Finite Group Databasebeta

← #34055691

Subgroups — #34055691

Order 19072.

400 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#23750177C211yesmaximal, characteristic, complemented
2C2×#18011898C211yesmaximal, characteristic
3#23750016C212yesmaximal, complemented
4#23749983C212yesmaximal, complemented
5#23749846C212yesmaximal, complemented
6#23749845C212yesmaximal, complemented
7#23749783C211yesmaximal, characteristic, complemented
8#23749780C211yesmaximal, characteristic, complemented
9#23749776C211yesmaximal, characteristic
10#23749679C211yesmaximal, characteristic, complemented
11#450×C149C211yesmaximal, characteristic
12C22×Dic298C2211yescharacteristic
13C2×Dic596C2212yescomplemented
14C2×#14558726C2212yescomplemented
15C2×#14558724C2212yes—
16#18011910C2211yescharacteristic, complemented
17#18011909C2211yescharacteristic, complemented
18#18011901—21no—
19#18011898—21no—
20#18011898—21no—
21#18011898C2212yes—
22#18011898C2212yes—
23#18011890C2211yescharacteristic
24#18011888—21no—
25Q8×Dic149C2211yescharacteristic, complemented
26#18011876—21no—
27#18011876C2211yescharacteristic, complemented
28#18011867—21no—
29C2×C22⋊Dic149—21no—
30C2×C22⋊Dic149—21no—
31C2×C4×Dic149C2211yescharacteristic
32#18011840C2212yescomplemented
33#18011836C2212yescomplemented
34#18011834C2212yescomplemented
35#18011832C2212yes—
36C22⋊#11924746C2211yescharacteristic
37#18011821C2212yescomplemented
38C22×D4×C149C2211yescharacteristic
39C2×SD16×C149C2212yes—
40C22⋊Q8×C149C2211yescharacteristic
41#102×C149C2212yes—
42C22⋊C8×C149C2211yescharacteristic
43C2×Dic298—21no—
44C2×Dic298—21no—
45C2×Dic298—42no—
46C2×Dic298C2311yescharacteristic
47C2×Dic298C2312yes—
48C22×Dic149—21no—
49C22×Dic149D411yescharacteristic
50C22×Dic149D411yescharacteristic
51Dic596—42no—
52#14558726—42no—
53#14558724—42no—
54C2×#11924746C2312yes—
55C4⋊Dic149—21no—
56C4⋊Dic149C2311yescharacteristic
57#14558715—21no—
58#14558715—21no—
59#14558715—21no—
60#14558715—21no—
61#14558715—21no—
62C22⋊Dic149—22no—
63C22⋊Dic149—22no—
64C22⋊Dic149—22no—
65C22⋊Dic149—22no—
66C22⋊Dic149—21no—
67C4×Dic149—21no—
68C4×Dic149—21no—
69C4×Dic149D411yescharacteristic
70C4×Dic149D411yescharacteristic
71C4×Dic149C2311yescharacteristic
72C4×Dic149C2311yescharacteristic
73C24×C149—21no—
74C2×Q8×C149C2311yescharacteristic
75C2×D4×C149—21no—
76C2×D4×C149—42no—
77C2×D4×C149C2312yes—
78C22×C4×C149C2311yescharacteristic
79SD16×C149—42no—
80C2×C8×C149C2312yes—
81C4⋊C4×C149—21no—
82C4⋊C4×C149C2311yescharacteristic
83C22⋊C4×C149—21no—
84Dic298—22no—
85Dic298—22no—
86Dic298—22no—
87Dic298—42no—
88Dic298—42no—
89C2×Dic149—22no—
90C2×Dic149—22no—
91C2×Dic149—21no—
92C2×Dic149—21no—
93C2×Dic149—41no—
94C2×Dic149—42no—
95C2×Dic149C2×D411yescharacteristic
96C2×Dic149C2×D411yescharacteristic
97C2×Dic149C2×D411yescharacteristic
98C2×Dic149C2×D411yescharacteristic
99#11924746—22no—
100C23×C149—22no—
101C23×C149—42no—
102C23×C149—42no—
103C23×C149—42no—
104C23×C149C2×D411yescharacteristic
105Q8×C149—22no—
106D4×C149—22no—
107D4×C149—22no—
108D4×C149—42no—
109C2×C4×C149—22no—
110C2×C4×C149—21no—
111C2×C4×C149—21no—
112C2×C4×C149C2×D411yescharacteristic
113C2×C4×C149C2411yescharacteristic
114C8×C149—22no—
115C22×C149—21no—
116C22×C149—21no—
117C22×C149—42no—
118C22×C149—42no—
119C22×C149—42no—
120C22×C149—84no—
121C22×C149#13712yes—
122C22×C149C22×D411yescharacteristic
123C4×C149—21no—
124C4×C149—41no—
125C4×C149—41no—
126C4×C149H(5,𝔽2)12yes—
127Dic149—21no—
128Dic149—21no—
129Dic149—21no—
130Dic149—21no—
131Dic149—42no—
132Dic149—41no—
133Dic149—41no—
134C2×C149—22no—
135C2×C149—82no—
136C2×C149#53511yescharacteristic
137C2×C149C2×#13711yescharacteristic
138C2×C149#57511yescharacteristic
139C149#321511yescharacteristic, complemented, Sylow
140#3215—1491nomaximal, Sylow
141#533—1491no—
142C2×#124—1491no—
143#497—1492no—
144#489—1492no—
145#464—1492no—
146#464—1492no—
147#451—1491no—
148#451—1491no—
149#450—1491no—
150#450D14911yescharacteristic
151#432—1491no—
152C22×Q8—1491no—
153C22×D4C2×D14911yescharacteristic
154C2×Q16—1492no—
155C2×Q16—1492no—
156C2×SD16—1492no—
157C2×SD16C2×D14912yes—
158C4⋊Q8—1491no—
159C4⋊Q8—1491no—
160#125—2981no—
161#124—2981no—
162#124—2981no—
163#124—1492no—
164#124—1492no—
165C22⋊Q8—2981no—
166C22⋊Q8—1491no—
167C22⋊Q8C2×D14911yescharacteristic
168C4×Q8—2981no—
169C4×Q8—1491no—
170C4×Q8—1491no—
171#117—2981no—
172C2×C22⋊C4—2981no—
173C2×C22⋊C4—2981no—
174C2×C42—1491no—
175C4⋊C8—1492no—
176#103—1492no—
177#103—1492no—
178#102—1492no—
179#102C2×D14912yes—
180C22⋊C8—1491no—
181C22⋊C8C2×D14911yescharacteristic
182#97—1492no—
183C24—21no—
184C2×Q8—2981no—
185C2×Q8—2981no—
186C2×Q8—5962no—
187C2×Q8—1492no—
188C2×Q8—1491no—
189C2×Q8C22×D14911yescharacteristic
190C2×D4—21no—
191C2×D4—42no—
192C2×D4C22×D14912yes—
193C22×C4—2981no—
194C22×C4—1491no—
195C22×C4—1491no—
196C22×C4C22×D14911yescharacteristic
197Q16—5962no—
198Q16—5962no—
199SD16—42no—
200SD16—5962no—
201C2×C8—1492no—
202C2×C8C22×D14912yes—
203C4⋊C4—21no—
204C4⋊C4—2981no—
205C4⋊C4—2981no—
206C4⋊C4—2981no—
207C4⋊C4—2981no—
208C4⋊C4—2981no—
209C4⋊C4—2981no—
210C4⋊C4—1491no—
211C4⋊C4C22×D14911yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—2981no—
214C22⋊C4—2982no—
215C22⋊C4—2982no—
216C22⋊C4—2982no—
217C22⋊C4—2982no—
218C42—2981no—
219C42—2981no—
220C42—1491no—
221C42—1491no—
222C42—1491no—
223C42—1491no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23D4×D14911yescharacteristic
229Q8—22no—
230Q8—2982no—
231Q8—2982no—
232Q8—2982no—
233Q8—5962no—
234Q8—5962no—
235D4—22no—
236D4—22no—
237D4—42no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—2982no—
242C2×C4—2981no—
243C2×C4—2981no—
244C2×C4—2982no—
245C2×C4—5961no—
246C2×C4—5962no—
247C2×C4—1491no—
248C2×C4—1491no—
249C2×C4—1491no—
250C2×C4—1491no—
251C2×C4D4×D14911yescharacteristic
252C2×C4C23×D14911yescharacteristic
253C8—22no—
254C8—2982no—
255C22—21no—
256C22—21no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—84no—
261C22#1801194012yes—
262C22C2×D4×D14911yescharacteristic
263C4—21no—
264C4—41no—
265C4—41no—
266C4—2981no—
267C4—2981no—
268C4—2981no—
269C4—2981no—
270C4—5961no—
271C4—5962no—
272C4—5961no—
273C4#1801195612yes—
274C2—22no—
275C2—82no—
276C2#2375019011yescharacteristic
277C2C2×#1801194011yescharacteristic
278C2#2375042111yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#450×C149C2
socleC22×C149C22×D4
prime radicalC22×D4×C149C22
hypercenter#450D149
hypocenterC149#3215
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. #23750177C2semidirect
3. #23750016C2semidirect
4. #23749983C2semidirect
5. #23749846C2semidirect
6. #23749845C2semidirect
7. #23749783C2semidirect
8. #23749780C2semidirect
10. #23749679C2semidirect
13. C2×Dic596C22semidirect
14. C2×#14558726C22semidirect
16. #18011910C22semidirect
17. #18011909C22semidirect
25. Q8×Dic149C22semidirect
27. #18011876C22semidirect
32. #18011840C22semidirect
33. #18011836C22semidirect
34. #18011834C22semidirect
37. #18011821C22semidirect
139. C149#3215semidirect

Maximal subgroups

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

subgroupindex
#237501772
C2×#180118982
#237500162
#237499832
#237498462
#237498452
#237497832
#237497802
#237497762
#237496792
#450×C1492
#3215149