Finite Group Databasebeta

← #34056554

Subgroups — #34056554

Order 19072.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750288C211yesmaximal, characteristic, complemented
2#23750276C211yesmaximal, characteristic
3#23749974C211yesmaximal, characteristic, complemented
4#23749957C211yesmaximal, characteristic, complemented
5#23749909C211yesmaximal, characteristic, complemented
6#23749856C211yesmaximal, characteristic, complemented
7#23749855C211yesmaximal, characteristic, complemented
8#23749829C211yesmaximal, characteristic, complemented
9#23749766C211yesmaximal, characteristic, complemented
10#23749765C211yesmaximal, characteristic, complemented
11#23749712C211yesmaximal, characteristic, complemented
12#23749688C211yesmaximal, characteristic, complemented
13C2×#18011834C211yesmaximal, characteristic, complemented
14#103×D149C211yesmaximal, characteristic, complemented
15#483×C149C211yesmaximal, characteristic, complemented
16C2×#14558736—21no—
17C22×Dic298C2211yescharacteristic
18C2×Q8×D149C2211yescharacteristic
19C2×#14558725C2211yescharacteristic
20C2×#14558725C2211yescharacteristic
21C2×#14558723C2211yescharacteristic
22C22×#11924746C2211yescharacteristic
23C2×C8×D149C2211yescharacteristic
24#18011910—21no—
25#18011910C2211yescharacteristic
26#18011909C2211yescharacteristic
27#18011896—21no—
28#18011894—21no—
29#18011891C2211yescharacteristic
30#18011889—21no—
31#18011889—21no—
32#18011888—21no—
33#18011886C2211yescharacteristic
34#18011885C2211yescharacteristic
35#18011876C2211yescharacteristic
36#18011873C2211yescharacteristic
37C4×C22⋊D149C2211yescharacteristic
38#18011871—21no—
39C4⋊C4×D149C2211yescharacteristic
40#18011842C2211yescharacteristic
41#18011840C2211yescharacteristic
42#18011836C2211yescharacteristic
43#18011836C2211yescharacteristic
44#18011835C2211yescharacteristic
45#18011834C2211yescharacteristic
46#18011834C2211yescharacteristic
47#18011834C2211yescharacteristic
48#18011834C2211yescharacteristic
49#18011831C2211yescharacteristic
50#18011823C2211yescharacteristic
51C8×Dic149C2211yescharacteristic
52C22⋊Q8×C149C2211yescharacteristic
53C2×C4⋊C4×C149C2211yescharacteristic
54#106×C149C2211yescharacteristic
55#106×C149C2211yescharacteristic
56#103×C149C2211yescharacteristic
57#103×C149C2211yescharacteristic
58C22⋊C8×C149C2211yescharacteristic
59#14558736—44no—
60C2×Dic298—21no—
61C2×Dic298—21no—
62C2×Dic298—22no—
63C2×Dic298—21no—
64C2×Dic298—41no—
65C2×Dic298C2311yescharacteristic
66C2×Dic298C2311yescharacteristic
67Q8×D149—22no—
68Q8×D149—41no—
69C2×D596—21no—
70C2×D596C2311yescharacteristic
71C2×C22⋊D149—21no—
72C2×C22⋊D149D411yescharacteristic
73C22×Dic149—21no—
74C2×C4×D149—21no—
75C2×C4×D149—21no—
76C2×C4×D149—21no—
77C2×C4×D149C2311yescharacteristic
78#14558725—22no—
79#14558725—41no—
80#14558723—41no—
81C2×#11924746—41no—
82C2×#11924746C2311yescharacteristic
83C2×#11924746C2311yescharacteristic
84C8×D149—41no—
85C4⋊Dic149—21no—
86C4⋊Dic149C2311yescharacteristic
87#14558715—21no—
88#14558715—21no—
89#14558715—21no—
90#14558715—21no—
91#14558715—21no—
92#14558715D411yescharacteristic
93C22⋊Dic149D411yescharacteristic
94#14558713—21no—
95#14558713—21no—
96#14558713—21no—
97#14558713—21no—
98#14558713D411yescharacteristic
99C4×Dic149—21no—
100C4×Dic149C2311yescharacteristic
101C2×Q8×C149C2311yescharacteristic
102C22×C4×C149—21no—
103C22×C4×C149C2311yescharacteristic
104C2×C8×C149C2311yescharacteristic
105C2×C8×C149C2311yescharacteristic
106C4⋊C4×C149—21no—
107C4⋊C4×C149—21no—
108C4⋊C4×C149C2311yescharacteristic
109C4⋊C4×C149C2311yescharacteristic
110C4⋊C4×C149C2311yescharacteristic
111C22⋊C4×C149—21no—
112C22×D149—21no—
113C22×D149C2×D411yescharacteristic
114Dic298—21no—
115Dic298—21no—
116Dic298—21no—
117Dic298—42no—
118Dic298—42no—
119Dic298—42no—
120Dic298#4112yescomplemented
121D596—21no—
122D596—21no—
123D596—42no—
124C22⋊D149—42no—
125C22⋊D149SD1614yescomplemented
126C2×Dic149—21no—
127C2×Dic149—21no—
128C2×Dic149—21no—
129C2×Dic149—42no—
130C2×Dic149C2×D411yescharacteristic
131C2×Dic149C2×D411yescharacteristic
132C2×Dic149C2×D411yescharacteristic
133C4×D149—21no—
134C4×D149—21no—
135C4×D149—42no—
136C4×D149—42no—
137C4×D149—42no—
138#11924746—21no—
139#11924746—21no—
140C23×C149C2×D411yescharacteristic
141Q8×C149—21no—
142Q8×C149—21no—
143C2×C4×C149—21no—
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×C149C2×D411yescharacteristic
151C2×C4×C149C2411yescharacteristic
152C8×C149—21no—
153C8×C149—21no—
154C22×C149—21no—
155C22×C149—21no—
156C22×C149C2×SD1612yes—
157C22×C149C22×D411yescharacteristic
158C2×D149—21no—
159C2×D149—21no—
160C2×D149—41no—
161C2×D149—41no—
162C2×D149—41no—
163C2×D149C2×SD1612yes—
164C4×C149—21no—
165C4×C149—41no—
166C4×C149—41no—
167C4×C149—41no—
168C4×C149—41no—
169C4×C149C2×#4111yescharacteristic
170C4×C149H(5,𝔽2)11yescharacteristic
171Dic149—21no—
172Dic149—22no—
173Dic149—21no—
174Dic149—41no—
175Dic149—41no—
176Dic149C2×SD1612yes—
177C2×C149—22no—
178C2×C149#54611yescharacteristic
179C2×C149C22×SD1611yescharacteristic
180C2×C149#57811yescharacteristic
181D149—22no—
182D149—81no—
183C149#329811yescharacteristic, complemented, Sylow
184#3298—1491nomaximal, Sylow
185D4×Q8—1491no—
186#547—1491no—
187#487—1491no—
188#483—1491no—
189#483D14911yescharacteristic, complemented
190#475—1491no—
191#465—1491no—
192#465—1491no—
193#461—1491no—
194#448—1491no—
195#448—1491no—
196C4×SD16—1491no—
197C8×D4—1491no—
198C2×#103—1491no—
199C2×#103—1491no—
200C2×#41—2981no—
201C22×Q8—1491no—
202C22×Q8—1491no—
203C2×SD16—1491no—
204C2×SD16—1491no—
205C2×SD16—1491no—
206C22×C8—1491no—
207C22×C8—1491no—
208C4⋊Q8—2981no—
209C4⋊Q8—1491no—
210C4⋊Q8—1491no—
211#123—2981no—
212#123—2981no—
213C22⋊Q8—2981no—
214C22⋊Q8—2981no—
215C22⋊Q8—2981no—
216C22⋊Q8—1491no—
217C22⋊Q8C2×D14911yescharacteristic
218#121—1491no—
219#121—1491no—
220C4×Q8—1491no—
221C4×D4—2981no—
222C4×D4—1491no—
223C4×D4—1491no—
224C2×C4⋊C4—1491no—
225C2×C4⋊C4C2×D14911yescharacteristic
226#106—1491no—
227#106C2×D14911yescharacteristic
228#106C2×D14911yescharacteristic
229C4⋊C8—1491no—
230#103—1491no—
231#103—1491no—
232#103—1491no—
233#103—1491no—
234#103—1491no—
235#103—1491no—
236#103—1491no—
237#103C2×D14911yescharacteristic
238#103C2×D14911yescharacteristic
239#102—1491no—
240C22⋊C8—1491no—
241C22⋊C8C2×D14911yescharacteristic
242C4×C8—1491no—
243#41—5964no—
244C2×Q8—2981no—
245C2×Q8—2981no—
246C2×Q8—2982no—
247C2×Q8—2981no—
248C2×Q8—2982no—
249C2×Q8—5961no—
250C2×Q8—5961no—
251C2×Q8—1491no—
252C2×Q8—1491no—
253C2×Q8C22×D14911yescharacteristic
254C2×D4—2981no—
255C2×D4—2981no—
256C2×D4—1491no—
257C2×D4—1491no—
258C22×C4—21no—
259C22×C4—2981no—
260C22×C4—2981no—
261C22×C4—2981no—
262C22×C4—2981no—
263C22×C4—1491no—
264C22×C4C22×D14911yescharacteristic
265SD16—2982no—
266SD16—5961no—
267SD16—5961no—
268C2×C8—5961no—
269C2×C8—5961no—
270C2×C8—1491no—
271C2×C8—1491no—
272C2×C8C22×D14911yescharacteristic
273C2×C8C22×D14911yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—2981no—
277C4⋊C4—2981no—
278C4⋊C4—2981no—
279C4⋊C4—2981no—
280C4⋊C4—2981no—
281C4⋊C4—2981no—
282C4⋊C4—1491no—
283C4⋊C4—1491no—
284C4⋊C4C22×D14911yescharacteristic
285C4⋊C4C22×D14911yescharacteristic
286C4⋊C4C22×D14911yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—2981no—
289C22⋊C4—2981no—
290C22⋊C4—2981no—
291C22⋊C4—2981no—
292C22⋊C4—1491no—
293C22⋊C4—1491no—
294C42—2981no—
295C42—1491no—
296C23—2981no—
297C23—1491no—
298C23D4×D14911yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—2981no—
302Q8—2981no—
303Q8—2981no—
304Q8—5962no—
305Q8—5962no—
306Q8—5962no—
307Q8—1492no—
308D4—2981no—
309D4—2981no—
310D4—5962no—
311D4—5962no—
312D4—1494no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—2981no—
321C2×C4—2981no—
322C2×C4—2981no—
323C2×C4—2981no—
324C2×C4—2981no—
325C2×C4—5962no—
326C2×C4—5962no—
327C2×C4—5962no—
328C2×C4—5962no—
329C2×C4—1491no—
330C2×C4—1491no—
331C2×C4—1491no—
332C2×C4D4×D14911yescharacteristic
333C2×C4C23×D14911yescharacteristic
334C8—21no—
335C8—21no—
336C8—2981no—
337C8—2981no—
338C22—21no—
339C22—21no—
340C22—2981no—
341C22—2981no—
342C22—5961no—
343C22—5961no—
344C22—5961no—
345C22—1492no—
346C22SD16×D14912yes—
347C22C2×D4×D14911yescharacteristic
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—2981no—
354C4—2981no—
355C4—2982no—
356C4—5961no—
357C4—5961no—
358C4—1492no—
359C4#41×D14911yescharacteristic
360C4#1801195711yescharacteristic
361C2—22no—
362C2—2982no—
363C2—11921no—
364C2#2375026711yescharacteristic
365C2C2×SD16×D14911yescharacteristic
366C2#2375044111yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#483×C149C2
socleC22×C149C22×D4
prime radical#23750276C2
hypercenter#483D149
hypocenterC149#3298
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #23750288C2semidirect
3. #23749974C2semidirect
4. #23749957C2semidirect
5. #23749909C2semidirect
6. #23749856C2semidirect
7. #23749855C2semidirect
8. #23749829C2semidirect
9. #23749766C2semidirect
10. #23749765C2semidirect
11. #23749712C2semidirect
12. #23749688C2semidirect
13. C2×#18011834C2semidirect
14. #103×D149C2semidirect
15. #483×C149C2semidirect
120. Dic298#41semidirect
125. C22⋊D149SD16semidirect
183. C149#3298semidirect
189. #483D149semidirect

Maximal subgroups

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

subgroupindex
#237502882
#237502762
#237499742
#237499572
#237499092
#237498562
#237498552
#237498292
#237497662
#237497652
#237497122
#237496882
C2×#180118342
#103×D1492
#483×C1492
#3298149