Finite Group Databasebeta

← #34055169

Subgroups — #34055169

Order 19072.

564 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 149 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×#18011942C211yesmaximal, characteristic, complemented
2C2×#18011936C211yesmaximal, characteristic, complemented
3C2×#18011931C211yesmaximal, characteristic, complemented
4C22×#14558723C211yesmaximal, characteristic, complemented
5#23750194C211yesmaximal, characteristic, complemented
6#23750017C212yesmaximal, complemented
7#23750015C212yesmaximal, complemented
8#23750007C212yesmaximal, complemented
9#23749999C212yesmaximal, complemented
10#23749581C211yesmaximal, characteristic, complemented
11#536×C149C211yesmaximal, characteristic, complemented
12C2×#14558736C2211yescharacteristic
13C22×D596C2211yescharacteristic
14#18011942—24no—
15#18011936—24no—
16#18011931—24no—
17C2×#14558726C2211yescharacteristic, complemented
18C2×#14558726C2212yescomplemented
19C2×#14558724C2211yescharacteristic, complemented
20C2×#14558723—24no—
21C2×#14558723C2212yes—
22C2×#14558723C2212yes—
23C2×#14558723C2212yes—
24C2×#14558723C2211yescharacteristic, complemented
25C2×#14558723C2212yescomplemented
26C2×#14558721C2212yes—
27C2×#14558721C2211yescharacteristic, complemented
28C2×#14558720C2211yescharacteristic
29C22×#11924746C2211yescharacteristic
30#18011908C2212yes—
31#18011899C2212yes—
32#18011886—22no—
33#18011881—22no—
34#18011822C2212yescomplemented
35C4×#11924746C2212yescomplemented
36C2×#41×C149C2211yescharacteristic
37C22×Q8×C149C2211yescharacteristic
38C4⋊Q8×C149C2212yescomplemented
39#124×C149C2212yescomplemented
40#123×C149—22no—
41C22⋊Q8×C149—22no—
42#117×C149C2211yescharacteristic
43C23×D149—21no—
44#14558736—22no—
45#14558736—22no—
46C2×Dic298C2311yescharacteristic
47C2×D596—22no—
48C2×D596—22no—
49C2×D596—21no—
50C2×D596—24no—
51C2×D596C2311yescharacteristic
52C2×D596C2312yes—
53C2×C22⋊D149—21no—
54C2×C4×D149—21no—
55#14558726—22no—
56#14558726—24no—
57#14558724—22no—
58#14558723—24no—
59#14558723—24no—
60#14558723—24no—
61#14558723—22no—
62#14558723—24no—
63#14558721—24no—
64#14558721—22no—
65#14558720D414yescomplemented
66C2×#11924746D414yescomplemented
67C2×#11924746C2311yescharacteristic
68C2×#11924746C2311yescharacteristic
69C2×#11924746C2312yes—
70#14558713—24no—
71#41×C149—22no—
72#41×C149—22no—
73C2×Q8×C149—21no—
74C2×Q8×C149—22no—
75C2×Q8×C149—22no—
76C2×Q8×C149C2312yes—
77C2×Q8×C149C2311yescharacteristic
78C2×D4×C149—21no—
79C2×D4×C149C2311yescharacteristic
80C22×C4×C149—21no—
81C22×C4×C149—21no—
82C22×C4×C149C2311yescharacteristic
83C4⋊C4×C149—24no—
84C4⋊C4×C149D412yes—
85C22⋊C4×C149—24no—
86C22⋊C4×C149D412yes—
87C42×C149C2312yes—
88C22×D149—21no—
89C22×D149—22no—
90C22×D149—42no—
91C22×D149—82no—
92Dic298—21no—
93Dic298—21no—
94D596—22no—
95D596—21no—
96D596—22no—
97D596—21no—
98D596—42no—
99D596—84no—
100C22⋊D149—42no—
101C2×Dic149—21no—
102C4×D149—42no—
103#11924746C2×D414yes—
104#11924746C2×D412yes—
105#11924746C2×D412yes—
106C23×C149—21no—
107C23×C149C2×D411yescharacteristic
108Q8×C149—22no—
109Q8×C149—21no—
110Q8×C149—22no—
111Q8×C149—21no—
112Q8×C149—42no—
113D4×C149—21no—
114D4×C149—21no—
115D4×C149—42no—
116C2×C4×C149—22no—
117C2×C4×C149—21no—
118C2×C4×C149—42no—
119C2×C4×C149—42no—
120C2×C4×C149C2×D412yes—
121C2×C4×C149C2×D412yes—
122C2×C4×C149C2×D414yes—
123C2×C4×C149C2×D411yescharacteristic
124C2×C4×C149C2411yescharacteristic
125C22×C149—22no—
126C22×C149—41no—
127C22×C149—42no—
128C22×C149#12712yes—
129C22×C149C22×D411yescharacteristic
130C2×D149—41no—
131C2×D149—42no—
132C2×D149—42no—
133C2×D149—44no—
134C2×D149—84no—
135C4×C149—24no—
136C4×C149—41no—
137C4×C149—42no—
138C4×C149#12712yes—
139C4×C149C22×D411yescharacteristic
140C4×C149C22×D411yescharacteristic
141Dic149—41no—
142C2×C149—22no—
143C2×C149—81no—
144C2×C149C2×#12711yescharacteristic
145C2×C149#57712yes—
146D149—81no—
147D149—82no—
148C149#315411yescharacteristic, complemented, Sylow
149#3154—1491nomaximal, Sylow
150C2×#137—1491no—
151C2×AGL(1,ℤ/8)—1491no—
152C2×#135—1491no—
153C22×SD16—1491no—
154#536D14911yescharacteristic, complemented
155#535—1491no—
156#497—1492no—
157#496—1492no—
158#495—1492no—
159#492—1492no—
160#405—1491no—
161C2×#41—1491no—
162C2×#41C2×D14911yescharacteristic
163C22×Q8C2×D14911yescharacteristic
164C22×D4—1491no—
165#137—2984no—
166AGL(1,ℤ/8)—2984no—
167#135—2984no—
168C2×Q16—1491no—
169C2×Q16—1492no—
170C2×SD16—2984no—
171C2×SD16—1492no—
172C2×SD16—1492no—
173C2×SD16—1492no—
174C2×SD16—1491no—
175C2×SD16—1492no—
176C2×SD16—1491no—
177C2×D8—1492no—
178C2×D8—1491no—
179C2×M16—1491no—
180C22×C8—1491no—
181C4⋊Q8C2×D14912yescomplemented
182#127—1492no—
183#124—1492no—
184#124C2×D14912yescomplemented
185#123—22no—
186C22⋊Q8—22no—
187#121—2982no—
188#120—2982no—
189#117C2×D14911yescharacteristic
190#97—1492no—
191C4×C8—1492no—
192C24—2981no—
193#41—22no—
194#41—22no—
195#41—2982no—
196#41—2982no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—1491no—
201C2×Q8C22×D14912yes—
202C2×Q8C22×D14911yescharacteristic
203C2×D4—21no—
204C2×D4—2981no—
205C2×D4—2982no—
206C2×D4—2982no—
207C2×D4—2981no—
208C2×D4—2984no—
209C2×D4—1492no—
210C2×D4—1491no—
211C2×D4C22×D14911yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—2981no—
215C22×C4C22×D14911yescharacteristic
216Q16—2982no—
217Q16—2984no—
218SD16—2982no—
219SD16—2984no—
220SD16—2984no—
221SD16—2982no—
222SD16—2984no—
223SD16—2984no—
224D8—2982no—
225D8—2984no—
226M16—1494no—
227C2×C8—1494no—
228C2×C8—1491no—
229C2×C8—1492no—
230C2×C8—1491no—
231C4⋊C4—24no—
232C4⋊C4C22⋊D14912yes—
233C22⋊C4—24no—
234C22⋊C4—2984no—
235C22⋊C4C22⋊D14912yes—
236C42C22×D14912yes—
237C23—21no—
238C23—2982no—
239C23—2981no—
240C23—5962no—
241C23—11922no—
242C23C2×C22⋊D14911yescharacteristic
243Q8—22no—
244Q8—21no—
245Q8—22no—
246Q8—21no—
247Q8—42no—
248Q8—2981no—
249Q8—2981no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—2982no—
254D4—2981no—
255D4—2982no—
256D4—2981no—
257D4—5962no—
258D4—5962no—
259D4—11924no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—2981no—
265C2×C4—5962no—
266C2×C4D4×D14912yes—
267C2×C4D4×D14912yes—
268C2×C4C2×C22⋊D14914yes—
269C2×C4C2×C22⋊D14911yescharacteristic
270C2×C4C23×D14911yescharacteristic
271C8—1492no—
272C8—1492no—
273C8—1494no—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—5961no—
278C22—5962no—
279C22—5962no—
280C22—5964no—
281C22—11924no—
282C22#1801190712yes—
283C22C22×C22⋊D14911yescharacteristic
284C4—24no—
285C4—41no—
286C4—42no—
287C4—5961no—
288C4#1801190712yes—
289C4C2×D4×D14911yescharacteristic
290C4C2×D4×D14911yescharacteristic
291C2—22no—
292C2—81no—
293C2—11922no—
294C2—11921no—
295C2C2×#1801190711yescharacteristic
296C2#2375043312yes—
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#536×C149C2
socleC22×C149C22×D4
hypercenter#536D149
hypocenterC149#3154
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#18011942C2semidirect
2. C2×#18011936C2semidirect
3. C2×#18011931C2semidirect
4. C22×#14558723C2semidirect
5. #23750194C2semidirect
6. #23750017C2semidirect
7. #23750015C2semidirect
8. #23750007C2semidirect
9. #23749999C2semidirect
10. #23749581C2semidirect
11. #536×C149C2semidirect
17. C2×#14558726C22semidirect
18. C2×#14558726C22semidirect
19. C2×#14558724C22semidirect
24. C2×#14558723C22semidirect
25. C2×#14558723C22semidirect
27. C2×#14558721C22semidirect
34. #18011822C22semidirect
35. C4×#11924746C22semidirect
38. C4⋊Q8×C149C22semidirect
39. #124×C149C22semidirect
65. #14558720D4semidirect
66. C2×#11924746D4semidirect
148. C149#3154semidirect
154. #536D149semidirect
181. C4⋊Q8C2×D149semidirect
184. #124C2×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119422
C2×#180119362
C2×#180119312
C22×#145587232
#237501942
#237500172
#237500152
#237500072
#237499992
#237495812
#536×C1492
#3154149