Finite Group Databasebeta

← #49170648

Subgroups — #49170648

Order 37504.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34036792C211yesmaximal, characteristic, complemented
2#34036780C211yesmaximal, characteristic
3#34036478C211yesmaximal, characteristic, complemented
4#34036461C211yesmaximal, characteristic, complemented
5#34036413C211yesmaximal, characteristic, complemented
6#34036360C211yesmaximal, characteristic, complemented
7#34036359C211yesmaximal, characteristic, complemented
8#34036333C211yesmaximal, characteristic, complemented
9#34036270C211yesmaximal, characteristic, complemented
10#34036269C211yesmaximal, characteristic, complemented
11#34036216C211yesmaximal, characteristic, complemented
12#34036192C211yesmaximal, characteristic, complemented
13C2×#22627547C211yesmaximal, characteristic, complemented
14#103×D293C211yesmaximal, characteristic, complemented
15#483×C293C211yesmaximal, characteristic, complemented
16C2×#17989802—21no—
17C22×Dic586C2211yescharacteristic
18C2×Q8×D293C2211yescharacteristic
19C2×#17989791C2211yescharacteristic
20C2×#17989791C2211yescharacteristic
21C2×#17989789C2211yescharacteristic
22C22×#14556873C2211yescharacteristic
23C2×C8×D293C2211yescharacteristic
24#22627623—21no—
25#22627623C2211yescharacteristic
26#22627622C2211yescharacteristic
27#22627609—21no—
28#22627607—21no—
29#22627604C2211yescharacteristic
30#22627602—21no—
31#22627602—21no—
32#22627601—21no—
33#22627599C2211yescharacteristic
34#22627598C2211yescharacteristic
35#22627589C2211yescharacteristic
36#22627586C2211yescharacteristic
37C4×C22⋊D293C2211yescharacteristic
38#22627584—21no—
39C4⋊C4×D293C2211yescharacteristic
40#22627555C2211yescharacteristic
41#22627553C2211yescharacteristic
42#22627549C2211yescharacteristic
43#22627549C2211yescharacteristic
44#22627548C2211yescharacteristic
45#22627547C2211yescharacteristic
46#22627547C2211yescharacteristic
47#22627547C2211yescharacteristic
48#22627547C2211yescharacteristic
49#22627544C2211yescharacteristic
50#22627536C2211yescharacteristic
51C8×Dic293C2211yescharacteristic
52C22⋊Q8×C293C2211yescharacteristic
53C2×C4⋊C4×C293C2211yescharacteristic
54#106×C293C2211yescharacteristic
55#106×C293C2211yescharacteristic
56#103×C293C2211yescharacteristic
57#103×C293C2211yescharacteristic
58C22⋊C8×C293C2211yescharacteristic
59#17989802—44no—
60C2×Dic586—21no—
61C2×Dic586—21no—
62C2×Dic586—22no—
63C2×Dic586—21no—
64C2×Dic586—41no—
65C2×Dic586C2311yescharacteristic
66C2×Dic586C2311yescharacteristic
67Q8×D293—22no—
68Q8×D293—41no—
69C2×D1172—21no—
70C2×D1172C2311yescharacteristic
71C2×C22⋊D293—21no—
72C2×C22⋊D293D411yescharacteristic
73C22×Dic293—21no—
74C2×C4×D293—21no—
75C2×C4×D293—21no—
76C2×C4×D293—21no—
77C2×C4×D293C2311yescharacteristic
78#17989791—22no—
79#17989791—41no—
80#17989789—41no—
81C2×#14556873—41no—
82C2×#14556873C2311yescharacteristic
83C2×#14556873C2311yescharacteristic
84C8×D293—41no—
85C4⋊Dic293—21no—
86C4⋊Dic293C2311yescharacteristic
87#17989781—21no—
88#17989781—21no—
89#17989781—21no—
90#17989781—21no—
91#17989781—21no—
92#17989781D411yescharacteristic
93C22⋊Dic293D411yescharacteristic
94#17989779—21no—
95#17989779—21no—
96#17989779—21no—
97#17989779—21no—
98#17989779D411yescharacteristic
99C4×Dic293—21no—
100C4×Dic293C2311yescharacteristic
101C2×Q8×C293C2311yescharacteristic
102C22×C4×C293—21no—
103C22×C4×C293C2311yescharacteristic
104C2×C8×C293C2311yescharacteristic
105C2×C8×C293C2311yescharacteristic
106C4⋊C4×C293—21no—
107C4⋊C4×C293—21no—
108C4⋊C4×C293C2311yescharacteristic
109C4⋊C4×C293C2311yescharacteristic
110C4⋊C4×C293C2311yescharacteristic
111C22⋊C4×C293—21no—
112C22×D293—21no—
113C22×D293C2×D411yescharacteristic
114Dic586—21no—
115Dic586—21no—
116Dic586—21no—
117Dic586—42no—
118Dic586—42no—
119Dic586—42no—
120Dic586#4112yescomplemented
121D1172—21no—
122D1172—21no—
123D1172—42no—
124C22⋊D293—42no—
125C22⋊D293SD1614yescomplemented
126C2×Dic293—21no—
127C2×Dic293—21no—
128C2×Dic293—21no—
129C2×Dic293—42no—
130C2×Dic293C2×D411yescharacteristic
131C2×Dic293C2×D411yescharacteristic
132C2×Dic293C2×D411yescharacteristic
133C4×D293—21no—
134C4×D293—21no—
135C4×D293—42no—
136C4×D293—42no—
137C4×D293—42no—
138#14556873—21no—
139#14556873—21no—
140C23×C293C2×D411yescharacteristic
141Q8×C293—21no—
142Q8×C293—21no—
143C2×C4×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—42no—
150C2×C4×C293C2×D411yescharacteristic
151C2×C4×C293C2411yescharacteristic
152C8×C293—21no—
153C8×C293—21no—
154C22×C293—21no—
155C22×C293—21no—
156C22×C293C2×SD1612yes—
157C22×C293C22×D411yescharacteristic
158C2×D293—21no—
159C2×D293—21no—
160C2×D293—41no—
161C2×D293—41no—
162C2×D293—41no—
163C2×D293C2×SD1612yes—
164C4×C293—21no—
165C4×C293—41no—
166C4×C293—41no—
167C4×C293—41no—
168C4×C293—41no—
169C4×C293C2×#4111yescharacteristic
170C4×C293H(5,𝔽2)11yescharacteristic
171Dic293—21no—
172Dic293—22no—
173Dic293—21no—
174Dic293—41no—
175Dic293—41no—
176Dic293C2×SD1612yes—
177C2×C293—22no—
178C2×C293#54611yescharacteristic
179C2×C293C22×SD1611yescharacteristic
180C2×C293#57811yescharacteristic
181D293—22no—
182D293—81no—
183C293#329811yescharacteristic, complemented, Sylow
184#3298—2931nomaximal, Sylow
185D4×Q8—2931no—
186#547—2931no—
187#487—2931no—
188#483—2931no—
189#483D29311yescharacteristic, complemented
190#475—2931no—
191#465—2931no—
192#465—2931no—
193#461—2931no—
194#448—2931no—
195#448—2931no—
196C4×SD16—2931no—
197C8×D4—2931no—
198C2×#103—2931no—
199C2×#103—2931no—
200C2×#41—5861no—
201C22×Q8—2931no—
202C22×Q8—2931no—
203C2×SD16—2931no—
204C2×SD16—2931no—
205C2×SD16—2931no—
206C22×C8—2931no—
207C22×C8—2931no—
208C4⋊Q8—2931no—
209C4⋊Q8—2931no—
210C4⋊Q8—5861no—
211#123—5861no—
212#123—5861no—
213C22⋊Q8—2931no—
214C22⋊Q8—5861no—
215C22⋊Q8—5861no—
216C22⋊Q8—5861no—
217C22⋊Q8C2×D29311yescharacteristic
218#121—2931no—
219#121—2931no—
220C4×Q8—2931no—
221C4×D4—2931no—
222C4×D4—2931no—
223C4×D4—5861no—
224C2×C4⋊C4—2931no—
225C2×C4⋊C4C2×D29311yescharacteristic
226#106—2931no—
227#106C2×D29311yescharacteristic
228#106C2×D29311yescharacteristic
229C4⋊C8—2931no—
230#103—2931no—
231#103—2931no—
232#103—2931no—
233#103—2931no—
234#103—2931no—
235#103—2931no—
236#103—2931no—
237#103C2×D29311yescharacteristic
238#103C2×D29311yescharacteristic
239#102—2931no—
240C22⋊C8—2931no—
241C22⋊C8C2×D29311yescharacteristic
242C4×C8—2931no—
243#41—11724no—
244C2×Q8—2931no—
245C2×Q8—2931no—
246C2×Q8—5861no—
247C2×Q8—5861no—
248C2×Q8—5862no—
249C2×Q8—5861no—
250C2×Q8—5862no—
251C2×Q8—11721no—
252C2×Q8—11721no—
253C2×Q8C22×D29311yescharacteristic
254C2×D4—2931no—
255C2×D4—2931no—
256C2×D4—5861no—
257C2×D4—5861no—
258C22×C4—21no—
259C22×C4—2931no—
260C22×C4—5861no—
261C22×C4—5861no—
262C22×C4—5861no—
263C22×C4—5861no—
264C22×C4C22×D29311yescharacteristic
265SD16—5862no—
266SD16—11721no—
267SD16—11721no—
268C2×C8—2931no—
269C2×C8—2931no—
270C2×C8—11721no—
271C2×C8—11721no—
272C2×C8C22×D29311yescharacteristic
273C2×C8C22×D29311yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—2931no—
277C4⋊C4—2931no—
278C4⋊C4—5861no—
279C4⋊C4—5861no—
280C4⋊C4—5861no—
281C4⋊C4—5861no—
282C4⋊C4—5861no—
283C4⋊C4—5861no—
284C4⋊C4C22×D29311yescharacteristic
285C4⋊C4C22×D29311yescharacteristic
286C4⋊C4C22×D29311yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—2931no—
289C22⋊C4—2931no—
290C22⋊C4—5861no—
291C22⋊C4—5861no—
292C22⋊C4—5861no—
293C22⋊C4—5861no—
294C42—2931no—
295C42—5861no—
296C23—2931no—
297C23—5861no—
298C23D4×D29311yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—2932no—
302Q8—5861no—
303Q8—5861no—
304Q8—5861no—
305Q8—11722no—
306Q8—11722no—
307Q8—11722no—
308D4—2934no—
309D4—5861no—
310D4—5861no—
311D4—11722no—
312D4—11722no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—2931no—
321C2×C4—2931no—
322C2×C4—2931no—
323C2×C4—5861no—
324C2×C4—5861no—
325C2×C4—5861no—
326C2×C4—5861no—
327C2×C4—5861no—
328C2×C4—11722no—
329C2×C4—11722no—
330C2×C4—11722no—
331C2×C4—11722no—
332C2×C4D4×D29311yescharacteristic
333C2×C4C23×D29311yescharacteristic
334C8—21no—
335C8—21no—
336C8—5861no—
337C8—5861no—
338C22—21no—
339C22—21no—
340C22—2932no—
341C22—5861no—
342C22—5861no—
343C22—11721no—
344C22—11721no—
345C22—11721no—
346C22SD16×D29312yes—
347C22C2×D4×D29311yescharacteristic
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—2932no—
354C4—5861no—
355C4—5862no—
356C4—5861no—
357C4—11721no—
358C4—11721no—
359C4#41×D29311yescharacteristic
360C4#2262767011yescharacteristic
361C2—22no—
362C2—23441no—
363C2—5862no—
364C2#3403677111yescharacteristic
365C2C2×SD16×D29311yescharacteristic
366C2#3403694511yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#483×C293C2
socleC22×C293C22×D4
prime radical#34036780C2
hypercenter#483D293
hypocenterC293#3298
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #34036792C2semidirect
3. #34036478C2semidirect
4. #34036461C2semidirect
5. #34036413C2semidirect
6. #34036360C2semidirect
7. #34036359C2semidirect
8. #34036333C2semidirect
9. #34036270C2semidirect
10. #34036269C2semidirect
11. #34036216C2semidirect
12. #34036192C2semidirect
13. C2×#22627547C2semidirect
14. #103×D293C2semidirect
15. #483×C293C2semidirect
120. Dic586#41semidirect
125. C22⋊D293SD16semidirect
183. C293#3298semidirect
189. #483D293semidirect

Maximal subgroups

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

subgroupindex
#340367922
#340367802
#340364782
#340364612
#340364132
#340363602
#340363592
#340363332
#340362702
#340362692
#340362162
#340361922
C2×#226275472
#103×D2932
#483×C2932
#3298293