Finite Group Databasebeta

← C37×#3326 #17992144

Subgroups — C37×#3326 #17992144

Order 4736.

392 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
1C37×#553C211yesmaximal, characteristic, complemented
2C37×#548C211yesmaximal, characteristic, complemented
3C37×#488C211yesmaximal, characteristic, complemented
4C37×#483C211yesmaximal, characteristic, complemented
5C37×#482C211yesmaximal, characteristic, complemented
6C37×#479C211yesmaximal, characteristic, complemented
7C37×#469C211yesmaximal, characteristic, complemented
8C37×#468C211yesmaximal, characteristic, complemented
9C37×#463C211yesmaximal, characteristic, complemented
10C37×#449C211yesmaximal, characteristic, complemented
11C37×#448C211yesmaximal, characteristic, complemented
12C4×SD16×C37C211yesmaximal, characteristic, complemented
13C37×#436C211yesmaximal, characteristic, complemented
14C37×#419C211yesmaximal, characteristic, complemented
15C37×#417C211yesmaximal, characteristic, complemented
16C2×#41×C37C2211yescharacteristic
17C22×Q8×C37C2211yescharacteristic
18C2×SD16×C37C2211yescharacteristic
19C2×SD16×C37C2211yescharacteristic
20C2×D8×C37C2211yescharacteristic
21C2×M16×C37C2211yescharacteristic
22C2×M16×C37C2211yescharacteristic
23#126×C37—21no—
24#125×C37C2211yescharacteristic
25#124×C37—21no—
26#124×C37C2211yescharacteristic
27#123×C37—21no—
28C22⋊Q8×C37—21no—
29C22⋊Q8×C37C2211yescharacteristic
30#121×C37—21no—
31#121×C37—21no—
32#121×C37C2211yescharacteristic
33#121×C37C2211yescharacteristic
34#121×C37C2211yescharacteristic
35C4×Q8×C37C2211yescharacteristic
36C4×D4×C37—21no—
37C4×D4×C37—21no—
38C4×D4×C37C2211yescharacteristic
39C4×D4×C37C2211yescharacteristic
40#117×C37C2211yescharacteristic
41C2×C4⋊C4×C37C2211yescharacteristic
42#107×C37C2211yescharacteristic
43#106×C37C2211yescharacteristic
44#106×C37C2211yescharacteristic
45C4⋊C8×C37C2211yescharacteristic
46#103×C37C2211yescharacteristic
47#103×C37C2211yescharacteristic
48#103×C37C2211yescharacteristic
49#103×C37C2211yescharacteristic
50#103×C37C2211yescharacteristic
51#103×C37C2211yescharacteristic
52#102×C37C2211yescharacteristic
53#102×C37C2211yescharacteristic
54#102×C37C2211yescharacteristic
55#102×C37C2211yescharacteristic
56C22⋊C8×C37C2211yescharacteristic
57C22⋊C8×C37C2211yescharacteristic
58C4×C8×C37C2211yescharacteristic
59#41×C37—22no—
60#41×C37—41no—
61C2×Q8×C37—21no—
62C2×Q8×C37—22no—
63C2×Q8×C37—41no—
64C2×Q8×C37C2311yescharacteristic
65C2×Q8×C37C2311yescharacteristic
66C2×D4×C37—21no—
67C2×D4×C37—21no—
68C2×D4×C37—21no—
69C2×D4×C37—21no—
70C2×D4×C37D411yescharacteristic
71C2×D4×C37C2311yescharacteristic
72C2×D4×C37C2311yescharacteristic
73C22×C4×C37—21no—
74C22×C4×C37—21no—
75C22×C4×C37—21no—
76C22×C4×C37—21no—
77C22×C4×C37C2311yescharacteristic
78C22×C4×C37C2311yescharacteristic
79SD16×C37—22no—
80SD16×C37—41no—
81D8×C37—41no—
82M16×C37—41no—
83M16×C37—41no—
84C2×C8×C37C2311yescharacteristic
85C2×C8×C37C2311yescharacteristic
86C2×C8×C37C2311yescharacteristic
87C2×C8×C37C2311yescharacteristic
88C4⋊C4×C37—21no—
89C4⋊C4×C37—21no—
90C4⋊C4×C37—21no—
91C4⋊C4×C37—21no—
92C4⋊C4×C37—21no—
93C4⋊C4×C37D411yescharacteristic
94C4⋊C4×C37C2311yescharacteristic
95C4⋊C4×C37C2311yescharacteristic
96C4⋊C4×C37C2311yescharacteristic
97C4⋊C4×C37C2311yescharacteristic
98C22⋊C4×C37—21no—
99C22⋊C4×C37—21no—
100C22⋊C4×C37—21no—
101C22⋊C4×C37—21no—
102C22⋊C4×C37—21no—
103C22⋊C4×C37—21no—
104C22⋊C4×C37—21no—
105C22⋊C4×C37—21no—
106C22⋊C4×C37D411yescharacteristic
107C22⋊C4×C37D411yescharacteristic
108C42×C37—21no—
109C42×C37—21no—
110C42×C37C2311yescharacteristic
111C23×C37—21no—
112C23×C37—21no—
113C23×C37C2×D411yescharacteristic
114C23×C37C2×D411yescharacteristic
115Q8×C37—21no—
116Q8×C37—21no—
117Q8×C37—21no—
118Q8×C37—42no—
119Q8×C37#4112yes—
120D4×C37—21no—
121D4×C37—21no—
122D4×C37—21no—
123D4×C37—21no—
124D4×C37—41no—
125D4×C37—42no—
126D4×C37—42no—
127D4×C37—42no—
128D4×C37—81no—
129C2×C4×C37—21no—
130C2×C4×C37—21no—
131C2×C4×C37—21no—
132C2×C4×C37—21no—
133C2×C4×C37—21no—
134C2×C4×C37—21no—
135C2×C4×C37—21no—
136C2×C4×C37—21no—
137C2×C4×C37—21no—
138C2×C4×C37—21no—
139C2×C4×C37—42no—
140C2×C4×C37—42no—
141C2×C4×C37—81no—
142C2×C4×C37—81no—
143C2×C4×C37C2×D411yescharacteristic
144C2×C4×C37C2×D411yescharacteristic
145C2×C4×C37C2×D411yescharacteristic
146C2×C4×C37C2×D411yescharacteristic
147C2×C4×C37C2411yescharacteristic
148C8×C37—21no—
149C8×C37—21no—
150C8×C37—21no—
151C8×C37—21no—
152C22×C37—21no—
153C22×C37—21no—
154C22×C37—21no—
155C22×C37—21no—
156C22×C37—21no—
157C22×C37—21no—
158C22×C37—41no—
159C22×C37—41no—
160C22×C37—41no—
161C22×C37—41no—
162C22×C37—41no—
163C22×C37—41no—
164C22×C37C22×D411yescharacteristic
165C4×C37—21no—
166C4×C37—21no—
167C4×C37—22no—
168C4×C37—21no—
169C4×C37—21no—
170C4×C37—41no—
171C4×C37—41no—
172C4×C37—41no—
173C4×C37—41no—
174C4×C37—41no—
175C4×C37C2×#4111yescharacteristic
176C4×C37H(5,𝔽2)11yescharacteristic
177#3326C3711yesmaximal, characteristic, direct product factor, Sylow
178C2×C37—41no—
179C2×C37—41no—
180C2×C37—81no—
181C2×C37—81no—
182C2×C37#54611yescharacteristic
183C2×C37#57511yescharacteristic
184C2×C37#57711yescharacteristic
185#553C2×C3711yescharacteristic, complemented
186#548C2×C3711yescharacteristic, complemented
187#488C2×C3711yescharacteristic, complemented
188#483C2×C3711yescharacteristic, complemented
189#482C2×C3711yescharacteristic, complemented
190#479C2×C3711yescharacteristic, complemented
191#469C2×C3711yescharacteristic, complemented
192#468C2×C3711yescharacteristic, complemented
193#463C2×C3711yescharacteristic, complemented
194#449C2×C3711yescharacteristic, complemented
195#448C2×C3711yescharacteristic, complemented
196C4×SD16C2×C3711yescharacteristic, complemented
197#436C2×C3711yescharacteristic, complemented
198#419C2×C3711yescharacteristic, complemented
199#417C2×C3711yescharacteristic, complemented
200C37#332611yescharacteristic, direct product factor, Sylow
201C2×#41C22×C3711yescharacteristic
202C22×Q8C22×C3711yescharacteristic
203C2×SD16C22×C3711yescharacteristic
204C2×SD16C22×C3711yescharacteristic
205C2×D8C22×C3711yescharacteristic
206C2×M16C22×C3711yescharacteristic
207C2×M16C22×C3711yescharacteristic
208#126—21no—
209#125C22×C3711yescharacteristic
210#124—21no—
211#124C22×C3711yescharacteristic
212#123—21no—
213C22⋊Q8—21no—
214C22⋊Q8C22×C3711yescharacteristic
215#121—21no—
216#121—21no—
217#121C22×C3711yescharacteristic
218#121C22×C3711yescharacteristic
219#121C22×C3711yescharacteristic
220C4×Q8C22×C3711yescharacteristic
221C4×D4—21no—
222C4×D4—21no—
223C4×D4C22×C3711yescharacteristic
224C4×D4C22×C3711yescharacteristic
225#117C22×C3711yescharacteristic
226C2×C4⋊C4C22×C3711yescharacteristic
227#107C22×C3711yescharacteristic
228#106C22×C3711yescharacteristic
229#106C22×C3711yescharacteristic
230C4⋊C8C22×C3711yescharacteristic
231#103C22×C3711yescharacteristic
232#103C22×C3711yescharacteristic
233#103C22×C3711yescharacteristic
234#103C22×C3711yescharacteristic
235#103C22×C3711yescharacteristic
236#103C22×C3711yescharacteristic
237#102C22×C3711yescharacteristic
238#102C22×C3711yescharacteristic
239#102C22×C3711yescharacteristic
240#102C22×C3711yescharacteristic
241C22⋊C8C22×C3711yescharacteristic
242C22⋊C8C22×C3711yescharacteristic
243C4×C8C22×C3711yescharacteristic
244#41—22no—
245#41—41no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C23×C3711yescharacteristic
250C2×Q8C23×C3711yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4D4×C3711yescharacteristic
256C2×D4C23×C3711yescharacteristic
257C2×D4C23×C3711yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4C23×C3711yescharacteristic
263C22×C4C23×C3711yescharacteristic
264SD16—22no—
265SD16—41no—
266D8—41no—
267M16—41no—
268M16—41no—
269C2×C8C23×C3711yescharacteristic
270C2×C8C23×C3711yescharacteristic
271C2×C8C23×C3711yescharacteristic
272C2×C8C23×C3711yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4D4×C3711yescharacteristic
279C4⋊C4C23×C3711yescharacteristic
280C4⋊C4C23×C3711yescharacteristic
281C4⋊C4C23×C3711yescharacteristic
282C4⋊C4C23×C3711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4D4×C3711yescharacteristic
292C22⋊C4D4×C3711yescharacteristic
293C42—21no—
294C42—21no—
295C42C23×C3711yescharacteristic
296C23—21no—
297C23—21no—
298C23C2×D4×C3711yescharacteristic
299C23C2×D4×C3711yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#41×C3712yes—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—41no—
310D4—42no—
311D4—42no—
312D4—42no—
313D4—81no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4C2×D4×C3711yescharacteristic
329C2×C4C2×D4×C3711yescharacteristic
330C2×C4C2×D4×C3711yescharacteristic
331C2×C4C2×D4×C3711yescharacteristic
332C2×C4C24×C3711yescharacteristic
333C8—21no—
334C8—21no—
335C8—21no—
336C8—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22C22×D4×C3711yescharacteristic
350C4—21no—
351C4—21no—
352C4—22no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4C2×#41×C3711yescharacteristic
361C4H(5,𝔽2)×C3711yescharacteristic
362C2—41no—
363C2—41no—
364C2—81no—
365C2—81no—
366C2C37×#54611yescharacteristic
367C2C37×#57511yescharacteristic
368C2C37×#57711yescharacteristic
subgroupG/H
center, socleC22×C37C22×D4
commutator, Frattini, cosocleC2×C4C24×C37
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C37 ◁ C4×D4×C37 ◁ G

Complements

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

NH ≅ G/Ntype
1. C37×#553C2semidirect
2. C37×#548C2semidirect
3. C37×#488C2semidirect
4. C37×#483C2semidirect
5. C37×#482C2semidirect
6. C37×#479C2semidirect
7. C37×#469C2semidirect
8. C37×#468C2semidirect
9. C37×#463C2semidirect
10. C37×#449C2semidirect
11. C37×#448C2semidirect
12. C4×SD16×C37C2semidirect
13. C37×#436C2semidirect
14. C37×#419C2semidirect
15. C37×#417C2semidirect
177. #3326C37direct (N × H)
185. #553C2×C37semidirect
186. #548C2×C37semidirect
187. #488C2×C37semidirect
188. #483C2×C37semidirect
189. #482C2×C37semidirect
190. #479C2×C37semidirect
191. #469C2×C37semidirect
192. #468C2×C37semidirect
193. #463C2×C37semidirect
194. #449C2×C37semidirect
195. #448C2×C37semidirect
196. C4×SD16C2×C37semidirect
197. #436C2×C37semidirect
198. #419C2×C37semidirect
199. #417C2×C37semidirect
200. C37#3326direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#5532
C37×#5482
C37×#4882
C37×#4832
C37×#4822
C37×#4792
C37×#4692
C37×#4682
C37×#4632
C37×#4492
C37×#4482
C4×SD16×C372
C37×#4362
C37×#4192
C37×#4172
#332637