Finite Group Databasebeta

← #49169007

Subgroups — #49169007

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#34036746C211yesmaximal, characteristic
2#34036607C211yesmaximal, characteristic, complemented
3#34036471C211yesmaximal, characteristic, complemented
4#34036450C211yesmaximal, characteristic
5#34036422C211yesmaximal, characteristic, complemented
6#34036412C211yesmaximal, characteristic, complemented
7#34036349C211yesmaximal, characteristic, complemented
8#34036349C211yesmaximal, characteristic, complemented
9#34036285C211yesmaximal, characteristic, complemented
10#34036233C211yesmaximal, characteristic, complemented
11#34036230C211yesmaximal, characteristic, complemented
12#34036221C211yesmaximal, characteristic, complemented
13SD16×Dic293C211yesmaximal, characteristic, complemented
14#34036187C211yesmaximal, characteristic, complemented
15#448×C293C211yesmaximal, characteristic
16C2×#17989792C2211yescharacteristic
17C2×#17989792C2211yescharacteristic
18#22627623C2211yescharacteristic
19#22627617—21no—
20#22627611—21no—
21#22627611C2211yescharacteristic
22#22627605—21no—
23#22627603C2211yescharacteristic
24#22627601—21no—
25Q8×Dic293—21no—
26Q8×Dic293—21no—
27Q8×Dic293C2211yescharacteristic
28Q8×Dic293C2211yescharacteristic
29Q8×Dic293C2211yescharacteristic
30Q8×Dic293C2211yescharacteristic
31#22627589C2211yescharacteristic
32D4×Dic293C2211yescharacteristic
33#22627583—21no—
34#22627583—21no—
35#22627583C2211yescharacteristic
36#22627583C2211yescharacteristic
37#22627558C2211yescharacteristic
38#22627557C2211yescharacteristic
39#22627556C2211yescharacteristic
40#22627555C2211yescharacteristic
41#22627553C2211yescharacteristic
42#22627553C2211yescharacteristic
43#22627549C2211yescharacteristic
44#22627549C2211yescharacteristic
45#22627548C2211yescharacteristic
46#22627548C2211yescharacteristic
47#22627545C2211yescharacteristic
48#22627545C2211yescharacteristic
49C22⋊#14556873C2211yescharacteristic
50#22627534C2211yescharacteristic
51C8×Dic293C2211yescharacteristic
52C22×Q8×C293C2211yescharacteristic
53C2×SD16×C293C2211yescharacteristic
54C2×SD16×C293C2211yescharacteristic
55#121×C293C2211yescharacteristic
56#103×C293C2211yescharacteristic
57#103×C293C2211yescharacteristic
58C22⋊C8×C293C2211yescharacteristic
59C2×Dic586—21no—
60C2×Dic586C2311yescharacteristic
61C22×Dic293—21no—
62#17989792—22no—
63#17989792—22no—
64C2×#14556873C2311yescharacteristic
65C2×#14556873C2311yescharacteristic
66C4⋊Dic293—21no—
67C4⋊Dic293—21no—
68C4⋊Dic293—21no—
69C4⋊Dic293—21no—
70C4⋊Dic293D411yescharacteristic
71C4⋊Dic293D411yescharacteristic
72C4⋊Dic293C2311yescharacteristic
73C4⋊Dic293C2311yescharacteristic
74C4⋊Dic293C2311yescharacteristic
75#17989781—21no—
76#17989781—21no—
77#17989781—21no—
78C22⋊Dic293—21no—
79C22⋊Dic293—21no—
80C22⋊Dic293—21no—
81C22⋊Dic293—21no—
82C22⋊Dic293—21no—
83C22⋊Dic293D411yescharacteristic
84C22⋊Dic293D411yescharacteristic
85C4×Dic293—21no—
86C4×Dic293—21no—
87C4×Dic293—21no—
88C4×Dic293—21no—
89C4×Dic293—21no—
90C4×Dic293—21no—
91C4×Dic293C2311yescharacteristic
92C4×Dic293C2311yescharacteristic
93C2×Q8×C293—22no—
94C2×Q8×C293—22no—
95C2×Q8×C293—21no—
96C2×Q8×C293C2311yescharacteristic
97C2×Q8×C293C2311yescharacteristic
98C2×D4×C293—21no—
99C2×D4×C293C2311yescharacteristic
100C22×C4×C293—21no—
101C22×C4×C293C2311yescharacteristic
102SD16×C293—22no—
103SD16×C293—22no—
104C2×C8×C293C2311yescharacteristic
105C2×C8×C293C2311yescharacteristic
106C4⋊C4×C293C2311yescharacteristic
107C22⋊C4×C293—21no—
108Dic586—21no—
109Dic586—21no—
110Dic586—81no—
111C2×Dic293—21no—
112C2×Dic293—21no—
113C2×Dic293—21no—
114C2×Dic293—21no—
115C2×Dic293—81no—
116C2×Dic293C2×D411yescharacteristic
117C2×Dic293C2×D411yescharacteristic
118C2×Dic293C2×D411yescharacteristic
119C2×Dic293C2×D411yescharacteristic
120#14556873—21no—
121#14556873—21no—
122C23×C293—21no—
123C23×C293C2×D411yescharacteristic
124Q8×C293—24no—
125Q8×C293—21no—
126Q8×C293—21no—
127Q8×C293#4112yes—
128Q8×C293#4112yes—
129D4×C293—21no—
130D4×C293—21no—
131D4×C293—81no—
132C2×C4×C293—24no—
133C2×C4×C293—21no—
134C2×C4×C293—21no—
135C2×C4×C293—21no—
136C2×C4×C293—21no—
137C2×C4×C293—21no—
138C2×C4×C293C2×D411yescharacteristic
139C2×C4×C293C2411yescharacteristic
140C8×C293—21no—
141C8×C293—21no—
142C22×C293—21no—
143C22×C293—21no—
144C22×C293—21no—
145C22×C293—41no—
146C22×C293—41no—
147C22×C293—41no—
148C22×C293C22×D411yescharacteristic
149C4×C293—22no—
150C4×C293—22no—
151C4×C293—21no—
152C4×C293—41no—
153C4×C293C2×#4111yescharacteristic
154C4×C293C2×#4111yescharacteristic
155Dic293—21no—
156Dic293—21no—
157Dic293—21no—
158Dic293—21no—
159Dic293—41no—
160Dic293—41no—
161Dic293—41no—
162Dic293—41no—
163C2×C293—41no—
164C2×C293—81no—
165C2×C293#52511yescharacteristic
166C2×C293#57711yescharacteristic
167C2×C293#57811yescharacteristic
168C293#312111yescharacteristic, complemented, Sylow
169#3121—2931nomaximal, Sylow
170#542—2931no—
171#519—2931no—
172#485—2931no—
173#482—2931no—
174#477—2931no—
175#475—2931no—
176#464—2931no—
177#464—2931no—
178#451—2931no—
179#448D29311yescharacteristic
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×SD16C2×D29311yescharacteristic
189C2×SD16C2×D29311yescharacteristic
190C4⋊Q8—2931no—
191#126—5861no—
192#124—2931no—
193#124—5861no—
194#123—5861no—
195C22⋊Q8—2931no—
196C22⋊Q8—5861no—
197#121C2×D29311yescharacteristic
198C4×Q8—2931no—
199C4×Q8—2931no—
200C4×Q8—2931no—
201C4×Q8—2931no—
202C4×Q8—2931no—
203C4×Q8—5861no—
204C4×Q8—5861no—
205C4×D4—2931no—
206#117—2931no—
207#117—2931no—
208#117—5861no—
209#117—5861no—
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#103C2×D29311yescharacteristic
221#103C2×D29311yescharacteristic
222#102—2931no—
223#102—2931no—
224C22⋊C8—2931no—
225C22⋊C8C2×D29311yescharacteristic
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—21no—
236C2×D4C22×D29311yescharacteristic
237C22×C4—21no—
238C22×C4—5861no—
239C22×C4C22×D29311yescharacteristic
240Q16—5862no—
241Q16—5862no—
242SD16—22no—
243SD16—22no—
244C2×C8—2931no—
245C2×C8—2931no—
246C2×C8C22×D29311yescharacteristic
247C2×C8C22×D29311yescharacteristic
248C4⋊C4—2931no—
249C4⋊C4—2931no—
250C4⋊C4—2931no—
251C4⋊C4—2931no—
252C4⋊C4—2931no—
253C4⋊C4—5861no—
254C4⋊C4—5861no—
255C4⋊C4—5861no—
256C4⋊C4—5861no—
257C4⋊C4—5861no—
258C4⋊C4—5861no—
259C4⋊C4—5861no—
260C4⋊C4C22×D29311yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—2931no—
263C22⋊C4—2931no—
264C22⋊C4—5861no—
265C22⋊C4—5861no—
266C22⋊C4—5861no—
267C22⋊C4—5861no—
268C22⋊C4—5861no—
269C42—2931no—
270C42—2931no—
271C42—5861no—
272C42—5861no—
273C42—5861no—
274C42—5861no—
275C42—5861no—
276C42—5861no—
277C23—21no—
278C23D4×D29311yescharacteristic
279Q8—24no—
280Q8—21no—
281Q8—21no—
282Q8—23441no—
283Q8—5861no—
284Q8—5861no—
285Q8#1798980112yes—
286Q8#1798980112yes—
287D4—21no—
288D4—21no—
289D4—81no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—2931no—
297C2×C4—2931no—
298C2×C4—2931no—
299C2×C4—2931no—
300C2×C4—23441no—
301C2×C4—5861no—
302C2×C4—5861no—
303C2×C4—5861no—
304C2×C4—5861no—
305C2×C4D4×D29311yescharacteristic
306C2×C4C23×D29311yescharacteristic
307C8—21no—
308C8—21no—
309C8—5861no—
310C8—5861no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22C2×D4×D29311yescharacteristic
318C4—22no—
319C4—22no—
320C4—21no—
321C4—41no—
322C4—5861no—
323C4—5861no—
324C4—5861no—
325C4—5861no—
326C4—11721no—
327C4—11721no—
328C4—11721no—
329C4—11721no—
330C4C2×#1798980111yescharacteristic
331C4C2×#1798980111yescharacteristic
332C2—41no—
333C2—81no—
334C2#3403663511yescharacteristic
335C2#3403693511yescharacteristic
336C2#3403694511yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#448×C293C2
socleC22×C293C22×D4
prime radical#121×C293C22
hypercenter#448D293
hypocenterC293#3121
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. #34036607C2semidirect
3. #34036471C2semidirect
5. #34036422C2semidirect
6. #34036412C2semidirect
7. #34036349C2semidirect
8. #34036349C2semidirect
9. #34036285C2semidirect
10. #34036233C2semidirect
11. #34036230C2semidirect
12. #34036221C2semidirect
13. SD16×Dic293C2semidirect
14. #34036187C2semidirect
168. C293#3121semidirect

Maximal subgroups

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

subgroupindex
#340367462
#340366072
#340364712
#340364502
#340364222
#340364122
#340363492
#340362852
#340362332
#340362302
#340362212
SD16×Dic2932
#340361872
#448×C2932
#3121293