Finite Group Databasebeta

← #49170666

Subgroups — #49170666

Order 37504.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34036788C211yesmaximal, characteristic
2#34036486C211yesmaximal, characteristic, complemented
3#34036470C211yesmaximal, characteristic, complemented
4#34036442C211yesmaximal, characteristic, complemented
5#34036433C211yesmaximal, characteristic, complemented
6#34036368C211yesmaximal, characteristic, complemented
7#34036360C211yesmaximal, characteristic, complemented
8#34036351C211yesmaximal, characteristic, complemented
9#34036289C211yesmaximal, characteristic, complemented
10#34036284C211yesmaximal, characteristic, complemented
11C4×#17989792C211yesmaximal, characteristic, complemented
12D4×#14556873C211yesmaximal, characteristic, complemented
13#34036123C211yesmaximal, characteristic, complemented
14C2×#22627547C211yesmaximal, characteristic, complemented
15#552×C293C211yesmaximal, characteristic, complemented
16C2×#17989802C2211yescharacteristic
17C22×Dic586C2211yescharacteristic
18C2×#17989792C2211yescharacteristic
19C2×#17989792C2211yescharacteristic
20C2×#17989789C2211yescharacteristic
21C22×#14556873C2211yescharacteristic
22C22×#14556873C2211yescharacteristic
23#22627612C2211yescharacteristic
24#22627610—21no—
25#22627603C2211yescharacteristic
26#22627602—21no—
27#22627598C2211yescharacteristic
28#22627596—21no—
29C4×Dic586C2211yescharacteristic
30#22627584—21no—
31#22627557C2211yescharacteristic
32#22627557C2211yescharacteristic
33#22627555C2211yescharacteristic
34C4⋊#14556873C2211yescharacteristic
35#22627548C2211yescharacteristic
36#22627547C2211yescharacteristic
37#22627547C2211yescharacteristic
38#22627547C2211yescharacteristic
39#22627547C2211yescharacteristic
40#22627547C2211yescharacteristic
41#22627547C2211yescharacteristic
42#22627544C2211yescharacteristic
43#22627544C2211yescharacteristic
44#22627544C2211yescharacteristic
45C22⋊#14556873C2211yescharacteristic
46C22⋊#14556873C2211yescharacteristic
47C4×#14556873C2211yescharacteristic
48#126×C293—21no—
49#125×C293—21no—
50#125×C293C2211yescharacteristic
51#123×C293—21no—
52C22⋊Q8×C293C2211yescharacteristic
53C22⋊Q8×C293C2211yescharacteristic
54C4×Q8×C293C2211yescharacteristic
55C4×D4×C293C2211yescharacteristic
56#117×C293—21no—
57#117×C293C2211yescharacteristic
58C2×C4⋊C4×C293C2211yescharacteristic
59#17989802—22no—
60#17989802—41no—
61C2×Dic586—22no—
62C2×Dic586—21no—
63C2×Dic586—41no—
64C2×Dic586C2311yescharacteristic
65C2×Dic586C2311yescharacteristic
66C2×D1172C2311yescharacteristic
67C2×C22⋊D293—21no—
68C2×C22⋊D293—21no—
69C22×Dic293—21no—
70C2×C4×D293—21no—
71#17989792—22no—
72#17989792—41no—
73#17989789—41no—
74C2×#14556873—41no—
75C2×#14556873—41no—
76C2×#14556873C2311yescharacteristic
77C2×#14556873C2311yescharacteristic
78C2×#14556873C2311yescharacteristic
79C2×#14556873C2311yescharacteristic
80C4⋊Dic293C2311yescharacteristic
81#17989781—21no—
82#17989781—21no—
83C22⋊Dic293—21no—
84#17989779—21no—
85#17989779—21no—
86#17989779—21no—
87C4×Dic293—21no—
88C2×Q8×C293C2311yescharacteristic
89C2×D4×C293D411yescharacteristic
90C22×C4×C293—21no—
91C22×C4×C293C2311yescharacteristic
92C22×C4×C293C2311yescharacteristic
93C4⋊C4×C293—21no—
94C4⋊C4×C293—21no—
95C4⋊C4×C293—21no—
96C4⋊C4×C293—21no—
97C4⋊C4×C293—21no—
98C4⋊C4×C293—21no—
99C4⋊C4×C293D411yescharacteristic
100C4⋊C4×C293C2311yescharacteristic
101C4⋊C4×C293C2311yescharacteristic
102C4⋊C4×C293C2311yescharacteristic
103C22⋊C4×C293—21no—
104C22⋊C4×C293—21no—
105C22⋊C4×C293—21no—
106C22⋊C4×C293D411yescharacteristic
107C22⋊C4×C293D411yescharacteristic
108C42×C293—21no—
109C42×C293—21no—
110C42×C293C2311yescharacteristic
111C22×D293—21no—
112Dic586—21no—
113Dic586—21no—
114Dic586—21no—
115Dic586—42no—
116Dic586#4112yes—
117D1172—21no—
118D1172—21no—
119C22⋊D293—42no—
120C22⋊D293—42no—
121C2×Dic293—21no—
122C2×Dic293—21no—
123C2×Dic293—21no—
124C2×Dic293—42no—
125C4×D293—42no—
126#14556873—21no—
127#14556873—21no—
128#14556873—21no—
129#14556873—21no—
130C23×C293C2×D411yescharacteristic
131C23×C293C2×D411yescharacteristic
132Q8×C293—21no—
133Q8×C293—21no—
134D4×C293—22no—
135C2×C4×C293—21no—
136C2×C4×C293—21no—
137C2×C4×C293—21no—
138C2×C4×C293—21no—
139C2×C4×C293—21no—
140C2×C4×C293—21no—
141C2×C4×C293—21no—
142C2×C4×C293—21no—
143C2×C4×C293—42no—
144C2×C4×C293C2×D411yescharacteristic
145C2×C4×C293C2×D411yescharacteristic
146C2×C4×C293C2×D411yescharacteristic
147C2×C4×C293C2×D411yescharacteristic
148C2×C4×C293C2411yescharacteristic
149C22×C293—21no—
150C22×C293—21no—
151C22×C293—21no—
152C22×C293—21no—
153C22×C293—21no—
154C22×C293#13512yes—
155C22×C293C22×D411yescharacteristic
156C2×D293—41no—
157C2×D293—41no—
158C2×D293—41no—
159C4×C293—21no—
160C4×C293—21no—
161C4×C293—21no—
162C4×C293—21no—
163C4×C293—41no—
164C4×C293—41no—
165C4×C293—41no—
166C4×C293—41no—
167C4×C293C2×#4111yescharacteristic
168C4×C293H(5,𝔽2)11yescharacteristic
169Dic293—22no—
170Dic293—41no—
171Dic293—41no—
172C2×C293—22no—
173C2×C293—41no—
174C2×C293#54611yescharacteristic
175C2×C293C2×#13511yescharacteristic
176C2×C293#57811yescharacteristic
177D293—81no—
178C293#330011yescharacteristic, complemented, Sylow
179#3300—2931nomaximal, Sylow
180#552D29311yescharacteristic, complemented
181#548—2931no—
182#488—2931no—
183#485—2931no—
184#480—2931no—
185#479—2931no—
186#467—2931no—
187#465—2931no—
188#464—2931no—
189#452—2931no—
190#451—2931no—
191C4×Q16—2931no—
192C8×D4—2931no—
193#416—2931no—
194C2×#103—2931no—
195C2×#41—2931no—
196C22×Q8—2931no—
197C2×Q16—2931no—
198C2×Q16—2931no—
199C2×SD16—2931no—
200C22×C8—2931no—
201C22×C8—2931no—
202#126—21no—
203#125—21no—
204#125C2×D29311yescharacteristic
205#124—2931no—
206#124—5861no—
207#123—21no—
208C22⋊Q8—2931no—
209C22⋊Q8—5861no—
210C22⋊Q8C2×D29311yescharacteristic
211C22⋊Q8C2×D29311yescharacteristic
212#121—2931no—
213#121—5861no—
214C4×Q8—2931no—
215C4×Q8C2×D29311yescharacteristic
216C4×D4—5861no—
217C4×D4C2×D29311yescharacteristic
218#117—21no—
219#117C2×D29311yescharacteristic
220C2×C4⋊C4C2×D29311yescharacteristic
221#107—2931no—
222#107—2931no—
223#106—2931no—
224C4⋊C8—2931no—
225#103—2931no—
226#103—2931no—
227#103—2931no—
228#103—2931no—
229#103—2931no—
230#103—2931no—
231#103—2931no—
232#102—2931no—
233#102—2931no—
234#102—2931no—
235C22⋊C8—2931no—
236C22⋊C8—2931no—
237C4×C8—2931no—
238#41—5862no—
239#41—11721no—
240C2×Q8—2931no—
241C2×Q8—2931no—
242C2×Q8—5862no—
243C2×Q8—5861no—
244C2×Q8—11721no—
245C2×Q8C22×D29311yescharacteristic
246C2×D4—2931no—
247C2×D4—5861no—
248C2×D4—5861no—
249C2×D4C22⋊D29311yescharacteristic
250C22×C4—21no—
251C22×C4—5861no—
252C22×C4—5861no—
253C22×C4C22×D29311yescharacteristic
254C22×C4C22×D29311yescharacteristic
255Q16—5862no—
256Q16—11721no—
257SD16—11721no—
258C2×C8—2931no—
259C2×C8—2931no—
260C2×C8—2931no—
261C2×C8—2931no—
262C2×C8—11721no—
263C2×C8—11721no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—2931no—
271C4⋊C4—5861no—
272C4⋊C4—5861no—
273C4⋊C4C22⋊D29311yescharacteristic
274C4⋊C4C22×D29311yescharacteristic
275C4⋊C4C22×D29311yescharacteristic
276C4⋊C4C22×D29311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—5861no—
281C22⋊C4—5861no—
282C22⋊C4—5861no—
283C22⋊C4—5861no—
284C22⋊C4C22⋊D29311yescharacteristic
285C22⋊C4C22⋊D29311yescharacteristic
286C42—21no—
287C42—21no—
288C42—5861no—
289C42C22×D29311yescharacteristic
290C23—5861no—
291C23C2×C22⋊D29311yescharacteristic
292C23C2×C22⋊D29311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—2932no—
296Q8—5861no—
297Q8—5861no—
298Q8—5861no—
299Q8—11722no—
300D4—22no—
301D4—5861no—
302D4—5861no—
303D4—11722no—
304D4—11722no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—5861no—
315C2×C4—5861no—
316C2×C4—5861no—
317C2×C4—11722no—
318C2×C4—11722no—
319C2×C4C2×C22⋊D29311yescharacteristic
320C2×C4C2×C22⋊D29311yescharacteristic
321C2×C4C2×C22⋊D29311yescharacteristic
322C2×C4C2×C22⋊D29311yescharacteristic
323C2×C4C23×D29311yescharacteristic
324C8—5861no—
325C8—5861no—
326C8—5861no—
327C8—5861no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—11721no—
334C22—11721no—
335C22—11721no—
336C22#2262764412yes—
337C22C22×C22⋊D29311yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—5862no—
347C4—11721no—
348C4—11721no—
349C4#41×D29311yescharacteristic
350C4#2262767011yescharacteristic
351C2—22no—
352C2—41no—
353C2—23441no—
354C2#3403676711yescharacteristic
355C2C2×#2262764411yescharacteristic
356C2#3403694411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#552×C293C2
socleC22×C293C22×D4
prime radical#34036788C2
hypercenter#552D293
hypocenterC293#3300
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #34036486C2semidirect
3. #34036470C2semidirect
4. #34036442C2semidirect
5. #34036433C2semidirect
6. #34036368C2semidirect
7. #34036360C2semidirect
8. #34036351C2semidirect
9. #34036289C2semidirect
10. #34036284C2semidirect
11. C4×#17989792C2semidirect
12. D4×#14556873C2semidirect
13. #34036123C2semidirect
14. C2×#22627547C2semidirect
15. #552×C293C2semidirect
178. C293#3300semidirect
180. #552D293semidirect

Maximal subgroups

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

subgroupindex
#340367882
#340364862
#340364702
#340364422
#340364332
#340363682
#340363602
#340363512
#340362892
#340362842
C4×#179897922
D4×#145568732
#340361232
C2×#226275472
#552×C2932
#3300293