Finite Group Databasebeta

← #34055712

Subgroups — #34055712

Order 19072.

490 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18011907C211yesmaximal, characteristic, complemented
2#23750161C211yesmaximal, characteristic, complemented
3#23750011C212yesmaximal, complemented
4#23749988C212yesmaximal, complemented
5#23749823C212yesmaximal, complemented
6#23749819C212yesmaximal, complemented
7#23749775C211yesmaximal, characteristic, complemented
8#23749761C211yesmaximal, characteristic, complemented
9#23749759C211yesmaximal, characteristic, complemented
10#23749679C211yesmaximal, characteristic, complemented
11#450×C149C211yesmaximal, characteristic, complemented
12C22×D596C2211yescharacteristic
13C22×C22⋊D149—21no—
14C22×C22⋊D149—21no—
15C2×#14558723C2212yescomplemented
16C2×D1192C2212yescomplemented
17C2×#14558721C2212yescomplemented
18#18011907—21no—
19#18011907—21no—
20#18011907C2212yes—
21#18011907C2212yes—
22#18011906—21no—
23#18011902C2211yescharacteristic, complemented
24#18011900C2211yescharacteristic, complemented
25#18011895—21no—
26#18011885C2211yescharacteristic
27Q8×Dic149C2211yescharacteristic, complemented
28#18011873C2211yescharacteristic, complemented
29#18011871—21no—
30#18011869—21no—
31C2×C4×Dic149C2211yescharacteristic
32#18011840C2212yescomplemented
33#18011833C2212yescomplemented
34#18011832C2212yescomplemented
35#18011831C2212yescomplemented
36C22⋊#11924746C2211yescharacteristic
37#18011821C2212yescomplemented
38C22×D4×C149C2211yescharacteristic
39C2×SD16×C149C2212yescomplemented
40C22⋊Q8×C149C2211yescharacteristic
41#102×C149C2212yescomplemented
42C22⋊C8×C149C2211yescharacteristic
43C23×D149—21no—
44C2×D596—21no—
45C2×D596—42no—
46C2×D596C2312yes—
47C2×D596C2311yescharacteristic
48C2×C22⋊D149—22no—
49C2×C22⋊D149—22no—
50C2×C22⋊D149—22no—
51C2×C22⋊D149—22no—
52C2×C22⋊D149—21no—
53C2×C22⋊D149—44no—
54C2×C22⋊D149—44no—
55C22×Dic149D411yescharacteristic
56C22×Dic149D411yescharacteristic
57C2×C4×D149—21no—
58#14558723—42no—
59D1192—42no—
60#14558721—42no—
61C2×#11924746C2312yes—
62C4⋊Dic149—21no—
63C4⋊Dic149—21no—
64C4⋊Dic149C2311yescharacteristic
65#14558715—21no—
66#14558713—21no—
67#14558713—21no—
68#14558713—21no—
69#14558713—21no—
70C4×Dic149—21no—
71C4×Dic149D411yescharacteristic
72C4×Dic149D411yescharacteristic
73C4×Dic149C2311yescharacteristic
74C4×Dic149C2311yescharacteristic
75C24×C149—21no—
76C2×Q8×C149C2311yescharacteristic
77C2×D4×C149—21no—
78C2×D4×C149—42no—
79C2×D4×C149C2312yes—
80C22×C4×C149C2311yescharacteristic
81SD16×C149—42no—
82C2×C8×C149C2312yes—
83C4⋊C4×C149—21no—
84C4⋊C4×C149C2311yescharacteristic
85C22⋊C4×C149—21no—
86C22×D149—21no—
87C22×D149—22no—
88C22×D149—42no—
89C22×D149—42no—
90C22×D149—42no—
91D596—22no—
92D596—22no—
93D596—22no—
94D596—42no—
95C22⋊D149—44no—
96C22⋊D149—44no—
97C22⋊D149—44no—
98C22⋊D149—42no—
99C22⋊D149—44no—
100C2×Dic149—22no—
101C2×Dic149—22no—
102C2×Dic149—21no—
103C2×Dic149C2×D411yescharacteristic
104C2×Dic149C2×D411yescharacteristic
105C2×Dic149C2×D411yescharacteristic
106C2×Dic149C2×D411yescharacteristic
107C4×D149—42no—
108#11924746—22no—
109C23×C149—22no—
110C23×C149—42no—
111C23×C149—42no—
112C23×C149—42no—
113C23×C149C2×D411yescharacteristic
114Q8×C149—22no—
115D4×C149—22no—
116D4×C149—22no—
117D4×C149—42no—
118C2×C4×C149—22no—
119C2×C4×C149—21no—
120C2×C4×C149—21no—
121C2×C4×C149C2×D411yescharacteristic
122C2×C4×C149C2411yescharacteristic
123C8×C149—22no—
124C22×C149—21no—
125C22×C149—21no—
126C22×C149—42no—
127C22×C149—42no—
128C22×C149—42no—
129C22×C149—84no—
130C22×C149AGL(1,ℤ/8)12yes—
131C22×C149C22×D411yescharacteristic
132C2×D149—42no—
133C2×D149—41no—
134C2×D149—41no—
135C2×D149—42no—
136C2×D149—42no—
137C2×D149—41no—
138C2×D149—84no—
139C4×C149—21no—
140C4×C149—41no—
141C4×C149—41no—
142C4×C149H(5,𝔽2)12yes—
143Dic149—21no—
144Dic149—21no—
145Dic149—21no—
146Dic149—41no—
147Dic149AGL(1,ℤ/8)12yes—
148C2×C149—22no—
149C2×C149—82no—
150C2×C149#53511yescharacteristic
151C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
153D149—81no—
154D149—82no—
155C149#321711yescharacteristic, complemented, Sylow
156#3217—1491nomaximal, Sylow
157C2×#127—1491no—
158#529—1491no—
159#496—1492no—
160#490—1492no—
161#460—1492no—
162#459—1492no—
163#450—1491no—
164#450D14911yescharacteristic, complemented
165#447—1491no—
166#447—1491no—
167#432—1491no—
168C22×D4—2981no—
169C22×D4—2981no—
170C22×D4—1491no—
171C22×D4C2×D14911yescharacteristic
172C2×SD16—1492no—
173C2×SD16C2×D14912yescomplemented
174C2×D8—1492no—
175C2×D8—1492no—
176#127—2981no—
177#127—2981no—
178#127—1492no—
179#127—1492no—
180#126—2981no—
181#125—1491no—
182#124—1491no—
183#123—2981no—
184C22⋊Q8C2×D14911yescharacteristic
185#121—1491no—
186C4×Q8—1491no—
187C4×D4—2981no—
188C4×D4—1491no—
189#117—2981no—
190C2×C42—1491no—
191C4⋊C8—1492no—
192#102—1492no—
193#102—1492no—
194#102—1492no—
195#102C2×D14912yescomplemented
196C22⋊C8—1491no—
197C22⋊C8C2×D14911yescharacteristic
198#97—1492no—
199C24—21no—
200C24—2981no—
201C2×Q8C22×D14911yescharacteristic
202C2×D4—21no—
203C2×D4—42no—
204C2×D4—2981no—
205C2×D4—2982no—
206C2×D4—2982no—
207C2×D4—2982no—
208C2×D4—2982no—
209C2×D4—2981no—
210C2×D4—5962no—
211C2×D4—5964no—
212C2×D4—5964no—
213C2×D4—1492no—
214C2×D4—1491no—
215C2×D4C22×D14912yes—
216C22×C4—2981no—
217C22×C4—1491no—
218C22×C4—1491no—
219C22×C4C22×D14911yescharacteristic
220SD16—42no—
221SD16—5962no—
222D8—5962no—
223D8—5962no—
224C2×C8—1492no—
225C2×C8C22×D14912yes—
226C4⋊C4—21no—
227C4⋊C4—2981no—
228C4⋊C4—2981no—
229C4⋊C4—2981no—
230C4⋊C4—1491no—
231C4⋊C4C22×D14911yescharacteristic
232C22⋊C4—21no—
233C22⋊C4—2981no—
234C22⋊C4—2981no—
235C22⋊C4—2981no—
236C22⋊C4—2981no—
237C42—2981no—
238C42—1491no—
239C42—1491no—
240C42—1491no—
241C42—1491no—
242C23—22no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—2981no—
247C23—2982no—
248C23—5962no—
249C23—5962no—
250C23—5962no—
251C23D4×D14911yescharacteristic
252Q8—22no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—2982no—
257D4—2982no—
258D4—2982no—
259D4—5962no—
260D4—5964no—
261D4—5964no—
262D4—5964no—
263D4—5962no—
264D4—5964no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—2982no—
269C2×C4—2982no—
270C2×C4—2981no—
271C2×C4—5962no—
272C2×C4—1491no—
273C2×C4—1491no—
274C2×C4—1491no—
275C2×C4—1491no—
276C2×C4D4×D14911yescharacteristic
277C2×C4C23×D14911yescharacteristic
278C8—22no—
279C8—2982no—
280C22—21no—
281C22—21no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—84no—
286C22—5961no—
287C22—5962no—
288C22—5962no—
289C22—5961no—
290C22—5962no—
291C22—5961no—
292C22—11924no—
293C22#1801193812yes—
294C22C2×D4×D14911yescharacteristic
295C4—21no—
296C4—41no—
297C4—41no—
298C4—2981no—
299C4—2981no—
300C4—2981no—
301C4—5961no—
302C4—1492no—
303C4#1801195612yes—
304C2—22no—
305C2—82no—
306C2—11922no—
307C2—11921no—
308C2#2375019011yescharacteristic
309C2AGL(1,ℤ/8)×D14911yescharacteristic
310C2C2×#1801193811yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#450×C149C2
socleC22×C149C22×D4
hypercenter#450D149
hypocenterC149#3217
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#18011907C2semidirect
2. #23750161C2semidirect
3. #23750011C2semidirect
4. #23749988C2semidirect
5. #23749823C2semidirect
6. #23749819C2semidirect
7. #23749775C2semidirect
8. #23749761C2semidirect
9. #23749759C2semidirect
10. #23749679C2semidirect
11. #450×C149C2semidirect
15. C2×#14558723C22semidirect
16. C2×D1192C22semidirect
17. C2×#14558721C22semidirect
23. #18011902C22semidirect
24. #18011900C22semidirect
27. Q8×Dic149C22semidirect
28. #18011873C22semidirect
32. #18011840C22semidirect
33. #18011833C22semidirect
34. #18011832C22semidirect
35. #18011831C22semidirect
37. #18011821C22semidirect
39. C2×SD16×C149C22semidirect
41. #102×C149C22semidirect
155. C149#3217semidirect
164. #450D149semidirect
173. C2×SD16C2×D149semidirect
195. #102C2×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119072
#237501612
#237500112
#237499882
#237498232
#237498192
#237497752
#237497612
#237497592
#237496792
#450×C1492
#3217149