Finite Group Databasebeta

← #34058687

Subgroups — #34058687

Order 19072.

1012 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
1#23750348C213yesmaximal, complemented
2#23750336C213yesmaximal, complemented
3#23750328C213yesmaximal, complemented
4#23750313C213yesmaximal, complemented
5Q8×D596C211yesmaximal, characteristic, complemented
6#23750287C211yesmaximal, characteristic, complemented
7#23750287C211yesmaximal, characteristic, complemented
8#23750249C213yesmaximal, complemented
9#23750238C213yesmaximal, complemented
10#23750162C213yesmaximal, complemented
11#23750160C213yesmaximal, complemented
12#23750107C211yesmaximal, characteristic, complemented
13#23750101C211yesmaximal, characteristic, complemented
14C4×Q8×D149C211yesmaximal, characteristic, complemented
15#557×C149C211yesmaximal, characteristic, complemented
16C2×#14558737C2211yescharacteristic
17C2×Q8×D149C2211yescharacteristic
18C2×Q8×D149C2211yescharacteristic
19#18011911C2213yes—
20#18011906C2216yes—
21#18011906C2213yes—
22#18011906C2213yes—
23#18011906C2216yes—
24#18011902C2213yes—
25#18011902C2216yes—
26#18011900C2213yes—
27#18011899C2213yes—
28#18011899C2213yes—
29#18011896C2213yes—
30#18011896C2211yescharacteristic
31#18011896C2216yes—
32#18011891C2213yes—
33#18011891C2213yes—
34#18011891C2213yes—
35#18011891C2213yes—
36#18011887C2213yes—
37#18011887C2216yes—
38#18011886C2213yes—
39#18011886C2213yes—
40#18011886C2213yes—
41Q8×Dic149C2211yescharacteristic
42C4×Dic298C2213yes—
43C4×Dic298C2213yes—
44#18011876C2213yes—
45#18011876C2213yes—
46C4×D596C2213yes—
47C4×D596C2213yes—
48C4×D596C2213yes—
49C4×D596C2213yes—
50#18011873C2213yes—
51#18011873C2216yes—
52#18011869C2216yes—
53#18011869C2213yes—
54#18011868C2213yes—
55#18011868C2213yes—
56C4⋊C4×D149C2213yes—
57C4⋊C4×D149C2213yes—
58C42×D149C2213yes—
59C4⋊Q8×C149C2213yes—
60#125×C149C2213yes—
61#125×C149C2213yes—
62C4×Q8×C149C2213yes—
63C4×Q8×C149C2211yescharacteristic
64C4×Q8×C149C2211yescharacteristic
65C4×Q8×C149C2211yescharacteristic
66#14558737—24no—
67C2×Dic298C2313yes—
68C2×Dic298C2313yes—
69Q8×D149—24no—
70Q8×D149—24no—
71C2×D596C2313yes—
72C2×D596C2313yes—
73C2×D596C2311yescharacteristic
74C2×D596C2313yes—
75C2×C4×D149C2313yes—
76C2×C4×D149C2313yes—
77C2×C4×D149C2313yes—
78C2×C4×D149C2311yescharacteristic
79C2×C4×D149C2313yes—
80C2×C4×D149C2316yes—
81C4⋊Dic149C2313yes—
82C4⋊Dic149C2316yes—
83C4⋊Dic149C2313yes—
84C4⋊Dic149C2313yes—
85C4⋊Dic149C2313yes—
86#14558715C2313yes—
87#14558715C2313yes—
88#14558715C2316yes—
89#14558715C2313yes—
90#14558715C2311yescharacteristic
91#14558713C2313yes—
92#14558713C2311yescharacteristic
93#14558713C2313yes—
94#14558713C2316yes—
95#14558713C2316yes—
96#14558713C2313yes—
97#14558713C2313yes—
98#14558713C2313yes—
99#14558713C2313yes—
100#14558713C2311yescharacteristic
101#14558713C2313yes—
102#14558713C2311yescharacteristic
103C4×Dic149C2313yes—
104C4×Dic149C2313yes—
105C4×Dic149C2316yes—
106C4×Dic149C2313yes—
107C2×Q8×C149C2313yes—
108C2×Q8×C149C2311yescharacteristic
109C4⋊C4×C149C2311yescharacteristic
110C4⋊C4×C149C2313yes—
111C4⋊C4×C149C2313yes—
112C4⋊C4×C149C2316yes—
113C4⋊C4×C149C2313yes—
114C4⋊C4×C149C2313yes—
115C4⋊C4×C149C2313yes—
116C42×C149C2313yes—
117C42×C149C2313yes—
118C42×C149C2313yes—
119C22×D149C2411yescharacteristic
120C22×D149C2413yes—
121C22×D149C2411yescharacteristic
122Dic298—26no—
123Dic298—26no—
124D596—26no—
125D596—22no—
126D596—26no—
127D596—43no—
128C2×Dic149C2413yes—
129C2×Dic149C2411yescharacteristic
130C2×Dic149C2413yes—
131C2×Dic149C2413yes—
132C2×Dic149C2411yescharacteristic
133C4×D149—22no—
134C4×D149—26no—
135C4×D149—26no—
136C4×D149—26no—
137C4×D149—46no—
138C4×D149—43no—
139Q8×C149—43no—
140Q8×C149#4114yes—
141C2×C4×C149C2413yes—
142C2×C4×C149C2413yes—
143C2×C4×C149C2411yescharacteristic
144C2×C4×C149C2411yescharacteristic
145C2×C4×C149C2413yes—
146C2×C4×C149C2411yescharacteristic
147C2×C4×C149C2413yes—
148C22×C149C2511yescharacteristic
149C2×D149—21no—
150C2×D149—21no—
151C2×D149—21no—
152C2×D149—23no—
153C2×D149—21no—
154C2×D149—23no—
155C2×D149—23no—
156C2×D149—21no—
157C2×D1492−1+412yes—
158C4×C149—21no—
159C4×C149—21no—
160C4×C149—23no—
161C4×C149—21no—
162C4×C149—23no—
163C4×C149—23no—
164C4×C149C2×#4116yes—
165Dic149—21no—
166Dic149—23no—
167Dic149—23no—
168Dic149—21no—
169Dic149—23no—
170C2×C149C22×#4111yescharacteristic
171C2×C149C2×2−1+411yescharacteristic
172C2×C149#58511yescharacteristic
173D149—22no—
174D149—41no—
175D149—43no—
176C149#351611yescharacteristic, complemented, Sylow
177#3516—1491nomaximal, Sylow
178#557D14911yescharacteristic, complemented
179#556—1493no—
180#554—1493no—
181#553—1493no—
182#552—1493no—
183D4×Q8—1491no—
184#548—1491no—
185#548—1491no—
186#542—1493no—
187#541—1493no—
188#529—1493no—
189#529—1493no—
190#520—1491no—
191#519—1491no—
192C2×C4×Q8—1491no—
193C2×#41—1491no—
194C22×Q8—1491no—
195C22×Q8—1491no—
196C4⋊Q8—1493no—
197C4⋊Q8C2×D14913yes—
198#126—1496no—
199#126—1493no—
200#126—1493no—
201#126—1496no—
202#125—1493no—
203#125—1496no—
204#125C2×D14913yes—
205#125C2×D14913yes—
206#124—1493no—
207#124—1493no—
208#124—1493no—
209#123—1491no—
210#123—1493no—
211#123—1496no—
212C22⋊Q8—1493no—
213C22⋊Q8—1496no—
214C22⋊Q8—1493no—
215C22⋊Q8—1493no—
216C22⋊Q8—1493no—
217C22⋊Q8—1493no—
218#121—1493no—
219#121—1493no—
220#121—1493no—
221C4×Q8—1491no—
222C4×Q8—1493no—
223C4×Q8—1493no—
224C4×Q8—1493no—
225C4×Q8—1493no—
226C4×Q8C2×D14911yescharacteristic
227C4×Q8C2×D14911yescharacteristic
228C4×Q8C2×D14911yescharacteristic
229C4×Q8C2×D14913yes—
230C4×D4—1493no—
231C4×D4—1496no—
232C4×D4—1493no—
233C4×D4—1493no—
234C4×D4—1493no—
235C4×D4—1493no—
236#117—1493no—
237#117—1496no—
238#117—1493no—
239#117—1493no—
240C2×C4⋊C4—1493no—
241C2×C4⋊C4—1493no—
242C2×C42—1493no—
243#41—2984no—
244C2×Q8—2984no—
245C2×Q8—2984no—
246C2×Q8—1493no—
247C2×Q8—1493no—
248C2×Q8C22×D14913yes—
249C2×Q8C22×D14911yescharacteristic
250C2×D4—1493no—
251C2×D4—1491no—
252C2×D4—1493no—
253C2×D4—1493no—
254C22×C4—1491no—
255C22×C4—1493no—
256C22×C4—1493no—
257C22×C4—1493no—
258C22×C4—1493no—
259C22×C4—1496no—
260C4⋊C4—1493no—
261C4⋊C4—1493no—
262C4⋊C4—1496no—
263C4⋊C4—1491no—
264C4⋊C4—1493no—
265C4⋊C4—1493no—
266C4⋊C4—1496no—
267C4⋊C4—1493no—
268C4⋊C4—1493no—
269C4⋊C4—1493no—
270C4⋊C4C22×D14913yes—
271C4⋊C4C22×D14913yes—
272C4⋊C4C22×D14913yes—
273C4⋊C4C22×D14913yes—
274C4⋊C4C22×D14916yes—
275C4⋊C4C22×D14911yescharacteristic
276C4⋊C4C22×D14913yes—
277C22⋊C4—1493no—
278C22⋊C4—1491no—
279C22⋊C4—1493no—
280C22⋊C4—1493no—
281C22⋊C4—1493no—
282C22⋊C4—1496no—
283C22⋊C4—1493no—
284C22⋊C4—1491no—
285C22⋊C4—1491no—
286C22⋊C4—1496no—
287C22⋊C4—1493no—
288C22⋊C4—1493no—
289C42—1493no—
290C42—1493no—
291C42—1493no—
292C42—1496no—
293C42C22×D14913yes—
294C42C22×D14913yes—
295C42C22×D14913yes—
296C23—1491no—
297C23—1493no—
298C23—1491no—
299Q8—43no—
300Q8—2986no—
301Q8—2986no—
302Q8#1455873714yes—
303D4—2986no—
304D4—2982no—
305D4—2986no—
306D4—5963no—
307C2×C4—2982no—
308C2×C4—2986no—
309C2×C4—2986no—
310C2×C4—2986no—
311C2×C4—5966no—
312C2×C4—5963no—
313C2×C4—1491no—
314C2×C4—1493no—
315C2×C4—1493no—
316C2×C4—1493no—
317C2×C4—1491no—
318C2×C4C23×D14911yescharacteristic
319C2×C4C23×D14913yes—
320C2×C4C23×D14913yes—
321C2×C4C23×D14911yescharacteristic
322C2×C4C23×D14911yescharacteristic
323C2×C4C23×D14913yes—
324C2×C4C23×D14913yes—
325C22—2981no—
326C22—2981no—
327C22—2981no—
328C22—2983no—
329C22—2981no—
330C22—2983no—
331C22—2983no—
332C22—2981no—
333C22—1492no—
334C22C24×D14911yescharacteristic
335C4—21no—
336C4—23no—
337C4—21no—
338C4—23no—
339C4—23no—
340C4—21no—
341C4—2981no—
342C4—2983no—
343C4—2983no—
344C4—2981no—
345C4—2983no—
346C4C2×#1455873716yes—
347C2—2982no—
348C2—5961no—
349C2—5963no—
350C2C22×#1455873711yescharacteristic
351C22−1+4×D14911yescharacteristic
352C2#2375046111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
Fitting#557×C149C2
hypercenter#557D149
hypocenterC149#3516
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #23750348C2semidirect
2. #23750336C2semidirect
3. #23750328C2semidirect
4. #23750313C2semidirect
5. Q8×D596C2semidirect
6. #23750287C2semidirect
7. #23750287C2semidirect
8. #23750249C2semidirect
9. #23750238C2semidirect
10. #23750162C2semidirect
11. #23750160C2semidirect
12. #23750107C2semidirect
13. #23750101C2semidirect
14. C4×Q8×D149C2semidirect
15. #557×C149C2semidirect
176. C149#3516semidirect
178. #557D149semidirect

Maximal subgroups

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

subgroupindex
#237503482
#237503362
#237503282
#237503132
Q8×D5962
#237502872
#237502492
#237502382
#237501622
#237501602
#237501072
#237501012
C4×Q8×D1492
#557×C1492
#3516149