Finite Group Databasebeta

← #34047815

Subgroups — #34047815

Order 19072.

496 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23749524C211yesmaximal, characteristic, complemented
2#23749518C211yesmaximal, characteristic
3#23749518C211yesmaximal, characteristic
4#23749518C211yesmaximal, characteristic
5#23749511C211yesmaximal, characteristic, complemented
6#23749511C211yesmaximal, characteristic, complemented
7#23749511C211yesmaximal, characteristic, complemented
8#23749500C211yesmaximal, characteristic
9C4⋊C4×Dic149C211yesmaximal, characteristic, complemented
10C4⋊C4×Dic149C211yesmaximal, characteristic, complemented
11C4×C4⋊Dic149C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic149C211yesmaximal, characteristic
13C22⋊C4×Dic149C211yesmaximal, characteristic
14#23749471C211yesmaximal, characteristic, complemented
15C2×#126×C149C211yesmaximal, characteristic
16C23×Dic149C2211yescharacteristic
17C2×C4⋊Dic149C2211yescharacteristic
18C2×C4⋊Dic149C2211yescharacteristic
19C2×C4⋊Dic149C2211yescharacteristic
20C2×C4⋊Dic149C2211yescharacteristic
21C2×C4⋊Dic149C2211yescharacteristic
22C2×C22⋊Dic149C2211yescharacteristic
23C2×C22⋊Dic149C2211yescharacteristic
24C2×C22⋊Dic149C2211yescharacteristic
25C2×C4×Dic149C2211yescharacteristic
26C2×C4×Dic149C2211yescharacteristic
27C2×C4×Dic149C2211yescharacteristic
28C2×C4×Dic149C2211yescharacteristic
29C2×C4×Dic149C2211yescharacteristic
30C2×C4×Dic149C2211yescharacteristic
31C2×C4×Dic149C2211yescharacteristic
32#18011818C2211yescharacteristic
33#18011818C2211yescharacteristic
34#18011818C2211yescharacteristic
35#18011818C2211yescharacteristic
36#18011818C2211yescharacteristic
37#18011818C2211yescharacteristic
38#18011818C2211yescharacteristic
39#18011818C2211yescharacteristic
40#18011818C2211yescharacteristic
41#18011818C2211yescharacteristic
42#18011818C2211yescharacteristic
43#18011818C2211yescharacteristic
44#126×C149C418yescomplemented
45C2×C4⋊C4×C149C2211yescharacteristic
46C2×C4⋊C4×C149C2211yescharacteristic
47C2×C4⋊C4×C149C2211yescharacteristic
48C2×C22⋊C4×C149C2211yescharacteristic
49C2×C22⋊C4×C149C2211yescharacteristic
50C2×C22⋊C4×C149C2211yescharacteristic
51C2×C42×C149C2211yescharacteristic
52C22×Dic149—22no—
53C22×Dic149—22no—
54C22×Dic149—41no—
55C22×Dic149C2311yescharacteristic
56C22×Dic149C2311yescharacteristic
57C22×Dic149C2311yescharacteristic
58C22×Dic149C2311yescharacteristic
59C22×Dic149C2311yescharacteristic
60C22×Dic149C2311yescharacteristic
61C22×Dic149C2311yescharacteristic
62C22×Dic149C2311yescharacteristic
63C4⋊Dic149—22no—
64C4⋊Dic149—22no—
65C4⋊Dic149—22no—
66C4⋊Dic149—22no—
67C4⋊Dic149—41no—
68C22⋊Dic149—22no—
69C22⋊Dic149—22no—
70C22⋊Dic149—41no—
71C4×Dic149—22no—
72C4×Dic149—22no—
73C4×Dic149—22no—
74C4×Dic149—22no—
75C4×Dic149—41no—
76C4×Dic149—41no—
77C4×Dic149—41no—
78C24×C149C2311yescharacteristic
79C22×C4×C149C2311yescharacteristic
80C22×C4×C149C2311yescharacteristic
81C22×C4×C149C2311yescharacteristic
82C22×C4×C149C2311yescharacteristic
83C22×C4×C149C2311yescharacteristic
84C22×C4×C149C2311yescharacteristic
85C4⋊C4×C149C2×C414yescomplemented
86C4⋊C4×C149C2×C414yescomplemented
87C4⋊C4×C149C2×C414yescomplemented
88C22⋊C4×C149C2×C414yes—
89C22⋊C4×C149C2×C414yes—
90C22⋊C4×C149C2×C414yes—
91C42×C149C2×C414yescomplemented
92C2×Dic149—21no—
93C2×Dic149—21no—
94C2×Dic149—21no—
95C2×Dic149—21no—
96C2×Dic149—21no—
97C2×Dic149—21no—
98C2×Dic149—21no—
99C2×Dic149—21no—
100C2×Dic149—21no—
101C2×Dic149—21no—
102C2×Dic149—21no—
103C2×Dic149—21no—
104C2×Dic149—21no—
105C2×Dic149—21no—
106C2×Dic149—21no—
107C2×Dic149—21no—
108C2×Dic149—21no—
109C2×Dic149—21no—
110C2×Dic149—21no—
111C2×Dic149—21no—
112C2×Dic149—44no—
113C2×Dic149#4112yes—
114C2×Dic149#4112yes—
115C2×Dic149#4112yes—
116C2×Dic149#4112yes—
117C23×C149—21no—
118C23×C149—21no—
119C23×C149—21no—
120C23×C149—21no—
121C23×C149—21no—
122C23×C149—21no—
123C23×C149C22×C412yes—
124C23×C149C2411yescharacteristic
125C2×C4×C149—21no—
126C2×C4×C149—21no—
127C2×C4×C149—21no—
128C2×C4×C149—21no—
129C2×C4×C149—21no—
130C2×C4×C149—21no—
131C2×C4×C149—21no—
132C2×C4×C149—21no—
133C2×C4×C149—21no—
134C2×C4×C149—21no—
135C2×C4×C149—21no—
136C2×C4×C149—21no—
137C2×C4×C149C22×C412yes—
138C2×C4×C149C22×C412yes—
139C2×C4×C149C22×C412yes—
140C2×C4×C149C22×C412yes—
141C2×C4×C149C22×C412yes—
142C2×C4×C149C22×C412yes—
143C22×C149—22no—
144C22×C149—22no—
145C22×C149—22no—
146C22×C149—41no—
147C22×C149—41no—
148C22×C149—41no—
149C22×C149—41no—
150C22×C149C23×C411yescharacteristic
151C22×C149C2×#4111yescharacteristic
152C22×C149C2×#4111yescharacteristic
153C22×C149C2×#4111yescharacteristic
154C22×C149C2×#4111yescharacteristic
155C22×C149H(5,𝔽2)11yescharacteristic
156C22×C1492−1+411yescharacteristic
157C4×C149—22no—
158C4×C149—22no—
159C4×C149—22no—
160C4×C149—22no—
161C4×C149—22no—
162C4×C149—22no—
163Dic149—22no—
164Dic149—22no—
165Dic149—22no—
166Dic149—22no—
167Dic149—41no—
168Dic149—41no—
169Dic149—41no—
170Dic149—41no—
171C2×C149—42no—
172C2×C149C4×#4111yescharacteristic
173C2×C149C4×#4111yescharacteristic
174C2×C149#52011yescharacteristic
175C2×C149#55111yescharacteristic
176C2×C149#55211yescharacteristic
177C2×C149#55311yescharacteristic
178C2×C149#55611yescharacteristic
179C149#237311yescharacteristic, complemented, Sylow
180#2373—1491nomaximal, Sylow
181C2×#126D14911yescharacteristic
182#389—1491no—
183#388—1491no—
184#388—1491no—
185#388—1491no—
186#387—1491no—
187#387—1491no—
188#387—1491no—
189#385—1491no—
190C4×C4⋊C4—1491no—
191C4×C4⋊C4—1491no—
192C4×C4⋊C4—1491no—
193C4×C22⋊C4—1491no—
194C4×C22⋊C4—1491no—
195#376—1491no—
196C23×C4—1491no—
197#126Dic14918yescomplemented
198C2×C4⋊C4—1491no—
199C2×C4⋊C4—1491no—
200C2×C4⋊C4—1491no—
201C2×C4⋊C4—1491no—
202C2×C4⋊C4—1491no—
203C2×C4⋊C4C2×D14911yescharacteristic
204C2×C4⋊C4C2×D14911yescharacteristic
205C2×C4⋊C4C2×D14911yescharacteristic
206C2×C22⋊C4—1491no—
207C2×C22⋊C4—1491no—
208C2×C22⋊C4—1491no—
209C2×C22⋊C4C2×D14911yescharacteristic
210C2×C22⋊C4C2×D14911yescharacteristic
211C2×C22⋊C4C2×D14911yescharacteristic
212C2×C42—1491no—
213C2×C42—1491no—
214C2×C42—1491no—
215C2×C42—1491no—
216C2×C42—1491no—
217C2×C42—1491no—
218C2×C42—1491no—
219C2×C42C2×D14911yescharacteristic
220#95—1491no—
221#95—1491no—
222#95—1491no—
223#95—1491no—
224#95—1491no—
225#95—1491no—
226#95—1491no—
227#95—1491no—
228#95—1491no—
229#95—1491no—
230#95—1491no—
231#95—1491no—
232C24C22×D14911yescharacteristic
233C22×C4—2982no—
234C22×C4—2982no—
235C22×C4—5961no—
236C22×C4—1491no—
237C22×C4—1491no—
238C22×C4—1491no—
239C22×C4—1491no—
240C22×C4—1491no—
241C22×C4—1491no—
242C22×C4—1491no—
243C22×C4—1491no—
244C22×C4C22×D14911yescharacteristic
245C22×C4C22×D14911yescharacteristic
246C22×C4C22×D14911yescharacteristic
247C22×C4C22×D14911yescharacteristic
248C22×C4C22×D14911yescharacteristic
249C22×C4C22×D14911yescharacteristic
250C4⋊C4—2982no—
251C4⋊C4—2982no—
252C4⋊C4—2982no—
253C4⋊C4—2982no—
254C4⋊C4—5961no—
255C4⋊C4C2×Dic14914yescomplemented
256C4⋊C4C2×Dic14914yescomplemented
257C4⋊C4C2×Dic14914yescomplemented
258C22⋊C4—2982no—
259C22⋊C4—2982no—
260C22⋊C4—5961no—
261C22⋊C4C2×Dic14914yes—
262C22⋊C4C2×Dic14914yes—
263C22⋊C4C2×Dic14914yes—
264C42—2982no—
265C42—2982no—
266C42—2982no—
267C42—2982no—
268C42—5961no—
269C42—5961no—
270C42—5961no—
271C42C2×Dic14914yescomplemented
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C22×Dic14912yes—
279C23C23×D14911yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—2981no—
293C2×C4—2981no—
294C2×C4—2981no—
295C2×C4—2981no—
296C2×C4—2981no—
297C2×C4—2981no—
298C2×C4—2981no—
299C2×C4—2981no—
300C2×C4—2981no—
301C2×C4—2981no—
302C2×C4—2981no—
303C2×C4—2981no—
304C2×C4—2981no—
305C2×C4—2981no—
306C2×C4—2981no—
307C2×C4—2981no—
308C2×C4—2981no—
309C2×C4—2981no—
310C2×C4—2981no—
311C2×C4—2981no—
312C2×C4—5964no—
313C2×C4—1492no—
314C2×C4—1492no—
315C2×C4—1492no—
316C2×C4—1492no—
317C2×C4C22×Dic14912yes—
318C2×C4C22×Dic14912yes—
319C2×C4C22×Dic14912yes—
320C2×C4C22×Dic14912yes—
321C2×C4C22×Dic14912yes—
322C2×C4C22×Dic14912yes—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22C23×Dic14911yescharacteristic
331C22#41×D14911yescharacteristic
332C22#41×D14911yescharacteristic
333C22#41×D14911yescharacteristic
334C22#41×D14911yescharacteristic
335C22#1801195711yescharacteristic
336C22#1801195811yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—22no—
342C4—22no—
343C4—2982no—
344C4—2982no—
345C4—2982no—
346C4—2982no—
347C4—5961no—
348C4—5961no—
349C4—5961no—
350C4—5961no—
351C2—42no—
352C2#41×Dic14911yescharacteristic
353C2#41×Dic14911yescharacteristic
354C2#2375010811yescharacteristic
355C2#2375030011yescharacteristic
356C2#2375031111yescharacteristic
357C2#2375032611yescharacteristic
358C2#2375034211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D149
commutatorC22×C149C23×C4
FittingC2×#126×C149C2
socle, cosocleC23×C149C24
prime radicalC24×C149C23
hypercenterC2×#126D149
hypocenterC149#2373
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #23749524C2semidirect
5. #23749511C2semidirect
6. #23749511C2semidirect
7. #23749511C2semidirect
9. C4⋊C4×Dic149C2semidirect
10. C4⋊C4×Dic149C2semidirect
11. C4×C4⋊Dic149C2semidirect
14. #23749471C2semidirect
44. #126×C149C4semidirect
85. C4⋊C4×C149C2×C4semidirect
86. C4⋊C4×C149C2×C4semidirect
87. C4⋊C4×C149C2×C4semidirect
91. C42×C149C2×C4semidirect
179. C149#2373semidirect
197. #126Dic149semidirect
255. C4⋊C4C2×Dic149semidirect
256. C4⋊C4C2×Dic149semidirect
257. C4⋊C4C2×Dic149semidirect
271. C42C2×Dic149semidirect

Maximal subgroups

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

subgroupindex
#237495242
#237495182
#237495112
#237495002
C4⋊C4×Dic1492
C4×C4⋊Dic1492
C22⋊C4×Dic1492
#237494712
C2×#126×C1492
#2373149