Finite Group Databasebeta

← #49169014

Subgroups — #49169014

Order 37504.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34036750C211yesmaximal, characteristic
2#34036470C211yesmaximal, characteristic, complemented
3#34036453C211yesmaximal, characteristic
4#34036424C211yesmaximal, characteristic, complemented
5#34036414C211yesmaximal, characteristic, complemented
6#34036351C211yesmaximal, characteristic, complemented
7#34036351C211yesmaximal, characteristic, complemented
8#34036285C211yesmaximal, characteristic, complemented
9#34036268C211yesmaximal, characteristic
10#34036234C211yesmaximal, characteristic, complemented
11#34036227C211yesmaximal, characteristic, complemented
12C4×#17989792C211yesmaximal, characteristic, complemented
13C4×#17989789C211yesmaximal, characteristic, complemented
14#34036190C211yesmaximal, characteristic, complemented
15#519×C293C211yesmaximal, characteristic, complemented
16C2×#17989792C2211yescharacteristic
17C2×#17989792C2211yescharacteristic
18C2×#17989789C2211yescharacteristic
19C2×#17989789C2211yescharacteristic
20#22627624C2211yescharacteristic
21#22627619—21no—
22#22627612C2211yescharacteristic
23#22627610—21no—
24#22627606—21no—
25#22627603C2211yescharacteristic
26#22627602—21no—
27#22627598C2211yescharacteristic
28C4×Dic586C2211yescharacteristic
29C4×D1172C2211yescharacteristic
30#22627557C2211yescharacteristic
31#22627557C2211yescharacteristic
32#22627555C2211yescharacteristic
33#22627555C2211yescharacteristic
34C4⋊#14556873C2211yescharacteristic
35C4⋊#14556873C2211yescharacteristic
36#22627548C2211yescharacteristic
37#22627548C2211yescharacteristic
38#22627548C2211yescharacteristic
39#22627548C2211yescharacteristic
40#22627547C2211yescharacteristic
41#22627547C2211yescharacteristic
42#22627544C2211yescharacteristic
43#22627544C2211yescharacteristic
44C22⋊#14556873C2211yescharacteristic
45C22⋊#14556873C2211yescharacteristic
46#22627535C2211yescharacteristic
47C4×#14556873C2211yescharacteristic
48C22×Q8×C293C2211yescharacteristic
49C4×Q8×C293—21no—
50C4×Q8×C293—21no—
51C4×Q8×C293C2211yescharacteristic
52C4×Q8×C293C2211yescharacteristic
53C4×Q8×C293C2211yescharacteristic
54C4×Q8×C293C2211yescharacteristic
55#117×C293—21no—
56#117×C293—21no—
57#117×C293C2211yescharacteristic
58#117×C293C2211yescharacteristic
59C2×Dic586—21no—
60C2×Dic586C2311yescharacteristic
61C2×D1172C2311yescharacteristic
62C2×C22⋊D293—21no—
63C2×C4×D293—21no—
64#17989792—22no—
65#17989792—22no—
66#17989789—22no—
67#17989789—22no—
68C2×#14556873C2311yescharacteristic
69C2×#14556873C2311yescharacteristic
70C2×#14556873C2311yescharacteristic
71C2×#14556873C2311yescharacteristic
72C4⋊Dic293—21no—
73C4⋊Dic293C2311yescharacteristic
74C4⋊Dic293C2311yescharacteristic
75#17989781—21no—
76#17989781—21no—
77C22⋊Dic293—21no—
78#17989779—21no—
79#17989779—21no—
80#17989779—21no—
81#17989779—21no—
82C4×Dic293—21no—
83C2×Q8×C293—22no—
84C2×Q8×C293—22no—
85C2×Q8×C293—21no—
86C2×Q8×C293C2311yescharacteristic
87C2×Q8×C293C2311yescharacteristic
88C22×C4×C293—21no—
89C22×C4×C293C2311yescharacteristic
90C4⋊C4×C293—21no—
91C4⋊C4×C293—21no—
92C4⋊C4×C293—21no—
93C4⋊C4×C293—21no—
94C4⋊C4×C293D411yescharacteristic
95C4⋊C4×C293D411yescharacteristic
96C4⋊C4×C293C2311yescharacteristic
97C4⋊C4×C293C2311yescharacteristic
98C22⋊C4×C293—21no—
99C22⋊C4×C293D411yescharacteristic
100C22⋊C4×C293D411yescharacteristic
101C42×C293—21no—
102C42×C293—21no—
103C42×C293—21no—
104C42×C293—21no—
105C42×C293—21no—
106C42×C293C2311yescharacteristic
107C42×C293C2311yescharacteristic
108C22×D293—21no—
109Dic586—21no—
110Dic586—21no—
111Dic586—81no—
112D1172—21no—
113D1172—21no—
114C22⋊D293—81no—
115C2×Dic293—21no—
116C2×Dic293—21no—
117C2×Dic293—21no—
118C4×D293—81no—
119#14556873—21no—
120#14556873—21no—
121#14556873—21no—
122#14556873—21no—
123C23×C293C2×D411yescharacteristic
124Q8×C293—24no—
125Q8×C293—21no—
126Q8×C293—21no—
127Q8×C293#4112yes—
128Q8×C293#4112yes—
129C2×C4×C293—24no—
130C2×C4×C293—21no—
131C2×C4×C293—21no—
132C2×C4×C293—21no—
133C2×C4×C293—21no—
134C2×C4×C293—21no—
135C2×C4×C293—21no—
136C2×C4×C293C2×D411yescharacteristic
137C2×C4×C293C2×D411yescharacteristic
138C2×C4×C293C2×D411yescharacteristic
139C2×C4×C293C2×D411yescharacteristic
140C2×C4×C293C2×D411yescharacteristic
141C2×C4×C293C2411yescharacteristic
142C22×C293—21no—
143C22×C293—21no—
144C22×C293—21no—
145C22×C293C22×D411yescharacteristic
146C2×D293—41no—
147C2×D293—41no—
148C2×D293—41no—
149C4×C293—22no—
150C4×C293—22no—
151C4×C293—21no—
152C4×C293—21no—
153C4×C293—21no—
154C4×C293—21no—
155C4×C293—21no—
156C4×C293—41no—
157C4×C293—41no—
158C4×C293C2×#4111yescharacteristic
159C4×C293C2×#4111yescharacteristic
160Dic293—41no—
161Dic293—41no—
162Dic293—41no—
163C2×C293—41no—
164C2×C293#52511yescharacteristic
165C2×C293#57711yescharacteristic
166C2×C293#57811yescharacteristic
167D293—81no—
168C293#312111yescharacteristic, complemented, Sylow
169#3121—2931nomaximal, Sylow
170#542—2931no—
171#519D29311yescharacteristic, complemented
172#485—2931no—
173#482—2931no—
174#477—2931no—
175#475—2931no—
176#464—2931no—
177#464—2931no—
178#451—2931no—
179#448—2931no—
180#441—2931no—
181#440—2931no—
182C4×Q16—2931no—
183C4×SD16—2931no—
184#433—2931no—
185C22×Q8C2×D29311yescharacteristic
186C2×Q16—2931no—
187C2×Q16—2931no—
188C2×SD16—2931no—
189C2×SD16—2931no—
190C4⋊Q8—2931no—
191#126—5861no—
192#124—2931no—
193#124—5861no—
194#123—5861no—
195C22⋊Q8—2931no—
196C22⋊Q8—5861no—
197#121—2931no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—2931no—
201C4×Q8C2×D29311yescharacteristic
202C4×Q8C2×D29311yescharacteristic
203C4×Q8C2×D29311yescharacteristic
204C4×Q8C2×D29311yescharacteristic
205C4×D4—2931no—
206#117—21no—
207#117—21no—
208#117C2×D29311yescharacteristic
209#117C2×D29311yescharacteristic
210#107—2931no—
211#107—2931no—
212#106—2931no—
213#106—2931no—
214C4⋊C8—2931no—
215C4⋊C8—2931no—
216#103—2931no—
217#103—2931no—
218#103—2931no—
219#103—2931no—
220#103—2931no—
221#103—2931no—
222#102—2931no—
223#102—2931no—
224C22⋊C8—2931no—
225C22⋊C8—2931no—
226#97—2931no—
227C4×C8—2931no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—21no—
231C2×Q8—2931no—
232C2×Q8—5861no—
233C2×Q8C22×D29311yescharacteristic
234C2×Q8C22×D29311yescharacteristic
235C2×D4—2931no—
236C2×D4—5861no—
237C22×C4—21no—
238C22×C4—5861no—
239C22×C4C22×D29311yescharacteristic
240Q16—5862no—
241Q16—5862no—
242SD16—5862no—
243SD16—5862no—
244C2×C8—2931no—
245C2×C8—2931no—
246C2×C8—2931no—
247C2×C8—2931no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—2931no—
253C4⋊C4—2931no—
254C4⋊C4—5861no—
255C4⋊C4—5861no—
256C4⋊C4—5861no—
257C4⋊C4C22⋊D29311yescharacteristic
258C4⋊C4C22⋊D29311yescharacteristic
259C4⋊C4C22×D29311yescharacteristic
260C4⋊C4C22×D29311yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—5861no—
263C22⋊C4—5861no—
264C22⋊C4—5861no—
265C22⋊C4—5861no—
266C22⋊C4—5861no—
267C22⋊C4C22⋊D29311yescharacteristic
268C22⋊C4C22⋊D29311yescharacteristic
269C42—21no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—5861no—
275C42C22×D29311yescharacteristic
276C42C22×D29311yescharacteristic
277C23—5861no—
278C23C2×C22⋊D29311yescharacteristic
279Q8—24no—
280Q8—21no—
281Q8—21no—
282Q8—23441no—
283Q8—5861no—
284Q8—5861no—
285Q8#1798980212yes—
286Q8#1798980212yes—
287D4—23441no—
288D4—5861no—
289D4—5861no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—23441no—
298C2×C4—5861no—
299C2×C4—5861no—
300C2×C4—5861no—
301C2×C4C2×C22⋊D29311yescharacteristic
302C2×C4C2×C22⋊D29311yescharacteristic
303C2×C4C2×C22⋊D29311yescharacteristic
304C2×C4C2×C22⋊D29311yescharacteristic
305C2×C4C2×C22⋊D29311yescharacteristic
306C2×C4C23×D29311yescharacteristic
307C8—5861no—
308C8—5861no—
309C8—5861no—
310C8—5861no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—11721no—
315C22—11721no—
316C22—11721no—
317C22C22×C22⋊D29311yescharacteristic
318C4—22no—
319C4—22no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—11721no—
328C4—11721no—
329C4—11721no—
330C4C2×#1798980211yescharacteristic
331C4C2×#1798980211yescharacteristic
332C2—41no—
333C2—23441no—
334C2#3403664011yescharacteristic
335C2#3403693711yescharacteristic
336C2#3403694411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#519×C293C2
socleC22×C293C22×D4
prime radical#22627598C22
hypercenter#519D293
hypocenterC293#3121
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. #34036470C2semidirect
4. #34036424C2semidirect
5. #34036414C2semidirect
6. #34036351C2semidirect
7. #34036351C2semidirect
8. #34036285C2semidirect
10. #34036234C2semidirect
11. #34036227C2semidirect
12. C4×#17989792C2semidirect
13. C4×#17989789C2semidirect
14. #34036190C2semidirect
15. #519×C293C2semidirect
168. C293#3121semidirect
171. #519D293semidirect

Maximal subgroups

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

subgroupindex
#340367502
#340364702
#340364532
#340364242
#340364142
#340363512
#340362852
#340362682
#340362342
#340362272
C4×#179897922
C4×#179897892
#340361902
#519×C2932
#3121293