Finite Group Databasebeta

← #34057003

Subgroups — #34057003

Order 19072.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750330C211yesmaximal, characteristic, complemented
2#23750274C211yesmaximal, characteristic
3#23749974C211yesmaximal, characteristic, complemented
4#23749955C211yesmaximal, characteristic, complemented
5#23749954C211yesmaximal, characteristic, complemented
6#23749913C211yesmaximal, characteristic, complemented
7#23749883C211yesmaximal, characteristic, complemented
8#23749882C211yesmaximal, characteristic, complemented
9#23749834C211yesmaximal, characteristic, complemented
10#23749788C211yesmaximal, characteristic, complemented
11#23749787C211yesmaximal, characteristic, complemented
12#23749716C211yesmaximal, characteristic, complemented
13#23749701C211yesmaximal, characteristic, complemented
14#23749623C211yesmaximal, characteristic, complemented
15#417×C149C211yesmaximal, characteristic, complemented
16C2×#14558735C2211yescharacteristic
17C2×Dic596C2211yescharacteristic
18C2×#14558726C2211yescharacteristic
19C2×#14558724C2211yescharacteristic
20C2×#14558719C2211yescharacteristic
21#18011910C2211yescharacteristic
22#18011909C2211yescharacteristic
23#18011901—21no—
24#18011894—21no—
25#18011892—21no—
26#18011891C2211yescharacteristic
27#18011890—21no—
28#18011890C2211yescharacteristic
29#18011890C2211yescharacteristic
30#18011889—21no—
31#18011887C2211yescharacteristic
32#18011883—21no—
33#18011876C2211yescharacteristic
34D4×Dic149C2211yescharacteristic
35C4×C22⋊D149—21no—
36C4×C22⋊D149C2211yescharacteristic
37C2×#14558715—21no—
38C4⋊C4×D149C2211yescharacteristic
39#18011843C2211yescharacteristic
40#18011842C2211yescharacteristic
41#18011842C2211yescharacteristic
42#18011840C2211yescharacteristic
43#18011836C2211yescharacteristic
44#18011836C2211yescharacteristic
45#18011835C2211yescharacteristic
46#18011834C2211yescharacteristic
47#18011834C2211yescharacteristic
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#14558735—22no—
60#14558735—41no—
61C2×Dic298—21no—
62C2×Dic298C2311yescharacteristic
63C2×Dic298C2311yescharacteristic
64C2×C22⋊D149—21no—
65C2×C22⋊D149D411yescharacteristic
66C22×Dic149—21no—
67C22×Dic149—21no—
68C2×C4×D149—21no—
69C2×C4×D149C2311yescharacteristic
70Dic596—41no—
71#14558726—41no—
72#14558724—22no—
73#14558719—41no—
74C2×#11924746C2311yescharacteristic
75C2×#11924746C2311yescharacteristic
76C4⋊Dic149—21no—
77C4⋊Dic149C2311yescharacteristic
78C4⋊Dic149C2311yescharacteristic
79#14558715—21no—
80#14558715—21no—
81#14558715—21no—
82#14558715—21no—
83#14558715—21no—
84#14558715—21no—
85#14558715—21no—
86#14558715—21no—
87#14558715D411yescharacteristic
88C22⋊Dic149—21no—
89C22⋊Dic149—21no—
90C22⋊Dic149—21no—
91C22⋊Dic149D411yescharacteristic
92#14558713—21no—
93#14558713—21no—
94#14558713D411yescharacteristic
95C4×Dic149—21no—
96C4×Dic149C2311yescharacteristic
97#41×C149—22no—
98#41×C149—41no—
99C2×Q8×C149C2311yescharacteristic
100C2×D4×C149—21no—
101C2×D4×C149C2311yescharacteristic
102C22×C4×C149—21no—
103C22×C4×C149—21no—
104C22×C4×C149C2311yescharacteristic
105M16×C149—41no—
106C2×C8×C149C2311yescharacteristic
107C2×C8×C149C2311yescharacteristic
108C4⋊C4×C149—21no—
109C4⋊C4×C149C2311yescharacteristic
110C4⋊C4×C149C2311yescharacteristic
111C22×D149C2×D411yescharacteristic
112Dic298—21no—
113Dic298—21no—
114Dic298—21no—
115Dic298—21no—
116Dic298—81no—
117C22⋊D149—22no—
118C22⋊D149—42no—
119C2×Dic149—21no—
120C2×Dic149—21no—
121C2×Dic149—21no—
122C2×Dic149—21no—
123C2×Dic149—42no—
124C2×Dic149—42no—
125C2×Dic149C2×D411yescharacteristic
126C2×Dic149C2×D411yescharacteristic
127C2×Dic149C2×D411yescharacteristic
128C4×D149—21no—
129C4×D149—21no—
130C4×D149—81no—
131#11924746—21no—
132#11924746—21no—
133C23×C149—21no—
134C23×C149C2×D411yescharacteristic
135Q8×C149—21no—
136Q8×C149—21no—
137D4×C149—21no—
138D4×C149—42no—
139D4×C149#4112yes—
140C2×C4×C149—21no—
141C2×C4×C149—21no—
142C2×C4×C149—21no—
143C2×C4×C149—21no—
144C2×C4×C149—21no—
145C2×C4×C149—42no—
146C2×C4×C149—81no—
147C2×C4×C149C2×D411yescharacteristic
148C2×C4×C149C2411yescharacteristic
149C8×C149—21no—
150C8×C149—21no—
151C22×C149—21no—
152C22×C149—22no—
153C22×C149—21no—
154C22×C149—21no—
155C22×C149—41no—
156C22×C149—41no—
157C22×C149C22×D411yescharacteristic
158C2×D149—21no—
159C2×D149—21no—
160C2×D149—21no—
161C4×C149—21no—
162C4×C149—41no—
163C4×C149—41no—
164C4×C149—41no—
165C4×C149C2×#4111yescharacteristic
166C4×C149H(5,𝔽2)11yescharacteristic
167Dic149—21no—
168Dic149—21no—
169Dic149—41no—
170Dic149—41no—
171Dic149—41no—
172Dic149—41no—
173Dic149#13712yes—
174C2×C149—41no—
175C2×C149—42no—
176C2×C149#54611yescharacteristic
177C2×C149C2×#13711yescharacteristic
178C2×C149#57711yescharacteristic
179D149—41no—
180C149#333111yescharacteristic, complemented, Sylow
181#3331—1491nomaximal, Sylow
182#554—1491no—
183#547—1491no—
184#487—1491no—
185#483—1491no—
186#483—1491no—
187#476—1491no—
188#470—1491no—
189#470—1491no—
190#462—1491no—
191#452—1491no—
192#452—1491no—
193C4×SD16—1491no—
194#436—1491no—
195#417—1491no—
196#417D14911yescharacteristic, complemented
197C2×#41—1491no—
198C2×#41C2×D14911yescharacteristic
199C2×Q16—1491no—
200C2×Q16—1491no—
201C2×SD16—1491no—
202C2×M16—1491no—
203C2×M16C2×D14911yescharacteristic
204C4⋊Q8—1491no—
205C4⋊Q8—1491no—
206#125—2981no—
207#123—2981no—
208#123—2981no—
209C22⋊Q8—2981no—
210C22⋊Q8—2981no—
211C22⋊Q8—1491no—
212C22⋊Q8—1491no—
213C22⋊Q8—1491no—
214C22⋊Q8—1491no—
215#121—2981no—
216C4×Q8—1491no—
217C4×D4—2981no—
218C4×D4—1491no—
219C4×D4—1491no—
220C2×C4⋊C4—2981no—
221C2×C4⋊C4—1491no—
222C2×C4⋊C4C2×D14911yescharacteristic
223#106—1491no—
224#106—1491no—
225#106—1491no—
226C4⋊C8—1491no—
227#103—1491no—
228#103—1491no—
229#103—1491no—
230#103—1491no—
231#103—1491no—
232#103C2×D14911yescharacteristic
233#103C2×D14911yescharacteristic
234#102—1491no—
235#102C2×D14911yescharacteristic
236#102C2×D14911yescharacteristic
237C22⋊C8—1491no—
238C22⋊C8—1491no—
239C4×C8—1491no—
240#41—22no—
241#41—41no—
242#41—2982no—
243#41—5961no—
244C2×Q8—2981no—
245C2×Q8—1491no—
246C2×Q8—1491no—
247C2×Q8C22×D14911yescharacteristic
248C2×D4—21no—
249C2×D4—2981no—
250C2×D4—1491no—
251C2×D4C22×D14911yescharacteristic
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—2981no—
255C22×C4—2981no—
256C22×C4—2981no—
257C22×C4—1491no—
258C22×C4C22×D14911yescharacteristic
259Q16—5961no—
260Q16—5961no—
261SD16—2982no—
262M16—41no—
263M16—5961no—
264C2×C8—1491no—
265C2×C8—1491no—
266C2×C8C22×D14911yescharacteristic
267C2×C8C22×D14911yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—2981no—
270C4⋊C4—2981no—
271C4⋊C4—2981no—
272C4⋊C4—2981no—
273C4⋊C4—2981no—
274C4⋊C4—2981no—
275C4⋊C4—2981no—
276C4⋊C4—2981no—
277C4⋊C4—2981no—
278C4⋊C4—1491no—
279C4⋊C4—1491no—
280C4⋊C4—1491no—
281C4⋊C4C22×D14911yescharacteristic
282C4⋊C4C22×D14911yescharacteristic
283C22⋊C4—2981no—
284C22⋊C4—2981no—
285C22⋊C4—2981no—
286C22⋊C4—2981no—
287C22⋊C4—2981no—
288C22⋊C4—1491no—
289C22⋊C4—1491no—
290C42—2981no—
291C42—1491no—
292C23—21no—
293C23—1491no—
294C23D4×D14911yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—2981no—
298Q8—2981no—
299Q8—2981no—
300Q8—2981no—
301Q8—11921no—
302D4—21no—
303D4—42no—
304D4—2982no—
305D4—5962no—
306D4#1455873612yes—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—81no—
314C2×C4—2981no—
315C2×C4—2981no—
316C2×C4—2981no—
317C2×C4—2981no—
318C2×C4—2981no—
319C2×C4—2981no—
320C2×C4—5962no—
321C2×C4—5962no—
322C2×C4—1491no—
323C2×C4—1491no—
324C2×C4—1491no—
325C2×C4—11921no—
326C2×C4D4×D14911yescharacteristic
327C2×C4C23×D14911yescharacteristic
328C8—21no—
329C8—21no—
330C8—2981no—
331C8—2981no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—2981no—
339C22—2981no—
340C22—2981no—
341C22C2×D4×D14911yescharacteristic
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—2981no—
347C4—2981no—
348C4—5961no—
349C4—5961no—
350C4—5961no—
351C4—5961no—
352C4—1492no—
353C4C2×#1455873611yescharacteristic
354C4#1801195611yescharacteristic
355C2—41no—
356C2—42no—
357C2—5961no—
358C2#2375026511yescharacteristic
359C2#137×D14911yescharacteristic
360C2#2375043511yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#417×C149C2
socleC22×C149C22×D4
prime radical#23750274C2
hypercenter#417D149
hypocenterC149#3331
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. #23750330C2semidirect
3. #23749974C2semidirect
4. #23749955C2semidirect
5. #23749954C2semidirect
6. #23749913C2semidirect
7. #23749883C2semidirect
8. #23749882C2semidirect
9. #23749834C2semidirect
10. #23749788C2semidirect
11. #23749787C2semidirect
12. #23749716C2semidirect
13. #23749701C2semidirect
14. #23749623C2semidirect
15. #417×C149C2semidirect
180. C149#3331semidirect
196. #417D149semidirect

Maximal subgroups

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

subgroupindex
#237503302
#237502742
#237499742
#237499552
#237499542
#237499132
#237498832
#237498822
#237498342
#237497882
#237497872
#237497162
#237497012
#237496232
#417×C1492
#3331149