Finite Group Databasebeta

← #34080663

Subgroups — #34080663

Order 19328.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18012769C211yesmaximal, characteristic, complemented
2C2×#18012758C211yesmaximal, characteristic, complemented
3#23752868C211yesmaximal, characteristic, complemented
4#23752861C211yesmaximal, characteristic, complemented
5#23752603C211yesmaximal, characteristic, complemented
6#23752456C211yesmaximal, characteristic, complemented
7#23752452C211yesmaximal, characteristic, complemented
8#23752425C211yesmaximal, characteristic, complemented
9#23752421C211yesmaximal, characteristic, complemented
10#23752375C211yesmaximal, characteristic, complemented
11#23752371C211yesmaximal, characteristic, complemented
12#23752370C211yesmaximal, characteristic, complemented
13#23752309C211yesmaximal, characteristic, complemented
14#23752286C211yesmaximal, characteristic, complemented
15#449×C151C211yesmaximal, characteristic, complemented
16C2×#14559003C2211yescharacteristic
17C2×#14559003C2211yescharacteristic
18C22×Dic302C2211yescharacteristic
19#18012769—24no—
20#18012758—24no—
21C2×Dic604C2211yescharacteristic
22C2×Dic604C2211yescharacteristic
23C2×#14558994C2211yescharacteristic
24C2×#14558994C2211yescharacteristic
25C2×#14558992C2211yescharacteristic
26C2×#14558992C2211yescharacteristic
27C2×#14558992C2211yescharacteristic
28C2×#14558988C2211yescharacteristic
29C2×C8×D151C2211yescharacteristic
30#18012737C2211yescharacteristic
31#18012726C2211yescharacteristic
32#18012725—21no—
33#18012722—21no—
34#18012720—21no—
35#18012719C2211yescharacteristic
36#18012718C2211yescharacteristic
37#18012716—21no—
38#18012715C2211yescharacteristic
39#18012711—21no—
40#18012711—21no—
41#18012704C2211yescharacteristic
42D4×Dic151—21no—
43D4×Dic151C2211yescharacteristic
44C4×C22⋊D151C2211yescharacteristic
45C2×Dic151⋊C4—21no—
46#18012672C2211yescharacteristic
47Dic151⋊C8C2211yescharacteristic
48#18012664C2211yescharacteristic
49#18012664C2211yescharacteristic
50#18012662C2211yescharacteristic
51#18012660C2211yescharacteristic
52(C2×D151)⋊C8C2211yescharacteristic
53#18012649C2211yescharacteristic
54C2×#41×C151C2211yescharacteristic
55C2×SD16×C151C2211yescharacteristic
56C2×D8×C151C2211yescharacteristic
57#121×C151C2211yescharacteristic
58#103×C151C2211yescharacteristic
59#102×C151C2211yescharacteristic
60C22⋊C8×C151C2211yescharacteristic
61#14559003—22no—
62#14559003—22no—
63#14559003—41no—
64#14559003—41no—
65C2×Dic302—22no—
66C2×Dic302—21no—
67C2×Dic302—41no—
68C2×Dic302C2311yescharacteristic
69C2×Dic302C2311yescharacteristic
70C2×Dic302C2311yescharacteristic
71C2×C22⋊D151—21no—
72C2×C22⋊D151—21no—
73C2×C22⋊D151D411yescharacteristic
74C22×Dic151—21no—
75C22×Dic151—21no—
76C22×Dic151—21no—
77C22×Dic151—21no—
78C2×C4×D151C2311yescharacteristic
79Dic604—22no—
80Dic604—41no—
81#14558994—22no—
82#14558994—22no—
83#14558992—22no—
84#14558992—22no—
85#14558992D414yescomplemented
86#14558988—22no—
87C2×C151⋊C8C2311yescharacteristic
88C2×C151⋊C8C2311yescharacteristic
89C8×D151—22no—
90C4⋊Dic151—21no—
91C4⋊Dic151C2311yescharacteristic
92Dic151⋊C4—21no—
93Dic151⋊C4—21no—
94Dic151⋊C4—21no—
95Dic151⋊C4—21no—
96Dic151⋊C4—21no—
97Dic151⋊C4D411yescharacteristic
98C22⋊Dic151—21no—
99C22⋊Dic151—21no—
100C22⋊Dic151—21no—
101C22⋊Dic151—21no—
102C22⋊Dic151D411yescharacteristic
103(C2×D151)⋊C4—21no—
104(C2×D151)⋊C4—21no—
105(C2×D151)⋊C4D411yescharacteristic
106C4×Dic151—21no—
107C4×Dic151C2311yescharacteristic
108#41×C151—22no—
109#41×C151—41no—
110C2×Q8×C151C2311yescharacteristic
111C2×D4×C151—21no—
112C2×D4×C151—21no—
113C2×D4×C151C2311yescharacteristic
114C2×D4×C151C2311yescharacteristic
115C22×C4×C151—21no—
116C22×C4×C151C2311yescharacteristic
117SD16×C151—41no—
118D8×C151—22no—
119C2×C8×C151C2311yescharacteristic
120C2×C8×C151C2311yescharacteristic
121C4⋊C4×C151C2311yescharacteristic
122C22⋊C4×C151—21no—
123C22×D151C2×D411yescharacteristic
124Dic302—21no—
125Dic302—21no—
126Dic302—21no—
127Dic302—21no—
128Dic302—21no—
129Dic302—42no—
130Dic302C2×D412yes—
131C22⋊D151—41no—
132C22⋊D151—42no—
133C22⋊D151—42no—
134C2×Dic151—21no—
135C2×Dic151—21no—
136C2×Dic151—21no—
137C2×Dic151—21no—
138C2×Dic151—42no—
139C2×Dic151—42no—
140C2×Dic151—42no—
141C2×Dic151—42no—
142C2×Dic151C2×D411yescharacteristic
143C2×Dic151C2×D411yescharacteristic
144C2×Dic151C2×D411yescharacteristic
145C4×D151—21no—
146C4×D151—21no—
147C151⋊C8—21no—
148C151⋊C8C2×D412yes—
149C23×C151—21no—
150C23×C151—21no—
151C23×C151C2×D411yescharacteristic
152Q8×C151—21no—
153Q8×C151—21no—
154D4×C151—21no—
155D4×C151—21no—
156D4×C151—21no—
157D4×C151—42no—
158D4×C151—42no—
159D4×C151C2×D412yes—
160C2×C4×C151—21no—
161C2×C4×C151—21no—
162C2×C4×C151—21no—
163C2×C4×C151—21no—
164C2×C4×C151—42no—
165C2×C4×C151C2×D411yescharacteristic
166C2×C4×C151C2411yescharacteristic
167C8×C151—21no—
168C8×C151—21no—
169C22×C151—21no—
170C22×C151—22no—
171C22×C151—21no—
172C22×C151—21no—
173C22×C151—41no—
174C22×C151—41no—
175C22×C151—41no—
176C22×C151—41no—
177C22×C151—41no—
178C22×C151C22×D411yescharacteristic
179C2×D151—21no—
180C2×D151—21no—
181C2×D151—21no—
182C4×C151—21no—
183C4×C151—41no—
184C4×C151—41no—
185C4×C151C22×D411yescharacteristic
186C4×C151H(5,𝔽2)11yescharacteristic
187Dic151—21no—
188Dic151—22no—
189Dic151—21no—
190Dic151—21no—
191Dic151—41no—
192Dic151—41no—
193Dic151—41no—
194C2×C151—41no—
195C2×C151—42no—
196C2×C151—81no—
197C2×C151D4211yescharacteristic
198C2×C151#57511yescharacteristic
199C2×C151#57811yescharacteristic
200D151—41no—
201C151#326911yescharacteristic, complemented, Sylow
202#3269—1511nomaximal, Sylow
203C2×#137—1511no—
204C2×#135—1511no—
205#548—1511no—
206#547—1511no—
207#497—1511no—
208#468—1511no—
209#467—1511no—
210#463—1511no—
211#462—1511no—
212#452—1511no—
213#452—1511no—
214#451—1511no—
215#449D15111yescharacteristic, complemented
216#440—1511no—
217#435—1511no—
218C2×#41—1511no—
219C2×#41—1511no—
220C2×#41C2×D15111yescharacteristic
221C22×Q8—1511no—
222#137—3024no—
223#135—3024no—
224C2×Q16—1511no—
225C2×Q16—1511no—
226C2×Q16—1511no—
227C2×Q16—1511no—
228C2×SD16—1511no—
229C2×SD16—1511no—
230C2×SD16—1511no—
231C2×SD16C2×D15111yescharacteristic
232C2×D8C2×D15111yescharacteristic
233C2×M16—1511no—
234C22×C8—1511no—
235C4⋊Q8—1511no—
236#124—3021no—
237#124—1511no—
238#123—3021no—
239#123—3021no—
240C22⋊Q8—3021no—
241C22⋊Q8—1511no—
242C22⋊Q8—1511no—
243C22⋊Q8—1511no—
244#121—3021no—
245#121—3021no—
246#121C2×D15111yescharacteristic
247C4×Q8—1511no—
248C4×D4—3021no—
249C4×D4—1511no—
250C4×D4—1511no—
251C2×C4⋊C4—3021no—
252#107—1511no—
253C4⋊C8—1511no—
254#103—1511no—
255#103—1511no—
256#103—1511no—
257#103C2×D15111yescharacteristic
258#102—1511no—
259#102C2×D15111yescharacteristic
260C22⋊C8—1511no—
261C22⋊C8C2×D15111yescharacteristic
262#97—1511no—
263#41—22no—
264#41—41no—
265#41—3022no—
266#41—3022no—
267#41—6041no—
268#41—6041no—
269C2×Q8—3022no—
270C2×Q8—3021no—
271C2×Q8—6041no—
272C2×Q8—1511no—
273C2×Q8—1511no—
274C2×Q8—1511no—
275C2×Q8C22×D15111yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—3021no—
279C2×D4—3021no—
280C2×D4—1511no—
281C2×D4C22×D15111yescharacteristic
282C2×D4C22×D15111yescharacteristic
283C22×C4—21no—
284C22×C4—3021no—
285C22×C4—3021no—
286C22×C4—3021no—
287C22×C4—3021no—
288C22×C4—1511no—
289C22×C4C22×D15111yescharacteristic
290Q16—3022no—
291Q16—3022no—
292Q16—3022no—
293Q16—6041no—
294SD16—41no—
295SD16—3022no—
296SD16—3022no—
297SD16—1514no—
298D8—22no—
299M16—3022no—
300C2×C8—3022no—
301C2×C8—1511no—
302C2×C8—1511no—
303C2×C8C22×D15111yescharacteristic
304C2×C8C22×D15111yescharacteristic
305C4⋊C4—3021no—
306C4⋊C4—3021no—
307C4⋊C4—3021no—
308C4⋊C4—3021no—
309C4⋊C4—3021no—
310C4⋊C4—3021no—
311C4⋊C4—1511no—
312C4⋊C4—1511no—
313C4⋊C4C22×D15111yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—3021no—
316C22⋊C4—3021no—
317C22⋊C4—3021no—
318C22⋊C4—3021no—
319C22⋊C4—3021no—
320C22⋊C4—3021no—
321C22⋊C4—1511no—
322C22⋊C4—1511no—
323C42—3021no—
324C42—1511no—
325C23—21no—
326C23—21no—
327C23—1511no—
328C23D4×D15111yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—3021no—
332Q8—3021no—
333Q8—3021no—
334Q8—3021no—
335Q8—3021no—
336Q8—6042no—
337Q8—1512no—
338D4—21no—
339D4—21no—
340D4—21no—
341D4—42no—
342D4—42no—
343D4—6041no—
344D4—6042no—
345D4—6042no—
346D4D4×D15112yes—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—42no—
352C2×C4—3021no—
353C2×C4—3021no—
354C2×C4—3021no—
355C2×C4—3021no—
356C2×C4—3021no—
357C2×C4—3021no—
358C2×C4—6042no—
359C2×C4—6042no—
360C2×C4—6042no—
361C2×C4—6042no—
362C2×C4—1511no—
363C2×C4—1511no—
364C2×C4—1511no—
365C2×C4D4×D15111yescharacteristic
366C2×C4C23×D15111yescharacteristic
367C8—21no—
368C8—21no—
369C8—3021no—
370C8—1512no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—22no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—3021no—
381C22—3021no—
382C22—3021no—
383C22C2×D4×D15111yescharacteristic
384C4—21no—
385C4—41no—
386C4—41no—
387C4—3021no—
388C4—3021no—
389C4—3021no—
390C4—3022no—
391C4—6041no—
392C4—6041no—
393C4—6041no—
394C4C2×D4×D15111yescharacteristic
395C4#1801278411yescharacteristic
396C2—41no—
397C2—42no—
398C2—81no—
399C2—6041no—
400C2#2375284411yescharacteristic
401C2#2375300811yescharacteristic
402C2#2375302611yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#449×C151C2
socleC22×C151C22×D4
hypercenter#449D151
hypocenterC151#3269
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#18012769C2semidirect
2. C2×#18012758C2semidirect
3. #23752868C2semidirect
4. #23752861C2semidirect
5. #23752603C2semidirect
6. #23752456C2semidirect
7. #23752452C2semidirect
8. #23752425C2semidirect
9. #23752421C2semidirect
10. #23752375C2semidirect
11. #23752371C2semidirect
12. #23752370C2semidirect
13. #23752309C2semidirect
14. #23752286C2semidirect
15. #449×C151C2semidirect
85. #14558992D4semidirect
201. C151#3269semidirect
215. #449D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127692
C2×#180127582
#237528682
#237528612
#237526032
#237524562
#237524522
#237524252
#237524212
#237523752
#237523712
#237523702
#237523092
#237522862
#449×C1512
#3269151