Finite Group Databasebeta

← C37×#19514 #22640767

Subgroups — C37×#19514 #22640767

Order 9472.

622 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3226C211yesmaximal, characteristic, complemented
2C37×#3214C211yesmaximal, characteristic
3C37×#3206C211yesmaximal, characteristic, complemented
4C37×#2981C211yesmaximal, characteristic, complemented
5C37×#1701C212yesmaximal, complemented
6C37×#1693C212yesmaximal, complemented
7C37×#1681C212yesmaximal, complemented
8C37×#1673C212yesmaximal, complemented
9C37×#1572C211yesmaximal, characteristic, complemented
10C37×#1556C211yesmaximal, characteristic, complemented
11C37×#1553C211yesmaximal, characteristic, complemented
12C37×#532C2211yescharacteristic
13C37×#532C2211yescharacteristic
14C2×C4⋊Q8×C37C2211yescharacteristic
15C37×#500C2211yescharacteristic
16C37×#500C2211yescharacteristic
17C37×#496—21no—
18C37×#492—21no—
19C37×#489—21no—
20C37×#487—21no—
21C37×#486C2211yescharacteristic
22C37×#486C2211yescharacteristic
23C37×#480—22no—
24C37×#474—22no—
25C37×#461C2212yes—
26C37×#461C2212yes—
27C37×#461C2212yes—
28C37×#459C2212yes—
29C37×#449—22no—
30C37×#448—22no—
31C37×#436C2212yes—
32C37×#435—22no—
33C8×D4×C37C2212yes—
34C37×#432C2211yescharacteristic
35C37×#432C2211yescharacteristic
36C37×#431C2211yescharacteristic
37C37×#431C2211yescharacteristic
38C37×#408—22no—
39C37×#335C2212yes—
40C37×#334C2212yes—
41C37×#332C2212yes—
42C37×#332C2212yes—
43C37×#332C2212yes—
44C37×#331C2212yes—
45C2×#41×C37—21no—
46C2×#41×C37—21no—
47C22×Q8×C37—21no—
48C2×SD16×C37—22no—
49C2×SD16×C37—22no—
50C2×SD16×C37—22no—
51C2×D8×C37—22no—
52C2×M16×C37—22no—
53C22×C8×C37—22no—
54C4⋊Q8×C37—21no—
55C4⋊Q8×C37—21no—
56C4⋊Q8×C37—21no—
57C4⋊Q8×C37D411yescharacteristic
58C4⋊Q8×C37C2311yescharacteristic
59C4⋊Q8×C37C2311yescharacteristic
60C4⋊Q8×C37C2311yescharacteristic
61#127×C37D411yescharacteristic
62#127×C37C2311yescharacteristic
63#124×C37—21no—
64#124×C37—21no—
65#121×C37—21no—
66#121×C37—21no—
67#121×C37—21no—
68#121×C37D412yes—
69C4×D4×C37—21no—
70C4×D4×C37—21no—
71C4×D4×C37—21no—
72C4×D4×C37C2312yes—
73C2×C4⋊C4×C37—21no—
74C2×C4⋊C4×C37—21no—
75C2×C42×C37C2311yescharacteristic
76#107×C37—22no—
77#107×C37—22no—
78C4⋊C8×C37D412yes—
79C4⋊C8×C37C2312yes—
80C4⋊C8×C37C2312yes—
81C4⋊C8×C37C2312yes—
82#103×C37—22no—
83#103×C37—22no—
84#103×C37—22no—
85#102×C37—22no—
86#102×C37—22no—
87#102×C37—22no—
88C22⋊C8×C37—21no—
89C22⋊C8×C37—21no—
90C22⋊C8×C37—21no—
91C22⋊C8×C37—22no—
92C22⋊C8×C37—22no—
93C22⋊C8×C37D412yes—
94#97×C37—21no—
95#97×C37—21no—
96C4×C8×C37—21no—
97C4×C8×C37C2311yescharacteristic
98C4×C8×C37C2311yescharacteristic
99#41×C37—82no—
100#41×C37—82no—
101C2×Q8×C37—21no—
102C2×Q8×C37—21no—
103C2×Q8×C37—21no—
104C2×Q8×C37—21no—
105C2×Q8×C37—41no—
106C2×Q8×C37—82no—
107C2×D4×C37—22no—
108C2×D4×C37—21no—
109C2×D4×C37—21no—
110C2×D4×C37—21no—
111C2×D4×C37—21no—
112C2×D4×C37—21no—
113C2×D4×C37—41no—
114C2×D4×C37C2×D412yes—
115C22×C4×C37—22no—
116C22×C4×C37—41no—
117C22×C4×C37—41no—
118C22×C4×C37C2×D411yescharacteristic
119C22×C4×C37C2×D411yescharacteristic
120C22×C4×C37#4111yescharacteristic
121SD16×C37—82no—
122SD16×C37—82no—
123SD16×C37—82no—
124D8×C37—82no—
125M16×C37—82no—
126C2×C8×C37—22no—
127C2×C8×C37—22no—
128C2×C8×C37—22no—
129C2×C8×C37—82no—
130C2×C8×C37C2×D412yes—
131C2×C8×C37C2×D412yes—
132C4⋊C4×C37—21no—
133C4⋊C4×C37—21no—
134C4⋊C4×C37—21no—
135C4⋊C4×C37—21no—
136C4⋊C4×C37—21no—
137C4⋊C4×C37—21no—
138C4⋊C4×C37—21no—
139C4⋊C4×C37—41no—
140C4⋊C4×C37—41no—
141C4⋊C4×C37C2×D412yes—
142C22⋊C4×C37—22no—
143C22⋊C4×C37—22no—
144C22⋊C4×C37—41no—
145C22⋊C4×C37—41no—
146C42×C37C2×D411yescharacteristic
147C42×C37C2×D411yescharacteristic
148C42×C37#4111yescharacteristic
149C42×C37C2411yescharacteristic
150C23×C37—22no—
151C23×C37—41no—
152C23×C37#12111yescharacteristic
153Q8×C37—42no—
154Q8×C37—42no—
155Q8×C37—42no—
156Q8×C37—81no—
157Q8×C37—82no—
158D4×C37—22no—
159D4×C37—22no—
160D4×C37—42no—
161D4×C37—42no—
162D4×C37—81no—
163D4×C37—81no—
164D4×C37—81no—
165D4×C37—82no—
166D4×C37—82no—
167C2×C4×C37—22no—
168C2×C4×C37—22no—
169C2×C4×C37—22no—
170C2×C4×C37—41no—
171C2×C4×C37—42no—
172C2×C4×C37—42no—
173C2×C4×C37—41no—
174C2×C4×C37—41no—
175C2×C4×C37—41no—
176C2×C4×C37—82no—
177C2×C4×C37—82no—
178C2×C4×C37#12111yescharacteristic
179C2×C4×C37#12111yescharacteristic
180C2×C4×C37#12111yescharacteristic
181C2×C4×C37C22×D411yescharacteristic
182C2×C4×C37C22×D411yescharacteristic
183C2×C4×C37C2×#4111yescharacteristic
184C8×C37—22no—
185C8×C37—22no—
186C8×C37—42no—
187C8×C37—42no—
188C8×C37—42no—
189#19514C3711yesmaximal, characteristic, direct product factor, Sylow
190C22×C37—21no—
191C22×C37—21no—
192C22×C37—21no—
193C22×C37—42no—
194C22×C37—42no—
195C22×C37—42no—
196C22×C37—81no—
197C22×C37—81no—
198C22×C37—81no—
199C22×C37C2×#12111yescharacteristic
200C4×C37—21no—
201C4×C37—21no—
202C4×C37—21no—
203C4×C37—21no—
204C4×C37—42no—
205C4×C37—81no—
206C4×C37—81no—
207C4×C37—81no—
208C4×C37#57612yes—
209C4×C37#57712yes—
210#3226C2×C3711yescharacteristic, complemented
211#3214C2×C3711yescharacteristic
212#3206C2×C3711yescharacteristic, complemented
213#2981C2×C3711yescharacteristic, complemented
214#1701C2×C3712yescomplemented
215#1693C2×C3712yescomplemented
216#1681C2×C3712yescomplemented
217#1673C2×C3712yescomplemented
218#1572C2×C3711yescharacteristic, complemented
219#1556C2×C3711yescharacteristic, complemented
220#1553C2×C3711yescharacteristic, complemented
221C2×C37—41no—
222C2×C37—82no—
223C2×C37—161no—
224C2×C37#304211yescharacteristic
225C2×C37#305811yescharacteristic
226C2×C37#306911yescharacteristic
227#532C22×C3711yescharacteristic
228#532C22×C3711yescharacteristic
229C2×C4⋊Q8C22×C3711yescharacteristic
230#500C22×C3711yescharacteristic
231#500C22×C3711yescharacteristic
232#496—21no—
233#492—21no—
234#489—21no—
235#487—21no—
236#486C22×C3711yescharacteristic
237#486C22×C3711yescharacteristic
238#480—22no—
239#474—22no—
240#461C22×C3712yes—
241#461C22×C3712yes—
242#461C22×C3712yes—
243#459C22×C3712yes—
244#449—22no—
245#448—22no—
246#436C22×C3712yes—
247#435—22no—
248C8×D4C22×C3712yes—
249#432C22×C3711yescharacteristic
250#432C22×C3711yescharacteristic
251#431C22×C3711yescharacteristic
252#431C22×C3711yescharacteristic
253#408—22no—
254#335C22×C3712yes—
255#334C22×C3712yes—
256#332C22×C3712yes—
257#332C22×C3712yes—
258#332C22×C3712yes—
259#331C22×C3712yes—
260C37#1951411yescharacteristic, direct product factor, Sylow
261C2×#41—21no—
262C2×#41—21no—
263C22×Q8—21no—
264C2×SD16—22no—
265C2×SD16—22no—
266C2×SD16—22no—
267C2×D8—22no—
268C2×M16—22no—
269C22×C8—22no—
270C4⋊Q8—21no—
271C4⋊Q8—21no—
272C4⋊Q8—21no—
273C4⋊Q8D4×C3711yescharacteristic
274C4⋊Q8C23×C3711yescharacteristic
275C4⋊Q8C23×C3711yescharacteristic
276C4⋊Q8C23×C3711yescharacteristic
277#127D4×C3711yescharacteristic
278#127C23×C3711yescharacteristic
279#124—21no—
280#124—21no—
281#121—21no—
282#121—21no—
283#121—21no—
284#121D4×C3712yes—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4C23×C3712yes—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C42C23×C3711yescharacteristic
292#107—22no—
293#107—22no—
294C4⋊C8D4×C3712yes—
295C4⋊C8C23×C3712yes—
296C4⋊C8C23×C3712yes—
297C4⋊C8C23×C3712yes—
298#103—22no—
299#103—22no—
300#103—22no—
301#102—22no—
302#102—22no—
303#102—22no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306C22⋊C8—21no—
307C22⋊C8—22no—
308C22⋊C8—22no—
309C22⋊C8D4×C3712yes—
310#97—21no—
311#97—21no—
312C4×C8—21no—
313C4×C8C23×C3711yescharacteristic
314C4×C8C23×C3711yescharacteristic
315#41—82no—
316#41—82no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—41no—
322C2×Q8—82no—
323C2×D4—22no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—41no—
330C2×D4C2×D4×C3712yes—
331C22×C4—22no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4C2×D4×C3711yescharacteristic
335C22×C4C2×D4×C3711yescharacteristic
336C22×C4#41×C3711yescharacteristic
337SD16—82no—
338SD16—82no—
339SD16—82no—
340D8—82no—
341M16—82no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—82no—
346C2×C8C2×D4×C3712yes—
347C2×C8C2×D4×C3712yes—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4C2×D4×C3712yes—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C42C2×D4×C3711yescharacteristic
363C42C2×D4×C3711yescharacteristic
364C42#41×C3711yescharacteristic
365C42C24×C3711yescharacteristic
366C23—22no—
367C23—41no—
368C23#121×C3711yescharacteristic
369Q8—42no—
370Q8—42no—
371Q8—42no—
372Q8—81no—
373Q8—82no—
374D4—22no—
375D4—22no—
376D4—42no—
377D4—42no—
378D4—81no—
379D4—81no—
380D4—81no—
381D4—82no—
382D4—82no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4#121×C3711yescharacteristic
395C2×C4#121×C3711yescharacteristic
396C2×C4#121×C3711yescharacteristic
397C2×C4C22×D4×C3711yescharacteristic
398C2×C4C22×D4×C3711yescharacteristic
399C2×C4C2×#41×C3711yescharacteristic
400C8—22no—
401C8—22no—
402C8—42no—
403C8—42no—
404C8—42no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22C2×#121×C3711yescharacteristic
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—42no—
420C4—81no—
421C4—81no—
422C4—81no—
423C4C37×#57612yes—
424C4C37×#57712yes—
425C2—41no—
426C2—82no—
427C2—161no—
428C2C37×#304211yescharacteristic
429C2C37×#305811yescharacteristic
430C2C37×#306911yescharacteristic
subgroupG/H
center, socleC22×C37C2×#121
commutator, Frattini, cosocleC42C24×C37
prime radicalC37×#3214C2
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×#3226C2semidirect
3. C37×#3206C2semidirect
4. C37×#2981C2semidirect
5. C37×#1701C2semidirect
6. C37×#1693C2semidirect
7. C37×#1681C2semidirect
8. C37×#1673C2semidirect
9. C37×#1572C2semidirect
10. C37×#1556C2semidirect
11. C37×#1553C2semidirect
189. #19514C37direct (N × H)
210. #3226C2×C37semidirect
212. #3206C2×C37semidirect
213. #2981C2×C37semidirect
214. #1701C2×C37semidirect
215. #1693C2×C37semidirect
216. #1681C2×C37semidirect
217. #1673C2×C37semidirect
218. #1572C2×C37semidirect
219. #1556C2×C37semidirect
220. #1553C2×C37semidirect
260. C37#19514direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#32262
C37×#32142
C37×#32062
C37×#29812
C37×#17012
C37×#16932
C37×#16812
C37×#16732
C37×#15722
C37×#15562
C37×#15532
#1951437