Finite Group Databasebeta

← #49168706

Subgroups — #49168706

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#34036636C211yesmaximal, characteristic
2C2×C4⋊C4×D293C211yesmaximal, characteristic, complemented
3#34036463C212yesmaximal, complemented
4#34036458C212yesmaximal, complemented
5#34036446C212yesmaximal, complemented
6#34036441C212yesmaximal, complemented
7#34036171C211yesmaximal, characteristic, complemented
8#34036169C211yesmaximal, characteristic, complemented
9#34036130C211yesmaximal, characteristic, complemented
10#34036092C211yesmaximal, characteristic, complemented
11#417×C293C211yesmaximal, characteristic, complemented
12C22×C4×D293—21no—
13C22×C4×D293C2211yescharacteristic
14C2×#17989786C2211yescharacteristic
15#22627605—21no—
16#22627600C2211yescharacteristic
17#22627600C2211yescharacteristic
18#22627595C2211yescharacteristic
19#22627595C2211yescharacteristic
20#22627592—21no—
21C4×C22⋊D293—21no—
22C4×C22⋊D293—21no—
23#22627583C2211yescharacteristic
24C2×C4⋊Dic293C2211yescharacteristic
25C2×#17989781—21no—
26C4⋊C4×D293—21no—
27C4⋊C4×D293—21no—
28C4⋊C4×D293C2212yes—
29C4⋊C4×D293C2212yes—
30#22627558C2212yescomplemented
31#22627557C2212yescomplemented
32#22627556C2212yescomplemented
33#22627555C2212yescomplemented
34#22627548C2212yescomplemented
35#22627545C2212yescomplemented
36#22627536C2212yes—
37#22627536C2212yes—
38C2×#41×C293C2211yescharacteristic
39C2×M16×C293C2211yescharacteristic
40C2×C4⋊C4×C293C2211yescharacteristic
41#103×C293C2212yescomplemented
42#102×C293C2212yescomplemented
43C23×D293D411yescharacteristic
44C2×C22⋊D293—21no—
45C2×C22⋊D293—21no—
46C22×Dic293—21no—
47C22×Dic293D411yescharacteristic
48C2×C4×D293—22no—
49C2×C4×D293—22no—
50C2×C4×D293—22no—
51C2×C4×D293—22no—
52C2×C4×D293—44no—
53C2×C4×D293D411yescharacteristic
54C2×C4×D293D411yescharacteristic
55C2×C4×D293C2311yescharacteristic
56C2×C4×D293C2311yescharacteristic
57#17989786—22no—
58C2×#14556873C2312yes—
59C4⋊Dic293—21no—
60C4⋊Dic293C2311yescharacteristic
61C4⋊Dic293C2311yescharacteristic
62C4⋊Dic293C2312yes—
63#17989781—21no—
64#17989781—21no—
65#17989781—22no—
66#17989781—22no—
67C22⋊Dic293—21no—
68C22⋊Dic293—21no—
69C22⋊Dic293—21no—
70#17989779—21no—
71#17989779—21no—
72C4×Dic293—21no—
73#41×C293—22no—
74#41×C293—22no—
75C2×Q8×C293C2311yescharacteristic
76C2×D4×C293—21no—
77C2×D4×C293C2311yescharacteristic
78C22×C4×C293—21no—
79C22×C4×C293—21no—
80C22×C4×C293C2311yescharacteristic
81M16×C293—22no—
82C2×C8×C293C2312yes—
83C4⋊C4×C293—21no—
84C4⋊C4×C293C2312yes—
85C22×D293—22no—
86C22×D293—22no—
87C22×D293—22no—
88C22×D293C2×D411yescharacteristic
89C22×D293C2×D411yescharacteristic
90C22⋊D293—42no—
91C22⋊D293—42no—
92C2×Dic293—22no—
93C2×Dic293—21no—
94C2×Dic293—21no—
95C2×Dic293—22no—
96C2×Dic293—42no—
97C2×Dic293C2×D411yescharacteristic
98C2×Dic293C2×D411yescharacteristic
99C4×D293—22no—
100C4×D293—22no—
101C4×D293—21no—
102C4×D293—21no—
103C4×D293—21no—
104C4×D293—21no—
105C4×D293—44no—
106C4×D293—44no—
107#14556873—22no—
108C23×C293—21no—
109C23×C293C2×D411yescharacteristic
110Q8×C293—21no—
111Q8×C293—21no—
112D4×C293—21no—
113D4×C293—21no—
114D4×C293—42no—
115C2×C4×C293—22no—
116C2×C4×C293—21no—
117C2×C4×C293—42no—
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—41no—
127C22×C293—41no—
128C22×C293—41no—
129C22×C293#12312yes—
130C22×C293C22×D411yescharacteristic
131C2×D293—28no—
132C2×D293—21no—
133C2×D293—21no—
134C2×D293—21no—
135C2×D293—21no—
136C2×D293AGL(1,ℤ/8)12yes—
137C2×D293#13712yes—
138C4×C293—41no—
139C4×C293—42no—
140C4×C293#12312yes—
141C4×C293C2×#4111yescharacteristic
142C4×C293C2×#4111yescharacteristic
143Dic293—21no—
144Dic293—21no—
145Dic293—42no—
146Dic293—41no—
147Dic293—41no—
148C2×C293—22no—
149C2×C293—81no—
150C2×C293C2×#12311yescharacteristic
151C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C293C2×#13711yescharacteristic
153D293—22no—
154D293—22no—
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#428—2931no—
165#417—2931no—
166#417D29311yescharacteristic, complemented
167#407—2931no—
168C2×#41C2×D29311yescharacteristic
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#117—2931no—
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#107—2932no—
191#106—2932no—
192#106—2932no—
193#103—2932no—
194#103C2×D29312yescomplemented
195#102—2932no—
196#102C2×D29312yescomplemented
197C22⋊C8—2932no—
198C22⋊C8—2932no—
199C24—2931no—
200#41—22no—
201#41—22no—
202C2×Q8C22×D29311yescharacteristic
203C2×D4—21no—
204C2×D4—5861no—
205C2×D4—5861no—
206C2×D4C22×D29311yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—2931no—
210C22×C4—2931no—
211C22×C4—2931no—
212C22×C4—2931no—
213C22×C4—2931no—
214C22×C4—5862no—
215C22×C4—5862no—
216C22×C4—5862no—
217C22×C4—5862no—
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—2931no—
228C4⋊C4—2931no—
229C4⋊C4—5862no—
230C4⋊C4—5862no—
231C4⋊C4—5861no—
232C4⋊C4—5861no—
233C4⋊C4—5861no—
234C4⋊C4C22×D29312yes—
235C22⋊C4—5861no—
236C22⋊C4—5861no—
237C22⋊C4—5861no—
238C22⋊C4—5861no—
239C22⋊C4—5861no—
240C42—5861no—
241C23—21no—
242C23—2931no—
243C23—2931no—
244C23—5862no—
245C23—5862no—
246C23—5862no—
247C23D4×D29311yescharacteristic
248Q8—21no—
249Q8—21no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—11722no—
254D4—11722no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
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—5861no—
270C2×C4—5862no—
271C2×C4—11724no—
272C2×C4—11724no—
273C2×C4—11722no—
274C2×C4D4×D29311yescharacteristic
275C2×C4#1798980112yes—
276C2×C4#1798980212yes—
277C2×C4C23×D29311yescharacteristic
278C8—22no—
279C8—5862no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—2932no—
286C22—2932no—
287C22—5868no—
288C22—5861no—
289C22—5861no—
290C22—5861no—
291C22—5861no—
292C22#2262760612yes—
293C22C2×D4×D29311yescharacteristic
294C4—41no—
295C4—42no—
296C4—5861no—
297C4—5861no—
298C4—11722no—
299C4—11721no—
300C4—11721no—
301C4#2262760612yes—
302C4C2×#1798980111yescharacteristic
303C4C2×#1798980211yescharacteristic
304C2—22no—
305C2—81no—
306C2—5862no—
307C2—5862no—
308C2C2×#2262760611yescharacteristic
309C2AGL(1,ℤ/8)×D29311yescharacteristic
310C2#137×D29311yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#417×C293C2
socleC22×C293C22×D4
prime radical#34036636C2
hypercenter#417D293
hypocenterC293#3092
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D293C2semidirect
3. #34036463C2semidirect
4. #34036458C2semidirect
5. #34036446C2semidirect
6. #34036441C2semidirect
7. #34036171C2semidirect
8. #34036169C2semidirect
9. #34036130C2semidirect
10. #34036092C2semidirect
11. #417×C293C2semidirect
30. #22627558C22semidirect
31. #22627557C22semidirect
32. #22627556C22semidirect
33. #22627555C22semidirect
34. #22627548C22semidirect
35. #22627545C22semidirect
41. #103×C293C22semidirect
42. #102×C293C22semidirect
155. C293#3092semidirect
166. #417D293semidirect
194. #103C2×D293semidirect
196. #102C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340366362
C2×C4⋊C4×D2932
#340364632
#340364582
#340364462
#340364412
#340361712
#340361692
#340361302
#340360922
#417×C2932
#3092293