Finite Group Databasebeta

← #49161954

Subgroups — #49161954

Order 37504.

520 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627610C211yesmaximal, characteristic
2#34036035C211yesmaximal, characteristic, complemented
3#34036033C211yesmaximal, characteristic, complemented
4#34036020C211yesmaximal, characteristic, complemented
5#34036019C211yesmaximal, characteristic, complemented
6#34036017C211yesmaximal, characteristic, complemented
7#34036006C211yesmaximal, characteristic
8#34036005C211yesmaximal, characteristic
9#34036004C211yesmaximal, characteristic, complemented
10#34036003C211yesmaximal, characteristic, complemented
11#34036003C211yesmaximal, characteristic, complemented
12#34036001C211yesmaximal, characteristic, complemented
13#34035994C211yesmaximal, characteristic, complemented
14#34035993C211yesmaximal, characteristic, complemented
15#388×C293C211yesmaximal, characteristic, complemented
16C22×Dic586C2211yescharacteristic
17C22×C22⋊D293C2211yescharacteristic
18C23×Dic293C2211yescharacteristic
19C22×C4×D293C2211yescharacteristic
20#22627610C418yescomplemented
21C2×C4⋊Dic293C2211yescharacteristic
22C2×C4⋊Dic293C2211yescharacteristic
23C2×#17989781C2211yescharacteristic
24C2×#17989781C2211yescharacteristic
25C2×#17989781C2211yescharacteristic
26C2×C22⋊Dic293C2211yescharacteristic
27C2×C22⋊Dic293C2211yescharacteristic
28C2×#17989779C2211yescharacteristic
29C2×#17989779C2211yescharacteristic
30C2×#17989779C2211yescharacteristic
31C2×#17989779C2211yescharacteristic
32C2×#17989779C2211yescharacteristic
33C2×C4×Dic293C2211yescharacteristic
34C2×C4×Dic293C2211yescharacteristic
35#22627531C2211yescharacteristic
36#22627531C2211yescharacteristic
37#22627531C2211yescharacteristic
38#22627531C2211yescharacteristic
39#22627531C2211yescharacteristic
40#22627531C2211yescharacteristic
41#22627531C2211yescharacteristic
42#22627531C2211yescharacteristic
43#22627531C2211yescharacteristic
44#22627531C2211yescharacteristic
45C2×C4⋊C4×C293C2211yescharacteristic
46C2×C22⋊C4×C293C2211yescharacteristic
47C2×C22⋊C4×C293C2211yescharacteristic
48C2×C22⋊C4×C293C2211yescharacteristic
49C2×C42×C293C2211yescharacteristic
50#95×C293C2211yescharacteristic
51#95×C293C2211yescharacteristic
52C23×D293C2311yescharacteristic
53C2×Dic586—41no—
54C2×Dic586—41no—
55C2×Dic586C2×C414yescomplemented
56C2×C22⋊D293—41no—
57C2×C22⋊D293—41no—
58C2×C22⋊D293C2×C414yes—
59C22×Dic293—41no—
60C22×Dic293—41no—
61C22×Dic293—41no—
62C22×Dic293C2311yescharacteristic
63C22×Dic293C2311yescharacteristic
64C22×Dic293C2311yescharacteristic
65C22×Dic293C2311yescharacteristic
66C22×Dic293C2311yescharacteristic
67C22×Dic293C2311yescharacteristic
68C22×Dic293C2311yescharacteristic
69C2×C4×D293—41no—
70C2×C4×D293—41no—
71C2×C4×D293—41no—
72C4⋊Dic293—41no—
73C4⋊Dic293—41no—
74#17989781—41no—
75#17989781—41no—
76#17989781—41no—
77C22⋊Dic293—41no—
78C22⋊Dic293C2×C414yescomplemented
79#17989779—41no—
80#17989779—41no—
81#17989779—41no—
82#17989779C2×C414yescomplemented
83#17989779C2×C414yescomplemented
84C4×Dic293—41no—
85C4×Dic293C2×C414yescomplemented
86C24×C293C2311yescharacteristic
87C22×C4×C293C2311yescharacteristic
88C22×C4×C293C2311yescharacteristic
89C22×C4×C293C2311yescharacteristic
90C22×C4×C293C2311yescharacteristic
91C22×C4×C293C2311yescharacteristic
92C22×C4×C293C2311yescharacteristic
93C4⋊C4×C293—41no—
94C22⋊C4×C293—41no—
95C22⋊C4×C293—41no—
96C22⋊C4×C293C2×C414yescomplemented
97C42×C293—41no—
98C22×D293—21no—
99C22×D293—21no—
100C22×D293—21no—
101C22×D293—21no—
102C22×D293—21no—
103C22×D293—21no—
104C22×D293C22×C412yes—
105Dic586—44no—
106C22⋊D293—44no—
107C2×Dic293—21no—
108C2×Dic293—21no—
109C2×Dic293—21no—
110C2×Dic293—21no—
111C2×Dic293—21no—
112C2×Dic293—21no—
113C2×Dic293—21no—
114C2×Dic293—21no—
115C2×Dic293—21no—
116C2×Dic293—21no—
117C2×Dic293—21no—
118C2×Dic293—21no—
119C2×Dic293—21no—
120C2×Dic293—21no—
121C2×Dic293—21no—
122C2×Dic293—21no—
123C2×Dic293—21no—
124C2×Dic293—21no—
125C2×Dic293—44no—
126C2×Dic293C22×C412yes—
127C2×Dic293C22×C412yes—
128C2×Dic293C22×C412yes—
129C4×D293—44no—
130C23×C293—21no—
131C23×C293—21no—
132C23×C293—21no—
133C23×C293—21no—
134C23×C293—21no—
135C23×C293—21no—
136C23×C293C22×C412yes—
137C23×C293C2411yescharacteristic
138C2×C4×C293—21no—
139C2×C4×C293—21no—
140C2×C4×C293—21no—
141C2×C4×C293—21no—
142C2×C4×C293—21no—
143C2×C4×C293—21no—
144C2×C4×C293—21no—
145C2×C4×C293—21no—
146C2×C4×C293—21no—
147C2×C4×C293—21no—
148C2×C4×C293—21no—
149C2×C4×C293—21no—
150C2×C4×C293—21no—
151C2×C4×C293—21no—
152C2×C4×C293—21no—
153C2×C4×C293—21no—
154C2×C4×C293C22×C412yes—
155C2×C4×C293C22×C412yes—
156C22×C293—22no—
157C22×C293—22no—
158C22×C293—22no—
159C22×C293—41no—
160C22×C293—41no—
161C22×C293—41no—
162C22×C293—41no—
163C22×C293C23×C411yescharacteristic
164C22×C293H(5,𝔽2)11yescharacteristic
165C22×C293H(5,𝔽2)11yescharacteristic
166C22×C293H(5,𝔽2)11yescharacteristic
167C22×C293H(5,𝔽2)11yescharacteristic
168C22×C2932−1+411yescharacteristic
169C22×C2932−1+411yescharacteristic
170C2×D293—22no—
171C2×D293—22no—
172C2×D293—22no—
173C2×D293—41no—
174C2×D293—41no—
175C2×D293—41no—
176C2×D293—41no—
177C4×C293—22no—
178C4×C293—22no—
179C4×C293—41no—
180C4×C293—41no—
181C4×C293—41no—
182C4×C293—41no—
183Dic293—22no—
184Dic293—22no—
185Dic293—22no—
186Dic293—41no—
187Dic293—41no—
188Dic293—41no—
189Dic293—41no—
190C2×C293—42no—
191C2×C293#51811yescharacteristic
192C2×C293#52011yescharacteristic
193C2×C293#52011yescharacteristic
194C2×C293H(3,𝔽4)11yescharacteristic
195C2×C293#56211yescharacteristic
196C2×C293#56211yescharacteristic
197C2×C293#56311yescharacteristic
198D293—42no—
199C293#237911yescharacteristic, complemented, Sylow
200#2379—2931nomaximal, Sylow
201C2×#124—2931no—
202#391—2931no—
203#391—2931no—
204#388—2931no—
205#388—2931no—
206#388—2931no—
207#388D29311yescharacteristic, complemented
208#386—2931no—
209#386—2931no—
210#385—2931no—
211#385—2931no—
212#385—2931no—
213#385—2931no—
214#382—2931no—
215#382—2931no—
216C22×Q8—2931no—
217C22×D4—2931no—
218C23×C4—2931no—
219C23×C4—2931no—
220#124—2938no—
221C2×C4⋊C4—2931no—
222C2×C4⋊C4—2931no—
223C2×C4⋊C4—2931no—
224C2×C4⋊C4—2931no—
225C2×C4⋊C4—2931no—
226C2×C4⋊C4C2×D29311yescharacteristic
227C2×C22⋊C4—2931no—
228C2×C22⋊C4—2931no—
229C2×C22⋊C4—2931no—
230C2×C22⋊C4—2931no—
231C2×C22⋊C4—2931no—
232C2×C22⋊C4—2931no—
233C2×C22⋊C4—2931no—
234C2×C22⋊C4C2×D29311yescharacteristic
235C2×C22⋊C4C2×D29311yescharacteristic
236C2×C22⋊C4C2×D29311yescharacteristic
237C2×C42—2931no—
238C2×C42—2931no—
239C2×C42C2×D29311yescharacteristic
240#95—2931no—
241#95—2931no—
242#95—2931no—
243#95—2931no—
244#95—2931no—
245#95—2931no—
246#95—2931no—
247#95—2931no—
248#95—2931no—
249#95—2931no—
250#95C2×D29311yescharacteristic
251#95C2×D29311yescharacteristic
252C24—2931no—
253C24C22×D29311yescharacteristic
254C2×Q8—2934no—
255C2×Q8—11721no—
256C2×Q8—11721no—
257C2×D4—2934no—
258C2×D4—11721no—
259C2×D4—11721no—
260C22×C4—2931no—
261C22×C4—2931no—
262C22×C4—2931no—
263C22×C4—2931no—
264C22×C4—2931no—
265C22×C4—2931no—
266C22×C4—2931no—
267C22×C4—11721no—
268C22×C4—11721no—
269C22×C4—11721no—
270C22×C4—11721no—
271C22×C4—11721no—
272C22×C4—11721no—
273C22×C4C22×D29311yescharacteristic
274C22×C4C22×D29311yescharacteristic
275C22×C4C22×D29311yescharacteristic
276C22×C4C22×D29311yescharacteristic
277C22×C4C22×D29311yescharacteristic
278C22×C4C22×D29311yescharacteristic
279C4⋊C4—41no—
280C4⋊C4—11721no—
281C4⋊C4—11721no—
282C4⋊C4—11721no—
283C4⋊C4—11721no—
284C4⋊C4—11721no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—2934no—
288C22⋊C4—2934no—
289C22⋊C4—2934no—
290C22⋊C4—11721no—
291C22⋊C4—11721no—
292C22⋊C4—11721no—
293C22⋊C4—11721no—
294C22⋊C4C4×D29314yescomplemented
295C42—41no—
296C42—2934no—
297C42—11721no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—2932no—
305C23—5861no—
306C23—5861no—
307C23—5861no—
308C23—5861no—
309C23—5861no—
310C23—5861no—
311C23C2×C4×D29312yes—
312C23C23×D29311yescharacteristic
313Q8—11724no—
314D4—11724no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—2932no—
332C2×C4—2932no—
333C2×C4—2932no—
334C2×C4—5861no—
335C2×C4—5861no—
336C2×C4—5861no—
337C2×C4—5861no—
338C2×C4—5861no—
339C2×C4—5861no—
340C2×C4—5861no—
341C2×C4—5861no—
342C2×C4—5861no—
343C2×C4—5861no—
344C2×C4—5861no—
345C2×C4—5861no—
346C2×C4—5861no—
347C2×C4—5861no—
348C2×C4—5861no—
349C2×C4—5861no—
350C2×C4—5861no—
351C2×C4—5861no—
352C2×C4—11724no—
353C2×C4—11724no—
354C2×C4C2×C4×D29312yes—
355C2×C4C2×C4×D29312yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—5862no—
364C22—5862no—
365C22—5862no—
366C22—11721no—
367C22—11721no—
368C22—11721no—
369C22—11721no—
370C22C22×C4×D29311yescharacteristic
371C22#2262766911yescharacteristic
372C22#2262766911yescharacteristic
373C22#2262767011yescharacteristic
374C22#2262767011yescharacteristic
375C22#2262767111yescharacteristic
376C22#2262767111yescharacteristic
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—5862no—
384C4—5862no—
385C4—5862no—
386C4—11721no—
387C4—11721no—
388C4—11721no—
389C4—11721no—
390C2—42no—
391C2—11722no—
392C2#3403660111yescharacteristic
393C2#3403660811yescharacteristic
394C2#3403660911yescharacteristic
395C2#3403687111yescharacteristic
396C2#3403687411yescharacteristic
397C2#3403687611yescharacteristic
398C2#3403688111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D293
commutatorC22×C293C23×C4
Fitting#388×C293C2
socle, cosocleC23×C293C24
prime radicalC22×C22⋊D293C22
hypercenter#388D293
hypocenterC293#2379
LCSG ▷ C22×C293 ▷ C293
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #34036035C2semidirect
3. #34036033C2semidirect
4. #34036020C2semidirect
5. #34036019C2semidirect
6. #34036017C2semidirect
9. #34036004C2semidirect
10. #34036003C2semidirect
11. #34036003C2semidirect
12. #34036001C2semidirect
13. #34035994C2semidirect
14. #34035993C2semidirect
15. #388×C293C2semidirect
20. #22627610C4semidirect
55. C2×Dic586C2×C4semidirect
78. C22⋊Dic293C2×C4semidirect
82. #17989779C2×C4semidirect
83. #17989779C2×C4semidirect
85. C4×Dic293C2×C4semidirect
96. C22⋊C4×C293C2×C4semidirect
199. C293#2379semidirect
207. #388D293semidirect
294. C22⋊C4C4×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276102
#340360352
#340360332
#340360202
#340360192
#340360172
#340360062
#340360052
#340360042
#340360032
#340360012
#340359942
#340359932
#388×C2932
#2379293