Finite Group Databasebeta

← #34056291

Subgroups — #34056291

Order 19072.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×#18011936C212yesmaximal, complemented
2D4×D596C211yesmaximal, characteristic, complemented
3#23749999C211yesmaximal, characteristic, complemented
4#23749873C212yesmaximal, complemented
5#23749827C211yesmaximal, characteristic, complemented
6#23749820C211yesmaximal, characteristic, complemented
7#23749775C212yesmaximal, complemented
8#23749770C212yesmaximal, complemented
9C4×#14558721C211yesmaximal, characteristic, complemented
10#23749703C211yesmaximal, characteristic, complemented
11#547×C149C211yesmaximal, characteristic, complemented
12C22×D596C2212yes—
13C2×D4×D149—21no—
14#18011936—28no—
15C2×#14558723C2212yes—
16C2×#14558723C2212yescomplemented
17C2×#14558723C2212yescomplemented
18C2×#14558721C2211yescharacteristic
19C2×#14558721C2212yescomplemented
20C2×#14558720C2212yescomplemented
21#18011908C2211yescharacteristic
22#18011886—21no—
23#18011885C2212yes—
24#18011881—22no—
25C4×D596C2211yescharacteristic
26#18011844C2211yescharacteristic, complemented
27C4⋊#11924746C2211yescharacteristic, complemented
28#18011835C2212yescomplemented
29#18011832C2211yescharacteristic, complemented
30#18011831C2211yescharacteristic
31#18011831C2212yescomplemented
32C22⋊#11924746C2212yescomplemented
33C4×#11924746C2211yescharacteristic, complemented
34C2×#41×C149C2212yescomplemented
35C4⋊Q8×C149C2211yescharacteristic, complemented
36#123×C149—22no—
37C22⋊Q8×C149C2212yescomplemented
38#121×C149—21no—
39C4×D4×C149C2211yescharacteristic
40C4×D4×C149C2211yescharacteristic, complemented
41C2×C4⋊C4×C149—21no—
42C23×D149—22no—
43C2×D596—24no—
44C2×D596—21no—
45C2×D596—22no—
46C2×D596—21no—
47C2×D596—42no—
48C2×D596C2312yes—
49C2×D596C2311yescharacteristic
50C2×C22⋊D149—22no—
51D4×D149—44no—
52C2×C4×D149—21no—
53#14558723—24no—
54#14558723—24no—
55#14558723—42no—
56#14558721—24no—
57#14558721D414yescomplemented
58#14558720—24no—
59C2×#11924746C2312yes—
60C2×#11924746C2311yescharacteristic
61C2×#11924746C2311yescharacteristic
62C4⋊Dic149C2311yescharacteristic
63#14558713—21no—
64#14558713—22no—
65#41×C149—24no—
66#41×C149—42no—
67C2×Q8×C149C2312yes—
68C2×D4×C149—22no—
69C2×D4×C149D411yescharacteristic
70C2×D4×C149C2311yescharacteristic
71C22×C4×C149—21no—
72C22×C4×C149—22no—
73C22×C4×C149C2312yes—
74C4⋊C4×C149—21no—
75C4⋊C4×C149—22no—
76C4⋊C4×C149—21no—
77C4⋊C4×C149D411yescharacteristic
78C4⋊C4×C149C2311yescharacteristic
79C22⋊C4×C149—22no—
80C22⋊C4×C149—21no—
81C22⋊C4×C149D412yes—
82C42×C149C2311yescharacteristic
83C22×D149—21no—
84C22×D149—22no—
85C22×D149—44no—
86C22×D149—44no—
87C22×D149—82no—
88D596—22no—
89D596—21no—
90D596—22no—
91D596—44no—
92D596—42no—
93D596—81no—
94D596C2×D412yes—
95C22⋊D149—44no—
96C2×Dic149—21no—
97C4×D149—42no—
98#11924746—22no—
99#11924746—21no—
100#11924746C2×D412yescomplemented
101C23×C149—21no—
102C23×C149C2×D412yes—
103Q8×C149—22no—
104Q8×C149—22no—
105D4×C149—22no—
106D4×C149—21no—
107D4×C149—44no—
108D4×C149C2×D412yescomplemented
109C2×C4×C149—22no—
110C2×C4×C149—22no—
111C2×C4×C149—21no—
112C2×C4×C149—22no—
113C2×C4×C149—44no—
114C2×C4×C149—42no—
115C2×C4×C149C2×D411yescharacteristic
116C2×C4×C149C2×D412yes—
117C2×C4×C149C2×D411yescharacteristic
118C2×C4×C149C2411yescharacteristic
119C22×C149—22no—
120C22×C149—22no—
121C22×C149—22no—
122C22×C149—22no—
123C22×C149—41no—
124C22×C149—41no—
125C22×C149C22×D411yescharacteristic
126C2×D149—22no—
127C2×D149—41no—
128C2×D149—41no—
129C2×D149—42no—
130C2×D149—42no—
131C2×D149—42no—
132C2×D149—88no—
133C4×C149—22no—
134C4×C149—21no—
135C4×C149—42no—
136C4×C149—41no—
137C4×C149AGL(1,ℤ/8)12yescomplemented
138C4×C149C22×D411yescharacteristic
139C4×C149H(5,𝔽2)11yescharacteristic
140Dic149—41no—
141C2×C149—42no—
142C2×C149—42no—
143C2×C149D4211yescharacteristic
144C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C149#57711yescharacteristic
146D149—42no—
147D149—82no—
148C149#327211yescharacteristic, complemented, Sylow
149#3272—1491nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1492no—
151#547D14911yescharacteristic, complemented
152D42—1491no—
153#492—1491no—
154#468—1492no—
155#461—1491no—
156#459—1491no—
157#450—1492no—
158#449—1492no—
159C4×D8—1491no—
160#436—1491no—
161C2×#41C2×D14912yescomplemented
162C22×D4—2981no—
163C22×D4—1492no—
164AGL(1,ℤ/8)—2988no—
165C2×SD16—1492no—
166C2×SD16—1492no—
167C2×SD16—1492no—
168C2×D8—1492no—
169C2×D8—1491no—
170C2×M16—1492no—
171C4⋊Q8C2×D14911yescharacteristic, complemented
172#127—1491no—
173#123—22no—
174C22⋊Q8C2×D14912yescomplemented
175#121—21no—
176#121—2981no—
177#121—1492no—
178#120—2982no—
179C4×D4—1491no—
180C4×D4C2×D14911yescharacteristic
181C4×D4C2×D14911yescharacteristic, complemented
182C2×C4⋊C4—21no—
183#107—1491no—
184C4⋊C8—1491no—
185#103—1492no—
186#102—1491no—
187#102—1491no—
188#102—1492no—
189C22⋊C8—1492no—
190C4×C8—1491no—
191C24—2982no—
192#41—24no—
193#41—42no—
194C2×Q8C22×D14912yes—
195C2×D4—22no—
196C2×D4—2984no—
197C2×D4—2982no—
198C2×D4—2982no—
199C2×D4—2981no—
200C2×D4—2981no—
201C2×D4—5962no—
202C2×D4—5964no—
203C2×D4—1492no—
204C2×D4—1491no—
205C2×D4C22⋊D14911yescharacteristic
206C2×D4C22×D14911yescharacteristic
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—2981no—
210C22×C4C22×D14912yes—
211SD16—2984no—
212SD16—2984no—
213SD16—5962no—
214D8—2984no—
215D8—1494no—
216M16—2984no—
217C2×C8—1491no—
218C2×C8—1492no—
219C2×C8—1491no—
220C4⋊C4—21no—
221C4⋊C4—22no—
222C4⋊C4—21no—
223C4⋊C4—1491no—
224C4⋊C4C22⋊D14911yescharacteristic
225C4⋊C4C22×D14911yescharacteristic
226C22⋊C4—22no—
227C22⋊C4—21no—
228C22⋊C4—2981no—
229C22⋊C4—2982no—
230C22⋊C4C22⋊D14912yes—
231C42C22×D14911yescharacteristic
232C23—21no—
233C23—2981no—
234C23—2982no—
235C23—5964no—
236C23—5964no—
237C23—11922no—
238C23C2×C22⋊D14912yes—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—21no—
243D4—44no—
244D4—2982no—
245D4—2981no—
246D4—2982no—
247D4—5964no—
248D4—5964no—
249D4—5962no—
250D4—1492no—
251D4—11921no—
252D4D4×D14912yescomplemented
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—2981no—
260C2×C4—5962no—
261C2×C4C2×C22⋊D14911yescharacteristic
262C2×C4C2×C22⋊D14912yes—
263C2×C4C2×C22⋊D14911yescharacteristic
264C2×C4C23×D14911yescharacteristic
265C8—2982no—
266C8—2981no—
267C8—1492no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—2982no—
275C22—5961no—
276C22—5961no—
277C22—5962no—
278C22—5962no—
279C22—5962no—
280C22—11928no—
281C22C22×C22⋊D14911yescharacteristic
282C4—21no—
283C4—22no—
284C4—41no—
285C4—42no—
286C4—5961no—
287C4#1801193612yescomplemented
288C4C2×D4×D14911yescharacteristic
289C4#1801195611yescharacteristic
290C2—42no—
291C2—42no—
292C2—5962no—
293C2—11922no—
294C2D4×C22⋊D14911yescharacteristic
295C2C2×#1801193611yescharacteristic
296C2#2375043311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#547×C149C2
socleC22×C149C22×D4
hypercenter#547D149
hypocenterC149#3272
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#18011936C2semidirect
2. D4×D596C2semidirect
3. #23749999C2semidirect
4. #23749873C2semidirect
5. #23749827C2semidirect
6. #23749820C2semidirect
7. #23749775C2semidirect
8. #23749770C2semidirect
9. C4×#14558721C2semidirect
10. #23749703C2semidirect
11. #547×C149C2semidirect
16. C2×#14558723C22semidirect
17. C2×#14558723C22semidirect
19. C2×#14558721C22semidirect
20. C2×#14558720C22semidirect
26. #18011844C22semidirect
27. C4⋊#11924746C22semidirect
28. #18011835C22semidirect
29. #18011832C22semidirect
31. #18011831C22semidirect
32. C22⋊#11924746C22semidirect
33. C4×#11924746C22semidirect
34. C2×#41×C149C22semidirect
35. C4⋊Q8×C149C22semidirect
37. C22⋊Q8×C149C22semidirect
40. C4×D4×C149C22semidirect
57. #14558721D4semidirect
100. #11924746C2×D4semidirect
108. D4×C149C2×D4semidirect
137. C4×C149AGL(1,ℤ/8)semidirect
148. C149#3272semidirect
151. #547D149semidirect
161. C2×#41C2×D149semidirect
171. C4⋊Q8C2×D149semidirect
174. C22⋊Q8C2×D149semidirect
181. C4×D4C2×D149semidirect
252. D4D4×D149semidirect
287. C4#18011936semidirect

Maximal subgroups

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

subgroupindex
C2×#180119362
D4×D5962
#237499992
#237498732
#237498272
#237498202
#237497752
#237497702
C4×#145587212
#237497032
#547×C1492
#3272149