Finite Group Databasebeta

← #34100610

Subgroups — #34100610

Order 20096.

304 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C2×#19097804C211yesmaximal, characteristic
2#23758039C211yesmaximal, characteristic
3#23757289C211yesmaximal, characteristic
4#23757283C211yesmaximal, characteristic, complemented
5C2×#19097689C211yesmaximal, characteristic, complemented
6#23757044C211yesmaximal, characteristic, complemented
7#338×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9C22×Dic314C2211yescharacteristic
10#19097804—22no—
11#19097804—22no—
12C2×Dic628C412yescomplemented
13C2×#14578943C412yescomplemented
14#19097779—22no—
15#19097777—22no—
16C2×#14578938C2211yescharacteristic
17M16×D157—22no—
18#19097730C2211yescharacteristic
19#19097697—22no—
20#19097696—22no—
21#19097689—22no—
22C2×M16×C157C2211yescharacteristic
23C2×M16×C157C2211yescharacteristic
24C2×M16×C157C2211yescharacteristic
25#14578954—21no—
26#14578954—21no—
27#14578954D412yes—
28#14578954D412yes—
29C2×Dic314—21no—
30C2×Dic314—21no—
31C2×Dic314—21no—
32C2×Dic314—21no—
33C2×Dic314—21no—
34C2×Dic314C2×C412yes—
35C2×Dic314C2×C411yescharacteristic
36C2×D628C2×C411yescharacteristic
37C2×C22⋊D157—21no—
38C22×Dic157—21no—
39C2×C4×D157—21no—
40Dic628—24no—
41#14578943—24no—
42#14578938—21no—
43#14578938D412yes—
44#14578937—24no—
45C2×#11929434—21no—
46C8×D157—24no—
47C4⋊Dic157—21no—
48C22⋊Dic157—21no—
49C4×Dic157—21no—
50C22×C4×C157C2311yescharacteristic
51M16×C157—21no—
52M16×C157—21no—
53M16×C157—21no—
54M16×C157—21no—
55M16×C157—21no—
56M16×C157—21no—
57C2×C8×C157—21no—
58C2×C8×C157C2×C412yes—
59C2×C8×C157D412yes—
60C22×D157—21no—
61Dic314—21no—
62Dic314—22no—
63Dic314—22no—
64Dic314—41no—
65Dic314—41no—
66Dic314C22⋊C412yes—
67D628—21no—
68D628C22⋊C412yes—
69C22⋊D157—22no—
70C22⋊D157—22no—
71C2×Dic157—21no—
72C2×Dic157—22no—
73C2×Dic157—41no—
74C2×Dic157—41no—
75C2×Dic157—41no—
76C4×D157—22no—
77C4×D157—22no—
78#11929434—22no—
79C23×C157#4111yescharacteristic
80C2×C4×C157C22×C411yescharacteristic
81C2×C4×C157C2×D411yescharacteristic
82C2×C4×C157C2×D411yescharacteristic
83C2×C4×C157C2×D411yescharacteristic
84C2×C4×C157C2×D411yescharacteristic
85C2×C4×C157#4111yescharacteristic
86C8×C157—22no—
87C8×C157—22no—
88C8×C157—41no—
89C22×C157—41no—
90C22×C157C4×D411yescharacteristic
91C22×C157#12111yescharacteristic
92C22×C157#12311yescharacteristic
93C2×D157—22no—
94C2×D157—81no—
95C4×C157C2×C22⋊C411yescharacteristic
96C4×C157C4×D411yescharacteristic
97C4×C157#12011yescharacteristic
98C4×C157#12111yescharacteristic
99Dic157—22no—
100Dic157—42no—
101Dic157—81no—
102C2×C157—21no—
103C2×C157—21no—
104C2×C157—21no—
105C2×C157#38611yescharacteristic
106D157—42no—
107C157#190211yescharacteristic, complemented, Sylow
108#1902—1571nomaximal, Sylow
109C2×#137—1571no—
110#568—1571no—
111#421—1571no—
112#419—1571no—
113C2×#101—1571no—
114#344—1571no—
115#338D15711yescharacteristic, complemented
116C2×#41—1571no—
117C22×Q8—1571no—
118#137—3142no—
119#137—3142no—
120C2×Q16—1572no—
121C2×SD16—1572no—
122#131—3142no—
123#131—3142no—
124C2×M16—3142no—
125C2×M16—1571no—
126C2×M16C2×D15711yescharacteristic
127C2×M16C2×D15711yescharacteristic
128C2×M16C2×D15711yescharacteristic
129#117—1571no—
130#104—3142no—
131#103—3142no—
132#101—3142no—
133#41—3141no—
134#41—3141no—
135#41—1572no—
136#41—1572no—
137C2×Q8—3141no—
138C2×Q8—3141no—
139C2×Q8—3141no—
140C2×Q8—3141no—
141C2×Q8—3141no—
142C2×Q8—1572no—
143C2×Q8—1571no—
144C2×D4—3141no—
145C2×D4—1571no—
146C22×C4—3141no—
147C22×C4—3141no—
148C22×C4C22×D15711yescharacteristic
149Q16—3144no—
150SD16—3144no—
151M16—21no—
152M16—21no—
153M16—21no—
154M16—21no—
155M16—21no—
156M16—21no—
157M16—3144no—
158M16—3141no—
159M16—1572no—
160C2×C8—21no—
161C2×C8—3144no—
162C2×C8—3141no—
163C2×C8C4×D15712yes—
164C2×C8D62812yes—
165C4⋊C4—3141no—
166C22⋊C4—3141no—
167C42—3141no—
168C23—3141no—
169C23#1457895511yescharacteristic
170Q8—3141no—
171Q8—3142no—
172Q8—3142no—
173Q8—6281no—
174Q8—6281no—
175Q8—1572no—
176D4—3142no—
177D4—3142no—
178D4—3141no—
179D4—1572no—
180C2×C4—3141no—
181C2×C4—3142no—
182C2×C4—3142no—
183C2×C4—3142no—
184C2×C4—6281no—
185C2×C4—6281no—
186C2×C4—6281no—
187C2×C4C2×C4×D15711yescharacteristic
188C2×C4D4×D15711yescharacteristic
189C2×C4D4×D15711yescharacteristic
190C2×C4D4×D15711yescharacteristic
191C2×C4C2×D62811yescharacteristic
192C2×C4#1457895411yescharacteristic
193C8—22no—
194C8—22no—
195C8—41no—
196C8—3142no—
197C22—41no—
198C22—3142no—
199C22—12561no—
200C22#1909773311yescharacteristic
201C22#1909774611yescharacteristic
202C22#1909775611yescharacteristic
203C4—3142no—
204C4—6282no—
205C4—12561no—
206C4C22⋊C4×D15711yescharacteristic
207C4C4×D62811yescharacteristic
208C4#1909774111yescharacteristic
209C4#1909774411yescharacteristic
210C2—21no—
211C2—21no—
212C2—21no—
213C2—6282no—
214C2#2375715111yescharacteristic
subgroupG/H
centerC2#23757151
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#338×C157C2
socleC2×C157#386
prime radicalC2×#14578954C22
cosocleC22×C4×C157C23
hypercenter#338D157
hypocenterC157#1902
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ #338

Complements

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

NH ≅ G/Ntype
4. #23757283C2semidirect
5. C2×#19097689C2semidirect
6. #23757044C2semidirect
7. #338×C157C2semidirect
12. C2×Dic628C4semidirect
13. C2×#14578943C4semidirect
107. C157#1902semidirect
115. #338D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978042
#237580392
#237572892
#237572832
C2×#190976892
#237570442
#338×C1572
#1902157