Finite Group Databasebeta

← #34058702

Subgroups — #34058702

Order 19072.

1010 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×Dic298C211yesmaximal, characteristic, complemented
2#23750348C213yesmaximal, complemented
3#23750347C213yesmaximal, complemented
4#23750343C213yesmaximal, complemented
5#23750338C213yesmaximal, complemented
6Q8×C22⋊D149C211yesmaximal, characteristic, complemented
7#23750287C211yesmaximal, characteristic, complemented
8#23750286C211yesmaximal, characteristic, complemented
9#23750247C213yesmaximal, complemented
10#23750246C213yesmaximal, complemented
11#23750179C213yesmaximal, complemented
12#23750153C213yesmaximal, complemented
13#23750101C211yesmaximal, characteristic, complemented
14C4×#14558737C211yesmaximal, characteristic, complemented
15#519×C149C211yesmaximal, characteristic, complemented
16C2×#14558737C2211yescharacteristic
17C2×Q8×D149C2211yescharacteristic
18#18011911C2213yes—
19#18011910C2213yes—
20#18011910C2213yes—
21#18011909C2213yes—
22#18011906C2213yes—
23#18011906C2216yes—
24#18011905C2216yes—
25#18011904C2213yes—
26#18011901C2213yes—
27#18011900C2213yes—
28#18011899C2213yes—
29#18011899C2216yes—
30#18011897C2211yescharacteristic
31#18011897C2216yes—
32#18011896C2213yes—
33#18011893C2213yes—
34#18011891C2213yes—
35#18011890C2213yes—
36#18011889C2213yes—
37#18011889C2216yes—
38#18011887C2213yes—
39#18011887C2213yes—
40#18011886C2213yes—
41#18011885C2213yes—
42#18011884C2213yes—
43Q8×Dic149C2211yescharacteristic
44Q8×Dic149C2211yescharacteristic
45C4×Dic298C2213yes—
46C4×Dic298C2213yes—
47C4×Dic298C2216yes—
48#18011876C2213yes—
49C4×D596C2213yes—
50C4×D596C2213yes—
51#18011873C2213yes—
52C4×C22⋊D149C2213yes—
53C4×C22⋊D149C2213yes—
54#18011869C2213yes—
55#18011869C2213yes—
56#18011868C2213yes—
57#18011868C2216yes—
58C42×D149C2213yes—
59C22×Q8×C149C2211yescharacteristic
60C4×Q8×C149C2213yes—
61C4×Q8×C149C2211yescharacteristic
62C4×Q8×C149C2213yes—
63C4×Q8×C149C2211yescharacteristic
64#117×C149C2213yes—
65#117×C149C2213yes—
66#14558737—24no—
67C2×Dic298C2313yes—
68C2×Dic298C2311yescharacteristic
69C2×Dic298C2316yes—
70Q8×D149—24no—
71C2×D596C2313yes—
72C2×D596C2313yes—
73C2×C22⋊D149C2311yescharacteristic
74C2×C22⋊D149C2313yes—
75C2×C4×D149C2313yes—
76C2×C4×D149C2313yes—
77C2×C4×D149C2313yes—
78C2×C4×D149C2313yes—
79C4⋊Dic149C2313yes—
80C4⋊Dic149C2313yes—
81C4⋊Dic149C2313yes—
82C4⋊Dic149C2313yes—
83#14558715C2313yes—
84#14558715C2313yes—
85#14558715C2316yes—
86#14558715C2313yes—
87#14558715C2313yes—
88#14558715C2313yes—
89#14558715C2313yes—
90C22⋊Dic149C2311yescharacteristic
91C22⋊Dic149C2313yes—
92#14558713C2313yes—
93#14558713C2316yes—
94#14558713C2316yes—
95#14558713C2311yescharacteristic
96#14558713C2313yes—
97#14558713C2313yes—
98#14558713C2316yes—
99#14558713C2313yes—
100#14558713C2311yescharacteristic
101C4×Dic149C2313yes—
102C4×Dic149C2316yes—
103C4×Dic149C2313yes—
104C4×Dic149C2313yes—
105C4×Dic149C2311yescharacteristic
106C2×Q8×C149—24no—
107C2×Q8×C149C2313yes—
108C2×Q8×C149C2311yescharacteristic
109C22×C4×C149C2313yes—
110C4⋊C4×C149C2313yes—
111C4⋊C4×C149C2313yes—
112C4⋊C4×C149C2313yes—
113C4⋊C4×C149C2313yes—
114C22⋊C4×C149C2313yes—
115C22⋊C4×C149C2311yescharacteristic
116C42×C149C2316yes—
117C42×C149C2313yes—
118C42×C149C2313yes—
119C22×D149C2411yescharacteristic
120C22×D149C2413yes—
121Dic298—26no—
122Dic298—22no—
123Dic298—46no—
124D596—26no—
125D596—43no—
126C22⋊D149—22no—
127C22⋊D149—43no—
128C2×Dic149C2411yescharacteristic
129C2×Dic149C2413yes—
130C2×Dic149C2413yes—
131C2×Dic149C2413yes—
132C2×Dic149C2411yescharacteristic
133C2×Dic149C2411yescharacteristic
134C4×D149—26no—
135C4×D149—26no—
136C4×D149—26no—
137C4×D149—43no—
138C23×C149C2411yescharacteristic
139Q8×C149—26no—
140Q8×C149#4114yes—
141C2×C4×C149—26no—
142C2×C4×C149C2411yescharacteristic
143C2×C4×C149C2413yes—
144C2×C4×C149C2413yes—
145C2×C4×C149C2411yescharacteristic
146C2×C4×C149C2413yes—
147C2×C4×C149C2413yes—
148C22×C149—21no—
149C22×C149—21no—
150C22×C149—21no—
151C22×C149C2511yescharacteristic
152C2×D149—21no—
153C2×D149—23no—
154C2×D149—23no—
155C2×D149—21no—
156C2×D149—23no—
157C2×D149—21no—
158C4×C149—23no—
159C4×C149—23no—
160C4×C149—21no—
161C4×C149—21no—
162C4×C149—23no—
163C4×C149C2×#4116yes—
164Dic149—21no—
165Dic149—23no—
166Dic149—21no—
167Dic149—23no—
168Dic149—23no—
169Dic1492−1+412yes—
170C2×C149—41no—
171C2×C149C22×#4111yescharacteristic
172C2×C149C2×2−1+411yescharacteristic
173C2×C149#58511yescharacteristic
174D149—41no—
175D149—43no—
176C149#351811yescharacteristic, complemented, Sylow
177#3518—1491nomaximal, Sylow
178Q82—1491no—
179#556—1493no—
180#556—1493no—
181#556—1493no—
182#555—1493no—
183D4×Q8—1491no—
184#548—1491no—
185#548—1491no—
186#542—1493no—
187#542—1493no—
188#533—1493no—
189#529—1493no—
190#519—1491no—
191#519D14911yescharacteristic, complemented
192C4×#41—1491no—
193C2×#41—1491no—
194C22×Q8—1491no—
195C22×Q8C2×D14911yescharacteristic
196C4⋊Q8—1493no—
197C4⋊Q8—1493no—
198C4⋊Q8—1493no—
199C4⋊Q8—1493no—
200#126—1493no—
201#126—1496no—
202#126—1493no—
203#126—1496no—
204#125—1493no—
205#124—1493no—
206#124—1491no—
207#124—1496no—
208#124—1493no—
209#124—1496no—
210#123—1493no—
211#123—1493no—
212C22⋊Q8—1493no—
213C22⋊Q8—1493no—
214C22⋊Q8—1493no—
215C22⋊Q8—1493no—
216C22⋊Q8—1496no—
217C22⋊Q8—1493no—
218#121—1493no—
219#121—1493no—
220#121—1493no—
221C4×Q8—1491no—
222C4×Q8—1491no—
223C4×Q8—1493no—
224C4×Q8—1493no—
225C4×Q8—1493no—
226C4×Q8—1496no—
227C4×Q8C2×D14911yescharacteristic
228C4×Q8C2×D14913yes—
229C4×Q8C2×D14911yescharacteristic
230C4×Q8C2×D14913yes—
231C4×D4—1493no—
232C4×D4—1493no—
233C4×D4—1493no—
234C4×D4—1493no—
235C4×D4—1493no—
236#117—1493no—
237#117—1493no—
238#117—1493no—
239#117—1496no—
240#117C2×D14913yes—
241#117C2×D14913yes—
242C2×C42—1493no—
243#41—2984no—
244C2×Q8—24no—
245C2×Q8—2984no—
246C2×Q8—1491no—
247C2×Q8—1493no—
248C2×Q8—1496no—
249C2×Q8C22×D14913yes—
250C2×Q8C22×D14911yescharacteristic
251C2×D4—1491no—
252C2×D4—1493no—
253C2×D4—1493no—
254C2×D4—1493no—
255C22×C4—1493no—
256C22×C4—1493no—
257C22×C4—1493no—
258C22×C4—1493no—
259C22×C4C22×D14913yes—
260C4⋊C4—1493no—
261C4⋊C4—1493no—
262C4⋊C4—1493no—
263C4⋊C4—1493no—
264C4⋊C4—1493no—
265C4⋊C4—1493no—
266C4⋊C4—1493no—
267C4⋊C4—1493no—
268C4⋊C4—1496no—
269C4⋊C4—1493no—
270C4⋊C4—1493no—
271C4⋊C4C22×D14913yes—
272C4⋊C4C22×D14913yes—
273C4⋊C4C22×D14913yes—
274C4⋊C4C22×D14913yes—
275C22⋊C4—1491no—
276C22⋊C4—1493no—
277C22⋊C4—1493no—
278C22⋊C4—1491no—
279C22⋊C4—1496no—
280C22⋊C4—1496no—
281C22⋊C4—1496no—
282C22⋊C4—1491no—
283C22⋊C4—1493no—
284C22⋊C4—1493no—
285C22⋊C4—1493no—
286C22⋊C4C22×D14913yes—
287C22⋊C4C22×D14911yescharacteristic
288C42—1491no—
289C42—1493no—
290C42—1493no—
291C42—1496no—
292C42—1493no—
293C42C22×D14913yes—
294C42C22×D14916yes—
295C42C22×D14913yes—
296C23—1491no—
297C23—1493no—
298C23C23×D14911yescharacteristic
299Q8—26no—
300Q8—2982no—
301Q8—2986no—
302Q8—5966no—
303Q8#1455873614yes—
304D4—2982no—
305D4—2986no—
306D4—5963no—
307D4—5963no—
308C2×C4—26no—
309C2×C4—2986no—
310C2×C4—2986no—
311C2×C4—2986no—
312C2×C4—5963no—
313C2×C4—1491no—
314C2×C4—1491no—
315C2×C4—1493no—
316C2×C4—1493no—
317C2×C4—1493no—
318C2×C4—1491no—
319C2×C4C23×D14913yes—
320C2×C4C23×D14911yescharacteristic
321C2×C4C23×D14911yescharacteristic
322C2×C4C23×D14913yes—
323C2×C4C23×D14913yes—
324C2×C4C23×D14913yes—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—2981no—
329C22—2983no—
330C22—2983no—
331C22—2981no—
332C22—2983no—
333C22—2981no—
334C22C24×D14911yescharacteristic
335C4—23no—
336C4—23no—
337C4—21no—
338C4—21no—
339C4—23no—
340C4—2981no—
341C4—2983no—
342C4—2981no—
343C4—2983no—
344C4—2983no—
345C4—1492no—
346C4C2×#1455873616yes—
347C2—41no—
348C2—5961no—
349C2—5963no—
350C2C22×#1455873611yescharacteristic
351C22−1+4×D14911yescharacteristic
352C2#2375046211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
Fitting#519×C149C2
hypercenter#519D149
hypocenterC149#3518
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
1. Q8×Dic298C2semidirect
2. #23750348C2semidirect
3. #23750347C2semidirect
4. #23750343C2semidirect
5. #23750338C2semidirect
6. Q8×C22⋊D149C2semidirect
7. #23750287C2semidirect
8. #23750286C2semidirect
9. #23750247C2semidirect
10. #23750246C2semidirect
11. #23750179C2semidirect
12. #23750153C2semidirect
13. #23750101C2semidirect
14. C4×#14558737C2semidirect
15. #519×C149C2semidirect
176. C149#3518semidirect
191. #519D149semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic2982
#237503482
#237503472
#237503432
#237503382
Q8×C22⋊D1492
#237502872
#237502862
#237502472
#237502462
#237501792
#237501532
#237501012
C4×#145587372
#519×C1492
#3518149