Finite Group Databasebeta

← #49168711

Subgroups — #49168711

Order 37504.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 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#34036641C211yesmaximal, characteristic
2C2×C4⋊C4×D293C211yesmaximal, characteristic, complemented
3#34036466C212yesmaximal, complemented
4#34036461C212yesmaximal, complemented
5#34036449C212yesmaximal, complemented
6#34036444C212yesmaximal, complemented
7#34036172C211yesmaximal, characteristic, complemented
8#34036131C211yesmaximal, characteristic, complemented
9#34036128C211yesmaximal, characteristic, complemented
10#34036092C211yesmaximal, characteristic, complemented
11#428×C293C211yesmaximal, characteristic, complemented
12C2×#17989802C2211yescharacteristic
13C22×C4×D293—21no—
14C22×C4×D293C2211yescharacteristic
15C2×#17989786C2211yescharacteristic
16#22627608—21no—
17#22627604C2211yescharacteristic
18#22627604C2211yescharacteristic
19#22627599C2211yescharacteristic
20#22627599C2211yescharacteristic
21#22627594—21no—
22C4×D1172—21no—
23C4×D1172—21no—
24C2×C4⋊Dic293C2211yescharacteristic
25C2×#17989781—21no—
26C4⋊C4×D293—21no—
27C4⋊C4×D293—21no—
28C4⋊C4×D293C2212yes—
29C4⋊C4×D293C2212yes—
30#22627557C2212yescomplemented
31#22627555C2212yescomplemented
32#22627549C2212yescomplemented
33#22627547C2212yescomplemented
34#22627546C2212yescomplemented
35#22627544C2212yescomplemented
36#22627536C2212yes—
37#22627536C2212yes—
38C2×M16×C293C2211yescharacteristic
39#117×C293C2211yescharacteristic
40C2×C4⋊C4×C293C2211yescharacteristic
41#107×C293C2212yescomplemented
42#106×C293C2212yescomplemented
43C23×D293D411yescharacteristic
44#17989802—22no—
45#17989802—22no—
46C2×Dic586C2311yescharacteristic
47C2×D1172—21no—
48C2×D1172—21no—
49C2×D1172C2311yescharacteristic
50C2×C22⋊D293—21no—
51C22×Dic293—21no—
52C22×Dic293D411yescharacteristic
53C2×C4×D293—22no—
54C2×C4×D293—22no—
55C2×C4×D293—21no—
56C2×C4×D293—22no—
57C2×C4×D293—22no—
58C2×C4×D293—44no—
59C2×C4×D293D411yescharacteristic
60C2×C4×D293D411yescharacteristic
61C2×C4×D293C2311yescharacteristic
62C2×C4×D293C2311yescharacteristic
63#17989786—22no—
64C2×#14556873C2312yes—
65C4⋊Dic293—21no—
66C4⋊Dic293—21no—
67C4⋊Dic293—21no—
68C4⋊Dic293C2312yes—
69#17989781—22no—
70#17989781—22no—
71#17989779—21no—
72#17989779—21no—
73#17989779—21no—
74#17989779—21no—
75C22×C4×C293—21no—
76C22×C4×C293C2311yescharacteristic
77M16×C293—22no—
78C2×C8×C293C2312yes—
79C4⋊C4×C293—21no—
80C4⋊C4×C293C2311yescharacteristic
81C4⋊C4×C293C2311yescharacteristic
82C4⋊C4×C293C2312yes—
83C22⋊C4×C293—21no—
84C42×C293—21no—
85C22×D293—22no—
86C22×D293—22no—
87C22×D293—22no—
88C22×D293—21no—
89C22×D293C2×D411yescharacteristic
90C22×D293C2×D411yescharacteristic
91Dic586—21no—
92Dic586—21no—
93D1172—21no—
94D1172—21no—
95D1172—42no—
96D1172—42no—
97C22⋊D293—42no—
98C2×Dic293—22no—
99C2×Dic293—21no—
100C2×Dic293—22no—
101C2×Dic293—42no—
102C2×Dic293C2×D411yescharacteristic
103C2×Dic293C2×D411yescharacteristic
104C4×D293—22no—
105C4×D293—22no—
106C4×D293—21no—
107C4×D293—21no—
108C4×D293—21no—
109C4×D293—21no—
110C4×D293—44no—
111C4×D293—44no—
112C4×D293—42no—
113#14556873—22no—
114C23×C293C2×D411yescharacteristic
115C2×C4×C293—21no—
116C2×C4×C293—21no—
117C2×C4×C293—22no—
118C2×C4×C293—42no—
119C2×C4×C293C2×D411yescharacteristic
120C2×C4×C293#4112yes—
121C2×C4×C293#4112yes—
122C2×C4×C293C2411yescharacteristic
123C8×C293—22no—
124C22×C293—21no—
125C22×C293—21no—
126C22×C293#12312yes—
127C22×C293C22×D411yescharacteristic
128C2×D293—28no—
129C2×D293—21no—
130C2×D293—21no—
131C2×D293—21no—
132C2×D293—21no—
133C2×D293—41no—
134C2×D293—41no—
135C2×D293—41no—
136C2×D293AGL(1,ℤ/8)12yes—
137C2×D293#13712yes—
138C4×C293—42no—
139C4×C293—41no—
140C4×C293—41no—
141C4×C293#12312yes—
142C4×C293C2×#4111yescharacteristic
143C4×C293C2×#4111yescharacteristic
144Dic293—21no—
145Dic293—21no—
146Dic293—41no—
147Dic293—42no—
148C2×C293—22no—
149C2×C293C2×#12311yescharacteristic
150C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C293C2×#13711yescharacteristic
152D293—22no—
153D293—22no—
154D293—81no—
155C293#309211yescharacteristic, complemented, Sylow
156#3092—2931nomaximal, Sylow
157#525—2931no—
158C22×C4⋊C4—2931no—
159#484—2932no—
160#483—2932no—
161#481—2932no—
162#480—2932no—
163#428—2931no—
164#428D29311yescharacteristic, complemented
165#417—2931no—
166#417—2931no—
167#407—2931no—
168C2×#41—2931no—
169C23×C4—2931no—
170C23×C4—5861no—
171C2×M16—2931no—
172C2×M16C2×D29311yescharacteristic
173#123—5861no—
174C22⋊Q8—2931no—
175C22⋊Q8—2931no—
176#121—2931no—
177#121—2931no—
178#120—5861no—
179C4×D4—5861no—
180C4×D4—5861no—
181#117C2×D29311yescharacteristic
182C2×C4⋊C4—2931no—
183C2×C4⋊C4—2932no—
184C2×C4⋊C4—2932no—
185C2×C4⋊C4—5861no—
186C2×C4⋊C4—5861no—
187C2×C4⋊C4—5861no—
188C2×C4⋊C4C2×D29311yescharacteristic
189#107—2932no—
190#107C2×D29312yescomplemented
191#106—2932no—
192#106C2×D29312yescomplemented
193#103—2932no—
194#103—2932no—
195#102—2932no—
196#102—2932no—
197C22⋊C8—2932no—
198C22⋊C8—2932no—
199C24—2931no—
200#41—5862no—
201#41—5862no—
202C2×Q8—2931no—
203C2×D4—2931no—
204C2×D4—5861no—
205C2×D4—5861no—
206C2×D4—5861no—
207C22×C4—21no—
208C22×C4—2931no—
209C22×C4—2931no—
210C22×C4—2931no—
211C22×C4—2931no—
212C22×C4—2931no—
213C22×C4—5862no—
214C22×C4—5862no—
215C22×C4—5862no—
216C22×C4—5862no—
217C22×C4—5861no—
218C22×C4—5861no—
219C22×C4—11724no—
220C22×C4C22×D29311yescharacteristic
221M16—22no—
222M16—5862no—
223C2×C8—2932no—
224C2×C8C22×D29312yes—
225C4⋊C4—21no—
226C4⋊C4—2932no—
227C4⋊C4—5862no—
228C4⋊C4—5862no—
229C4⋊C4—5861no—
230C4⋊C4—5861no—
231C4⋊C4—5861no—
232C4⋊C4C22×D29312yes—
233C4⋊C4C22×D29311yescharacteristic
234C4⋊C4C22×D29311yescharacteristic
235C22⋊C4—21no—
236C22⋊C4—5861no—
237C22⋊C4—5861no—
238C22⋊C4—5861no—
239C22⋊C4—5861no—
240C42—21no—
241C23—2931no—
242C23—2931no—
243C23—5862no—
244C23—5862no—
245C23—5862no—
246C23—5861no—
247C23D4×D29311yescharacteristic
248Q8—5861no—
249Q8—5861no—
250D4—5861no—
251D4—5861no—
252D4—11722no—
253D4—11722no—
254D4—11722no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—2931no—
260C2×C4—2931no—
261C2×C4—5862no—
262C2×C4—5862no—
263C2×C4—5862no—
264C2×C4—5861no—
265C2×C4—5861no—
266C2×C4—5861no—
267C2×C4—5861no—
268C2×C4—5861no—
269C2×C4—5862no—
270C2×C4—11724no—
271C2×C4—11724no—
272C2×C4—11722no—
273C2×C4—11722no—
274C2×C4D4×D29311yescharacteristic
275C2×C4#1798980212yes—
276C2×C4#1798980312yes—
277C2×C4C23×D29311yescharacteristic
278C8—22no—
279C8—5862no—
280C22—21no—
281C22—21no—
282C22—2932no—
283C22—2932no—
284C22—5868no—
285C22—5861no—
286C22—5861no—
287C22—5861no—
288C22—5861no—
289C22—11721no—
290C22—11721no—
291C22—11721no—
292C22#2262760912yes—
293C22C2×D4×D29311yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—5861no—
298C4—5861no—
299C4—11721no—
300C4—11722no—
301C4#2262760912yes—
302C4C2×#1798980211yescharacteristic
303C4C2×#1798980311yescharacteristic
304C2—22no—
305C2—23441no—
306C2—5862no—
307C2—5862no—
308C2C2×#2262760911yescharacteristic
309C2AGL(1,ℤ/8)×D29311yescharacteristic
310C2#137×D29311yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#428×C293C2
socleC22×C293C22×D4
prime radical#34036641C2
hypercenter#428D293
hypocenterC293#3092
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D293C2semidirect
3. #34036466C2semidirect
4. #34036461C2semidirect
5. #34036449C2semidirect
6. #34036444C2semidirect
7. #34036172C2semidirect
8. #34036131C2semidirect
9. #34036128C2semidirect
10. #34036092C2semidirect
11. #428×C293C2semidirect
30. #22627557C22semidirect
31. #22627555C22semidirect
32. #22627549C22semidirect
33. #22627547C22semidirect
34. #22627546C22semidirect
35. #22627544C22semidirect
41. #107×C293C22semidirect
42. #106×C293C22semidirect
155. C293#3092semidirect
164. #428D293semidirect
190. #107C2×D293semidirect
192. #106C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340366412
C2×C4⋊C4×D2932
#340364662
#340364612
#340364492
#340364442
#340361722
#340361312
#340361282
#340360922
#428×C2932
#3092293