Finite Group Databasebeta

← #34055689

Subgroups — #34055689

Order 19072.

400 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 111 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#23750180C211yesmaximal, characteristic, complemented
2#124×D149C211yesmaximal, characteristic
3#23750017C211yesmaximal, characteristic, complemented
4#23749985C211yesmaximal, characteristic, complemented
5#23749984C211yesmaximal, characteristic, complemented
6#23749849C212yesmaximal, complemented
7#23749845C212yesmaximal, complemented
8#23749779C212yesmaximal, complemented
9#23749774C212yesmaximal
10#23749680C211yesmaximal, characteristic, complemented
11#497×C149C211yesmaximal, characteristic, complemented
12C2×Q8×D149C2211yescharacteristic
13C2×D4×D149C2211yescharacteristic
14C2×#14558726C2212yescomplemented
15C2×#14558723C2212yes—
16#18011911C2211yescharacteristic, complemented
17#18011902—21no—
18#18011900C2211yescharacteristic
19#18011899C2211yescharacteristic
20#18011898C2211yescharacteristic
21#18011897—21no—
22#18011897—21no—
23#18011891—21no—
24#18011887C2212yes—
25Q8×Dic149C2212yescomplemented
26#18011876—21no—
27#18011869—21no—
28C22⋊C4×D149—21no—
29C22⋊C4×D149—21no—
30C42×D149C2211yescharacteristic
31#18011840C2212yescomplemented
32#18011836C2212yescomplemented
33#18011835C2212yescomplemented
34#18011833C2212yes—
35#18011832C2212yes—
36#18011823C2212yes—
37#18011822C2211yescharacteristic, complemented
38C2×Q16×C149C2212yescomplemented
39C2×SD16×C149C2212yes—
40C4⋊Q8×C149C2211yescharacteristic, complemented
41#124×C149C2211yescharacteristic
42#97×C149C2211yescharacteristic, complemented
43C23×D149—21no—
44C2×Dic298—21no—
45C2×Dic298—21no—
46C2×Dic298C2311yescharacteristic
47Q8×D149—42no—
48C2×D596C2311yescharacteristic
49C2×C22⋊D149—21no—
50D4×D149—42no—
51C2×C4×D149—21no—
52C2×C4×D149D411yescharacteristic
53C2×C4×D149D411yescharacteristic
54C2×C4×D149C2311yescharacteristic
55#14558726—42no—
56#14558723—42no—
57C2×#11924746C2312yes—
58C4⋊Dic149—22no—
59C4⋊Dic149—21no—
60C4⋊Dic149C2312yes—
61#14558715—22no—
62C22⋊Dic149—21no—
63C22⋊Dic149—21no—
64#14558713—22no—
65#14558713—21no—
66#14558713—21no—
67#14558713—21no—
68#14558713—21no—
69C4×Dic149—22no—
70C4×Dic149D411yescharacteristic
71C4×Dic149D411yescharacteristic
72C4×Dic149C2311yescharacteristic
73C2×Q8×C149C2312yes—
74C2×Q8×C149C2311yescharacteristic
75C2×D4×C149C2311yescharacteristic
76Q16×C149—42no—
77SD16×C149—42no—
78C2×C8×C149C2312yes—
79C4⋊C4×C149—21no—
80C4⋊C4×C149—21no—
81C22⋊C4×C149—21no—
82C22⋊C4×C149—21no—
83C42×C149C2311yescharacteristic
84C22×D149—21no—
85C22×D149—42no—
86C22×D149—42no—
87C22×D149—42no—
88C22×D149C2×D411yescharacteristic
89Dic298—22no—
90Dic298—42no—
91Dic298—42no—
92D596—22no—
93C22⋊D149—42no—
94C2×Dic149—21no—
95C2×Dic149—22no—
96C2×Dic149C2×D411yescharacteristic
97C2×Dic149C2×D411yescharacteristic
98C2×Dic149C2×D411yescharacteristic
99C4×D149—22no—
100C4×D149—22no—
101C4×D149—41no—
102C4×D149—42no—
103#11924746—22no—
104C23×C149—21no—
105Q8×C149—22no—
106Q8×C149—22no—
107Q8×C149—22no—
108D4×C149—22no—
109C2×C4×C149—21no—
110C2×C4×C149—22no—
111C2×C4×C149C2×D411yescharacteristic
112C2×C4×C149C2×D411yescharacteristic
113C2×C4×C149C2411yescharacteristic
114C8×C149—22no—
115C22×C149—41no—
116C22×C149—41no—
117C22×C149—41no—
118C22×C149C22×D411yescharacteristic
119C2×D149—21no—
120C2×D149—21no—
121C2×D149—41no—
122C2×D149—41no—
123C2×D149—41no—
124C2×D149—84no—
125C2×D149#13712yes—
126C4×C149—21no—
127C4×C149—21no—
128C4×C149—42no—
129C4×C149—41no—
130C4×C149H(5,𝔽2)12yes—
131Dic149—21no—
132Dic149—21no—
133Dic149—21no—
134Dic149—41no—
135Dic149—42no—
136C2×C149—81no—
137C2×C149#53511yescharacteristic
138C2×C149C2×#13711yescharacteristic
139C2×C149#57511yescharacteristic
140D149—22no—
141D149—81no—
142C149#321511yescharacteristic, complemented, Sylow
143#3215—1491nomaximal, Sylow
144#533—1491no—
145C2×#124—1491no—
146#497—1491no—
147#497D14911yescharacteristic, complemented
148#489—1491no—
149#489—1491no—
150#464—1492no—
151#464—1492no—
152#451—1492no—
153#450—1492no—
154#432—1491no—
155C22×Q8—1491no—
156C22×D4—1491no—
157C2×Q16—1492no—
158C2×Q16C2×D14912yescomplemented
159C2×SD16—1492no—
160C2×SD16C2×D14912yes—
161C4⋊Q8—1491no—
162C4⋊Q8C2×D14911yescharacteristic, complemented
163#125—2981no—
164#124—2981no—
165#124—2981no—
166#124—1491no—
167#124—1491no—
168#124—1491no—
169#124C2×D14911yescharacteristic
170C22⋊Q8—2981no—
171C22⋊Q8—1492no—
172C4×Q8—2981no—
173C4×Q8—1492no—
174#117—2981no—
175C2×C22⋊C4—2981no—
176C2×C22⋊C4—2981no—
177C2×C42—1491no—
178C4⋊C8—1492no—
179#103—1492no—
180#103—1492no—
181#102—1492no—
182#102—1492no—
183C22⋊C8—1492no—
184#97—1491no—
185#97C2×D14911yescharacteristic, complemented
186C24—2981no—
187C2×Q8—2981no—
188C2×Q8—2981no—
189C2×Q8—5962no—
190C2×Q8—1491no—
191C2×Q8C22×D14911yescharacteristic
192C2×Q8C22×D14912yes—
193C2×D4—2981no—
194C2×D4—5962no—
195C2×D4—1491no—
196C2×D4C22×D14911yescharacteristic
197C22×C4—2981no—
198C22×C4—1491no—
199C22×C4—1491no—
200C22×C4—1491no—
201Q16—42no—
202Q16—5962no—
203SD16—42no—
204SD16—5962no—
205C2×C8—1492no—
206C2×C8C22×D14912yes—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—2982no—
210C4⋊C4—2982no—
211C4⋊C4—2981no—
212C4⋊C4—1492no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—2982no—
216C22⋊C4—2981no—
217C22⋊C4—2981no—
218C22⋊C4—2981no—
219C22⋊C4—2981no—
220C22⋊C4—2981no—
221C22⋊C4—2981no—
222C42—2982no—
223C42—1491no—
224C42—1491no—
225C42—1491no—
226C42C22×D14911yescharacteristic
227C23—21no—
228C23—2981no—
229C23—5962no—
230C23—5962no—
231C23—5962no—
232C23—1491no—
233Q8—22no—
234Q8—22no—
235Q8—22no—
236Q8—2982no—
237Q8—5962no—
238Q8—5962no—
239D4—22no—
240D4—2982no—
241D4—5962no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—2982no—
245C2×C4—2982no—
246C2×C4—2982no—
247C2×C4—2981no—
248C2×C4—5961no—
249C2×C4—5962no—
250C2×C4—1491no—
251C2×C4—1491no—
252C2×C4—1491no—
253C2×C4D4×D14911yescharacteristic
254C2×C4D4×D14911yescharacteristic
255C2×C4C23×D14911yescharacteristic
256C8—22no—
257C8—2982no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—2981no—
262C22—2981no—
263C22—5961no—
264C22—5961no—
265C22—5961no—
266C22—1492no—
267C22—11924no—
268C22C2×D4×D14911yescharacteristic
269C4—21no—
270C4—21no—
271C4—41no—
272C4—42no—
273C4—2981no—
274C4—2981no—
275C4—2981no—
276C4—5962no—
277C4—5961no—
278C4#1801195612yes—
279C2—81no—
280C2—2982no—
281C2—11921no—
282C2#2375018811yescharacteristic
283C2#137×D14911yescharacteristic
284C2#2375042411yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#497×C149C2
socleC22×C149C22×D4
prime radicalC2×D4×D149C22
hypercenter#497D149
hypocenterC149#3215
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #23750180C2semidirect
3. #23750017C2semidirect
4. #23749985C2semidirect
5. #23749984C2semidirect
6. #23749849C2semidirect
7. #23749845C2semidirect
8. #23749779C2semidirect
10. #23749680C2semidirect
11. #497×C149C2semidirect
14. C2×#14558726C22semidirect
16. #18011911C22semidirect
25. Q8×Dic149C22semidirect
31. #18011840C22semidirect
32. #18011836C22semidirect
33. #18011835C22semidirect
37. #18011822C22semidirect
38. C2×Q16×C149C22semidirect
40. C4⋊Q8×C149C22semidirect
42. #97×C149C22semidirect
142. C149#3215semidirect
147. #497D149semidirect
158. C2×Q16C2×D149semidirect
162. C4⋊Q8C2×D149semidirect
185. #97C2×D149semidirect

Maximal subgroups

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

subgroupindex
#237501802
#124×D1492
#237500172
#237499852
#237499842
#237498492
#237498452
#237497792
#237497742
#237496802
#497×C1492
#3215149