Finite Group Databasebeta

← #49170633

Subgroups — #49170633

Order 37504.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34036827C211yesmaximal, characteristic, complemented
2#34036770C211yesmaximal, characteristic, complemented
3#34036485C211yesmaximal, characteristic, complemented
4#34036456C211yesmaximal, characteristic, complemented
5#34036432C211yesmaximal, characteristic, complemented
6#34036366C211yesmaximal, characteristic, complemented
7#34036360C211yesmaximal, characteristic, complemented
8#34036347C211yesmaximal, characteristic, complemented
9#34036277C211yesmaximal, characteristic, complemented
10#34036265C211yesmaximal, characteristic, complemented
11#34036211C211yesmaximal, characteristic, complemented
12#34036192C211yesmaximal, characteristic, complemented
13#34036124C211yesmaximal, characteristic, complemented
14C2×#22627544C211yesmaximal, characteristic, complemented
15#484×C293C211yesmaximal, characteristic, complemented
16C2×#17989803C2211yescharacteristic
17C22×D1172C2211yescharacteristic
18C2×#17989789C2211yescharacteristic, complemented
19C2×D2344C2211yescharacteristic
20C2×D2344C2211yescharacteristic
21C22×#14556873C2211yescharacteristic
22C2×C8×D293C2211yescharacteristic, complemented
23#22627619—21no—
24#22627615C2211yescharacteristic, complemented
25#22627613C2211yescharacteristic
26#22627609—21no—
27#22627609—21no—
28#22627599—21no—
29#22627599C2211yescharacteristic
30#22627599C2211yescharacteristic, complemented
31#22627598—21no—
32#22627598C2211yescharacteristic
33#22627594—21no—
34#22627586C2211yescharacteristic
35#22627586C2211yescharacteristic, complemented
36C4×C22⋊D293—21no—
37C4×C22⋊D293C2211yescharacteristic
38#22627582C2211yescharacteristic, complemented
39C22⋊C4×D293—21no—
40#22627555C2211yescharacteristic, complemented
41#22627553C2211yescharacteristic, complemented
42#22627548C2211yescharacteristic, complemented
43#22627546C2211yescharacteristic
44#22627546C2211yescharacteristic
45#22627544C2211yescharacteristic
46#22627544C2211yescharacteristic
47#22627544C2211yescharacteristic
48#22627544C2211yescharacteristic
49#22627544C2211yescharacteristic, complemented
50#22627536C2211yescharacteristic, complemented
51C8×Dic293C2211yescharacteristic, complemented
52C22⋊Q8×C293C2211yescharacteristic
53C2×C4⋊C4×C293C2211yescharacteristic
54#107×C293C2211yescharacteristic, complemented
55#107×C293C2211yescharacteristic, complemented
56#103×C293C2211yescharacteristic, complemented
57#103×C293C2211yescharacteristic, complemented
58C22⋊C8×C293C2211yescharacteristic
59C23×D293—21no—
60#17989803—22no—
61#17989803—41no—
62C2×D1172—22no—
63C2×D1172—21no—
64C2×D1172—21no—
65C2×D1172—21no—
66C2×D1172—21no—
67C2×D1172—41no—
68C2×D1172C2311yescharacteristic
69C2×D1172C2311yescharacteristic
70C2×D1172C2311yescharacteristic
71C2×C22⋊D293—21no—
72C2×C22⋊D293D411yescharacteristic
73C2×C4×D293—21no—
74C2×C4×D293—21no—
75C2×C4×D293—21no—
76C2×C4×D293C2311yescharacteristic
77#17989789—41no—
78D2344—22no—
79D2344—41no—
80C2×#14556873—41no—
81C2×#14556873C2311yescharacteristic
82C2×#14556873C2311yescharacteristic
83C8×D293—41no—
84C4⋊Dic293—21no—
85C4⋊Dic293C2311yescharacteristic
86#17989781—21no—
87#17989781D411yescharacteristic
88C22⋊Dic293D411yescharacteristic
89#17989779—21no—
90#17989779—21no—
91#17989779—21no—
92#17989779—21no—
93#17989779—21no—
94#17989779—21no—
95#17989779—21no—
96#17989779—21no—
97#17989779D411yescharacteristic
98C4×Dic293—21no—
99C4×Dic293C2311yescharacteristic
100C2×Q8×C293C2311yescharacteristic
101C22×C4×C293—21no—
102C22×C4×C293C2311yescharacteristic
103C2×C8×C293C2311yescharacteristic
104C2×C8×C293C2311yescharacteristic
105C4⋊C4×C293—21no—
106C4⋊C4×C293—21no—
107C4⋊C4×C293C2311yescharacteristic
108C4⋊C4×C293C2311yescharacteristic
109C4⋊C4×C293C2311yescharacteristic
110C22⋊C4×C293—21no—
111C22×D293—21no—
112C22×D293—21no—
113C22×D293—21no—
114C22×D293—42no—
115C22×D293—42no—
116C22×D293—81no—
117C22×D293C2×D411yescharacteristic
118D1172—21no—
119D1172—21no—
120D1172—21no—
121D1172—21no—
122D1172—21no—
123D1172—42no—
124D1172—42no—
125D1172—81no—
126D1172—81no—
127D1172#4112yes—
128C22⋊D293—22no—
129C22⋊D293—42no—
130C2×Dic293—21no—
131C2×Dic293C2×D411yescharacteristic
132C2×Dic293C2×D411yescharacteristic
133C2×Dic293C2×D411yescharacteristic
134C4×D293—21no—
135C4×D293—21no—
136C4×D293—42no—
137C4×D293—81no—
138C4×D293—81no—
139#14556873—21no—
140#14556873—21no—
141C23×C293C2×D411yescharacteristic
142Q8×C293—21no—
143Q8×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—81no—
151C2×C4×C293C2×D411yescharacteristic
152C2×C4×C293C2411yescharacteristic
153C8×C293—21no—
154C8×C293—21no—
155C22×C293—21no—
156C22×C293—21no—
157C22×C293#13512yes—
158C22×C293C22×D411yescharacteristic
159C2×D293—21no—
160C2×D293—22no—
161C2×D293—21no—
162C2×D293—21no—
163C2×D293—41no—
164C2×D293—41no—
165C2×D293—41no—
166C2×D293—41no—
167C2×D293—41no—
168C2×D293—41no—
169C2×D293—41no—
170C2×D293—41no—
171C2×D293—84no—
172C4×C293—21no—
173C4×C293—41no—
174C4×C293—41no—
175C4×C293—41no—
176C4×C293—41no—
177C4×C293C2×#4111yescharacteristic
178C4×C293H(5,𝔽2)11yescharacteristic
179Dic293—21no—
180Dic293—21no—
181Dic293—21no—
182Dic293—41no—
183C2×C293—22no—
184C2×C293#54611yescharacteristic
185C2×C293C2×#13511yescharacteristic
186C2×C293#57611yescharacteristic
187D293—41no—
188D293—42no—
189D293—81no—
190D293—81no—
191C293#329711yescharacteristic, complemented, Sylow
192#3297—2931nomaximal, Sylow
193#553—2931no—
194#546—2931no—
195#488—2931no—
196#484D29311yescharacteristic, complemented
197#482—2931no—
198#478—2931no—
199#466—2931no—
200#465—2931no—
201#463—2931no—
202#449—2931no—
203#447—2931no—
204C4×D8—2931no—
205C8×D4—2931no—
206#416—2931no—
207C2×#102—2931no—
208C2×#41—2931no—
209C22×D4—2931no—
210C2×SD16—2931no—
211C2×D8—2931no—
212C2×D8—2931no—
213C22×C8—2931no—
214C22×C8—2931no—
215#126—5861no—
216#125—2931no—
217#124—2931no—
218#123—5861no—
219#123—5861no—
220C22⋊Q8C2×D29311yescharacteristic
221#121—2931no—
222#121—2931no—
223#121—2931no—
224#121—5861no—
225#121—5861no—
226#120—5861no—
227C4×D4—2931no—
228C4×D4—2931no—
229C4×D4—2931no—
230C4×D4—5861no—
231#117—2931no—
232C2×C4⋊C4C2×D29311yescharacteristic
233C2×C22⋊C4—5861no—
234#107C2×D29311yescharacteristic, complemented
235#107C2×D29311yescharacteristic, complemented
236#106—2931no—
237C4⋊C8—2931no—
238#103—2931no—
239#103C2×D29311yescharacteristic, complemented
240#103C2×D29311yescharacteristic, complemented
241#102—2931no—
242#102—2931no—
243#102—2931no—
244#102—2931no—
245#102—2931no—
246#102—2931no—
247#102—2931no—
248C22⋊C8—2931no—
249C22⋊C8C2×D29311yescharacteristic
250C4×C8—2931no—
251C24—5861no—
252#41—5862no—
253#41—11721no—
254C2×Q8C22×D29311yescharacteristic
255C2×D4—2931no—
256C2×D4—2931no—
257C2×D4—2931no—
258C2×D4—2931no—
259C2×D4—5862no—
260C2×D4—5861no—
261C2×D4—5861no—
262C2×D4—5861no—
263C2×D4—5861no—
264C2×D4—5861no—
265C2×D4—11721no—
266C22×C4—21no—
267C22×C4—2931no—
268C22×C4—5861no—
269C22×C4—5861no—
270C22×C4—5861no—
271C22×C4C22×D29311yescharacteristic
272SD16—11721no—
273D8—5862no—
274D8—11721no—
275C2×C8—2931no—
276C2×C8—2931no—
277C2×C8—11721no—
278C2×C8—11721no—
279C2×C8C22×D29311yescharacteristic
280C2×C8C22×D29311yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—2931no—
284C4⋊C4—2931no—
285C4⋊C4—5861no—
286C4⋊C4—5861no—
287C4⋊C4C22×D29311yescharacteristic
288C4⋊C4C22×D29311yescharacteristic
289C4⋊C4C22×D29311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—2931no—
292C22⋊C4—2931no—
293C22⋊C4—5861no—
294C22⋊C4—5861no—
295C22⋊C4—5861no—
296C22⋊C4—5861no—
297C22⋊C4—5861no—
298C22⋊C4—5861no—
299C22⋊C4—5861no—
300C22⋊C4—5861no—
301C42—2931no—
302C42—5861no—
303C23—2931no—
304C23—23441no—
305C23—5861no—
306C23—5861no—
307C23—5861no—
308C23—11722no—
309C23—11722no—
310C23D4×D29311yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—2932no—
314D4—23441no—
315D4—23441no—
316D4—5862no—
317D4—5861no—
318D4—5861no—
319D4—5861no—
320D4—5861no—
321D4—5861no—
322D4—11722no—
323D4—11722no—
324D4—11722no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—81no—
332C2×C4—2931no—
333C2×C4—2931no—
334C2×C4—2931no—
335C2×C4—23441no—
336C2×C4—23441no—
337C2×C4—5861no—
338C2×C4—5861no—
339C2×C4—5861no—
340C2×C4—11722no—
341C2×C4D4×D29311yescharacteristic
342C2×C4C23×D29311yescharacteristic
343C8—21no—
344C8—21no—
345C8—5861no—
346C8—5861no—
347C22—21no—
348C22—21no—
349C22—23444no—
350C22—5861no—
351C22—5862no—
352C22—5861no—
353C22—5861no—
354C22—11721no—
355C22—11721no—
356C22—11721no—
357C22—11721no—
358C22—11721no—
359C22—11721no—
360C22—11721no—
361C22—11721no—
362C22#2262764712yes—
363C22C2×D4×D29311yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—5861no—
370C4—5861no—
371C4—5861no—
372C4—11721no—
373C4#41×D29311yescharacteristic
374C4#2262767011yescharacteristic
375C2—22no—
376C2—23441no—
377C2—23441no—
378C2—11722no—
379C2—11721no—
380C2#3403677111yescharacteristic
381C2C2×#2262764711yescharacteristic
382C2#3403693311yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#484×C293C2
socleC22×C293C22×D4
hypercenter#484D293
hypocenterC293#3297
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #34036827C2semidirect
2. #34036770C2semidirect
3. #34036485C2semidirect
4. #34036456C2semidirect
5. #34036432C2semidirect
6. #34036366C2semidirect
7. #34036360C2semidirect
8. #34036347C2semidirect
9. #34036277C2semidirect
10. #34036265C2semidirect
11. #34036211C2semidirect
12. #34036192C2semidirect
13. #34036124C2semidirect
14. C2×#22627544C2semidirect
15. #484×C293C2semidirect
18. C2×#17989789C22semidirect
22. C2×C8×D293C22semidirect
24. #22627615C22semidirect
30. #22627599C22semidirect
35. #22627586C22semidirect
38. #22627582C22semidirect
40. #22627555C22semidirect
41. #22627553C22semidirect
42. #22627548C22semidirect
49. #22627544C22semidirect
50. #22627536C22semidirect
51. C8×Dic293C22semidirect
54. #107×C293C22semidirect
55. #107×C293C22semidirect
56. #103×C293C22semidirect
57. #103×C293C22semidirect
191. C293#3297semidirect
196. #484D293semidirect
234. #107C2×D293semidirect
235. #107C2×D293semidirect
239. #103C2×D293semidirect
240. #103C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340368272
#340367702
#340364852
#340364562
#340364322
#340363662
#340363602
#340363472
#340362772
#340362652
#340362112
#340361922
#340361242
C2×#226275442
#484×C2932
#3297293