Finite Group Databasebeta

← #49170676

Subgroups — #49170676

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#34036821C211yesmaximal, characteristic, complemented
2#34036790C211yesmaximal, characteristic
3#34036485C211yesmaximal, characteristic, complemented
4#34036469C211yesmaximal, characteristic, complemented
5#34036443C211yesmaximal, characteristic, complemented
6#34036438C211yesmaximal, characteristic, complemented
7#34036367C211yesmaximal, characteristic, complemented
8#34036360C211yesmaximal, characteristic, complemented
9#34036353C211yesmaximal, characteristic, complemented
10#34036290C211yesmaximal, characteristic, complemented
11Q16×Dic293C211yesmaximal, characteristic, complemented
12#34036192C211yesmaximal, characteristic, complemented
13#34036124C211yesmaximal, characteristic, complemented
14C2×#22627548C211yesmaximal, characteristic, complemented
15#451×C293C211yesmaximal, characteristic, complemented
16C2×#17989803C2211yescharacteristic
17C2×#17989789C2211yescharacteristic
18C22×#14556873C2211yescharacteristic
19C2×C8×D293C2211yescharacteristic
20#22627619—21no—
21#22627615—21no—
22#22627615C2211yescharacteristic
23#22627613—21no—
24#22627613C2211yescharacteristic
25#22627608—21no—
26#22627600—21no—
27#22627600C2211yescharacteristic
28#22627600C2211yescharacteristic
29#22627598—21no—
30#22627598C2211yescharacteristic
31Q8×Dic293C2211yescharacteristic
32Q8×Dic293C2211yescharacteristic
33C4×C22⋊D293—21no—
34C4×C22⋊D293C2211yescharacteristic
35#22627582C2211yescharacteristic
36#22627580—21no—
37C2×C4⋊Dic293C2211yescharacteristic
38#22627558C2211yescharacteristic
39#22627558C2211yescharacteristic
40#22627555C2211yescharacteristic
41#22627553C2211yescharacteristic
42#22627548C2211yescharacteristic
43#22627548C2211yescharacteristic
44#22627548C2211yescharacteristic
45#22627548C2211yescharacteristic
46#22627548C2211yescharacteristic
47#22627546C2211yescharacteristic
48#22627546C2211yescharacteristic
49#22627544C2211yescharacteristic
50#22627536C2211yescharacteristic
51C8×Dic293C2211yescharacteristic
52C22×Q8×C293C2211yescharacteristic
53C2×Q16×C293C2211yescharacteristic
54C2×Q16×C293C2211yescharacteristic
55C22⋊Q8×C293C2211yescharacteristic
56#103×C293C2211yescharacteristic
57#103×C293C2211yescharacteristic
58C22⋊C8×C293C2211yescharacteristic
59#17989803—22no—
60#17989803—41no—
61C2×D1172—21no—
62C2×D1172C2311yescharacteristic
63C2×C22⋊D293—21no—
64C2×C22⋊D293D411yescharacteristic
65C22×Dic293—21no—
66C2×C4×D293—21no—
67C2×C4×D293C2311yescharacteristic
68#17989789—41no—
69C2×#14556873—41no—
70C2×#14556873C2311yescharacteristic
71C2×#14556873C2311yescharacteristic
72C8×D293—41no—
73C4⋊Dic293—21no—
74C4⋊Dic293—21no—
75C4⋊Dic293—21no—
76C4⋊Dic293—21no—
77C4⋊Dic293C2311yescharacteristic
78C4⋊Dic293C2311yescharacteristic
79C4⋊Dic293C2311yescharacteristic
80#17989781—21no—
81#17989781—21no—
82#17989781—21no—
83#17989781D411yescharacteristic
84C22⋊Dic293D411yescharacteristic
85#17989779—21no—
86#17989779—21no—
87#17989779—21no—
88#17989779—21no—
89#17989779—21no—
90#17989779—21no—
91#17989779D411yescharacteristic
92C4×Dic293—21no—
93C4×Dic293—21no—
94C4×Dic293—21no—
95C4×Dic293C2311yescharacteristic
96C2×Q8×C293—21no—
97C2×Q8×C293—22no—
98C2×Q8×C293—41no—
99C2×Q8×C293C2311yescharacteristic
100C2×Q8×C293C2311yescharacteristic
101C2×Q8×C293C2311yescharacteristic
102C22×C4×C293—21no—
103C22×C4×C293C2311yescharacteristic
104Q16×C293—22no—
105Q16×C293—41no—
106C2×C8×C293C2311yescharacteristic
107C2×C8×C293C2311yescharacteristic
108C4⋊C4×C293—21no—
109C4⋊C4×C293C2311yescharacteristic
110C22⋊C4×C293—21no—
111C22×D293—21no—
112C22×D293C2×D411yescharacteristic
113D1172—21no—
114D1172—21no—
115D1172—42no—
116C22⋊D293—22no—
117C22⋊D293—42no—
118C2×Dic293—21no—
119C2×Dic293—21no—
120C2×Dic293—21no—
121C2×Dic293—42no—
122C2×Dic293C2×D411yescharacteristic
123C2×Dic293C2×D411yescharacteristic
124C2×Dic293C2×D411yescharacteristic
125C4×D293—21no—
126C4×D293—21no—
127C4×D293—42no—
128#14556873—21no—
129#14556873—21no—
130C23×C293C2×D411yescharacteristic
131Q8×C293—21no—
132Q8×C293—21no—
133Q8×C293—21no—
134Q8×C293—21no—
135Q8×C293—21no—
136Q8×C293—42no—
137Q8×C293#4112yes—
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—42no—
145C2×C4×C293C2×D411yescharacteristic
146C2×C4×C293C2411yescharacteristic
147C8×C293—21no—
148C8×C293—21no—
149C22×C293—21no—
150C22×C293—21no—
151C22×C293#13512yes—
152C22×C293C22×D411yescharacteristic
153C2×D293—21no—
154C2×D293—21no—
155C2×D293—21no—
156C2×D293—41no—
157C2×D293—41no—
158C2×D293—41no—
159C4×C293—21no—
160C4×C293—22no—
161C4×C293—41no—
162C4×C293—41no—
163C4×C293—41no—
164C4×C293C2×#4111yescharacteristic
165C4×C293H(5,𝔽2)11yescharacteristic
166Dic293—21no—
167Dic293—21no—
168Dic293—21no—
169Dic293—41no—
170Dic293—41no—
171Dic293—41no—
172C2×C293—22no—
173C2×C293#54611yescharacteristic
174C2×C293C2×#13511yescharacteristic
175C2×C293#57811yescharacteristic
176D293—41no—
177D293—81no—
178C293#330011yescharacteristic, complemented, Sylow
179#3300—2931nomaximal, Sylow
180#552—2931no—
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#451D29311yescharacteristic, complemented
191C4×Q16—2931no—
192C8×D4—2931no—
193#416—2931no—
194C2×#103—2931no—
195C2×#41—2931no—
196C22×Q8C2×D29311yescharacteristic
197C2×Q16C2×D29311yescharacteristic
198C2×Q16C2×D29311yescharacteristic
199C2×SD16—2931no—
200C22×C8—2931no—
201C22×C8—2931no—
202#126—5861no—
203#125—2931no—
204#125—5861no—
205#124—2931no—
206#124—5861no—
207#123—5861no—
208C22⋊Q8—2931no—
209C22⋊Q8—2931no—
210C22⋊Q8—5861no—
211C22⋊Q8C2×D29311yescharacteristic
212#121—2931no—
213#121—5861no—
214C4×Q8—2931no—
215C4×Q8—2931no—
216C4×D4—2931no—
217C4×D4—5861no—
218#117—2931no—
219#117—5861no—
220C2×C4⋊C4—2931no—
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#103C2×D29311yescharacteristic
231#103C2×D29311yescharacteristic
232#102—2931no—
233#102—2931no—
234#102—2931no—
235C22⋊C8—2931no—
236C22⋊C8C2×D29311yescharacteristic
237C4×C8—2931no—
238#41—5862no—
239#41—11721no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8C22×D29311yescharacteristic
244C2×Q8C22×D29311yescharacteristic
245C2×Q8C22×D29311yescharacteristic
246C2×D4—2931no—
247C2×D4—2931no—
248C2×D4—5861no—
249C2×D4—5861no—
250C22×C4—21no—
251C22×C4—2931no—
252C22×C4—5861no—
253C22×C4—5861no—
254C22×C4C22×D29311yescharacteristic
255Q16—22no—
256Q16—41no—
257SD16—11721no—
258C2×C8—2931no—
259C2×C8—2931no—
260C2×C8—11721no—
261C2×C8—11721no—
262C2×C8C22×D29311yescharacteristic
263C2×C8C22×D29311yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—2931no—
266C4⋊C4—2931no—
267C4⋊C4—2931no—
268C4⋊C4—2931no—
269C4⋊C4—5861no—
270C4⋊C4—5861no—
271C4⋊C4—5861no—
272C4⋊C4—5861no—
273C4⋊C4—5861no—
274C4⋊C4—5861no—
275C4⋊C4—5861no—
276C4⋊C4C22×D29311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—2931no—
279C22⋊C4—2931no—
280C22⋊C4—5861no—
281C22⋊C4—5861no—
282C22⋊C4—5861no—
283C22⋊C4—5861no—
284C22⋊C4—5861no—
285C22⋊C4—5861no—
286C42—2931no—
287C42—5861no—
288C42—5861no—
289C42—5861no—
290C23—2931no—
291C23—5861no—
292C23D4×D29311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—42no—
299Q8#1798980112yes—
300D4—5862no—
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—42no—
312C2×C4—2931no—
313C2×C4—2931no—
314C2×C4—2931no—
315C2×C4—5861no—
316C2×C4—5861no—
317C2×C4—5861no—
318C2×C4—5861no—
319C2×C4—5861no—
320C2×C4—11722no—
321C2×C4—11722no—
322C2×C4D4×D29311yescharacteristic
323C2×C4C23×D29311yescharacteristic
324C8—21no—
325C8—21no—
326C8—5861no—
327C8—5861no—
328C22—21no—
329C22—21no—
330C22—5861no—
331C22—5861no—
332C22—5861no—
333C22—11721no—
334C22—11721no—
335C22—11721no—
336C22#2262764712yes—
337C22C2×D4×D29311yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—5861no—
344C4—5861no—
345C4—5861no—
346C4—11721no—
347C4—11721no—
348C4—11721no—
349C4C2×#1798980111yescharacteristic
350C4#2262766911yescharacteristic
351C2—22no—
352C2—23441no—
353C2—11721no—
354C2#3403676611yescharacteristic
355C2C2×#2262764711yescharacteristic
356C2#3403694511yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#451×C293C2
socleC22×C293C22×D4
prime radical#34036790C2
hypercenter#451D293
hypocenterC293#3300
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. #34036821C2semidirect
3. #34036485C2semidirect
4. #34036469C2semidirect
5. #34036443C2semidirect
6. #34036438C2semidirect
7. #34036367C2semidirect
8. #34036360C2semidirect
9. #34036353C2semidirect
10. #34036290C2semidirect
11. Q16×Dic293C2semidirect
12. #34036192C2semidirect
13. #34036124C2semidirect
14. C2×#22627548C2semidirect
15. #451×C293C2semidirect
178. C293#3300semidirect
190. #451D293semidirect

Maximal subgroups

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

subgroupindex
#340368212
#340367902
#340364852
#340364692
#340364432
#340364382
#340363672
#340363602
#340363532
#340362902
Q16×Dic2932
#340361922
#340361242
C2×#226275482
#451×C2932
#3300293