Finite Group Databasebeta

← #34099263

Subgroups — #34099263

Order 20096.

256 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 61 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#23757645C211yesmaximal, characteristic, complemented
2#23757568C211yesmaximal, characteristic, complemented
3#23757563C211yesmaximal, characteristic, complemented
4#23757017C211yesmaximal, characteristic, complemented
5Q8⋊#11929434C211yesmaximal, characteristic, complemented
6D4⋊#11929434C211yesmaximal, characteristic, complemented
7#550×C157C211yesmaximal, characteristic
8C2×#14578942—22no—
9#19097771C2211yescharacteristic
10#19097704—21no—
11#19097702—21no—
12C4⋊#11929434C2211yescharacteristic
13C4⋊#11929434C2211yescharacteristic
14#19097695—21no—
15#19097692—21no—
16C4×#11929434C2211yescharacteristic
17C2×#41×C157—21no—
18#127×C157—21no—
19#127×C157C2211yescharacteristic
20#121×C157—22no—
21#121×C157—21no—
22C4×Q8×C157C2211yescharacteristic
23C4×D4×C157—21no—
24C4×D4×C157C2211yescharacteristic
25C2×Dic314—21no—
26#14578942—44no—
27C2×#11929434—21no—
28C2×#11929434—21no—
29C2×#11929434—21no—
30C4⋊Dic157—21no—
31C4⋊Dic157—21no—
32#41×C157—42no—
33#41×C157—42no—
34C2×Q8×C157D411yescharacteristic
35C2×D4×C157—22no—
36C2×D4×C157—22no—
37C2×D4×C157—21no—
38C2×D4×C157—21no—
39C2×D4×C157D411yescharacteristic
40C2×D4×C157D412yes—
41C22×C4×C157—22no—
42C22×C4×C157—21no—
43C4⋊C4×C157—21no—
44C4⋊C4×C157D411yescharacteristic
45C4⋊C4×C157D411yescharacteristic
46C22⋊C4×C157—22no—
47C22⋊C4×C157—21no—
48C42×C157—21no—
49C42×C157C2311yescharacteristic
50Dic314—42no—
51C2×Dic157—41no—
52#11929434—22no—
53#11929434—41no—
54#11929434—41no—
55C23×C157—21no—
56C23×C157—22no—
57Q8×C157—21no—
58Q8×C157SD1612yescomplemented
59D4×C157—21no—
60D4×C157—21no—
61D4×C157—42no—
62D4×C157—44no—
63D4×C157—41no—
64D4×C157—41no—
65D4×C157—41no—
66D4×C157—41no—
67D4×C157—82no—
68C2×C4×C157—21no—
69C2×C4×C157—21no—
70C2×C4×C157—21no—
71C2×C4×C157—44no—
72C2×C4×C157—41no—
73C2×C4×C157—41no—
74C2×C4×C157C2×D411yescharacteristic
75C2×C4×C157C2×D411yescharacteristic
76C2×C4×C157C2×D411yescharacteristic
77C22×C157—42no—
78C22×C157—41no—
79C22×C157—41no—
80C22×C157—42no—
81C22×C157—42no—
82C22×C157—41no—
83C22×C157#12011yescharacteristic
84C4×C157—22no—
85C4×C157—21no—
86C4×C157—41no—
87C4×C157—41no—
88C4×C157C2×SD1611yescharacteristic
89C4×C157#13511yescharacteristic
90C4×C157AGL(1,ℤ/8)11yescharacteristic
91C4×C157#13711yescharacteristic
92Dic157—81no—
93C2×C157—81no—
94C2×C157—82no—
95C2×C157#45011yescharacteristic
96C2×C157#45211yescharacteristic
97C2×C157#45311yescharacteristic
98C157#163111yescharacteristic, complemented, Sylow
99#1631—1571nomaximal, Sylow
100#550D15711yescharacteristic
101#492—1571no—
102#477—1571no—
103#476—1571no—
104#331—1571no—
105Q8⋊C8—1571no—
106D4⋊C8—1571no—
107C2×#41—21no—
108C2×SD16—3142no—
109C4⋊Q8—1571no—
110#127—21no—
111#127C2×D15711yescharacteristic
112#121—22no—
113#121—21no—
114C4×Q8C2×D15711yescharacteristic
115C4×D4—21no—
116C4×D4C2×D15711yescharacteristic
117#107—3141no—
118#106—3141no—
119C4⋊C8—1571no—
120C4⋊C8—1571no—
121#103—3141no—
122#102—3141no—
123C4×C8—1571no—
124#41—42no—
125#41—42no—
126C2×Q8—3141no—
127C2×Q8C22⋊D15711yescharacteristic
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4C22⋊D15711yescharacteristic
133C2×D4C22⋊D15712yes—
134C22×C4—22no—
135C22×C4—21no—
136SD16—6284no—
137C2×C8—3141no—
138C2×C8—3141no—
139C2×C8—3141no—
140C4⋊C4—21no—
141C4⋊C4—3141no—
142C4⋊C4—3141no—
143C4⋊C4C22⋊D15711yescharacteristic
144C4⋊C4C22⋊D15711yescharacteristic
145C22⋊C4—22no—
146C22⋊C4—21no—
147C42—21no—
148C42C22×D15711yescharacteristic
149C23—21no—
150C23—22no—
151Q8—21no—
152Q8—6282no—
153Q8#1457894212yescomplemented
154D4—21no—
155D4—21no—
156D4—42no—
157D4—44no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—41no—
162D4—82no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—44no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—6281no—
170C2×C4C2×C22⋊D15711yescharacteristic
171C2×C4C2×C22⋊D15711yescharacteristic
172C2×C4C2×C22⋊D15711yescharacteristic
173C8—3142no—
174C8—6281no—
175C8—6281no—
176C22—42no—
177C22—41no—
178C22—41no—
179C22—42no—
180C22—42no—
181C22—41no—
182C22#1909774011yescharacteristic
183C4—22no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4—12561no—
188C4C2×#1457894211yescharacteristic
189C4#1909779111yescharacteristic
190C4#1909779711yescharacteristic
191C4#1909780111yescharacteristic
192C2—81no—
193C2—82no—
194C2#2375742411yescharacteristic
195C2#2375743511yescharacteristic
196C2#2375744111yescharacteristic
subgroupG/H
centerC22#19097740
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting, prime radical#550×C157C2
socleC22×C157#120
hypercenter#550D157
hypocenterC157#1631
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #23757645C2semidirect
2. #23757568C2semidirect
3. #23757563C2semidirect
4. #23757017C2semidirect
5. Q8⋊#11929434C2semidirect
6. D4⋊#11929434C2semidirect
58. Q8×C157SD16semidirect
98. C157#1631semidirect
153. Q8#14578942semidirect

Maximal subgroups

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

subgroupindex
#237576452
#237575682
#237575632
#237570172
Q8⋊#119294342
D4⋊#119294342
#550×C1572
#1631157