Finite Group Databasebeta

← #49170687

Subgroups — #49170687

Order 37504.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34036825C211yesmaximal, characteristic, complemented
2#34036483C211yesmaximal, characteristic, complemented
3#34036467C211yesmaximal, characteristic, complemented
4#34036445C211yesmaximal, characteristic, complemented
5#34036436C211yesmaximal, characteristic, complemented
6#34036363C211yesmaximal, characteristic, complemented
7#34036360C211yesmaximal, characteristic, complemented
8#34036341C211yesmaximal, characteristic, complemented
9#34036274C211yesmaximal, characteristic, complemented
10#34036268C211yesmaximal, characteristic, complemented
11C4×#17989789C211yesmaximal, characteristic, complemented
12D4×#14556873C211yesmaximal, characteristic, complemented
13#34036125C211yesmaximal, characteristic, complemented
14C2×#22627548C211yesmaximal, characteristic, complemented
15#548×C293C211yesmaximal, characteristic, complemented
16C2×#17989789C2211yescharacteristic
17C2×#17989789C2211yescharacteristic
18C2×#17989787C2211yescharacteristic
19C22×#14556873C2211yescharacteristic
20C22×#14556873C2211yescharacteristic
21#22627619—21no—
22#22627616C2211yescharacteristic
23#22627606—21no—
24#22627598C2211yescharacteristic
25#22627598C2211yescharacteristic
26#22627596—21no—
27C4×D1172C2211yescharacteristic
28#22627584—21no—
29#22627583C2211yescharacteristic
30C2×C4⋊Dic293C2211yescharacteristic
31#22627557C2211yescharacteristic
32#22627555C2211yescharacteristic
33#22627555C2211yescharacteristic
34C4⋊#14556873C2211yescharacteristic
35#22627548C2211yescharacteristic
36#22627548C2211yescharacteristic
37#22627548C2211yescharacteristic
38#22627548C2211yescharacteristic
39#22627548C2211yescharacteristic
40#22627548C2211yescharacteristic
41#22627545C2211yescharacteristic
42#22627545C2211yescharacteristic
43#22627545C2211yescharacteristic
44#22627544C2211yescharacteristic
45C22⋊#14556873C2211yescharacteristic
46C22⋊#14556873C2211yescharacteristic
47C4×#14556873C2211yescharacteristic
48C2×#41×C293C2211yescharacteristic
49C22×Q8×C293C2211yescharacteristic
50#124×C293—21no—
51#124×C293C2211yescharacteristic
52C22⋊Q8×C293—21no—
53C22⋊Q8×C293C2211yescharacteristic
54#121×C293—21no—
55#121×C293C2211yescharacteristic
56C4×Q8×C293C2211yescharacteristic
57C4×D4×C293—21no—
58C4×D4×C293C2211yescharacteristic
59C2×D1172C2311yescharacteristic
60C2×C22⋊D293—21no—
61C2×C22⋊D293—21no—
62C22×Dic293—21no—
63C2×C4×D293—21no—
64#17989789—22no—
65#17989789—41no—
66#17989787—41no—
67C2×#14556873—41no—
68C2×#14556873—41no—
69C2×#14556873C2311yescharacteristic
70C2×#14556873C2311yescharacteristic
71C2×#14556873C2311yescharacteristic
72C2×#14556873C2311yescharacteristic
73C4⋊Dic293—21no—
74C4⋊Dic293C2311yescharacteristic
75C4⋊Dic293C2311yescharacteristic
76C4⋊Dic293C2311yescharacteristic
77#17989781—21no—
78#17989781—21no—
79C22⋊Dic293—21no—
80#17989779—21no—
81#17989779—21no—
82#17989779—21no—
83C4×Dic293—21no—
84#41×C293—22no—
85#41×C293—41no—
86C2×Q8×C293—21no—
87C2×Q8×C293—22no—
88C2×Q8×C293—41no—
89C2×Q8×C293C2311yescharacteristic
90C2×Q8×C293C2311yescharacteristic
91C2×D4×C293—21no—
92C2×D4×C293—21no—
93C2×D4×C293D411yescharacteristic
94C2×D4×C293C2311yescharacteristic
95C22×C4×C293—21no—
96C22×C4×C293—21no—
97C22×C4×C293C2311yescharacteristic
98C22×C4×C293C2311yescharacteristic
99C4⋊C4×C293—21no—
100C4⋊C4×C293—21no—
101C4⋊C4×C293D411yescharacteristic
102C4⋊C4×C293C2311yescharacteristic
103C22⋊C4×C293—21no—
104C22⋊C4×C293—21no—
105C22⋊C4×C293—21no—
106C22⋊C4×C293—21no—
107C22⋊C4×C293D411yescharacteristic
108C22⋊C4×C293D411yescharacteristic
109C42×C293—21no—
110C42×C293C2311yescharacteristic
111C22×D293—21no—
112D1172—21no—
113D1172—21no—
114C22⋊D293—42no—
115C22⋊D293—81no—
116C2×Dic293—21no—
117C2×Dic293—21no—
118C2×Dic293—21no—
119C2×Dic293—81no—
120C4×D293—81no—
121#14556873—21no—
122#14556873—21no—
123#14556873—21no—
124#14556873—21no—
125C23×C293—21no—
126C23×C293C2×D411yescharacteristic
127C23×C293C2×D411yescharacteristic
128Q8×C293—21no—
129Q8×C293—21no—
130Q8×C293—21no—
131Q8×C293—42no—
132Q8×C293#4112yes—
133D4×C293—22no—
134D4×C293—21no—
135D4×C293—21no—
136D4×C293—42no—
137D4×C293—42no—
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—21no—
145C2×C4×C293—42no—
146C2×C4×C293—42no—
147C2×C4×C293C2×D411yescharacteristic
148C2×C4×C293C2×D411yescharacteristic
149C2×C4×C293C2×D411yescharacteristic
150C2×C4×C293C2×D411yescharacteristic
151C2×C4×C293C2411yescharacteristic
152C22×C293—21no—
153C22×C293—21no—
154C22×C293—21no—
155C22×C293—21no—
156C22×C293—21no—
157C22×C293—41no—
158C22×C293—41no—
159C22×C293—41no—
160C22×C293#13512yes—
161C22×C293C22×D411yescharacteristic
162C2×D293—41no—
163C2×D293—41no—
164C2×D293—41no—
165C4×C293—21no—
166C4×C293—21no—
167C4×C293—22no—
168C4×C293—21no—
169C4×C293—21no—
170C4×C293—41no—
171C4×C293—41no—
172C4×C293C2×#4111yescharacteristic
173C4×C293H(5,𝔽2)11yescharacteristic
174Dic293—41no—
175Dic293—41no—
176Dic293—41no—
177C2×C293—22no—
178C2×C293—41no—
179C2×C293—81no—
180C2×C293#54611yescharacteristic
181C2×C293C2×#13511yescharacteristic
182C2×C293#57711yescharacteristic
183D293—81no—
184C293#330111yescharacteristic, complemented, Sylow
185#3301—2931nomaximal, Sylow
186#553—2931no—
187#548D29311yescharacteristic, complemented
188#488—2931no—
189#485—2931no—
190#481—2931no—
191#479—2931no—
192#466—2931no—
193#465—2931no—
194#463—2931no—
195#449—2931no—
196#448—2931no—
197C4×SD16—2931no—
198C8×D4—2931no—
199#416—2931no—
200C2×#103—2931no—
201C2×#41C2×D29311yescharacteristic
202C22×Q8C2×D29311yescharacteristic
203C2×SD16—2931no—
204C2×SD16—2931no—
205C2×D8—2931no—
206C22×C8—2931no—
207C22×C8—2931no—
208#126—5861no—
209#125—2931no—
210#124—21no—
211#124C2×D29311yescharacteristic
212#123—5861no—
213C22⋊Q8—21no—
214C22⋊Q8C2×D29311yescharacteristic
215#121—21no—
216#121—2931no—
217#121—2931no—
218#121—5861no—
219#121C2×D29311yescharacteristic
220C4×Q8C2×D29311yescharacteristic
221C4×D4—21no—
222C4×D4—2931no—
223C4×D4—5861no—
224C4×D4C2×D29311yescharacteristic
225#117—2931no—
226C2×C4⋊C4—2931no—
227#107—2931no—
228#106—2931no—
229#106—2931no—
230C4⋊C8—2931no—
231#103—2931no—
232#103—2931no—
233#103—2931no—
234#103—2931no—
235#103—2931no—
236#103—2931no—
237#102—2931no—
238#102—2931no—
239#102—2931no—
240#102—2931no—
241C22⋊C8—2931no—
242C22⋊C8—2931no—
243C4×C8—2931no—
244#41—22no—
245#41—41no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C22×D29311yescharacteristic
250C2×Q8C22×D29311yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—2931no—
254C2×D4—5861no—
255C2×D4—5861no—
256C2×D4C22⋊D29311yescharacteristic
257C2×D4C22×D29311yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—5861no—
261C22×C4—5861no—
262C22×C4C22×D29311yescharacteristic
263C22×C4C22×D29311yescharacteristic
264SD16—5862no—
265SD16—11721no—
266D8—11721no—
267C2×C8—2931no—
268C2×C8—2931no—
269C2×C8—2931no—
270C2×C8—2931no—
271C2×C8—11721no—
272C2×C8—11721no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—2931no—
276C4⋊C4—2931no—
277C4⋊C4—2931no—
278C4⋊C4—5861no—
279C4⋊C4—5861no—
280C4⋊C4—5861no—
281C4⋊C4C22⋊D29311yescharacteristic
282C4⋊C4C22×D29311yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—5861no—
288C22⋊C4—5861no—
289C22⋊C4—5861no—
290C22⋊C4—5861no—
291C22⋊C4C22⋊D29311yescharacteristic
292C22⋊C4C22⋊D29311yescharacteristic
293C42—21no—
294C42—5861no—
295C42C22×D29311yescharacteristic
296C23—21no—
297C23—5861no—
298C23C2×C22⋊D29311yescharacteristic
299C23C2×C22⋊D29311yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#1798980112yes—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—42no—
310D4—23441no—
311D4—5861no—
312D4—5861no—
313D4—11722no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—23441no—
324C2×C4—23441no—
325C2×C4—5861no—
326C2×C4—5861no—
327C2×C4—5861no—
328C2×C4C2×C22⋊D29311yescharacteristic
329C2×C4C2×C22⋊D29311yescharacteristic
330C2×C4C2×C22⋊D29311yescharacteristic
331C2×C4C2×C22⋊D29311yescharacteristic
332C2×C4C23×D29311yescharacteristic
333C8—5861no—
334C8—5861no—
335C8—5861no—
336C8—5861no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—11721no—
346C22—11721no—
347C22—11721no—
348C22#2262764412yes—
349C22C22×C22⋊D29311yescharacteristic
350C4—21no—
351C4—21no—
352C4—22no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—11721no—
358C4—11721no—
359C4—11721no—
360C4C2×#1798980111yescharacteristic
361C4#2262766911yescharacteristic
362C2—22no—
363C2—41no—
364C2—81no—
365C2—23441no—
366C2#3403677311yescharacteristic
367C2C2×#2262764411yescharacteristic
368C2#3403693711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#548×C293C2
socleC22×C293C22×D4
hypercenter#548D293
hypocenterC293#3301
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #34036825C2semidirect
2. #34036483C2semidirect
3. #34036467C2semidirect
4. #34036445C2semidirect
5. #34036436C2semidirect
6. #34036363C2semidirect
7. #34036360C2semidirect
8. #34036341C2semidirect
9. #34036274C2semidirect
10. #34036268C2semidirect
11. C4×#17989789C2semidirect
12. D4×#14556873C2semidirect
13. #34036125C2semidirect
14. C2×#22627548C2semidirect
15. #548×C293C2semidirect
184. C293#3301semidirect
187. #548D293semidirect

Maximal subgroups

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

subgroupindex
#340368252
#340364832
#340364672
#340364452
#340364362
#340363632
#340363602
#340363412
#340362742
#340362682
C4×#179897892
D4×#145568732
#340361252
C2×#226275482
#548×C2932
#3301293