Finite Group Databasebeta

← C37×#9265 #22630518

Subgroups — C37×#9265 #22630518

Order 9472.

438 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3211C211yesmaximal, characteristic
2C37×#2968C211yesmaximal, characteristic, complemented
3C37×#1532C211yesmaximal, characteristic, complemented
4C2×C37×#329C211yesmaximal, characteristic, complemented
5C37×#1469C211yesmaximal, characteristic, complemented
6C37×#1302C211yesmaximal, characteristic, complemented
7C37×#1281C211yesmaximal, characteristic, complemented
8C37×#529C2211yescharacteristic
9C2×#124×C37C2211yescharacteristic
10C37×#495—21no—
11C37×#488—21no—
12C37×#464—21no—
13C37×#463—21no—
14C37×#451C411yescharacteristic
15C37×#450C411yescharacteristic
16C37×#448C411yescharacteristic
17C37×#447C411yescharacteristic
18C37×#446—21no—
19C37×#435—21no—
20C37×#435—21no—
21C8×D4×C37—21no—
22C37×#433—21no—
23C37×#432C2211yescharacteristic
24C37×#431C2211yescharacteristic
25C37×#431C2211yescharacteristic
26C37×#405—21no—
27C2×#97×C37C2211yescharacteristic
28C2×#97×C37C2211yescharacteristic
29C37×#330—21no—
30C37×#329—21no—
31C37×#329—21no—
32C37×#329—21no—
33C37×#323—21no—
34C37×#323—21no—
35C22×Q8×C37C2×C411yescharacteristic
36C22×D4×C37C2×C411yescharacteristic
37C2×Q16×C37—21no—
38C2×SD16×C37—21no—
39C2×SD16×C37—21no—
40C2×D8×C37—21no—
41C2×M16×C37—21no—
42C22×C8×C37—21no—
43C22×C8×C37—21no—
44C22×C8×C37—21no—
45#126×C37—21no—
46#125×C37D411yescharacteristic
47#124×C37—21no—
48#124×C37—21no—
49#124×C37—21no—
50#124×C37—21no—
51#124×C37D411yescharacteristic
52#123×C37—21no—
53C22⋊Q8×C37C2×C411yescharacteristic
54#121×C37C2×C411yescharacteristic
55C4×Q8×C37D411yescharacteristic
56C4×D4×C37—21no—
57C4×D4×C37D411yescharacteristic
58#117×C37—21no—
59C2×C22⋊C4×C37—21no—
60C2×C22⋊C4×C37—21no—
61C2×C42×C37C2311yescharacteristic
62C4⋊C8×C37—21no—
63C4⋊C8×C37—21no—
64C4⋊C8×C37—21no—
65C4⋊C8×C37—21no—
66C4⋊C8×C37—21no—
67#103×C37—21no—
68#103×C37—21no—
69#102×C37—21no—
70#102×C37—21no—
71C22⋊C8×C37—21no—
72C22⋊C8×C37—21no—
73C22⋊C8×C37—21no—
74C22⋊C8×C37C2×C411yescharacteristic
75C22⋊C8×C37C2×C411yescharacteristic
76C22⋊C8×C37D411yescharacteristic
77C22⋊C8×C37D411yescharacteristic
78#97×C37—21no—
79#97×C37—21no—
80#97×C37—21no—
81#97×C37—21no—
82#97×C37—21no—
83#97×C37D411yescharacteristic
84#97×C37D411yescharacteristic
85C4×C8×C37—21no—
86C4×C8×C37—21no—
87C4×C8×C37—21no—
88C24×C37—21no—
89C2×Q8×C37—21no—
90C2×Q8×C37—21no—
91C2×Q8×C37—81no—
92C2×Q8×C37C22⋊C411yescharacteristic
93C2×D4×C37—21no—
94C2×D4×C37—21no—
95C2×D4×C37—21no—
96C2×D4×C37—81no—
97C2×D4×C37C22⋊C411yescharacteristic
98C22×C4×C37—21no—
99C22×C4×C37—21no—
100C22×C4×C37C22×C411yescharacteristic
101C22×C4×C37C2×D411yescharacteristic
102C22×C4×C37#4111yescharacteristic
103Q16×C37—81no—
104SD16×C37—81no—
105SD16×C37—81no—
106D8×C37—81no—
107M16×C37—42no—
108C2×C8×C37—21no—
109C2×C8×C37—22no—
110C2×C8×C37—21no—
111C2×C8×C37—21no—
112C2×C8×C37—21no—
113C2×C8×C37—21no—
114C2×C8×C37—21no—
115C2×C8×C37—42no—
116C2×C8×C37—42no—
117C2×C8×C37—41no—
118C2×C8×C37—41no—
119C2×C8×C37—41no—
120C4⋊C4×C37—21no—
121C4⋊C4×C37—21no—
122C4⋊C4×C37—21no—
123C4⋊C4×C37—21no—
124C4⋊C4×C37C22⋊C411yescharacteristic
125C4⋊C4×C37C22⋊C411yescharacteristic
126C22⋊C4×C37—21no—
127C22⋊C4×C37—21no—
128C22⋊C4×C37—21no—
129C22⋊C4×C37—21no—
130C22⋊C4×C37—21no—
131C22⋊C4×C37—41no—
132C22⋊C4×C37—41no—
133C22⋊C4×C37—41no—
134C22⋊C4×C37—41no—
135C42×C37—21no—
136C42×C37C2×D411yescharacteristic
137C42×C37C2×D411yescharacteristic
138C42×C37C2×D411yescharacteristic
139C42×C37#4111yescharacteristic
140C23×C37—21no—
141C23×C37—41no—
142C23×C37—41no—
143C23×C37—41no—
144C23×C37—81no—
145C23×C37—81no—
146C23×C37C4×D411yescharacteristic
147Q8×C37—41no—
148Q8×C37—41no—
149Q8×C37—41no—
150Q8×C37—81no—
151D4×C37—41no—
152D4×C37—41no—
153D4×C37—41no—
154D4×C37—41no—
155D4×C37—41no—
156D4×C37—81no—
157C2×C4×C37—21no—
158C2×C4×C37—21no—
159C2×C4×C37—21no—
160C2×C4×C37—21no—
161C2×C4×C37—21no—
162C2×C4×C37—41no—
163C2×C4×C37—41no—
164C2×C4×C37—41no—
165C2×C4×C37—41no—
166C2×C4×C37—41no—
167C2×C4×C37—81no—
168C2×C4×C37C2×C22⋊C411yescharacteristic
169C2×C4×C37C4×D411yescharacteristic
170C2×C4×C37#12011yescharacteristic
171C2×C4×C37#12111yescharacteristic
172C2×C4×C37#12111yescharacteristic
173C2×C4×C37#12311yescharacteristic
174C8×C37—22no—
175C8×C37—42no—
176C8×C37—41no—
177C8×C37—41no—
178C8×C37—41no—
179C8×C37—41no—
180C8×C37—41no—
181C8×C37—81no—
182#9265C3711yesmaximal, characteristic, direct product factor, Sylow
183C22×C37—21no—
184C22×C37—21no—
185C22×C37—41no—
186C22×C37—41no—
187C22×C37—41no—
188C22×C37—81no—
189C22×C37—81no—
190C22×C37—81no—
191C22×C37—161no—
192C22×C37—161no—
193C22×C37#38611yescharacteristic
194C22×C37#44311yescharacteristic
195C22×C37#44411yescharacteristic
196C4×C37—21no—
197C4×C37—21no—
198C4×C37—21no—
199C4×C37—21no—
200C4×C37—21no—
201C4×C37—21no—
202C4×C37—41no—
203C4×C37—41no—
204C4×C37—41no—
205C4×C37—81no—
206#3211C2×C3711yescharacteristic
207#2968C2×C3711yescharacteristic, complemented
208#1532C2×C3711yescharacteristic, complemented
209C2×#329C2×C3711yescharacteristic, complemented
210#1469C2×C3711yescharacteristic, complemented
211#1302C2×C3711yescharacteristic, complemented
212#1281C2×C3711yescharacteristic, complemented
213C2×C37—21no—
214C2×C37—21no—
215C2×C37—81no—
216C2×C37—161no—
217C2×C37#187611yescharacteristic
218C2×C37#190611yescharacteristic
219C2×C37#191411yescharacteristic
220#529C22×C3711yescharacteristic
221C2×#124C22×C3711yescharacteristic
222#495—21no—
223#488—21no—
224#464—21no—
225#463—21no—
226#451C4×C3711yescharacteristic
227#450C4×C3711yescharacteristic
228#448C4×C3711yescharacteristic
229#447C4×C3711yescharacteristic
230#446—21no—
231#435—21no—
232#435—21no—
233C8×D4—21no—
234#433—21no—
235#432C22×C3711yescharacteristic
236#431C22×C3711yescharacteristic
237#431C22×C3711yescharacteristic
238#405—21no—
239C2×#97C22×C3711yescharacteristic
240C2×#97C22×C3711yescharacteristic
241#330—21no—
242#329—21no—
243#329—21no—
244#329—21no—
245#323—21no—
246#323—21no—
247C37#926511yescharacteristic, direct product factor, Sylow
248C22×Q8C2×C4×C3711yescharacteristic
249C22×D4C2×C4×C3711yescharacteristic
250C2×Q16—21no—
251C2×SD16—21no—
252C2×SD16—21no—
253C2×D8—21no—
254C2×M16—21no—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8—21no—
258#126—21no—
259#125D4×C3711yescharacteristic
260#124—21no—
261#124—21no—
262#124—21no—
263#124—21no—
264#124D4×C3711yescharacteristic
265#123—21no—
266C22⋊Q8C2×C4×C3711yescharacteristic
267#121C2×C4×C3711yescharacteristic
268C4×Q8D4×C3711yescharacteristic
269C4×D4—21no—
270C4×D4D4×C3711yescharacteristic
271#117—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C42C23×C3711yescharacteristic
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280#103—21no—
281#103—21no—
282#102—21no—
283#102—21no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8C2×C4×C3711yescharacteristic
288C22⋊C8C2×C4×C3711yescharacteristic
289C22⋊C8D4×C3711yescharacteristic
290C22⋊C8D4×C3711yescharacteristic
291#97—21no—
292#97—21no—
293#97—21no—
294#97—21no—
295#97—21no—
296#97D4×C3711yescharacteristic
297#97D4×C3711yescharacteristic
298C4×C8—21no—
299C4×C8—21no—
300C4×C8—21no—
301C24—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—81no—
305C2×Q8C22⋊C4×C3711yescharacteristic
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—81no—
310C2×D4C22⋊C4×C3711yescharacteristic
311C22×C4—21no—
312C22×C4—21no—
313C22×C4C22×C4×C3711yescharacteristic
314C22×C4C2×D4×C3711yescharacteristic
315C22×C4#41×C3711yescharacteristic
316Q16—81no—
317SD16—81no—
318SD16—81no—
319D8—81no—
320M16—42no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4C22⋊C4×C3711yescharacteristic
338C4⋊C4C22⋊C4×C3711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C42—21no—
349C42C2×D4×C3711yescharacteristic
350C42C2×D4×C3711yescharacteristic
351C42C2×D4×C3711yescharacteristic
352C42#41×C3711yescharacteristic
353C23—21no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—81no—
358C23—81no—
359C23C4×D4×C3711yescharacteristic
360Q8—41no—
361Q8—41no—
362Q8—41no—
363Q8—81no—
364D4—41no—
365D4—41no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—81no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—81no—
381C2×C4C2×C22⋊C4×C3711yescharacteristic
382C2×C4C4×D4×C3711yescharacteristic
383C2×C4#120×C3711yescharacteristic
384C2×C4#121×C3711yescharacteristic
385C2×C4#121×C3711yescharacteristic
386C2×C4#123×C3711yescharacteristic
387C8—22no—
388C8—42no—
389C8—41no—
390C8—41no—
391C8—41no—
392C8—41no—
393C8—41no—
394C8—81no—
395C22—21no—
396C22—21no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—161no—
404C22—161no—
405C22C37×#38611yescharacteristic
406C22C37×#44311yescharacteristic
407C22C37×#44411yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—81no—
418C2—21no—
419C2—21no—
420C2—81no—
421C2—161no—
422C2C37×#187611yescharacteristic
423C2C37×#190611yescharacteristic
424C2C37×#191411yescharacteristic
subgroupG/H
center, socleC22×C37#386
commutatorC22×C4C22×C4×C37
Frattini, cosocleC2×C42C23×C37
prime radicalC37×#447C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C37 ◁ C2×C42×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C37×#2968C2semidirect
3. C37×#1532C2semidirect
4. C2×C37×#329C2semidirect
5. C37×#1469C2semidirect
6. C37×#1302C2semidirect
7. C37×#1281C2semidirect
182. #9265C37direct (N × H)
207. #2968C2×C37semidirect
208. #1532C2×C37semidirect
209. C2×#329C2×C37semidirect
210. #1469C2×C37semidirect
211. #1302C2×C37semidirect
212. #1281C2×C37semidirect
247. C37#9265direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#32112
C37×#29682
C37×#15322
C2×C37×#3292
C37×#14692
C37×#13022
C37×#12812
#926537