Finite Group Databasebeta

← #49169252

Subgroups — #49169252

Order 37504.

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

normal subgroups only

This group has 572 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×#22627652C211yesmaximal, characteristic, complemented
3C2×#22627649C211yesmaximal, characteristic, complemented
4#34036675C211yesmaximal, characteristic, complemented
5C2×#22627613C211yesmaximal, characteristic, complemented
6#34036522C212yesmaximal, complemented
7#34036515C212yesmaximal, complemented
8#34036514C212yesmaximal, complemented
9#34036510C212yesmaximal, complemented
10M16×Dic293C211yesmaximal, characteristic, complemented
11C2×#137×C293C211yesmaximal, characteristic, complemented
12C2×#17989802C2211yescharacteristic
13C22×D1172C2211yescharacteristic
14#22627655—24no—
15#22627652—24no—
16#22627649—24no—
17C2×#17989792C2212yescomplemented
18C2×#17989791C2212yes—
19C2×#17989789C2212yes—
20C2×#17989789C2212yescomplemented
21C2×D2344C2212yes—
22C2×#17989787C2212yes—
23C2×#17989786C2211yescharacteristic
24#22627622C2211yescharacteristic, complemented
25#22627620C2211yescharacteristic, complemented
26#22627613—22no—
27#22627613C2212yes—
28#22627613C2212yes—
29#22627613C2211yescharacteristic, complemented
30#22627611C2211yescharacteristic, complemented
31#22627596—22no—
32C4×C22⋊D293—22no—
33C2×#17989779—22no—
34C2×C4×Dic293C2211yescharacteristic
35#22627534C2212yescomplemented
36C8×Dic293C2212yescomplemented
37C2×#41×C293C2211yescharacteristic
38C22×Q8×C293C2211yescharacteristic
39#137×C293—24no—
40C2×Q16×C293C2212yescomplemented
41C2×SD16×C293C2212yescomplemented
42C2×M16×C293C2211yescharacteristic
43C23×D293—21no—
44#17989802—22no—
45#17989802—22no—
46C2×Dic586C2311yescharacteristic
47C2×D1172—22no—
48C2×D1172—22no—
49C2×D1172—21no—
50C2×D1172C2312yes—
51C2×D1172C2311yescharacteristic
52C2×C22⋊D293—22no—
53C2×C22⋊D293—21no—
54C22×Dic293D412yes—
55C2×C4×D293—21no—
56#17989792—24no—
57#17989791—24no—
58#17989789—24no—
59#17989789—24no—
60D2344—24no—
61#17989787—24no—
62#17989786D414yescomplemented
63C2×#14556873C2312yes—
64#17989781—22no—
65C22⋊Dic293—22no—
66#17989779—22no—
67#17989779—24no—
68#17989779—24no—
69C4×Dic293D412yes—
70C4×Dic293C2311yescharacteristic
71C4×Dic293C2311yescharacteristic
72#41×C293—22no—
73#41×C293—22no—
74C2×Q8×C293—21no—
75C2×Q8×C293—22no—
76C2×Q8×C293—22no—
77C2×Q8×C293C2312yes—
78C2×Q8×C293C2311yescharacteristic
79C2×D4×C293—21no—
80C2×D4×C293C2311yescharacteristic
81C22×C4×C293—21no—
82C22×C4×C293—21no—
83C22×C4×C293C2311yescharacteristic
84Q16×C293—24no—
85SD16×C293—24no—
86M16×C293D414yescomplemented
87C2×C8×C293C2312yes—
88C22×D293—22no—
89C22×D293—21no—
90C22×D293—44no—
91C22×D293—42no—
92Dic586—21no—
93Dic586—21no—
94D1172—22no—
95D1172—21no—
96D1172—22no—
97D1172—21no—
98D1172—42no—
99C22⋊D293—44no—
100C22⋊D293—42no—
101C2×Dic293—21no—
102C2×Dic293—42no—
103C2×Dic293C2×D412yes—
104C2×Dic293C2×D412yes—
105C4×D293—42no—
106#14556873C2×D414yes—
107C23×C293—21no—
108C23×C293C2×D411yescharacteristic
109Q8×C293—22no—
110Q8×C293—21no—
111Q8×C293—22no—
112Q8×C293—21no—
113Q8×C293—42no—
114D4×C293—21no—
115D4×C293—21no—
116D4×C293—42no—
117C2×C4×C293—21no—
118C2×C4×C293—22no—
119C2×C4×C293—42no—
120C2×C4×C293—42no—
121C2×C4×C293C2×D411yescharacteristic
122C2×C4×C293C2×D412yes—
123C2×C4×C293C2×D412yes—
124C2×C4×C293C2411yescharacteristic
125C8×C293C2×D414yes—
126C22×C293—22no—
127C22×C293—41no—
128C22×C293—42no—
129C22×C293#12712yes—
130C22×C293C22×D411yescharacteristic
131C2×D293—41no—
132C2×D293—42no—
133C2×D293—42no—
134C2×D293—44no—
135C2×D293—84no—
136C4×C293—41no—
137C4×C293—42no—
138C4×C293#12712yes—
139C4×C293C22×D411yescharacteristic
140C4×C293C22×D411yescharacteristic
141Dic293—22no—
142Dic293—22no—
143Dic293—41no—
144C2×C293—22no—
145C2×C293—81no—
146C2×C293C2×#12711yescharacteristic
147C2×C293#57512yes—
148D293—81no—
149D293—82no—
150C293#315311yescharacteristic, complemented, Sylow
151#3153—2931nomaximal, Sylow
152C2×#137—2931no—
153C2×#137D29311yescharacteristic, complemented
154C2×AGL(1,ℤ/8)—2931no—
155C2×AGL(1,ℤ/8)—2931no—
156#532—2931no—
157C2×#124—2931no—
158#497—2932no—
159#496—2932no—
160#495—2932no—
161#495—2932no—
162C4×M16—2931no—
163C2×#41—2931no—
164C2×#41C2×D29311yescharacteristic
165C22×Q8C2×D29311yescharacteristic
166C22×D4—2931no—
167#137—24no—
168#137—5864no—
169AGL(1,ℤ/8)—5864no—
170AGL(1,ℤ/8)—5864no—
171C2×Q16—2932no—
172C2×Q16C2×D29312yescomplemented
173C2×SD16—2932no—
174C2×SD16—2932no—
175C2×SD16—2932no—
176C2×SD16C2×D29312yescomplemented
177C2×D8—2932no—
178C2×D8—2932no—
179C2×M16—2931no—
180C2×M16C2×D29311yescharacteristic
181C4⋊Q8—2931no—
182#127—2931no—
183#124—2932no—
184#124—2932no—
185#124—2931no—
186#124—2931no—
187#124—5862no—
188#121—5862no—
189C4×D4—5862no—
190C2×C22⋊C4—5862no—
191C2×C42—2931no—
192#97—2932no—
193C4×C8—2932no—
194C24—5861no—
195#41—22no—
196#41—22no—
197#41—5862no—
198#41—5862no—
199C2×Q8—21no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—2931no—
203C2×Q8C22×D29312yes—
204C2×Q8C22×D29311yescharacteristic
205C2×D4—21no—
206C2×D4—2932no—
207C2×D4—2931no—
208C2×D4—5862no—
209C2×D4—5861no—
210C2×D4—5862no—
211C2×D4—5861no—
212C2×D4—5862no—
213C2×D4C22×D29311yescharacteristic
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—2932no—
217C22×C4—5861no—
218C22×C4C22×D29311yescharacteristic
219Q16—24no—
220Q16—5864no—
221SD16—24no—
222SD16—5864no—
223SD16—5864no—
224SD16—5864no—
225D8—5864no—
226D8—5864no—
227M16—2934no—
228M16C22⋊D29314yescomplemented
229C2×C8—2932no—
230C2×C8C22×D29312yes—
231C4⋊C4—5862no—
232C22⋊C4—5864no—
233C22⋊C4—5864no—
234C22⋊C4—5862no—
235C22⋊C4—5862no—
236C42—2932no—
237C42—2931no—
238C42—2931no—
239C23—21no—
240C23—5862no—
241C23—5861no—
242C23—11724no—
243C23—11722no—
244C23D4×D29311yescharacteristic
245Q8—21no—
246Q8—22no—
247Q8—22no—
248Q8—21no—
249Q8—42no—
250Q8—5861no—
251Q8—5861no—
252D4—21no—
253D4—21no—
254D4—42no—
255D4—5862no—
256D4—5861no—
257D4—5861no—
258D4—5862no—
259D4—11724no—
260D4—11722no—
261D4—11722no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—2932no—
267C2×C4—2932no—
268C2×C4—5861no—
269C2×C4—11722no—
270C2×C4—11722no—
271C2×C4D4×D29311yescharacteristic
272C2×C4D4×D29312yes—
273C2×C4C2×C22⋊D29312yes—
274C2×C4C23×D29311yescharacteristic
275C8—2934no—
276C8C2×C22⋊D29314yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—23444no—
281C22—11721no—
282C22—11722no—
283C22—11722no—
284C22—11724no—
285C22#2262762012yes—
286C22C2×D4×D29311yescharacteristic
287C4—41no—
288C4—42no—
289C4—5862no—
290C4—5862no—
291C4—11721no—
292C4#2262762012yes—
293C4C2×D4×D29311yescharacteristic
294C4C22×C22⋊D29311yescharacteristic
295C2—22no—
296C2—81no—
297C2—23442no—
298C2—23441no—
299C2C2×#2262762011yescharacteristic
300C2#3403692812yes—
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
FittingC2×#137×C293C2
socleC22×C293C22×D4
hypercenterC2×#137D293
hypocenterC293#3153
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#22627655C2semidirect
2. C2×#22627652C2semidirect
3. C2×#22627649C2semidirect
4. #34036675C2semidirect
5. C2×#22627613C2semidirect
6. #34036522C2semidirect
7. #34036515C2semidirect
8. #34036514C2semidirect
9. #34036510C2semidirect
10. M16×Dic293C2semidirect
11. C2×#137×C293C2semidirect
17. C2×#17989792C22semidirect
20. C2×#17989789C22semidirect
24. #22627622C22semidirect
25. #22627620C22semidirect
29. #22627613C22semidirect
30. #22627611C22semidirect
35. #22627534C22semidirect
36. C8×Dic293C22semidirect
40. C2×Q16×C293C22semidirect
41. C2×SD16×C293C22semidirect
62. #17989786D4semidirect
86. M16×C293D4semidirect
150. C293#3153semidirect
153. C2×#137D293semidirect
172. C2×Q16C2×D293semidirect
176. C2×SD16C2×D293semidirect
228. M16C22⋊D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276552
C2×#226276522
C2×#226276492
#340366752
C2×#226276132
#340365222
#340365152
#340365142
#340365102
M16×Dic2932
C2×#137×C2932
#3153293