Finite Group Databasebeta

← #34081105

Subgroups — #34081105

Order 19328.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23752873C211yesmaximal, characteristic, complemented
2#23752862C211yesmaximal, characteristic, complemented
3#23752573C211yesmaximal, characteristic, complemented
4#23752546C211yesmaximal, characteristic, complemented
5#23752526C211yesmaximal, characteristic, complemented
6#23752505C211yesmaximal, characteristic, complemented
7#23752455C211yesmaximal, characteristic, complemented
8#23752446C211yesmaximal, characteristic, complemented
9#23752429C211yesmaximal, characteristic, complemented
10#23752356C211yesmaximal, characteristic, complemented
11#23752313C211yesmaximal, characteristic, complemented
12#23752280C211yesmaximal, characteristic, complemented
13#23752212C211yesmaximal, characteristic, complemented
14C2×#18012663C211yesmaximal, characteristic, complemented
15#448×C151C211yesmaximal, characteristic, complemented
16C2×#14559005C2211yescharacteristic
17C2×#14559003—21no—
18C2×#14558989C2211yescharacteristic
19C2×#14558987C2211yescharacteristic
20C22×C151⋊C8C2211yescharacteristic
21#18012738C2211yescharacteristic
22#18012728—21no—
23#18012728C2211yescharacteristic
24#18012721—21no—
25#18012720—21no—
26#18012716—21no—
27#18012715—21no—
28#18012715C2211yescharacteristic
29#18012713—21no—
30#18012713C2211yescharacteristic
31#18012710C2211yescharacteristic
32Q8×Dic151C2211yescharacteristic
33D4×Dic151C2211yescharacteristic
34C4×C22⋊D151—21no—
35C4×C22⋊D151C2211yescharacteristic
36C2×C4⋊Dic151C2211yescharacteristic
37C4⋊C4×D151C2211yescharacteristic
38#18012673C2211yescharacteristic
39#18012673C2211yescharacteristic
40#18012672C2211yescharacteristic
41Dic151⋊C8C2211yescharacteristic
42#18012663C2211yescharacteristic
43#18012663C2211yescharacteristic
44#18012663C2211yescharacteristic
45#18012663C2211yescharacteristic
46#18012661C2211yescharacteristic
47#18012661C2211yescharacteristic
48#18012660C2211yescharacteristic
49#18012659C2211yescharacteristic
50(C2×D151)⋊C8C2211yescharacteristic
51#18012649C2211yescharacteristic
52C22×Q8×C151C2211yescharacteristic
53C2×SD16×C151C2211yescharacteristic
54C2×SD16×C151C2211yescharacteristic
55#121×C151C2211yescharacteristic
56#103×C151C2211yescharacteristic
57#103×C151C2211yescharacteristic
58C22⋊C8×C151C2211yescharacteristic
59#14559005—22no—
60#14559005—41no—
61#14559003—44no—
62C2×Dic302—21no—
63C2×D604—21no—
64C2×D604C2311yescharacteristic
65C2×C22⋊D151—21no—
66C2×C22⋊D151—21no—
67C2×C22⋊D151D411yescharacteristic
68C22×Dic151—21no—
69C22×Dic151—21no—
70C2×C4×D151—21no—
71C2×C4×D151—21no—
72C2×C4×D151C2311yescharacteristic
73#14558989—41no—
74#14558987—41no—
75C2×C151⋊C8—41no—
76C2×C151⋊C8C2311yescharacteristic
77C2×C151⋊C8C2311yescharacteristic
78C4⋊Dic151—21no—
79C4⋊Dic151—21no—
80C4⋊Dic151C2311yescharacteristic
81C4⋊Dic151C2311yescharacteristic
82C4⋊Dic151C2311yescharacteristic
83Dic151⋊C4—21no—
84Dic151⋊C4—21no—
85Dic151⋊C4—21no—
86Dic151⋊C4D411yescharacteristic
87C22⋊Dic151—21no—
88C22⋊Dic151—21no—
89C22⋊Dic151D411yescharacteristic
90(C2×D151)⋊C4—21no—
91(C2×D151)⋊C4—21no—
92(C2×D151)⋊C4—21no—
93(C2×D151)⋊C4—21no—
94(C2×D151)⋊C4D411yescharacteristic
95C4×Dic151—21no—
96C4×Dic151C2311yescharacteristic
97C2×Q8×C151—21no—
98C2×Q8×C151—22no—
99C2×Q8×C151—41no—
100C2×Q8×C151C2311yescharacteristic
101C2×Q8×C151C2311yescharacteristic
102C2×D4×C151—21no—
103C2×D4×C151C2311yescharacteristic
104C22×C4×C151—21no—
105C22×C4×C151C2311yescharacteristic
106SD16×C151—22no—
107SD16×C151—41no—
108C2×C8×C151C2311yescharacteristic
109C2×C8×C151C2311yescharacteristic
110C4⋊C4×C151C2311yescharacteristic
111C22⋊C4×C151—21no—
112C22×D151—21no—
113C22×D151C2×D411yescharacteristic
114Dic302—42no—
115D604—21no—
116D604—21no—
117D604—42no—
118C22⋊D151—22no—
119C22⋊D151—42no—
120C22⋊D151—42no—
121C2×Dic151—21no—
122C2×Dic151—21no—
123C2×Dic151—21no—
124C2×Dic151—42no—
125C2×Dic151—42no—
126C2×Dic151C2×D411yescharacteristic
127C2×Dic151C2×D411yescharacteristic
128C2×Dic151C2×D411yescharacteristic
129C4×D151—21no—
130C4×D151—21no—
131C4×D151—42no—
132C4×D151—42no—
133C151⋊C8—21no—
134C151⋊C8—21no—
135C23×C151—21no—
136C23×C151C2×D411yescharacteristic
137Q8×C151—21no—
138Q8×C151—21no—
139Q8×C151—21no—
140Q8×C151—42no—
141Q8×C151#4112yescomplemented
142D4×C151—21no—
143D4×C151—21no—
144D4×C151—42no—
145C2×C4×C151—21no—
146C2×C4×C151—21no—
147C2×C4×C151—21no—
148C2×C4×C151—21no—
149C2×C4×C151—21no—
150C2×C4×C151—42no—
151C2×C4×C151C2×D411yescharacteristic
152C2×C4×C151C2411yescharacteristic
153C8×C151—21no—
154C8×C151—21no—
155C22×C151—21no—
156C22×C151—21no—
157C22×C151—41no—
158C22×C151—41no—
159C22×C151—41no—
160C22×C151AGL(1,ℤ/8)12yes—
161C22×C151C22×D411yescharacteristic
162C2×D151—21no—
163C2×D151—21no—
164C2×D151—21no—
165C2×D151—41no—
166C2×D151—41no—
167C2×D151—41no—
168C4×C151—21no—
169C4×C151—22no—
170C4×C151—41no—
171C4×C151—41no—
172C4×C151C2×#4111yescharacteristic
173C4×C151H(5,𝔽2)11yescharacteristic
174Dic151—21no—
175Dic151—21no—
176Dic151—21no—
177Dic151—41no—
178Dic151—41no—
179Dic151—41no—
180C2×C151—22no—
181C2×C151—81no—
182C2×C151#54611yescharacteristic
183C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C151#57811yescharacteristic
185D151—41no—
186D151—81no—
187C151#330811yescharacteristic, complemented, Sylow
188#3308—1511nomaximal, Sylow
189#548—1511no—
190#547—1511no—
191#489—1511no—
192#484—1511no—
193#480—1511no—
194#477—1511no—
195#468—1511no—
196#466—1511no—
197#463—1511no—
198#449—1511no—
199#448D15111yescharacteristic, complemented
200#440—1511no—
201#435—1511no—
202#417—1511no—
203C2×#103—1511no—
204C2×#41—3021no—
205C2×#41—1511no—
206C22×Q8C2×D15111yescharacteristic
207C2×SD16C2×D15111yescharacteristic
208C2×SD16C2×D15111yescharacteristic
209C2×D8—1511no—
210C2×M16—1511no—
211C22×C8—1511no—
212C4⋊Q8—1511no—
213#124—3021no—
214#124—1511no—
215#123—3021no—
216#123—3021no—
217C22⋊Q8—3021no—
218C22⋊Q8—3021no—
219C22⋊Q8—1511no—
220#121—3021no—
221#121—1511no—
222#121—1511no—
223#121C2×D15111yescharacteristic
224C4×Q8—1511no—
225C4×D4—3021no—
226C4×D4—1511no—
227C4×D4—1511no—
228C2×C4⋊C4—1511no—
229C2×C4⋊C4—1511no—
230#107—1511no—
231#107—1511no—
232#107—1511no—
233C4⋊C8—1511no—
234#103—1511no—
235#103—1511no—
236#103—1511no—
237#103—1511no—
238#103C2×D15111yescharacteristic
239#103C2×D15111yescharacteristic
240#102—1511no—
241#102—1511no—
242#102—1511no—
243#102—1511no—
244C22⋊C8—1511no—
245C22⋊C8C2×D15111yescharacteristic
246#97—1511no—
247#41—3022no—
248#41—6041no—
249#41—6044no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—3021no—
254C2×Q8C22×D15111yescharacteristic
255C2×Q8C22×D15111yescharacteristic
256C2×D4—21no—
257C2×D4—3021no—
258C2×D4—3021no—
259C2×D4—3021no—
260C2×D4—1511no—
261C2×D4—1511no—
262C2×D4C22×D15111yescharacteristic
263C22×C4—21no—
264C22×C4—3021no—
265C22×C4—3021no—
266C22×C4—3021no—
267C22×C4—3021no—
268C22×C4—1511no—
269C22×C4C22×D15111yescharacteristic
270SD16—22no—
271SD16—41no—
272D8—6041no—
273M16—6041no—
274C2×C8—6041no—
275C2×C8—1511no—
276C2×C8—1511no—
277C2×C8C22×D15111yescharacteristic
278C2×C8C22×D15111yescharacteristic
279C4⋊C4—3021no—
280C4⋊C4—3021no—
281C4⋊C4—3021no—
282C4⋊C4—3021no—
283C4⋊C4—3021no—
284C4⋊C4—1511no—
285C4⋊C4—1511no—
286C4⋊C4—1511no—
287C4⋊C4—1511no—
288C4⋊C4C22×D15111yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—3021no—
291C22⋊C4—3021no—
292C22⋊C4—3021no—
293C22⋊C4—3021no—
294C22⋊C4—3021no—
295C22⋊C4—3021no—
296C22⋊C4—1511no—
297C22⋊C4—1511no—
298C42—3021no—
299C42—1511no—
300C23—21no—
301C23—3021no—
302C23—1511no—
303C23D4×D15111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—6042no—
309Q8#1455900312yescomplemented
310D4—21no—
311D4—21no—
312D4—42no—
313D4—3022no—
314D4—3021no—
315D4—3021no—
316D4—6042no—
317D4—6042no—
318D4—6042no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—3021no—
326C2×C4—3021no—
327C2×C4—3021no—
328C2×C4—3021no—
329C2×C4—3021no—
330C2×C4—6042no—
331C2×C4—6042no—
332C2×C4—6042no—
333C2×C4—6042no—
334C2×C4—1511no—
335C2×C4—1511no—
336C2×C4—1511no—
337C2×C4D4×D15111yescharacteristic
338C2×C4C23×D15111yescharacteristic
339C8—21no—
340C8—21no—
341C8—3021no—
342C8—3021no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—3021no—
349C22—3021no—
350C22—3021no—
351C22—6041no—
352C22—6041no—
353C22—6041no—
354C22#1801276612yes—
355C22C2×D4×D15111yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—3021no—
361C4—3021no—
362C4—3021no—
363C4—6041no—
364C4—6041no—
365C4—6041no—
366C4C2×#1455900311yescharacteristic
367C4#1801278411yescharacteristic
368C2—22no—
369C2—81no—
370C2—6041no—
371C2—12081no—
372C2#2375284911yescharacteristic
373C2C2×#1801276611yescharacteristic
374C2#2375302811yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#448×C151C2
socleC22×C151C22×D4
hypercenter#448D151
hypocenterC151#3308
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. #23752873C2semidirect
2. #23752862C2semidirect
3. #23752573C2semidirect
4. #23752546C2semidirect
5. #23752526C2semidirect
6. #23752505C2semidirect
7. #23752455C2semidirect
8. #23752446C2semidirect
9. #23752429C2semidirect
10. #23752356C2semidirect
11. #23752313C2semidirect
12. #23752280C2semidirect
13. #23752212C2semidirect
14. C2×#18012663C2semidirect
15. #448×C151C2semidirect
141. Q8×C151#41semidirect
187. C151#3308semidirect
199. #448D151semidirect
309. Q8#14559003semidirect

Maximal subgroups

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

subgroupindex
#237528732
#237528622
#237525732
#237525462
#237525262
#237525052
#237524552
#237524462
#237524292
#237523562
#237523132
#237522802
#237522122
C2×#180126632
#448×C1512
#3308151