Finite Group Databasebeta

← #34080126

Subgroups — #34080126

Order 19328.

490 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 115 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×#18012735C211yesmaximal, characteristic, complemented
2#23752743C211yesmaximal, characteristic, complemented
3#23752600C212yesmaximal, complemented
4#23752575C212yesmaximal, complemented
5#23752411C212yesmaximal, complemented
6#23752405C212yesmaximal, complemented
7#23752365C211yesmaximal, characteristic, complemented
8#23752363C211yesmaximal, characteristic, complemented
9#23752345C211yesmaximal, characteristic, complemented
10#23752266C211yesmaximal, characteristic, complemented
11#447×C151C211yesmaximal, characteristic, complemented
12C22×D604C2211yescharacteristic
13C22×C22⋊D151—21no—
14C22×C22⋊D151—21no—
15C2×#14558993C2212yescomplemented
16C2×#14558992C2212yescomplemented
17C2×#14558989C2212yescomplemented
18#18012735—21no—
19#18012735—21no—
20#18012735C2212yes—
21#18012735C2212yes—
22#18012732—21no—
23#18012730C2211yescharacteristic, complemented
24#18012726C2211yescharacteristic, complemented
25#18012720—21no—
26#18012718C2211yescharacteristic
27#18012704C2211yescharacteristic, complemented
28D4×Dic151—21no—
29D4×Dic151C2211yescharacteristic, complemented
30#18012695—21no—
31C2×C4×Dic151C2211yescharacteristic
32Dic151⋊C8C2212yescomplemented
33#18012661C2212yescomplemented
34#18012660C2212yescomplemented
35#18012659C2212yescomplemented
36C22⋊(C151⋊C8)C2211yescharacteristic
37#18012649C2212yescomplemented
38C22×D4×C151C2211yescharacteristic
39C2×D8×C151C2212yescomplemented
40#121×C151C2211yescharacteristic
41#102×C151C2212yescomplemented
42C22⋊C8×C151C2211yescharacteristic
43C23×D151—21no—
44C2×Dic302C2311yescharacteristic
45C2×D604—21no—
46C2×D604—42no—
47C2×D604C2312yes—
48C2×C22⋊D151—22no—
49C2×C22⋊D151—22no—
50C2×C22⋊D151—22no—
51C2×C22⋊D151—22no—
52C2×C22⋊D151—44no—
53C2×C22⋊D151—44no—
54C22×Dic151—21no—
55C22×Dic151D411yescharacteristic
56C22×Dic151D411yescharacteristic
57#14558993—42no—
58#14558992—42no—
59#14558989—42no—
60C2×C151⋊C8C2312yes—
61C4⋊Dic151—21no—
62C4⋊Dic151C2311yescharacteristic
63Dic151⋊C4—21no—
64Dic151⋊C4—21no—
65Dic151⋊C4—21no—
66C22⋊Dic151—21no—
67C22⋊Dic151—21no—
68C22⋊Dic151—21no—
69C22⋊Dic151—21no—
70C4×Dic151—21no—
71C4×Dic151D411yescharacteristic
72C4×Dic151D411yescharacteristic
73C4×Dic151C2311yescharacteristic
74C4×Dic151C2311yescharacteristic
75C24×C151—21no—
76C2×D4×C151—21no—
77C2×D4×C151—21no—
78C2×D4×C151—42no—
79C2×D4×C151C2311yescharacteristic
80C2×D4×C151C2312yes—
81C22×C4×C151C2311yescharacteristic
82D8×C151—42no—
83C2×C8×C151C2312yes—
84C4⋊C4×C151C2311yescharacteristic
85C22⋊C4×C151—21no—
86C22×D151—22no—
87C22×D151—42no—
88C22×D151—42no—
89C22×D151—42no—
90Dic302—22no—
91D604—22no—
92D604—22no—
93D604—42no—
94C22⋊D151—44no—
95C22⋊D151—44no—
96C22⋊D151—44no—
97C22⋊D151—44no—
98C2×Dic151—22no—
99C2×Dic151—22no—
100C2×Dic151—21no—
101C2×Dic151—21no—
102C2×Dic151—42no—
103C2×Dic151C2×D411yescharacteristic
104C2×Dic151C2×D411yescharacteristic
105C2×Dic151C2×D411yescharacteristic
106C2×Dic151C2×D411yescharacteristic
107C151⋊C8—22no—
108C23×C151—22no—
109C23×C151—21no—
110C23×C151—42no—
111C23×C151—42no—
112C23×C151—42no—
113C23×C151C2×D411yescharacteristic
114D4×C151—22no—
115D4×C151—22no—
116D4×C151—22no—
117D4×C151—42no—
118D4×C151—42no—
119C2×C4×C151—22no—
120C2×C4×C151—21no—
121C2×C4×C151C2×D411yescharacteristic
122C2×C4×C151C2411yescharacteristic
123C8×C151—22no—
124C22×C151—21no—
125C22×C151—21no—
126C22×C151—42no—
127C22×C151—41no—
128C22×C151—41no—
129C22×C151—42no—
130C22×C151—42no—
131C22×C151—41no—
132C22×C151—84no—
133C22×C151AGL(1,ℤ/8)12yes—
134C22×C151C22×D411yescharacteristic
135C2×D151—42no—
136C2×D151—42no—
137C2×D151—42no—
138C2×D151—84no—
139C4×C151—21no—
140C4×C151—41no—
141C4×C151H(5,𝔽2)12yes—
142Dic151—21no—
143Dic151—21no—
144Dic151—21no—
145Dic151—41no—
146Dic151—41no—
147Dic151AGL(1,ℤ/8)12yes—
148C2×C151—22no—
149C2×C151—81no—
150C2×C151—82no—
151C2×C151#53511yescharacteristic
152C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
153C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
154D151—82no—
155C151#321711yescharacteristic, complemented, Sylow
156#3217—1511nomaximal, Sylow
157C2×#127—1511no—
158#529—1511no—
159#496—1512no—
160#490—1512no—
161#460—1512no—
162#459—1512no—
163#450—1511no—
164#450—1511no—
165#447—1511no—
166#447D15111yescharacteristic, complemented
167#432—1511no—
168C22×D4—3021no—
169C22×D4—3021no—
170C22×D4—1511no—
171C22×D4C2×D15111yescharacteristic
172C2×SD16—1512no—
173C2×SD16—1512no—
174C2×D8—1512no—
175C2×D8C2×D15112yescomplemented
176#127—3021no—
177#127—3021no—
178#127—1512no—
179#127—1512no—
180#126—3021no—
181#125—1511no—
182#124—1511no—
183#123—3021no—
184C22⋊Q8—1511no—
185#121C2×D15111yescharacteristic
186C4×Q8—1511no—
187C4×D4—3021no—
188C4×D4—1511no—
189#117—3021no—
190C2×C42—1511no—
191C4⋊C8—1512no—
192#102—1512no—
193#102—1512no—
194#102—1512no—
195#102C2×D15112yescomplemented
196C22⋊C8—1511no—
197C22⋊C8C2×D15111yescharacteristic
198#97—1512no—
199C24—21no—
200C24—3021no—
201C2×Q8—1511no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—42no—
205C2×D4—3022no—
206C2×D4—3022no—
207C2×D4—3022no—
208C2×D4—3022no—
209C2×D4—3021no—
210C2×D4—6042no—
211C2×D4—6044no—
212C2×D4—6044no—
213C2×D4—1512no—
214C2×D4C22×D15112yes—
215C2×D4C22×D15111yescharacteristic
216C22×C4—3021no—
217C22×C4—1511no—
218C22×C4—1511no—
219C22×C4C22×D15111yescharacteristic
220SD16—6042no—
221SD16—6042no—
222D8—42no—
223D8—6042no—
224C2×C8—1512no—
225C2×C8C22×D15112yes—
226C4⋊C4—3021no—
227C4⋊C4—3021no—
228C4⋊C4—3021no—
229C4⋊C4—3021no—
230C4⋊C4—1511no—
231C4⋊C4C22×D15111yescharacteristic
232C22⋊C4—21no—
233C22⋊C4—3021no—
234C22⋊C4—3021no—
235C22⋊C4—3021no—
236C22⋊C4—3021no—
237C42—3021no—
238C42—1511no—
239C42—1511no—
240C42—1511no—
241C42—1511no—
242C23—21no—
243C23—22no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—3022no—
248C23—6042no—
249C23—6042no—
250C23—6042no—
251C23D4×D15111yescharacteristic
252Q8—3022no—
253D4—22no—
254D4—22no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—3022no—
259D4—3022no—
260D4—6042no—
261D4—6044no—
262D4—6044no—
263D4—6044no—
264D4—6044no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—3022no—
268C2×C4—3022no—
269C2×C4—3021no—
270C2×C4—3021no—
271C2×C4—6042no—
272C2×C4—1511no—
273C2×C4—1511no—
274C2×C4—1511no—
275C2×C4—1511no—
276C2×C4D4×D15111yescharacteristic
277C2×C4C23×D15111yescharacteristic
278C8—22no—
279C8—3022no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—42no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—84no—
289C22—6042no—
290C22—6042no—
291C22—6042no—
292C22—12084no—
293C22#1801276312yes—
294C22C2×D4×D15111yescharacteristic
295C4—21no—
296C4—41no—
297C4—3021no—
298C4—3021no—
299C4—3021no—
300C4—6041no—
301C4—6041no—
302C4—1512no—
303C4#1801278412yes—
304C2—22no—
305C2—82no—
306C2—81no—
307C2—12082no—
308C2#2375277711yescharacteristic
309C2AGL(1,ℤ/8)×D15111yescharacteristic
310C2C2×#1801276311yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#447×C151C2
socleC22×C151C22×D4
hypercenter#447D151
hypocenterC151#3217
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#18012735C2semidirect
2. #23752743C2semidirect
3. #23752600C2semidirect
4. #23752575C2semidirect
5. #23752411C2semidirect
6. #23752405C2semidirect
7. #23752365C2semidirect
8. #23752363C2semidirect
9. #23752345C2semidirect
10. #23752266C2semidirect
11. #447×C151C2semidirect
15. C2×#14558993C22semidirect
16. C2×#14558992C22semidirect
17. C2×#14558989C22semidirect
23. #18012730C22semidirect
24. #18012726C22semidirect
27. #18012704C22semidirect
29. D4×Dic151C22semidirect
32. Dic151⋊C8C22semidirect
33. #18012661C22semidirect
34. #18012660C22semidirect
35. #18012659C22semidirect
37. #18012649C22semidirect
39. C2×D8×C151C22semidirect
41. #102×C151C22semidirect
155. C151#3217semidirect
166. #447D151semidirect
175. C2×D8C2×D151semidirect
195. #102C2×D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127352
#237527432
#237526002
#237525752
#237524112
#237524052
#237523652
#237523632
#237523452
#237522662
#447×C1512
#3217151