Finite Group Databasebeta

← C37×#29697 #22650950

Subgroups — C37×#29697 #22650950

Order 9472.

1012 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C37×#542C211yesmaximal, characteristic
2C2×C37×#475C211yesmaximal, characteristic, complemented
3C2×C37×#476C211yesmaximal, characteristic, complemented
4C37×#2628C211yesmaximal, characteristic, complemented
5C37×#2058C212yesmaximal, complemented
6C37×#2042C212yesmaximal, complemented
7C37×#2035C212yesmaximal, complemented
8C37×#2034C212yesmaximal
9C37×#1805C211yesmaximal, characteristic, complemented
10C37×#1804C211yesmaximal, characteristic, complemented
11C37×#1798C211yesmaximal, characteristic, complemented
12C37×#542—24no—
13C37×#542—24no—
14C2×C4⋊Q8×C37C2211yescharacteristic
15C2×C4⋊Q8×C37C2211yescharacteristic
16C2×#126×C37—21no—
17C2×#124×C37—21no—
18C2×#124×C37C2211yescharacteristic
19C2×#123×C37—21no—
20C2×C22⋊Q8×C37—21no—
21C2×C22⋊Q8×C37C2211yescharacteristic
22C2×#121×C37C2211yescharacteristic
23C2×C4×Q8×C37C2211yescharacteristic
24C2×C4×D4×C37C2211yescharacteristic
25C2×#117×C37C2211yescharacteristic
26C37×#476—24no—
27C37×#475—24no—
28C2×#106×C37C2212yes—
29C2×#106×C37C2212yes—
30C2×C4⋊C8×C37C2211yescharacteristic
31C2×C4⋊C8×C37C2211yescharacteristic
32C2×#103×C37C2212yes—
33C2×#103×C37C2212yes—
34C2×#102×C37C2212yes—
35C2×#102×C37C2212yes—
36C2×C22⋊C8×C37C2212yes—
37C37×#400—21no—
38C37×#399—21no—
39C37×#395—21no—
40C37×#393—21no—
41C37×#389C2212yes—
42C37×#382C2211yescharacteristic
43C37×#380C2212yes—
44C37×#340C2212yes—
45C37×#340C2212yes—
46C37×#336C2212yes—
47C22×Q8×C37—21no—
48C22×Q8×C37—21no—
49C22×Q8×C37C2311yescharacteristic
50C22×D4×C37—21no—
51C22×D4×C37C2311yescharacteristic
52C23×C4×C37—21no—
53C23×C4×C37C2311yescharacteristic
54C22×C8×C37C2312yes—
55C22×C8×C37C2312yes—
56C4⋊Q8×C37—22no—
57C4⋊Q8×C37—22no—
58C4⋊Q8×C37—22no—
59C4⋊Q8×C37—22no—
60#126×C37—28no—
61#124×C37—28no—
62#124×C37—22no—
63#124×C37—22no—
64#123×C37—28no—
65C22⋊Q8×C37—28no—
66C22⋊Q8×C37—22no—
67C22⋊Q8×C37—22no—
68#121×C37—22no—
69#121×C37—22no—
70C4×Q8×C37—22no—
71C4×Q8×C37D414yes—
72C4×D4×C37—22no—
73C4×D4×C37D414yes—
74#117×C37—22no—
75#117×C37—22no—
76C2×C4⋊C4×C37—21no—
77C2×C4⋊C4×C37—21no—
78C2×C4⋊C4×C37—21no—
79C2×C4⋊C4×C37—21no—
80C2×C4⋊C4×C37—22no—
81C2×C4⋊C4×C37D411yescharacteristic
82C2×C4⋊C4×C37D411yescharacteristic
83C2×C4⋊C4×C37C2312yes—
84C2×C4⋊C4×C37C2312yes—
85C2×C4⋊C4×C37C2311yescharacteristic
86C2×C4⋊C4×C37C2311yescharacteristic
87C2×C22⋊C4×C37—21no—
88C2×C22⋊C4×C37—21no—
89C2×C22⋊C4×C37—21no—
90C2×C22⋊C4×C37—21no—
91C2×C22⋊C4×C37—21no—
92C2×C22⋊C4×C37—22no—
93C2×C22⋊C4×C37D411yescharacteristic
94C2×C22⋊C4×C37D411yescharacteristic
95C2×C42×C37—21no—
96C2×C42×C37C2311yescharacteristic
97C2×C42×C37C2311yescharacteristic
98#106×C37—24no—
99#106×C37—24no—
100C4⋊C8×C37—22no—
101C4⋊C8×C37—22no—
102#103×C37—24no—
103#103×C37—42no—
104#102×C37—24no—
105#102×C37—42no—
106C22⋊C8×C37—42no—
107#95×C37—22no—
108#95×C37—21no—
109#95×C37—21no—
110#95×C37—22no—
111C24×C37—21no—
112C24×C37C2×D411yescharacteristic
113C2×Q8×C37—24no—
114C2×Q8×C37—21no—
115C2×Q8×C37—21no—
116C2×Q8×C37—21no—
117C2×Q8×C37—21no—
118C2×Q8×C37—21no—
119C2×Q8×C37—42no—
120C2×Q8×C37—82no—
121C2×Q8×C37—82no—
122C2×Q8×C37C2×D412yes—
123C2×D4×C37—24no—
124C2×D4×C37—21no—
125C2×D4×C37—21no—
126C2×D4×C37—21no—
127C2×D4×C37—21no—
128C2×D4×C37—21no—
129C2×D4×C37—82no—
130C2×D4×C37C2×D412yes—
131C22×C4×C37—21no—
132C22×C4×C37—21no—
133C22×C4×C37—21no—
134C22×C4×C37—21no—
135C22×C4×C37—21no—
136C22×C4×C37—21no—
137C22×C4×C37—21no—
138C22×C4×C37—21no—
139C22×C4×C37—21no—
140C22×C4×C37—24no—
141C22×C4×C37—22no—
142C22×C4×C37—22no—
143C22×C4×C37—82no—
144C22×C4×C37C2×D411yescharacteristic
145C22×C4×C37C2×D411yescharacteristic
146C22×C4×C37C2×D411yescharacteristic
147C22×C4×C37C2×D411yescharacteristic
148C22×C4×C37C2×D411yescharacteristic
149C22×C4×C37C2411yescharacteristic
150C2×C8×C37—22no—
151C2×C8×C37—22no—
152C2×C8×C37—22no—
153C2×C8×C37—22no—
154C2×C8×C37—22no—
155C2×C8×C37—22no—
156C4⋊C4×C37—24no—
157C4⋊C4×C37—24no—
158C4⋊C4×C37—24no—
159C4⋊C4×C37—22no—
160C4⋊C4×C37—22no—
161C4⋊C4×C37—21no—
162C4⋊C4×C37—21no—
163C4⋊C4×C37—22no—
164C4⋊C4×C37—22no—
165C4⋊C4×C37—22no—
166C4⋊C4×C37—22no—
167C4⋊C4×C37—42no—
168C4⋊C4×C37—82no—
169C4⋊C4×C37C2×D412yes—
170C4⋊C4×C37C2×D412yes—
171C22⋊C4×C37—24no—
172C22⋊C4×C37—24no—
173C22⋊C4×C37—24no—
174C22⋊C4×C37—24no—
175C22⋊C4×C37—24no—
176C22⋊C4×C37—22no—
177C22⋊C4×C37—22no—
178C22⋊C4×C37—82no—
179C42×C37—24no—
180C42×C37—21no—
181C42×C37—21no—
182C42×C37C2×D412yes—
183C42×C37C2×D412yes—
184C23×C37—21no—
185C23×C37—22no—
186C23×C37—21no—
187C23×C37—21no—
188C23×C37—22no—
189C23×C37—22no—
190C23×C37—41no—
191C23×C37—41no—
192C23×C37—42no—
193C23×C37—42no—
194C23×C37C22×D411yescharacteristic
195Q8×C37—22no—
196Q8×C37—22no—
197Q8×C37—41no—
198Q8×C37—41no—
199Q8×C37—84no—
200Q8×C37—84no—
201D4×C37—22no—
202D4×C37—22no—
203D4×C37—41no—
204D4×C37—41no—
205D4×C37—84no—
206C2×C4×C37—22no—
207C2×C4×C37—22no—
208C2×C4×C37—21no—
209C2×C4×C37—22no—
210C2×C4×C37—21no—
211C2×C4×C37—22no—
212C2×C4×C37—21no—
213C2×C4×C37—22no—
214C2×C4×C37—21no—
215C2×C4×C37—21no—
216C2×C4×C37—22no—
217C2×C4×C37—22no—
218C2×C4×C37—41no—
219C2×C4×C37—41no—
220C2×C4×C37—42no—
221C2×C4×C37—42no—
222C2×C4×C37—41no—
223C2×C4×C37—41no—
224C2×C4×C37—41no—
225C2×C4×C37—41no—
226C2×C4×C37—42no—
227C2×C4×C37—42no—
228C2×C4×C37—42no—
229C2×C4×C37—42no—
230C2×C4×C37—42no—
231C2×C4×C37—42no—
232C2×C4×C37—84no—
233C2×C4×C37#12012yes—
234C2×C4×C37#12012yes—
235C2×C4×C37C22×D411yescharacteristic
236C2×C4×C37C22×D411yescharacteristic
237C2×C4×C37H(5,𝔽2)11yescharacteristic
238C2×C4×C372−1+411yescharacteristic
239C2×C4×C372−1+411yescharacteristic
240C2×C4×C372−1+411yescharacteristic
241C8×C37—42no—
242C8×C37—42no—
243#29697C3711yesmaximal, characteristic, direct product factor, Sylow
244C22×C37—41no—
245C22×C37—42no—
246C22×C37—42no—
247C22×C37—42no—
248C22×C37—42no—
249C22×C37—41no—
250C22×C37—41no—
251C22×C37—42no—
252C22×C37—82no—
253C22×C37—82no—
254C22×C37C2×#12011yescharacteristic
255C22×C37#53611yescharacteristic
256C22×C37#53711yescharacteristic
257C22×C37#57712yes—
258C22×C37#57712yes—
259C4×C37—22no—
260C4×C37—22no—
261C4×C37—41no—
262C4×C37—42no—
263C4×C37—41no—
264C4×C37—42no—
265C4×C37—42no—
266C4×C37—41no—
267C4×C37—82no—
268C4×C37—82no—
269C4×C37#53611yescharacteristic
270C4×C37#53611yescharacteristic
271C4×C37#53711yescharacteristic
272C4×C37#53711yescharacteristic
273C2×#542C2×C3711yescharacteristic
274C2×#475C2×C3711yescharacteristic, complemented
275C2×#476C2×C3711yescharacteristic, complemented
276#2628C2×C3711yescharacteristic, complemented
277#2058C2×C3712yescomplemented
278#2042C2×C3712yescomplemented
279#2035C2×C3712yescomplemented
280#2034C2×C3712yes—
281#1805C2×C3711yescharacteristic, complemented
282#1804C2×C3711yescharacteristic, complemented
283#1798C2×C3711yescharacteristic, complemented
284C2×C37—81no—
285C2×C37—82no—
286C2×C37#261611yescharacteristic
287C2×C37#300911yescharacteristic
288C2×C37#301011yescharacteristic
289C2×C37#324412yes—
290C2×C37#326312yes—
291#542—24no—
292#542—24no—
293C2×C4⋊Q8C22×C3711yescharacteristic
294C2×C4⋊Q8C22×C3711yescharacteristic
295C2×#126—21no—
296C2×#124—21no—
297C2×#124C22×C3711yescharacteristic
298C2×#123—21no—
299C2×C22⋊Q8—21no—
300C2×C22⋊Q8C22×C3711yescharacteristic
301C2×#121C22×C3711yescharacteristic
302C2×C4×Q8C22×C3711yescharacteristic
303C2×C4×D4C22×C3711yescharacteristic
304C2×#117C22×C3711yescharacteristic
305#476—24no—
306#475—24no—
307C2×#106C22×C3712yes—
308C2×#106C22×C3712yes—
309C2×C4⋊C8C22×C3711yescharacteristic
310C2×C4⋊C8C22×C3711yescharacteristic
311C2×#103C22×C3712yes—
312C2×#103C22×C3712yes—
313C2×#102C22×C3712yes—
314C2×#102C22×C3712yes—
315C2×C22⋊C8C22×C3712yes—
316#400—21no—
317#399—21no—
318#395—21no—
319#393—21no—
320#389C22×C3712yes—
321#382C22×C3711yescharacteristic
322#380C22×C3712yes—
323#340C22×C3712yes—
324#340C22×C3712yes—
325#336C22×C3712yes—
326C37#2969711yescharacteristic, direct product factor, Sylow
327C22×Q8—21no—
328C22×Q8—21no—
329C22×Q8C23×C3711yescharacteristic
330C22×D4—21no—
331C22×D4C23×C3711yescharacteristic
332C23×C4—21no—
333C23×C4C23×C3711yescharacteristic
334C22×C8C23×C3712yes—
335C22×C8C23×C3712yes—
336C4⋊Q8—22no—
337C4⋊Q8—22no—
338C4⋊Q8—22no—
339C4⋊Q8—22no—
340#126—28no—
341#124—28no—
342#124—22no—
343#124—22no—
344#123—28no—
345C22⋊Q8—28no—
346C22⋊Q8—22no—
347C22⋊Q8—22no—
348#121—22no—
349#121—22no—
350C4×Q8—22no—
351C4×Q8D4×C3714yes—
352C4×D4—22no—
353C4×D4D4×C3714yes—
354#117—22no—
355#117—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4D4×C3711yescharacteristic
362C2×C4⋊C4D4×C3711yescharacteristic
363C2×C4⋊C4C23×C3712yes—
364C2×C4⋊C4C23×C3712yes—
365C2×C4⋊C4C23×C3711yescharacteristic
366C2×C4⋊C4C23×C3711yescharacteristic
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4D4×C3711yescharacteristic
374C2×C22⋊C4D4×C3711yescharacteristic
375C2×C42—21no—
376C2×C42C23×C3711yescharacteristic
377C2×C42C23×C3711yescharacteristic
378#106—24no—
379#106—24no—
380C4⋊C8—22no—
381C4⋊C8—22no—
382#103—24no—
383#103—42no—
384#102—24no—
385#102—42no—
386C22⋊C8—42no—
387#95—22no—
388#95—21no—
389#95—21no—
390#95—22no—
391C24—21no—
392C24C2×D4×C3711yescharacteristic
393C2×Q8—24no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—42no—
400C2×Q8—82no—
401C2×Q8—82no—
402C2×Q8C2×D4×C3712yes—
403C2×D4—24no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—82no—
410C2×D4C2×D4×C3712yes—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—82no—
424C22×C4C2×D4×C3711yescharacteristic
425C22×C4C2×D4×C3711yescharacteristic
426C22×C4C2×D4×C3711yescharacteristic
427C22×C4C2×D4×C3711yescharacteristic
428C22×C4C2×D4×C3711yescharacteristic
429C22×C4C24×C3711yescharacteristic
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—42no—
448C4⋊C4—82no—
449C4⋊C4C2×D4×C3712yes—
450C4⋊C4C2×D4×C3712yes—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—82no—
459C42—24no—
460C42—21no—
461C42—21no—
462C42C2×D4×C3712yes—
463C42C2×D4×C3712yes—
464C23—21no—
465C23—22no—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—22no—
470C23—41no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23C22×D4×C3711yescharacteristic
475Q8—22no—
476Q8—22no—
477Q8—41no—
478Q8—41no—
479Q8—84no—
480Q8—84no—
481D4—22no—
482D4—22no—
483D4—41no—
484D4—41no—
485D4—84no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—84no—
513C2×C4#120×C3712yes—
514C2×C4#120×C3712yes—
515C2×C4C22×D4×C3711yescharacteristic
516C2×C4C22×D4×C3711yescharacteristic
517C2×C4H(5,𝔽2)×C3711yescharacteristic
518C2×C42−1+4×C3711yescharacteristic
519C2×C42−1+4×C3711yescharacteristic
520C2×C42−1+4×C3711yescharacteristic
521C8—42no—
522C8—42no—
523C22—41no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—41no—
529C22—41no—
530C22—42no—
531C22—82no—
532C22—82no—
533C22C2×#120×C3711yescharacteristic
534C22C37×#53611yescharacteristic
535C22C37×#53711yescharacteristic
536C22C37×#57712yes—
537C22C37×#57712yes—
538C4—22no—
539C4—22no—
540C4—41no—
541C4—42no—
542C4—41no—
543C4—42no—
544C4—42no—
545C4—41no—
546C4—82no—
547C4—82no—
548C4C37×#53611yescharacteristic
549C4C37×#53611yescharacteristic
550C4C37×#53711yescharacteristic
551C4C37×#53711yescharacteristic
552C2—81no—
553C2—82no—
554C2C37×#261611yescharacteristic
555C2C37×#300911yescharacteristic
556C2C37×#301011yescharacteristic
557C2C37×#324412yes—
558C2C37×#326312yes—
subgroupG/H
center, socleC23×C37C22×D4
commutator, Frattini, cosocleC22×C4C24×C37
prime radicalC2×#121×C37C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C37 ◁ C2×#117×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C37×#475C2semidirect
3. C2×C37×#476C2semidirect
4. C37×#2628C2semidirect
5. C37×#2058C2semidirect
6. C37×#2042C2semidirect
7. C37×#2035C2semidirect
9. C37×#1805C2semidirect
10. C37×#1804C2semidirect
11. C37×#1798C2semidirect
243. #29697C37direct (N × H)
274. C2×#475C2×C37semidirect
275. C2×#476C2×C37semidirect
276. #2628C2×C37semidirect
277. #2058C2×C37semidirect
278. #2042C2×C37semidirect
279. #2035C2×C37semidirect
281. #1805C2×C37semidirect
282. #1804C2×C37semidirect
283. #1798C2×C37semidirect
326. C37#29697direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C37×#5422
C2×C37×#4752
C2×C37×#4762
C37×#26282
C37×#20582
C37×#20422
C37×#20352
C37×#20342
C37×#18052
C37×#18042
C37×#17982
#2969737