Finite Group Databasebeta

← C37×#12996 #22634249

Subgroups — C37×#12996 #22634249

Order 9472.

512 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C37×#472C211yesmaximal, characteristic
2C37×#2913C211yesmaximal, characteristic, complemented
3C37×#2140C211yesmaximal, characteristic, complemented
4C2×C37×#357C211yesmaximal, characteristic, complemented
5C2×C37×#350C211yesmaximal, characteristic, complemented
6C37×#1387C211yesmaximal, characteristic, complemented
7C37×#1382C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C37—21no—
9C22×D8×C37C2211yescharacteristic
10C2×#131×C37C2211yescharacteristic
11C37×#520—21no—
12C2×M32×C37C2211yescharacteristic
13C22×C16×C37C2211yescharacteristic
14C37×#472—22no—
15C37×#472C414yescomplemented
16C2×#108×C37C2211yescharacteristic
17C37×#428—21no—
18C37×#428—22no—
19C37×#427—21no—
20C37×#427C2211yescharacteristic
21C2×#107×C37—21no—
22C2×#107×C37C2211yescharacteristic
23C2×#100×C37—21no—
24C37×#357—22no—
25C37×#357—21no—
26C37×#357—21no—
27C37×#350—22no—
28C2×#41×C37D411yescharacteristic
29C22×D4×C37—21no—
30AGL(1,ℤ/8)×C37—24no—
31AGL(1,ℤ/8)×C37—42no—
32C2×SD16×C37—41no—
33C2×D8×C37—21no—
34C2×D8×C37—21no—
35C2×D8×C37—21no—
36C2×D8×C37—21no—
37C2×D8×C37—21no—
38C2×D8×C37—41no—
39C2×D8×C37C2×C412yescomplemented
40#131×C37—21no—
41#131×C37—21no—
42#131×C37C2×C412yes—
43#131×C37D412yes—
44C2×M16×C37—21no—
45C2×M16×C37—21no—
46C2×M16×C37D411yescharacteristic
47C22×C8×C37—21no—
48C22×C8×C37C2311yescharacteristic
49C4×Q8×C37—41no—
50C4×D4×C37—42no—
51C4×D4×C37—41no—
52#117×C37—22no—
53#117×C37—21no—
54C2×C4⋊C4×C37—22no—
55C2×C4⋊C4×C37—21no—
56M32×C37—21no—
57M32×C37—21no—
58C2×C16×C37—21no—
59C2×C16×C37—21no—
60C2×C16×C37—21no—
61C2×C16×C37—21no—
62#108×C37—21no—
63#108×C37C2×C412yescomplemented
64#107×C37—22no—
65#107×C37—21no—
66#107×C37D411yescharacteristic
67#107×C37D412yes—
68#107×C37D411yescharacteristic
69#106×C37—22no—
70#106×C37—21no—
71#100×C37—24no—
72C24×C37—41no—
73#41×C37—21no—
74#41×C37—21no—
75#41×C37C22⋊C412yes—
76#41×C37SD1612yes—
77C2×Q8×C37D811yescharacteristic
78C2×D4×C37—21no—
79C2×D4×C37—22no—
80C2×D4×C37—41no—
81C2×D4×C37—41no—
82C2×D4×C37—41no—
83C2×D4×C37—41no—
84C2×D4×C37—41no—
85C2×D4×C37D811yescharacteristic
86C22×C4×C37—21no—
87C22×C4×C37—42no—
88C22×C4×C37—41no—
89C22×C4×C37C2×D411yescharacteristic
90SD16×C37—44no—
91D8×C37—22no—
92D8×C37—22no—
93D8×C37—44no—
94D8×C37—41no—
95D8×C37—41no—
96M16×C37—21no—
97M16×C37—22no—
98M16×C37—22no—
99M16×C37—22no—
100M16×C37—41no—
101M16×C37C22⋊C412yes—
102C2×C8×C37—21no—
103C2×C8×C37—22no—
104C2×C8×C37—22no—
105C2×C8×C37—41no—
106C2×C8×C37C22×C411yescharacteristic
107C2×C8×C37C2×D411yescharacteristic
108C2×C8×C37C2×D411yescharacteristic
109C2×C8×C37C2×D411yescharacteristic
110C2×C8×C37#4111yescharacteristic
111C2×C8×C37#4111yescharacteristic
112C4⋊C4×C37—21no—
113C4⋊C4×C37—22no—
114C4⋊C4×C37—21no—
115C4⋊C4×C37—42no—
116C4⋊C4×C37—41no—
117C22⋊C4×C37—42no—
118C22⋊C4×C37—41no—
119C42×C37—42no—
120C42×C37—41no—
121C16×C37—41no—
122C16×C37—41no—
123C23×C37—21no—
124C23×C37—42no—
125C23×C37—81no—
126C23×C37—81no—
127C23×C37—161no—
128C23×C37—161no—
129C23×C37AGL(1,ℤ/8)11yescharacteristic
130Q8×C37—21no—
131Q8×C37#10212yes—
132D4×C37—22no—
133D4×C37—22no—
134D4×C37—21no—
135D4×C37—42no—
136D4×C37—42no—
137D4×C37—81no—
138D4×C37—81no—
139D4×C37#10212yes—
140C2×C4×C37—21no—
141C2×C4×C37—22no—
142C2×C4×C37—22no—
143C2×C4×C37—41no—
144C2×C4×C37—42no—
145C2×C4×C37—41no—
146C2×C4×C37—161no—
147C2×C4×C37—162no—
148C2×C4×C37C2×C22⋊C411yescharacteristic
149C2×C4×C37#12011yescharacteristic
150C2×C4×C37#12311yescharacteristic
151C2×C4×C37C2×D811yescharacteristic
152C2×C4×C37C2×SD1611yescharacteristic
153C2×C4×C37#13711yescharacteristic
154C8×C37—22no—
155C8×C37—42no—
156C8×C37C4×D411yescharacteristic
157C8×C37C4×D411yescharacteristic
158C8×C37#12111yescharacteristic
159C8×C37#12111yescharacteristic
160#12996C3711yesmaximal, characteristic, direct product factor, Sylow
161C22×C37—21no—
162C22×C37—21no—
163C22×C37—22no—
164C22×C37—41no—
165C22×C37—41no—
166C22×C37—82no—
167C22×C37—162no—
168C22×C37—162no—
169C22×C37—161no—
170C22×C37—161no—
171C22×C37#41711yescharacteristic
172C22×C37#44711yescharacteristic
173C22×C37#48111yescharacteristic
174C4×C37—22no—
175C4×C37—82no—
176C4×C37—81no—
177C4×C37—81no—
178C4×C37#38611yescharacteristic
179C4×C37C2×#10211yescharacteristic
180C4×C37#44811yescharacteristic
181C4×C37#48011yescharacteristic
182C2×#472C2×C3711yescharacteristic
183#2913C2×C3711yescharacteristic, complemented
184#2140C2×C3711yescharacteristic, complemented
185C2×#357C2×C3711yescharacteristic, complemented
186C2×#350C2×C3711yescharacteristic, complemented
187#1387C2×C3711yescharacteristic, complemented
188#1382C2×C3711yescharacteristic, complemented
189C2×C37—21no—
190C2×C37—21no—
191C2×C37—42no—
192C2×C37—162no—
193C2×C37#187411yescharacteristic
194C2×C37#219211yescharacteristic
195C2×C37#219311yescharacteristic
196C2×AGL(1,ℤ/8)—21no—
197C22×D8C22×C3711yescharacteristic
198C2×#131C22×C3711yescharacteristic
199#520—21no—
200C2×M32C22×C3711yescharacteristic
201C22×C16C22×C3711yescharacteristic
202#472—22no—
203#472C4×C3714yescomplemented
204C2×#108C22×C3711yescharacteristic
205#428—21no—
206#428—22no—
207#427—21no—
208#427C22×C3711yescharacteristic
209C2×#107—21no—
210C2×#107C22×C3711yescharacteristic
211C2×#100—21no—
212#357—22no—
213#357—21no—
214#357—21no—
215#350—22no—
216C37#1299611yescharacteristic, direct product factor, Sylow
217C2×#41D4×C3711yescharacteristic
218C22×D4—21no—
219AGL(1,ℤ/8)—24no—
220AGL(1,ℤ/8)—42no—
221C2×SD16—41no—
222C2×D8—21no—
223C2×D8—21no—
224C2×D8—21no—
225C2×D8—21no—
226C2×D8—21no—
227C2×D8—41no—
228C2×D8C2×C4×C3712yescomplemented
229#131—21no—
230#131—21no—
231#131C2×C4×C3712yes—
232#131D4×C3712yes—
233C2×M16—21no—
234C2×M16—21no—
235C2×M16D4×C3711yescharacteristic
236C22×C8—21no—
237C22×C8C23×C3711yescharacteristic
238C4×Q8—41no—
239C4×D4—42no—
240C4×D4—41no—
241#117—22no—
242#117—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245M32—21no—
246M32—21no—
247C2×C16—21no—
248C2×C16—21no—
249C2×C16—21no—
250C2×C16—21no—
251#108—21no—
252#108C2×C4×C3712yescomplemented
253#107—22no—
254#107—21no—
255#107D4×C3711yescharacteristic
256#107D4×C3712yes—
257#107D4×C3711yescharacteristic
258#106—22no—
259#106—21no—
260#100—24no—
261C24—41no—
262#41—21no—
263#41—21no—
264#41C22⋊C4×C3712yes—
265#41SD16×C3712yes—
266C2×Q8D8×C3711yescharacteristic
267C2×D4—21no—
268C2×D4—22no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4D8×C3711yescharacteristic
275C22×C4—21no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4C2×D4×C3711yescharacteristic
279SD16—44no—
280D8—22no—
281D8—22no—
282D8—44no—
283D8—41no—
284D8—41no—
285M16—21no—
286M16—22no—
287M16—22no—
288M16—22no—
289M16—41no—
290M16C22⋊C4×C3712yes—
291C2×C8—21no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—41no—
295C2×C8C22×C4×C3711yescharacteristic
296C2×C8C2×D4×C3711yescharacteristic
297C2×C8C2×D4×C3711yescharacteristic
298C2×C8C2×D4×C3711yescharacteristic
299C2×C8#41×C3711yescharacteristic
300C2×C8#41×C3711yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C22⋊C4—42no—
307C22⋊C4—41no—
308C42—42no—
309C42—41no—
310C16—41no—
311C16—41no—
312C23—21no—
313C23—42no—
314C23—81no—
315C23—81no—
316C23—161no—
317C23—161no—
318C23AGL(1,ℤ/8)×C3711yescharacteristic
319Q8—21no—
320Q8#102×C3712yes—
321D4—22no—
322D4—22no—
323D4—21no—
324D4—42no—
325D4—42no—
326D4—81no—
327D4—81no—
328D4#102×C3712yes—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—161no—
336C2×C4—162no—
337C2×C4C2×C22⋊C4×C3711yescharacteristic
338C2×C4#120×C3711yescharacteristic
339C2×C4#123×C3711yescharacteristic
340C2×C4C2×D8×C3711yescharacteristic
341C2×C4C2×SD16×C3711yescharacteristic
342C2×C4#137×C3711yescharacteristic
343C8—22no—
344C8—42no—
345C8C4×D4×C3711yescharacteristic
346C8C4×D4×C3711yescharacteristic
347C8#121×C3711yescharacteristic
348C8#121×C3711yescharacteristic
349C22—21no—
350C22—21no—
351C22—22no—
352C22—41no—
353C22—41no—
354C22—82no—
355C22—162no—
356C22—162no—
357C22—161no—
358C22—161no—
359C22C37×#41711yescharacteristic
360C22C37×#44711yescharacteristic
361C22C37×#48111yescharacteristic
362C4—22no—
363C4—82no—
364C4—81no—
365C4—81no—
366C4C37×#38611yescharacteristic
367C4C2×#102×C3711yescharacteristic
368C4C37×#44811yescharacteristic
369C4C37×#48011yescharacteristic
370C2—21no—
371C2—21no—
372C2—42no—
373C2—162no—
374C2C37×#187411yescharacteristic
375C2C37×#219211yescharacteristic
376C2C37×#219311yescharacteristic
subgroupG/H
center, socleC22×C37#447
commutatorC2×C8C22×C4×C37
Frattini, cosocleC22×C8C23×C37
prime radicalC2×C37×#472C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C37 ◁ C22×C4×C37 ◁ C2×#131×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C37×#2913C2semidirect
3. C37×#2140C2semidirect
4. C2×C37×#357C2semidirect
5. C2×C37×#350C2semidirect
6. C37×#1387C2semidirect
7. C37×#1382C2semidirect
15. C37×#472C4semidirect
39. C2×D8×C37C2×C4semidirect
63. #108×C37C2×C4semidirect
160. #12996C37direct (N × H)
183. #2913C2×C37semidirect
184. #2140C2×C37semidirect
185. C2×#357C2×C37semidirect
186. C2×#350C2×C37semidirect
187. #1387C2×C37semidirect
188. #1382C2×C37semidirect
203. #472C4×C37semidirect
216. C37#12996direct (N × H)
228. C2×D8C2×C4×C37semidirect
252. #108C2×C4×C37semidirect

Maximal subgroups

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

subgroupindex
C2×C37×#4722
C37×#29132
C37×#21402
C2×C37×#3572
C2×C37×#3502
C37×#13872
C37×#13822
#1299637