Finite Group Databasebeta

← #34078481

Subgroups — #34078481

Order 19328.

736 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 145 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#23753048C211yesmaximal, characteristic, complemented
2C2×#18012765C212yesmaximal, complemented
3#23752384C214yesmaximal, complemented
4#23752379C214yesmaximal, complemented
5#23752230C212yesmaximal, complemented
6#23752196C211yesmaximal, characteristic, complemented
7#535×C151C211yesmaximal, characteristic, complemented
8#18012787C2212yescomplemented
9#18012786—22no—
10#18012785—22no—
11#18012784—22no—
12#18012784C2212yescomplemented
13C2×#14559004—22no—
14C2×#14559004C2212yes—
15#41×D151—24no—
16#18012765—24no—
17#18012765C2218yescomplemented
18C2×#14558992—22no—
19C2×#14558989—22no—
20C2×#14558988C2212yescomplemented
21#18012667C2218yescomplemented
22#18012658C2212yescomplemented
23#18012656C2212yescomplemented
24C2×#41×C151C2211yescharacteristic
25C22×D4×C151C2211yescharacteristic, complemented
26#127×C151C2212yescomplemented
27#124×C151C2212yescomplemented
28#121×C151—22no—
29#120×C151—22no—
30#117×C151C2211yescharacteristic, complemented
31#14559005—24no—
32#14559004—24no—
33#14559004—24no—
34#14559004—22no—
35#14559004—24no—
36#14559004—22no—
37#14559004D414yescomplemented
38#14559004C2314yes—
39#14559003—28no—
40#14559003—24no—
41C2×Dic302—22no—
42C2×Dic302D412yescomplemented
43Q8×D151—24no—
44C2×D604—22no—
45C2×D604D412yescomplemented
46C2×C22⋊D151—24no—
47C2×C22⋊D151—22no—
48D4×D151—28no—
49D4×D151—24no—
50C2×C4×D151—24no—
51C2×C4×D151—22no—
52#14558992—28no—
53#14558989—28no—
54#14558988—22no—
55#14558988C2314yes—
56C2×C151⋊C8—22no—
57C24×C151—21no—
58#41×C151—22no—
59#41×C151—22no—
60C2×Q8×C151C2311yescharacteristic
61C2×D4×C151—24no—
62C2×D4×C151—22no—
63C2×D4×C151—22no—
64C2×D4×C151—21no—
65C2×D4×C151D412yes—
66C2×D4×C151C2312yes—
67C2×D4×C151C2311yescharacteristic
68C22×C4×C151D412yes—
69C22×C4×C151C2311yescharacteristic
70C4⋊C4×C151—21no—
71C22⋊C4×C151—21no—
72C22⋊C4×C151—24no—
73C42×C151C2312yes—
74C22×D151—24no—
75C22×D151—22no—
76Dic302—24no—
77Dic302—22no—
78Dic302C2×D414yes—
79D604—24no—
80D604—22no—
81D604C2×D414yes—
82C22⋊D151—28no—
83C22⋊D151—24no—
84C22⋊D151—24no—
85C2×Dic151—24no—
86C2×Dic151—22no—
87C4×D151—28no—
88C4×D151—24no—
89C4×D151—24no—
90C151⋊C8—24no—
91C23×C151—22no—
92C23×C151—42no—
93C23×C151—82no—
94C23×C151C2×D412yes—
95C23×C151C2×D411yescharacteristic
96Q8×C151—22no—
97D4×C151—24no—
98D4×C151—22no—
99D4×C151—24no—
100D4×C151—41no—
101D4×C151—41no—
102D4×C151—84no—
103C2×C4×C151—22no—
104C2×C4×C151—22no—
105C2×C4×C151—22no—
106C2×C4×C151C2×D412yes—
107C2×C4×C151C2×D412yes—
108C2×C4×C151C2×D412yes—
109C2×C4×C151C2×D411yescharacteristic
110C2×C4×C151C2411yescharacteristic
111C22×C151—22no—
112C22×C151—41no—
113C22×C151—42no—
114C22×C151—44no—
115C22×C151—42no—
116C22×C151—84no—
117C22×C151#12012yes—
118C22×C151C22×D411yescharacteristic
119C2×D151—24no—
120C2×D151—82no—
121C2×D151—84no—
122C4×C151—22no—
123C4×C151—42no—
124C4×C151#12011yescharacteristic
125C4×C151#12011yescharacteristic
126C4×C151C22×D412yes—
127Dic151—24no—
128C2×C151—21no—
129C2×C151—22no—
130C2×C151—42no—
131C2×C151—82no—
132C2×C151C2×#12011yescharacteristic
133D151—44no—
134C151#302111yescharacteristic, complemented, Sylow
135#3021—1511nomaximal, Sylow
136#585—1511no—
137C2×AGL(1,ℤ/8)—1512no—
138#535D15111yescharacteristic, complemented
139#454—1514no—
140#453—1514no—
141#421—1512no—
142#413—1511no—
1432−1+4—3022no—
1442−1+4—1512no—
145H(5,𝔽2)—3022no—
146H(5,𝔽2)—3022no—
147H(5,𝔽2)—1512no—
148C2×#41—3024no—
149C2×#41—3022no—
150C2×#41—1512no—
151C2×#41C2×D15111yescharacteristic
152C22×D4C2×D15111yescharacteristic, complemented
153AGL(1,ℤ/8)—3024no—
154AGL(1,ℤ/8)—1518no—
155C2×SD16—3022no—
156C2×D8—3022no—
157C2×M16—1512no—
158#127C2×D15112yescomplemented
159#124C2×D15112yescomplemented
160#121—22no—
161#120—22no—
162#117C2×D15111yescharacteristic, complemented
163#104—1518no—
164#101—1512no—
165#100—1512no—
166C24—21no—
167#41—22no—
168#41—22no—
169#41—3024no—
170#41—3024no—
171#41—3028no—
172#41—3024no—
173#41—3024no—
174#41—3022no—
175#41—3022no—
176#41—3024no—
177#41—1514no—
178#41—1514no—
179C2×Q8—3022no—
180C2×Q8—3024no—
181C2×Q8—1512no—
182C2×Q8C22×D15111yescharacteristic
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—21no—
187C2×D4—3022no—
188C2×D4—3028no—
189C2×D4—3024no—
190C2×D4—3024no—
191C2×D4—3022no—
192C2×D4—1512no—
193C2×D4C22⋊D15112yes—
194C2×D4C22×D15112yes—
195C2×D4C22×D15111yescharacteristic
196C22×C4—3024no—
197C22×C4—3022no—
198C22×C4C22⋊D15112yes—
199C22×C4C22×D15111yescharacteristic
200SD16—3028no—
201D8—3028no—
202M16—3022no—
203M16—1514no—
204C2×C8—3022no—
205C4⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—24no—
208C42C22×D15112yes—
209C23—22no—
210C23—42no—
211C23—82no—
212C23—3024no—
213C23—3022no—
214C23C2×C22⋊D15112yes—
215C23C2×C22⋊D15111yescharacteristic
216Q8—22no—
217Q8—3024no—
218Q8—3022no—
219Q8—1514no—
220D4—24no—
221D4—22no—
222D4—24no—
223D4—41no—
224D4—41no—
225D4—84no—
226D4—3024no—
227D4—3028no—
228D4—3024no—
229D4—3024no—
230D4—3022no—
231D4—1514no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—3028no—
236C2×C4—3024no—
237C2×C4—3022no—
238C2×C4—3024no—
239C2×C4—3024no—
240C2×C4D4×D15112yes—
241C2×C4D4×D15112yes—
242C2×C4C2×C22⋊D15112yes—
243C2×C4C2×C22⋊D15111yescharacteristic
244C2×C4C23×D15111yescharacteristic
245C8—3024no—
246C22—22no—
247C22—41no—
248C22—42no—
249C22—44no—
250C22—42no—
251C22—84no—
252C22—3024no—
253C22—12082no—
254C22—12084no—
255C22#1801270712yes—
256C22C22×C22⋊D15111yescharacteristic
257C4—22no—
258C4—42no—
259C4—3024no—
260C4#1801270711yescharacteristic
261C4#1801270711yescharacteristic
262C4C2×D4×D15112yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—82no—
267C2—6044no—
268C2C2×#1801270711yescharacteristic
subgroupG/H
centerC2C2×#18012707
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#535×C151C2
socleC2×C151C2×#120
hypercenter#535D151
hypocenterC151#3021
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #23753048C2semidirect
2. C2×#18012765C2semidirect
3. #23752384C2semidirect
4. #23752379C2semidirect
5. #23752230C2semidirect
6. #23752196C2semidirect
7. #535×C151C2semidirect
8. #18012787C22semidirect
12. #18012784C22semidirect
17. #18012765C22semidirect
20. C2×#14558988C22semidirect
21. #18012667C22semidirect
22. #18012658C22semidirect
23. #18012656C22semidirect
25. C22×D4×C151C22semidirect
26. #127×C151C22semidirect
27. #124×C151C22semidirect
30. #117×C151C22semidirect
37. #14559004D4semidirect
42. C2×Dic302D4semidirect
45. C2×D604D4semidirect
134. C151#3021semidirect
138. #535D151semidirect
152. C22×D4C2×D151semidirect
158. #127C2×D151semidirect
159. #124C2×D151semidirect
162. #117C2×D151semidirect

Maximal subgroups

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

subgroupindex
#237530482
C2×#180127652
#237523842
#237523792
#237522302
#237521962
#535×C1512
#3021151