Finite Group Databasebeta

← #34113167

Subgroups — #34113167

Order 20096.

350 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 123 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#23758002C211yesmaximal, characteristic, complemented
2#23757668C212yesmaximal
3#23757652C211yesmaximal, characteristic, complemented
4#23757492C212yesmaximal
5#23757483C212yesmaximal, complemented
6#23757480C211yesmaximal, characteristic, complemented
7#23757478C211yesmaximal, characteristic
8#23757404C211yesmaximal, characteristic, complemented
9#23757378C212yesmaximal, complemented
10C4×#14578942C211yesmaximal, characteristic
11#556×C157C211yesmaximal, characteristic
12C2×#14578944C2212yes—
13C2×#14578944C2212yes—
14C2×#14578944C2212yes—
15C2×#14578942C2211yescharacteristic
16C2×#14578942C2212yes—
17#19097771C2211yescharacteristic
18#19097771C2212yes—
19#19097762—21no—
20#19097762—22no—
21Q8×Dic157—21no—
22C4×Dic314C2211yescharacteristic
23C4×Dic314C2212yes—
24#19097702C2211yescharacteristic
25C4⋊#11929434C2211yescharacteristic
26C4⋊#11929434C2212yes—
27#19097695C2212yes—
28#19097694C2211yescharacteristic
29#19097694C2212yes—
30#19097692C2211yescharacteristic
31#19097682C2212yes—
32C4×#11929434C2211yescharacteristic
33C4⋊Q8×C157C2211yescharacteristic
34#126×C157—22no—
35#124×C157C2212yes—
36C22⋊Q8×C157—21no—
37C4×Q8×C157C2212yes—
38C4×Q8×C157C2211yescharacteristic
39C4×D4×C157C2211yescharacteristic
40#117×C157—21no—
41C2×Dic314C2312yes—
42C2×Dic314C2311yescharacteristic
43#14578944—24no—
44#14578944—42no—
45#14578944—42no—
46#14578942—22no—
47#14578942—42no—
48C2×#11929434C2311yescharacteristic
49C2×#11929434C2312yes—
50C2×#11929434C2311yescharacteristic
51C4⋊Dic157—22no—
52C4⋊Dic157—21no—
53C4⋊Dic157—22no—
54C4⋊Dic157—21no—
55C4⋊Dic157C2311yescharacteristic
56#14578933—22no—
57#14578933—21no—
58C4×Dic157—22no—
59C4×Dic157—21no—
60C2×Q8×C157D411yescharacteristic
61C2×Q8×C157C2312yes—
62C2×D4×C157C2311yescharacteristic
63C22×C4×C157—21no—
64C4⋊C4×C157—21no—
65C4⋊C4×C157—22no—
66C4⋊C4×C157—21no—
67C4⋊C4×C157D411yescharacteristic
68C4⋊C4×C157D412yes—
69C4⋊C4×C157C2311yescharacteristic
70C22⋊C4×C157—22no—
71C22⋊C4×C157—21no—
72C22⋊C4×C157—22no—
73C42×C157—22no—
74C42×C157C2311yescharacteristic
75C42×C157C2312yes—
76Dic314—22no—
77Dic314—21no—
78Dic314—21no—
79Dic314—22no—
80C2×Dic157—22no—
81C2×Dic157—21no—
82C2×Dic157—21no—
83#11929434—21no—
84#11929434—22no—
85#11929434#4112yes—
86C23×C157—21no—
87Q8×C157—22no—
88Q8×C157—22no—
89Q8×C157—22no—
90D4×C157—21no—
91D4×C157—21no—
92C2×C4×C157—21no—
93C2×C4×C157—22no—
94C2×C4×C157—42no—
95C2×C4×C157C2×D412yes—
96C2×C4×C157C2×D411yescharacteristic
97C2×C4×C157C2×D412yes—
98C2×C4×C157C2×D411yescharacteristic
99C2×C4×C157C2411yescharacteristic
100C22×C157—41no—
101C22×C157—41no—
102C22×C157—41no—
103C22×C157C22×D411yescharacteristic
104C4×C157—22no—
105C4×C157—22no—
106C4×C157—21no—
107C4×C157—42no—
108C4×C157—41no—
109C4×C157#13712yes—
110C4×C157C2×#4111yescharacteristic
111C4×C157H(5,𝔽2)11yescharacteristic
112Dic157—42no—
113Dic157—41no—
114Dic157—41no—
115C2×C157—81no—
116C2×C157#55011yescharacteristic
117C2×C157C2×#13711yescharacteristic
118C2×C157#57811yescharacteristic
119C157#339811yescharacteristic, complemented, Sylow
120#3398—1571nomaximal, Sylow
121#557—1571no—
122#556D15711yescharacteristic
123#497—1572no—
124#494—1571no—
125#464—1572no—
126#462—1571no—
127#462—1572no—
128#461—1571no—
129#446—1571no—
130#441—1572no—
131C4×SD16—1571no—
132C2×Q16—1572no—
133C2×Q16—1572no—
134C2×Q16—1572no—
135C2×SD16—1571no—
136C2×SD16—1572no—
137C4⋊Q8—1571no—
138C4⋊Q8—1572no—
139C4⋊Q8C2×D15711yescharacteristic
140#126—22no—
141#125—3141no—
142#125—3142no—
143#124C2×D15712yes—
144C22⋊Q8—21no—
145C4×Q8—3141no—
146C4×Q8—1571no—
147C4×Q8—1572no—
148C4×Q8C2×D15712yes—
149C4×Q8C2×D15711yescharacteristic
150C4×D4C2×D15711yescharacteristic
151#117—21no—
152#106—1571no—
153C4⋊C8—1571no—
154C4⋊C8—1572no—
155#103—1572no—
156#103—1571no—
157#103—1572no—
158#102—1571no—
159#97—1572no—
160C4×C8—1571no—
161C2×Q8—1571no—
162C2×Q8—1572no—
163C2×Q8C22⋊D15711yescharacteristic
164C2×Q8C22×D15712yes—
165C2×D4C22×D15711yescharacteristic
166C22×C4—21no—
167Q16—3144no—
168Q16—6282no—
169Q16—6282no—
170SD16—3142no—
171SD16—6282no—
172C2×C8—1571no—
173C2×C8—1572no—
174C2×C8—1571no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—3142no—
179C4⋊C4—3141no—
180C4⋊C4—3142no—
181C4⋊C4—3141no—
182C4⋊C4—3142no—
183C4⋊C4—3141no—
184C4⋊C4—1571no—
185C4⋊C4C22⋊D15711yescharacteristic
186C4⋊C4C22⋊D15712yes—
187C4⋊C4C22×D15711yescharacteristic
188C22⋊C4—22no—
189C22⋊C4—21no—
190C22⋊C4—22no—
191C42—22no—
192C42—3142no—
193C42—3141no—
194C42C22×D15711yescharacteristic
195C42C22×D15712yes—
196C23—21no—
197Q8—22no—
198Q8—22no—
199Q8—22no—
200Q8—3141no—
201Q8—3142no—
202Q8—3142no—
203Q8—3141no—
204D4—21no—
205D4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—3141no—
210C2×C4—3141no—
211C2×C4—3142no—
212C2×C4C2×C22⋊D15712yes—
213C2×C4C2×C22⋊D15712yes—
214C2×C4C2×C22⋊D15711yescharacteristic
215C2×C4C2×C22⋊D15711yescharacteristic
216C2×C4C23×D15711yescharacteristic
217C8—3142no—
218C8—3141no—
219C8—1572no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22C22×C22⋊D15711yescharacteristic
224C4—22no—
225C4—21no—
226C4—22no—
227C4—42no—
228C4—41no—
229C4—6281no—
230C4—6281no—
231C4—6282no—
232C4#1909780112yes—
233C4#41×D15711yescharacteristic
234C4#1909781711yescharacteristic
235C2—81no—
236C2#2375794211yescharacteristic
237C2C2×#1909780111yescharacteristic
238C2#2375808811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#556×C157C2
socleC22×C157C22×D4
prime radicalC2×D4×C157C23
hypercenter#556D157
hypocenterC157#3398
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #23758002C2semidirect
3. #23757652C2semidirect
5. #23757483C2semidirect
6. #23757480C2semidirect
8. #23757404C2semidirect
9. #23757378C2semidirect
119. C157#3398semidirect

Maximal subgroups

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

subgroupindex
#237580022
#237576682
#237576522
#237574922
#237574832
#237574802
#237574782
#237574042
#237573782
C4×#145789422
#556×C1572
#3398157