Finite Group Databasebeta

← C37×#19485 #22640738

Subgroups — C37×#19485 #22640738

Order 9472.

604 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3209C211yesmaximal, characteristic, complemented
2C37×#3107C211yesmaximal, characteristic
3C37×#3079C211yesmaximal, characteristic
4C37×#2990C211yesmaximal, characteristic
5C37×#1700C212yesmaximal, complemented
6C37×#1690C212yesmaximal, complemented
7C37×#1679C212yesmaximal, complemented
8C37×#1669C212yesmaximal, complemented
9C37×#1574C211yesmaximal, characteristic, complemented
10C37×#1561C211yesmaximal, characteristic, complemented
11C37×#1546C211yesmaximal, characteristic, complemented
12C37×#533C2211yescharacteristic
13C2×C4⋊Q8×C37C2211yescharacteristic
14C4×#41×C37C2211yescharacteristic
15C37×#494C2212yes—
16C37×#487C2212yes—
17C37×#485—21no—
18C37×#477—21no—
19C37×#476—21no—
20C37×#475C2211yescharacteristic
21C37×#475C2212yes—
22C37×#474C2211yescharacteristic
23C37×#474C2212yes—
24C37×#462—21no—
25C37×#462—21no—
26C37×#462C2211yescharacteristic
27C37×#461C2211yescharacteristic
28C37×#452—21no—
29C37×#451—22no—
30C37×#446C2212yes—
31C4×Q16×C37—21no—
32C4×SD16×C37—21no—
33C37×#435—22no—
34C8×D4×C37C2212yes—
35C37×#432—21no—
36C37×#431C2211yescharacteristic
37C37×#431C2211yescharacteristic
38C37×#428—21no—
39C37×#423C2211yescharacteristic
40C37×#423C2211yescharacteristic
41C2×C4⋊C8×C37—21no—
42C37×#417—21no—
43C37×#335C2212yes—
44C37×#334C2212yes—
45C37×#332C2212yes—
46C37×#332C2212yes—
47C37×#332C2212yes—
48C37×#332C2212yes—
49C2×#41×C37D411yescharacteristic
50C22×Q8×C37—21no—
51C2×Q16×C37—21no—
52C2×Q16×C37—22no—
53C2×Q16×C37—22no—
54C2×SD16×C37—21no—
55C2×M16×C37—21no—
56C2×M16×C37—21no—
57C22×C8×C37—22no—
58C4⋊Q8×C37—21no—
59C4⋊Q8×C37—21no—
60C4⋊Q8×C37—21no—
61C4⋊Q8×C37C2312yes—
62C4⋊Q8×C37C2311yescharacteristic
63C4⋊Q8×C37C2311yescharacteristic
64#125×C37—21no—
65C22⋊Q8×C37—21no—
66C22⋊Q8×C37—21no—
67C4×Q8×C37—21no—
68C4×Q8×C37—21no—
69C4×Q8×C37D411yescharacteristic
70C4×Q8×C37C2311yescharacteristic
71C4×D4×C37—21no—
72C4×D4×C37—21no—
73C4×D4×C37D411yescharacteristic
74C4×D4×C37C2311yescharacteristic
75#117×C37—21no—
76#117×C37—21no—
77#117×C37D411yescharacteristic
78C2×C4⋊C4×C37—21no—
79C2×C4⋊C4×C37—21no—
80C2×C42×C37—21no—
81C2×C42×C37C2311yescharacteristic
82#107×C37—22no—
83#107×C37—21no—
84#106×C37—22no—
85#106×C37—21no—
86C4⋊C8×C37—22no—
87C4⋊C8×C37—21no—
88C4⋊C8×C37—21no—
89C4⋊C8×C37D411yescharacteristic
90C4⋊C8×C37D411yescharacteristic
91C4⋊C8×C37C2312yes—
92C4⋊C8×C37C2311yescharacteristic
93C4⋊C8×C37C2312yes—
94C4⋊C8×C37C2311yescharacteristic
95#103×C37—22no—
96#103×C37—21no—
97#103×C37—22no—
98#103×C37—22no—
99#103×C37—21no—
100#103×C37—21no—
101#102×C37—22no—
102#102×C37—21no—
103C22⋊C8×C37—21no—
104C22⋊C8×C37—22no—
105C22⋊C8×C37D412yes—
106#97×C37—22no—
107C4×C8×C37—21no—
108C4×C8×C37C2312yes—
109#41×C37—22no—
110#41×C37—41no—
111C2×Q8×C37—21no—
112C2×Q8×C37—22no—
113C2×Q8×C37—21no—
114C2×Q8×C37—41no—
115C2×Q8×C37—81no—
116C2×Q8×C37—81no—
117C2×Q8×C37C2×D411yescharacteristic
118C2×D4×C37—21no—
119C2×D4×C37C2×D411yescharacteristic
120C22×C4×C37—21no—
121C22×C4×C37—21no—
122C22×C4×C37—21no—
123C22×C4×C37—41no—
124C22×C4×C37C2×D411yescharacteristic
125C22×C4×C37C2×D411yescharacteristic
126C22×C4×C37#4111yescharacteristic
127Q16×C37—42no—
128Q16×C37—82no—
129Q16×C37—82no—
130SD16×C37—42no—
131M16×C37—42no—
132M16×C37—81no—
133C2×C8×C37—21no—
134C2×C8×C37—21no—
135C2×C8×C37—22no—
136C2×C8×C37—22no—
137C2×C8×C37—82no—
138C2×C8×C37C2×D412yes—
139C2×C8×C37C2×D412yes—
140C4⋊C4×C37—21no—
141C4⋊C4×C37—21no—
142C4⋊C4×C37—22no—
143C4⋊C4×C37—21no—
144C4⋊C4×C37—21no—
145C4⋊C4×C37—22no—
146C4⋊C4×C37—21no—
147C4⋊C4×C37—21no—
148C4⋊C4×C37—41no—
149C4⋊C4×C37—41no—
150C4⋊C4×C37—41no—
151C4⋊C4×C37—41no—
152C4⋊C4×C37C2×D411yescharacteristic
153C4⋊C4×C37C2×D411yescharacteristic
154C22⋊C4×C37—21no—
155C22⋊C4×C37—21no—
156C22⋊C4×C37—21no—
157C22⋊C4×C37—41no—
158C42×C37—21no—
159C42×C37—21no—
160C42×C37—21no—
161C42×C37—41no—
162C42×C37C2×D411yescharacteristic
163C42×C37C2×D411yescharacteristic
164C42×C37#4111yescharacteristic
165C42×C37C2411yescharacteristic
166C23×C37—21no—
167C23×C37#12111yescharacteristic
168Q8×C37—21no—
169Q8×C37—21no—
170Q8×C37—41no—
171Q8×C37—41no—
172Q8×C37—41no—
173Q8×C37—42no—
174Q8×C37—41no—
175Q8×C37—42no—
176Q8×C37—82no—
177D4×C37—21no—
178D4×C37—42no—
179D4×C37#13712yes—
180C2×C4×C37—21no—
181C2×C4×C37—21no—
182C2×C4×C37—21no—
183C2×C4×C37—21no—
184C2×C4×C37—22no—
185C2×C4×C37—21no—
186C2×C4×C37—21no—
187C2×C4×C37—21no—
188C2×C4×C37—42no—
189C2×C4×C37—41no—
190C2×C4×C37—41no—
191C2×C4×C37—42no—
192C2×C4×C37—41no—
193C2×C4×C37—41no—
194C2×C4×C37—42no—
195C2×C4×C37—82no—
196C2×C4×C37#12111yescharacteristic
197C2×C4×C37#12111yescharacteristic
198C2×C4×C37#12111yescharacteristic
199C2×C4×C37C22×D411yescharacteristic
200C2×C4×C37C22×D411yescharacteristic
201C2×C4×C37C2×#4111yescharacteristic
202C8×C37—22no—
203C8×C37—22no—
204C8×C37—41no—
205C8×C37—42no—
206C8×C37—41no—
207C8×C37—42no—
208#19485C3711yesmaximal, characteristic, direct product factor, Sylow
209C22×C37—21no—
210C22×C37—21no—
211C22×C37—22no—
212C22×C37—21no—
213C22×C37—41no—
214C22×C37—41no—
215C22×C37C2×#12111yescharacteristic
216C4×C37—21no—
217C4×C37—21no—
218C4×C37—21no—
219C4×C37—41no—
220C4×C37—41no—
221C4×C37—41no—
222C4×C37—82no—
223C4×C37—81no—
224C4×C37—81no—
225C4×C37#47312yes—
226C4×C37C2×#13711yescharacteristic
227C4×C37#57511yescharacteristic
228C4×C37#57611yescharacteristic
229C4×C37#57711yescharacteristic
230#3209C2×C3711yescharacteristic, complemented
231#3107C2×C3711yescharacteristic
232#3079C2×C3711yescharacteristic
233#2990C2×C3711yescharacteristic
234#1700C2×C3712yescomplemented
235#1690C2×C3712yescomplemented
236#1679C2×C3712yescomplemented
237#1669C2×C3712yescomplemented
238#1574C2×C3711yescharacteristic, complemented
239#1561C2×C3711yescharacteristic, complemented
240#1546C2×C3711yescharacteristic, complemented
241C2×C37—41no—
242C2×C37—42no—
243C2×C37#304511yescharacteristic
244C2×C37#305611yescharacteristic
245C2×C37C2×#47311yescharacteristic
246#533C22×C3711yescharacteristic
247C2×C4⋊Q8C22×C3711yescharacteristic
248C4×#41C22×C3711yescharacteristic
249#494C22×C3712yes—
250#487C22×C3712yes—
251#485—21no—
252#477—21no—
253#476—21no—
254#475C22×C3712yes—
255#475C22×C3711yescharacteristic
256#474C22×C3712yes—
257#474C22×C3711yescharacteristic
258#462—21no—
259#462—21no—
260#462C22×C3711yescharacteristic
261#461C22×C3711yescharacteristic
262#452—21no—
263#451—22no—
264#446C22×C3712yes—
265C4×Q16—21no—
266C4×SD16—21no—
267#435—22no—
268C8×D4C22×C3712yes—
269#432—21no—
270#431C22×C3711yescharacteristic
271#431C22×C3711yescharacteristic
272#428—21no—
273#423C22×C3711yescharacteristic
274#423C22×C3711yescharacteristic
275C2×C4⋊C8—21no—
276#417—21no—
277#335C22×C3712yes—
278#334C22×C3712yes—
279#332C22×C3712yes—
280#332C22×C3712yes—
281#332C22×C3712yes—
282#332C22×C3712yes—
283C37#1948511yescharacteristic, direct product factor, Sylow
284C2×#41D4×C3711yescharacteristic
285C22×Q8—21no—
286C2×Q16—21no—
287C2×Q16—22no—
288C2×Q16—22no—
289C2×SD16—21no—
290C2×M16—21no—
291C2×M16—21no—
292C22×C8—22no—
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8C23×C3712yes—
297C4⋊Q8C23×C3711yescharacteristic
298C4⋊Q8C23×C3711yescharacteristic
299#125—21no—
300C22⋊Q8—21no—
301C22⋊Q8—21no—
302C4×Q8—21no—
303C4×Q8—21no—
304C4×Q8D4×C3711yescharacteristic
305C4×Q8C23×C3711yescharacteristic
306C4×D4—21no—
307C4×D4—21no—
308C4×D4D4×C3711yescharacteristic
309C4×D4C23×C3711yescharacteristic
310#117—21no—
311#117—21no—
312#117D4×C3711yescharacteristic
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C42—21no—
316C2×C42C23×C3711yescharacteristic
317#107—22no—
318#107—21no—
319#106—22no—
320#106—21no—
321C4⋊C8—22no—
322C4⋊C8—21no—
323C4⋊C8—21no—
324C4⋊C8D4×C3711yescharacteristic
325C4⋊C8D4×C3711yescharacteristic
326C4⋊C8C23×C3712yes—
327C4⋊C8C23×C3711yescharacteristic
328C4⋊C8C23×C3712yes—
329C4⋊C8C23×C3711yescharacteristic
330#103—22no—
331#103—21no—
332#103—22no—
333#103—22no—
334#103—21no—
335#103—21no—
336#102—22no—
337#102—21no—
338C22⋊C8—21no—
339C22⋊C8—22no—
340C22⋊C8D4×C3712yes—
341#97—22no—
342C4×C8—21no—
343C4×C8C23×C3712yes—
344#41—22no—
345#41—41no—
346C2×Q8—21no—
347C2×Q8—22no—
348C2×Q8—21no—
349C2×Q8—41no—
350C2×Q8—81no—
351C2×Q8—81no—
352C2×Q8C2×D4×C3711yescharacteristic
353C2×D4—21no—
354C2×D4C2×D4×C3711yescharacteristic
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—41no—
359C22×C4C2×D4×C3711yescharacteristic
360C22×C4C2×D4×C3711yescharacteristic
361C22×C4#41×C3711yescharacteristic
362Q16—42no—
363Q16—82no—
364Q16—82no—
365SD16—42no—
366M16—42no—
367M16—81no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—82no—
373C2×C8C2×D4×C3712yes—
374C2×C8C2×D4×C3712yes—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—22no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4C2×D4×C3711yescharacteristic
388C4⋊C4C2×D4×C3711yescharacteristic
389C22⋊C4—21no—
390C22⋊C4—21no—
391C22⋊C4—21no—
392C22⋊C4—41no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—41no—
397C42C2×D4×C3711yescharacteristic
398C42C2×D4×C3711yescharacteristic
399C42#41×C3711yescharacteristic
400C42C24×C3711yescharacteristic
401C23—21no—
402C23#121×C3711yescharacteristic
403Q8—21no—
404Q8—21no—
405Q8—41no—
406Q8—41no—
407Q8—41no—
408Q8—42no—
409Q8—41no—
410Q8—42no—
411Q8—82no—
412D4—21no—
413D4—42no—
414D4#137×C3712yes—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—82no—
431C2×C4#121×C3711yescharacteristic
432C2×C4#121×C3711yescharacteristic
433C2×C4#121×C3711yescharacteristic
434C2×C4C22×D4×C3711yescharacteristic
435C2×C4C22×D4×C3711yescharacteristic
436C2×C4C2×#41×C3711yescharacteristic
437C8—22no—
438C8—22no—
439C8—41no—
440C8—42no—
441C8—41no—
442C8—42no—
443C22—21no—
444C22—21no—
445C22—22no—
446C22—21no—
447C22—41no—
448C22—41no—
449C22C2×#121×C3711yescharacteristic
450C4—21no—
451C4—21no—
452C4—21no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4—82no—
457C4—81no—
458C4—81no—
459C4C37×#47312yes—
460C4C2×#137×C3711yescharacteristic
461C4C37×#57511yescharacteristic
462C4C37×#57611yescharacteristic
463C4C37×#57711yescharacteristic
464C2—41no—
465C2—42no—
466C2C37×#304511yescharacteristic
467C2C37×#305611yescharacteristic
468C2C2×C37×#47311yescharacteristic
subgroupG/H
center, socleC22×C37C2×#121
commutator, Frattini, cosocleC42C24×C37
prime radicalC2×#41×C37D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C37 ◁ C2×C42×C37 ◁ G

Complements

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

NH ≅ G/Ntype
1. C37×#3209C2semidirect
5. C37×#1700C2semidirect
6. C37×#1690C2semidirect
7. C37×#1679C2semidirect
8. C37×#1669C2semidirect
9. C37×#1574C2semidirect
10. C37×#1561C2semidirect
11. C37×#1546C2semidirect
208. #19485C37direct (N × H)
230. #3209C2×C37semidirect
234. #1700C2×C37semidirect
235. #1690C2×C37semidirect
236. #1679C2×C37semidirect
237. #1669C2×C37semidirect
238. #1574C2×C37semidirect
239. #1561C2×C37semidirect
240. #1546C2×C37semidirect
283. C37#19485direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#32092
C37×#31072
C37×#30792
C37×#29902
C37×#17002
C37×#16902
C37×#16792
C37×#16692
C37×#15742
C37×#15612
C37×#15462
#1948537