Finite Group Databasebeta

← C37×#36700 #22657953

Subgroups — C37×#36700 #22657953

Order 9472.

1356 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#2839C214yesmaximal, complemented
2C37×#2836C214yesmaximal
3C37×#2833C214yesmaximal, complemented
4C37×#2831C214yesmaximal
5C37×#2402C211yesmaximal, characteristic, complemented
6C37×#2402C214yesmaximal, complemented
7C37×#2401C214yesmaximal
8C37×#2399C211yesmaximal, characteristic, complemented
9C37×#2396C212yesmaximal
10C37×#2298C211yesmaximal, characteristic
11C37×#2295C212yesmaximal, complemented
12C2×C4×Q8×C37C2212yes—
13C2×#117×C37C2211yescharacteristic
14C37×#401C2214yes—
15C37×#400C2212yes—
16C37×#400C2214yes—
17C37×#400C2214yes—
18C37×#400C2214yes—
19C37×#400C2214yes—
20C37×#400C2214yes—
21C37×#400C2214yes—
22C37×#400C2214yes—
23C37×#400C2214yes—
24C37×#399C2212yes—
25C37×#398C2214yes—
26C37×#398C2214yes—
27C37×#398C2214yes—
28C37×#398C2214yes—
29C37×#398C2214yes—
30C37×#398C2214yes—
31C37×#398C2214yes—
32C37×#397C2214yes—
33C37×#397C2214yes—
34C37×#397C2214yes—
35C37×#396C2214yes—
36C37×#396C2214yes—
37C37×#395C2212yes—
38C37×#395C2214yes—
39C37×#393C2212yes—
40C37×#391C2214yes—
41C37×#389C2214yes—
42C37×#388C2214yes—
43C37×#387C2214yes—
44C37×#387C2214yes—
45C37×#387C2214yes—
46C4×C4⋊C4×C37C2214yes—
47C4×C4⋊C4×C37C2212yes—
48C4×C4⋊C4×C37C2214yes—
49C4×C4⋊C4×C37C2214yes—
50C4×C4⋊C4×C37C2214yes—
51C4×C4⋊C4×C37C2212yes—
52C4×C22⋊C4×C37C2214yes—
53C4×C22⋊C4×C37C2212yes—
54C37×#376C2211yescharacteristic
55C37×#376C2214yes—
56C43×C37C2211yescharacteristic
57C22×Q8×C37C2312yes—
58C23×C4×C37C2311yescharacteristic
59C4×Q8×C37—28no—
60#117×C37—24no—
61C2×C4⋊C4×C37C2314yes—
62C2×C4⋊C4×C37C2312yes—
63C2×C4⋊C4×C37C2312yes—
64C2×C4⋊C4×C37C2314yes—
65C2×C4⋊C4×C37C2314yes—
66C2×C4⋊C4×C37C2314yes—
67C2×C4⋊C4×C37C2314yes—
68C2×C4⋊C4×C37C2314yes—
69C2×C4⋊C4×C37C2314yes—
70C2×C4⋊C4×C37C2314yes—
71C2×C4⋊C4×C37C2314yes—
72C2×C4⋊C4×C37C2314yes—
73C2×C4⋊C4×C37C2314yes—
74C2×C22⋊C4×C37C2312yes—
75C2×C22⋊C4×C37C2314yes—
76C2×C22⋊C4×C37C2314yes—
77C2×C22⋊C4×C37C2314yes—
78C2×C42×C37C2312yes—
79C2×C42×C37C2312yes—
80C2×C42×C37C2314yes—
81C2×C42×C37C2314yes—
82C2×C42×C37C2311yescharacteristic
83C2×C42×C37C2314yes—
84C2×C42×C37C2311yescharacteristic
85#95×C37C2314yes—
86#95×C37C2314yes—
87#95×C37C2314yes—
88#95×C37C2314yes—
89#95×C37C2314yes—
90#95×C37C2314yes—
91#95×C37C2314yes—
92#95×C37C2314yes—
93#95×C37C2312yes—
94#95×C37C2312yes—
95#95×C37C2314yes—
96#95×C37C2314yes—
97#95×C37C2314yes—
98#95×C37C2314yes—
99#95×C37C2314yes—
100#95×C37C2314yes—
101#95×C37C2314yes—
102#95×C37C2314yes—
103#95×C37C2314yes—
104C24×C37C2411yescharacteristic
105C2×Q8×C37—24no—
106C2×Q8×C37—44no—
107C22×C4×C37—22no—
108C22×C4×C37—24no—
109C22×C4×C37C2411yescharacteristic
110C22×C4×C37C2414yes—
111C22×C4×C37C2414yes—
112C22×C4×C37C2414yes—
113C22×C4×C37C2414yes—
114C22×C4×C37C2414yes—
115C22×C4×C37C2412yes—
116C22×C4×C37C2414yes—
117C22×C4×C37C2412yes—
118C22×C4×C37C2411yescharacteristic
119C4⋊C4×C37—24no—
120C4⋊C4×C37—24no—
121C4⋊C4×C37—28no—
122C4⋊C4×C37—44no—
123C4⋊C4×C37—44no—
124C4⋊C4×C37—44no—
125C4⋊C4×C37—44no—
126C4⋊C4×C37—44no—
127C4⋊C4×C37—44no—
128C4⋊C4×C37—44no—
129C4⋊C4×C37—44no—
130C4⋊C4×C37—44no—
131C4⋊C4×C37—44no—
132C22⋊C4×C37—24no—
133C22⋊C4×C37—44no—
134C22⋊C4×C37—44no—
135C22⋊C4×C37—44no—
136C42×C37—22no—
137C42×C37—28no—
138C42×C37—24no—
139C42×C37—24no—
140C42×C37—28no—
141C42×C37—44no—
142C42×C37#4114yes—
143C23×C37—21no—
144C23×C37—22no—
145C23×C37—24no—
146C23×C37C2511yescharacteristic
147Q8×C37—48no—
148C2×C4×C37—24no—
149C2×C4×C37—22no—
150C2×C4×C37—24no—
151C2×C4×C37—24no—
152C2×C4×C37—21no—
153C2×C4×C37—24no—
154C2×C4×C37—24no—
155C2×C4×C37—24no—
156C2×C4×C37—24no—
157C2×C4×C37—22no—
158C2×C4×C37—24no—
159C2×C4×C37—24no—
160C2×C4×C37—24no—
161C2×C4×C37—24no—
162C2×C4×C37—24no—
163C2×C4×C37—24no—
164C2×C4×C37—24no—
165C2×C4×C37—24no—
166C2×C4×C37—24no—
167C2×C4×C37—24no—
168C2×C4×C37—24no—
169C2×C4×C37—24no—
170C2×C4×C37—24no—
171C2×C4×C37—44no—
172C2×C4×C37C2×#4114yes—
173C2×C4×C37C2×#4112yes—
174C2×C4×C372−1+414yes—
175#36700C3711yesmaximal, characteristic, direct product factor, Sylow
176C22×C37—41no—
177C22×C37—42no—
178C22×C37—44no—
179C22×C37C22×#4111yescharacteristic
180C22×C37C2×2−1+412yes—
181C22×C37#58514yes—
182C4×C37—24no—
183C4×C37—41no—
184C4×C37—42no—
185C4×C37—44no—
186C4×C37—44no—
187C4×C37—44no—
188C4×C37—44no—
189C4×C37—44no—
190C4×C37—44no—
191C4×C37#54114yes—
192#2839C2×C3714yescomplemented
193#2836C2×C3714yes—
194#2833C2×C3714yescomplemented
195#2831C2×C3714yes—
196#2402C2×C3711yescharacteristic, complemented
197#2402C2×C3714yescomplemented
198#2401C2×C3714yes—
199#2399C2×C3711yescharacteristic, complemented
200#2396C2×C3712yes—
201#2298C2×C3711yescharacteristic
202#2295C2×C3712yescomplemented
203C2×C37—81no—
204C2×C37C2×#54111yescharacteristic
205C2×C37#345512yes—
206C2×C37#353114yes—
207C2×C4×Q8C22×C3712yes—
208C2×#117C22×C3711yescharacteristic
209#401C22×C3714yes—
210#400C22×C3714yes—
211#400C22×C3714yes—
212#400C22×C3714yes—
213#400C22×C3714yes—
214#400C22×C3714yes—
215#400C22×C3714yes—
216#400C22×C3714yes—
217#400C22×C3712yes—
218#400C22×C3714yes—
219#399C22×C3712yes—
220#398C22×C3714yes—
221#398C22×C3714yes—
222#398C22×C3714yes—
223#398C22×C3714yes—
224#398C22×C3714yes—
225#398C22×C3714yes—
226#398C22×C3714yes—
227#397C22×C3714yes—
228#397C22×C3714yes—
229#397C22×C3714yes—
230#396C22×C3714yes—
231#396C22×C3714yes—
232#395C22×C3714yes—
233#395C22×C3712yes—
234#393C22×C3712yes—
235#391C22×C3714yes—
236#389C22×C3714yes—
237#388C22×C3714yes—
238#387C22×C3714yes—
239#387C22×C3714yes—
240#387C22×C3714yes—
241C4×C4⋊C4C22×C3714yes—
242C4×C4⋊C4C22×C3712yes—
243C4×C4⋊C4C22×C3714yes—
244C4×C4⋊C4C22×C3714yes—
245C4×C4⋊C4C22×C3714yes—
246C4×C4⋊C4C22×C3712yes—
247C4×C22⋊C4C22×C3714yes—
248C4×C22⋊C4C22×C3712yes—
249#376C22×C3711yescharacteristic
250#376C22×C3714yes—
251C43C22×C3711yescharacteristic
252C37#3670011yescharacteristic, direct product factor, Sylow
253C22×Q8C23×C3712yes—
254C23×C4C23×C3711yescharacteristic
255C4×Q8—28no—
256#117—24no—
257C2×C4⋊C4C23×C3712yes—
258C2×C4⋊C4C23×C3714yes—
259C2×C4⋊C4C23×C3712yes—
260C2×C4⋊C4C23×C3714yes—
261C2×C4⋊C4C23×C3714yes—
262C2×C4⋊C4C23×C3714yes—
263C2×C4⋊C4C23×C3714yes—
264C2×C4⋊C4C23×C3714yes—
265C2×C4⋊C4C23×C3714yes—
266C2×C4⋊C4C23×C3714yes—
267C2×C4⋊C4C23×C3714yes—
268C2×C4⋊C4C23×C3714yes—
269C2×C4⋊C4C23×C3714yes—
270C2×C22⋊C4C23×C3712yes—
271C2×C22⋊C4C23×C3714yes—
272C2×C22⋊C4C23×C3714yes—
273C2×C22⋊C4C23×C3714yes—
274C2×C42C23×C3711yescharacteristic
275C2×C42C23×C3714yes—
276C2×C42C23×C3712yes—
277C2×C42C23×C3712yes—
278C2×C42C23×C3714yes—
279C2×C42C23×C3714yes—
280C2×C42C23×C3711yescharacteristic
281#95C23×C3714yes—
282#95C23×C3714yes—
283#95C23×C3714yes—
284#95C23×C3714yes—
285#95C23×C3714yes—
286#95C23×C3714yes—
287#95C23×C3714yes—
288#95C23×C3714yes—
289#95C23×C3712yes—
290#95C23×C3712yes—
291#95C23×C3714yes—
292#95C23×C3714yes—
293#95C23×C3714yes—
294#95C23×C3714yes—
295#95C23×C3714yes—
296#95C23×C3714yes—
297#95C23×C3714yes—
298#95C23×C3714yes—
299#95C23×C3714yes—
300C24C24×C3711yescharacteristic
301C2×Q8—24no—
302C2×Q8—44no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4C24×C3711yescharacteristic
306C22×C4C24×C3714yes—
307C22×C4C24×C3714yes—
308C22×C4C24×C3714yes—
309C22×C4C24×C3714yes—
310C22×C4C24×C3714yes—
311C22×C4C24×C3712yes—
312C22×C4C24×C3714yes—
313C22×C4C24×C3712yes—
314C22×C4C24×C3711yescharacteristic
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—28no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C22⋊C4—24no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C42—22no—
333C42—28no—
334C42—24no—
335C42—24no—
336C42—28no—
337C42—44no—
338C42#41×C3714yes—
339C23—21no—
340C23—22no—
341C23—24no—
342C23C25×C3711yescharacteristic
343Q8—48no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—21no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—44no—
368C2×C4C2×#41×C3714yes—
369C2×C4C2×#41×C3712yes—
370C2×C42−1+4×C3714yes—
371C22—41no—
372C22—42no—
373C22—44no—
374C22C22×#41×C3711yescharacteristic
375C22C2×2−1+4×C3712yes—
376C22C37×#58514yes—
377C4—24no—
378C4—41no—
379C4—44no—
380C4—42no—
381C4—44no—
382C4—44no—
383C4—44no—
384C4—44no—
385C4—44no—
386C4C37×#54114yes—
387C2—81no—
388C2C2×C37×#54111yescharacteristic
389C2C37×#345512yes—
390C2C37×#353114yes—
subgroupG/H
centerC22×C4×C37C24
commutator, Frattini, cosocleC23C25×C37
socleC23×C37C25
prime radicalC24×C37C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C37 ◁ G

Complements

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

NH ≅ G/Ntype
1. C37×#2839C2semidirect
3. C37×#2833C2semidirect
5. C37×#2402C2semidirect
6. C37×#2402C2semidirect
8. C37×#2399C2semidirect
11. C37×#2295C2semidirect
175. #36700C37direct (N × H)
192. #2839C2×C37semidirect
194. #2833C2×C37semidirect
196. #2402C2×C37semidirect
197. #2402C2×C37semidirect
199. #2399C2×C37semidirect
202. #2295C2×C37semidirect
252. C37#36700direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#28392
C37×#28362
C37×#28332
C37×#28312
C37×#24022
C37×#24012
C37×#23992
C37×#23962
C37×#22982
C37×#22952
#3670037