Finite Group Databasebeta

← C37×#9754 #22631007

Subgroups — C37×#9754 #22631007

Order 9472.

416 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
1C37×#3131C211yesmaximal, characteristic
2C37×#1770C211yesmaximal, characteristic, complemented
3C37×#1564C211yesmaximal, characteristic, complemented
4C37×#1503C211yesmaximal, characteristic, complemented
5C37×#1502C211yesmaximal, characteristic
6C37×#1481C211yesmaximal, characteristic, complemented
7C37×#1480C211yesmaximal, characteristic
8C37×#533C2211yescharacteristic
9C37×#532C2211yescharacteristic
10C37×#488—21no—
11C37×#488—21no—
12C37×#487C411yescharacteristic, complemented
13C37×#487C411yescharacteristic, complemented
14C37×#486C411yescharacteristic, complemented
15C37×#486C411yescharacteristic, complemented
16C37×#431C2211yescharacteristic
17C37×#431C2211yescharacteristic
18C2×C4⋊C8×C37C2211yescharacteristic
19C37×#416—21no—
20C37×#416—21no—
21C2×#97×C37—21no—
22C2×C4×C8×C37C2211yescharacteristic
23C4×C4⋊C4×C37C2211yescharacteristic
24C37×#336—21no—
25C37×#335—21no—
26C37×#334—21no—
27C37×#328—22no—
28C37×#327—22no—
29C37×Q8⋊C8—22no—
30C37×D4⋊C8—22no—
31C2×#41×C37—21no—
32C22×C8×C37—21no—
33C22×C8×C37D411yescharacteristic
34C22×C8×C37D411yescharacteristic
35C4⋊Q8×C37C2×C411yescharacteristic
36C4⋊Q8×C37C2×C411yescharacteristic
37C4⋊Q8×C37C2×C411yescharacteristic
38#127×C37C2×C411yescharacteristic
39#125×C37—21no—
40#124×C37—21no—
41C22⋊Q8×C37—21no—
42C22⋊Q8×C37—21no—
43#121×C37—21no—
44#121×C37—21no—
45C4×Q8×C37—21no—
46C4×Q8×C37—21no—
47C4×D4×C37—21no—
48C4×D4×C37—21no—
49#117×C37—21no—
50C2×C4⋊C4×C37—21no—
51C2×C42×C37—21no—
52C2×C42×C37C2311yescharacteristic
53C4⋊C8×C37—21no—
54C4⋊C8×C37—21no—
55C4⋊C8×C37—21no—
56C4⋊C8×C37D412yes—
57#103×C37—21no—
58#103×C37—21no—
59#103×C37—21no—
60#103×C37—21no—
61#102×C37—21no—
62#102×C37—21no—
63#102×C37—21no—
64#102×C37—21no—
65C22⋊C8×C37—21no—
66C22⋊C8×C37—21no—
67#97×C37—42no—
68C4×C8×C37—21no—
69C4×C8×C37—21no—
70C4×C8×C37C2×C411yescharacteristic
71C4×C8×C37C2×C411yescharacteristic
72C4×C8×C37D411yescharacteristic
73C4×C8×C37D411yescharacteristic
74#95×C37D412yes—
75#41×C37—41no—
76#41×C37—41no—
77#41×C37—41no—
78#41×C37—41no—
79C2×Q8×C37—21no—
80C2×Q8×C37—21no—
81C2×Q8×C37—21no—
82C2×D4×C37—21no—
83C2×D4×C37—21no—
84C2×D4×C37—21no—
85C2×D4×C37—41no—
86C22×C4×C37—22no—
87C22×C4×C37—41no—
88C22×C4×C37C2×D411yescharacteristic
89C22×C4×C37C2×D411yescharacteristic
90C22×C4×C37#4111yescharacteristic
91C2×C8×C37—21no—
92C2×C8×C37—21no—
93C2×C8×C37—21no—
94C2×C8×C37—21no—
95C2×C8×C37—22no—
96C2×C8×C37—41no—
97C2×C8×C37—41no—
98C2×C8×C37—41no—
99C2×C8×C37—41no—
100C2×C8×C37C22⋊C411yescharacteristic
101C2×C8×C37C22⋊C411yescharacteristic
102C2×C8×C37C22⋊C411yescharacteristic
103C2×C8×C37C22⋊C411yescharacteristic
104C4⋊C4×C37—21no—
105C4⋊C4×C37—21no—
106C4⋊C4×C37—21no—
107C4⋊C4×C37—21no—
108C4⋊C4×C37—21no—
109C4⋊C4×C37—21no—
110C4⋊C4×C37—42no—
111C4⋊C4×C37—41no—
112C4⋊C4×C37—41no—
113C22⋊C4×C37—41no—
114C22⋊C4×C37—41no—
115C22⋊C4×C37—41no—
116C42×C37—41no—
117C42×C37—41no—
118C42×C37—41no—
119C42×C37C22×C411yescharacteristic
120C42×C37C2×D411yescharacteristic
121C42×C37C2×D411yescharacteristic
122C42×C37#4111yescharacteristic
123C23×C37—41no—
124C23×C37#12111yescharacteristic
125Q8×C37—41no—
126Q8×C37—41no—
127Q8×C37—42no—
128Q8×C37—81no—
129D4×C37—41no—
130D4×C37—41no—
131D4×C37—42no—
132D4×C37—81no—
133D4×C37—81no—
134D4×C37—81no—
135C2×C4×C37—22no—
136C2×C4×C37—41no—
137C2×C4×C37—42no—
138C2×C4×C37—42no—
139C2×C4×C37—41no—
140C2×C4×C37—41no—
141C2×C4×C37—41no—
142C2×C4×C37—42no—
143C2×C4×C37—81no—
144C2×C4×C37—81no—
145C2×C4×C37C2×C22⋊C411yescharacteristic
146C2×C4×C37C4×D411yescharacteristic
147C2×C4×C37C4×D411yescharacteristic
148C2×C4×C37#12111yescharacteristic
149C2×C4×C37#12411yescharacteristic
150C2×C4×C37#12711yescharacteristic
151C2×C4×C37#13511yescharacteristic
152C2×C4×C37#13511yescharacteristic
153C2×C4×C37AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C37#13711yescharacteristic
155C8×C37—21no—
156C8×C37—21no—
157C8×C37—21no—
158C8×C37—21no—
159C8×C37—42no—
160C8×C37—81no—
161C8×C37—81no—
162#9754C3711yesmaximal, characteristic, direct product factor, Sylow
163C22×C37—21no—
164C22×C37—21no—
165C22×C37—81no—
166C22×C37—81no—
167C22×C37—81no—
168C22×C37#39011yescharacteristic
169C22×C37#46311yescharacteristic
170C22×C37#46411yescharacteristic
171C4×C37—21no—
172C4×C37—21no—
173C4×C37—21no—
174C4×C37—82no—
175C4×C37—81no—
176C4×C37—81no—
177C4×C37—81no—
178C4×C37#41611yescharacteristic
179C4×C37#41711yescharacteristic
180C4×C37#44412yes—
181C4×C37#46311yescharacteristic
182C4×C37#46411yescharacteristic
183#3131C2×C3711yescharacteristic
184#1770C2×C3711yescharacteristic, complemented
185#1564C2×C3711yescharacteristic, complemented
186#1503C2×C3711yescharacteristic, complemented
187#1502C2×C3711yescharacteristic
188#1481C2×C3711yescharacteristic, complemented
189#1480C2×C3711yescharacteristic
190C2×C37—21no—
191C2×C37—21no—
192C2×C37—161no—
193C2×C37#197511yescharacteristic
194C2×C37#197611yescharacteristic
195C2×C37#198311yescharacteristic
196#533C22×C3711yescharacteristic
197#532C22×C3711yescharacteristic
198#488—21no—
199#488—21no—
200#487C4×C3711yescharacteristic, complemented
201#487C4×C3711yescharacteristic, complemented
202#486C4×C3711yescharacteristic, complemented
203#486C4×C3711yescharacteristic, complemented
204#431C22×C3711yescharacteristic
205#431C22×C3711yescharacteristic
206C2×C4⋊C8C22×C3711yescharacteristic
207#416—21no—
208#416—21no—
209C2×#97—21no—
210C2×C4×C8C22×C3711yescharacteristic
211C4×C4⋊C4C22×C3711yescharacteristic
212#336—21no—
213#335—21no—
214#334—21no—
215#328—22no—
216#327—22no—
217Q8⋊C8—22no—
218D4⋊C8—22no—
219C37#975411yescharacteristic, direct product factor, Sylow
220C2×#41—21no—
221C22×C8—21no—
222C22×C8D4×C3711yescharacteristic
223C22×C8D4×C3711yescharacteristic
224C4⋊Q8C2×C4×C3711yescharacteristic
225C4⋊Q8C2×C4×C3711yescharacteristic
226C4⋊Q8C2×C4×C3711yescharacteristic
227#127C2×C4×C3711yescharacteristic
228#125—21no—
229#124—21no—
230C22⋊Q8—21no—
231C22⋊Q8—21no—
232#121—21no—
233#121—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×D4—21no—
237C4×D4—21no—
238#117—21no—
239C2×C4⋊C4—21no—
240C2×C42—21no—
241C2×C42C23×C3711yescharacteristic
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8D4×C3712yes—
246#103—21no—
247#103—21no—
248#103—21no—
249#103—21no—
250#102—21no—
251#102—21no—
252#102—21no—
253#102—21no—
254C22⋊C8—21no—
255C22⋊C8—21no—
256#97—42no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8C2×C4×C3711yescharacteristic
260C4×C8C2×C4×C3711yescharacteristic
261C4×C8D4×C3711yescharacteristic
262C4×C8D4×C3711yescharacteristic
263#95D4×C3712yes—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4C2×D4×C3711yescharacteristic
278C22×C4C2×D4×C3711yescharacteristic
279C22×C4#41×C3711yescharacteristic
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—22no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8C22⋊C4×C3711yescharacteristic
290C2×C8C22⋊C4×C3711yescharacteristic
291C2×C8C22⋊C4×C3711yescharacteristic
292C2×C8C22⋊C4×C3711yescharacteristic
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C42—41no—
306C42—41no—
307C42—41no—
308C42C22×C4×C3711yescharacteristic
309C42C2×D4×C3711yescharacteristic
310C42C2×D4×C3711yescharacteristic
311C42#41×C3711yescharacteristic
312C23—41no—
313C23#121×C3711yescharacteristic
314Q8—41no—
315Q8—41no—
316Q8—42no—
317Q8—81no—
318D4—41no—
319D4—41no—
320D4—42no—
321D4—81no—
322D4—81no—
323D4—81no—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4C2×C22⋊C4×C3711yescharacteristic
335C2×C4C4×D4×C3711yescharacteristic
336C2×C4C4×D4×C3711yescharacteristic
337C2×C4#121×C3711yescharacteristic
338C2×C4#124×C3711yescharacteristic
339C2×C4#127×C3711yescharacteristic
340C2×C4#135×C3711yescharacteristic
341C2×C4#135×C3711yescharacteristic
342C2×C4AGL(1,ℤ/8)×C3711yescharacteristic
343C2×C4#137×C3711yescharacteristic
344C8—21no—
345C8—21no—
346C8—21no—
347C8—21no—
348C8—42no—
349C8—81no—
350C8—81no—
351C22—21no—
352C22—21no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22C37×#39011yescharacteristic
357C22C37×#46311yescharacteristic
358C22C37×#46411yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—82no—
363C4—81no—
364C4—81no—
365C4—81no—
366C4C37×#41611yescharacteristic
367C4C37×#41711yescharacteristic
368C4C37×#44412yes—
369C4C37×#46311yescharacteristic
370C4C37×#46411yescharacteristic
371C2—21no—
372C2—21no—
373C2—161no—
374C2C37×#197511yescharacteristic
375C2C37×#197611yescharacteristic
376C2C37×#198311yescharacteristic
subgroupG/H
centerC2×C4×C37#121
commutatorC42C22×C4×C37
Frattini, cosocleC2×C42C23×C37
socleC22×C37#390
prime radicalC37×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C37 ◁ C2×C42×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C37×#1770C2semidirect
3. C37×#1564C2semidirect
4. C37×#1503C2semidirect
6. C37×#1481C2semidirect
12. C37×#487C4semidirect
13. C37×#487C4semidirect
14. C37×#486C4semidirect
15. C37×#486C4semidirect
162. #9754C37direct (N × H)
184. #1770C2×C37semidirect
185. #1564C2×C37semidirect
186. #1503C2×C37semidirect
188. #1481C2×C37semidirect
200. #487C4×C37semidirect
201. #487C4×C37semidirect
202. #486C4×C37semidirect
203. #486C4×C37semidirect
219. C37#9754direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#31312
C37×#17702
C37×#15642
C37×#15032
C37×#15022
C37×#14812
C37×#14802
#975437