Finite Group Databasebeta

← #34102149

Subgroups — #34102149

Order 20096.

226 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 45 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#23757462C212yesmaximal
2#23757450C211yesmaximal, characteristic
3#23757068C211yesmaximal, characteristic, complemented
4#23757063C212yesmaximal, complemented
5#563×C157C211yesmaximal, characteristic, complemented
6#19097816C2211yescharacteristic
7#19097802—21no—
8#19097754—22no—
9#19097699—21no—
10#19097688C2211yescharacteristic
11#19097686C2212yes—
12#19097685—22no—
13C4⋊Q8×C157—21no—
14#126×C157—22no—
15#125×C157—21no—
16#124×C157C2211yescharacteristic
17#123×C157C2212yes—
18C22⋊Q8×C157—22no—
19#14578954—21no—
20#14578953—41no—
21C2×C22⋊D157—22no—
22D4×D157—41no—
23#14578944—41no—
24#14578941—41no—
25#14578938—21no—
26#14578933—42no—
27C22⋊Dic157—22no—
28#14578931—42no—
29C2×Q8×C157—21no—
30C2×Q8×C157D411yescharacteristic
31C2×D4×C157C2311yescharacteristic
32C22×C4×C157D412yes—
33C4⋊C4×C157—21no—
34C4⋊C4×C157—22no—
35C4⋊C4×C157—22no—
36C4⋊C4×C157—21no—
37C4⋊C4×C157—22no—
38C22⋊C4×C157—22no—
39C22⋊C4×C157—22no—
40C22⋊C4×C157D412yes—
41C42×C157—21no—
42C42×C157D411yescharacteristic
43C22×D157—22no—
44Dic314—21no—
45D628—21no—
46C22⋊D157—42no—
47C22⋊D157—41no—
48C2×Dic157—22no—
49C2×Dic157—42no—
50C4×D157—41no—
51#11929434—41no—
52C23×C157C2×D412yes—
53Q8×C157—41no—
54Q8×C157—81no—
55D4×C157—41no—
56C2×C4×C157—21no—
57C2×C4×C157—22no—
58C2×C4×C157—21no—
59C2×C4×C157—22no—
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—42no—
69C4×C157—41no—
70C4×C157—42no—
71C4×C157—41no—
72Dic157—41no—
73Dic157—82no—
74C2×C157—21no—
75C2×C157—42no—
76C2×C157B(4,𝔽2)11yescharacteristic
77D157—81no—
78C157#220111yescharacteristic, complemented, Sylow
79#2201—1571nomaximal, Sylow
80#563D15711yescharacteristic, complemented
81#458—1572no—
82#455—1571no—
83#354—1571no—
84#352—1572no—
85H(5,𝔽2)—1571no—
86#137—3141no—
87C4⋊Q8—21no—
88#126—22no—
89#125—21no—
90#124C2×D15711yescharacteristic
91#123—3142no—
92#123C2×D15712yes—
93C22⋊Q8—22no—
94#104—3141no—
95#100—1571no—
96#99—3142no—
97#99—1572no—
98#41—3141no—
99#41—6281no—
100C2×Q8—21no—
101C2×Q8C22⋊D15711yescharacteristic
102C2×D4—3142no—
103C2×D4—6281no—
104C2×D4C22×D15711yescharacteristic
105C22×C4C22⋊D15712yes—
106Q16—6281no—
107SD16—6281no—
108M16—3141no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—6282no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—3142no—
118C22⋊C4—6282no—
119C22⋊C4C22⋊D15712yes—
120C42—21no—
121C42C22⋊D15711yescharacteristic
122C23—3142no—
123C23C2×C22⋊D15712yes—
124Q8—41no—
125Q8—81no—
126Q8—3141no—
127D4—41no—
128D4—3141no—
129D4—6282no—
130D4—6281no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—3142no—
137C2×C4—6281no—
138C2×C4—6282no—
139C2×C4C2×C22⋊D15711yescharacteristic
140C8—6281no—
141C22—22no—
142C22—42no—
143C22—6281no—
144C22—12562no—
145C22#1909774011yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—6281no—
152C4—12562no—
153C2—21no—
154C2—42no—
155C2—12561no—
156C2#2375746011yescharacteristic
subgroupG/H
centerC2#23757460
commutator, cosocleC2×D4×C157C23
FrattiniC2×D4C22×D157
Fitting#563×C157C2
socleC2×C157B(4,𝔽2)
prime radical#19097816C22
hypercenter#563D157
hypocenterC157#2201
LCSG ▷ C2×D4×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #563

Complements

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

NH ≅ G/Ntype
3. #23757068C2semidirect
4. #23757063C2semidirect
5. #563×C157C2semidirect
78. C157#2201semidirect
80. #563D157semidirect

Maximal subgroups

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

subgroupindex
#237574622
#237574502
#237570682
#237570632
#563×C1572
#2201157