Finite Group Databasebeta

← #49169170

Subgroups — #49169170

Order 37504.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#34036684C211yesmaximal, characteristic, complemented
2#124×D293C211yesmaximal, characteristic
3#34036491C211yesmaximal, characteristic, complemented
4#34036489C211yesmaximal, characteristic, complemented
5#34036488C211yesmaximal, characteristic, complemented
6#34036485C212yesmaximal, complemented
7#34036482C212yesmaximal, complemented
8#34036136C212yesmaximal, complemented
9#34036132C212yesmaximal
10#34036081C211yesmaximal, characteristic, complemented
11#489×C293C211yesmaximal, characteristic, complemented
12C2×Q8×D293C2211yescharacteristic
13C2×D4×D293C2211yescharacteristic
14C2×#17989785C2212yes—
15#22627624C2211yescharacteristic, complemented
16#22627615C2212yescomplemented
17#22627613C2211yescharacteristic
18#22627612C2211yescharacteristic
19#22627611C2211yescharacteristic
20#22627610—21no—
21#22627610—21no—
22#22627604—21no—
23#22627600—21no—
24Q8×Dic293—21no—
25#22627589—21no—
26#22627582C2212yes—
27C22⋊C4×D293—21no—
28C22⋊C4×D293—21no—
29C42×D293C2211yescharacteristic
30#22627549C2212yescomplemented
31#22627548C2212yescomplemented
32#22627547C2212yescomplemented
33#22627546C2212yes—
34#22627545C2212yes—
35#22627544C2212yes—
36#22627535C2211yescharacteristic, complemented
37C8×Dic293C2212yescomplemented
38C4⋊Q8×C293C2211yescharacteristic, complemented
39#124×C293C2211yescharacteristic
40#103×C293C2212yescomplemented
41#102×C293C2212yes—
42#97×C293C2211yescharacteristic, complemented
43C23×D293—21no—
44C2×Dic586—21no—
45C2×Dic586—21no—
46C2×Dic586C2311yescharacteristic
47Q8×D293—24no—
48C2×D1172C2311yescharacteristic
49C2×C22⋊D293—21no—
50D4×D293—24no—
51C2×C4×D293—21no—
52C2×C4×D293D411yescharacteristic
53C2×C4×D293D411yescharacteristic
54C2×C4×D293C2311yescharacteristic
55#17989785—24no—
56C2×#14556873C2312yes—
57C4⋊Dic293—22no—
58C4⋊Dic293—21no—
59C4⋊Dic293C2312yes—
60#17989781—22no—
61C22⋊Dic293—21no—
62C22⋊Dic293—21no—
63#17989779—22no—
64#17989779—21no—
65#17989779—21no—
66#17989779—21no—
67#17989779—21no—
68C4×Dic293—22no—
69C4×Dic293D411yescharacteristic
70C4×Dic293D411yescharacteristic
71C4×Dic293C2311yescharacteristic
72C2×Q8×C293—21no—
73C2×Q8×C293C2311yescharacteristic
74C2×D4×C293C2311yescharacteristic
75C2×C8×C293C2312yes—
76C4⋊C4×C293—21no—
77C4⋊C4×C293C2312yes—
78C22⋊C4×C293—21no—
79C22⋊C4×C293—21no—
80C42×C293C2311yescharacteristic
81C22×D293—21no—
82C22×D293—42no—
83C22×D293—42no—
84C22×D293—42no—
85C22×D293C2×D411yescharacteristic
86Dic586—22no—
87Dic586—42no—
88Dic586—42no—
89D1172—22no—
90C22⋊D293—42no—
91C2×Dic293—22no—
92C2×Dic293—21no—
93C2×Dic293C2×D411yescharacteristic
94C2×Dic293C2×D411yescharacteristic
95C2×Dic293C2×D411yescharacteristic
96C4×D293—41no—
97C4×D293—41no—
98C4×D293—42no—
99C4×D293#4114yes—
100#14556873—22no—
101C23×C293—21no—
102Q8×C293—22no—
103Q8×C293—42no—
104D4×C293—22no—
105C2×C4×C293—21no—
106C2×C4×C293—22no—
107C2×C4×C293C2×D411yescharacteristic
108C2×C4×C293C2×D411yescharacteristic
109C2×C4×C293C2411yescharacteristic
110C8×C293—22no—
111C22×C293—41no—
112C22×C293—41no—
113C22×C293—41no—
114C22×C293C22×D411yescharacteristic
115C2×D293—21no—
116C2×D293—21no—
117C2×D293—41no—
118C2×D293—41no—
119C2×D293—41no—
120C2×D293—84no—
121C2×D293#12412yes—
122C4×C293—21no—
123C4×C293—21no—
124C4×C293—41no—
125C4×C293—42no—
126C4×C293C2×#4112yes—
127Dic293—21no—
128Dic293—21no—
129Dic293—41no—
130Dic293—42no—
131Dic293#12412yes—
132C2×C293—81no—
133C2×C293C2×#12411yescharacteristic
134C2×C293#57511yescharacteristic
135C2×C293#57511yescharacteristic
136D293—22no—
137D293—81no—
138C293#314011yescharacteristic, complemented, Sylow
139#3140—2931nomaximal, Sylow
140#533—2931no—
141C2×#124—2931no—
142#489—2931no—
143#489—2931no—
144#489—2931no—
145#489D29311yescharacteristic, complemented
146#488—2932no—
147#488—2932no—
148#419—2932no—
149#418—2932no—
150C4×M16—2931no—
151C22×Q8—2931no—
152C22×D4—2931no—
153C2×M16—2932no—
154C4⋊Q8—2931no—
155C4⋊Q8C2×D29311yescharacteristic, complemented
156#125—2932no—
157#124—2931no—
158#124—2931no—
159#124—2931no—
160#124—5861no—
161#124—5861no—
162#124C2×D29311yescharacteristic
163C22⋊Q8—5861no—
164C22⋊Q8—5861no—
165C4×Q8—5861no—
166C4×Q8—5861no—
167#117—2932no—
168C2×C22⋊C4—5861no—
169C2×C22⋊C4—5861no—
170C2×C42—2931no—
171#103—2932no—
172#103—2932no—
173#103—2932no—
174#103C2×D29312yescomplemented
175#102—2932no—
176#102—2932no—
177#102—2932no—
178#102C2×D29312yes—
179#97—2931no—
180#97C2×D29311yescharacteristic, complemented
181C4×C8—2932no—
182C24—5861no—
183C2×Q8—21no—
184C2×Q8—2931no—
185C2×Q8—5861no—
186C2×Q8—5864no—
187C2×Q8—5861no—
188C2×Q8C22×D29311yescharacteristic
189C2×D4—2931no—
190C2×D4—5864no—
191C2×D4—5861no—
192C2×D4C22×D29311yescharacteristic
193C22×C4—2931no—
194C22×C4—2931no—
195C22×C4—2931no—
196C22×C4—5861no—
197M16—5864no—
198C2×C8—2932no—
199C2×C8C22×D29312yes—
200C4⋊C4—21no—
201C4⋊C4—2932no—
202C4⋊C4—5862no—
203C4⋊C4—5862no—
204C4⋊C4—5861no—
205C4⋊C4C22×D29312yes—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—5861no—
209C22⋊C4—5861no—
210C22⋊C4—5862no—
211C22⋊C4—5861no—
212C22⋊C4—5861no—
213C22⋊C4—5861no—
214C22⋊C4—5861no—
215C42—2931no—
216C42—2931no—
217C42—2931no—
218C42—5862no—
219C42C22×D29311yescharacteristic
220C23—21no—
221C23—2931no—
222C23—5861no—
223C23—11722no—
224C23—11722no—
225C23—11722no—
226Q8—22no—
227Q8—42no—
228Q8—5862no—
229Q8—11722no—
230Q8—11722no—
231D4—22no—
232D4—5862no—
233D4—11722no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—2931no—
237C2×C4—2934no—
238C2×C4—2931no—
239C2×C4—2931no—
240C2×C4—5862no—
241C2×C4—5861no—
242C2×C4—11721no—
243C2×C4—11721no—
244C2×C4—11722no—
245C2×C4D4×D29311yescharacteristic
246C2×C4D4×D29311yescharacteristic
247C2×C4C23×D29311yescharacteristic
248C8—22no—
249C8—5862no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22—2932no—
254C22—23444no—
255C22—5861no—
256C22—5861no—
257C22—11721no—
258C22—11721no—
259C22—11721no—
260C22C2×D4×D29311yescharacteristic
261C4—21no—
262C4—21no—
263C4—41no—
264C4—42no—
265C4—2932no—
266C4—5861no—
267C4—5861no—
268C4—11721no—
269C4—11722no—
270C4#41×D29312yes—
271C2—81no—
272C2—23441no—
273C2—5862no—
274C2#124×D29311yescharacteristic
275C2#3403692511yescharacteristic
276C2#3403692811yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#489×C293C2
socleC22×C293C22×D4
prime radicalC2×D4×D293C22
hypercenter#489D293
hypocenterC293#3140
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #34036684C2semidirect
3. #34036491C2semidirect
4. #34036489C2semidirect
5. #34036488C2semidirect
6. #34036485C2semidirect
7. #34036482C2semidirect
8. #34036136C2semidirect
10. #34036081C2semidirect
11. #489×C293C2semidirect
15. #22627624C22semidirect
16. #22627615C22semidirect
30. #22627549C22semidirect
31. #22627548C22semidirect
32. #22627547C22semidirect
36. #22627535C22semidirect
37. C8×Dic293C22semidirect
38. C4⋊Q8×C293C22semidirect
40. #103×C293C22semidirect
42. #97×C293C22semidirect
138. C293#3140semidirect
145. #489D293semidirect
155. C4⋊Q8C2×D293semidirect
174. #103C2×D293semidirect
180. #97C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340366842
#124×D2932
#340364912
#340364892
#340364882
#340364852
#340364822
#340361362
#340361322
#340360812
#489×C2932
#3140293