Finite Group Databasebeta

← C37×#9206 #22630459

Subgroups — C37×#9206 #22630459

Order 9472.

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

normal subgroups only

This group has 578 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
1C37×#2618C211yesmaximal, characteristic
2C37×#2313C211yesmaximal, characteristic, complemented
3C37×#1519C212yesmaximal
4C2×C37×#324C211yesmaximal, characteristic, complemented
5C37×#1289C212yesmaximal, complemented
6C2×#123×C37—21no—
7C2×#121×C37C2211yescharacteristic
8C22×C4⋊C4×C37C2211yescharacteristic
9C2×C22⋊C8×C37C2212yes—
10C37×#398C412yescomplemented
11C37×#396—22no—
12C37×#394C412yescomplemented
13C37×#387—24no—
14C37×#386—21no—
15C37×#381—21no—
16C37×#380C2211yescharacteristic
17C37×#380C2212yes—
18C4×C22⋊C4×C37—21no—
19C37×#327—24no—
20C37×#324—22no—
21C22×D4×C37—21no—
22C22×D4×C37C2×C411yescharacteristic
23C23×C4×C37D411yescharacteristic
24C23×C4×C37D411yescharacteristic
25C23×C4×C37C2311yescharacteristic
26C22×C8×C37—22no—
27#123×C37—44no—
28#121×C37—22no—
29#121×C37—22no—
30C2×C4⋊C4×C37—22no—
31C2×C4⋊C4×C37—22no—
32C2×C4⋊C4×C37—22no—
33C2×C4⋊C4×C37—22no—
34C2×C4⋊C4×C37—21no—
35C2×C4⋊C4×C37C2×C411yescharacteristic
36C2×C4⋊C4×C37C2×C412yes—
37C2×C4⋊C4×C37D412yes—
38C2×C22⋊C4×C37—21no—
39C2×C22⋊C4×C37—22no—
40C2×C22⋊C4×C37—22no—
41C2×C22⋊C4×C37—21no—
42C2×C22⋊C4×C37—22no—
43C2×C42×C37—22no—
44C22⋊C8×C37—22no—
45C22⋊C8×C37—22no—
46#95×C37—22no—
47#95×C37—24no—
48#95×C37—22no—
49#95×C37C2×C412yes—
50#95×C37D414yes—
51C24×C37—21no—
52C24×C37C2×D411yescharacteristic
53C2×D4×C37—22no—
54C2×D4×C37—41no—
55C2×D4×C37—41no—
56C2×D4×C37—42no—
57C2×D4×C37—42no—
58C2×D4×C37—81no—
59C22×C4×C37—21no—
60C22×C4×C37—21no—
61C22×C4×C37—21no—
62C22×C4×C37—22no—
63C22×C4×C37—22no—
64C22×C4×C37—21no—
65C22×C4×C37—21no—
66C22×C4×C37—22no—
67C22×C4×C37—24no—
68C22×C4×C37—81no—
69C22×C4×C37—81no—
70C22×C4×C37C22⋊C412yes—
71C22×C4×C37C22⋊C412yes—
72C22×C4×C37C22×C411yescharacteristic
73C22×C4×C37C2×D412yes—
74C22×C4×C37C2×D411yescharacteristic
75C22×C4×C37#4112yes—
76C2×C8×C37—42no—
77C2×C8×C37—82no—
78C4⋊C4×C37—24no—
79C4⋊C4×C37—24no—
80C4⋊C4×C37—22no—
81C4⋊C4×C37—44no—
82C4⋊C4×C37—82no—
83C4⋊C4×C37—81no—
84C22⋊C4×C37—44no—
85C22⋊C4×C37—42no—
86C22⋊C4×C37—42no—
87C22⋊C4×C37—42no—
88C22⋊C4×C37—42no—
89C22⋊C4×C37—82no—
90C42×C37—82no—
91C23×C37—41no—
92C23×C37—41no—
93C23×C37—42no—
94C23×C37—42no—
95C23×C37—41no—
96C23×C37—41no—
97C23×C37—41no—
98C23×C37C2×C22⋊C411yescharacteristic
99C23×C37#12011yescharacteristic
100C23×C37#12311yescharacteristic
101C23×C37#13512yes—
102C23×C37AGL(1,ℤ/8)12yes—
103D4×C37—82no—
104D4×C37—82no—
105D4×C37—82no—
106C2×C4×C37—22no—
107C2×C4×C37—22no—
108C2×C4×C37—22no—
109C2×C4×C37—22no—
110C2×C4×C37—41no—
111C2×C4×C37—42no—
112C2×C4×C37—42no—
113C2×C4×C37—41no—
114C2×C4×C37—42no—
115C2×C4×C37—42no—
116C2×C4×C37—42no—
117C2×C4×C37—42no—
118C2×C4×C37—44no—
119C2×C4×C37—42no—
120C2×C4×C37—44no—
121C2×C4×C37—42no—
122C2×C4×C37—41no—
123C2×C4×C37—41no—
124C2×C4×C37—42no—
125C2×C4×C37—41no—
126C2×C4×C37—44no—
127C2×C4×C37—81no—
128C2×C4×C37C4×D412yes—
129C2×C4×C37#12112yes—
130C8×C37—82no—
131#9206C3711yesmaximal, characteristic, direct product factor, Sylow
132C22×C37—21no—
133C22×C37—21no—
134C22×C37—21no—
135C22×C37—21no—
136C22×C37—21no—
137C22×C37—21no—
138C22×C37—42no—
139C22×C37—42no—
140C22×C37—82no—
141C22×C37—82no—
142C22×C37—81no—
143C22×C37—81no—
144C22×C37—81no—
145C22×C37#38611yescharacteristic
146C22×C37#41611yescharacteristic
147C22×C37#41811yescharacteristic
148C22×C37#44912yes—
149C22×C37#48212yes—
150C4×C37—22no—
151C4×C37—22no—
152C4×C37—42no—
153C4×C37—42no—
154C4×C37—82no—
155C4×C37—84no—
156C4×C37—81no—
157#2618C2×C3711yescharacteristic
158#2313C2×C3711yescharacteristic, complemented
159#1519C2×C3712yes—
160C2×#324C2×C3711yescharacteristic, complemented
161#1289C2×C3712yescomplemented
162C2×C37—21no—
163C2×C37—22no—
164C2×C37—21no—
165C2×C37—22no—
166C2×C37—161no—
167C2×C37#187511yescharacteristic
168C2×C37#191211yescharacteristic
169C2×C37#191311yescharacteristic
170C2×#123—21no—
171C2×#121C22×C3711yescharacteristic
172C22×C4⋊C4C22×C3711yescharacteristic
173C2×C22⋊C8C22×C3712yes—
174#398C4×C3712yescomplemented
175#396—22no—
176#394C4×C3712yescomplemented
177#387—24no—
178#386—21no—
179#381—21no—
180#380C22×C3711yescharacteristic
181#380C22×C3712yes—
182C4×C22⋊C4—21no—
183#327—24no—
184#324—22no—
185C37#920611yescharacteristic, direct product factor, Sylow
186C22×D4—21no—
187C22×D4C2×C4×C3711yescharacteristic
188C23×C4D4×C3711yescharacteristic
189C23×C4D4×C3711yescharacteristic
190C23×C4C23×C3711yescharacteristic
191C22×C8—22no—
192#123—44no—
193#121—22no—
194#121—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4C2×C4×C3711yescharacteristic
201C2×C4⋊C4C2×C4×C3712yes—
202C2×C4⋊C4D4×C3712yes—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C42—22no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211#95—22no—
212#95—24no—
213#95—22no—
214#95C2×C4×C3712yes—
215#95D4×C3714yes—
216C24—21no—
217C24C2×D4×C3711yescharacteristic
218C2×D4—22no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—81no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—81no—
234C22×C4—81no—
235C22×C4C22⋊C4×C3712yes—
236C22×C4C22⋊C4×C3712yes—
237C22×C4C22×C4×C3711yescharacteristic
238C22×C4C2×D4×C3712yes—
239C22×C4C2×D4×C3711yescharacteristic
240C22×C4#41×C3712yes—
241C2×C8—42no—
242C2×C8—82no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—22no—
246C4⋊C4—44no—
247C4⋊C4—82no—
248C4⋊C4—81no—
249C22⋊C4—44no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—82no—
255C42—82no—
256C23—41no—
257C23—41no—
258C23—42no—
259C23—42no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23C2×C22⋊C4×C3711yescharacteristic
264C23#120×C3711yescharacteristic
265C23#123×C3711yescharacteristic
266C23#135×C3712yes—
267C23AGL(1,ℤ/8)×C3712yes—
268D4—82no—
269D4—82no—
270D4—82no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—44no—
292C2×C4—81no—
293C2×C4C4×D4×C3712yes—
294C2×C4#121×C3712yes—
295C8—82no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—42no—
304C22—82no—
305C22—82no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22C37×#38611yescharacteristic
310C22C37×#41611yescharacteristic
311C22C37×#41811yescharacteristic
312C22C37×#44912yes—
313C22C37×#48212yes—
314C4—22no—
315C4—22no—
316C4—42no—
317C4—42no—
318C4—82no—
319C4—84no—
320C4—81no—
321C2—21no—
322C2—22no—
323C2—21no—
324C2—22no—
325C2—161no—
326C2C37×#187511yescharacteristic
327C2C37×#191211yescharacteristic
328C2C37×#191311yescharacteristic
subgroupG/H
center, socleC22×C37#386
commutatorC22×C4C22×C4×C37
Frattini, cosocleC23×C4C23×C37
prime radicalC2×#121×C37C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C37 ◁ C23×C4×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C37×#2313C2semidirect
4. C2×C37×#324C2semidirect
5. C37×#1289C2semidirect
10. C37×#398C4semidirect
12. C37×#394C4semidirect
131. #9206C37direct (N × H)
158. #2313C2×C37semidirect
160. C2×#324C2×C37semidirect
161. #1289C2×C37semidirect
174. #398C4×C37semidirect
176. #394C4×C37semidirect
185. C37#9206direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#26182
C37×#23132
C37×#15192
C2×C37×#3242
C37×#12892
#920637