Finite Group Databasebeta

← C37×#33804 #22655057

Subgroups — C37×#33804 #22655057

Order 9472.

496 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3168C211yesmaximal, characteristic
2C2×C37×#500C211yesmaximal, characteristic, complemented
3C37×#2255C212yesmaximal, complemented
4C37×#2243C212yesmaximal, complemented
5C37×#2235C212yesmaximal, complemented
6C37×#2232C212yesmaximal
7C37×#2227C212yesmaximal, complemented
8C37×#2226C212yesmaximal, complemented
9C37×#2163C211yesmaximal, characteristic, complemented
10C2×C4⋊Q8×C37—21no—
11C37×#529—21no—
12C37×#500—21no—
13C37×#500—21no—
14C37×#500C2212yes—
15C37×#500C2212yes—
16C37×#499C2211yescharacteristic
17C37×#495C2211yescharacteristic
18C37×#488—21no—
19C37×#467C2211yescharacteristic
20C37×#466C2211yescharacteristic
21C37×#465—21no—
22C4×Q16×C37C2211yescharacteristic
23C4×SD16×C37—21no—
24C4×D8×C37C2211yescharacteristic
25C2×#107×C37—21no—
26C2×#107×C37C2212yes—
27C2×C4×C8×C37C2211yescharacteristic
28C37×#367C2214yes—
29C37×#366C2214yes—
30C37×C4⋊C16C2212yes—
31C37×#358C2214yes—
32C37×#357C2214yes—
33C37×C22⋊C16C2212yes—
34C37×#346C2212yes—
35C22×Q8×C37—41no—
36C2×Q16×C37C2311yescharacteristic
37C2×SD16×C37—21no—
38C2×D8×C37C2311yescharacteristic
39C22×C8×C37D411yescharacteristic
40C22×C8×C37C2311yescharacteristic
41C4⋊Q8×C37—22no—
42C4⋊Q8×C37—22no—
43C4⋊Q8×C37—41no—
44C4⋊Q8×C37—41no—
45#126×C37—41no—
46#125×C37—21no—
47#124×C37—21no—
48#123×C37—41no—
49C22⋊Q8×C37—21no—
50#121×C37—21no—
51C4×Q8×C37—21no—
52C4×D4×C37—21no—
53C4×D4×C37—41no—
54#117×C37—41no—
55C2×C4⋊C4×C37—22no—
56C2×C4⋊C4×C37—41no—
57C2×C42×C37D411yescharacteristic
58C2×C16×C37C2314yes—
59#107×C37—22no—
60#107×C37—22no—
61#107×C37—22no—
62#107×C37C2314yes—
63#107×C37C2311yescharacteristic
64#107×C37C2311yescharacteristic
65#106×C37—21no—
66#103×C37—21no—
67#103×C37—21no—
68#102×C37—21no—
69#102×C37—21no—
70C4×C8×C37D411yescharacteristic
71C4×C8×C37D411yescharacteristic
72C4×C8×C37C2311yescharacteristic
73C4×C8×C37C2311yescharacteristic
74C2×Q8×C37—21no—
75C2×Q8×C37—42no—
76C2×Q8×C37—42no—
77C2×Q8×C37—84no—
78C2×D4×C37—21no—
79C2×D4×C37—41no—
80C22×C4×C37—42no—
81C22×C4×C37—41no—
82C22×C4×C37D811yescharacteristic
83C22×C4×C37D811yescharacteristic
84C22×C4×C37C2×D411yescharacteristic
85Q16×C37—22no—
86SD16×C37—42no—
87D8×C37—22no—
88C2×C8×C37—22no—
89C2×C8×C37—22no—
90C2×C8×C37C2×D411yescharacteristic
91C2×C8×C37C2×D411yescharacteristic
92C2×C8×C37C2×D411yescharacteristic
93C2×C8×C37C2411yescharacteristic
94C4⋊C4×C37—21no—
95C4⋊C4×C37—21no—
96C4⋊C4×C37—24no—
97C4⋊C4×C37—42no—
98C4⋊C4×C37—42no—
99C4⋊C4×C37—42no—
100C4⋊C4×C37—41no—
101C4⋊C4×C37—41no—
102C4⋊C4×C37—41no—
103C4⋊C4×C37—41no—
104C22⋊C4×C37—41no—
105C22⋊C4×C37—41no—
106C22⋊C4×C37—41no—
107C22⋊C4×C37—41no—
108C22⋊C4×C37—41no—
109C42×C37—41no—
110C42×C37D811yescharacteristic
111C42×C37D811yescharacteristic
112C42×C37C2×D411yescharacteristic
113C42×C37C2×D411yescharacteristic
114C16×C37—24no—
115C23×C37—41no—
116C23×C37C2×D811yescharacteristic
117Q8×C37—41no—
118Q8×C37—41no—
119Q8×C37—84no—
120Q8×C37—84no—
121D4×C37—41no—
122D4×C37—41no—
123D4×C37—82no—
124C2×C4×C37—22no—
125C2×C4×C37—21no—
126C2×C4×C37—21no—
127C2×C4×C37—22no—
128C2×C4×C37—41no—
129C2×C4×C37—41no—
130C2×C4×C37—41no—
131C2×C4×C37—44no—
132C2×C4×C37—84no—
133C2×C4×C37—82no—
134C2×C4×C37C2×D811yescharacteristic
135C2×C4×C37C2×D811yescharacteristic
136C2×C4×C37C2×D811yescharacteristic
137C2×C4×C37C2×D811yescharacteristic
138C2×C4×C37C2×D811yescharacteristic
139C2×C4×C37C22×D411yescharacteristic
140C8×C37—21no—
141C8×C37—21no—
142C8×C37#13712yes—
143C8×C372−1+412yes—
144#33804C3711yesmaximal, characteristic, direct product factor, Sylow
145C22×C37—21no—
146C22×C37—21no—
147C22×C37—81no—
148C22×C37—81no—
149C22×C37—81no—
150C22×C37#50912yes—
151C22×C37C22×D811yescharacteristic
152C4×C37—21no—
153C4×C37—21no—
154C4×C37—21no—
155C4×C37—21no—
156C4×C37—84no—
157C4×C37—81no—
158C4×C37—81no—
159C4×C37—81no—
160C4×C37#51012yes—
161C4×C37#53611yescharacteristic
162C4×C37C2×#13711yescharacteristic
163#3168C2×C3711yescharacteristic
164C2×#500C2×C3711yescharacteristic, complemented
165#2255C2×C3712yescomplemented
166#2243C2×C3712yescomplemented
167#2235C2×C3712yescomplemented
168#2232C2×C3712yes—
169#2227C2×C3712yescomplemented
170#2226C2×C3712yescomplemented
171#2163C2×C3711yescharacteristic, complemented
172C2×C37—22no—
173C2×C37—161no—
174C2×C37#322611yescharacteristic
175C2×C37C2×#50911yescharacteristic
176C2×C37C2×#51011yescharacteristic
177C2×C4⋊Q8—21no—
178#529—21no—
179#500—21no—
180#500—21no—
181#500C22×C3712yes—
182#500C22×C3712yes—
183#499C22×C3711yescharacteristic
184#495C22×C3711yescharacteristic
185#488—21no—
186#467C22×C3711yescharacteristic
187#466C22×C3711yescharacteristic
188#465—21no—
189C4×Q16C22×C3711yescharacteristic
190C4×SD16—21no—
191C4×D8C22×C3711yescharacteristic
192C2×#107—21no—
193C2×#107C22×C3712yes—
194C2×C4×C8C22×C3711yescharacteristic
195#367C22×C3714yes—
196#366C22×C3714yes—
197C4⋊C16C22×C3712yes—
198#358C22×C3714yes—
199#357C22×C3714yes—
200C22⋊C16C22×C3712yes—
201#346C22×C3712yes—
202C37#3380411yescharacteristic, direct product factor, Sylow
203C22×Q8—41no—
204C2×Q16C23×C3711yescharacteristic
205C2×SD16—21no—
206C2×D8C23×C3711yescharacteristic
207C22×C8D4×C3711yescharacteristic
208C22×C8C23×C3711yescharacteristic
209C4⋊Q8—22no—
210C4⋊Q8—22no—
211C4⋊Q8—41no—
212C4⋊Q8—41no—
213#126—41no—
214#125—21no—
215#124—21no—
216#123—41no—
217C22⋊Q8—21no—
218#121—21no—
219C4×Q8—21no—
220C4×D4—21no—
221C4×D4—41no—
222#117—41no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—41no—
225C2×C42D4×C3711yescharacteristic
226C2×C16C23×C3714yes—
227#107—22no—
228#107—22no—
229#107—22no—
230#107C23×C3714yes—
231#107C23×C3711yescharacteristic
232#107C23×C3711yescharacteristic
233#106—21no—
234#103—21no—
235#103—21no—
236#102—21no—
237#102—21no—
238C4×C8D4×C3711yescharacteristic
239C4×C8D4×C3711yescharacteristic
240C4×C8C23×C3711yescharacteristic
241C4×C8C23×C3711yescharacteristic
242C2×Q8—21no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—84no—
246C2×D4—21no—
247C2×D4—41no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4D8×C3711yescharacteristic
251C22×C4D8×C3711yescharacteristic
252C22×C4C2×D4×C3711yescharacteristic
253Q16—22no—
254SD16—42no—
255D8—22no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8C2×D4×C3711yescharacteristic
259C2×C8C2×D4×C3711yescharacteristic
260C2×C8C2×D4×C3711yescharacteristic
261C2×C8C24×C3711yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—24no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C42—41no—
278C42D8×C3711yescharacteristic
279C42D8×C3711yescharacteristic
280C42C2×D4×C3711yescharacteristic
281C42C2×D4×C3711yescharacteristic
282C16—24no—
283C23—41no—
284C23C2×D8×C3711yescharacteristic
285Q8—41no—
286Q8—41no—
287Q8—84no—
288Q8—84no—
289D4—41no—
290D4—41no—
291D4—82no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—44no—
300C2×C4—84no—
301C2×C4—82no—
302C2×C4C2×D8×C3711yescharacteristic
303C2×C4C2×D8×C3711yescharacteristic
304C2×C4C2×D8×C3711yescharacteristic
305C2×C4C2×D8×C3711yescharacteristic
306C2×C4C2×D8×C3711yescharacteristic
307C2×C4C22×D4×C3711yescharacteristic
308C8—21no—
309C8—21no—
310C8#137×C3712yes—
311C82−1+4×C3712yes—
312C22—21no—
313C22—21no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22C37×#50912yes—
318C22C22×D8×C3711yescharacteristic
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—84no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4C37×#51012yes—
328C4C37×#53611yescharacteristic
329C4C2×#137×C3711yescharacteristic
330C2—22no—
331C2—161no—
332C2C37×#322611yescharacteristic
333C2C2×C37×#50911yescharacteristic
334C2C2×C37×#51011yescharacteristic
subgroupG/H
center, socleC22×C37C22×D8
commutator, Frattini, cosocleC2×C8C24×C37
prime radicalC37×#466C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C37 ◁ C2×C42×C37 ◁ C2×C4×C8×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C37×#500C2semidirect
3. C37×#2255C2semidirect
4. C37×#2243C2semidirect
5. C37×#2235C2semidirect
7. C37×#2227C2semidirect
8. C37×#2226C2semidirect
9. C37×#2163C2semidirect
144. #33804C37direct (N × H)
164. C2×#500C2×C37semidirect
165. #2255C2×C37semidirect
166. #2243C2×C37semidirect
167. #2235C2×C37semidirect
169. #2227C2×C37semidirect
170. #2226C2×C37semidirect
171. #2163C2×C37semidirect
202. C37#33804direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#31682
C2×C37×#5002
C37×#22552
C37×#22432
C37×#22352
C37×#22322
C37×#22272
C37×#22262
C37×#21632
#3380437