Finite Group Databasebeta

← #34113166

Subgroups — #34113166

Order 20096.

350 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 115 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#23758001C211yesmaximal, characteristic, complemented
2#23757997C211yesmaximal, characteristic
3#23757666C212yesmaximal
4#23757652C211yesmaximal, characteristic, complemented
5#23757498C212yesmaximal
6#23757481C212yesmaximal, complemented
7#23757479C211yesmaximal, characteristic
8#23757400C211yesmaximal, characteristic, complemented
9#23757377C212yesmaximal, complemented
10C4×#14578941C211yesmaximal, characteristic
11#462×C157C211yesmaximal, characteristic, complemented
12C2×#14578944C2212yes—
13C2×#14578944C2212yes—
14C2×#14578943C2212yes—
15C2×#14578941C2211yescharacteristic
16#19097771C2211yescharacteristic
17#19097769C2212yes—
18#19097766—22no—
19#19097763—21no—
20#19097761—22no—
21#19097760C2212yes—
22#19097751—21no—
23Q8×Dic157C2212yes—
24C4×Dic314—21no—
25C4×Dic314C2211yescharacteristic
26#19097736C2212yes—
27C4×D628C2211yescharacteristic
28#19097729—21no—
29#19097702C2211yescharacteristic
30#19097700C2212yes—
31#19097696C2212yes—
32#19097695C2211yescharacteristic
33#19097691C2211yescharacteristic
34#19097681C2212yes—
35C4×#11929434C2211yescharacteristic
36C2×Q16×C157C2212yes—
37C4⋊Q8×C157C2211yescharacteristic
38C4×Q8×C157C2211yescharacteristic
39C4⋊C8×C157C2211yescharacteristic
40#103×C157C2212yes—
41C2×Dic314D411yescharacteristic
42C2×Dic314C2312yes—
43C2×D628C2311yescharacteristic
44C2×C4×D157—21no—
45#14578944—24no—
46#14578944—42no—
47#14578943—42no—
48#14578941—22no—
49C2×#11929434C2311yescharacteristic
50C2×#11929434C2311yescharacteristic
51C4⋊Dic157—22no—
52C4⋊Dic157—21no—
53C4⋊Dic157—21no—
54C4⋊Dic157D411yescharacteristic
55C4⋊Dic157C2311yescharacteristic
56#14578933—22no—
57#14578933—22no—
58#14578933—22no—
59#14578933D412yes—
60#14578931—22no—
61#14578931—22no—
62#14578931—21no—
63C4×Dic157—22no—
64C4×Dic157—22no—
65C4×Dic157C2312yes—
66C2×Q8×C157C2312yes—
67C2×Q8×C157C2311yescharacteristic
68Q16×C157—42no—
69C2×C8×C157C2312yes—
70C4⋊C4×C157—21no—
71C4⋊C4×C157—21no—
72C4⋊C4×C157—21no—
73C4⋊C4×C157C2311yescharacteristic
74C42×C157—21no—
75C42×C157C2311yescharacteristic
76C22×D157—21no—
77Dic314—22no—
78Dic314—22no—
79Dic314—22no—
80D628—21no—
81D628—21no—
82C2×Dic157—21no—
83C2×Dic157—22no—
84C2×Dic157C2×D412yes—
85C2×Dic157C2×D412yes—
86C4×D157—42no—
87#11929434—21no—
88#11929434#4112yes—
89Q8×C157—22no—
90Q8×C157—21no—
91Q8×C157—22no—
92Q8×C157—21no—
93C2×C4×C157—21no—
94C2×C4×C157—21no—
95C2×C4×C157—22no—
96C2×C4×C157C2×D411yescharacteristic
97C2×C4×C157C2×D411yescharacteristic
98C2×C4×C157C2411yescharacteristic
99C8×C157—22no—
100C22×C157C22×D411yescharacteristic
101C2×D157—41no—
102C2×D157—41no—
103C2×D157—41no—
104C4×C157—21no—
105C4×C157—41no—
106C4×C157—42no—
107C4×C157—41no—
108C4×C157#13712yes—
109C4×C157C2×#4111yescharacteristic
110C4×C157H(5,𝔽2)11yescharacteristic
111Dic157—22no—
112Dic157—22no—
113Dic157—42no—
114Dic157—41no—
115C2×C157#55011yescharacteristic
116C2×C157C2×#13711yescharacteristic
117C2×C157#57811yescharacteristic
118D157—81no—
119C157#339811yescharacteristic, complemented, Sylow
120#3398—1571nomaximal, Sylow
121#557—1571no—
122#556—1571no—
123#497—1572no—
124#494—1571no—
125#464—1572no—
126#462—1572no—
127#462D15711yescharacteristic, complemented
128#461—1571no—
129#446—1571no—
130#441—1572no—
131C4×SD16—1571no—
132C2×Q16—1572no—
133C2×Q16—1572no—
134C2×Q16C2×D15712yes—
135C2×SD16—1571no—
136C2×SD16—1572no—
137C4⋊Q8—1572no—
138C4⋊Q8—1571no—
139C4⋊Q8C2×D15711yescharacteristic
140#126—3142no—
141#125—3141no—
142#125—3142no—
143#124—1572no—
144C22⋊Q8—3141no—
145C4×Q8—3141no—
146C4×Q8—1572no—
147C4×Q8—1572no—
148C4×Q8—1571no—
149C4×Q8C2×D15711yescharacteristic
150C4×D4—1571no—
151#117—3141no—
152#106—1571no—
153C4⋊C8—1572no—
154C4⋊C8C2×D15711yescharacteristic
155#103—1571no—
156#103—1572no—
157#103C2×D15712yes—
158#102—1571no—
159#97—1572no—
160C4×C8—1571no—
161C2×Q8—1572no—
162C2×Q8—1571no—
163C2×Q8C22×D15711yescharacteristic
164C2×Q8C22×D15712yes—
165C2×D4—1571no—
166C22×C4—3141no—
167Q16—42no—
168Q16—3144no—
169Q16—6282no—
170SD16—3142no—
171SD16—6282no—
172C2×C8—1571no—
173C2×C8—1571no—
174C2×C8C22×D15712yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—3142no—
179C4⋊C4—3142no—
180C4⋊C4—3142no—
181C4⋊C4—3142no—
182C4⋊C4—3141no—
183C4⋊C4—3141no—
184C4⋊C4—1571no—
185C4⋊C4—1571no—
186C4⋊C4—1572no—
187C4⋊C4C22×D15711yescharacteristic
188C22⋊C4—3142no—
189C22⋊C4—3142no—
190C22⋊C4—3141no—
191C42—21no—
192C42—3142no—
193C42—3142no—
194C42—1572no—
195C42C22×D15711yescharacteristic
196C23—3141no—
197Q8—21no—
198Q8—22no—
199Q8—21no—
200Q8—22no—
201Q8—3142no—
202Q8—3142no—
203Q8—3142no—
204D4—3141no—
205D4—3141no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—3141no—
210C2×C4—3142no—
211C2×C4—6282no—
212C2×C4—1572no—
213C2×C4—1572no—
214C2×C4D4×D15711yescharacteristic
215C2×C4D4×D15711yescharacteristic
216C2×C4C23×D15711yescharacteristic
217C8—22no—
218C8—3141no—
219C8—1572no—
220C22—6281no—
221C22—6281no—
222C22—6281no—
223C22C2×D4×D15711yescharacteristic
224C4—21no—
225C4—41no—
226C4—41no—
227C4—42no—
228C4—3142no—
229C4—3142no—
230C4—6282no—
231C4—6281no—
232C4#1909780312yes—
233C4#41×D15711yescharacteristic
234C4#1909781611yescharacteristic
235C2—12561no—
236C2#2375794111yescharacteristic
237C2C2×#1909780311yescharacteristic
238C2#2375808911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#462×C157C2
socleC22×C157C22×D4
prime radicalC2×D628C23
hypercenter#462D157
hypocenterC157#3398
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. #23758001C2semidirect
4. #23757652C2semidirect
6. #23757481C2semidirect
8. #23757400C2semidirect
9. #23757377C2semidirect
11. #462×C157C2semidirect
119. C157#3398semidirect
127. #462D157semidirect

Maximal subgroups

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

subgroupindex
#237580012
#237579972
#237576662
#237576522
#237574982
#237574812
#237574792
#237574002
#237573772
C4×#145789412
#462×C1572
#3398157