Finite Group Databasebeta

← C37×#62002 #22683255

Subgroups — C37×#62002 #22683255

Order 9472.

1872 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3503C211yesmaximal, characteristic, complemented
2C37×#3483C211yesmaximal, characteristic
3C37×#3392C212yesmaximal, complemented
4C37×#3363C212yesmaximal, complemented
5C37×#3329C216yesmaximal, complemented
6C37×#3278C216yesmaximal, complemented
7C37×#3254C213yesmaximal, complemented
8C37×#3206C213yesmaximal, complemented
9C37×#3107C213yesmaximal, complemented
10C37×#3039C213yesmaximal, complemented
11C37×#2991C211yesmaximal, characteristic, complemented
12C2×#137×C37C2216yes—
13Q82×C37—21no—
14Q82×C37C2211yescharacteristic
15C37×#557C2212yes—
16C37×#555—23no—
17C37×#555C2213yes—
18C37×#554—23no—
19C37×#550—21no—
20C37×#550C2213yes—
21C37×#550C2211yescharacteristic
22D4×Q8×C37C2216yes—
23C37×#544—23no—
24C37×#533C2213yes—
25C37×#532—23no—
26C37×#532C2213yes—
27C2×C4⋊Q8×C37C2213yes—
28C37×#520—21no—
29C4×#41×C37—21no—
30C4×#41×C37C2213yes—
31C4×#41×C37C2211yescharacteristic
32C37×#492C2216yes—
33C37×#487C2216yes—
34C37×#485C2216yes—
35C37×#477C2216yes—
36C37×#475C2211yescharacteristic
37C37×#474C2213yes—
38C37×#468C22112yes—
39C37×#462C2216yes—
40C37×#462C2213yes—
41C37×#461C2216yes—
42C37×#461C2213yes—
43C37×#461C2216yes—
44C37×#460C2216yes—
45C37×#452C2216yes—
46C37×#448C22112yes—
47C8×Q8×C37C2212yes—
48C4×Q16×C37C2216yes—
49C4×SD16×C37C2216yes—
50C37×#436C2216yes—
51C37×#431C2213yes—
52C37×#423C2213yes—
53C37×#419C2216yes—
54C2×#41×C37—21no—
55C2×#41×C37—23no—
56C2×#41×C37D411yescharacteristic
57C2×#41×C37C2313yes—
58C22×Q8×C37C2316yes—
59#137×C37—412no—
60C2×Q16×C37C2316yes—
61C2×SD16×C37C23112yes—
62C2×SD16×C37C2316yes—
63C2×M16×C37C2316yes—
64C4⋊Q8×C37—23no—
65C4⋊Q8×C37—26no—
66C4⋊Q8×C37—23no—
67C4⋊Q8×C37—23no—
68C4⋊Q8×C37C2313yes—
69C4⋊Q8×C37C2316yes—
70C4⋊Q8×C37C2313yes—
71C4⋊Q8×C37C2311yescharacteristic
72#127×C37—23no—
73#127×C37C2313yes—
74#125×C37—26no—
75#125×C37—23no—
76#124×C37—26no—
77#124×C37—23no—
78C22⋊Q8×C37—23no—
79C22⋊Q8×C37—212no—
80C22⋊Q8×C37C2316yes—
81#121×C37—23no—
82#121×C37—23no—
83#121×C37—23no—
84#121×C37—26no—
85#121×C37C2316yes—
86C4×Q8×C37—22no—
87C4×Q8×C37—21no—
88C4×Q8×C37—23no—
89C4×Q8×C37D411yescharacteristic
90C4×Q8×C37C2316yes—
91C4×Q8×C37C2313yes—
92C4×Q8×C37C2311yescharacteristic
93C4×Q8×C37C2311yescharacteristic
94C4×D4×C37—26no—
95C4×D4×C37—23no—
96C4×D4×C37—23no—
97C4×D4×C37—23no—
98C4×D4×C37—23no—
99C4×D4×C37D413yes—
100C4×D4×C37C2313yes—
101C4×D4×C37C2313yes—
102C4×D4×C37C2313yes—
103#117×C37—23no—
104#117×C37—23no—
105#117×C37—23no—
106#117×C37D413yes—
107#117×C37C2313yes—
108C2×C4⋊C4×C37—23no—
109C2×C4⋊C4×C37—23no—
110C2×C42×C37—23no—
111C2×C42×C37C2313yes—
112#107×C37C2316yes—
113#106×C37C2312yes—
114C4⋊C8×C37C2313yes—
115C4⋊C8×C37C2316yes—
116C4⋊C8×C37C2311yescharacteristic
117#103×C37C2316yes—
118#103×C37C23112yes—
119#103×C37C2312yes—
120#103×C37C2316yes—
121#102×C37C2316yes—
122C22⋊C8×C37C2316yes—
123C4×C8×C37C2316yes—
124#41×C37—24no—
125#41×C37—26no—
126#41×C37—43no—
127#41×C37—44no—
128#41×C37—86no—
129C2×Q8×C37—21no—
130C2×Q8×C37—26no—
131C2×Q8×C37—23no—
132C2×Q8×C37—46no—
133C2×Q8×C37—46no—
134C2×Q8×C37C2×D411yescharacteristic
135C2×Q8×C37C2416yes—
136C2×Q8×C37C2411yescharacteristic
137C2×D4×C37—23no—
138C2×D4×C37—26no—
139C2×D4×C37—23no—
140C2×D4×C37—23no—
141C2×D4×C37C2×D413yes—
142C2×D4×C37C2413yes—
143C22×C4×C37—26no—
144C22×C4×C37—23no—
145C22×C4×C37—23no—
146C22×C4×C37—23no—
147C22×C4×C37C2×D413yes—
148C22×C4×C37C2×D413yes—
149C22×C4×C37C2413yes—
150Q16×C37—212no—
151SD16×C37—46no—
152SD16×C37—412no—
153M16×C37—46no—
154C2×C8×C37C2412yes—
155C2×C8×C37C2416yes—
156C4⋊C4×C37—23no—
157C4⋊C4×C37—23no—
158C4⋊C4×C37—26no—
159C4⋊C4×C37—26no—
160C4⋊C4×C37—21no—
161C4⋊C4×C37—21no—
162C4⋊C4×C37—23no—
163C4⋊C4×C37—23no—
164C4⋊C4×C37—26no—
165C4⋊C4×C37—26no—
166C4⋊C4×C37C2×D413yes—
167C4⋊C4×C37C2×D413yes—
168C4⋊C4×C37C2412yes—
169C4⋊C4×C37C2411yescharacteristic
170C4⋊C4×C37C2413yes—
171C22⋊C4×C37—26no—
172C22⋊C4×C37—23no—
173C22⋊C4×C37—26no—
174C22⋊C4×C37—23no—
175C22⋊C4×C37—23no—
176C22⋊C4×C37C2×D416yes—
177C42×C37—21no—
178C42×C37—23no—
179C42×C37—23no—
180C42×C37—26no—
181C42×C37C2×D413yes—
182C42×C37C2×D413yes—
183C42×C37C2413yes—
184C42×C37C2411yescharacteristic
185C23×C37—23no—
186C23×C37C22×D413yes—
187Q8×C37—21no—
188Q8×C37—26no—
189Q8×C37—26no—
190Q8×C37—42no—
191Q8×C37—86no—
192Q8×C37—83no—
193Q8×C37#13714yescomplemented
194Q8×C37H(5,𝔽2)12yes—
195D4×C37—26no—
196D4×C37—23no—
197D4×C37—23no—
198D4×C37—46no—
199D4×C37—46no—
200D4×C37—83no—
201D4×C37—86no—
202C2×C4×C37—21no—
203C2×C4×C37—26no—
204C2×C4×C37—26no—
205C2×C4×C37—23no—
206C2×C4×C37—23no—
207C2×C4×C37—21no—
208C2×C4×C37—22no—
209C2×C4×C37—23no—
210C2×C4×C37—26no—
211C2×C4×C37—46no—
212C2×C4×C37—46no—
213C2×C4×C37—46no—
214C2×C4×C37—86no—
215C2×C4×C37C22×D413yes—
216C2×C4×C37C22×D411yescharacteristic
217C2×C4×C37C22×D413yes—
218C2×C4×C37C22×D413yes—
219C2×C4×C37C22×D411yescharacteristic
220C2×C4×C37C2511yescharacteristic
221C8×C37—26no—
222C8×C37—22no—
223#62002C3711yesmaximal, characteristic, direct product factor, Sylow
224C22×C37—23no—
225C22×C37—23no—
226C22×C37—23no—
227C22×C37—43no—
228C22×C37—43no—
229C22×C37—43no—
230C22×C37C23×D411yescharacteristic
231C4×C37—23no—
232C4×C37—22no—
233C4×C37—21no—
234C4×C37—23no—
235C4×C37—46no—
236C4×C37—41no—
237C4×C37—43no—
238C4×C37—42no—
239C4×C37C2×#13716yes—
240C4×C37#57712yes—
241C4×C37C2×H(5,𝔽2)11yescharacteristic
242C4×C37#58511yescharacteristic
243#3503C2×C3711yescharacteristic, complemented
244#3483C2×C3711yescharacteristic
245#3392C2×C3712yescomplemented
246#3363C2×C3712yescomplemented
247#3329C2×C3716yescomplemented
248#3278C2×C3716yescomplemented
249#3254C2×C3713yescomplemented
250#3206C2×C3713yescomplemented
251#3107C2×C3713yescomplemented
252#3039C2×C3713yescomplemented
253#2991C2×C3711yescharacteristic, complemented
254C2×C37—43no—
255C2×C37—83no—
256C2×C37#348811yescharacteristic
257C2×C37C22×#13711yescharacteristic
258C2×C37C2×#57711yescharacteristic
259C2×#137C22×C3716yes—
260Q82—21no—
261Q82C22×C3711yescharacteristic
262#557C22×C3712yes—
263#555—23no—
264#555C22×C3713yes—
265#554—23no—
266#550—21no—
267#550C22×C3713yes—
268#550C22×C3711yescharacteristic
269D4×Q8C22×C3716yes—
270#544—23no—
271#533C22×C3713yes—
272#532—23no—
273#532C22×C3713yes—
274C2×C4⋊Q8C22×C3713yes—
275#520—21no—
276C4×#41—21no—
277C4×#41C22×C3713yes—
278C4×#41C22×C3711yescharacteristic
279#492C22×C3716yes—
280#487C22×C3716yes—
281#485C22×C3716yes—
282#477C22×C3716yes—
283#475C22×C3711yescharacteristic
284#474C22×C3713yes—
285#468C22×C37112yes—
286#462C22×C3713yes—
287#462C22×C3716yes—
288#461C22×C3716yes—
289#461C22×C3713yes—
290#461C22×C3716yes—
291#460C22×C3716yes—
292#452C22×C3716yes—
293#448C22×C37112yes—
294C8×Q8C22×C3712yes—
295C4×Q16C22×C3716yes—
296C4×SD16C22×C3716yes—
297#436C22×C3716yes—
298#431C22×C3713yes—
299#423C22×C3713yes—
300#419C22×C3716yes—
301C37#6200211yescharacteristic, direct product factor, Sylow
302C2×#41—21no—
303C2×#41—23no—
304C2×#41D4×C3711yescharacteristic
305C2×#41C23×C3713yes—
306C22×Q8C23×C3716yes—
307#137—412no—
308C2×Q16C23×C3716yes—
309C2×SD16C23×C37112yes—
310C2×SD16C23×C3716yes—
311C2×M16C23×C3716yes—
312C4⋊Q8—23no—
313C4⋊Q8—26no—
314C4⋊Q8—23no—
315C4⋊Q8—23no—
316C4⋊Q8C23×C3713yes—
317C4⋊Q8C23×C3716yes—
318C4⋊Q8C23×C3713yes—
319C4⋊Q8C23×C3711yescharacteristic
320#127—23no—
321#127C23×C3713yes—
322#125—26no—
323#125—23no—
324#124—26no—
325#124—23no—
326C22⋊Q8—23no—
327C22⋊Q8—212no—
328C22⋊Q8C23×C3716yes—
329#121—23no—
330#121—23no—
331#121—23no—
332#121—26no—
333#121C23×C3716yes—
334C4×Q8—21no—
335C4×Q8—22no—
336C4×Q8—23no—
337C4×Q8D4×C3711yescharacteristic
338C4×Q8C23×C3711yescharacteristic
339C4×Q8C23×C3713yes—
340C4×Q8C23×C3716yes—
341C4×Q8C23×C3711yescharacteristic
342C4×D4—26no—
343C4×D4—23no—
344C4×D4—23no—
345C4×D4—23no—
346C4×D4—23no—
347C4×D4D4×C3713yes—
348C4×D4C23×C3713yes—
349C4×D4C23×C3713yes—
350C4×D4C23×C3713yes—
351#117—23no—
352#117—23no—
353#117—23no—
354#117D4×C3713yes—
355#117C23×C3713yes—
356C2×C4⋊C4—23no—
357C2×C4⋊C4—23no—
358C2×C42—23no—
359C2×C42C23×C3713yes—
360#107C23×C3716yes—
361#106C23×C3712yes—
362C4⋊C8C23×C3713yes—
363C4⋊C8C23×C3716yes—
364C4⋊C8C23×C3711yescharacteristic
365#103C23×C3716yes—
366#103C23×C37112yes—
367#103C23×C3712yes—
368#103C23×C3716yes—
369#102C23×C3716yes—
370C22⋊C8C23×C3716yes—
371C4×C8C23×C3716yes—
372#41—24no—
373#41—26no—
374#41—43no—
375#41—44no—
376#41—86no—
377C2×Q8—21no—
378C2×Q8—26no—
379C2×Q8—23no—
380C2×Q8—46no—
381C2×Q8—46no—
382C2×Q8C2×D4×C3711yescharacteristic
383C2×Q8C24×C3716yes—
384C2×Q8C24×C3711yescharacteristic
385C2×D4—23no—
386C2×D4—26no—
387C2×D4—23no—
388C2×D4—23no—
389C2×D4C2×D4×C3713yes—
390C2×D4C24×C3713yes—
391C22×C4—26no—
392C22×C4—23no—
393C22×C4—23no—
394C22×C4—23no—
395C22×C4C2×D4×C3713yes—
396C22×C4C2×D4×C3713yes—
397C22×C4C24×C3713yes—
398Q16—212no—
399SD16—46no—
400SD16—412no—
401M16—46no—
402C2×C8C24×C3712yes—
403C2×C8C24×C3716yes—
404C4⋊C4—23no—
405C4⋊C4—23no—
406C4⋊C4—26no—
407C4⋊C4—26no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—23no—
411C4⋊C4—23no—
412C4⋊C4—26no—
413C4⋊C4—26no—
414C4⋊C4C2×D4×C3713yes—
415C4⋊C4C2×D4×C3713yes—
416C4⋊C4C24×C3712yes—
417C4⋊C4C24×C3711yescharacteristic
418C4⋊C4C24×C3713yes—
419C22⋊C4—26no—
420C22⋊C4—23no—
421C22⋊C4—26no—
422C22⋊C4—23no—
423C22⋊C4—23no—
424C22⋊C4C2×D4×C3716yes—
425C42—21no—
426C42—23no—
427C42—23no—
428C42—26no—
429C42C2×D4×C3713yes—
430C42C2×D4×C3713yes—
431C42C24×C3713yes—
432C42C24×C3711yescharacteristic
433C23—23no—
434C23C22×D4×C3713yes—
435Q8—21no—
436Q8—26no—
437Q8—26no—
438Q8—42no—
439Q8—86no—
440Q8—83no—
441Q8#137×C3714yescomplemented
442Q8H(5,𝔽2)×C3712yes—
443D4—26no—
444D4—23no—
445D4—23no—
446D4—46no—
447D4—46no—
448D4—83no—
449D4—86no—
450C2×C4—21no—
451C2×C4—26no—
452C2×C4—26no—
453C2×C4—23no—
454C2×C4—23no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—23no—
458C2×C4—26no—
459C2×C4—46no—
460C2×C4—46no—
461C2×C4—46no—
462C2×C4—86no—
463C2×C4C22×D4×C3713yes—
464C2×C4C22×D4×C3711yescharacteristic
465C2×C4C22×D4×C3713yes—
466C2×C4C22×D4×C3713yes—
467C2×C4C22×D4×C3711yescharacteristic
468C2×C4C25×C3711yescharacteristic
469C8—26no—
470C8—22no—
471C22—23no—
472C22—23no—
473C22—23no—
474C22—43no—
475C22—43no—
476C22—43no—
477C22C23×D4×C3711yescharacteristic
478C4—23no—
479C4—22no—
480C4—21no—
481C4—23no—
482C4—46no—
483C4—41no—
484C4—43no—
485C4—42no—
486C4C2×#137×C3716yes—
487C4C37×#57712yes—
488C4C2×H(5,𝔽2)×C3711yescharacteristic
489C4C37×#58511yescharacteristic
490C2—43no—
491C2—83no—
492C2C37×#348811yescharacteristic
493C2C22×#137×C3711yescharacteristic
494C2C2×C37×#57711yescharacteristic
subgroupG/H
center, socleC22×C37C23×D4
commutator, Frattini, cosocleC2×C4C25×C37
prime radicalC37×#3483C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C37 ◁ C4×#41×C37 ◁ G

Complements

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

NH ≅ G/Ntype
1. C37×#3503C2semidirect
3. C37×#3392C2semidirect
4. C37×#3363C2semidirect
5. C37×#3329C2semidirect
6. C37×#3278C2semidirect
7. C37×#3254C2semidirect
8. C37×#3206C2semidirect
9. C37×#3107C2semidirect
10. C37×#3039C2semidirect
11. C37×#2991C2semidirect
193. Q8×C37#137semidirect
223. #62002C37direct (N × H)
243. #3503C2×C37semidirect
245. #3392C2×C37semidirect
246. #3363C2×C37semidirect
247. #3329C2×C37semidirect
248. #3278C2×C37semidirect
249. #3254C2×C37semidirect
250. #3206C2×C37semidirect
251. #3107C2×C37semidirect
252. #3039C2×C37semidirect
253. #2991C2×C37semidirect
301. C37#62002direct (N × H)
441. Q8#137×C37semidirect

Maximal subgroups

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

subgroupindex
C37×#35032
C37×#34832
C37×#33922
C37×#33632
C37×#33292
C37×#32782
C37×#32542
C37×#32062
C37×#31072
C37×#30392
C37×#29912
#6200237