Finite Group Databasebeta

← #34078588

Subgroups — #34078588

Order 19328.

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

normal subgroups only

This group has 584 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
1C2×#18012764C211yesmaximal, characteristic, complemented
2C2×#18012735C211yesmaximal, characteristic, complemented
3#41×Dic151C211yesmaximal, characteristic, complemented
4#23752410C214yesmaximal, complemented
5#23752405C214yesmaximal, complemented
6#23752235C211yesmaximal, characteristic, complemented
7#23752218C211yesmaximal, characteristic, complemented
8#23752217C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C151C211yesmaximal, characteristic, complemented
10C22×D604C2211yescharacteristic
11C22×C22⋊D151—22no—
12#18012764—24no—
13C2×#14558991C2212yescomplemented
14C2×#14558989C2212yescomplemented
15C2×#14558988C2211yescharacteristic
16#18012735—21no—
17#18012735—21no—
18#18012735C2214yes—
19Q8×Dic151C2212yescomplemented
20D4×Dic151—22no—
21D4×Dic151C2212yescomplemented
22#18012698—21no—
23#18012698C2211yescharacteristic
24C2×C4×Dic151—21no—
25C2×C4×Dic151C2211yescharacteristic
26Dic151⋊C8C2214yescomplemented
27#18012661C2214yescomplemented
28#18012660C2214yescomplemented
29C2×#41×C151C2211yescharacteristic
30C22×D4×C151C2211yescharacteristic
31AGL(1,ℤ/8)×C151—24no—
32C2×SD16×C151C2212yescomplemented
33C2×D8×C151C2212yescomplemented
34C2×M16×C151C2211yescharacteristic
35C23×D151—21no—
36C2×D604—22no—
37C2×D604—21no—
38C2×D604—22no—
39C2×D604C2312yes—
40C2×C22⋊D151—24no—
41C2×C22⋊D151—24no—
42C2×C22⋊D151—48no—
43C22×Dic151—22no—
44C22×Dic151D412yes—
45#14558991—24no—
46#14558989—24no—
47#14558988—22no—
48C2×C151⋊C8C2312yes—
49C4⋊Dic151—22no—
50C4⋊Dic151C2312yes—
51C22⋊Dic151—22no—
52C22⋊Dic151—21no—
53C4×Dic151—22no—
54C4×Dic151—21no—
55C4×Dic151D411yescharacteristic
56C4×Dic151D411yescharacteristic
57C4×Dic151C2312yes—
58C24×C151—21no—
59#41×C151—22no—
60#41×C151D414yescomplemented
61C2×Q8×C151C2311yescharacteristic
62C2×D4×C151—21no—
63C2×D4×C151—22no—
64C2×D4×C151—21no—
65C2×D4×C151—22no—
66C2×D4×C151C2312yes—
67C2×D4×C151C2311yescharacteristic
68C22×C4×C151—21no—
69C22×C4×C151C2311yescharacteristic
70SD16×C151—24no—
71D8×C151—24no—
72M16×C151—22no—
73C2×C8×C151C2312yes—
74C22×D151—22no—
75C22×D151—44no—
76C22×D151—42no—
77D604—22no—
78D604—22no—
79D604—42no—
80C22⋊D151—48no—
81C22⋊D151—48no—
82C2×Dic151—24no—
83C2×Dic151—22no—
84C2×Dic151—44no—
85C2×Dic151C2×D412yes—
86C2×Dic151C2×D412yes—
87C151⋊C8—22no—
88C23×C151—21no—
89C23×C151—22no—
90C23×C151—44no—
91C23×C151—42no—
92C23×C151C2×D411yescharacteristic
93Q8×C151—21no—
94Q8×C151C2×D412yescomplemented
95D4×C151—24no—
96D4×C151—22no—
97D4×C151—21no—
98D4×C151—22no—
99D4×C151—42no—
100D4×C151C2×D412yescomplemented
101C2×C4×C151—21no—
102C2×C4×C151—24no—
103C2×C4×C151C2×D411yescharacteristic
104C2×C4×C151C2×D412yes—
105C2×C4×C151#4112yes—
106C2×C4×C151C2411yescharacteristic
107C8×C151—22no—
108C22×C151—22no—
109C22×C151—22no—
110C22×C151—42no—
111C22×C151—42no—
112C22×C151—44no—
113C22×C151—84no—
114C22×C151#12112yes—
115C22×C151C22×D411yescharacteristic
116C2×D151—42no—
117C2×D151—44no—
118C2×D151—84no—
119C4×C151—22no—
120C4×C151#12112yes—
121C4×C151C22×D411yescharacteristic
122C4×C151C2×#4111yescharacteristic
123Dic151—22no—
124Dic151—42no—
125Dic151AGL(1,ℤ/8)14yescomplemented
126C2×C151—22no—
127C2×C151—42no—
128C2×C151—82no—
129C2×C151C2×#12111yescharacteristic
130C2×C151C2×AGL(1,ℤ/8)12yes—
131D151—82no—
132C151#303811yescharacteristic, complemented, Sylow
133#3038—1511nomaximal, Sylow
134C2×AGL(1,ℤ/8)—1511no—
135C2×AGL(1,ℤ/8)D15111yescharacteristic, complemented
136C2×#127—1511no—
137C4×#41—1511no—
138#460—1514no—
139#459—1514no—
140#423—1511no—
141#418—1511no—
142#418—1511no—
143C2×#41C2×D15111yescharacteristic
144C22×D4—3022no—
145C22×D4—1511no—
146C22×D4C2×D15111yescharacteristic
147AGL(1,ℤ/8)—24no—
148AGL(1,ℤ/8)—3024no—
149C2×SD16—1512no—
150C2×SD16C2×D15112yescomplemented
151C2×D8—1512no—
152C2×D8C2×D15112yescomplemented
153C2×M16—1511no—
154C2×M16C2×D15111yescharacteristic
155#127—3021no—
156#127—3021no—
157#127—1514no—
158C4×Q8—1512no—
159C4×D4—3022no—
160C4×D4—1512no—
161#117—3021no—
162#117—1511no—
163C2×C42—3021no—
164C2×C42—1511no—
165C4⋊C8—1514no—
166#102—1514no—
167#102—1514no—
168C24—21no—
169C24—3021no—
170#41—22no—
171#41C22⋊D15114yescomplemented
172C2×Q8C22×D15111yescharacteristic
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—3024no—
178C2×D4—3024no—
179C2×D4—3021no—
180C2×D4—3022no—
181C2×D4—3022no—
182C2×D4—6048no—
183C2×D4—1512no—
184C2×D4C22×D15112yes—
185C2×D4C22×D15111yescharacteristic
186C22×C4—21no—
187C22×C4—3022no—
188C22×C4—1512no—
189C22×C4C22×D15111yescharacteristic
190SD16—24no—
191SD16—3024no—
192D8—24no—
193D8—3024no—
194M16—22no—
195M16—3022no—
196C2×C8—1512no—
197C2×C8C22×D15112yes—
198C4⋊C4—3022no—
199C4⋊C4—1512no—
200C22⋊C4—3021no—
201C22⋊C4—3022no—
202C42—3022no—
203C42—3021no—
204C42—1511no—
205C42—1511no—
206C42—1512no—
207C23—21no—
208C23—22no—
209C23—44no—
210C23—42no—
211C23—3022no—
212C23—6044no—
213C23—6042no—
214C23D4×D15111yescharacteristic
215Q8—21no—
216Q8C2×C22⋊D15112yescomplemented
217D4—24no—
218D4—22no—
219D4—21no—
220D4—22no—
221D4—42no—
222D4—3022no—
223D4—3022no—
224D4—6048no—
225D4—6048no—
226D4—6042no—
227D4C2×C22⋊D15112yescomplemented
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—3024no—
231C2×C4—3022no—
232C2×C4—6044no—
233C2×C4—1512no—
234C2×C4—1512no—
235C2×C4D4×D15111yescharacteristic
236C2×C4C2×C22⋊D15112yes—
237C2×C4#1455900312yes—
238C2×C4C23×D15111yescharacteristic
239C8—22no—
240C8—3022no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—44no—
245C22—42no—
246C22—84no—
247C22—6044no—
248C22—6042no—
249C22—12084no—
250C22#1801271112yes—
251C22C2×D4×D15111yescharacteristic
252C4—22no—
253C4—3022no—
254C4—6042no—
255C4—1514no—
256C4#1801271112yes—
257C4C22×C22⋊D15111yescharacteristic
258C4C2×#1455900311yescharacteristic
259C2—22no—
260C2—42no—
261C2—82no—
262C2—12082no—
263C2C2×#1801271111yescharacteristic
264C2AGL(1,ℤ/8)×D15112yes—
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
FittingC2×AGL(1,ℤ/8)×C151C2
socleC22×C151C22×D4
hypercenterC2×AGL(1,ℤ/8)D151
hypocenterC151#3038
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#18012764C2semidirect
2. C2×#18012735C2semidirect
3. #41×Dic151C2semidirect
4. #23752410C2semidirect
5. #23752405C2semidirect
6. #23752235C2semidirect
7. #23752218C2semidirect
8. #23752217C2semidirect
9. C2×AGL(1,ℤ/8)×C151C2semidirect
13. C2×#14558991C22semidirect
14. C2×#14558989C22semidirect
19. Q8×Dic151C22semidirect
21. D4×Dic151C22semidirect
26. Dic151⋊C8C22semidirect
27. #18012661C22semidirect
28. #18012660C22semidirect
32. C2×SD16×C151C22semidirect
33. C2×D8×C151C22semidirect
60. #41×C151D4semidirect
94. Q8×C151C2×D4semidirect
100. D4×C151C2×D4semidirect
125. Dic151AGL(1,ℤ/8)semidirect
132. C151#3038semidirect
135. C2×AGL(1,ℤ/8)D151semidirect
150. C2×SD16C2×D151semidirect
152. C2×D8C2×D151semidirect
171. #41C22⋊D151semidirect
216. Q8C2×C22⋊D151semidirect
227. D4C2×C22⋊D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127642
C2×#180127352
#41×Dic1512
#237524102
#237524052
#237522352
#237522182
#237522172
C2×AGL(1,ℤ/8)×C1512
#3038151