Finite Group Databasebeta

← #34100519

Subgroups — #34100519

Order 20096.

332 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 89 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#23757819C211yesmaximal, characteristic
2#23757289C212yesmaximal, complemented
3#23757247C211yesmaximal, characteristic, complemented
4#23757034C212yesmaximal, complemented
5#568×C157C211yesmaximal, characteristic, complemented
6C2×#14578954C2211yescharacteristic
7#19097769C411yescharacteristic, complemented
8#19097767C411yescharacteristic, complemented
9#19097760C411yescharacteristic, complemented
10#19097758C411yescharacteristic, complemented
11#19097743—22no—
12C4×C22⋊D157—22no—
13#19097730C2212yes—
14C2×C4×Dic157C2211yescharacteristic
15#19097697—24no—
16#19097686—22no—
17C2×#41×C157C2211yescharacteristic
18#131×C157—24no—
19C2×M16×C157—22no—
20C2×M16×C157C2212yes—
21#14578954—22no—
22#14578954—22no—
23C2×Dic314C2×C411yescharacteristic
24C2×D628C2×C411yescharacteristic
25C2×C22⋊D157—22no—
26C2×C22⋊D157—21no—
27C22×Dic157—21no—
28C2×C4×D157—21no—
29C4⋊Dic157—22no—
30#14578933—22no—
31C22⋊Dic157—22no—
32C22⋊Dic157—22no—
33#14578931—22no—
34C4×Dic157—21no—
35C4×Dic157—22no—
36C4×Dic157C2×C411yescharacteristic
37C4×Dic157C2×C411yescharacteristic
38#41×C157—22no—
39#41×C157—22no—
40C2×Q8×C157C2×C411yescharacteristic
41C2×D4×C157C2×C411yescharacteristic
42C2×D4×C157D412yes—
43C22×C4×C157D412yes—
44C22×C4×C157C2311yescharacteristic
45M16×C157—24no—
46M16×C157—22no—
47M16×C157D414yescomplemented
48C2×C8×C157—24no—
49C2×C8×C157—22no—
50C22×D157—21no—
51Dic314—22no—
52D628—22no—
53C22⋊D157—44no—
54C22⋊D157—42no—
55C2×Dic157—21no—
56C2×Dic157—21no—
57C2×Dic157—21no—
58C2×Dic157—42no—
59C2×Dic157—42no—
60C4×D157—41no—
61C4×D157—41no—
62C23×C157C22⋊C412yes—
63C23×C157C2×D411yescharacteristic
64Q8×C157—22no—
65D4×C157—24no—
66D4×C157—22no—
67C2×C4×C157—22no—
68C2×C4×C157—22no—
69C2×C4×C157C22⋊C412yes—
70C2×C4×C157C22×C411yescharacteristic
71C2×C4×C157C2×D412yes—
72C2×C4×C157C2×D411yescharacteristic
73C2×C4×C157#4112yes—
74C8×C157—24no—
75C22×C157—22no—
76C22×C157—41no—
77C22×C157—42no—
78C22×C157C2×C22⋊C411yescharacteristic
79C22×C157#12112yes—
80C2×D157—42no—
81C2×D157—41no—
82C4×C157—22no—
83C4×C157C4×D412yes—
84C4×C157#12011yescharacteristic
85C4×C157#12311yescharacteristic
86Dic157—22no—
87Dic157—41no—
88Dic157—41no—
89Dic157—82no—
90C2×C157—21no—
91C2×C157—22no—
92C2×C157—42no—
93C2×C157#38611yescharacteristic
94D157—81no—
95C157#188411yescharacteristic, complemented, Sylow
96#1884—1571nomaximal, Sylow
97#568D15711yescharacteristic, complemented
98#532—1571no—
99#421—1572no—
100#410—1571no—
101#339—1572no—
102C2×#41—1571no—
103C2×#41C2×D15711yescharacteristic
104#131—24no—
105C2×M16—22no—
106C2×M16C2×D15712yes—
107C4⋊Q8—1571no—
108#127—1571no—
109#124—1571no—
110#124—1571no—
111#121—3142no—
112C4×D4—3142no—
113#117—1572no—
114C2×C42—1571no—
115#104—3144no—
116#99—3142no—
117#41—22no—
118#41—22no—
119#41—3142no—
120#41—3142no—
121C2×Q8—1571no—
122C2×Q8C4×D15711yescharacteristic
123C2×D4—3141no—
124C2×D4—3142no—
125C2×D4—1571no—
126C2×D4C4×D15711yescharacteristic
127C2×D4D62812yes—
128C22×C4—3141no—
129C22×C4—3141no—
130C22×C4C22⋊D15712yes—
131C22×C4C22×D15711yescharacteristic
132M16—24no—
133M16—22no—
134M16C22⋊D15714yescomplemented
135C2×C8—24no—
136C2×C8—22no—
137C4⋊C4—3142no—
138C4⋊C4—3142no—
139C22⋊C4—3142no—
140C22⋊C4—3142no—
141C22⋊C4—3142no—
142C42—3141no—
143C42—3142no—
144C42—1571no—
145C42—1571no—
146C23—3141no—
147C23#1457893112yes—
148C23C2×D62811yescharacteristic
149Q8—22no—
150Q8—3142no—
151D4—24no—
152D4—22no—
153D4—3142no—
154D4—6284no—
155D4—6282no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—3141no—
159C2×C4—3141no—
160C2×C4—3141no—
161C2×C4—6282no—
162C2×C4—6281no—
163C2×C4—6281no—
164C2×C4—6282no—
165C2×C4#1457893112yes—
166C2×C4C2×C4×D15711yescharacteristic
167C2×C4C2×C22⋊D15712yes—
168C2×C4C2×C22⋊D15711yescharacteristic
169C2×C4#1457895412yes—
170C8—24no—
171C22—22no—
172C22—41no—
173C22—42no—
174C22—6282no—
175C22—6281no—
176C22C2×#1457893111yescharacteristic
177C22#1909774512yes—
178C4—22no—
179C4—3142no—
180C4—6281no—
181C4—6281no—
182C4—12562no—
183C4C4×C22⋊D15712yes—
184C4#1909774011yescharacteristic
185C4#1909775411yescharacteristic
186C2—21no—
187C2—22no—
188C2—42no—
189C2—12561no—
190C2#2375715211yescharacteristic
subgroupG/H
centerC4#19097740
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#568×C157C2
socleC2×C157#386
prime radical#23757819C2
cosocleC22×C4×C157C23
hypercenter#568D157
hypocenterC157#1884
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #23757289C2semidirect
3. #23757247C2semidirect
4. #23757034C2semidirect
5. #568×C157C2semidirect
7. #19097769C4semidirect
8. #19097767C4semidirect
9. #19097760C4semidirect
10. #19097758C4semidirect
47. M16×C157D4semidirect
95. C157#1884semidirect
97. #568D157semidirect
134. M16C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
#237578192
#237572892
#237572472
#237570342
#568×C1572
#1884157