Finite Group Databasebeta

← #34056572

Subgroups — #34056572

Order 19072.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750284C211yesmaximal, characteristic
2#23749982C211yesmaximal, characteristic, complemented
3#23749966C211yesmaximal, characteristic, complemented
4#23749938C211yesmaximal, characteristic, complemented
5#23749929C211yesmaximal, characteristic, complemented
6#23749864C211yesmaximal, characteristic, complemented
7#23749856C211yesmaximal, characteristic, complemented
8#23749847C211yesmaximal, characteristic, complemented
9#23749785C211yesmaximal, characteristic, complemented
10#23749780C211yesmaximal, characteristic, complemented
11C4×#14558726C211yesmaximal, characteristic, complemented
12D4×#11924746C211yesmaximal, characteristic, complemented
13#23749619C211yesmaximal, characteristic, complemented
14C2×#18011834C211yesmaximal, characteristic, complemented
15#552×C149C211yesmaximal, characteristic, complemented
16C2×#14558736C2211yescharacteristic
17C22×Dic298C2211yescharacteristic
18C2×#14558726C2211yescharacteristic
19C2×#14558726C2211yescharacteristic
20C2×#14558723C2211yescharacteristic
21C22×#11924746C2211yescharacteristic
22C22×#11924746C2211yescharacteristic
23#18011899C2211yescharacteristic
24#18011897—21no—
25#18011890C2211yescharacteristic
26#18011889—21no—
27#18011885C2211yescharacteristic
28#18011883—21no—
29C4×Dic298C2211yescharacteristic
30#18011871—21no—
31#18011844C2211yescharacteristic
32#18011844C2211yescharacteristic
33#18011842C2211yescharacteristic
34C4⋊#11924746C2211yescharacteristic
35#18011835C2211yescharacteristic
36#18011834C2211yescharacteristic
37#18011834C2211yescharacteristic
38#18011834C2211yescharacteristic
39#18011834C2211yescharacteristic
40#18011834C2211yescharacteristic
41#18011834C2211yescharacteristic
42#18011831C2211yescharacteristic
43#18011831C2211yescharacteristic
44#18011831C2211yescharacteristic
45C22⋊#11924746C2211yescharacteristic
46C22⋊#11924746C2211yescharacteristic
47C4×#11924746C2211yescharacteristic
48#126×C149—21no—
49#125×C149—21no—
50#125×C149C2211yescharacteristic
51#123×C149—21no—
52C22⋊Q8×C149C2211yescharacteristic
53C22⋊Q8×C149C2211yescharacteristic
54C4×Q8×C149C2211yescharacteristic
55C4×D4×C149C2211yescharacteristic
56#117×C149—21no—
57#117×C149C2211yescharacteristic
58C2×C4⋊C4×C149C2211yescharacteristic
59#14558736—22no—
60#14558736—41no—
61C2×Dic298—22no—
62C2×Dic298—21no—
63C2×Dic298—41no—
64C2×Dic298C2311yescharacteristic
65C2×Dic298C2311yescharacteristic
66C2×D596C2311yescharacteristic
67C2×C22⋊D149—21no—
68C2×C22⋊D149—21no—
69C22×Dic149—21no—
70C2×C4×D149—21no—
71#14558726—22no—
72#14558726—41no—
73#14558723—41no—
74C2×#11924746—41no—
75C2×#11924746—41no—
76C2×#11924746C2311yescharacteristic
77C2×#11924746C2311yescharacteristic
78C2×#11924746C2311yescharacteristic
79C2×#11924746C2311yescharacteristic
80C4⋊Dic149C2311yescharacteristic
81#14558715—21no—
82#14558715—21no—
83C22⋊Dic149—21no—
84#14558713—21no—
85#14558713—21no—
86#14558713—21no—
87C4×Dic149—21no—
88C2×Q8×C149C2311yescharacteristic
89C2×D4×C149D411yescharacteristic
90C22×C4×C149—21no—
91C22×C4×C149C2311yescharacteristic
92C22×C4×C149C2311yescharacteristic
93C4⋊C4×C149—21no—
94C4⋊C4×C149—21no—
95C4⋊C4×C149—21no—
96C4⋊C4×C149—21no—
97C4⋊C4×C149—21no—
98C4⋊C4×C149—21no—
99C4⋊C4×C149D411yescharacteristic
100C4⋊C4×C149C2311yescharacteristic
101C4⋊C4×C149C2311yescharacteristic
102C4⋊C4×C149C2311yescharacteristic
103C22⋊C4×C149—21no—
104C22⋊C4×C149—21no—
105C22⋊C4×C149—21no—
106C22⋊C4×C149D411yescharacteristic
107C22⋊C4×C149D411yescharacteristic
108C42×C149—21no—
109C42×C149—21no—
110C42×C149C2311yescharacteristic
111C22×D149—21no—
112Dic298—21no—
113Dic298—21no—
114Dic298—21no—
115Dic298—42no—
116Dic298#4112yes—
117D596—21no—
118D596—21no—
119C22⋊D149—42no—
120C22⋊D149—42no—
121C2×Dic149—21no—
122C2×Dic149—21no—
123C2×Dic149—21no—
124C2×Dic149—42no—
125C4×D149—42no—
126#11924746—21no—
127#11924746—21no—
128#11924746—21no—
129#11924746—21no—
130C23×C149C2×D411yescharacteristic
131C23×C149C2×D411yescharacteristic
132Q8×C149—21no—
133Q8×C149—21no—
134D4×C149—22no—
135C2×C4×C149—21no—
136C2×C4×C149—21no—
137C2×C4×C149—21no—
138C2×C4×C149—21no—
139C2×C4×C149—21no—
140C2×C4×C149—21no—
141C2×C4×C149—21no—
142C2×C4×C149—21no—
143C2×C4×C149—42no—
144C2×C4×C149C2×D411yescharacteristic
145C2×C4×C149C2×D411yescharacteristic
146C2×C4×C149C2×D411yescharacteristic
147C2×C4×C149C2×D411yescharacteristic
148C2×C4×C149C2411yescharacteristic
149C22×C149—21no—
150C22×C149—21no—
151C22×C149—21no—
152C22×C149—21no—
153C22×C149—21no—
154C22×C149#13512yes—
155C22×C149C22×D411yescharacteristic
156C2×D149—41no—
157C2×D149—41no—
158C2×D149—41no—
159C4×C149—21no—
160C4×C149—21no—
161C4×C149—21no—
162C4×C149—21no—
163C4×C149—41no—
164C4×C149—41no—
165C4×C149—41no—
166C4×C149—41no—
167C4×C149C2×#4111yescharacteristic
168C4×C149H(5,𝔽2)11yescharacteristic
169Dic149—22no—
170Dic149—41no—
171Dic149—41no—
172C2×C149—22no—
173C2×C149—41no—
174C2×C149#54611yescharacteristic
175C2×C149C2×#13511yescharacteristic
176C2×C149#57811yescharacteristic
177D149—81no—
178C149#330011yescharacteristic, complemented, Sylow
179#3300—1491nomaximal, Sylow
180#552D14911yescharacteristic, complemented
181#548—1491no—
182#488—1491no—
183#485—1491no—
184#480—1491no—
185#479—1491no—
186#467—1491no—
187#465—1491no—
188#464—1491no—
189#452—1491no—
190#451—1491no—
191C4×Q16—1491no—
192C8×D4—1491no—
193#416—1491no—
194C2×#103—1491no—
195C2×#41—1491no—
196C22×Q8—1491no—
197C2×Q16—1491no—
198C2×Q16—1491no—
199C2×SD16—1491no—
200C22×C8—1491no—
201C22×C8—1491no—
202#126—21no—
203#125—21no—
204#125C2×D14911yescharacteristic
205#124—2981no—
206#124—1491no—
207#123—21no—
208C22⋊Q8—2981no—
209C22⋊Q8—1491no—
210C22⋊Q8C2×D14911yescharacteristic
211C22⋊Q8C2×D14911yescharacteristic
212#121—2981no—
213#121—1491no—
214C4×Q8—1491no—
215C4×Q8C2×D14911yescharacteristic
216C4×D4—2981no—
217C4×D4C2×D14911yescharacteristic
218#117—21no—
219#117C2×D14911yescharacteristic
220C2×C4⋊C4C2×D14911yescharacteristic
221#107—1491no—
222#107—1491no—
223#106—1491no—
224C4⋊C8—1491no—
225#103—1491no—
226#103—1491no—
227#103—1491no—
228#103—1491no—
229#103—1491no—
230#103—1491no—
231#103—1491no—
232#102—1491no—
233#102—1491no—
234#102—1491no—
235C22⋊C8—1491no—
236C22⋊C8—1491no—
237C4×C8—1491no—
238#41—2982no—
239#41—5961no—
240C2×Q8—2982no—
241C2×Q8—2981no—
242C2×Q8—5961no—
243C2×Q8—1491no—
244C2×Q8—1491no—
245C2×Q8C22×D14911yescharacteristic
246C2×D4—2981no—
247C2×D4—2981no—
248C2×D4—1491no—
249C2×D4C22⋊D14911yescharacteristic
250C22×C4—21no—
251C22×C4—2981no—
252C22×C4—2981no—
253C22×C4C22×D14911yescharacteristic
254C22×C4C22×D14911yescharacteristic
255Q16—2982no—
256Q16—5961no—
257SD16—5961no—
258C2×C8—5961no—
259C2×C8—5961no—
260C2×C8—1491no—
261C2×C8—1491no—
262C2×C8—1491no—
263C2×C8—1491no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—2981no—
271C4⋊C4—2981no—
272C4⋊C4—1491no—
273C4⋊C4C22⋊D14911yescharacteristic
274C4⋊C4C22×D14911yescharacteristic
275C4⋊C4C22×D14911yescharacteristic
276C4⋊C4C22×D14911yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—2981no—
281C22⋊C4—2981no—
282C22⋊C4—2981no—
283C22⋊C4—2981no—
284C22⋊C4C22⋊D14911yescharacteristic
285C22⋊C4C22⋊D14911yescharacteristic
286C42—21no—
287C42—21no—
288C42—2981no—
289C42C22×D14911yescharacteristic
290C23—2981no—
291C23C2×C22⋊D14911yescharacteristic
292C23C2×C22⋊D14911yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—2981no—
296Q8—2981no—
297Q8—2981no—
298Q8—5962no—
299Q8—1492no—
300D4—22no—
301D4—2981no—
302D4—2981no—
303D4—5962no—
304D4—5962no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—2981no—
315C2×C4—2981no—
316C2×C4—2981no—
317C2×C4—5962no—
318C2×C4—5962no—
319C2×C4C2×C22⋊D14911yescharacteristic
320C2×C4C2×C22⋊D14911yescharacteristic
321C2×C4C2×C22⋊D14911yescharacteristic
322C2×C4C2×C22⋊D14911yescharacteristic
323C2×C4C23×D14911yescharacteristic
324C8—2981no—
325C8—2981no—
326C8—2981no—
327C8—2981no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—5961no—
334C22—5961no—
335C22—5961no—
336C22#1801193112yes—
337C22C22×C22⋊D14911yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—2982no—
347C4—5961no—
348C4—5961no—
349C4#41×D14911yescharacteristic
350C4#1801195711yescharacteristic
351C2—22no—
352C2—41no—
353C2—11921no—
354C2#2375026311yescharacteristic
355C2C2×#1801193111yescharacteristic
356C2#2375044011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#552×C149C2
socleC22×C149C22×D4
prime radical#23750284C2
hypercenter#552D149
hypocenterC149#3300
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #23749982C2semidirect
3. #23749966C2semidirect
4. #23749938C2semidirect
5. #23749929C2semidirect
6. #23749864C2semidirect
7. #23749856C2semidirect
8. #23749847C2semidirect
9. #23749785C2semidirect
10. #23749780C2semidirect
11. C4×#14558726C2semidirect
12. D4×#11924746C2semidirect
13. #23749619C2semidirect
14. C2×#18011834C2semidirect
15. #552×C149C2semidirect
178. C149#3300semidirect
180. #552D149semidirect

Maximal subgroups

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

subgroupindex
#237502842
#237499822
#237499662
#237499382
#237499292
#237498642
#237498562
#237498472
#237497852
#237497802
C4×#145587262
D4×#119247462
#237496192
C2×#180118342
#552×C1492
#3300149