Finite Group Databasebeta

← C37×#18221 #22639474

Subgroups — C37×#18221 #22639474

Order 9472.

890 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3502C211yesmaximal, characteristic
2C37×#3104C212yesmaximal
3C37×#2581C211yesmaximal, characteristic
4C37×#1617C214yesmaximal
5C37×#1605C214yesmaximal
6C37×#1518C212yesmaximal
7C37×#1461C211yesmaximal, characteristic
8Q82×C37—22no—
9C37×#557—22no—
10C37×#557—24no—
11C37×#544—21no—
12C37×#544C2214yes—
13C37×#533—22no—
14C37×#533C2212yes—
15C37×#519—21no—
16C2×C4×Q8×C37C2211yescharacteristic
17C2×C4×Q8×C37C2211yescharacteristic
18C37×#484C2214yes—
19C37×#477—22no—
20C37×#462—22no—
21C37×#451C2214yes—
22C4×Q16×C37—24no—
23C37×#431C2212yes—
24C37×#400C2212yes—
25C37×#395C2212yes—
26C37×#391—21no—
27C37×#389—21no—
28C37×#387—22no—
29C37×#382C2211yescharacteristic
30C37×#328C2218yes—
31C37×#324C2214yes—
32C22×Q8×C37D411yescharacteristic
33C22×Q8×C37C2311yescharacteristic
34C2×Q16×C37—24no—
35C4⋊Q8×C37—22no—
36C4⋊Q8×C37—22no—
37C4⋊Q8×C37—24no—
38C4⋊Q8×C37—28no—
39C4⋊Q8×C37D412yes—
40C4⋊Q8×C37D412yes—
41#125×C37—24no—
42#125×C37—24no—
43#125×C37—28no—
44#125×C37—22no—
45C22⋊Q8×C37—22no—
46C22⋊Q8×C37—24no—
47C22⋊Q8×C37C2314yes—
48C4×Q8×C37—24no—
49C4×Q8×C37—24no—
50C4×Q8×C37—24no—
51C4×Q8×C37—24no—
52C4×Q8×C37—22no—
53C4×Q8×C37—22no—
54C4×Q8×C37—22no—
55C4×Q8×C37—22no—
56C4×Q8×C37—22no—
57C4×Q8×C37D414yes—
58#117×C37—24no—
59#117×C37—22no—
60C2×C4⋊C4×C37—21no—
61C2×C4⋊C4×C37—21no—
62C2×C4⋊C4×C37—22no—
63C2×C4⋊C4×C37D411yescharacteristic
64C2×C4⋊C4×C37C2311yescharacteristic
65C2×C4⋊C4×C37C2312yes—
66C2×C42×C37—21no—
67C2×C42×C37D412yes—
68C2×C42×C37C2311yescharacteristic
69#107×C37—24no—
70C4⋊C8×C37—22no—
71#103×C37—24no—
72#103×C37—24no—
73C22⋊C8×C37C2314yes—
74C4×C8×C37—22no—
75#95×C37—21no—
76#95×C37—22no—
77#95×C37—22no—
78#95×C37—22no—
79#95×C37C2312yes—
80C2×Q8×C37—24no—
81C2×Q8×C37—22no—
82C2×Q8×C37—22no—
83C2×Q8×C37—22no—
84C2×Q8×C37—22no—
85C2×Q8×C37—42no—
86C2×Q8×C37Q1614yescomplemented
87C2×Q8×C37C2×D414yes—
88C22×C4×C37—21no—
89C22×C4×C37—22no—
90C22×C4×C37—21no—
91C22×C4×C37C2×D412yes—
92C22×C4×C37C2×D412yes—
93C22×C4×C37C2×D411yescharacteristic
94C22×C4×C37C2×D411yescharacteristic
95C22×C4×C37C2411yescharacteristic
96Q16×C37—48no—
97C2×C8×C37—24no—
98C4⋊C4×C37—24no—
99C4⋊C4×C37—24no—
100C4⋊C4×C37—24no—
101C4⋊C4×C37—28no—
102C4⋊C4×C37—22no—
103C4⋊C4×C37—22no—
104C4⋊C4×C37—22no—
105C4⋊C4×C37—22no—
106C4⋊C4×C37—24no—
107C4⋊C4×C37—24no—
108C4⋊C4×C37—24no—
109C4⋊C4×C37—22no—
110C4⋊C4×C37—44no—
111C4⋊C4×C37—41no—
112C4⋊C4×C37—41no—
113C4⋊C4×C37—42no—
114C4⋊C4×C37C2×D414yes—
115C22⋊C4×C37—24no—
116C42×C37—28no—
117C42×C37—22no—
118C42×C37—22no—
119C42×C37—24no—
120C42×C37—42no—
121C42×C37C2×D412yes—
122C42×C37C2×D412yes—
123C23×C37C22×D411yescharacteristic
124Q8×C37—28no—
125Q8×C37—24no—
126Q8×C37—44no—
127Q8×C37—44no—
128Q8×C37—42no—
129Q8×C37—84no—
130C2×C4×C37—22no—
131C2×C4×C37—22no—
132C2×C4×C37—22no—
133C2×C4×C37—24no—
134C2×C4×C37—24no—
135C2×C4×C37—22no—
136C2×C4×C37—41no—
137C2×C4×C37—41no—
138C2×C4×C37—41no—
139C2×C4×C37—41no—
140C2×C4×C37—42no—
141C2×C4×C37—42no—
142C2×C4×C37—42no—
143C2×C4×C37—42no—
144C2×C4×C37—41no—
145C2×C4×C37—42no—
146C2×C4×C37#12011yescharacteristic
147C2×C4×C37#12011yescharacteristic
148C2×C4×C37#12012yes—
149C2×C4×C37C2×Q1614yes—
150C2×C4×C37C2×Q1612yes—
151C2×C4×C37#13712yes—
152C2×C4×C37C22×D412yes—
153C8×C37—44no—
154#18221C3711yesmaximal, characteristic, direct product factor, Sylow
155C22×C37—21no—
156C22×C37—21no—
157C22×C37C2×#12011yescharacteristic
158C22×C37C22×Q1611yescharacteristic
159C22×C37C2×#13711yescharacteristic
160C4×C37—24no—
161C4×C37—21no—
162C4×C37—21no—
163C4×C37—22no—
164C4×C37—44no—
165C4×C37—42no—
166C4×C37—42no—
167C4×C37—44no—
168C4×C37—82no—
169C4×C37—81no—
170C4×C37#45114yes—
171#3502C2×C3711yescharacteristic
172#3104C2×C3712yes—
173#2581C2×C3711yescharacteristic
174#1617C2×C3714yes—
175#1605C2×C3714yes—
176#1518C2×C3712yes—
177#1461C2×C3711yescharacteristic
178C2×C37—21no—
179C2×C37—21no—
180C2×C37C2×#45111yescharacteristic
181C2×C37#302011yescharacteristic
182C2×C37#302511yescharacteristic
183Q82—22no—
184#557—22no—
185#557—24no—
186#544—21no—
187#544C22×C3714yes—
188#533—22no—
189#533C22×C3712yes—
190#519—21no—
191C2×C4×Q8C22×C3711yescharacteristic
192C2×C4×Q8C22×C3711yescharacteristic
193#484C22×C3714yes—
194#477—22no—
195#462—22no—
196#451C22×C3714yes—
197C4×Q16—24no—
198#431C22×C3712yes—
199#400C22×C3712yes—
200#395C22×C3712yes—
201#391—21no—
202#389—21no—
203#387—22no—
204#382C22×C3711yescharacteristic
205#328C22×C3718yes—
206#324C22×C3714yes—
207C37#1822111yescharacteristic, direct product factor, Sylow
208C22×Q8D4×C3711yescharacteristic
209C22×Q8C23×C3711yescharacteristic
210C2×Q16—24no—
211C4⋊Q8—22no—
212C4⋊Q8—22no—
213C4⋊Q8—24no—
214C4⋊Q8—28no—
215C4⋊Q8D4×C3712yes—
216C4⋊Q8D4×C3712yes—
217#125—24no—
218#125—24no—
219#125—28no—
220#125—22no—
221C22⋊Q8—22no—
222C22⋊Q8—24no—
223C22⋊Q8C23×C3714yes—
224C4×Q8—24no—
225C4×Q8—24no—
226C4×Q8—24no—
227C4×Q8—24no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8D4×C3714yes—
234#117—24no—
235#117—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4D4×C3711yescharacteristic
240C2×C4⋊C4C23×C3711yescharacteristic
241C2×C4⋊C4C23×C3712yes—
242C2×C42—21no—
243C2×C42D4×C3712yes—
244C2×C42C23×C3711yescharacteristic
245#107—24no—
246C4⋊C8—22no—
247#103—24no—
248#103—24no—
249C22⋊C8C23×C3714yes—
250C4×C8—22no—
251#95—21no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95C23×C3712yes—
256C2×Q8—24no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—42no—
262C2×Q8Q16×C3714yescomplemented
263C2×Q8C2×D4×C3714yes—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4C2×D4×C3712yes—
268C22×C4C2×D4×C3712yes—
269C22×C4C2×D4×C3711yescharacteristic
270C22×C4C2×D4×C3711yescharacteristic
271C22×C4C24×C3711yescharacteristic
272Q16—48no—
273C2×C8—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—28no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—44no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4C2×D4×C3714yes—
291C22⋊C4—24no—
292C42—28no—
293C42—22no—
294C42—22no—
295C42—24no—
296C42—42no—
297C42C2×D4×C3712yes—
298C42C2×D4×C3712yes—
299C23C22×D4×C3711yescharacteristic
300Q8—28no—
301Q8—24no—
302Q8—44no—
303Q8—42no—
304Q8—44no—
305Q8—84no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4#120×C3711yescharacteristic
323C2×C4#120×C3711yescharacteristic
324C2×C4#120×C3712yes—
325C2×C4C2×Q16×C3714yes—
326C2×C4C2×Q16×C3712yes—
327C2×C4#137×C3712yes—
328C2×C4C22×D4×C3712yes—
329C8—44no—
330C22—21no—
331C22—21no—
332C22C2×#120×C3711yescharacteristic
333C22C22×Q16×C3711yescharacteristic
334C22C2×#137×C3711yescharacteristic
335C4—24no—
336C4—21no—
337C4—21no—
338C4—22no—
339C4—44no—
340C4—42no—
341C4—42no—
342C4—44no—
343C4—82no—
344C4—81no—
345C4C37×#45114yes—
346C2—21no—
347C2—21no—
348C2C2×C37×#45111yescharacteristic
349C2C37×#302011yescharacteristic
350C2C37×#302511yescharacteristic
subgroupG/H
center, socleC22×C37C2×#120
commutator, Frattini, cosocleC22×C4C24×C37
prime radicalC23×C37C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C37 ◁ C2×C42×C37 ◁ G

Complements

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

NH ≅ G/Ntype
86. C2×Q8×C37Q16semidirect
154. #18221C37direct (N × H)
207. C37#18221direct (N × H)
262. C2×Q8Q16×C37semidirect

Maximal subgroups

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

subgroupindex
C37×#35022
C37×#31042
C37×#25812
C37×#16172
C37×#16052
C37×#15182
C37×#14612
#1822137