Finite Group Databasebeta

← #49169263

Subgroups — #49169263

Order 37504.

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×#22627655C211yesmaximal, characteristic, complemented
2C2×#22627649C211yesmaximal, characteristic, complemented
3C2×#22627644C211yesmaximal, characteristic, complemented
4C22×#17989789C211yesmaximal, characteristic, complemented
5#34036698C211yesmaximal, characteristic, complemented
6#34036521C212yesmaximal, complemented
7#34036519C212yesmaximal, complemented
8#34036511C212yesmaximal, complemented
9#34036503C212yesmaximal, complemented
10#34036085C211yesmaximal, characteristic, complemented
11#536×C293C211yesmaximal, characteristic, complemented
12C2×#17989802C2211yescharacteristic
13C22×D1172C2211yescharacteristic
14#22627655—24no—
15#22627649—24no—
16#22627644—24no—
17C2×#17989792C2212yescomplemented
18C2×#17989792C2211yescharacteristic, complemented
19C2×#17989790C2211yescharacteristic, complemented
20C2×#17989789—24no—
21C2×#17989789C2212yes—
22C2×#17989789C2212yes—
23C2×#17989789C2212yes—
24C2×#17989789C2211yescharacteristic, complemented
25C2×#17989789C2212yescomplemented
26C2×#17989787C2212yes—
27C2×#17989787C2211yescharacteristic, complemented
28C2×#17989786C2211yescharacteristic
29C22×#14556873C2211yescharacteristic
30#22627621C2212yes—
31#22627612C2212yes—
32#22627599—22no—
33#22627594—22no—
34#22627535C2212yescomplemented
35C4×#14556873C2212yescomplemented
36C2×#41×C293C2211yescharacteristic
37C22×Q8×C293C2211yescharacteristic
38C4⋊Q8×C293C2212yescomplemented
39#124×C293C2212yescomplemented
40#123×C293—22no—
41C22⋊Q8×C293—22no—
42#117×C293C2211yescharacteristic
43C23×D293—21no—
44#17989802—22no—
45#17989802—22no—
46C2×Dic586C2311yescharacteristic
47C2×D1172—22no—
48C2×D1172—22no—
49C2×D1172—21no—
50C2×D1172—24no—
51C2×D1172C2312yes—
52C2×D1172C2311yescharacteristic
53C2×C22⋊D293—21no—
54C2×C4×D293—21no—
55#17989792—22no—
56#17989792—24no—
57#17989790—22no—
58#17989789—24no—
59#17989789—24no—
60#17989789—24no—
61#17989789—22no—
62#17989789—24no—
63#17989787—24no—
64#17989787—22no—
65#17989786D414yescomplemented
66C2×#14556873D414yescomplemented
67C2×#14556873C2311yescharacteristic
68C2×#14556873C2311yescharacteristic
69C2×#14556873C2312yes—
70#17989779—24no—
71#41×C293—22no—
72#41×C293—22no—
73C2×Q8×C293—21no—
74C2×Q8×C293—22no—
75C2×Q8×C293—22no—
76C2×Q8×C293C2312yes—
77C2×Q8×C293C2311yescharacteristic
78C2×D4×C293—21no—
79C2×D4×C293C2311yescharacteristic
80C22×C4×C293—21no—
81C22×C4×C293—21no—
82C22×C4×C293C2311yescharacteristic
83C4⋊C4×C293—24no—
84C4⋊C4×C293D412yes—
85C22⋊C4×C293—24no—
86C22⋊C4×C293D412yes—
87C42×C293C2312yes—
88C22×D293—22no—
89C22×D293—21no—
90C22×D293—42no—
91C22×D293—82no—
92Dic586—21no—
93Dic586—21no—
94D1172—22no—
95D1172—21no—
96D1172—22no—
97D1172—21no—
98D1172—42no—
99D1172—84no—
100C22⋊D293—42no—
101C2×Dic293—21no—
102C4×D293—42no—
103#14556873C2×D412yes—
104#14556873C2×D412yes—
105#14556873C2×D414yes—
106C23×C293—21no—
107C23×C293C2×D411yescharacteristic
108Q8×C293—22no—
109Q8×C293—22no—
110Q8×C293—21no—
111Q8×C293—21no—
112Q8×C293—42no—
113D4×C293—21no—
114D4×C293—21no—
115D4×C293—42no—
116C2×C4×C293—22no—
117C2×C4×C293—21no—
118C2×C4×C293—42no—
119C2×C4×C293—42no—
120C2×C4×C293C2×D412yes—
121C2×C4×C293C2×D412yes—
122C2×C4×C293C2×D414yes—
123C2×C4×C293C2×D411yescharacteristic
124C2×C4×C293C2411yescharacteristic
125C22×C293—22no—
126C22×C293—41no—
127C22×C293—42no—
128C22×C293#12712yes—
129C22×C293C22×D411yescharacteristic
130C2×D293—41no—
131C2×D293—42no—
132C2×D293—42no—
133C2×D293—44no—
134C2×D293—84no—
135C4×C293—24no—
136C4×C293—41no—
137C4×C293—42no—
138C4×C293#12712yes—
139C4×C293C22×D411yescharacteristic
140C4×C293C22×D411yescharacteristic
141Dic293—41no—
142C2×C293—22no—
143C2×C293—81no—
144C2×C293C2×#12711yescharacteristic
145C2×C293#57712yes—
146D293—81no—
147D293—82no—
148C293#315411yescharacteristic, complemented, Sylow
149#3154—2931nomaximal, Sylow
150C2×#137—2931no—
151C2×AGL(1,ℤ/8)—2931no—
152C2×#135—2931no—
153C22×SD16—2931no—
154#536D29311yescharacteristic, complemented
155#535—2931no—
156#497—2932no—
157#496—2932no—
158#495—2932no—
159#492—2932no—
160#405—2931no—
161C2×#41—2931no—
162C2×#41C2×D29311yescharacteristic
163C22×Q8C2×D29311yescharacteristic
164C22×D4—2931no—
165#137—5864no—
166AGL(1,ℤ/8)—5864no—
167#135—5864no—
168C2×Q16—2931no—
169C2×Q16—2932no—
170C2×SD16—2932no—
171C2×SD16—2932no—
172C2×SD16—2932no—
173C2×SD16—2931no—
174C2×SD16—2932no—
175C2×SD16—2931no—
176C2×SD16—5864no—
177C2×D8—2932no—
178C2×D8—2931no—
179C2×M16—2931no—
180C22×C8—2931no—
181C4⋊Q8C2×D29312yescomplemented
182#127—2932no—
183#124—2932no—
184#124C2×D29312yescomplemented
185#123—22no—
186C22⋊Q8—22no—
187#121—5862no—
188#120—5862no—
189#117C2×D29311yescharacteristic
190#97—2932no—
191C4×C8—2932no—
192C24—5861no—
193#41—22no—
194#41—22no—
195#41—5862no—
196#41—5862no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—2931no—
201C2×Q8C22×D29312yes—
202C2×Q8C22×D29311yescharacteristic
203C2×D4—21no—
204C2×D4—2932no—
205C2×D4—2931no—
206C2×D4—5861no—
207C2×D4—5862no—
208C2×D4—5862no—
209C2×D4—5864no—
210C2×D4—5861no—
211C2×D4C22×D29311yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—5861no—
215C22×C4C22×D29311yescharacteristic
216Q16—5862no—
217Q16—5864no—
218SD16—5862no—
219SD16—5864no—
220SD16—5864no—
221SD16—5862no—
222SD16—5864no—
223SD16—5864no—
224D8—5862no—
225D8—5864no—
226M16—2934no—
227C2×C8—2934no—
228C2×C8—2931no—
229C2×C8—2932no—
230C2×C8—2931no—
231C4⋊C4—24no—
232C4⋊C4C22⋊D29312yes—
233C22⋊C4—24no—
234C22⋊C4—5864no—
235C22⋊C4C22⋊D29312yes—
236C42C22×D29312yes—
237C23—21no—
238C23—23442no—
239C23—5862no—
240C23—5861no—
241C23—11722no—
242C23C2×C22⋊D29311yescharacteristic
243Q8—22no—
244Q8—21no—
245Q8—22no—
246Q8—21no—
247Q8—42no—
248Q8—5861no—
249Q8—5861no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—23444no—
254D4—5862no—
255D4—5861no—
256D4—5862no—
257D4—5861no—
258D4—11722no—
259D4—11722no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—5861no—
265C2×C4—11722no—
266C2×C4D4×D29312yes—
267C2×C4D4×D29312yes—
268C2×C4C2×C22⋊D29314yes—
269C2×C4C2×C22⋊D29311yescharacteristic
270C2×C4C23×D29311yescharacteristic
271C8—2932no—
272C8—2932no—
273C8—2934no—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—23444no—
278C22—11721no—
279C22—11722no—
280C22—11722no—
281C22—11724no—
282C22#2262762012yes—
283C22C22×C22⋊D29311yescharacteristic
284C4—24no—
285C4—41no—
286C4—42no—
287C4—11721no—
288C4#2262762012yes—
289C4C2×D4×D29311yescharacteristic
290C4C2×D4×D29311yescharacteristic
291C2—22no—
292C2—81no—
293C2—23442no—
294C2—23441no—
295C2C2×#2262762011yescharacteristic
296C2#3403693712yes—
subgroupG/H
centerC22C22×C22⋊D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#536×C293C2
socleC22×C293C22×D4
hypercenter#536D293
hypocenterC293#3154
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#22627655C2semidirect
2. C2×#22627649C2semidirect
3. C2×#22627644C2semidirect
4. C22×#17989789C2semidirect
5. #34036698C2semidirect
6. #34036521C2semidirect
7. #34036519C2semidirect
8. #34036511C2semidirect
9. #34036503C2semidirect
10. #34036085C2semidirect
11. #536×C293C2semidirect
17. C2×#17989792C22semidirect
18. C2×#17989792C22semidirect
19. C2×#17989790C22semidirect
24. C2×#17989789C22semidirect
25. C2×#17989789C22semidirect
27. C2×#17989787C22semidirect
34. #22627535C22semidirect
35. C4×#14556873C22semidirect
38. C4⋊Q8×C293C22semidirect
39. #124×C293C22semidirect
65. #17989786D4semidirect
66. C2×#14556873D4semidirect
148. C293#3154semidirect
154. #536D293semidirect
181. C4⋊Q8C2×D293semidirect
184. #124C2×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276552
C2×#226276492
C2×#226276442
C22×#179897892
#340366982
#340365212
#340365192
#340365112
#340365032
#340360852
#536×C2932
#3154293