Finite Group Databasebeta

← #34099246

Subgroups — #34099246

Order 20096.

216 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 57 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#23757621C211yesmaximal, characteristic, complemented
2#23757473C211yesmaximal, characteristic, complemented
3#23757468C211yesmaximal, characteristic
4#23757019C211yesmaximal, characteristic, complemented
5Q8⋊#11929434C211yesmaximal, characteristic, complemented
6D4⋊#11929434C211yesmaximal, characteristic, complemented
7#556×C157C211yesmaximal, characteristic, complemented
8C2×#14578941—21no—
9C2×#14578939—21no—
10#19097768C2211yescharacteristic
11C4⋊#11929434C2211yescharacteristic
12C4⋊#11929434C2211yescharacteristic
13#19097691—22no—
14#19097691—21no—
15#19097691—21no—
16C4×#11929434C2211yescharacteristic
17C4⋊Q8×C157C2211yescharacteristic
18#126×C157—22no—
19#124×C157—21no—
20C22⋊Q8×C157—21no—
21C4×Q8×C157—21no—
22C4×Q8×C157C2211yescharacteristic
23C4×D4×C157C2211yescharacteristic
24#117×C157—21no—
25C2×D628—21no—
26C2×D628—21no—
27C2×D628—21no—
28#14578941—42no—
29#14578939—42no—
30C2×#11929434—21no—
31C2×#11929434—21no—
32C2×#11929434—21no—
33C2×Q8×C157—21no—
34C2×Q8×C157D411yescharacteristic
35C2×D4×C157D411yescharacteristic
36C22×C4×C157—21no—
37C4⋊C4×C157—22no—
38C4⋊C4×C157—21no—
39C4⋊C4×C157—21no—
40C4⋊C4×C157D411yescharacteristic
41C4⋊C4×C157D411yescharacteristic
42C4⋊C4×C157D412yes—
43C22⋊C4×C157—22no—
44C22⋊C4×C157—22no—
45C22⋊C4×C157—21no—
46C42×C157—22no—
47C42×C157—21no—
48C42×C157C2311yescharacteristic
49C22×D157—41no—
50D628—42no—
51D628—41no—
52D628—41no—
53D628—41no—
54D628—41no—
55#11929434—41no—
56#11929434—41no—
57#11929434—41no—
58C23×C157—21no—
59Q8×C157—21no—
60Q8×C157—21no—
61Q8×C157—41no—
62Q8×C157—41no—
63D4×C157—21no—
64D4×C157—21no—
65C2×C4×C157—21no—
66C2×C4×C157—21no—
67C2×C4×C157—21no—
68C2×C4×C157—22no—
69C2×C4×C157—41no—
70C2×C4×C157—41no—
71C2×C4×C157C2×D411yescharacteristic
72C2×C4×C157C2×D411yescharacteristic
73C2×C4×C157C2×D411yescharacteristic
74C22×C157—41no—
75C22×C157—41no—
76C22×C157—41no—
77C22×C157#12011yescharacteristic
78C2×D157—81no—
79C2×D157—81no—
80C2×D157—81no—
81C4×C157—21no—
82C4×C157—41no—
83C4×C157—41no—
84C4×C157—42no—
85C4×C157—41no—
86C4×C157#13511yescharacteristic
87C4×C157#13511yescharacteristic
88C4×C157AGL(1,ℤ/8)11yescharacteristic
89C4×C157AGL(1,ℤ/8)11yescharacteristic
90C2×C157—81no—
91C2×C157#44911yescharacteristic
92C2×C157#44911yescharacteristic
93C2×C157#45411yescharacteristic
94D157—161no—
95C157#162811yescharacteristic, complemented, Sylow
96#1628—1571nomaximal, Sylow
97#556D15711yescharacteristic, complemented
98#486—1571no—
99#460—1571no—
100#459—1571no—
101#332—1571no—
102Q8⋊C8—1571no—
103D4⋊C8—1571no—
104C2×SD16—3141no—
105C2×D8—3141no—
106C4⋊Q8C2×D15711yescharacteristic
107#127—1571no—
108#126—22no—
109#124—21no—
110C22⋊Q8—21no—
111C4×Q8—21no—
112C4×Q8C2×D15711yescharacteristic
113C4×D4C2×D15711yescharacteristic
114#117—21no—
115C4⋊C8—1571no—
116C4⋊C8—1571no—
117#102—3141no—
118#102—3141no—
119#102—3142no—
120C4×C8—1571no—
121C2×Q8—21no—
122C2×Q8C22⋊D15711yescharacteristic
123C2×D4—3141no—
124C2×D4—3141no—
125C2×D4—3141no—
126C2×D4C22⋊D15711yescharacteristic
127C22×C4—21no—
128SD16—6282no—
129D8—6282no—
130C2×C8—3141no—
131C2×C8—3141no—
132C2×C8—3141no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4C22⋊D15711yescharacteristic
137C4⋊C4C22⋊D15711yescharacteristic
138C4⋊C4C22⋊D15712yes—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—21no—
142C42—22no—
143C42—21no—
144C42C22×D15711yescharacteristic
145C23—21no—
146C23—6281no—
147Q8—21no—
148Q8—21no—
149Q8—41no—
150Q8—41no—
151D4—21no—
152D4—21no—
153D4—6282no—
154D4—6281no—
155D4—6281no—
156D4—6281no—
157D4—6281no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4C2×C22⋊D15711yescharacteristic
165C2×C4C2×C22⋊D15711yescharacteristic
166C2×C4C2×C22⋊D15711yescharacteristic
167C8—6281no—
168C8—6281no—
169C8—6281no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—12561no—
174C22—12561no—
175C22—12561no—
176C22#1909774011yescharacteristic
177C4—21no—
178C4—41no—
179C4—41no—
180C4—42no—
181C4—41no—
182C4#1909779111yescharacteristic
183C4#1909779111yescharacteristic
184C4#1909779611yescharacteristic
185C4#1909779611yescharacteristic
186C2—81no—
187C2—25121no—
188C2#2375741811yescharacteristic
189C2#2375741811yescharacteristic
190C2#2375744611yescharacteristic
subgroupG/H
centerC22#19097740
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#556×C157C2
socleC22×C157#120
prime radical#23757468C2
hypercenter#556D157
hypocenterC157#1628
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #23757621C2semidirect
2. #23757473C2semidirect
4. #23757019C2semidirect
5. Q8⋊#11929434C2semidirect
6. D4⋊#11929434C2semidirect
7. #556×C157C2semidirect
95. C157#1628semidirect
97. #556D157semidirect

Maximal subgroups

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

subgroupindex
#237576212
#237574732
#237574682
#237570192
Q8⋊#119294342
D4⋊#119294342
#556×C1572
#1628157