Finite Group Databasebeta

← #49170692

Subgroups — #49170692

Order 37504.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34036829C211yesmaximal, characteristic, complemented
2#34036791C211yesmaximal, characteristic, complemented
3#34036484C211yesmaximal, characteristic, complemented
4#34036473C211yesmaximal, characteristic, complemented
5#34036449C211yesmaximal, characteristic, complemented
6#34036436C211yesmaximal, characteristic, complemented
7#34036364C211yesmaximal, characteristic, complemented
8#34036357C211yesmaximal, characteristic, complemented
9#34036344C211yesmaximal, characteristic, complemented
10#34036273C211yesmaximal, characteristic, complemented
11#34036266C211yesmaximal, characteristic, complemented
12C8×D1172C211yesmaximal, characteristic, complemented
13#34036124C211yesmaximal, characteristic, complemented
14#103×D293C211yesmaximal, characteristic, complemented
15C4×SD16×C293C211yesmaximal, characteristic, complemented
16C2×#17989803C2211yescharacteristic
17C2×Q8×D293C2211yescharacteristic
18C2×#17989790C2211yescharacteristic
19C2×#17989787C2211yescharacteristic
20C2×C8×D293C2211yescharacteristic
21C2×C8×D293C2211yescharacteristic
22#22627617—21no—
23#22627616C2211yescharacteristic
24#22627612—21no—
25#22627612C2211yescharacteristic
26#22627607—21no—
27#22627604—21no—
28#22627604C2211yescharacteristic
29#22627599—21no—
30#22627599C2211yescharacteristic
31#22627597—21no—
32#22627595C2211yescharacteristic
33#22627595C2211yescharacteristic
34C4×D1172—21no—
35C4×D1172C2211yescharacteristic
36C4×C22⋊D293—21no—
37#22627582C2211yescharacteristic
38C4⋊C4×D293C2211yescharacteristic
39#22627557C2211yescharacteristic
40#22627555C2211yescharacteristic
41C4⋊#14556873C2211yescharacteristic
42#22627549C2211yescharacteristic
43#22627549C2211yescharacteristic
44#22627548C2211yescharacteristic
45#22627548C2211yescharacteristic
46#22627547C2211yescharacteristic
47#22627546C2211yescharacteristic
48#22627546C2211yescharacteristic
49#22627544C2211yescharacteristic
50#22627536C2211yescharacteristic
51#22627536C2211yescharacteristic
52C2×SD16×C293C2211yescharacteristic
53C4×Q8×C293C2211yescharacteristic
54C4×D4×C293C2211yescharacteristic
55#106×C293C2211yescharacteristic
56#103×C293C2211yescharacteristic
57#102×C293C2211yescharacteristic
58C4×C8×C293C2211yescharacteristic
59#17989803—22no—
60#17989803—41no—
61C2×Dic586—21no—
62C2×Dic586C2311yescharacteristic
63Q8×D293—22no—
64Q8×D293—41no—
65C2×D1172—21no—
66C2×D1172—21no—
67C2×D1172D411yescharacteristic
68C2×D1172C2311yescharacteristic
69C2×C22⋊D293—21no—
70C2×C22⋊D293—21no—
71C2×C4×D293—21no—
72C2×C4×D293—21no—
73C2×C4×D293—21no—
74C2×C4×D293C2311yescharacteristic
75C2×C4×D293C2311yescharacteristic
76#17989790—41no—
77#17989787—41no—
78C2×#14556873C2311yescharacteristic
79C2×#14556873C2311yescharacteristic
80C8×D293—41no—
81C8×D293—41no—
82C4⋊Dic293—21no—
83C4⋊Dic293D411yescharacteristic
84C4⋊Dic293C2311yescharacteristic
85C4⋊Dic293C2311yescharacteristic
86#17989781—21no—
87#17989781—21no—
88#17989781—21no—
89C22⋊Dic293—21no—
90C22⋊Dic293—21no—
91#17989779—21no—
92#17989779—21no—
93#17989779—21no—
94#17989779—21no—
95#17989779—21no—
96#17989779D411yescharacteristic
97#17989779D411yescharacteristic
98C4×Dic293—21no—
99C2×Q8×C293C2311yescharacteristic
100C2×D4×C293C2311yescharacteristic
101C22×C4×C293—21no—
102SD16×C293—22no—
103C2×C8×C293C2311yescharacteristic
104C2×C8×C293C2311yescharacteristic
105C4⋊C4×C293—21no—
106C4⋊C4×C293C2311yescharacteristic
107C4⋊C4×C293C2311yescharacteristic
108C22⋊C4×C293—21no—
109C42×C293—21no—
110C42×C293C2311yescharacteristic
111C22×D293—21no—
112C22×D293C2×D411yescharacteristic
113C22×D293C2×D411yescharacteristic
114Dic586—21no—
115Dic586—21no—
116Dic586—42no—
117D1172—22no—
118D1172—21no—
119D1172—21no—
120D1172—42no—
121D1172—42no—
122C22⋊D293—42no—
123C22⋊D293—81no—
124C2×Dic293—21no—
125C2×Dic293—21no—
126C2×Dic293—21no—
127C2×Dic293C2×D411yescharacteristic
128C2×Dic293C2×D411yescharacteristic
129C4×D293—21no—
130C4×D293—21no—
131C4×D293—21no—
132C4×D293—21no—
133C4×D293—42no—
134C4×D293—42no—
135C4×D293—81no—
136#14556873—21no—
137#14556873—21no—
138C23×C293—21no—
139Q8×C293—21no—
140Q8×C293#4112yes—
141D4×C293—21no—
142D4×C293—21no—
143C2×C4×C293—21no—
144C2×C4×C293—21no—
145C2×C4×C293—21no—
146C2×C4×C293—81no—
147C2×C4×C293C2×D411yescharacteristic
148C2×C4×C293C2×D411yescharacteristic
149C2×C4×C293C2411yescharacteristic
150C8×C293—21no—
151C8×C293—21no—
152C22×C293—41no—
153C22×C293—41no—
154C22×C293—41no—
155C22×C293C22×D411yescharacteristic
156C2×D293—21no—
157C2×D293—21no—
158C2×D293—21no—
159C2×D293—21no—
160C2×D293—21no—
161C2×D293—41no—
162C2×D293—41no—
163C2×D293—41no—
164C2×D293#13512yes—
165C4×C293—22no—
166C4×C293—21no—
167C4×C293—21no—
168C4×C293—41no—
169C4×C293—41no—
170C4×C293C2×#4111yescharacteristic
171C4×C293H(5,𝔽2)11yescharacteristic
172Dic293—21no—
173Dic293—21no—
174Dic293—41no—
175Dic293—41no—
176Dic293—41no—
177C2×C293—81no—
178C2×C293#54611yescharacteristic
179C2×C293C2×#13511yescharacteristic
180C2×C293#57711yescharacteristic
181D293—22no—
182D293—41no—
183D293—81no—
184C293#330111yescharacteristic, complemented, Sylow
185#3301—2931nomaximal, Sylow
186#553—2931no—
187#548—2931no—
188#488—2931no—
189#485—2931no—
190#481—2931no—
191#479—2931no—
192#466—2931no—
193#465—2931no—
194#463—2931no—
195#449—2931no—
196#448—2931no—
197C4×SD16D29311yescharacteristic, complemented
198C8×D4—2931no—
199#416—2931no—
200C2×#103—2931no—
201C2×#41—2931no—
202C22×Q8—2931no—
203C2×SD16—2931no—
204C2×SD16C2×D29311yescharacteristic
205C2×D8—2931no—
206C22×C8—2931no—
207C22×C8—2931no—
208#126—5861no—
209#125—2931no—
210#124—2931no—
211#124—5861no—
212#123—5861no—
213C22⋊Q8—2931no—
214C22⋊Q8—5861no—
215#121—2931no—
216#121—2931no—
217#121—2931no—
218#121—5861no—
219#121—5861no—
220C4×Q8C2×D29311yescharacteristic
221C4×D4—2931no—
222C4×D4—5861no—
223C4×D4—5861no—
224C4×D4C2×D29311yescharacteristic
225#117—2931no—
226C2×C4⋊C4—2931no—
227#107—2931no—
228#106—2931no—
229#106C2×D29311yescharacteristic
230C4⋊C8—2931no—
231#103—2931no—
232#103—2931no—
233#103—2931no—
234#103—2931no—
235#103—2931no—
236#103C2×D29311yescharacteristic
237#102—2931no—
238#102—2931no—
239#102—2931no—
240#102C2×D29311yescharacteristic
241C22⋊C8—2931no—
242C22⋊C8—2931no—
243C4×C8C2×D29311yescharacteristic
244#41—5862no—
245#41—11721no—
246C2×Q8—2931no—
247C2×Q8—5861no—
248C2×Q8—5862no—
249C2×Q8—11721no—
250C2×Q8C22×D29311yescharacteristic
251C2×D4—2931no—
252C2×D4—2931no—
253C2×D4—5861no—
254C2×D4—5861no—
255C2×D4—5861no—
256C2×D4—5861no—
257C2×D4C22×D29311yescharacteristic
258C22×C4—21no—
259C22×C4—2931no—
260C22×C4—2931no—
261C22×C4—5861no—
262C22×C4—5861no—
263C22×C4—5861no—
264SD16—22no—
265SD16—11721no—
266D8—11721no—
267C2×C8—2931no—
268C2×C8—2931no—
269C2×C8—11721no—
270C2×C8—11721no—
271C2×C8C22×D29311yescharacteristic
272C2×C8C22×D29311yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—2931no—
275C4⋊C4—2931no—
276C4⋊C4—2931no—
277C4⋊C4—5861no—
278C4⋊C4—5861no—
279C4⋊C4—5861no—
280C4⋊C4—5861no—
281C4⋊C4C22×D29311yescharacteristic
282C4⋊C4C22×D29311yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—2931no—
285C22⋊C4—2931no—
286C22⋊C4—5861no—
287C22⋊C4—5861no—
288C22⋊C4—5861no—
289C22⋊C4—5861no—
290C22⋊C4—5861no—
291C22⋊C4—5861no—
292C22⋊C4—5861no—
293C42—21no—
294C42—5861no—
295C42C22×D29311yescharacteristic
296C23—21no—
297C23—2931no—
298C23—2931no—
299C23—5861no—
300Q8—21no—
301Q8—5861no—
302Q8—5861no—
303Q8—11722no—
304Q8#1798980212yes—
305D4—21no—
306D4—21no—
307D4—23441no—
308D4—5862no—
309D4—5861no—
310D4—5861no—
311D4—11722no—
312D4—11722no—
313D4—11722no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—81no—
318C2×C4—2931no—
319C2×C4—2931no—
320C2×C4—23441no—
321C2×C4—5861no—
322C2×C4—5861no—
323C2×C4—5861no—
324C2×C4—5861no—
325C2×C4—5861no—
326C2×C4—5861no—
327C2×C4—5861no—
328C2×C4—11722no—
329C2×C4—11722no—
330C2×C4D4×D29311yescharacteristic
331C2×C4D4×D29311yescharacteristic
332C2×C4C23×D29311yescharacteristic
333C8—21no—
334C8—21no—
335C8—5861no—
336C8—5861no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—2932no—
341C22—5861no—
342C22—5861no—
343C22—5861no—
344C22—5861no—
345C22—5861no—
346C22—11721no—
347C22—11721no—
348C22—11721no—
349C22C2×D4×D29311yescharacteristic
350C4—22no—
351C4—21no—
352C4—21no—
353C4—41no—
354C4—41no—
355C4—5861no—
356C4—5861no—
357C4—11721no—
358C4—11721no—
359C4—11721no—
360C4C2×#1798980211yescharacteristic
361C4#2262767011yescharacteristic
362C2—81no—
363C2—23441no—
364C2—5862no—
365C2—11721no—
366C2#3403677511yescharacteristic
367C2#135×D29311yescharacteristic
368C2#3403693511yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
FittingC4×SD16×C293C2
socleC22×C293C22×D4
hypercenterC4×SD16D293
hypocenterC293#3301
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #34036829C2semidirect
2. #34036791C2semidirect
3. #34036484C2semidirect
4. #34036473C2semidirect
5. #34036449C2semidirect
6. #34036436C2semidirect
7. #34036364C2semidirect
8. #34036357C2semidirect
9. #34036344C2semidirect
10. #34036273C2semidirect
11. #34036266C2semidirect
12. C8×D1172C2semidirect
13. #34036124C2semidirect
14. #103×D293C2semidirect
15. C4×SD16×C293C2semidirect
184. C293#3301semidirect
197. C4×SD16D293semidirect

Maximal subgroups

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

subgroupindex
#340368292
#340367912
#340364842
#340364732
#340364492
#340364362
#340363642
#340363572
#340363442
#340362732
#340362662
C8×D11722
#340361242
#103×D2932
C4×SD16×C2932
#3301293