Finite Group Databasebeta

← C37×#17398 #22638651

Subgroups — C37×#17398 #22638651

Order 9472.

1034 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#126×C37C211yesmaximal, characteristic
2C37×#2920C212yesmaximal, complemented
3C37×#1473C218yesmaximal, complemented
4C37×#1463C212yesmaximal, complemented
5C37×#1462C212yesmaximal
6C2×#126×C37—22no—
7C2×#126×C37—22no—
8C2×#126×C37—22no—
9C2×#126×C37—22no—
10C2×#126×C37C414yes—
11C2×#126×C37C2214yes—
12C22×C4⋊C4×C37C411yescharacteristic
13C22×C4⋊C4×C37C411yescharacteristic
14C22×C4⋊C4×C37C2211yescharacteristic
15C22×C22⋊C4×C37C411yescharacteristic
16C22×C22⋊C4×C37C411yescharacteristic
17C22×C22⋊C4×C37C2211yescharacteristic
18C22×C42×C37C2211yescharacteristic
19C37×#432—22no—
20C37×#432C2214yescomplemented
21C37×#431—22no—
22C37×#431C2214yescomplemented
23C37×#423—22no—
24C37×#407C2214yes—
25C4×M16×C37—22no—
26C37×#324C2218yescomplemented
27C37×#323C2218yes—
28C25×C37C2×C411yescharacteristic
29C23×C4×C37C2×C411yescharacteristic
30C23×C4×C37C2×C411yescharacteristic
31C23×C4×C37C2×C411yescharacteristic
32C23×C4×C37C2×C411yescharacteristic
33C23×C4×C37C2×C411yescharacteristic
34C23×C4×C37C2311yescharacteristic
35C2×M16×C37—24no—
36#126×C37—24no—
37#126×C37—24no—
38#126×C37—24no—
39#126×C37—24no—
40#126×C37—24no—
41#126×C37—24no—
42#126×C37—24no—
43#126×C37—24no—
44C2×C4⋊C4×C37—22no—
45C2×C4⋊C4×C37—22no—
46C2×C4⋊C4×C37—21no—
47C2×C4⋊C4×C37—21no—
48C2×C4⋊C4×C37—21no—
49C2×C4⋊C4×C37—21no—
50C2×C4⋊C4×C37—22no—
51C2×C4⋊C4×C37—22no—
52C2×C4⋊C4×C37C2×C412yes—
53C2×C4⋊C4×C37C2×C414yes—
54C2×C4⋊C4×C37C2×C414yes—
55C2×C4⋊C4×C37C2312yes—
56C2×C22⋊C4×C37—22no—
57C2×C22⋊C4×C37—21no—
58C2×C22⋊C4×C37—21no—
59C2×C22⋊C4×C37—21no—
60C2×C22⋊C4×C37—21no—
61C2×C22⋊C4×C37—22no—
62C2×C22⋊C4×C37—22no—
63C2×C22⋊C4×C37—22no—
64C2×C22⋊C4×C37C2×C412yes—
65C2×C22⋊C4×C37C2×C414yes—
66C2×C22⋊C4×C37C2×C414yes—
67C2×C22⋊C4×C37C2312yes—
68C2×C42×C37C2×C412yes—
69C2×C42×C37D412yes—
70C2×C42×C37D412yes—
71C2×C42×C37D412yes—
72C2×C42×C37D412yes—
73C2×C42×C37C2312yes—
74C4⋊C8×C37—24no—
75C4⋊C8×C37—24no—
76C22⋊C8×C37—24no—
77C22⋊C8×C37C2318yes—
78#97×C37—24no—
79C4×C8×C37—24no—
80C24×C37—21no—
81C24×C37—21no—
82C24×C37—42no—
83C24×C37—42no—
84C24×C37—42no—
85C24×C37C22×C411yescharacteristic
86C24×C37C22×C412yes—
87C22×C4×C37—21no—
88C22×C4×C37—21no—
89C22×C4×C37—21no—
90C22×C4×C37—21no—
91C22×C4×C37—21no—
92C22×C4×C37—21no—
93C22×C4×C37—42no—
94C22×C4×C37—42no—
95C22×C4×C37—42no—
96C22×C4×C37—42no—
97C22×C4×C37—42no—
98C22×C4×C37—42no—
99C22×C4×C37C22⋊C412yes—
100C22×C4×C37C22⋊C411yescharacteristic
101C22×C4×C37C22⋊C412yes—
102C22×C4×C37C22⋊C411yescharacteristic
103C22×C4×C37C22⋊C411yescharacteristic
104C22×C4×C37C22⋊C411yescharacteristic
105C22×C4×C37C22×C412yes—
106C22×C4×C37C22×C412yes—
107C22×C4×C37C22×C412yes—
108C22×C4×C37C22×C412yes—
109C22×C4×C37C22×C411yescharacteristic
110C22×C4×C37C22×C412yes—
111C22×C4×C37C2×D412yes—
112C22×C4×C37C2×D412yes—
113C22×C4×C37C2411yescharacteristic
114M16×C37—48no—
115C2×C8×C37—28no—
116C4⋊C4×C37—24no—
117C4⋊C4×C37—24no—
118C4⋊C4×C37—24no—
119C4⋊C4×C37—24no—
120C4⋊C4×C37—24no—
121C4⋊C4×C37—24no—
122C22⋊C4×C37—24no—
123C22⋊C4×C37—24no—
124C22⋊C4×C37—24no—
125C22⋊C4×C37—24no—
126C22⋊C4×C37—24no—
127C22⋊C4×C37—24no—
128C42×C37C22⋊C412yes—
129C42×C37C22⋊C412yes—
130C42×C37C22⋊C412yes—
131C42×C37C22⋊C412yes—
132C42×C37C2×D412yes—
133C42×C37C2×D412yes—
134C42×C37C2×D412yes—
135C42×C37C2×D412yes—
136C23×C37—22no—
137C23×C37—22no—
138C23×C37—22no—
139C23×C37—22no—
140C23×C37—24no—
141C23×C37—42no—
142C23×C37—42no—
143C23×C37—42no—
144C23×C37—42no—
145C23×C37—42no—
146C23×C37—42no—
147C23×C37—42no—
148C23×C37—42no—
149C23×C37—88no—
150C23×C37C2×C22⋊C411yescharacteristic
151C23×C37C2×C22⋊C411yescharacteristic
152C23×C37C23×C411yescharacteristic
153C2×C4×C37—24no—
154C2×C4×C37—24no—
155C2×C4×C37—24no—
156C2×C4×C37—22no—
157C2×C4×C37—22no—
158C2×C4×C37—22no—
159C2×C4×C37—24no—
160C2×C4×C37—42no—
161C2×C4×C37—41no—
162C2×C4×C37—41no—
163C2×C4×C37—41no—
164C2×C4×C37—41no—
165C2×C4×C37—41no—
166C2×C4×C37—42no—
167C2×C4×C37—42no—
168C2×C4×C37—41no—
169C2×C4×C37—42no—
170C2×C4×C37—42no—
171C2×C4×C37—42no—
172C2×C4×C37—42no—
173C2×C4×C37—42no—
174C2×C4×C37—42no—
175C2×C4×C37—42no—
176C2×C4×C37—42no—
177C2×C4×C37C2×C22⋊C412yes—
178C2×C4×C37C2×C22⋊C412yes—
179C2×C4×C37C2×C22⋊C412yes—
180C2×C4×C37C2×C22⋊C412yes—
181C2×C4×C37C2×C22⋊C412yes—
182C2×C4×C37C22×D412yes—
183C8×C37—48no—
184#17398C3711yesmaximal, characteristic, direct product factor, Sylow
185C22×C37—28no—
186C22×C37—21no—
187C22×C37—21no—
188C22×C37—21no—
189C22×C37—21no—
190C22×C37—21no—
191C22×C37—21no—
192C22×C37—42no—
193C22×C37—42no—
194C22×C37—42no—
195C22×C37—84no—
196C22×C37—84no—
197C22×C37—84no—
198C22×C37#41012yes—
199C22×C37#41312yes—
200C22×C37C22×C22⋊C411yescharacteristic
201C22×C37#56812yes—
202C4×C37—22no—
203C4×C37—22no—
204C4×C37—22no—
205C4×C37—22no—
206C4×C37—22no—
207C4×C37—22no—
208C4×C37—42no—
209C4×C37—42no—
210C4×C37—42no—
211C22×#126C2×C3711yescharacteristic
212#2920C2×C3712yescomplemented
213#1473C2×C3718yescomplemented
214#1463C2×C3712yescomplemented
215#1462C2×C3712yes—
216C2×C37—22no—
217C2×C37—22no—
218C2×C37—22no—
219C2×C37—82no—
220C2×C37#287911yescharacteristic
221C2×C37C2×#41011yescharacteristic
222C2×C37C2×#41311yescharacteristic
223C2×#126—22no—
224C2×#126—22no—
225C2×#126—22no—
226C2×#126—22no—
227C2×#126C4×C3714yes—
228C2×#126C22×C3714yes—
229C22×C4⋊C4C4×C3711yescharacteristic
230C22×C4⋊C4C4×C3711yescharacteristic
231C22×C4⋊C4C22×C3711yescharacteristic
232C22×C22⋊C4C4×C3711yescharacteristic
233C22×C22⋊C4C4×C3711yescharacteristic
234C22×C22⋊C4C22×C3711yescharacteristic
235C22×C42C22×C3711yescharacteristic
236#432—22no—
237#432C22×C3714yescomplemented
238#431—22no—
239#431C22×C3714yescomplemented
240#423—22no—
241#407C22×C3714yes—
242C4×M16—22no—
243#324C22×C3718yescomplemented
244#323C22×C3718yes—
245C37#1739811yescharacteristic, direct product factor, Sylow
246C25C2×C4×C3711yescharacteristic
247C23×C4C2×C4×C3711yescharacteristic
248C23×C4C2×C4×C3711yescharacteristic
249C23×C4C2×C4×C3711yescharacteristic
250C23×C4C2×C4×C3711yescharacteristic
251C23×C4C2×C4×C3711yescharacteristic
252C23×C4C23×C3711yescharacteristic
253C2×M16—24no—
254#126—24no—
255#126—24no—
256#126—24no—
257#126—24no—
258#126—24no—
259#126—24no—
260#126—24no—
261#126—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4C2×C4×C3714yes—
271C2×C4⋊C4C2×C4×C3714yes—
272C2×C4⋊C4C2×C4×C3712yes—
273C2×C4⋊C4C23×C3712yes—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4C2×C4×C3714yes—
283C2×C22⋊C4C2×C4×C3714yes—
284C2×C22⋊C4C2×C4×C3712yes—
285C2×C22⋊C4C23×C3712yes—
286C2×C42C2×C4×C3712yes—
287C2×C42D4×C3712yes—
288C2×C42D4×C3712yes—
289C2×C42D4×C3712yes—
290C2×C42D4×C3712yes—
291C2×C42C23×C3712yes—
292C4⋊C8—24no—
293C4⋊C8—24no—
294C22⋊C8—24no—
295C22⋊C8C23×C3718yes—
296#97—24no—
297C4×C8—24no—
298C24—21no—
299C24—21no—
300C24—42no—
301C24—42no—
302C24—42no—
303C24C22×C4×C3711yescharacteristic
304C24C22×C4×C3712yes—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4C22⋊C4×C3712yes—
318C22×C4C22⋊C4×C3711yescharacteristic
319C22×C4C22⋊C4×C3712yes—
320C22×C4C22⋊C4×C3711yescharacteristic
321C22×C4C22⋊C4×C3711yescharacteristic
322C22×C4C22⋊C4×C3711yescharacteristic
323C22×C4C22×C4×C3712yes—
324C22×C4C22×C4×C3712yes—
325C22×C4C22×C4×C3712yes—
326C22×C4C22×C4×C3712yes—
327C22×C4C22×C4×C3711yescharacteristic
328C22×C4C22×C4×C3712yes—
329C22×C4C2×D4×C3712yes—
330C22×C4C2×D4×C3712yes—
331C22×C4C24×C3711yescharacteristic
332M16—48no—
333C2×C8—28no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C42C22⋊C4×C3712yes—
347C42C22⋊C4×C3712yes—
348C42C22⋊C4×C3712yes—
349C42C22⋊C4×C3712yes—
350C42C2×D4×C3712yes—
351C42C2×D4×C3712yes—
352C42C2×D4×C3712yes—
353C42C2×D4×C3712yes—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—24no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—88no—
368C23C2×C22⋊C4×C3711yescharacteristic
369C23C2×C22⋊C4×C3711yescharacteristic
370C23C23×C4×C3711yescharacteristic
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4C2×C22⋊C4×C3712yes—
396C2×C4C2×C22⋊C4×C3712yes—
397C2×C4C2×C22⋊C4×C3712yes—
398C2×C4C2×C22⋊C4×C3712yes—
399C2×C4C2×C22⋊C4×C3712yes—
400C2×C4C22×D4×C3712yes—
401C8—48no—
402C22—28no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—84no—
413C22—84no—
414C22—84no—
415C22C37×#41012yes—
416C22C37×#41312yes—
417C22C22×C22⋊C4×C3711yescharacteristic
418C22C37×#56812yes—
419C4—22no—
420C4—22no—
421C4—22no—
422C4—22no—
423C4—22no—
424C4—22no—
425C4—42no—
426C4—42no—
427C4—42no—
428C2—22no—
429C2—22no—
430C2—22no—
431C2—82no—
432C2C37×#287911yescharacteristic
433C2C2×C37×#41011yescharacteristic
434C2C2×C37×#41311yescharacteristic
subgroupG/H
center, socleC22×C37C22×C22⋊C4
commutatorC23C23×C4×C37
Frattini, cosocleC22×C4C24×C37
prime radicalC25×C37C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C37 ◁ C22×C42×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C37×#2920C2semidirect
3. C37×#1473C2semidirect
4. C37×#1463C2semidirect
20. C37×#432C22semidirect
22. C37×#431C22semidirect
26. C37×#324C22semidirect
184. #17398C37direct (N × H)
212. #2920C2×C37semidirect
213. #1473C2×C37semidirect
214. #1463C2×C37semidirect
237. #432C22×C37semidirect
239. #431C22×C37semidirect
243. #324C22×C37semidirect
245. C37#17398direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#126×C372
C37×#29202
C37×#14732
C37×#14632
C37×#14622
#1739837