Finite Group Databasebeta

← #34098667

Subgroups — #34098667

Order 20096.

274 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 63 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⋊Q8×D157C211yesmaximal, characteristic
2#23757235C211yesmaximal, characteristic
3#23757135C211yesmaximal, characteristic
4#23757011C211yesmaximal, characteristic, complemented
5#23757010C211yesmaximal, characteristic, complemented
6#23757009C211yesmaximal, characteristic, complemented
7#328×C157C211yesmaximal, characteristic, complemented
8C2×Q8×D157C411yescharacteristic, complemented
9C22×C4×D157C2211yescharacteristic
10C2×#14578937—21no—
11#19097751C411yescharacteristic, complemented
12#19097750C2211yescharacteristic
13#19097749—21no—
14#19097748—21no—
15#19097747C411yescharacteristic, complemented
16C4⋊C4×D157—21no—
17C4⋊C4×D157C411yescharacteristic, complemented
18C2×#14578931—21no—
19C22⋊C4×D157—21no—
20C22⋊#11929434C2211yescharacteristic
21#19097683—21no—
22#19097678—21no—
23#19097678C2211yescharacteristic
24C22⋊Q8×C157C2211yescharacteristic
25C22⋊C8×C157C2211yescharacteristic
26#95×C157C2211yescharacteristic
27C23×D157D411yescharacteristic
28C2×Dic314—21no—
29C2×Dic314C2×C411yescharacteristic
30Q8×D157—42no—
31C22×Dic157—21no—
32C22×Dic157D411yescharacteristic
33C2×C4×D157—21no—
34C2×C4×D157—21no—
35C2×C4×D157—42no—
36C2×C4×D157C2×C411yescharacteristic
37C2×C4×D157C2×C411yescharacteristic
38C2×C4×D157D411yescharacteristic
39C2×C4×D157D411yescharacteristic
40#14578937—42no—
41C2×#11929434—21no—
42C4⋊Dic157—21no—
43C4⋊Dic157C2×C411yescharacteristic
44#14578933—21no—
45#14578933—21no—
46#14578933—21no—
47C22⋊Dic157—21no—
48#14578931—21no—
49#14578931—21no—
50#14578931—42no—
51C2×Q8×C157C2×C411yescharacteristic
52C22×C4×C157—21no—
53C22×C4×C157C2311yescharacteristic
54C2×C8×C157—21no—
55C4⋊C4×C157—21no—
56C4⋊C4×C157C2×C411yescharacteristic
57C22⋊C4×C157—21no—
58C22×D157—21no—
59C22×D157—21no—
60C22×D157—22no—
61C22×D157—41no—
62C22×D157C22⋊C411yescharacteristic
63C22×D157C22⋊C411yescharacteristic
64Dic314—42no—
65Dic314—41no—
66C2×Dic157—21no—
67C2×Dic157—21no—
68C2×Dic157—41no—
69C2×Dic157—41no—
70C2×Dic157—41no—
71C2×Dic157—41no—
72C2×Dic157C22⋊C411yescharacteristic
73C2×Dic157C22⋊C411yescharacteristic
74C4×D157—22no—
75C4×D157—22no—
76C4×D157—41no—
77C4×D157—41no—
78C4×D157—42no—
79C4×D157—42no—
80#11929434—41no—
81C23×C157C2×D411yescharacteristic
82Q8×C157—41no—
83C2×C4×C157—21no—
84C2×C4×C157—21no—
85C2×C4×C157—41no—
86C2×C4×C157—41no—
87C2×C4×C157—41no—
88C2×C4×C157—41no—
89C2×C4×C157C22×C411yescharacteristic
90C2×C4×C157C2×D411yescharacteristic
91C8×C157—41no—
92C22×C157—21no—
93C22×C157—21no—
94C22×C157C2×C22⋊C411yescharacteristic
95C22×C157#13711yescharacteristic
96C22×C157#13711yescharacteristic
97C2×D157—21no—
98C2×D157—21no—
99C2×D157—21no—
100C2×D157—21no—
101C2×D157—21no—
102C2×D157—42no—
103C2×D157—42no—
104C2×D157#9912yes—
105C4×C157—21no—
106C4×C157—21no—
107C4×C157—41no—
108C4×C157—41no—
109C4×C157—81no—
110Dic157—21no—
111Dic157—21no—
112Dic157—41no—
113Dic157—41no—
114Dic157—81no—
115C2×C157—21no—
116C2×C157—21no—
117C2×C157C2×#9911yescharacteristic
118C2×C157#41911yescharacteristic
119C2×C157#42111yescharacteristic
120D157—22no—
121D157—41no—
122C157#151111yescharacteristic, complemented, Sylow
123#1511—1571nomaximal, Sylow
124C2×C22⋊Q8—1571no—
125#407—1571no—
126#381—1571no—
127#328—1571no—
128#328—1571no—
129#328—1571no—
130#328D15711yescharacteristic, complemented
131C22×Q8—1571no—
132C23×C4—1571no—
133C2×M16—3141no—
134C22⋊Q8—3141no—
135C22⋊Q8—3141no—
136C22⋊Q8—1571no—
137C22⋊Q8—1571no—
138C22⋊Q8—1571no—
139C22⋊Q8C2×D15711yescharacteristic
140C2×C4⋊C4—3141no—
141C2×C4⋊C4—1571no—
142C2×C22⋊C4—3141no—
143C2×C22⋊C4—3141no—
144C22⋊C8—3141no—
145C22⋊C8—1571no—
146C22⋊C8C2×D15711yescharacteristic
147#95—3141no—
148#95—1571no—
149#95C2×D15711yescharacteristic
150C24—1571no—
151C2×Q8—3141no—
152C2×Q8—6282no—
153C2×Q8—1571no—
154C2×Q8C4×D15711yescharacteristic
155C22×C4—21no—
156C22×C4—3141no—
157C22×C4—3141no—
158C22×C4—3141no—
159C22×C4—6282no—
160C22×C4—1571no—
161C22×C4—1571no—
162C22×C4—1571no—
163C22×C4—1571no—
164C22×C4—1571no—
165C22×C4C22×D15711yescharacteristic
166M16—6282no—
167C2×C8—21no—
168C2×C8—3141no—
169C4⋊C4—21no—
170C4⋊C4—3141no—
171C4⋊C4—3141no—
172C4⋊C4—3141no—
173C4⋊C4—3141no—
174C4⋊C4—1571no—
175C4⋊C4C4×D15711yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—3141no—
178C22⋊C4—3141no—
179C22⋊C4—3141no—
180C22⋊C4—6282no—
181C23—3142no—
182C23—3141no—
183C23—3141no—
184C23—6281no—
185C23—1571no—
186C23—1571no—
187C23D4×D15711yescharacteristic
188Q8—41no—
189Q8—6281no—
190Q8—6282no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—3142no—
198C2×C4—3142no—
199C2×C4—3141no—
200C2×C4—3141no—
201C2×C4—6281no—
202C2×C4—6281no—
203C2×C4—6282no—
204C2×C4—6282no—
205C2×C4—6281no—
206C2×C4—6281no—
207C2×C4—6281no—
208C2×C4—6281no—
209C2×C4—1571no—
210C2×C4—1571no—
211C2×C4C2×C4×D15711yescharacteristic
212C2×C4D4×D15711yescharacteristic
213C8—41no—
214C8—6281no—
215C22—21no—
216C22—21no—
217C22—3141no—
218C22—3141no—
219C22—3141no—
220C22—3141no—
221C22—3141no—
222C22—6282no—
223C22—6282no—
224C22—1572no—
225C22C22⋊C4×D15711yescharacteristic
226C22#1909780011yescharacteristic
227C22#1909780311yescharacteristic
228C4—21no—
229C4—21no—
230C4—41no—
231C4—41no—
232C4—81no—
233C4—3141no—
234C4—3141no—
235C4—6281no—
236C4—6281no—
237C4—12561no—
238C2—21no—
239C2—21no—
240C2—3142no—
241C2—6281no—
242C2#99×D15711yescharacteristic
243C2#2375728011yescharacteristic
244C2#2375728811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D157
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#328×C157C2
socleC22×C157C2×C22⋊C4
prime radicalC23×D157D4
cosocleC22×C4×C157C23
hypercenter#328D157
hypocenterC157#1511
LCSG ▷ C2×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
4. #23757011C2semidirect
5. #23757010C2semidirect
6. #23757009C2semidirect
7. #328×C157C2semidirect
8. C2×Q8×D157C4semidirect
11. #19097751C4semidirect
15. #19097747C4semidirect
17. C4⋊C4×D157C4semidirect
122. C157#1511semidirect
130. #328D157semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1572
#237572352
#237571352
#237570112
#237570102
#237570092
#328×C1572
#1511157