Finite Group Databasebeta

← #34058770

Subgroups — #34058770

Order 19072.

1104 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#23750358C213yesmaximal, complemented
2Q8×Dic298C211yesmaximal, characteristic, complemented
3#23750330C213yesmaximal, complemented
4#23750294C211yesmaximal, characteristic, complemented
5#23750287C211yesmaximal, characteristic, complemented
6#23750282C211yesmaximal, characteristic, complemented
7#23750280C213yesmaximal, complemented
8#23750276C213yesmaximal, complemented
9#23750246C213yesmaximal, complemented
10#23750240C213yesmaximal, complemented
11#23750172C213yesmaximal, complemented
12#23750153C213yesmaximal, complemented
13#23750107C211yesmaximal, characteristic, complemented
14C4×Q8×D149C211yesmaximal, characteristic, complemented
15#520×C149C211yesmaximal, characteristic, complemented
16C2×#14558737C2211yescharacteristic
17C2×#14558736C2216yes—
18C2×Q8×D149C2211yescharacteristic
19#18011911C2213yes—
20#18011910C2213yes—
21#18011910C2213yes—
22#18011907C2213yes—
23#18011906C2213yes—
24#18011904C2213yes—
25#18011901C2213yes—
26#18011899C2213yes—
27#18011898C2213yes—
28#18011897C2213yes—
29#18011897C2216yes—
30#18011896C2213yes—
31#18011894C2216yes—
32#18011893C2213yes—
33#18011893C2216yes—
34#18011891C2213yes—
35#18011890C2216yes—
36#18011890C2213yes—
37#18011889C2211yescharacteristic
38#18011889C2216yes—
39#18011886C2213yes—
40#18011885C2213yes—
41#18011884C2213yes—
42#18011882C2213yes—
43#18011882C2213yes—
44Q8×Dic149C2211yescharacteristic
45Q8×Dic149C2211yescharacteristic
46C4×Dic298C2213yes—
47C4×Dic298C2213yes—
48#18011876C2213yes—
49C4×D596C2213yes—
50C4×D596C2213yes—
51#18011873C2213yes—
52C4×C22⋊D149C2213yes—
53C4×C22⋊D149C2213yes—
54C4×C22⋊D149C2216yes—
55#18011868C2213yes—
56C4⋊C4×D149C2213yes—
57C4⋊C4×D149C2213yes—
58C42×D149C2213yes—
59C2×#41×C149C2211yescharacteristic
60C4×Q8×C149C2211yescharacteristic
61C4×Q8×C149C2211yescharacteristic
62C4×D4×C149C2213yes—
63C4×D4×C149C2213yes—
64#117×C149C2213yes—
65C2×C4⋊C4×C149C2213yes—
66#14558737—24no—
67#14558736—412no—
68C2×Dic298C2313yes—
69C2×Dic298C2311yescharacteristic
70C2×Dic298C2316yes—
71Q8×D149—24no—
72C2×D596C2313yes—
73C2×D596C2313yes—
74C2×C22⋊D149C2313yes—
75C2×C22⋊D149C2313yes—
76C2×C22⋊D149C2316yes—
77C2×C4×D149C2313yes—
78C2×C4×D149C2313yes—
79C2×C4×D149C2313yes—
80C2×C4×D149C2311yescharacteristic
81C2×C4×D149C2316yes—
82C4⋊Dic149C2313yes—
83C4⋊Dic149C2313yes—
84C4⋊Dic149C2313yes—
85C4⋊Dic149C2313yes—
86#14558715C2311yescharacteristic
87#14558715C2313yes—
88#14558715C2313yes—
89#14558715C2313yes—
90#14558715C2313yes—
91#14558715C2316yes—
92#14558715C2311yescharacteristic
93C22⋊Dic149C2313yes—
94C22⋊Dic149C2313yes—
95C22⋊Dic149C2316yes—
96#14558713C2313yes—
97#14558713C2313yes—
98#14558713C2313yes—
99#14558713C2313yes—
100#14558713C2311yescharacteristic
101#14558713C2316yes—
102#14558713C2311yescharacteristic
103C4×Dic149C2313yes—
104C4×Dic149C2313yes—
105C4×Dic149C2313yes—
106C4×Dic149C2313yes—
107#41×C149—24no—
108C2×Q8×C149C2311yescharacteristic
109C2×D4×C149C2313yes—
110C22×C4×C149C2313yes—
111C22×C4×C149C2316yes—
112C4⋊C4×C149C2311yescharacteristic
113C4⋊C4×C149C2313yes—
114C4⋊C4×C149C2313yes—
115C4⋊C4×C149C2313yes—
116C22⋊C4×C149C2313yes—
117C22⋊C4×C149C2313yes—
118C42×C149C2313yes—
119C42×C149C2313yes—
120C22×D149C2413yes—
121C22×D149C2411yescharacteristic
122Dic298—26no—
123Dic298—22no—
124Dic298—46no—
125D596—26no—
126D596—26no—
127C22⋊D149—26no—
128C22⋊D149—26no—
129C22⋊D149—46no—
130C2×Dic149C2411yescharacteristic
131C2×Dic149C2413yes—
132C2×Dic149C2413yes—
133C2×Dic149C2413yes—
134C2×Dic149C2411yescharacteristic
135C2×Dic149C2411yescharacteristic
136C4×D149—26no—
137C4×D149—26no—
138C4×D149—26no—
139C4×D149—22no—
140C4×D149—46no—
141C23×C149C2413yes—
142Q8×C149#4114yescomplemented
143D4×C149—26no—
144C2×C4×C149—26no—
145C2×C4×C149—46no—
146C2×C4×C149C2413yes—
147C2×C4×C149C2411yescharacteristic
148C2×C4×C149C2413yes—
149C2×C4×C149C2411yescharacteristic
150C2×C4×C149C2411yescharacteristic
151C2×C4×C149C2413yes—
152C22×C149—23no—
153C22×C149—23no—
154C22×C149—23no—
155C22×C149C2511yescharacteristic
156C2×D149—21no—
157C2×D149—21no—
158C2×D149—23no—
159C2×D149—23no—
160C2×D149—23no—
161C2×D149H(5,𝔽2)12yes—
162C4×C149—21no—
163C4×C149—23no—
164C4×C149—23no—
165C4×C149—21no—
166C4×C149—21no—
167C4×C149C2×#4116yes—
168Dic149—21no—
169Dic149—23no—
170Dic149—23no—
171Dic149—23no—
172Dic149—21no—
173Dic1492−1+412yes—
174C2×C149—43no—
175C2×C149C22×#4111yescharacteristic
176C2×C149C2×H(5,𝔽2)11yescharacteristic
177C2×C149C2×2−1+411yescharacteristic
178D149—22no—
179D149—43no—
180C149#352211yescharacteristic, complemented, Sylow
181#3522—1491nomaximal, Sylow
182#559—1493no—
183Q82—1491no—
184#554—1493no—
185#550—1491no—
186#548—1491no—
187#548—1491no—
188#547—1493no—
189#547—1493no—
190#542—1493no—
191#542—1493no—
192#532—1493no—
193#529—1493no—
194#520—1491no—
195#520D14911yescharacteristic, complemented
196C2×C4×Q8—1491no—
197C2×#41—1491no—
198C2×#41—1496no—
199C2×#41C2×D14911yescharacteristic
200C22×Q8—1491no—
201C4⋊Q8—1493no—
202C4⋊Q8—1493no—
203C4⋊Q8—1493no—
204#127—1493no—
205#126—1493no—
206#126—1493no—
207#125—1493no—
208#124—1493no—
209#124—1493no—
210#124—1496no—
211#124—1493no—
212#123—1496no—
213#123—1493no—
214#123—1493no—
215#123—1496no—
216C22⋊Q8—1496no—
217C22⋊Q8—1493no—
218C22⋊Q8—1491no—
219C22⋊Q8—1496no—
220C22⋊Q8—1493no—
221#121—1493no—
222#121—1493no—
223#121—1493no—
224#121—1493no—
225#121—1493no—
226C4×Q8—1491no—
227C4×Q8—1491no—
228C4×Q8—1493no—
229C4×Q8—1493no—
230C4×Q8—1493no—
231C4×Q8C2×D14911yescharacteristic
232C4×Q8C2×D14911yescharacteristic
233C4×D4—1493no—
234C4×D4—1493no—
235C4×D4—1496no—
236C4×D4—1493no—
237C4×D4—1493no—
238C4×D4—1493no—
239C4×D4C2×D14913yes—
240C4×D4C2×D14913yes—
241#117—1493no—
242#117C2×D14913yes—
243C2×C4⋊C4—1493no—
244C2×C4⋊C4—1493no—
245C2×C4⋊C4C2×D14913yes—
246C2×C42—1493no—
247#41—24no—
248#41—2984no—
249#41—59612no—
250C2×Q8—2984no—
251C2×Q8—1491no—
252C2×Q8—1493no—
253C2×Q8—1496no—
254C2×Q8C22×D14911yescharacteristic
255C2×D4—1493no—
256C2×D4—1493no—
257C2×D4—1496no—
258C2×D4—1493no—
259C2×D4—1493no—
260C2×D4C22×D14913yes—
261C22×C4—1491no—
262C22×C4—1493no—
263C22×C4—1493no—
264C22×C4—1493no—
265C22×C4—1496no—
266C22×C4C22×D14913yes—
267C22×C4C22×D14916yes—
268C4⋊C4—1493no—
269C4⋊C4—1493no—
270C4⋊C4—1493no—
271C4⋊C4—1496no—
272C4⋊C4—1491no—
273C4⋊C4—1491no—
274C4⋊C4—1493no—
275C4⋊C4—1493no—
276C4⋊C4—1493no—
277C4⋊C4—1493no—
278C4⋊C4—1493no—
279C4⋊C4C22×D14913yes—
280C4⋊C4C22×D14913yes—
281C4⋊C4C22×D14913yes—
282C4⋊C4C22×D14911yescharacteristic
283C22⋊C4—1493no—
284C22⋊C4—1496no—
285C22⋊C4—1493no—
286C22⋊C4—1493no—
287C22⋊C4—1493no—
288C22⋊C4—1491no—
289C22⋊C4—1491no—
290C22⋊C4—1493no—
291C22⋊C4—1496no—
292C22⋊C4—1493no—
293C22⋊C4C22×D14913yes—
294C22⋊C4C22×D14913yes—
295C42—1493no—
296C42—1493no—
297C42—1493no—
298C42—1493no—
299C42C22×D14913yes—
300C42C22×D14913yes—
301C23—1493no—
302C23—1491no—
303C23C23×D14913yes—
304Q8—2982no—
305Q8—2986no—
306Q8—5966no—
307Q8#1455873614yescomplemented
308D4—26no—
309D4—2986no—
310D4—2986no—
311D4—2986no—
312D4—2986no—
313D4—5966no—
314C2×C4—26no—
315C2×C4—46no—
316C2×C4—2986no—
317C2×C4—2982no—
318C2×C4—2986no—
319C2×C4—2986no—
320C2×C4—5966no—
321C2×C4—1491no—
322C2×C4—1493no—
323C2×C4—1493no—
324C2×C4—1493no—
325C2×C4—1491no—
326C2×C4—1491no—
327C2×C4C23×D14911yescharacteristic
328C2×C4C23×D14913yes—
329C2×C4C23×D14911yescharacteristic
330C2×C4C23×D14913yes—
331C2×C4C23×D14911yescharacteristic
332C2×C4C23×D14913yes—
333C22—23no—
334C22—23no—
335C22—23no—
336C22—2981no—
337C22—2981no—
338C22—2983no—
339C22—2983no—
340C22—2983no—
341C22—1492no—
342C22C24×D14911yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—21no—
348C4—2981no—
349C4—2983no—
350C4—2983no—
351C4—2983no—
352C4—2981no—
353C4—1492no—
354C4C2×#1455873616yes—
355C2—43no—
356C2—2982no—
357C2—5963no—
358C2C22×#1455873611yescharacteristic
359C2H(5,𝔽2)×D14911yescharacteristic
360C22−1+4×D14911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
Fitting#520×C149C2
hypercenter#520D149
hypocenterC149#3522
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #23750358C2semidirect
2. Q8×Dic298C2semidirect
3. #23750330C2semidirect
4. #23750294C2semidirect
5. #23750287C2semidirect
6. #23750282C2semidirect
7. #23750280C2semidirect
8. #23750276C2semidirect
9. #23750246C2semidirect
10. #23750240C2semidirect
11. #23750172C2semidirect
12. #23750153C2semidirect
13. #23750107C2semidirect
14. C4×Q8×D149C2semidirect
15. #520×C149C2semidirect
142. Q8×C149#41semidirect
180. C149#3522semidirect
195. #520D149semidirect
307. Q8#14558736semidirect

Maximal subgroups

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

subgroupindex
#237503582
Q8×Dic2982
#237503302
#237502942
#237502872
#237502822
#237502802
#237502762
#237502462
#237502402
#237501722
#237501532
#237501072
C4×Q8×D1492
#520×C1492
#3522149