Finite Group Databasebeta

← #17997339

Subgroups — #17997339

Order 4736.

460 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11924534C211yesmaximal, characteristic
2#14557440C211yesmaximal, characteristic
3#14557437C211yesmaximal, characteristic
4#14557437C211yesmaximal, characteristic
5#14557430C211yesmaximal, characteristic
6#14557428C211yesmaximal, characteristic
7#14557420C211yesmaximal, characteristic
8#14557417C211yesmaximal, characteristic
9#14557415C211yesmaximal, characteristic
10#14557414C211yesmaximal, characteristic
11#14557397C211yesmaximal, characteristic
12C4×#10596936C211yesmaximal, characteristic
13#14557384C211yesmaximal, characteristic
14#14557378C211yesmaximal, characteristic
15C37×#391C211yesmaximal, characteristic
16C22×Dic74C2211yescharacteristic
17#11924534C418yescomplemented
18C2×C4⋊Dic37C2211yescharacteristic
19C2×#10596936C2211yescharacteristic
20C2×#10596936C2211yescharacteristic
21C2×#10596936C2211yescharacteristic
22C2×#10596936C2211yescharacteristic
23C2×#10596936C2211yescharacteristic
24C2×#10596936C2211yescharacteristic
25C2×#10596936C2211yescharacteristic
26C2×#10596936C2211yescharacteristic
27C2×C4×Dic37C2211yescharacteristic
28C2×C4×Dic37C2211yescharacteristic
29C2×C4×Dic37C2211yescharacteristic
30C2×C4×Dic37C2211yescharacteristic
31C2×C4×Dic37C2211yescharacteristic
32C2×C4×Dic37C2211yescharacteristic
33#11924443C2211yescharacteristic
34#11924443C2211yescharacteristic
35#11924443C2211yescharacteristic
36#11924443C2211yescharacteristic
37#11924443C2211yescharacteristic
38#11924443C2211yescharacteristic
39#11924443C2211yescharacteristic
40#11924443C2211yescharacteristic
41#11924443C2211yescharacteristic
42#11924443C2211yescharacteristic
43#11924443C2211yescharacteristic
44#11924443C2211yescharacteristic
45C22×Q8×C37C2211yescharacteristic
46C2×C4⋊C4×C37C2211yescharacteristic
47C2×C42×C37C2211yescharacteristic
48#95×C37C2211yescharacteristic
49#95×C37C2211yescharacteristic
50#95×C37C2211yescharacteristic
51#95×C37C2211yescharacteristic
52C2×Q8×C37—41no—
53C2×Q8×C37—41no—
54C2×Q8×C37C2×C414yes—
55C22×C4×C37C2311yescharacteristic
56C22×C4×C37C2311yescharacteristic
57C22×C4×C37C2311yescharacteristic
58C22×C4×C37C2311yescharacteristic
59C22×C4×C37C2311yescharacteristic
60C22×C4×C37C2311yescharacteristic
61C22×C4×C37C2311yescharacteristic
62C22×Dic37C2311yescharacteristic
63C22×Dic37C2311yescharacteristic
64C22×Dic37C2311yescharacteristic
65C22×Dic37C2311yescharacteristic
66C22×Dic37C2311yescharacteristic
67C22×Dic37C2311yescharacteristic
68C22×Dic37C2311yescharacteristic
69C22×Dic37C2311yescharacteristic
70C2×Dic74—41no—
71C2×Dic74—41no—
72C2×Dic74C2×C414yes—
73C4⋊C4×C37—41no—
74C42×C37—41no—
75C4⋊Dic37—41no—
76#10596936—22no—
77#10596936—22no—
78#10596936—41no—
79#10596936—41no—
80#10596936C2×C414yes—
81#10596936C2×C414yes—
82#10596936C2×C414yes—
83#10596936C2×C414yes—
84C4×Dic37—22no—
85C4×Dic37—41no—
86C4×Dic37—41no—
87C4×Dic37—41no—
88C4×Dic37C2×C414yes—
89C4×Dic37Q814yescomplemented
90C23×C37C2411yescharacteristic
91Q8×C37—44no—
92C2×C4×C37—21no—
93C2×C4×C37—21no—
94C2×C4×C37—21no—
95C2×C4×C37—21no—
96C2×C4×C37—21no—
97C2×C4×C37—21no—
98C2×C4×C37—21no—
99C2×C4×C37—21no—
100C2×C4×C37—21no—
101C2×C4×C37—21no—
102C2×C4×C37—21no—
103C2×C4×C37—21no—
104C2×C4×C37—21no—
105C2×C4×C37—21no—
106C2×C4×C37—21no—
107C2×C4×C37—21no—
108C2×C4×C37—21no—
109C2×C4×C37C22×C412yes—
110C2×C4×C37C22×C412yes—
111C2×C4×C37C22×C412yes—
112C2×C4×C37C2×Q812yes—
113C2×Dic37—21no—
114C2×Dic37—21no—
115C2×Dic37—21no—
116C2×Dic37—21no—
117C2×Dic37—21no—
118C2×Dic37—21no—
119C2×Dic37—21no—
120C2×Dic37—21no—
121C2×Dic37—21no—
122C2×Dic37—21no—
123C2×Dic37—21no—
124C2×Dic37—21no—
125C2×Dic37—21no—
126C2×Dic37—21no—
127C2×Dic37—21no—
128C2×Dic37—21no—
129C2×Dic37—21no—
130C2×Dic37C22×C412yes—
131C2×Dic37C22×C412yes—
132C2×Dic37C22×C412yes—
133C2×Dic37C22×C412yes—
134C2×Dic37C2×Q812yes—
135C2×Dic37C2×Q812yes—
136C2×Dic37#4112yes—
137Dic74—44no—
138C22×C37C23×C411yescharacteristic
139C22×C37C22×Q811yescharacteristic
140C22×C37C2×#4111yescharacteristic
141C22×C37H(5,𝔽2)11yescharacteristic
142C22×C37H(5,𝔽2)11yescharacteristic
143C22×C372−1+411yescharacteristic
144C22×C372−1+411yescharacteristic
145C4×C37—22no—
146C4×C37—22no—
147C4×C37—22no—
148C4×C37—22no—
149C4×C37—41no—
150C4×C37—41no—
151C4×C37—41no—
152Dic37—22no—
153Dic37—22no—
154Dic37—22no—
155Dic37—22no—
156Dic37—41no—
157Dic37—41no—
158Dic37—41no—
159Dic37C4×Q814yes—
160#2331—371nomaximal, Sylow
161C2×C37C2×C4×Q811yescharacteristic
162C2×C37#51811yescharacteristic
163C2×C37#51911yescharacteristic
164C2×C37#54211yescharacteristic
165C2×C37#54211yescharacteristic
166C2×C37#54411yescharacteristic
167C2×C37#54411yescharacteristic
168C2×C4⋊Q8—371no—
169#391—371no—
170#391—371no—
171#391—371no—
172#391D3711yescharacteristic
173#389—371no—
174#389—371no—
175#387—371no—
176#387—371no—
177#387—371no—
178#387—371no—
179#382—371no—
180C4×C4⋊C4—371no—
181C4×C4⋊C4—371no—
182#376—371no—
183C37#233111yescharacteristic, complemented, Sylow
184C22×Q8—371no—
185C22×Q8C2×D3711yescharacteristic
186C4⋊Q8—378no—
187C2×C4⋊C4—371no—
188C2×C4⋊C4—371no—
189C2×C4⋊C4—371no—
190C2×C4⋊C4—371no—
191C2×C4⋊C4—371no—
192C2×C4⋊C4—371no—
193C2×C4⋊C4—371no—
194C2×C4⋊C4—371no—
195C2×C4⋊C4—371no—
196C2×C4⋊C4C2×D3711yescharacteristic
197C2×C42—371no—
198C2×C42—371no—
199C2×C42—371no—
200C2×C42—371no—
201C2×C42—371no—
202C2×C42—371no—
203C2×C42C2×D3711yescharacteristic
204#95—371no—
205#95—371no—
206#95—371no—
207#95—371no—
208#95—371no—
209#95—371no—
210#95—371no—
211#95—371no—
212#95—371no—
213#95—371no—
214#95—371no—
215#95—371no—
216#95C2×D3711yescharacteristic
217#95C2×D3711yescharacteristic
218#95C2×D3711yescharacteristic
219#95C2×D3711yescharacteristic
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—374no—
223C2×Q8—1481no—
224C2×Q8—1481no—
225C2×Q8C4×D3714yes—
226C22×C4—371no—
227C22×C4—371no—
228C22×C4—371no—
229C22×C4—371no—
230C22×C4—371no—
231C22×C4—371no—
232C22×C4—371no—
233C22×C4—371no—
234C22×C4C22×D3711yescharacteristic
235C22×C4C22×D3711yescharacteristic
236C22×C4C22×D3711yescharacteristic
237C22×C4C22×D3711yescharacteristic
238C22×C4C22×D3711yescharacteristic
239C22×C4C22×D3711yescharacteristic
240C22×C4C22×D3711yescharacteristic
241C4⋊C4—41no—
242C4⋊C4—374no—
243C4⋊C4—374no—
244C4⋊C4—374no—
245C4⋊C4—374no—
246C4⋊C4—742no—
247C4⋊C4—742no—
248C4⋊C4—1481no—
249C4⋊C4—1481no—
250C4⋊C4—1481no—
251C42—41no—
252C42—374no—
253C42—374no—
254C42—742no—
255C42—1481no—
256C42—1481no—
257C42—1481no—
258C23C23×D3711yescharacteristic
259Q8—44no—
260Q8—1484no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—372no—
279C2×C4—372no—
280C2×C4—372no—
281C2×C4—372no—
282C2×C4—372no—
283C2×C4—372no—
284C2×C4—372no—
285C2×C4—741no—
286C2×C4—741no—
287C2×C4—741no—
288C2×C4—741no—
289C2×C4—741no—
290C2×C4—741no—
291C2×C4—741no—
292C2×C4—741no—
293C2×C4—741no—
294C2×C4—741no—
295C2×C4—741no—
296C2×C4—741no—
297C2×C4—741no—
298C2×C4—741no—
299C2×C4—741no—
300C2×C4—741no—
301C2×C4—741no—
302C2×C4C2×C4×D3712yes—
303C2×C4C2×C4×D3712yes—
304C2×C4C2×C4×D3712yes—
305C2×C4Q8×D3712yes—
306C22C22×C4×D3711yescharacteristic
307C22C2×Q8×D3711yescharacteristic
308C22#41×D3711yescharacteristic
309C22#1192458111yescharacteristic
310C22#1192458111yescharacteristic
311C22#1192458311yescharacteristic
312C22#1192458411yescharacteristic
313C4—22no—
314C4—22no—
315C4—22no—
316C4—22no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—374no—
321C4—742no—
322C4—742no—
323C4—742no—
324C4—742no—
325C4—1481no—
326C4—1481no—
327C4—1481no—
328C2C4×Q8×D3711yescharacteristic
329C2#1455800611yescharacteristic
330C2#1455800911yescharacteristic
331C2#1455814711yescharacteristic
332C2#1455815311yescharacteristic
333C2#1455816011yescharacteristic
334C2#1455816211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D37
commutatorC22×C37C23×C4
FittingC37×#391C2
socle, prime radical, cosocleC23×C37C24
hypercenter#391D37
hypocenterC37#2331
LCSG ▷ C22×C37 ▷ C37
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
17. #11924534C4semidirect
89. C4×Dic37Q8semidirect
183. C37#2331semidirect

Maximal subgroups

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

subgroupindex
C2×#119245342
#145574402
#145574372
#145574302
#145574282
#145574202
#145574172
#145574152
#145574142
#145573972
C4×#105969362
#145573842
#145573782
C37×#3912
#233137