Finite Group Databasebeta

← #34058516

Subgroups — #34058516

Order 19072.

988 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750354C211yesmaximal, characteristic, complemented
2#23750351C211yesmaximal, characteristic, complemented
3#23750351C211yesmaximal, characteristic, complemented
4#23750336C213yesmaximal, complemented
5#23750335C213yesmaximal, complemented
6#23750331C213yesmaximal, complemented
7#23750331C213yesmaximal, complemented
8#23750255C213yesmaximal, complemented
9#23750252C213yesmaximal, complemented
10#23750179C213yesmaximal, complemented
11#125×D149C213yesmaximal, complemented
12#23750107C211yesmaximal, characteristic
13#23750107C211yesmaximal, characteristic
14C4×#14558737C211yesmaximal, characteristic
15#557×C149C211yesmaximal, characteristic, complemented
16C2×#14558737C2211yescharacteristic
17#18011911C2213yes—
18#18011910C2213yes—
19#18011910C2213yes—
20#18011910C2213yes—
21#18011909C2213yes—
22#18011903C2213yes—
23#18011903C2213yes—
24#18011902C2213yes—
25#18011902C2213yes—
26#18011902C2213yes—
27#18011902C2213yes—
28#18011901C2213yes—
29#18011901C2216yes—
30#18011901C2211yescharacteristic
31#18011891C2213yes—
32#18011891C2216yes—
33#18011889C2213yes—
34#18011889C2213yes—
35#18011889C2216yes—
36#18011889C2216yes—
37#18011887C2213yes—
38#18011887C2216yes—
39Q8×Dic149C2211yescharacteristic
40Q8×Dic149C2211yescharacteristic
41Q8×Dic149C2211yescharacteristic
42C4×Dic298C2213yes—
43C4×Dic298C2213yes—
44C4×Dic298C2213yes—
45C4×D596C2213yes—
46C4×D596C2213yes—
47C4×D596C2213yes—
48#18011873C2213yes—
49#18011873C2213yes—
50#18011873C2213yes—
51#18011869C2213yes—
52#18011868C2213yes—
53#18011868C2213yes—
54C4⋊C4×D149C2213yes—
55C4⋊C4×D149C2213yes—
56C4⋊C4×D149C2216yes—
57C4⋊C4×D149C2216yes—
58C42×D149C2213yes—
59C4⋊Q8×C149C2213yes—
60#125×C149C2213yes—
61#125×C149C2213yes—
62C4×Q8×C149C2213yes—
63C4×Q8×C149C2211yescharacteristic
64C4×Q8×C149C2211yescharacteristic
65C4×Q8×C149C2211yescharacteristic
66#14558737Q818yescomplemented
67C2×Dic298C2313yes—
68C2×Dic298C2313yes—
69C2×Dic298C2313yes—
70C2×D596C2313yes—
71C2×C4×D149C2313yes—
72C2×C4×D149C2313yes—
73C2×C4×D149C2316yes—
74C2×C4×D149C2316yes—
75C2×C4×D149C2313yes—
76C4⋊Dic149C2313yes—
77C4⋊Dic149C2313yes—
78C4⋊Dic149C2313yes—
79C4⋊Dic149C2313yes—
80C4⋊Dic149C2316yes—
81C4⋊Dic149C2313yes—
82C4⋊Dic149C2311yescharacteristic
83#14558715C2311yescharacteristic
84#14558715C2313yes—
85#14558715C2313yes—
86#14558715C2313yes—
87#14558715C2311yescharacteristic
88#14558715C2313yes—
89#14558715C2316yes—
90#14558715C2316yes—
91#14558715C2313yes—
92#14558715C2313yes—
93#14558715C2311yescharacteristic
94#14558715C2311yescharacteristic
95#14558713C2313yes—
96#14558713C2313yes—
97#14558713C2313yes—
98#14558713C2313yes—
99#14558713C2316yes—
100C4×Dic149C2313yes—
101C4×Dic149C2313yes—
102C4×Dic149C2313yes—
103C4×Dic149C2311yescharacteristic
104C4×Dic149C2313yes—
105C2×Q8×C149C2313yes—
106C2×Q8×C149C2311yescharacteristic
107C4⋊C4×C149C2313yes—
108C4⋊C4×C149C2313yes—
109C4⋊C4×C149C2313yes—
110C4⋊C4×C149C2311yescharacteristic
111C4⋊C4×C149C2313yes—
112C4⋊C4×C149C2313yes—
113C4⋊C4×C149C2316yes—
114C42×C149C2313yes—
115C42×C149C2313yes—
116C42×C149C2313yes—
117C22×D149C2413yes—
118Dic298—43no—
119Dic298—43no—
120Dic298—43no—
121D596C2×Q8112yes—
122C2×Dic149C2411yescharacteristic
123C2×Dic149C2413yes—
124C2×Dic149C2413yes—
125C2×Dic149C2413yes—
126C2×Dic149C2411yescharacteristic
127C2×Dic149C2411yescharacteristic
128C2×Dic149C2411yescharacteristic
129C4×D149—43no—
130C4×D149—46no—
131C4×D149—46no—
132C4×D149—43no—
133C4×D149C2×Q8112yes—
134Q8×C149—43no—
135Q8×C149C2×Q814yes—
136C2×C4×C149C2411yescharacteristic
137C2×C4×C149C2413yes—
138C2×C4×C149C2413yes—
139C2×C4×C149C2413yes—
140C2×C4×C149C2411yescharacteristic
141C2×C4×C149C2411yescharacteristic
142C2×C4×C149C2413yes—
143C22×C149C2511yescharacteristic
144C2×D149—23no—
145C2×D149—23no—
146C2×D149C22×Q816yes—
147C4×C149—21no—
148C4×C149—23no—
149C4×C149—23no—
150C4×C149—21no—
151C4×C149—23no—
152C4×C149—21no—
153C4×C149C22×Q816yes—
154Dic149—21no—
155Dic149—23no—
156Dic149—23no—
157Dic149—23no—
158Dic149—21no—
159Dic149—21no—
160Dic149C22×Q812yes—
161C2×C149C23×Q811yescharacteristic
162C2×C149#58511yescharacteristic
163C2×C149#58511yescharacteristic
164D149—26no—
165C149#350411yescharacteristic, complemented, Sylow
166#3504—1491nomaximal, Sylow
167#557—1491no—
168#557—1491no—
169#557—1491no—
170#557D14911yescharacteristic, complemented
171#554—1493no—
172#554—1493no—
173#554—1493no—
174#554—1493no—
175#544—1493no—
176#544—1493no—
177#533—1493no—
178C2×#125—1493no—
179#520—1491no—
180#520—1491no—
181C4×#41—1491no—
182C2×#41—1491no—
183C4⋊Q8—1493no—
184C4⋊Q8—1493no—
185C4⋊Q8—1493no—
186C4⋊Q8—1493no—
187C4⋊Q8—1493no—
188C4⋊Q8C2×D14913yes—
189#125—1493no—
190#125—1496no—
191#125—1493no—
192#125—1493no—
193#125—1491no—
194#125—1493no—
195#125—1493no—
196#125—1493no—
197#125—1493no—
198#125C2×D14913yes—
199#125C2×D14913yes—
200C22⋊Q8—1493no—
201C22⋊Q8—1496no—
202C22⋊Q8—1493no—
203C22⋊Q8—1493no—
204C22⋊Q8—1493no—
205C22⋊Q8—1496no—
206C22⋊Q8—1496no—
207C22⋊Q8—1496no—
208C4×Q8—1491no—
209C4×Q8—1491no—
210C4×Q8—1491no—
211C4×Q8—1493no—
212C4×Q8—1493no—
213C4×Q8—1493no—
214C4×Q8C2×D14911yescharacteristic
215C4×Q8C2×D14911yescharacteristic
216C4×Q8C2×D14911yescharacteristic
217C4×Q8C2×D14913yes—
218C4×D4—1493no—
219C4×D4—1493no—
220C4×D4—1493no—
221C4×D4—1493no—
222C4×D4—1493no—
223C4×D4—1493no—
224#117—1493no—
225#117—1493no—
226#117—1493no—
227C2×C4⋊C4—1493no—
228C2×C4⋊C4—1493no—
229C2×C4⋊C4—1496no—
230C2×C4⋊C4—1496no—
231C2×C42—1493no—
232#41—1498no—
233C2×Q8—1493no—
234C2×Q8—1493no—
235C2×Q8—1493no—
236C2×Q8C22×D14913yes—
237C2×Q8C22×D14911yescharacteristic
238C2×D4—1493no—
239C22×C4—1493no—
240C22×C4—1493no—
241C22×C4—1496no—
242C22×C4—1496no—
243C22×C4—1493no—
244C4⋊C4—1493no—
245C4⋊C4—1493no—
246C4⋊C4—1493no—
247C4⋊C4—1493no—
248C4⋊C4—1493no—
249C4⋊C4—1493no—
250C4⋊C4—1493no—
251C4⋊C4—1496no—
252C4⋊C4—1496no—
253C4⋊C4—1496no—
254C4⋊C4—1491no—
255C4⋊C4—1491no—
256C4⋊C4—1493no—
257C4⋊C4—1491no—
258C4⋊C4—1491no—
259C4⋊C4—1491no—
260C4⋊C4—1493no—
261C4⋊C4—1493no—
262C4⋊C4—1493no—
263C4⋊C4C22×D14913yes—
264C4⋊C4C22×D14913yes—
265C4⋊C4C22×D14913yes—
266C4⋊C4C22×D14916yes—
267C4⋊C4C22×D14913yes—
268C4⋊C4C22×D14911yescharacteristic
269C4⋊C4C22×D14913yes—
270C22⋊C4—1493no—
271C22⋊C4—1493no—
272C22⋊C4—1496no—
273C22⋊C4—1493no—
274C22⋊C4—1493no—
275C42—1491no—
276C42—1493no—
277C42—1493no—
278C42—1493no—
279C42—1493no—
280C42C22×D14913yes—
281C42C22×D14913yes—
282C42C22×D14913yes—
283C23—1493no—
284Q8—43no—
285Q8—5963no—
286Q8—5963no—
287Q8—5963no—
288Q8Q8×D14914yes—
289D4—14912no—
290C2×C4—5966no—
291C2×C4—5963no—
292C2×C4—5963no—
293C2×C4—5966no—
294C2×C4—1491no—
295C2×C4—1491no—
296C2×C4—1491no—
297C2×C4—1493no—
298C2×C4—1493no—
299C2×C4—1493no—
300C2×C4—1491no—
301C2×C4—14912no—
302C2×C4C23×D14911yescharacteristic
303C2×C4C23×D14913yes—
304C2×C4C23×D14911yescharacteristic
305C2×C4C23×D14911yescharacteristic
306C2×C4C23×D14913yes—
307C2×C4C23×D14913yes—
308C2×C4C23×D14913yes—
309C22—2983no—
310C22—2983no—
311C22—1496no—
312C22C24×D14911yescharacteristic
313C4—23no—
314C4—23no—
315C4—21no—
316C4—21no—
317C4—23no—
318C4—21no—
319C4—2981no—
320C4—2983no—
321C4—2981no—
322C4—2983no—
323C4—2983no—
324C4—2981no—
325C4—1492no—
326C4C2×Q8×D14916yes—
327C2—2986no—
328C2C22×Q8×D14911yescharacteristic
329C2#2375046111yescharacteristic
330C2#2375046211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
Fitting#557×C149C2
prime radicalC2×#14558737C22
hypercenter#557D149
hypocenterC149#3504
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #23750354C2semidirect
2. #23750351C2semidirect
3. #23750351C2semidirect
4. #23750336C2semidirect
5. #23750335C2semidirect
6. #23750331C2semidirect
7. #23750331C2semidirect
8. #23750255C2semidirect
9. #23750252C2semidirect
10. #23750179C2semidirect
11. #125×D149C2semidirect
15. #557×C149C2semidirect
66. #14558737Q8semidirect
165. C149#3504semidirect
170. #557D149semidirect

Maximal subgroups

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

subgroupindex
#237503542
#237503512
#237503362
#237503352
#237503312
#237502552
#237502522
#237501792
#125×D1492
#237501072
C4×#145587372
#557×C1492
#3504149