Finite Group Databasebeta

← #34109790

Subgroups — #34109790

Order 20096.

460 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C22×#14578944C211yesmaximal, characteristic, complemented
2C2×#19097779C211yesmaximal, characteristic
3#23757849C211yesmaximal, characteristic
4#23757847C211yesmaximal, characteristic
5#23757530C214yesmaximal, complemented
6#23757513C214yesmaximal, complemented
7#23757309C211yesmaximal, characteristic, complemented
8#23757283C211yesmaximal, characteristic, complemented
9#419×C157C211yesmaximal, characteristic, complemented
10C2×#14578954C2211yescharacteristic
11C22×Dic314C2211yescharacteristic
12C2×#14578944—24no—
13C2×#14578944C2212yes—
14C2×#14578944C2212yes—
15#19097779—24no—
16C2×#14578937C2212yes—
17C22×#11929434C2211yescharacteristic
18C2×C8×D157C2212yes—
19#19097771—21no—
20#19097769—21no—
21#19097760—21no—
22#19097759—21no—
23#19097755—22no—
24#19097754—22no—
25#19097751C2214yes—
26#19097747C2214yes—
27#19097730C2211yescharacteristic
28#19097704C2212yes—
29#19097702C2212yes—
30#19097696C2214yes—
31C22×Q8×C157C2211yescharacteristic
32C2×M16×C157C2211yescharacteristic
33#117×C157C2211yescharacteristic
34#103×C157C2214yes—
35#14578954—22no—
36#14578954—22no—
37C2×Dic314—21no—
38C2×Dic314—22no—
39C2×Dic314—22no—
40C2×Dic314D411yescharacteristic
41C2×Dic314C2312yes—
42C2×D628D411yescharacteristic
43C2×C22⋊D157D412yes—
44C22×Dic157—21no—
45C2×C4×D157C2312yes—
46#14578944—24no—
47#14578944—24no—
48#14578937—24no—
49C2×#11929434D414yes—
50C2×#11929434C2311yescharacteristic
51C2×#11929434C2311yescharacteristic
52C8×D157—24no—
53C4⋊Dic157—24no—
54C4⋊Dic157C2312yes—
55#14578933—24no—
56C22⋊Dic157—21no—
57#14578931—24no—
58#14578931—24no—
59C4×Dic157—21no—
60C2×Q8×C157—21no—
61C2×Q8×C157—22no—
62C2×Q8×C157—22no—
63C2×Q8×C157C2312yes—
64C22×C4×C157—21no—
65C22×C4×C157C2311yescharacteristic
66M16×C157—22no—
67C2×C8×C157C2312yes—
68C4⋊C4×C157C2312yes—
69C22⋊C4×C157—21no—
70C42×C157—21no—
71C22×D157C2×D412yes—
72Dic314—22no—
73Dic314—22no—
74Dic314—22no—
75Dic314—42no—
76D628—22no—
77C22⋊D157—24no—
78C2×Dic157—22no—
79C2×Dic157—22no—
80C2×Dic157—42no—
81C2×Dic157C2×D412yes—
82C4×D157—22no—
83C4×D157—22no—
84#11929434C2×D412yes—
85#11929434C2×D412yes—
86C23×C157C2×D411yescharacteristic
87Q8×C157—22no—
88Q8×C157—22no—
89Q8×C157—42no—
90C2×C4×C157—22no—
91C2×C4×C157—22no—
92C2×C4×C157—42no—
93C2×C4×C157C2×D411yescharacteristic
94C2×C4×C157C2×D412yes—
95C2×C4×C157#4112yes—
96C2×C4×C157C2411yescharacteristic
97C8×C157—22no—
98C22×C157—22no—
99C22×C157#12112yes—
100C22×C157C22×D411yescharacteristic
101C2×D157—22no—
102C2×D157—24no—
103C4×C157—42no—
104C4×C157—42no—
105C4×C157#12112yes—
106C4×C157C22×D411yescharacteristic
107C4×C157C2×#4111yescharacteristic
108Dic157—22no—
109Dic157—42no—
110Dic157—42no—
111C2×C157—22no—
112C2×C157C2×#12111yescharacteristic
113C2×C157#57812yes—
114D157—42no—
115C157#305911yescharacteristic, complemented, Sylow
116#3059—1571nomaximal, Sylow
117C22×Q16—1571no—
118C2×#131—1571no—
119#536—1571no—
120#536—1571no—
121#470—1574no—
122#467—1574no—
123#427—1571no—
124#419—1571no—
125#419D15711yescharacteristic, complemented
126C2×#41—1571no—
127C22×Q8—1571no—
128C22×Q8C2×D15711yescharacteristic
129C2×Q16—3144no—
130C2×Q16—1572no—
131C2×Q16—1572no—
132#131—3144no—
133C2×M16—1572no—
134C2×M16C2×D15711yescharacteristic
135C22×C8—1572no—
136C22×C8—1571no—
137C4⋊Q8—3141no—
138C4⋊Q8—3141no—
139#124—3141no—
140#124—3141no—
141#123—3142no—
142#123—3142no—
143C22⋊Q8—1574no—
144C22⋊Q8—1574no—
145#117—1571no—
146#117C2×D15711yescharacteristic
147#107—1572no—
148#106—1572no—
149#103—1574no—
150#103C2×D15714yes—
151#41—3142no—
152#41—3142no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—3141no—
157C2×Q8—3142no—
158C2×Q8—3142no—
159C2×Q8—1572no—
160C2×Q8—1571no—
161C2×Q8C22×D15712yes—
162C2×D4—1572no—
163C2×D4—1571no—
164C22×C4—21no—
165C22×C4—3141no—
166C22×C4—1572no—
167C22×C4C22×D15711yescharacteristic
168Q16—3144no—
169Q16—3144no—
170M16—22no—
171M16—3144no—
172C2×C8—3144no—
173C2×C8—1574no—
174C2×C8—1571no—
175C2×C8—1571no—
176C2×C8C22×D15712yes—
177C4⋊C4—3144no—
178C4⋊C4—3144no—
179C4⋊C4—1572no—
180C4⋊C4C22×D15712yes—
181C22⋊C4—21no—
182C22⋊C4—3144no—
183C22⋊C4—3144no—
184C22⋊C4—3141no—
185C42—21no—
186C42—3141no—
187C23—1572no—
188C23D4×D15711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—3142no—
193Q8—3142no—
194Q8—3142no—
195Q8—6282no—
196D4—3144no—
197D4—3142no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—3142no—
202C2×C4—3142no—
203C2×C4—3142no—
204C2×C4—3142no—
205C2×C4—6282no—
206C2×C4—1572no—
207C2×C4D4×D15711yescharacteristic
208C2×C4D4×D15712yes—
209C2×C4#1457895412yes—
210C2×C4C23×D15711yescharacteristic
211C8—22no—
212C8—1572no—
213C8—1572no—
214C22—22no—
215C22—3144no—
216C22—3142no—
217C22#1909774412yes—
218C22C2×D4×D15711yescharacteristic
219C4—42no—
220C4—42no—
221C4—3142no—
222C4—6282no—
223C4—6282no—
224C4#1909774412yes—
225C4C2×D4×D15711yescharacteristic
226C4C2×#1457895411yescharacteristic
227C2—22no—
228C2—6282no—
229C2C2×#1909774411yescharacteristic
230C2#2375808912yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#419×C157C2
socleC22×C157C22×D4
prime radicalC2×#14578954C22
hypercenter#419D157
hypocenterC157#3059
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×#14578944C2semidirect
5. #23757530C2semidirect
6. #23757513C2semidirect
7. #23757309C2semidirect
8. #23757283C2semidirect
9. #419×C157C2semidirect
115. C157#3059semidirect
125. #419D157semidirect

Maximal subgroups

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

subgroupindex
C22×#145789442
C2×#190977792
#237578492
#237578472
#237575302
#237575132
#237573092
#237572832
#419×C1572
#3059157