Finite Group Databasebeta

← #34112781

Subgroups — #34112781

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#23757983C211yesmaximal, characteristic, complemented
2#23757627C212yesmaximal, complemented
3#23757623C211yesmaximal, characteristic, complemented
4#23757580C212yesmaximal, complemented
5#23757563C211yesmaximal, characteristic, complemented
6#23757485C212yesmaximal
7#23757475C211yesmaximal, characteristic, complemented
8Q8×#11929434C211yesmaximal, characteristic, complemented
9C4×#14578941C212yesmaximal, complemented
10C4×#14578939C211yesmaximal, characteristic
11#556×C157C211yesmaximal, characteristic, complemented
12C2×#14578941C2212yes—
13C2×#14578939C2211yescharacteristic
14#19097771C2211yescharacteristic
15#19097763C2212yes—
16#19097749—22no—
17#19097747—21no—
18C4×D628C2211yescharacteristic
19C4×D628C2212yes—
20C4⋊C4×D157—21no—
21#19097704C2211yescharacteristic
22#19097702C2212yes—
23C4⋊#11929434C2212yes—
24C4⋊#11929434C2211yescharacteristic
25#19097695C2212yes—
26#19097695C2212yes—
27#19097695C2212yes—
28#19097692C2211yescharacteristic
29#19097692C2212yes—
30#19097691C2211yescharacteristic
31C4×#11929434C2211yescharacteristic
32C4×#11929434C2212yes—
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×Dic314—21no—
42C2×D628C2311yescharacteristic
43C2×C4×D157—21no—
44C2×C4×D157—22no—
45#14578941—24no—
46#14578939—22no—
47C2×#11929434C2311yescharacteristic
48C2×#11929434C2312yes—
49C2×#11929434C2311yescharacteristic
50C4⋊Dic157—21no—
51C4⋊Dic157C2312yes—
52C4⋊Dic157C2311yescharacteristic
53#14578933—22no—
54#14578933—22no—
55#14578931—22no—
56#14578931—21no—
57C2×Q8×C157D411yescharacteristic
58C2×Q8×C157C2312yes—
59C2×D4×C157C2311yescharacteristic
60C22×C4×C157—21no—
61C4⋊C4×C157—21no—
62C4⋊C4×C157—22no—
63C4⋊C4×C157—21no—
64C4⋊C4×C157D411yescharacteristic
65C4⋊C4×C157D412yes—
66C4⋊C4×C157C2311yescharacteristic
67C22⋊C4×C157—22no—
68C22⋊C4×C157—21no—
69C22⋊C4×C157—22no—
70C42×C157—22no—
71C42×C157C2311yescharacteristic
72C42×C157C2312yes—
73C22×D157—21no—
74Dic314—81no—
75D628—21no—
76D628#4112yes—
77C2×Dic157—22no—
78C2×Dic157—21no—
79C4×D157—42no—
80C4×D157—82no—
81#11929434—21no—
82#11929434—22no—
83#11929434—21no—
84C23×C157—21no—
85Q8×C157—22no—
86Q8×C157—22no—
87Q8×C157—22no—
88D4×C157—21no—
89D4×C157—21no—
90C2×C4×C157—21no—
91C2×C4×C157—22no—
92C2×C4×C157—42no—
93C2×C4×C157C2×D412yes—
94C2×C4×C157C2×D411yescharacteristic
95C2×C4×C157C2×D412yes—
96C2×C4×C157C2×D411yescharacteristic
97C2×C4×C157C2411yescharacteristic
98C22×C157—41no—
99C22×C157—41no—
100C22×C157—41no—
101C22×C157C22×D411yescharacteristic
102C2×D157—22no—
103C2×D157—41no—
104C2×D157—41no—
105C4×C157—22no—
106C4×C157—22no—
107C4×C157—21no—
108C4×C157—42no—
109C4×C157—41no—
110C4×C157#13512yes—
111C4×C157C2×#4111yescharacteristic
112C4×C1572−1+411yescharacteristic
113Dic157—41no—
114Dic157—42no—
115C2×C157—81no—
116C2×C157#54811yescharacteristic
117C2×C157C2×#13511yescharacteristic
118C2×C157#57711yescharacteristic
119D157—42no—
120C157#336011yescharacteristic, complemented, Sylow
121#3360—1571nomaximal, Sylow
122#556D15711yescharacteristic, complemented
123#554—1571no—
124#488—1572no—
125#487—1571no—
126#479—1572no—
127#476—1571no—
128#463—1572no—
129#461—1571no—
130C8×Q8—1571no—
131C4×SD16—1572no—
132C4×D8—1571no—
133C2×SD16—1572no—
134C2×D8—1571no—
135C4⋊Q8—1571no—
136C4⋊Q8C2×D15711yescharacteristic
137#126—22no—
138#125—1572no—
139#124C2×D15712yes—
140C22⋊Q8—21no—
141C22⋊Q8—3141no—
142C22⋊Q8—3142no—
143C4×Q8C2×D15712yes—
144C4×Q8C2×D15711yescharacteristic
145C4×D4—1571no—
146C4×D4—1572no—
147C4×D4C2×D15711yescharacteristic
148#117—21no—
149C2×C4⋊C4—3141no—
150#107—1571no—
151#106—1572no—
152C4⋊C8—1571no—
153C4⋊C8—1572no—
154#103—1572no—
155#103—1572no—
156#103—1572no—
157#102—1571no—
158#102—1572no—
159#102—1571no—
160C4×C8—1571no—
161C4×C8—1572no—
162C2×Q8—3141no—
163C2×Q8C22⋊D15711yescharacteristic
164C2×Q8C22×D15712yes—
165C2×D4—1571no—
166C2×D4C22×D15711yescharacteristic
167C22×C4—21no—
168C22×C4—3141no—
169C22×C4—3142no—
170SD16—3144no—
171D8—3142no—
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—1572no—
182C4⋊C4—1571no—
183C4⋊C4C22⋊D15711yescharacteristic
184C4⋊C4C22⋊D15712yes—
185C4⋊C4C22×D15711yescharacteristic
186C22⋊C4—22no—
187C22⋊C4—21no—
188C22⋊C4—22no—
189C22⋊C4—3141no—
190C22⋊C4—3142no—
191C42—22no—
192C42C22×D15711yescharacteristic
193C42C22×D15712yes—
194C23—21no—
195C23—3141no—
196Q8—22no—
197Q8—22no—
198Q8—22no—
199Q8—12561no—
200D4—21no—
201D4—21no—
202D4—3141no—
203D4—1572no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—3141no—
208C2×C4—3142no—
209C2×C4—6282no—
210C2×C4—12562no—
211C2×C4C2×C22⋊D15712yes—
212C2×C4C2×C22⋊D15712yes—
213C2×C4C2×C22⋊D15711yescharacteristic
214C2×C4C2×C22⋊D15711yescharacteristic
215C2×C4C23×D15711yescharacteristic
216C8—3141no—
217C8—3142no—
218C8—3141no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—3142no—
223C22—6281no—
224C22—6281no—
225C22C22×C22⋊D15711yescharacteristic
226C4—22no—
227C4—21no—
228C4—22no—
229C4—42no—
230C4—41no—
231C4—6282no—
232C4—6281no—
233C4#1909779112yes—
234C4#41×D15711yescharacteristic
235C4#1909781811yescharacteristic
236C2—81no—
237C2—6282no—
238C2#2375793011yescharacteristic
239C2C2×#1909779111yescharacteristic
240C2#2375808111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#556×C157C2
socleC22×C157C22×D4
prime radicalC2×#14578939C22
hypercenter#556D157
hypocenterC157#3360
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. #23757983C2semidirect
2. #23757627C2semidirect
3. #23757623C2semidirect
4. #23757580C2semidirect
5. #23757563C2semidirect
7. #23757475C2semidirect
8. Q8×#11929434C2semidirect
9. C4×#14578941C2semidirect
11. #556×C157C2semidirect
120. C157#3360semidirect
122. #556D157semidirect

Maximal subgroups

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

subgroupindex
#237579832
#237576272
#237576232
#237575802
#237575632
#237574852
#237574752
Q8×#119294342
C4×#145789412
C4×#145789392
#556×C1572
#3360157