Finite Group Databasebeta

← #34056992

Subgroups — #34056992

Order 19072.

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#23750276C211yesmaximal, characteristic, complemented
2#23750274C211yesmaximal, characteristic, complemented
3#23749974C211yesmaximal, characteristic, complemented
4#23749960C211yesmaximal, characteristic, complemented
5#23749959C211yesmaximal, characteristic, complemented
6#23749903C211yesmaximal, characteristic, complemented
7#23749874C211yesmaximal, characteristic, complemented
8#23749871C211yesmaximal, characteristic, complemented
9#23749829C211yesmaximal, characteristic, complemented
10#23749771C211yesmaximal, characteristic, complemented
11#23749770C211yesmaximal, characteristic, complemented
12#23749705C211yesmaximal, characteristic, complemented
13#23749701C211yesmaximal, characteristic, complemented
14#23749623C211yesmaximal, characteristic, complemented
15#417×C149C211yesmaximal, characteristic, complemented
16C2×#14558736—21no—
17C2×#14558735C2211yescharacteristic
18C2×#14558725C2211yescharacteristic
19C2×#14558723C2211yescharacteristic
20C2×#14558721C2211yescharacteristic
21C2×#14558719C2211yescharacteristic
22#18011910C2211yescharacteristic
23#18011909C2211yescharacteristic
24#18011896—21no—
25#18011894—21no—
26#18011894—21no—
27#18011892—21no—
28#18011890C2211yescharacteristic
29#18011889—21no—
30#18011887C2211yescharacteristic
31#18011886C2211yescharacteristic
32#18011885C2211yescharacteristic
33#18011883—21no—
34D4×Dic149C2211yescharacteristic
35#18011873C2211yescharacteristic
36C4×C22⋊D149C2211yescharacteristic
37C2×#14558715—21no—
38C4⋊C4×D149C2211yescharacteristic
39#18011845C2211yescharacteristic
40#18011844C2211yescharacteristic
41#18011844C2211yescharacteristic
42#18011840C2211yescharacteristic
43#18011836C2211yescharacteristic
44#18011835C2211yescharacteristic
45#18011834C2211yescharacteristic
46#18011834C2211yescharacteristic
47#18011833C2211yescharacteristic
48#18011832C2211yescharacteristic
49C22⋊#11924746C2211yescharacteristic
50#18011823C2211yescharacteristic
51C8×Dic149C2211yescharacteristic
52C2×#41×C149C2211yescharacteristic
53C2×M16×C149C2211yescharacteristic
54C2×C4⋊C4×C149C2211yescharacteristic
55#103×C149C2211yescharacteristic
56#103×C149C2211yescharacteristic
57#102×C149C2211yescharacteristic
58#102×C149C2211yescharacteristic
59#14558736—44no—
60#14558735—22no—
61#14558735—41no—
62C2×Dic298—21no—
63C2×Dic298C2311yescharacteristic
64C2×D596—21no—
65C2×D596C2311yescharacteristic
66C2×C22⋊D149—21no—
67C2×C22⋊D149—21no—
68C2×C22⋊D149D411yescharacteristic
69C22×Dic149—21no—
70C22×Dic149—21no—
71C2×C4×D149—21no—
72C2×C4×D149—21no—
73C2×C4×D149C2311yescharacteristic
74#14558725—41no—
75#14558723—41no—
76#14558721—22no—
77#14558719—41no—
78C2×#11924746C2311yescharacteristic
79C2×#11924746C2311yescharacteristic
80C4⋊Dic149C2311yescharacteristic
81C4⋊Dic149C2311yescharacteristic
82#14558715—21no—
83#14558715—21no—
84#14558715—21no—
85#14558715—21no—
86#14558715—21no—
87#14558715—21no—
88#14558715D411yescharacteristic
89C22⋊Dic149—21no—
90C22⋊Dic149—21no—
91C22⋊Dic149D411yescharacteristic
92#14558713—21no—
93#14558713—21no—
94#14558713—21no—
95#14558713—21no—
96#14558713D411yescharacteristic
97C4×Dic149C2311yescharacteristic
98#41×C149—22no—
99#41×C149—41no—
100C2×Q8×C149C2311yescharacteristic
101C2×D4×C149—21no—
102C2×D4×C149C2311yescharacteristic
103C22×C4×C149—21no—
104C22×C4×C149—21no—
105C22×C4×C149C2311yescharacteristic
106M16×C149—41no—
107C2×C8×C149C2311yescharacteristic
108C2×C8×C149C2311yescharacteristic
109C4⋊C4×C149—21no—
110C4⋊C4×C149C2311yescharacteristic
111C4⋊C4×C149C2311yescharacteristic
112C22×D149—21no—
113C22×D149C2×D411yescharacteristic
114Dic298—21no—
115Dic298—21no—
116Dic298—42no—
117D596—21no—
118D596—21no—
119D596—42no—
120C22⋊D149—22no—
121C22⋊D149—42no—
122C22⋊D149—42no—
123C2×Dic149—21no—
124C2×Dic149—21no—
125C2×Dic149—21no—
126C2×Dic149—42no—
127C2×Dic149—42no—
128C2×Dic149C2×D411yescharacteristic
129C2×Dic149C2×D411yescharacteristic
130C2×Dic149C2×D411yescharacteristic
131C4×D149—21no—
132C4×D149—21no—
133C4×D149—42no—
134C4×D149—42no—
135#11924746—21no—
136#11924746—21no—
137C23×C149—21no—
138C23×C149C2×D411yescharacteristic
139Q8×C149—21no—
140Q8×C149—21no—
141D4×C149—21no—
142D4×C149—42no—
143D4×C149#4112yescomplemented
144C2×C4×C149—21no—
145C2×C4×C149—21no—
146C2×C4×C149—21no—
147C2×C4×C149—21no—
148C2×C4×C149—21no—
149C2×C4×C149—42no—
150C2×C4×C149—42no—
151C2×C4×C149C2×D411yescharacteristic
152C2×C4×C149C2411yescharacteristic
153C8×C149—21no—
154C8×C149—21no—
155C22×C149—21no—
156C22×C149—22no—
157C22×C149—21no—
158C22×C149—21no—
159C22×C149—41no—
160C22×C149—41no—
161C22×C149C22×D411yescharacteristic
162C2×D149—21no—
163C2×D149—21no—
164C2×D149—21no—
165C2×D149—41no—
166C2×D149—41no—
167C2×D149—41no—
168C4×C149—21no—
169C4×C149—41no—
170C4×C149—41no—
171C4×C149—41no—
172C4×C149C2×#4111yescharacteristic
173C4×C149H(5,𝔽2)11yescharacteristic
174Dic149—21no—
175Dic149—21no—
176Dic149—41no—
177Dic149—41no—
178Dic149—41no—
179Dic149AGL(1,ℤ/8)12yes—
180C2×C149—41no—
181C2×C149—42no—
182C2×C149#54611yescharacteristic
183C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C149#57811yescharacteristic
185D149—41no—
186D149—81no—
187C149#333011yescharacteristic, complemented, Sylow
188#3330—1491nomaximal, Sylow
189#547—1491no—
190#547—1491no—
191#487—1491no—
192#484—1491no—
193#484—1491no—
194#474—1491no—
195#468—1491no—
196#468—1491no—
197#461—1491no—
198#449—1491no—
199#449—1491no—
200C4×D8—1491no—
201#436—1491no—
202#417—1491no—
203#417D14911yescharacteristic, complemented
204C2×#41—2981no—
205C2×#41—1491no—
206C2×#41C2×D14911yescharacteristic
207C2×SD16—1491no—
208C2×SD16—1491no—
209C2×D8—1491no—
210C2×M16—1491no—
211C2×M16C2×D14911yescharacteristic
212C4⋊Q8—1491no—
213C4⋊Q8—1491no—
214#123—2981no—
215#123—2981no—
216#123—2981no—
217#123—2981no—
218C22⋊Q8—2981no—
219C22⋊Q8—1491no—
220C22⋊Q8—1491no—
221#121—2981no—
222#121—1491no—
223#121—1491no—
224C4×D4—1491no—
225C4×D4—1491no—
226C4×D4—1491no—
227C2×C4⋊C4—2981no—
228C2×C4⋊C4—1491no—
229C2×C4⋊C4C2×D14911yescharacteristic
230#107—1491no—
231#107—1491no—
232#107—1491no—
233C4⋊C8—1491no—
234#103—1491no—
235#103—1491no—
236#103—1491no—
237#103—1491no—
238#103C2×D14911yescharacteristic
239#103C2×D14911yescharacteristic
240#102—1491no—
241#102—1491no—
242#102C2×D14911yescharacteristic
243#102C2×D14911yescharacteristic
244C22⋊C8—1491no—
245C22⋊C8—1491no—
246C4×C8—1491no—
247#41—22no—
248#41—41no—
249#41—2982no—
250#41—5964no—
251#41—5961no—
252C2×Q8—2981no—
253C2×Q8—1491no—
254C2×Q8C22×D14911yescharacteristic
255C2×D4—21no—
256C2×D4—2981no—
257C2×D4—2981no—
258C2×D4—2981no—
259C2×D4—1491no—
260C2×D4—1491no—
261C2×D4C22×D14911yescharacteristic
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—2981no—
265C22×C4—2981no—
266C22×C4—2981no—
267C22×C4—2981no—
268C22×C4—1491no—
269C22×C4C22×D14911yescharacteristic
270SD16—5961no—
271SD16—5961no—
272D8—2982no—
273M16—41no—
274M16—5961no—
275C2×C8—1491no—
276C2×C8—1491no—
277C2×C8C22×D14911yescharacteristic
278C2×C8C22×D14911yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—2981no—
281C4⋊C4—2981no—
282C4⋊C4—2981no—
283C4⋊C4—2981no—
284C4⋊C4—2981no—
285C4⋊C4—2981no—
286C4⋊C4—1491no—
287C4⋊C4—1491no—
288C4⋊C4—1491no—
289C4⋊C4C22×D14911yescharacteristic
290C4⋊C4C22×D14911yescharacteristic
291C22⋊C4—2981no—
292C22⋊C4—2981no—
293C22⋊C4—2981no—
294C22⋊C4—2981no—
295C22⋊C4—2981no—
296C22⋊C4—2981no—
297C22⋊C4—1491no—
298C22⋊C4—1491no—
299C42—1491no—
300C23—21no—
301C23—2981no—
302C23—1491no—
303C23D4×D14911yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—2981no—
307Q8—2981no—
308Q8—5962no—
309D4—21no—
310D4—42no—
311D4—2982no—
312D4—2981no—
313D4—2981no—
314D4—5962no—
315D4—5962no—
316D4—5962no—
317D4#1455873612yescomplemented
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—2981no—
326C2×C4—2981no—
327C2×C4—2981no—
328C2×C4—2981no—
329C2×C4—2981no—
330C2×C4—5962no—
331C2×C4—5962no—
332C2×C4—5962no—
333C2×C4—5962no—
334C2×C4—1491no—
335C2×C4—1491no—
336C2×C4—1491no—
337C2×C4D4×D14911yescharacteristic
338C2×C4C23×D14911yescharacteristic
339C8—21no—
340C8—21no—
341C8—2981no—
342C8—2981no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—22no—
347C22—41no—
348C22—41no—
349C22—2981no—
350C22—2981no—
351C22—2981no—
352C22—5961no—
353C22—5961no—
354C22—5961no—
355C22C2×D4×D14911yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—2981no—
361C4—2981no—
362C4—5961no—
363C4—5961no—
364C4—5961no—
365C4—1492no—
366C4C2×#1455873611yescharacteristic
367C4#1801195611yescharacteristic
368C2—41no—
369C2—42no—
370C2—5961no—
371C2—11921no—
372C2#2375026511yescharacteristic
373C2AGL(1,ℤ/8)×D14911yescharacteristic
374C2#2375044111yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#417×C149C2
socleC22×C149C22×D4
hypercenter#417D149
hypocenterC149#3330
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #23750276C2semidirect
2. #23750274C2semidirect
3. #23749974C2semidirect
4. #23749960C2semidirect
5. #23749959C2semidirect
6. #23749903C2semidirect
7. #23749874C2semidirect
8. #23749871C2semidirect
9. #23749829C2semidirect
10. #23749771C2semidirect
11. #23749770C2semidirect
12. #23749705C2semidirect
13. #23749701C2semidirect
14. #23749623C2semidirect
15. #417×C149C2semidirect
143. D4×C149#41semidirect
187. C149#3330semidirect
203. #417D149semidirect
317. D4#14558736semidirect

Maximal subgroups

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

subgroupindex
#237502762
#237502742
#237499742
#237499602
#237499592
#237499032
#237498742
#237498712
#237498292
#237497712
#237497702
#237497052
#237497012
#237496232
#417×C1492
#3330149