Finite Group Databasebeta

← #34057257

Subgroups — #34057257

Order 19072.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750322C211yesmaximal, characteristic
2#23750317C211yesmaximal, characteristic, complemented
3#23750026C211yesmaximal, characteristic, complemented
4#23749963C211yesmaximal, characteristic, complemented
5#23749954C211yesmaximal, characteristic, complemented
6#23749947C211yesmaximal, characteristic, complemented
7#23749941C211yesmaximal, characteristic, complemented
8#23749934C211yesmaximal, characteristic, complemented
9#23749927C211yesmaximal, characteristic, complemented
10#23749868C211yesmaximal, characteristic, complemented
11SD16×Dic149C211yesmaximal, characteristic, complemented
12#23749701C211yesmaximal, characteristic, complemented
13#23749667C211yesmaximal, characteristic, complemented
14#23749666C211yesmaximal, characteristic, complemented
15#468×C149C211yesmaximal, characteristic, complemented
16C2×#14558719C2211yescharacteristic
17#18011906—21no—
18#18011904—21no—
19#18011902—21no—
20#18011902C2211yescharacteristic
21#18011902C2211yescharacteristic
22#18011895—21no—
23#18011892—21no—
24#18011887C2211yescharacteristic
25#18011882—21no—
26#18011882C2211yescharacteristic
27Q8×Dic149C2211yescharacteristic
28D4×Dic149C2211yescharacteristic
29C4×C22⋊D149C2211yescharacteristic
30#18011871—21no—
31#18011870C2211yescharacteristic
32#18011869C2211yescharacteristic
33#18011867—21no—
34C2×C4⋊Dic149C2211yescharacteristic
35C4⋊C4×D149C2211yescharacteristic
36#18011845C2211yescharacteristic
37#18011845C2211yescharacteristic
38#18011844C2211yescharacteristic
39#18011843C2211yescharacteristic
40#18011843C2211yescharacteristic
41#18011842C2211yescharacteristic
42#18011842C2211yescharacteristic
43#18011842C2211yescharacteristic
44#18011840C2211yescharacteristic
45#18011835C2211yescharacteristic
46#18011835C2211yescharacteristic
47#18011832C2211yescharacteristic
48#18011832C2211yescharacteristic
49C22⋊#11924746C2211yescharacteristic
50#18011823C2211yescharacteristic
51C8×Dic149C2211yescharacteristic
52C2×SD16×C149C2211yescharacteristic
53C2×M16×C149C2211yescharacteristic
54C22⋊Q8×C149C2211yescharacteristic
55#121×C149C2211yescharacteristic
56#107×C149C2211yescharacteristic
57#103×C149C2211yescharacteristic
58#102×C149C2211yescharacteristic
59C2×C22⋊D149—21no—
60C2×C22⋊D149D411yescharacteristic
61C22×Dic149—21no—
62C22×Dic149—21no—
63C2×C4×D149—21no—
64C2×C4×D149C2311yescharacteristic
65#14558719—22no—
66C2×#11924746C2311yescharacteristic
67C2×#11924746C2311yescharacteristic
68C4⋊Dic149—21no—
69C4⋊Dic149—21no—
70C4⋊Dic149—21no—
71C4⋊Dic149—21no—
72C4⋊Dic149C2311yescharacteristic
73C4⋊Dic149C2311yescharacteristic
74C4⋊Dic149C2311yescharacteristic
75C4⋊Dic149C2311yescharacteristic
76#14558715—21no—
77#14558715—21no—
78#14558715—21no—
79#14558715—21no—
80#14558715D411yescharacteristic
81C22⋊Dic149—21no—
82C22⋊Dic149—21no—
83C22⋊Dic149—21no—
84C22⋊Dic149D411yescharacteristic
85#14558713—21no—
86#14558713—21no—
87#14558713D411yescharacteristic
88C4×Dic149—21no—
89C4×Dic149—21no—
90C4×Dic149—21no—
91C4×Dic149C2311yescharacteristic
92C2×Q8×C149C2311yescharacteristic
93C2×D4×C149—21no—
94C2×D4×C149C2311yescharacteristic
95C22×C4×C149C2311yescharacteristic
96SD16×C149—22no—
97M16×C149—22no—
98C2×C8×C149C2311yescharacteristic
99C2×C8×C149C2311yescharacteristic
100C4⋊C4×C149—21no—
101C4⋊C4×C149C2311yescharacteristic
102C4⋊C4×C149C2311yescharacteristic
103C22⋊C4×C149—21no—
104C22⋊C4×C149—21no—
105C22×D149C2×D411yescharacteristic
106C22⋊D149—41no—
107C22⋊D149—42no—
108C2×Dic149—21no—
109C2×Dic149—21no—
110C2×Dic149—21no—
111C2×Dic149—21no—
112C2×Dic149—42no—
113C2×Dic149—81no—
114C2×Dic149C2×D411yescharacteristic
115C2×Dic149C2×D411yescharacteristic
116C2×Dic149C2×D411yescharacteristic
117C4×D149—21no—
118C4×D149—21no—
119C4×D149—81no—
120#11924746—21no—
121#11924746—21no—
122C23×C149—21no—
123C23×C149C2×D411yescharacteristic
124Q8×C149—21no—
125Q8×C149—21no—
126D4×C149—21no—
127D4×C149—21no—
128D4×C149—81no—
129C2×C4×C149—21no—
130C2×C4×C149—21no—
131C2×C4×C149—21no—
132C2×C4×C149—21no—
133C2×C4×C149—21no—
134C2×C4×C149C2×D411yescharacteristic
135C2×C4×C149C2411yescharacteristic
136C8×C149—21no—
137C8×C149#4112yes—
138C22×C149—21no—
139C22×C149—21no—
140C22×C149—21no—
141C22×C149—41no—
142C22×C149—41no—
143C22×C149—41no—
144C22×C149C22×D411yescharacteristic
145C2×D149—21no—
146C2×D149—21no—
147C2×D149—21no—
148C4×C149—21no—
149C4×C149—41no—
150C4×C149—41no—
151C4×C149—41no—
152C4×C149C2×#4111yescharacteristic
153C4×C1492−1+411yescharacteristic
154Dic149—21no—
155Dic149—21no—
156Dic149—21no—
157Dic149—41no—
158Dic149—41no—
159Dic149—41no—
160Dic149—41no—
161C2×C149—41no—
162C2×C149—81no—
163C2×C149#54711yescharacteristic
164C2×C149#57511yescharacteristic
165C2×C149#57711yescharacteristic
166D149—41no—
167C149#335111yescharacteristic, complemented, Sylow
168#3351—1491nomaximal, Sylow
169#553—1491no—
170#552—1491no—
171#499—1491no—
172#485—1491no—
173#483—1491no—
174#482—1491no—
175#481—1491no—
176#479—1491no—
177#478—1491no—
178#468—1491no—
179#468D14911yescharacteristic, complemented
180C4×SD16—1491no—
181#436—1491no—
182#428—1491no—
183#428—1491no—
184C2×SD16C2×D14911yescharacteristic
185C2×M16—1491no—
186C2×M16C2×D14911yescharacteristic
187#126—2981no—
188#126—2981no—
189#125—2981no—
190#125—1491no—
191#125—1491no—
192#123—2981no—
193#123—2981no—
194C22⋊Q8—1491no—
195C22⋊Q8C2×D14911yescharacteristic
196#121—2981no—
197#121—1491no—
198#121C2×D14911yescharacteristic
199C4×Q8—1491no—
200C4×D4—2981no—
201C4×D4—1491no—
202C4×D4—1491no—
203#117—2981no—
204#117—1491no—
205#117—1491no—
206C2×C4⋊C4—1491no—
207C2×C4⋊C4—1491no—
208#107—1491no—
209#107—1491no—
210#107—1491no—
211#107C2×D14911yescharacteristic
212#106—1491no—
213#106—1491no—
214#106—1491no—
215#106—1491no—
216#106—1491no—
217C4⋊C8—1491no—
218#103—1491no—
219#103—1491no—
220#103C2×D14911yescharacteristic
221#102—1491no—
222#102—1491no—
223#102C2×D14911yescharacteristic
224C22⋊C8—1491no—
225C22⋊C8—1491no—
226C4×C8—1491no—
227C2×Q8C22×D14911yescharacteristic
228C2×D4—21no—
229C2×D4—2981no—
230C2×D4—1491no—
231C2×D4C22×D14911yescharacteristic
232C22×C4—2981no—
233C22×C4—2981no—
234C22×C4—2981no—
235C22×C4—1491no—
236C22×C4C22×D14911yescharacteristic
237SD16—22no—
238M16—22no—
239M16—2982no—
240C2×C8—1491no—
241C2×C8—1491no—
242C2×C8C22×D14911yescharacteristic
243C2×C8C22×D14911yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—2981no—
246C4⋊C4—2981no—
247C4⋊C4—2981no—
248C4⋊C4—2981no—
249C4⋊C4—2981no—
250C4⋊C4—2981no—
251C4⋊C4—2981no—
252C4⋊C4—2981no—
253C4⋊C4—1491no—
254C4⋊C4—1491no—
255C4⋊C4—1491no—
256C4⋊C4—1491no—
257C4⋊C4—1491no—
258C4⋊C4C22×D14911yescharacteristic
259C4⋊C4C22×D14911yescharacteristic
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—2981no—
263C22⋊C4—2981no—
264C22⋊C4—2981no—
265C22⋊C4—2981no—
266C22⋊C4—2981no—
267C22⋊C4—1491no—
268C22⋊C4—1491no—
269C42—2981no—
270C42—2981no—
271C42—2981no—
272C42—1491no—
273C23—21no—
274C23—1491no—
275C23D4×D14911yescharacteristic
276Q8—21no—
277Q8—21no—
278D4—21no—
279D4—21no—
280D4—81no—
281D4—5961no—
282D4—5962no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—2981no—
289C2×C4—2981no—
290C2×C4—2981no—
291C2×C4—2981no—
292C2×C4—2981no—
293C2×C4—2981no—
294C2×C4—5962no—
295C2×C4—1491no—
296C2×C4—1491no—
297C2×C4—1491no—
298C2×C4—11921no—
299C2×C4—11921no—
300C2×C4D4×D14911yescharacteristic
301C2×C4C23×D14911yescharacteristic
302C8—21no—
303C8—2981no—
304C8—2981no—
305C8#1455873512yes—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—2981no—
313C22—2981no—
314C22—2981no—
315C22C2×D4×D14911yescharacteristic
316C4—21no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—2981no—
321C4—2981no—
322C4—2981no—
323C4—5961no—
324C4—5961no—
325C4—5961no—
326C4—5961no—
327C4C2×#1455873511yescharacteristic
328C4#1801195811yescharacteristic
329C2—41no—
330C2—81no—
331C2—5961no—
332C2#2375027511yescharacteristic
333C2#2375042411yescharacteristic
334C2#2375043511yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#468×C149C2
socleC22×C149C22×D4
prime radical#23750322C2
hypercenter#468D149
hypocenterC149#3351
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
2. #23750317C2semidirect
3. #23750026C2semidirect
4. #23749963C2semidirect
5. #23749954C2semidirect
6. #23749947C2semidirect
7. #23749941C2semidirect
8. #23749934C2semidirect
9. #23749927C2semidirect
10. #23749868C2semidirect
11. SD16×Dic149C2semidirect
12. #23749701C2semidirect
13. #23749667C2semidirect
14. #23749666C2semidirect
15. #468×C149C2semidirect
167. C149#3351semidirect
179. #468D149semidirect

Maximal subgroups

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

subgroupindex
#237503222
#237503172
#237500262
#237499632
#237499542
#237499472
#237499412
#237499342
#237499272
#237498682
SD16×Dic1492
#237497012
#237496672
#237496662
#468×C1492
#3351149