Finite Group Databasebeta

← #49170300

Subgroups — #49170300

Order 37504.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627657C211yesmaximal, characteristic, complemented
2C22×#17989791C211yesmaximal, characteristic, complemented
3D4×Dic586C211yesmaximal, characteristic, complemented
4#34036775C211yesmaximal, characteristic
5#34036523C211yesmaximal, characteristic, complemented
6#34036375C211yesmaximal, characteristic, complemented
7#34036370C211yesmaximal, characteristic, complemented
8#34036352C211yesmaximal, characteristic, complemented
9#34036335C211yesmaximal, characteristic, complemented
10#34036293C211yesmaximal, characteristic, complemented
11#34036287C211yesmaximal, characteristic, complemented
12#34036282C211yesmaximal, characteristic, complemented
13#34036270C211yesmaximal, characteristic, complemented
14#34036231C211yesmaximal, characteristic, complemented
15#435×C293C211yesmaximal, characteristic, complemented
16C2×#17989802C2211yescharacteristic
17C22×Dic586C2211yescharacteristic
18C22×Dic586C2211yescharacteristic, complemented
19C22×D1172C2211yescharacteristic
20#22627657—24no—
21C2×Dic1172C2211yescharacteristic, complemented
22C2×Dic1172C2211yescharacteristic, complemented
23C2×Dic1172C2211yescharacteristic, complemented
24C2×#17989791—24no—
25C2×#17989791C2211yescharacteristic
26C2×#17989791C2211yescharacteristic
27C2×#17989791C2211yescharacteristic
28C2×#17989791C2211yescharacteristic
29C2×#17989791C2211yescharacteristic
30C2×#17989791C2211yescharacteristic
31#22627624C2211yescharacteristic, complemented
32#22627623—21no—
33#22627612C2211yescharacteristic
34#22627609—21no—
35#22627603C2211yescharacteristic
36#22627603C2211yescharacteristic
37#22627603C2211yescharacteristic, complemented
38#22627601—21no—
39#22627601—21no—
40#22627598C2211yescharacteristic
41#22627597—21no—
42#22627592—21no—
43C4×Dic586C2211yescharacteristic, complemented
44C4×D1172C2211yescharacteristic
45D4×Dic293—21no—
46C22⋊C4×D293—21no—
47#22627558C2211yescharacteristic, complemented
48#22627549C2211yescharacteristic, complemented
49#22627549C2211yescharacteristic, complemented
50#22627549C2211yescharacteristic, complemented
51#22627549C2211yescharacteristic, complemented
52#22627546C2211yescharacteristic
53#22627546C2211yescharacteristic
54C2×M16×C293C2211yescharacteristic, complemented
55C22×C8×C293C2211yescharacteristic
56C4×D4×C293C2211yescharacteristic
57C4⋊C8×C293C2211yescharacteristic, complemented
58C22⋊C8×C293C2211yescharacteristic
59C22⋊C8×C293C2211yescharacteristic, complemented
60#97×C293C2211yescharacteristic, complemented
61C23×D293—21no—
62#17989802—22no—
63#17989802—41no—
64C2×Dic586—22no—
65C2×Dic586—22no—
66C2×Dic586—21no—
67C2×Dic586—21no—
68C2×Dic586—41no—
69C2×Dic586—41no—
70C2×Dic586C2311yescharacteristic
71C2×Dic586C2311yescharacteristic
72C2×Dic586C2311yescharacteristic
73C2×Dic586C2311yescharacteristic
74C2×D1172—22no—
75C2×D1172—21no—
76C2×D1172—41no—
77C2×D1172C2311yescharacteristic
78C2×D1172C2311yescharacteristic
79C2×C22⋊D293—21no—
80C2×C22⋊D293—21no—
81C22×Dic293—21no—
82C22×Dic293—21no—
83C2×C4×D293—21no—
84C2×C4×D293—21no—
85Dic1172—22no—
86Dic1172—22no—
87Dic1172—41no—
88#17989791—22no—
89#17989791—22no—
90#17989791—22no—
91#17989791—22no—
92#17989791—41no—
93#17989791D414yescomplemented
94C4⋊Dic293—21no—
95C4⋊Dic293—21no—
96C4⋊Dic293C2311yescharacteristic
97C4⋊Dic293C2311yescharacteristic
98#17989781—21no—
99#17989781—21no—
100#17989781—21no—
101#17989781—21no—
102C22⋊Dic293—21no—
103C22⋊Dic293—21no—
104C22⋊Dic293—21no—
105#17989779—21no—
106#17989779—21no—
107#17989779—21no—
108#17989779—21no—
109C4×Dic293—21no—
110C2×D4×C293D411yescharacteristic
111C22×C4×C293C2311yescharacteristic
112C22×C4×C293C2311yescharacteristic
113M16×C293—22no—
114C2×C8×C293—22no—
115C2×C8×C293C2311yescharacteristic
116C2×C8×C293C2311yescharacteristic
117C2×C8×C293C2311yescharacteristic
118C2×C8×C293C2311yescharacteristic
119C4⋊C4×C293D411yescharacteristic
120C22⋊C4×C293D411yescharacteristic
121C22⋊C4×C293D411yescharacteristic
122C42×C293C2311yescharacteristic
123C22×D293—21no—
124C22×D293—21no—
125C22×D293—42no—
126C22×D293—42no—
127C22×D293—81no—
128Dic586—21no—
129Dic586—21no—
130Dic586—21no—
131Dic586—21no—
132Dic586—21no—
133Dic586—21no—
134Dic586—21no—
135Dic586—42no—
136Dic586—42no—
137Dic586C2×D412yes—
138D1172—21no—
139D1172—21no—
140D1172—21no—
141D1172—42no—
142D1172C2×D412yes—
143C22⋊D293—42no—
144C22⋊D293—81no—
145C2×Dic293—21no—
146C2×Dic293—21no—
147C2×Dic293—21no—
148C2×Dic293—21no—
149C2×Dic293—21no—
150C2×Dic293—21no—
151C2×Dic293—42no—
152C2×Dic293—42no—
153C4×D293—42no—
154C4×D293—81no—
155C23×C293C2×D411yescharacteristic
156C23×C293C2×D411yescharacteristic
157D4×C293—22no—
158C2×C4×C293—21no—
159C2×C4×C293—21no—
160C2×C4×C293—21no—
161C2×C4×C293—21no—
162C2×C4×C293C2×D411yescharacteristic
163C2×C4×C293C2×D411yescharacteristic
164C2×C4×C293C2×D411yescharacteristic
165C2×C4×C293C2×D411yescharacteristic
166C2×C4×C293C2411yescharacteristic
167C8×C293—21no—
168C8×C293—21no—
169C8×C293—21no—
170C8×C293C2×D412yes—
171C22×C293—21no—
172C22×C293—21no—
173C22×C293—21no—
174C22×C293—21no—
175C22×C293—21no—
176C22×C293#13712yescomplemented
177C22×C293C22×D411yescharacteristic
178C2×D293—22no—
179C2×D293—41no—
180C2×D293—41no—
181C2×D293—41no—
182C2×D293—41no—
183C2×D293—41no—
184C2×D293—84no—
185C4×C293—21no—
186C4×C293—21no—
187C4×C293—21no—
188C4×C293—21no—
189C4×C293C22×D411yescharacteristic
190C4×C293H(5,𝔽2)11yescharacteristic
191Dic293—22no—
192Dic293—41no—
193Dic293—41no—
194Dic293—41no—
195Dic293—41no—
196Dic293—41no—
197C2×C293—22no—
198C2×C293—41no—
199C2×C293D4211yescharacteristic
200C2×C293C2×#13711yescharacteristic
201C2×C293#57711yescharacteristic
202D293—42no—
203D293—81no—
204C293#326611yescharacteristic, complemented, Sylow
205#3266—2931nomaximal, Sylow
206C2×#137—2931no—
207C22×SD16—2931no—
208D4×Q8—2931no—
209#546—2931no—
210#497—2931no—
211#468—2931no—
212#467—2931no—
213#464—2931no—
214#461—2931no—
215#452—2931no—
216#451—2931no—
217#450—2931no—
218#448—2931no—
219#440—2931no—
220#435D29311yescharacteristic, complemented
221C2×#41—2931no—
222C22×Q8—2931no—
223C22×Q8—2931no—
224C22×D4—2931no—
225#137—5864no—
226C2×Q16—2931no—
227C2×Q16—2931no—
228C2×Q16—2931no—
229C2×SD16—2931no—
230C2×SD16—2931no—
231C2×SD16—2931no—
232C2×SD16—2931no—
233C2×SD16—2931no—
234C2×SD16—2931no—
235C2×SD16—5864no—
236C2×M16C2×D29311yescharacteristic, complemented
237C22×C8C2×D29311yescharacteristic
238C4⋊Q8—2931no—
239C4⋊Q8—5861no—
240#124—2931no—
241#123—5861no—
242C22⋊Q8—2931no—
243C22⋊Q8—2931no—
244C22⋊Q8—2931no—
245C22⋊Q8—5861no—
246C22⋊Q8—5861no—
247#121—2931no—
248#121—5861no—
249#120—5861no—
250C4×Q8—2931no—
251C4×D4—2931no—
252C4×D4—5861no—
253C4×D4C2×D29311yescharacteristic
254C2×C22⋊C4—5861no—
255#107—2931no—
256C4⋊C8C2×D29311yescharacteristic, complemented
257#103—2931no—
258#103—2931no—
259#103—2931no—
260#103—2931no—
261#102—2931no—
262#102—2931no—
263C22⋊C8C2×D29311yescharacteristic
264C22⋊C8C2×D29311yescharacteristic, complemented
265#97C2×D29311yescharacteristic, complemented
266C24—5861no—
267#41—5862no—
268#41—11721no—
269C2×Q8—2931no—
270C2×Q8—2931no—
271C2×Q8—2931no—
272C2×Q8—2931no—
273C2×Q8—5862no—
274C2×Q8—5862no—
275C2×Q8—5861no—
276C2×Q8—5861no—
277C2×Q8—11721no—
278C2×Q8—11721no—
279C2×D4—2931no—
280C2×D4—2931no—
281C2×D4—5862no—
282C2×D4—5861no—
283C2×D4—5861no—
284C2×D4—5861no—
285C2×D4—11721no—
286C2×D4D117211yescharacteristic
287C22×C4—5861no—
288C22×C4—5861no—
289C22×C4—5861no—
290C22×C4—5861no—
291C22×C4C22×D29311yescharacteristic
292C22×C4C22×D29311yescharacteristic
293Q16—5862no—
294Q16—5862no—
295Q16—11721no—
296SD16—2934no—
297SD16—5862no—
298SD16—5862no—
299SD16—5862no—
300SD16—5862no—
301SD16—11721no—
302M16—22no—
303C2×C8—22no—
304C2×C8C22×D29311yescharacteristic
305C2×C8C22×D29311yescharacteristic
306C2×C8C22×D29311yescharacteristic
307C2×C8C22×D29311yescharacteristic
308C4⋊C4—2931no—
309C4⋊C4—2931no—
310C4⋊C4—5861no—
311C4⋊C4—5861no—
312C4⋊C4—5861no—
313C4⋊C4—5861no—
314C4⋊C4—5861no—
315C4⋊C4—5861no—
316C4⋊C4D117211yescharacteristic
317C22⋊C4—5861no—
318C22⋊C4—5861no—
319C22⋊C4—5861no—
320C22⋊C4—5861no—
321C22⋊C4—5861no—
322C22⋊C4—5861no—
323C22⋊C4—5861no—
324C22⋊C4D117211yescharacteristic
325C22⋊C4D117211yescharacteristic
326C42—5861no—
327C42C22×D29311yescharacteristic
328C23—23441no—
329C23—5861no—
330C23—5861no—
331C23—11722no—
332C23—11722no—
333C23C2×D117211yescharacteristic
334C23C2×D117211yescharacteristic
335Q8—2932no—
336Q8—5861no—
337Q8—5861no—
338Q8—5861no—
339Q8—5861no—
340Q8—5861no—
341Q8—5861no—
342Q8—5861no—
343Q8—11722no—
344Q8—11722no—
345D4—22no—
346D4—2932no—
347D4—23441no—
348D4—5861no—
349D4—5861no—
350D4—5861no—
351D4—11722no—
352D4—11722no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—23441no—
358C2×C4—5861no—
359C2×C4—5861no—
360C2×C4—5861no—
361C2×C4—5861no—
362C2×C4—5861no—
363C2×C4—5861no—
364C2×C4—11722no—
365C2×C4—11722no—
366C2×C4—11722no—
367C2×C4C2×D117211yescharacteristic
368C2×C4C2×D117211yescharacteristic
369C2×C4C2×D117211yescharacteristic
370C2×C4C2×D117211yescharacteristic
371C2×C4C23×D29311yescharacteristic
372C8—21no—
373C8—21no—
374C8—21no—
375C8D4×D29312yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—23444no—
382C22—5862no—
383C22—11721no—
384C22—11721no—
385C22—11721no—
386C22—11721no—
387C22—11721no—
388C22#2262765712yescomplemented
389C22C22×D117211yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—5862no—
395C4—11721no—
396C4—11721no—
397C4—11721no—
398C4—11721no—
399C4—11721no—
400C4C2×D4×D29311yescharacteristic
401C4#2262767011yescharacteristic
402C2—22no—
403C2—41no—
404C2—23441no—
405C2—11722no—
406C2D4×D117211yescharacteristic
407C2C2×#2262765711yescharacteristic
408C2#3403694111yescharacteristic
subgroupG/H
centerC22C22×D1172
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#435×C293C2
socleC22×C293C22×D4
prime radical#34036775C2
hypercenter#435D293
hypocenterC293#3266
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#22627657C2semidirect
2. C22×#17989791C2semidirect
3. D4×Dic586C2semidirect
5. #34036523C2semidirect
6. #34036375C2semidirect
7. #34036370C2semidirect
8. #34036352C2semidirect
9. #34036335C2semidirect
10. #34036293C2semidirect
11. #34036287C2semidirect
12. #34036282C2semidirect
13. #34036270C2semidirect
14. #34036231C2semidirect
15. #435×C293C2semidirect
18. C22×Dic586C22semidirect
21. C2×Dic1172C22semidirect
22. C2×Dic1172C22semidirect
23. C2×Dic1172C22semidirect
31. #22627624C22semidirect
37. #22627603C22semidirect
43. C4×Dic586C22semidirect
47. #22627558C22semidirect
48. #22627549C22semidirect
49. #22627549C22semidirect
50. #22627549C22semidirect
51. #22627549C22semidirect
54. C2×M16×C293C22semidirect
57. C4⋊C8×C293C22semidirect
59. C22⋊C8×C293C22semidirect
60. #97×C293C22semidirect
93. #17989791D4semidirect
176. C22×C293#137semidirect
204. C293#3266semidirect
220. #435D293semidirect
236. C2×M16C2×D293semidirect
256. C4⋊C8C2×D293semidirect
264. C22⋊C8C2×D293semidirect
265. #97C2×D293semidirect
388. C22#22627657semidirect

Maximal subgroups

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

subgroupindex
C2×#226276572
C22×#179897912
D4×Dic5862
#340367752
#340365232
#340363752
#340363702
#340363522
#340363352
#340362932
#340362872
#340362822
#340362702
#340362312
#435×C2932
#3266293