Finite Group Databasebeta

← #34055514

Subgroups — #34055514

Order 19072.

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

normal subgroups only

This group has 410 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×#18011891C211yesmaximal, characteristic
2C22⋊Q8×D149C211yesmaximal, characteristic
3#23749957C212yesmaximal, complemented
4#23749955C211yesmaximal, characteristic, complemented
5#23749884C212yesmaximal, complemented
6#23749882C212yesmaximal, complemented
7#23749783C211yesmaximal, characteristic, complemented
8#23749782C212yesmaximal, complemented
9#23749780C211yesmaximal, characteristic, complemented
10#23749587C211yesmaximal, characteristic
11#483×C149C211yesmaximal, characteristic, complemented
12C22×Dic298C2211yescharacteristic
13C2×Q8×D149C2211yescharacteristic
14C22×C4×D149C2211yescharacteristic
15C2×Dic596C2212yescomplemented
16C2×#14558726C2212yescomplemented
17C2×#14558719C2212yes—
18#18011891—21no—
19#18011891—21no—
20#18011891C2212yes—
21#18011891C2211yescharacteristic
22#18011891C2212yes—
23#18011890C2211yescharacteristic
24#18011889—21no—
25#18011888—21no—
26#18011887C2211yescharacteristic
27C2×C4⋊Dic149—21no—
28C4⋊C4×D149—21no—
29C4⋊C4×D149C2211yescharacteristic
30C2×#14558713—21no—
31C22⋊C4×D149—21no—
32#18011842C2212yescomplemented
33#18011836C2212yescomplemented
34#18011834C2212yescomplemented
35#18011834C2212yescomplemented
36C22⋊#11924746C2211yescharacteristic
37#18011823C2212yes—
38C22⋊Q8×C149C2211yescharacteristic
39C2×C4⋊C4×C149C2211yescharacteristic
40#106×C149C2212yescomplemented
41#103×C149C2212yescomplemented
42C22⋊C8×C149C2211yescharacteristic
43C23×D149D411yescharacteristic
44C2×Dic298—21no—
45C2×Dic298—21no—
46C2×Dic298—42no—
47C2×Dic298C2312yes—
48C2×Dic298C2311yescharacteristic
49Q8×D149—42no—
50C22×Dic149—21no—
51C22×Dic149D411yescharacteristic
52C2×C4×D149—21no—
53C2×C4×D149—21no—
54C2×C4×D149—42no—
55C2×C4×D149D411yescharacteristic
56C2×C4×D149D411yescharacteristic
57C2×C4×D149C2311yescharacteristic
58C2×C4×D149C2311yescharacteristic
59Dic596—42no—
60#14558726—42no—
61#14558719—42no—
62C2×#11924746C2312yes—
63C4⋊Dic149—21no—
64C4⋊Dic149—22no—
65C4⋊Dic149—22no—
66C4⋊Dic149C2311yescharacteristic
67#14558715—21no—
68#14558715—21no—
69#14558715—21no—
70C22⋊Dic149—21no—
71#14558713—21no—
72#14558713—21no—
73#14558713—22no—
74#14558713—22no—
75C2×Q8×C149C2311yescharacteristic
76C22×C4×C149—21no—
77C22×C4×C149C2311yescharacteristic
78C2×C8×C149C2312yes—
79C4⋊C4×C149—21no—
80C4⋊C4×C149—21no—
81C4⋊C4×C149C2312yes—
82C4⋊C4×C149C2311yescharacteristic
83C22⋊C4×C149—21no—
84C22×D149—22no—
85C22×D149—22no—
86C22×D149—41no—
87C22×D149C2×D411yescharacteristic
88C22×D149C2×D411yescharacteristic
89Dic298—22no—
90Dic298—22no—
91Dic298—22no—
92Dic298—42no—
93Dic298—42no—
94C2×Dic149—21no—
95C2×Dic149—21no—
96C2×Dic149—22no—
97C2×Dic149—41no—
98C2×Dic149—42no—
99C2×Dic149C2×D411yescharacteristic
100C2×Dic149C2×D411yescharacteristic
101C4×D149—22no—
102C4×D149—22no—
103C4×D149—41no—
104C4×D149—41no—
105C4×D149—42no—
106C4×D149—42no—
107#11924746—22no—
108C23×C149C2×D411yescharacteristic
109Q8×C149—22no—
110C2×C4×C149—22no—
111C2×C4×C149—21no—
112C2×C4×C149—21no—
113C2×C4×C149—22no—
114C2×C4×C149—42no—
115C2×C4×C149C2×D411yescharacteristic
116C2×C4×C149C2411yescharacteristic
117C8×C149—22no—
118C22×C149—21no—
119C22×C149—21no—
120C22×C149#13712yes—
121C22×C149C22×D411yescharacteristic
122C2×D149—21no—
123C2×D149—21no—
124C2×D149—21no—
125C2×D149—21no—
126C2×D149—21no—
127C2×D149—44no—
128C2×D149#13712yes—
129C4×C149—21no—
130C4×C149—41no—
131C4×C149—42no—
132C4×C149—41no—
133C4×C149H(5,𝔽2)12yes—
134Dic149—21no—
135Dic149—21no—
136Dic149—41no—
137Dic149—42no—
138Dic149—41no—
139C2×C149—22no—
140C2×C149#53411yescharacteristic
141C2×C149C2×#13711yescharacteristic
142C2×C149C2×#13711yescharacteristic
143D149—22no—
144D149—41no—
145C149#319511yescharacteristic, complemented, Sylow
146#3195—1491nomaximal, Sylow
147C2×C22⋊Q8—1491no—
148C2×C22⋊Q8—1491no—
149#483—1491no—
150#483—1492no—
151#483D14911yescharacteristic, complemented
152#470—1492no—
153#470—1492no—
154#451—1491no—
155#451—1492no—
156#451—1491no—
157#407—1491no—
158C22×Q8—1491no—
159C22×Q8—1491no—
160C23×C4—1491no—
161C2×Q16—1492no—
162C2×Q16—1492no—
163C2×M16—1492no—
164C22⋊Q8—2981no—
165C22⋊Q8—2981no—
166C22⋊Q8—2981no—
167C22⋊Q8—2981no—
168C22⋊Q8—1491no—
169C22⋊Q8—1491no—
170C22⋊Q8—1492no—
171C22⋊Q8—1491no—
172C22⋊Q8—1492no—
173C22⋊Q8C2×D14911yescharacteristic
174C2×C4⋊C4—2981no—
175C2×C4⋊C4—2981no—
176C2×C4⋊C4—1491no—
177C2×C4⋊C4C2×D14911yescharacteristic
178C2×C22⋊C4—2981no—
179C2×C22⋊C4—2981no—
180#106—1492no—
181#106C2×D14912yescomplemented
182#103—1492no—
183#103—1492no—
184#103—1492no—
185#103C2×D14912yescomplemented
186C22⋊C8—1491no—
187C22⋊C8—1492no—
188C22⋊C8C2×D14911yescharacteristic
189C24—1491no—
190C2×Q8—2981no—
191C2×Q8—2981no—
192C2×Q8—5962no—
193C2×Q8—5962no—
194C2×Q8—1492no—
195C2×Q8—1491no—
196C2×Q8C22×D14911yescharacteristic
197C22×C4—21no—
198C22×C4—2981no—
199C22×C4—2981no—
200C22×C4—2981no—
201C22×C4—5962no—
202C22×C4—1491no—
203C22×C4—1491no—
204C22×C4—1491no—
205C22×C4—1491no—
206C22×C4—1491no—
207C22×C4C22×D14911yescharacteristic
208Q16—5962no—
209Q16—5962no—
210M16—5962no—
211C2×C8—1492no—
212C2×C8C22×D14912yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—2981no—
216C4⋊C4—2981no—
217C4⋊C4—2982no—
218C4⋊C4—2982no—
219C4⋊C4—2981no—
220C4⋊C4—2981no—
221C4⋊C4—1491no—
222C4⋊C4C22×D14911yescharacteristic
223C4⋊C4C22×D14912yes—
224C22⋊C4—21no—
225C22⋊C4—2981no—
226C22⋊C4—2981no—
227C22⋊C4—2982no—
228C22⋊C4—2982no—
229C22⋊C4—2981no—
230C23—2982no—
231C23—2982no—
232C23—5961no—
233C23—1491no—
234C23—1491no—
235C23D4×D14911yescharacteristic
236Q8—22no—
237Q8—2982no—
238Q8—2982no—
239Q8—2982no—
240Q8—5962no—
241Q8—5962no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—2982no—
248C2×C4—2982no—
249C2×C4—2981no—
250C2×C4—2981no—
251C2×C4—2982no—
252C2×C4—5961no—
253C2×C4—5961no—
254C2×C4—5962no—
255C2×C4—5962no—
256C2×C4—5961no—
257C2×C4—5962no—
258C2×C4—1491no—
259C2×C4—1491no—
260C2×C4D4×D14911yescharacteristic
261C2×C4C23×D14911yescharacteristic
262C8—22no—
263C8—2982no—
264C22—21no—
265C22—21no—
266C22—2981no—
267C22—2981no—
268C22—2981no—
269C22—2981no—
270C22—2981no—
271C22—5964no—
272C22—1492no—
273C22#1801194012yes—
274C22C2×D4×D14911yescharacteristic
275C4—21no—
276C4—41no—
277C4—42no—
278C4—41no—
279C4—2981no—
280C4—2981no—
281C4—5961no—
282C4—5961no—
283C4—5962no—
284C4#1801195712yes—
285C2—22no—
286C2—2982no—
287C2—5961no—
288C2#2375018611yescharacteristic
289C2#137×D14911yescharacteristic
290C2C2×#1801194011yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#483×C149C2
socleC22×C149C22×D4
prime radicalC23×D149D4
hypercenter#483D149
hypocenterC149#3195
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. #23749957C2semidirect
4. #23749955C2semidirect
5. #23749884C2semidirect
6. #23749882C2semidirect
7. #23749783C2semidirect
8. #23749782C2semidirect
9. #23749780C2semidirect
11. #483×C149C2semidirect
15. C2×Dic596C22semidirect
16. C2×#14558726C22semidirect
32. #18011842C22semidirect
33. #18011836C22semidirect
34. #18011834C22semidirect
35. #18011834C22semidirect
40. #106×C149C22semidirect
41. #103×C149C22semidirect
145. C149#3195semidirect
151. #483D149semidirect
181. #106C2×D149semidirect
185. #103C2×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180118912
C22⋊Q8×D1492
#237499572
#237499552
#237498842
#237498822
#237497832
#237497822
#237497802
#237495872
#483×C1492
#3195149