Finite Group Databasebeta

← #34099307

Subgroups — #34099307

Order 20096.

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

normal subgroups only

This group has 194 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#23757996C211yesmaximal, characteristic
2#23757623C211yesmaximal, characteristic, complemented
3#23757560C211yesmaximal, characteristic
4#23757018C211yesmaximal, characteristic, complemented
5Q8⋊#11929434C211yesmaximal, characteristic, complemented
6#23757000C211yesmaximal, characteristic
7#477×C157C211yesmaximal, characteristic, complemented
8#19097771C2211yescharacteristic
9#19097766—22no—
10#19097759—21no—
11#19097747—21no—
12C4×Dic314—21no—
13C4×D628C2211yescharacteristic
14#19097729—21no—
15#19097702—21no—
16C4⋊#11929434C2211yescharacteristic
17#19097695—22no—
18#19097691—21no—
19C4×#11929434C2211yescharacteristic
20C4⋊Q8×C157C2211yescharacteristic
21C4×Q8×C157C2211yescharacteristic
22#107×C157—21no—
23C4⋊C8×C157C2211yescharacteristic
24#103×C157—21no—
25C2×Dic314—21no—
26C2×D628D411yescharacteristic
27C2×C4×D157—21no—
28C2×#11929434—21no—
29C2×#11929434—21no—
30C4⋊Dic157—21no—
31C4⋊Dic157D412yes—
32C4⋊Dic157D411yescharacteristic
33#14578933—22no—
34#14578931—22no—
35#14578931—22no—
36#14578931—21no—
37C4×Dic157—22no—
38C2×Q8×C157—21no—
39C2×Q8×C157D411yescharacteristic
40C2×C8×C157—21no—
41C4⋊C4×C157—21no—
42C4⋊C4×C157—21no—
43C4⋊C4×C157—21no—
44C4⋊C4×C157D411yescharacteristic
45C42×C157—21no—
46C42×C157C2311yescharacteristic
47C22×D157—21no—
48Dic314—41no—
49Dic314—41no—
50D628—21no—
51D628—21no—
52C2×Dic157—21no—
53C2×Dic157—22no—
54C4×D157—41no—
55C4×D157—41no—
56#11929434—41no—
57#11929434—41no—
58Q8×C157—21no—
59Q8×C157—21no—
60Q8×C157—42no—
61C2×C4×C157—21no—
62C2×C4×C157—21no—
63C2×C4×C157—41no—
64C2×C4×C157C2×D411yescharacteristic
65C2×C4×C157C2×D411yescharacteristic
66C2×C4×C157C2×D411yescharacteristic
67C8×C157—41no—
68C22×C157#12011yescharacteristic
69C2×D157—41no—
70C2×D157—41no—
71C2×D157—41no—
72C4×C157—21no—
73C4×C157—41no—
74C4×C157—41no—
75C4×C157—81no—
76C4×C157#13511yescharacteristic
77C4×C157#13511yescharacteristic
78C4×C157#13711yescharacteristic
79C4×C157#13711yescharacteristic
80Dic157—41no—
81Dic157—42no—
82C2×C157#45211yescharacteristic
83C2×C157#45211yescharacteristic
84C2×C157#45411yescharacteristic
85D157—81no—
86C157#163811yescharacteristic, complemented, Sylow
87#1638—1571nomaximal, Sylow
88#556—1571no—
89#487—1571no—
90#477D15711yescharacteristic, complemented
91#476—1571no—
92#332—1571no—
93Q8⋊C8—1571no—
94D4⋊C8—1571no—
95C4⋊Q8—1571no—
96C4⋊Q8C2×D15711yescharacteristic
97#126—3142no—
98#124—3141no—
99C22⋊Q8—3141no—
100C4×Q8—3141no—
101C4×Q8C2×D15711yescharacteristic
102C4×D4—1571no—
103#117—3141no—
104#107—21no—
105#106—3141no—
106C4⋊C8—1571no—
107C4⋊C8C2×D15711yescharacteristic
108#103—21no—
109#103—3142no—
110#102—3141no—
111C4×C8—1571no—
112C2×Q8—21no—
113C2×Q8—3141no—
114C2×Q8C22⋊D15711yescharacteristic
115C2×D4—1571no—
116C22×C4—3141no—
117C2×C8—21no—
118C2×C8—3141no—
119C2×C8—3141no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—3142no—
124C4⋊C4—3141no—
125C4⋊C4—1572no—
126C4⋊C4—1571no—
127C4⋊C4C22⋊D15711yescharacteristic
128C22⋊C4—3142no—
129C22⋊C4—3142no—
130C22⋊C4—3141no—
131C42—21no—
132C42—3142no—
133C42C22×D15711yescharacteristic
134C23—3141no—
135Q8—21no—
136Q8—21no—
137Q8—42no—
138Q8—6281no—
139Q8—6281no—
140D4—3141no—
141D4—3141no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—3141no—
146C2×C4—3142no—
147C2×C4—6281no—
148C2×C4—6281no—
149C2×C4D4×D15711yescharacteristic
150C2×C4D4×D15711yescharacteristic
151C2×C4C2×C22⋊D15711yescharacteristic
152C8—41no—
153C8—6281no—
154C8—6281no—
155C22—6281no—
156C22—6281no—
157C22—6281no—
158C22#1909773911yescharacteristic
159C4—21no—
160C4—41no—
161C4—41no—
162C4—81no—
163C4—6281no—
164C4—6282no—
165C4#1909779111yescharacteristic
166C4#1909779411yescharacteristic
167C4#1909780211yescharacteristic
168C4#1909780311yescharacteristic
169C2—12561no—
170C2#2375743311yescharacteristic
171C2#2375743411yescharacteristic
172C2#2375744411yescharacteristic
subgroupG/H
centerC22#19097739
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#477×C157C2
socleC22×C157#120
prime radicalC2×D628D4
hypercenter#477D157
hypocenterC157#1638
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #23757623C2semidirect
4. #23757018C2semidirect
5. Q8⋊#11929434C2semidirect
7. #477×C157C2semidirect
86. C157#1638semidirect
90. #477D157semidirect

Maximal subgroups

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

subgroupindex
#237579962
#237576232
#237575602
#237570182
Q8⋊#119294342
#237570002
#477×C1572
#1638157