Finite Group Databasebeta

← C37×#12797 #22634050

Subgroups — C37×#12797 #22634050

Order 9472.

680 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#127×C37C211yesmaximal, characteristic
2C2×C37×#353C212yesmaximal
3C37×#1799C212yesmaximal, complemented
4C37×#1393C212yesmaximal, complemented
5C2×#127×C37C2211yescharacteristic
6C2×C4×D4×C37—22no—
7C2×C4×D4×C37C411yescharacteristic, complemented
8C2×C4×D4×C37C2211yescharacteristic
9C2×#99×C37C2214yes—
10C37×#390—22no—
11C37×#390C411yescharacteristic, complemented
12C37×#390C2211yescharacteristic
13C37×#384C411yescharacteristic, complemented
14C37×#380—22no—
15C43×C37C411yescharacteristic, complemented
16C37×#353—24no—
17C37×#342—22no—
18C22×D4×C37—22no—
19C22×D4×C37C2×C411yescharacteristic
20C22×D4×C37C2311yescharacteristic
21C23×C4×C37—21no—
22C23×C4×C37D412yes—
23#127×C37—22no—
24#127×C37D412yes—
25#127×C37Q812yes—
26C4×D4×C37—41no—
27C4×D4×C37—41no—
28C4×D4×C37—48no—
29C4×D4×C37—41no—
30C4×D4×C37—41no—
31C2×C4⋊C4×C37—22no—
32C2×C4⋊C4×C37—42no—
33C2×C4⋊C4×C37C2×C411yescharacteristic
34C2×C4⋊C4×C37C2×C411yescharacteristic
35C2×C22⋊C4×C37—22no—
36C2×C22⋊C4×C37—21no—
37C2×C22⋊C4×C37—22no—
38C2×C22⋊C4×C37—24no—
39C2×C22⋊C4×C37D412yes—
40C2×C42×C37—21no—
41C2×C42×C37—21no—
42C2×C42×C37C2×C411yescharacteristic
43C2×C42×C37C2×C411yescharacteristic
44C2×C42×C37C2×C411yescharacteristic
45#99×C37—24no—
46#99×C37—24no—
47#95×C37—42no—
48C24×C37—21no—
49C24×C37C2×D412yes—
50C2×D4×C37—24no—
51C2×D4×C37—24no—
52C2×D4×C37—21no—
53C2×D4×C37—41no—
54C2×D4×C37—41no—
55C2×D4×C37—41no—
56C2×D4×C37—41no—
57C2×D4×C37—44no—
58C2×D4×C37—44no—
59C2×D4×C37C4⋊C412yescomplemented
60C2×D4×C37C2×D411yescharacteristic
61C2×D4×C37C2×Q811yescharacteristic
62C2×D4×C37#4112yes—
63C22×C4×C37—21no—
64C22×C4×C37—21no—
65C22×C4×C37—22no—
66C22×C4×C37—22no—
67C22×C4×C37—21no—
68C22×C4×C37—44no—
69C22×C4×C37—44no—
70C22×C4×C37—41no—
71C22×C4×C37—41no—
72C22×C4×C37—44no—
73C22×C4×C37C22⋊C411yescharacteristic
74C22×C4×C37C22⋊C412yes—
75C22×C4×C37C22⋊C411yescharacteristic
76C22×C4×C37C22×C411yescharacteristic
77C4⋊C4×C37—41no—
78C4⋊C4×C37—41no—
79C4⋊C4×C37—44no—
80C4⋊C4×C37—84no—
81C22⋊C4×C37—42no—
82C22⋊C4×C37—44no—
83C22⋊C4×C37—44no—
84C22⋊C4×C37—42no—
85C22⋊C4×C37—44no—
86C22⋊C4×C37—44no—
87C42×C37—24no—
88C42×C37—21no—
89C42×C37—21no—
90C42×C37—21no—
91C42×C37—41no—
92C42×C37—42no—
93C42×C37C4⋊C412yescomplemented
94C23×C37—22no—
95C23×C37—22no—
96C23×C37—22no—
97C23×C37—42no—
98C23×C37—42no—
99C23×C37—42no—
100C23×C37—42no—
101C23×C37—41no—
102C23×C37—41no—
103C23×C37C2×C22⋊C411yescharacteristic
104C23×C37#12112yes—
105C23×C37C22⋊Q812yes—
106D4×C37—41no—
107D4×C37—41no—
108D4×C37—42no—
109D4×C37—42no—
110D4×C37—48no—
111D4×C37—48no—
112D4×C37—42no—
113C2×C4×C37—22no—
114C2×C4×C37—22no—
115C2×C4×C37—22no—
116C2×C4×C37—21no—
117C2×C4×C37—21no—
118C2×C4×C37—22no—
119C2×C4×C37—22no—
120C2×C4×C37—21no—
121C2×C4×C37—22no—
122C2×C4×C37—41no—
123C2×C4×C37—41no—
124C2×C4×C37—44no—
125C2×C4×C37—41no—
126C2×C4×C37—42no—
127C2×C4×C37—42no—
128C2×C4×C37—41no—
129C2×C4×C37—82no—
130C2×C4×C37—82no—
131C2×C4×C37—82no—
132C2×C4×C37—84no—
133C2×C4×C37—84no—
134C2×C4×C37—84no—
135C2×C4×C37#9911yescharacteristic
136C2×C4×C37#9911yescharacteristic
137C2×C4×C37C2×C4⋊C411yescharacteristic
138C2×C4×C37#11711yescharacteristic
139#12797C3711yesmaximal, characteristic, direct product factor, Sylow
140C22×C37—21no—
141C22×C37—21no—
142C22×C37—22no—
143C22×C37—22no—
144C22×C37—42no—
145C22×C37—42no—
146C22×C37—44no—
147C22×C37—42no—
148C22×C37—42no—
149C22×C37—42no—
150C22×C37—82no—
151C22×C37—82no—
152C22×C37—81no—
153C22×C37—81no—
154C22×C37#38011yescharacteristic
155C22×C37C2×#9911yescharacteristic
156C22×C37#41011yescharacteristic
157C4×C37—24no—
158C4×C37—21no—
159C4×C37—21no—
160C4×C37—21no—
161C4×C37—42no—
162C4×C37—42no—
163C4×C37—41no—
164C4×C37—81no—
165C4×C37—164no—
166C4×C37#35312yescomplemented
167C4×#127C2×C3711yescharacteristic
168C2×#353C2×C3712yes—
169#1799C2×C3712yescomplemented
170#1393C2×C3712yescomplemented
171C2×C37—21no—
172C2×C37—21no—
173C2×C37—42no—
174C2×C37—42no—
175C2×C37—82no—
176C2×C37#179911yescharacteristic
177C2×C37C2×#35311yescharacteristic
178C2×C37#212611yescharacteristic
179C2×#127C22×C3711yescharacteristic
180C2×C4×D4—22no—
181C2×C4×D4C4×C3711yescharacteristic, complemented
182C2×C4×D4C22×C3711yescharacteristic
183C2×#99C22×C3714yes—
184#390—22no—
185#390C4×C3711yescharacteristic, complemented
186#390C22×C3711yescharacteristic
187#384C4×C3711yescharacteristic, complemented
188#380—22no—
189C43C4×C3711yescharacteristic, complemented
190#353—24no—
191#342—22no—
192C37#1279711yescharacteristic, direct product factor, Sylow
193C22×D4—22no—
194C22×D4C2×C4×C3711yescharacteristic
195C22×D4C23×C3711yescharacteristic
196C23×C4—21no—
197C23×C4D4×C3712yes—
198#127—22no—
199#127D4×C3712yes—
200#127Q8×C3712yes—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4—48no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4C2×C4×C3711yescharacteristic
209C2×C4⋊C4C2×C4×C3711yescharacteristic
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4D4×C3712yes—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42C2×C4×C3711yescharacteristic
218C2×C42C2×C4×C3711yescharacteristic
219C2×C42C2×C4×C3711yescharacteristic
220#99—24no—
221#99—24no—
222#95—42no—
223C24—21no—
224C24C2×D4×C3712yes—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—21no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4C4⋊C4×C3712yescomplemented
235C2×D4C2×D4×C3711yescharacteristic
236C2×D4C2×Q8×C3711yescharacteristic
237C2×D4#41×C3712yes—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—44no—
248C22×C4C22⋊C4×C3711yescharacteristic
249C22×C4C22⋊C4×C3711yescharacteristic
250C22×C4C22⋊C4×C3712yes—
251C22×C4C22×C4×C3711yescharacteristic
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—44no—
255C4⋊C4—84no—
256C22⋊C4—42no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—42no—
260C22⋊C4—44no—
261C22⋊C4—44no—
262C42—24no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—41no—
267C42—42no—
268C42C4⋊C4×C3712yescomplemented
269C23—22no—
270C23—22no—
271C23—22no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—41no—
277C23—41no—
278C23C2×C22⋊C4×C3711yescharacteristic
279C23#121×C3712yes—
280C23C22⋊Q8×C3712yes—
281D4—41no—
282D4—41no—
283D4—42no—
284D4—42no—
285D4—48no—
286D4—48no—
287D4—42no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—44no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—84no—
308C2×C4—84no—
309C2×C4—84no—
310C2×C4#99×C3711yescharacteristic
311C2×C4#99×C3711yescharacteristic
312C2×C4C2×C4⋊C4×C3711yescharacteristic
313C2×C4#117×C3711yescharacteristic
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—44no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—82no—
325C22—82no—
326C22—81no—
327C22—81no—
328C22C37×#38011yescharacteristic
329C22C2×#99×C3711yescharacteristic
330C22C37×#41011yescharacteristic
331C4—24no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—42no—
336C4—42no—
337C4—41no—
338C4—81no—
339C4—164no—
340C4C37×#35312yescomplemented
341C2—21no—
342C2—21no—
343C2—42no—
344C2—42no—
345C2—82no—
346C2C37×#179911yescharacteristic
347C2C2×C37×#35311yescharacteristic
348C2C37×#212611yescharacteristic
subgroupG/H
center, socleC22×C37C2×#99
commutatorC22×C4C22×C4×C37
Frattini, cosocleC22×D4C23×C37
prime radicalC2×#127×C37C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C37 ◁ C22×C4×C37 ◁ C2×C4×D4×C37 ◁ G

Complements

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

NH ≅ G/Ntype
3. C37×#1799C2semidirect
4. C37×#1393C2semidirect
7. C2×C4×D4×C37C4semidirect
11. C37×#390C4semidirect
13. C37×#384C4semidirect
15. C43×C37C4semidirect
59. C2×D4×C37C4⋊C4semidirect
93. C42×C37C4⋊C4semidirect
139. #12797C37direct (N × H)
166. C4×C37#353semidirect
169. #1799C2×C37semidirect
170. #1393C2×C37semidirect
181. C2×C4×D4C4×C37semidirect
185. #390C4×C37semidirect
187. #384C4×C37semidirect
189. C43C4×C37semidirect
192. C37#12797direct (N × H)
234. C2×D4C4⋊C4×C37semidirect
268. C42C4⋊C4×C37semidirect
340. C4C37×#353semidirect

Maximal subgroups

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

subgroupindex
C4×#127×C372
C2×C37×#3532
C37×#17992
C37×#13932
#1279737