Finite Group Databasebeta

← #34112789

Subgroups — #34112789

Order 20096.

350 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 117 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#23757997C211yesmaximal, characteristic, complemented
2#23757980C211yesmaximal, characteristic, complemented
3#23757629C212yesmaximal, complemented
4#23757623C211yesmaximal, characteristic, complemented
5#23757578C212yesmaximal, complemented
6#23757564C211yesmaximal, characteristic, complemented
7#23757490C212yesmaximal
8#23757397C211yesmaximal, characteristic, complemented
9SD16×Dic157C212yesmaximal, complemented
10C4×#14578939C211yesmaximal, characteristic
11#461×C157C211yesmaximal, characteristic, complemented
12C2×#14578939C2211yescharacteristic
13#19097771C2211yescharacteristic
14#19097766—22no—
15#19097762C2212yes—
16#19097760C2212yes—
17#19097751—21no—
18#19097750—21no—
19#19097748—22no—
20Q8×Dic157C2212yes—
21C4×Dic314C2211yescharacteristic
22C4×D628C2211yescharacteristic
23D4×Dic157C2212yes—
24#19097729—21no—
25C2×C4⋊Dic157—21no—
26#19097704C2211yescharacteristic
27#19097703C2212yes—
28#19097700C2212yes—
29#19097695C2212yes—
30#19097695C2212yes—
31#19097693C2212yes—
32#19097692C2211yescharacteristic
33#19097691C2211yescharacteristic
34C4×#11929434C2211yescharacteristic
35C8×Dic157C2212yes—
36C2×SD16×C157C2212yes—
37C4⋊Q8×C157C2211yescharacteristic
38C4×D4×C157C2211yescharacteristic
39C4⋊C8×C157C2211yescharacteristic
40#103×C157C2212yes—
41C2×Dic314—21no—
42C2×Dic314D411yescharacteristic
43C2×D628C2311yescharacteristic
44C22×Dic157—22no—
45C2×C4×D157—21no—
46#14578939—22no—
47C2×#11929434C2311yescharacteristic
48C2×#11929434C2311yescharacteristic
49C4⋊Dic157—22no—
50C4⋊Dic157—22no—
51C4⋊Dic157—21no—
52C4⋊Dic157D411yescharacteristic
53C4⋊Dic157C2311yescharacteristic
54C4⋊Dic157C2312yes—
55#14578933—22no—
56#14578933D412yes—
57C22⋊Dic157—22no—
58#14578931—22no—
59#14578931—22no—
60#14578931—21no—
61C4×Dic157—22no—
62C4×Dic157C2312yes—
63C2×Q8×C157C2312yes—
64C2×D4×C157C2311yescharacteristic
65C22×C4×C157—21no—
66SD16×C157—24no—
67C2×C8×C157C2312yes—
68C4⋊C4×C157—21no—
69C4⋊C4×C157—21no—
70C4⋊C4×C157C2311yescharacteristic
71C22⋊C4×C157—21no—
72C42×C157C2311yescharacteristic
73C22×D157—21no—
74Dic314—22no—
75Dic314—81no—
76D628—21no—
77D628—21no—
78C2×Dic157—21no—
79C2×Dic157—22no—
80C2×Dic157—82no—
81C2×Dic157C2×D412yes—
82C2×Dic157C2×D412yes—
83C4×D157—42no—
84#11929434—21no—
85#11929434—21no—
86C23×C157—21no—
87Q8×C157—22no—
88Q8×C157—22no—
89D4×C157—21no—
90D4×C157#4112yes—
91C2×C4×C157—21no—
92C2×C4×C157—22no—
93C2×C4×C157—42no—
94C2×C4×C157C2×D411yescharacteristic
95C2×C4×C157C2×D411yescharacteristic
96C2×C4×C157C2411yescharacteristic
97C8×C157—22no—
98C22×C157—22no—
99C22×C157—41no—
100C22×C157—41no—
101C22×C157C22×D411yescharacteristic
102C2×D157—41no—
103C2×D157—41no—
104C2×D157—41no—
105C4×C157—21no—
106C4×C157—42no—
107C4×C157—41no—
108C4×C157#13512yes—
109C4×C157C2×#4111yescharacteristic
110C4×C1572−1+411yescharacteristic
111Dic157—22no—
112Dic157—22no—
113Dic157—41no—
114Dic157—42no—
115C2×C157—42no—
116C2×C157#54811yescharacteristic
117C2×C157C2×#13511yescharacteristic
118C2×C157#57711yescharacteristic
119D157—81no—
120C157#336011yescharacteristic, complemented, Sylow
121#3360—1571nomaximal, Sylow
122#556—1571no—
123#554—1571no—
124#488—1572no—
125#487—1571no—
126#479—1572no—
127#476—1571no—
128#463—1572no—
129#461D15711yescharacteristic, complemented
130C8×Q8—1571no—
131C4×SD16—1572no—
132C4×D8—1571no—
133C2×SD16C2×D15712yes—
134C2×D8—1571no—
135C4⋊Q8—1571no—
136C4⋊Q8C2×D15711yescharacteristic
137#126—3142no—
138#125—1572no—
139#124—1572no—
140C22⋊Q8—3142no—
141C22⋊Q8—3141no—
142C22⋊Q8—3141no—
143C4×Q8—1572no—
144C4×Q8—1571no—
145C4×D4—1572no—
146C4×D4—1571no—
147C4×D4C2×D15711yescharacteristic
148#117—3141no—
149C2×C4⋊C4—3141no—
150#107—1571no—
151#106—1572no—
152C4⋊C8—1572no—
153C4⋊C8C2×D15711yescharacteristic
154#103—1572no—
155#103—1572no—
156#103C2×D15712yes—
157#102—1571no—
158#102—1572no—
159#102—1571no—
160C4×C8—1572no—
161C4×C8—1571no—
162C2×Q8—3141no—
163C2×Q8—1571no—
164C2×Q8C22×D15712yes—
165C2×D4—1571no—
166C2×D4C22×D15711yescharacteristic
167C22×C4—21no—
168C22×C4—3141no—
169C22×C4—3142no—
170SD16—24no—
171D8—3142no—
172C2×C8—1571no—
173C2×C8—1571no—
174C2×C8C22×D15712yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—3142no—
178C4⋊C4—3142no—
179C4⋊C4—3142no—
180C4⋊C4—3141no—
181C4⋊C4—1572no—
182C4⋊C4—1571no—
183C4⋊C4—1571no—
184C4⋊C4—1572no—
185C4⋊C4C22×D15711yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—3142no—
188C22⋊C4—3142no—
189C22⋊C4—3142no—
190C22⋊C4—3141no—
191C42—3142no—
192C42—1572no—
193C42C22×D15711yescharacteristic
194C23—21no—
195C23—3141no—
196Q8—22no—
197Q8—22no—
198Q8—3142no—
199Q8—12561no—
200D4—21no—
201D4—3141no—
202D4—3141no—
203D4#1457895312yes—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—3141no—
208C2×C4—3142no—
209C2×C4—6282no—
210C2×C4—1572no—
211C2×C4—1572no—
212C2×C4—12562no—
213C2×C4D4×D15711yescharacteristic
214C2×C4D4×D15711yescharacteristic
215C2×C4C23×D15711yescharacteristic
216C8—22no—
217C8—3141no—
218C8—3141no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—6281no—
223C22—6281no—
224C22—6281no—
225C22C2×D4×D15711yescharacteristic
226C4—21no—
227C4—42no—
228C4—41no—
229C4—3142no—
230C4—3142no—
231C4—6282no—
232C4—6281no—
233C4#1909779212yes—
234C4C2×#1457895311yescharacteristic
235C4#1909781811yescharacteristic
236C2—42no—
237C2—12561no—
238C2#2375792911yescharacteristic
239C2C2×#1909779211yescharacteristic
240C2#2375808211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#461×C157C2
socleC22×C157C22×D4
prime radicalC2×#14578939C22
hypercenter#461D157
hypocenterC157#3360
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #23757997C2semidirect
2. #23757980C2semidirect
3. #23757629C2semidirect
4. #23757623C2semidirect
5. #23757578C2semidirect
6. #23757564C2semidirect
8. #23757397C2semidirect
9. SD16×Dic157C2semidirect
11. #461×C157C2semidirect
120. C157#3360semidirect
129. #461D157semidirect

Maximal subgroups

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

subgroupindex
#237579972
#237579802
#237576292
#237576232
#237575782
#237575642
#237574902
#237573972
SD16×Dic1572
C4×#145789392
#461×C1572
#3360157