Finite Group Databasebeta

← C37×#33646 #22654899

Subgroups — C37×#33646 #22654899

Order 9472.

628 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×#3050C211yesmaximal, characteristic
2C22×#107×C37C211yesmaximal, characteristic, complemented
3C37×#2236C212yesmaximal, complemented
4C37×#2235C212yesmaximal, complemented
5C37×#2233C212yesmaximal, complemented
6C37×#2232C212yesmaximal, complemented
7C37×#2159C212yesmaximal, complemented
8C37×#2139C212yesmaximal, complemented
9C37×#2112C211yesmaximal, characteristic, complemented
10C2×#135×C37C2211yescharacteristic
11C23×C8×C37C2211yescharacteristic
12C37×#525—21no—
13C22×C4⋊C4×C37—21no—
14C2×M32×C37C2212yes—
15C37×#467C2211yescharacteristic
16C37×#467C2211yescharacteristic
17C37×#466C2211yescharacteristic
18C37×#466C2211yescharacteristic
19C37×#465—21no—
20C37×#465—21no—
21C37×#427C2211yescharacteristic
22C2×#107×C37—21no—
23C2×#107×C37—21no—
24C2×#107×C37—21no—
25C2×#107×C37C2212yes—
26C2×#107×C37C2212yes—
27C2×#107×C37C2212yes—
28C37×#416—21no—
29C37×#367C2214yescomplemented
30C37×#366C2214yescomplemented
31C37×#358C2214yescomplemented
32C37×#357C2214yescomplemented
33C37×C22⋊C16C2212yes—
34C37×C22⋊C16C2212yes—
35C2×#41×C37—21no—
36C23×C4×C37—41no—
37C23×C4×C37D411yescharacteristic
38#135×C37—24no—
39C2×Q16×C37C2311yescharacteristic
40C2×SD16×C37—21no—
41C2×D8×C37C2311yescharacteristic
42C22×C8×C37—24no—
43C22×C8×C37D411yescharacteristic
44C22×C8×C37D411yescharacteristic
45C22×C8×C37D411yescharacteristic
46C22×C8×C37C2311yescharacteristic
47C22×C8×C37C2311yescharacteristic
48C22×C8×C37C2311yescharacteristic
49#123×C37—41no—
50C22⋊Q8×C37—21no—
51C22⋊Q8×C37—21no—
52#121×C37—21no—
53#121×C37—21no—
54#120×C37—41no—
55C4×D4×C37—41no—
56C4×D4×C37—41no—
57#117×C37—21no—
58C2×C4⋊C4×C37—22no—
59C2×C4⋊C4×C37—22no—
60C2×C4⋊C4×C37—22no—
61C2×C4⋊C4×C37—41no—
62C2×C4⋊C4×C37—41no—
63C2×C4⋊C4×C37—41no—
64M32×C37—24no—
65C2×C16×C37C2314yes—
66#107×C37—22no—
67#107×C37—22no—
68#107×C37—22no—
69#107×C37C2314yes—
70#107×C37C2311yescharacteristic
71#107×C37C2311yescharacteristic
72#106×C37—21no—
73#103×C37—21no—
74#103×C37—21no—
75#102×C37—21no—
76#102×C37—21no—
77C24×C37D811yescharacteristic
78#41×C37—42no—
79#41×C37—42no—
80C2×Q8×C37—21no—
81C2×D4×C37—21no—
82C2×D4×C37—41no—
83C2×D4×C37—41no—
84C2×D4×C37—41no—
85C22×C4×C37—22no—
86C22×C4×C37—22no—
87C22×C4×C37—42no—
88C22×C4×C37—42no—
89C22×C4×C37—42no—
90C22×C4×C37—41no—
91C22×C4×C37—84no—
92C22×C4×C37D811yescharacteristic
93C22×C4×C37D811yescharacteristic
94C22×C4×C37D811yescharacteristic
95C22×C4×C37C2×D411yescharacteristic
96C22×C4×C37C2×D411yescharacteristic
97C22×C4×C37C2×D411yescharacteristic
98Q16×C37—22no—
99SD16×C37—42no—
100D8×C37—22no—
101C2×C8×C37—22no—
102C2×C8×C37—22no—
103C2×C8×C37—22no—
104C2×C8×C37—22no—
105C2×C8×C37—22no—
106C2×C8×C37C2×D411yescharacteristic
107C2×C8×C37C2×D411yescharacteristic
108C2×C8×C37C2×D411yescharacteristic
109C2×C8×C37#4114yes—
110C2×C8×C37C2411yescharacteristic
111C4⋊C4×C37—21no—
112C4⋊C4×C37—21no—
113C4⋊C4×C37—24no—
114C4⋊C4×C37—42no—
115C4⋊C4×C37—42no—
116C4⋊C4×C37—42no—
117C4⋊C4×C37—41no—
118C4⋊C4×C37—41no—
119C22⋊C4×C37—41no—
120C22⋊C4×C37—41no—
121C22⋊C4×C37—41no—
122C22⋊C4×C37—41no—
123C22⋊C4×C37—41no—
124C42×C37—41no—
125C16×C37—24no—
126C23×C37—22no—
127C23×C37—22no—
128C23×C37—22no—
129C23×C37—41no—
130C23×C37C2×D811yescharacteristic
131C23×C37C2×D811yescharacteristic
132C23×C37C2×D811yescharacteristic
133Q8×C37—41no—
134Q8×C37—41no—
135D4×C37—41no—
136D4×C37—41no—
137D4×C37—82no—
138D4×C37—82no—
139D4×C37—82no—
140C2×C4×C37—22no—
141C2×C4×C37—21no—
142C2×C4×C37—21no—
143C2×C4×C37—21no—
144C2×C4×C37—21no—
145C2×C4×C37—22no—
146C2×C4×C37—22no—
147C2×C4×C37—41no—
148C2×C4×C37—41no—
149C2×C4×C37—41no—
150C2×C4×C37—44no—
151C2×C4×C37—84no—
152C2×C4×C37—84no—
153C2×C4×C37—84no—
154C2×C4×C37—82no—
155C2×C4×C37#12312yes—
156C2×C4×C37C2×D811yescharacteristic
157C2×C4×C37C2×D811yescharacteristic
158C2×C4×C37C2×D811yescharacteristic
159C2×C4×C37#13712yes—
160C2×C4×C37C22×D411yescharacteristic
161C8×C37—21no—
162C8×C37—21no—
163C8×C37#12312yes—
164C8×C37C2×#4112yes—
165#33646C3711yesmaximal, characteristic, direct product factor, Sylow
166C22×C37—28no—
167C22×C37—21no—
168C22×C37—21no—
169C22×C37—21no—
170C22×C37—21no—
171C22×C37—21no—
172C22×C37—21no—
173C22×C37—81no—
174C22×C37—81no—
175C22×C37—81no—
176C22×C37#48012yes—
177C22×C37#50912yes—
178C22×C37#51012yes—
179C22×C37C22×D811yescharacteristic
180C4×C37—21no—
181C4×C37—21no—
182C4×C37—81no—
183C4×C37—84no—
184C4×C37—81no—
185C4×C37—81no—
186C4×C37#48012yes—
187C4×C37C2×#12311yescharacteristic
188C4×C37C2×#13711yescharacteristic
189#3050C2×C3711yescharacteristic
190C22×#107C2×C3711yescharacteristic, complemented
191#2236C2×C3712yescomplemented
192#2235C2×C3712yescomplemented
193#2233C2×C3712yescomplemented
194#2232C2×C3712yescomplemented
195#2159C2×C3712yescomplemented
196#2139C2×C3712yescomplemented
197#2112C2×C3711yescharacteristic, complemented
198C2×C37—22no—
199C2×C37—22no—
200C2×C37—22no—
201C2×C37—161no—
202C2×C37C2×#48011yescharacteristic
203C2×C37C2×#50911yescharacteristic
204C2×C37C2×#51011yescharacteristic
205C2×#135C22×C3711yescharacteristic
206C23×C8C22×C3711yescharacteristic
207#525—21no—
208C22×C4⋊C4—21no—
209C2×M32C22×C3712yes—
210#467C22×C3711yescharacteristic
211#467C22×C3711yescharacteristic
212#466C22×C3711yescharacteristic
213#466C22×C3711yescharacteristic
214#465—21no—
215#465—21no—
216#427C22×C3711yescharacteristic
217C2×#107—21no—
218C2×#107—21no—
219C2×#107—21no—
220C2×#107C22×C3712yes—
221C2×#107C22×C3712yes—
222C2×#107C22×C3712yes—
223#416—21no—
224#367C22×C3714yescomplemented
225#366C22×C3714yescomplemented
226#358C22×C3714yescomplemented
227#357C22×C3714yescomplemented
228C22⋊C16C22×C3712yes—
229C22⋊C16C22×C3712yes—
230C37#3364611yescharacteristic, direct product factor, Sylow
231C2×#41—21no—
232C23×C4—41no—
233C23×C4D4×C3711yescharacteristic
234#135—24no—
235C2×Q16C23×C3711yescharacteristic
236C2×SD16—21no—
237C2×D8C23×C3711yescharacteristic
238C22×C8—24no—
239C22×C8D4×C3711yescharacteristic
240C22×C8D4×C3711yescharacteristic
241C22×C8D4×C3711yescharacteristic
242C22×C8C23×C3711yescharacteristic
243C22×C8C23×C3711yescharacteristic
244C22×C8C23×C3711yescharacteristic
245#123—41no—
246C22⋊Q8—21no—
247C22⋊Q8—21no—
248#121—21no—
249#121—21no—
250#120—41no—
251C4×D4—41no—
252C4×D4—41no—
253#117—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260M32—24no—
261C2×C16C23×C3714yes—
262#107—22no—
263#107—22no—
264#107—22no—
265#107C23×C3714yes—
266#107C23×C3711yescharacteristic
267#107C23×C3711yescharacteristic
268#106—21no—
269#103—21no—
270#103—21no—
271#102—21no—
272#102—21no—
273C24D8×C3711yescharacteristic
274#41—42no—
275#41—42no—
276C2×Q8—21no—
277C2×D4—21no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—84no—
288C22×C4D8×C3711yescharacteristic
289C22×C4D8×C3711yescharacteristic
290C22×C4D8×C3711yescharacteristic
291C22×C4C2×D4×C3711yescharacteristic
292C22×C4C2×D4×C3711yescharacteristic
293C22×C4C2×D4×C3711yescharacteristic
294Q16—22no—
295SD16—42no—
296D8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8C2×D4×C3711yescharacteristic
303C2×C8C2×D4×C3711yescharacteristic
304C2×C8C2×D4×C3711yescharacteristic
305C2×C8#41×C3714yes—
306C2×C8C24×C3711yescharacteristic
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—24no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C42—41no—
321C16—24no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—41no—
326C23C2×D8×C3711yescharacteristic
327C23C2×D8×C3711yescharacteristic
328C23C2×D8×C3711yescharacteristic
329Q8—41no—
330Q8—41no—
331D4—41no—
332D4—41no—
333D4—82no—
334D4—82no—
335D4—82no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—44no—
347C2×C4—84no—
348C2×C4—84no—
349C2×C4—84no—
350C2×C4—82no—
351C2×C4#123×C3712yes—
352C2×C4C2×D8×C3711yescharacteristic
353C2×C4C2×D8×C3711yescharacteristic
354C2×C4C2×D8×C3711yescharacteristic
355C2×C4#137×C3712yes—
356C2×C4C22×D4×C3711yescharacteristic
357C8—21no—
358C8—21no—
359C8#123×C3712yes—
360C8C2×#41×C3712yes—
361C22—28no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22C37×#48012yes—
372C22C37×#50912yes—
373C22C37×#51012yes—
374C22C22×D8×C3711yescharacteristic
375C4—21no—
376C4—21no—
377C4—84no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4C37×#48012yes—
382C4C2×#123×C3711yescharacteristic
383C4C2×#137×C3711yescharacteristic
384C2—22no—
385C2—22no—
386C2—22no—
387C2—161no—
388C2C2×C37×#48011yescharacteristic
389C2C2×C37×#50911yescharacteristic
390C2C2×C37×#51011yescharacteristic
subgroupG/H
center, socleC22×C37C22×D8
commutator, Frattini, cosocleC2×C8C24×C37
prime radicalC37×#3050C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C37 ◁ C23×C4×C37 ◁ C23×C8×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×#107×C37C2semidirect
3. C37×#2236C2semidirect
4. C37×#2235C2semidirect
5. C37×#2233C2semidirect
6. C37×#2232C2semidirect
7. C37×#2159C2semidirect
8. C37×#2139C2semidirect
9. C37×#2112C2semidirect
29. C37×#367C22semidirect
30. C37×#366C22semidirect
31. C37×#358C22semidirect
32. C37×#357C22semidirect
165. #33646C37direct (N × H)
190. C22×#107C2×C37semidirect
191. #2236C2×C37semidirect
192. #2235C2×C37semidirect
193. #2233C2×C37semidirect
194. #2232C2×C37semidirect
195. #2159C2×C37semidirect
196. #2139C2×C37semidirect
197. #2112C2×C37semidirect
224. #367C22×C37semidirect
225. #366C22×C37semidirect
226. #358C22×C37semidirect
227. #357C22×C37semidirect
230. C37#33646direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#30502
C22×#107×C372
C37×#22362
C37×#22352
C37×#22332
C37×#22322
C37×#21592
C37×#21392
C37×#21122
#3364637