.

226 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 41 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#23758023C211yesmaximal, characteristic, complemented
2#23757464C212yesmaximal
3#23757450C211yesmaximal, characteristic
4#23757062C212yesmaximal, complemented
5#354×C157C211yesmaximal, characteristic, complemented
6#19097816C2211yescharacteristic
7#19097802—21no—
8#19097769—21no—
9#19097764—22no—
10#19097763—21no—
11#19097755—22no—
12#19097752C2212yes—
13#19097748—22no—
14#19097699—21no—
15#19097688C2211yescharacteristic
16#19097685—22no—
17#124×C157C2211yescharacteristic
18#99×C157C2212yes—
19#14578954—21no—
20#14578953—41no—
21C2×Dic314—21no—
22C2×C22⋊D157—22no—
23D4×D157—41no—
24C22×Dic157D412yes—
25#14578944—41no—
26#14578941—41no—
27#14578938—21no—
28C4⋊Dic157—21no—
29C4⋊Dic157—21no—
30C4⋊Dic157—42no—
31#14578933—22no—
32#14578933—22no—
33#14578933—22no—
34C22⋊Dic157—22no—
35C22⋊Dic157D412yes—
36#14578931—42no—
37C4×Dic157—21no—
38C2×Q8×C157D411yescharacteristic
39C2×D4×C157C2311yescharacteristic
40C22⋊C4×C157—22no—
41C22⋊C4×C157—22no—
42C42×C157D411yescharacteristic
43C22×D157—22no—
44Dic314—21no—
45Dic314—81no—
46D628—21no—
47C22⋊D157—42no—
48C22⋊D157—41no—
49C2×Dic157—22no—
50C2×Dic157—22no—
51C2×Dic157—22no—
52C2×Dic157—42no—
53C4×D157—41no—
54#11929434—41no—
55C23×C157C2×D412yes—
56Q8×C157—41no—
57D4×C157—41no—
58C2×C4×C157—21no—
59C2×C4×C157—21no—
60C2×C4×C157—42no—
61C2×C4×C157C2×D411yescharacteristic
62C22×C157—22no—
63C22×C157—42no—
64C22×C157#12011yescharacteristic
65C2×D157—41no—
66C2×D157—82no—
67C4×C157—21no—
68C4×C157—41no—
69C4×C157—41no—
70C4×C157—82no—
71Dic157—42no—
72Dic157—41no—
73Dic157—42no—
74C2×C157—21no—
75C2×C157—42no—
76C2×C157B(4,𝔽2)11yescharacteristic
77D157—81no—
78C157#220111yescharacteristic, complemented, Sylow
79#2201—1571nomaximal, Sylow
80#563—1571no—
81#458—1572no—
82#455—1571no—
83#354D15711yescharacteristic, complemented
84#352—1572no—
85H(5,𝔽2)—1571no—
86#137—3141no—
87C4⋊Q8—3141no—
88#126—3142no—
89#125—3141no—
90#124C2×D15711yescharacteristic
91#123—3142no—
92#123—1572no—
93C22⋊Q8—3142no—
94#104—3141no—
95#100—1571no—
96#99—3142no—
97#99C2×D15712yes—
98#41—3141no—
99#41—6281no—
100C2×Q8—3141no—
101C2×Q8D62811yescharacteristic
102C2×D4—3142no—
103C2×D4—6281no—
104C2×D4C22×D15711yescharacteristic
105C22×C4—1572no—
106Q16—6281no—
107SD16—6281no—
108M16—3141no—
109C4⋊C4—3141no—
110C4⋊C4—3142no—
111C4⋊C4—3142no—
112C4⋊C4—3141no—
113C4⋊C4—3142no—
114C4⋊C4—6282no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—3142no—
118C22⋊C4—6282no—
119C22⋊C4—1572no—
120C42—3141no—
121C42D62811yescharacteristic
122C23—3142no—
123C23D4×D15712yes—
124Q8—41no—
125Q8—3141no—
126Q8—12561no—
127D4—41no—
128D4—3141no—
129D4—6282no—
130D4—6281no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—42no—
134C2×C4—3142no—
135C2×C4—3142no—
136C2×C4—3142no—
137C2×C4—6281no—
138C2×C4—6282no—
139C2×C4C2×D62811yescharacteristic
140C8—6281no—
141C22—22no—
142C22—42no—
143C22—6281no—
144C22—12562no—
145C22#1909774111yescharacteristic
146C4—21no—
147C4—41no—
148C4—41no—
149C4—82no—
150C4—6282no—
151C4—6281no—
152C4—6282no—
153C2—21no—
154C2—42no—
155C2—12561no—
156C2#2375746111yescharacteristic
subgroupG/H
centerC2#23757461
commutator, cosocleC2×D4×C157C23
FrattiniC2×D4C22×D157
Fitting#354×C157C2
socleC2×C157B(4,𝔽2)
prime radical#19097816C22
hypercenter#354D157
hypocenterC157#2201
LCSG ▷ C2×D4×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
1. #23758023C2semidirect
4. #23757062C2semidirect
5. #354×C157C2semidirect
78. C157#2201semidirect
83. #354D157semidirect

Maximal subgroups

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

subgroupindex
#237580232
#237574642
#237574502
#237570622
#354×C1572
#2201157