Finite Group Databasebeta

← #49168179

Subgroups — #49168179

Order 37504.

766 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 145 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#34036965C211yesmaximal, characteristic, complemented
2#34036641C212yesmaximal, complemented
3#34036636C211yesmaximal, characteristic, complemented
4#34036320C214yesmaximal, complemented
5#34036317C214yesmaximal, complemented
6#34036102C212yesmaximal, complemented
7#410×C293C211yesmaximal, characteristic, complemented
8#22627672—21no—
9#22627671—22no—
10#22627670—22no—
11#22627669—21no—
12#22627669C2214yes—
13C2×#17989802—22no—
14C2×#17989802C2212yes—
15#41×D293—24no—
16C22×C4×D293C2211yescharacteristic, complemented
17#22627608C2214yescomplemented
18#22627605C2212yescomplemented
19#22627604—22no—
20#22627600—21no—
21#22627599—22no—
22#22627595—21no—
23#22627594C2214yescomplemented
24#22627592C2212yescomplemented
25C4×D1172—24no—
26C4×C22⋊D293—22no—
27#22627583C2211yescharacteristic, complemented
28#22627538C2218yescomplemented
29C2×#41×C293C2211yescharacteristic
30#117×C293C2212yescomplemented
31#99×C293C2214yescomplemented
32C23×D293C2311yescharacteristic
33#17989803—24no—
34#17989802—24no—
35#17989802—24no—
36#17989802—22no—
37#17989802—22no—
38#17989802—24no—
39#17989802—22no—
40#17989802—22no—
41#17989801—24no—
42#17989801—28no—
43C2×Dic586—22no—
44C2×Dic586D412yescomplemented
45Q8×D293—24no—
46C2×D1172—22no—
47C2×D1172—24no—
48C2×D1172D412yescomplemented
49C2×C22⋊D293—24no—
50C2×C22⋊D293—22no—
51C2×C22⋊D293C2314yes—
52D4×D293—24no—
53D4×D293—28no—
54C22×Dic293C2311yescharacteristic
55C2×C4×D293—24no—
56C2×C4×D293—21no—
57C2×C4×D293—22no—
58C2×C4×D293—21no—
59C2×C4×D293—22no—
60C2×C4×D293D414yescomplemented
61C4⋊Dic293—24no—
62C4⋊Dic293—21no—
63#17989781—22no—
64C22⋊Dic293—22no—
65C22⋊Dic293C2312yes—
66#17989779—24no—
67#17989779—24no—
68#17989779—22no—
69C4×Dic293—21no—
70#41×C293—22no—
71#41×C293—22no—
72C2×Q8×C293D411yescharacteristic, complemented
73C2×D4×C293D411yescharacteristic, complemented
74C2×D4×C293C2312yes—
75C22×C4×C293D412yescomplemented
76C22×C4×C293C2311yescharacteristic
77C4⋊C4×C293—22no—
78C22⋊C4×C293C2314yes—
79C42×C293—22no—
80C22×D293—21no—
81C22×D293—22no—
82C22×D293—24no—
83C22×D293—42no—
84C22×D293C2×D414yes—
85Dic586—24no—
86Dic586—22no—
87Dic586—22no—
88D1172—24no—
89D1172—22no—
90D1172—22no—
91D1172—48no—
92C22⋊D293—28no—
93C22⋊D293—24no—
94C22⋊D293—24no—
95C22⋊D293—44no—
96C2×Dic293—24no—
97C2×Dic293—22no—
98C2×Dic293—22no—
99C2×Dic293—21no—
100C2×Dic293C2×D414yes—
101C4×D293—28no—
102C4×D293—24no—
103C4×D293—24no—
104C4×D293—41no—
105C4×D293—42no—
106C4×D293—41no—
107C23×C293C2×D412yes—
108C23×C293C2411yescharacteristic
109Q8×C293—22no—
110D4×C293—24no—
111D4×C293—22no—
112C2×C4×C293—22no—
113C2×C4×C293—22no—
114C2×C4×C293—24no—
115C2×C4×C293C2×D412yes—
116C2×C4×C293C2×D412yes—
117C2×C4×C293C2×D412yes—
118C2×C4×C293C2×D411yescharacteristic
119C2×C4×C293C2×D411yescharacteristic
120C22×C293—22no—
121C22×C293—41no—
122C22×C293—42no—
123C22×C293C22×D412yes—
124C22×C293C22×D411yescharacteristic
125C2×D293—22no—
126C2×D293—24no—
127C2×D293—24no—
128C2×D293—41no—
129C2×D293—42no—
130C2×D293—41no—
131C2×D293—44no—
132C2×D293—84no—
133C4×C293—22no—
134C4×C293—44no—
135C4×C293#12011yescharacteristic
136C4×C293#12012yes—
137C4×C293#12011yescharacteristic
138Dic293—24no—
139Dic293—42no—
140Dic293—41no—
141C2×C293—21no—
142C2×C293—22no—
143C2×C293—42no—
144C2×C293C2×#12011yescharacteristic
145D293—42no—
146D293—44no—
147C293#302511yescharacteristic, complemented, Sylow
148#3025—2931nomaximal, Sylow
149#585—2931no—
150#525—2931no—
151#525—2932no—
152#458—2934no—
153B(4,𝔽2)—2934no—
154#410—2932no—
155#410D29311yescharacteristic, complemented
1562−1+4—5862no—
1572−1+4—5861no—
158H(5,𝔽2)—2934no—
159H(5,𝔽2)—5862no—
160H(5,𝔽2)—5861no—
161C2×#41—2932no—
162C2×#41—5864no—
163C2×#41—5862no—
164C2×#41C2×D29311yescharacteristic
165C23×C4—2931no—
166#123—2932no—
167#123—2934no—
168C22⋊Q8—5862no—
169C22⋊Q8—5861no—
170#121—5862no—
171#121—5861no—
172#120—2932no—
173#120—2934no—
174C4×D4—5864no—
175C4×D4—5862no—
176#117—2931no—
177#117C2×D29312yescomplemented
178#99—2938no—
179#99C2×D29314yescomplemented
180C24—2931no—
181#41—22no—
182#41—22no—
183#41—5864no—
184#41—5864no—
185#41—5868no—
186#41—5864no—
187#41—5864no—
188#41—5862no—
189#41—5862no—
190#41—5862no—
191#41—5862no—
192#41—5864no—
193C2×Q8—2932no—
194C2×Q8—5862no—
195C2×Q8—5864no—
196C2×Q8D117211yescharacteristic, complemented
197C2×D4—2934no—
198C2×D4—2932no—
199C2×D4—5862no—
200C2×D4—5864no—
201C2×D4—5868no—
202C2×D4—5864no—
203C2×D4—5864no—
204C2×D4—5862no—
205C2×D4D117211yescharacteristic, complemented
206C2×D4C22×D29312yes—
207C22×C4—2934no—
208C22×C4—2931no—
209C22×C4—5864no—
210C22×C4—5862no—
211C22×C4—5862no—
212C22×C4—5861no—
213C22×C4—5861no—
214C22×C4D117212yescomplemented
215C22×C4C22×D29311yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—5864no—
218C4⋊C4—5861no—
219C4⋊C4—5862no—
220C22⋊C4—2932no—
221C22⋊C4—5864no—
222C22⋊C4—5864no—
223C22⋊C4—5862no—
224C22⋊C4—5862no—
225C22⋊C4C22×D29314yes—
226C42—22no—
227C42—5861no—
228C23—2934no—
229C23—5864no—
230C23—5862no—
231C23—5861no—
232C23—11722no—
233C23C2×D117212yes—
234C23C23×D29311yescharacteristic
235Q8—22no—
236Q8—5864no—
237Q8—5862no—
238Q8—5862no—
239D4—24no—
240D4—22no—
241D4—5864no—
242D4—5868no—
243D4—5864no—
244D4—5864no—
245D4—5862no—
246D4—5862no—
247D4—11728no—
248D4—11724no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—2934no—
253C2×C4—5868no—
254C2×C4—5864no—
255C2×C4—5864no—
256C2×C4—5864no—
257C2×C4—5862no—
258C2×C4—5862no—
259C2×C4—5861no—
260C2×C4—11721no—
261C2×C4—11722no—
262C2×C4—11721no—
263C2×C4D4×D29312yes—
264C2×C4D4×D29312yes—
265C2×C4C2×D117212yes—
266C2×C4C2×D117211yescharacteristic
267C2×C4C2×D117211yescharacteristic
268C22—22no—
269C22—41no—
270C22—42no—
271C22—23444no—
272C22—5864no—
273C22—5862no—
274C22—5864no—
275C22—11724no—
276C22—11722no—
277C22—11721no—
278C22—11721no—
279C22C2×D4×D29312yes—
280C22C22×D117211yescharacteristic
281C4—22no—
282C4—44no—
283C4—5864no—
284C4—11722no—
285C4—11721no—
286C4#2262759411yescharacteristic
287C4#2262759412yes—
288C4#2262759411yescharacteristic
289C2—22no—
290C2—21no—
291C2—42no—
292C2—11724no—
293C2—11722no—
294C2C2×#2262759411yescharacteristic
subgroupG/H
centerC4#22627594
commutator, cosocleC23×C293C24
FrattiniC23C23×D293
Fitting#410×C293C2
socleC2×C293C2×#120
hypercenter#410D293
hypocenterC293#3025
LCSG ▷ C23×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #34036965C2semidirect
2. #34036641C2semidirect
3. #34036636C2semidirect
4. #34036320C2semidirect
5. #34036317C2semidirect
6. #34036102C2semidirect
7. #410×C293C2semidirect
16. C22×C4×D293C22semidirect
17. #22627608C22semidirect
18. #22627605C22semidirect
23. #22627594C22semidirect
24. #22627592C22semidirect
27. #22627583C22semidirect
28. #22627538C22semidirect
30. #117×C293C22semidirect
31. #99×C293C22semidirect
44. C2×Dic586D4semidirect
48. C2×D1172D4semidirect
60. C2×C4×D293D4semidirect
72. C2×Q8×C293D4semidirect
73. C2×D4×C293D4semidirect
75. C22×C4×C293D4semidirect
147. C293#3025semidirect
155. #410D293semidirect
177. #117C2×D293semidirect
179. #99C2×D293semidirect
196. C2×Q8D1172semidirect
205. C2×D4D1172semidirect
214. C22×C4D1172semidirect

Maximal subgroups

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

subgroupindex
#340369652
#340366412
#340366362
#340363202
#340363172
#340361022
#410×C2932
#3025293