Finite Group Databasebeta

← #34080176

Subgroups — #34080176

Order 19328.

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

normal subgroups only

This group has 396 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#23752781C211yesmaximal, characteristic
2#23752604C211yesmaximal, characteristic, complemented
3#23752586C211yesmaximal, characteristic, complemented
4#23752578C211yesmaximal, characteristic, complemented
5#23752569C211yesmaximal, characteristic, complemented
6#23752434C212yesmaximal, complemented
7#23752412C212yesmaximal, complemented
8#23752372C212yesmaximal, complemented
9#23752362C212yesmaximal, complemented
10#23752273C211yesmaximal, characteristic, complemented
11#541×C151C211yesmaximal, characteristic, complemented
12C2×#14559004C2211yescharacteristic
13C22×D604C2211yescharacteristic
14C2×#14558994C2212yescomplemented
15C2×#14558991C2212yes—
16C2×#14558991C2212yes—
17C2×#14558991C2212yescomplemented
18#18012736C2211yescharacteristic
19#18012736C2211yescharacteristic
20#18012727C2211yescharacteristic
21#18012727C2211yescharacteristic
22#18012714—22no—
23#18012709—22no—
24C4⋊(C151⋊C8)C2212yescomplemented
25#18012662C2212yescomplemented
26#18012659C2212yes—
27#18012659C2212yes—
28#18012659C2212yescomplemented
29C22⋊(C151⋊C8)C2212yes—
30#18012650C2211yescharacteristic, complemented
31C4×C151⋊C8C2211yescharacteristic, complemented
32C4⋊Q8×C151C2211yescharacteristic, complemented
33#126×C151—22no—
34#125×C151—22no—
35#125×C151C2211yescharacteristic, complemented
36C22⋊Q8×C151C2212yes—
37C4×Q8×C151C2212yescomplemented
38#117×C151C2211yescharacteristic
39C23×D151—21no—
40#14559004—42no—
41C2×Dic302C2311yescharacteristic
42C2×D604—21no—
43C2×D604—22no—
44C2×D604—22no—
45C2×D604—42no—
46C2×D604C2311yescharacteristic
47C2×D604C2311yescharacteristic
48C2×D604C2311yescharacteristic
49C2×C22⋊D151—21no—
50C2×C4×D151—21no—
51#14558994—42no—
52#14558991—42no—
53#14558991—42no—
54#14558991—42no—
55C2×C151⋊C8C2312yes—
56C2×C151⋊C8C2312yes—
57(C2×D151)⋊C4—22no—
58(C2×D151)⋊C4—22no—
59C2×Q8×C151C2312yes—
60C22×C4×C151C2311yescharacteristic
61C4⋊C4×C151—22no—
62C4⋊C4×C151—22no—
63C4⋊C4×C151—22no—
64C4⋊C4×C151—22no—
65C4⋊C4×C151D412yes—
66C4⋊C4×C151C2312yes—
67C22⋊C4×C151—22no—
68C22⋊C4×C151D412yes—
69C42×C151—22no—
70C42×C151C2311yescharacteristic
71C42×C151C2311yescharacteristic
72C22×D151—21no—
73C22×D151—21no—
74C22×D151—21no—
75C22×D151—42no—
76C22×D151—82no—
77Dic302—22no—
78D604—22no—
79D604—22no—
80D604—22no—
81D604—42no—
82D604—82no—
83D604—82no—
84C22⋊D151—42no—
85C2×Dic151—21no—
86C4×D151—42no—
87C151⋊C8—22no—
88C151⋊C8—22no—
89C23×C151C2×D411yescharacteristic
90Q8×C151—22no—
91Q8×C151—22no—
92C2×C4×C151—22no—
93C2×C4×C151—22no—
94C2×C4×C151—22no—
95C2×C4×C151C2×D412yes—
96C2×C4×C151C2×D412yes—
97C2×C4×C151C2×D411yescharacteristic
98C2×C4×C151C2411yescharacteristic
99C22×C151—21no—
100C22×C151—22no—
101C22×C151C22×D411yescharacteristic
102C2×D151—41no—
103C2×D151—41no—
104C2×D151—41no—
105C2×D151—42no—
106C2×D151—42no—
107C2×D151—42no—
108C2×D151—84no—
109C4×C151—21no—
110C4×C151—22no—
111C4×C151—22no—
112C4×C151—42no—
113C4×C151—42no—
114C4×C151H(5,𝔽2)12yes—
115Dic151—41no—
116C2×C151—41no—
117C2×C151#53511yescharacteristic
118C2×C151#57712yes—
119D151—81no—
120D151—81no—
121D151—81no—
122C151#322211yescharacteristic, complemented, Sylow
123#3222—1511nomaximal, Sylow
124#541D15111yescharacteristic, complemented
125#535—1511no—
126#497—1511no—
127#492—1511no—
128#490—1511no—
129#488—1511no—
130#464—1512no—
131#460—1512no—
132#452—1512no—
133#450—1512no—
134#433—1511no—
135C2×#41—1511no—
136C22×D4—1511no—
137C2×Q16—1512no—
138C2×SD16—1512no—
139C2×SD16—1512no—
140C2×SD16—1512no—
141C4⋊Q8C2×D15111yescharacteristic, complemented
142#127—1511no—
143#127—1511no—
144#126—22no—
145#125—22no—
146#125C2×D15111yescharacteristic, complemented
147#124—1511no—
148#124—1511no—
149C22⋊Q8C2×D15112yes—
150#121—3022no—
151#120—3022no—
152C4×Q8C2×D15112yescomplemented
153#117C2×D15111yescharacteristic
154C4⋊C8—1512no—
155#103—1512no—
156#102—1512no—
157#102—1512no—
158#102—1512no—
159C22⋊C8—1512no—
160#97—1511no—
161C4×C8—1511no—
162C24—3021no—
163#41—6042no—
164C2×Q8—1511no—
165C2×Q8C22×D15112yes—
166C2×D4—3021no—
167C2×D4—3021no—
168C2×D4—3022no—
169C2×D4—3022no—
170C2×D4—6042no—
171C2×D4—1511no—
172C2×D4—1511no—
173C2×D4—1511no—
174C22×C4—3021no—
175C22×C4C22×D15111yescharacteristic
176Q16—6042no—
177SD16—6042no—
178SD16—6042no—
179SD16—6042no—
180C2×C8—1512no—
181C2×C8—1512no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4C22⋊D15112yes—
187C4⋊C4C22×D15112yes—
188C22⋊C4—22no—
189C22⋊C4—3022no—
190C22⋊C4—3022no—
191C22⋊C4C22⋊D15112yes—
192C42—22no—
193C42C22×D15111yescharacteristic
194C42C22×D15111yescharacteristic
195C23—3021no—
196C23—3021no—
197C23—3021no—
198C23—6042no—
199C23—12082no—
200C23C2×C22⋊D15111yescharacteristic
201Q8—22no—
202Q8—22no—
203Q8—3022no—
204D4—3022no—
205D4—3022no—
206D4—3022no—
207D4—6042no—
208D4—6042no—
209D4—12082no—
210D4—12082no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—3021no—
215C2×C4—6042no—
216C2×C4C2×C22⋊D15112yes—
217C2×C4C2×C22⋊D15112yes—
218C2×C4C2×C22⋊D15111yescharacteristic
219C2×C4C23×D15111yescharacteristic
220C8—3022no—
221C8—3022no—
222C22—22no—
223C22—21no—
224C22—6042no—
225C22—6042no—
226C22—6042no—
227C22—6041no—
228C22—6041no—
229C22—6041no—
230C22—12084no—
231C22C22×C22⋊D15111yescharacteristic
232C4—22no—
233C4—22no—
234C4—21no—
235C4—42no—
236C4—42no—
237C4—6041no—
238C4#1801278512yes—
239C2—41no—
240C2—12081no—
241C2—12081no—
242C2—12081no—
243C2#2375277611yescharacteristic
244C2#2375302012yes—
subgroupG/H
centerC22C22×C22⋊D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#541×C151C2
socleC22×C151C22×D4
prime radical#23752781C2
hypercenter#541D151
hypocenterC151#3222
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #23752604C2semidirect
3. #23752586C2semidirect
4. #23752578C2semidirect
5. #23752569C2semidirect
6. #23752434C2semidirect
7. #23752412C2semidirect
8. #23752372C2semidirect
9. #23752362C2semidirect
10. #23752273C2semidirect
11. #541×C151C2semidirect
14. C2×#14558994C22semidirect
17. C2×#14558991C22semidirect
24. C4⋊(C151⋊C8)C22semidirect
25. #18012662C22semidirect
28. #18012659C22semidirect
30. #18012650C22semidirect
31. C4×C151⋊C8C22semidirect
32. C4⋊Q8×C151C22semidirect
35. #125×C151C22semidirect
37. C4×Q8×C151C22semidirect
122. C151#3222semidirect
124. #541D151semidirect
141. C4⋊Q8C2×D151semidirect
146. #125C2×D151semidirect
152. C4×Q8C2×D151semidirect

Maximal subgroups

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

subgroupindex
#237527812
#237526042
#237525862
#237525782
#237525692
#237524342
#237524122
#237523722
#237523622
#237522732
#541×C1512
#3222151