Finite Group Databasebeta

← C37×#9966 #22631219

Subgroups — C37×#9966 #22631219

Order 9472.

1020 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1020 conjugacy classes of subgroups — too many to draw. Only its 138 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
1C2×C37×#450C212yesmaximal
2C37×#2659C211yesmaximal, characteristic, complemented
3C2×C37×#328C212yesmaximal, complemented
4C37×#1438C211yesmaximal, characteristic
5C37×#1280C211yesmaximal, characteristic, complemented
6C23×D4×C37C2211yescharacteristic
7C22×SD16×C37—22no—
8C2×C22⋊Q8×C37C2212yes—
9C22×C22⋊C4×C37—22no—
10C37×#450—24no—
11C37×#450—24no—
12C2×#102×C37—22no—
13C2×C22⋊C8×C37C2212yes—
14C37×#398—22no—
15C37×#395—22no—
16C37×#380—22no—
17C2×#95×C37C2211yescharacteristic
18C2×#95×C37C2211yescharacteristic
19C37×#328—24no—
20C25×C37—21no—
21C22×Q8×C37D412yes—
22C22×D4×C37—22no—
23C22×D4×C37—22no—
24C22×D4×C37—21no—
25C22×D4×C37—44no—
26C22×D4×C37D412yes—
27C23×C4×C37—21no—
28C23×C4×C37D412yes—
29C23×C4×C37C2311yescharacteristic
30C2×SD16×C37—28no—
31C2×SD16×C37—48no—
32C22×C8×C37—22no—
33C22⋊Q8×C37—22no—
34C22⋊Q8×C37—22no—
35C22⋊Q8×C37—22no—
36C22⋊Q8×C37—22no—
37C2×C4⋊C4×C37—22no—
38C2×C4⋊C4×C37—22no—
39C2×C4⋊C4×C37—22no—
40C2×C4⋊C4×C37D412yes—
41C2×C22⋊C4×C37—22no—
42C2×C22⋊C4×C37—44no—
43C2×C22⋊C4×C37—44no—
44C2×C22⋊C4×C37—44no—
45#102×C37—28no—
46C22⋊C8×C37—22no—
47C22⋊C8×C37D414yes—
48#95×C37—24no—
49#95×C37—22no—
50#95×C37—22no—
51C24×C37—24no—
52C24×C37—22no—
53C24×C37—42no—
54C24×C37—42no—
55C24×C37—42no—
56C24×C37—41no—
57C24×C37—41no—
58C24×C37—42no—
59C24×C37—42no—
60C24×C37C2×D411yescharacteristic
61C2×Q8×C37—22no—
62C2×Q8×C37—22no—
63C2×Q8×C37—44no—
64C2×D4×C37—28no—
65C2×D4×C37—24no—
66C2×D4×C37—44no—
67C2×D4×C37—44no—
68C2×D4×C37—48no—
69C2×D4×C37—44no—
70C2×D4×C37—42no—
71C22×C4×C37—22no—
72C22×C4×C37—22no—
73C22×C4×C37—22no—
74C22×C4×C37—22no—
75C22×C4×C37—22no—
76C22×C4×C37—42no—
77C22×C4×C37—42no—
78C22×C4×C37—42no—
79C22×C4×C37—42no—
80C22×C4×C37—41no—
81C22×C4×C37—41no—
82C22×C4×C37—42no—
83C22×C4×C37SD1614yescomplemented
84C22×C4×C37C2×D412yes—
85C22×C4×C37C2×D411yescharacteristic
86C22×C4×C37C2×D412yes—
87C22×C4×C37#4111yescharacteristic
88SD16×C37—416no—
89C2×C8×C37—24no—
90C2×C8×C37—44no—
91C4⋊C4×C37—22no—
92C4⋊C4×C37—22no—
93C4⋊C4×C37—42no—
94C4⋊C4×C37—42no—
95C4⋊C4×C37—82no—
96C4⋊C4×C37—82no—
97C22⋊C4×C37—416no—
98C22⋊C4×C37—42no—
99C22⋊C4×C37—42no—
100C23×C37—21no—
101C23×C37—21no—
102C23×C37—22no—
103C23×C37—24no—
104C23×C37—44no—
105C23×C37—44no—
106C23×C37—42no—
107C23×C37—44no—
108C23×C37—44no—
109C23×C37—44no—
110C23×C37—44no—
111C23×C37—42no—
112C23×C37—44no—
113C23×C37—44no—
114C23×C37—82no—
115C23×C37—82no—
116C23×C37—82no—
117C23×C37—82no—
118C23×C37—82no—
119C23×C37—82no—
120C23×C37—81no—
121C23×C37—81no—
122C23×C37—81no—
123C23×C37—81no—
124C23×C37#12011yescharacteristic
125C23×C37#12111yescharacteristic
126C23×C37#12711yescharacteristic
127C23×C37C2×SD1612yes—
128C23×C37#13712yes—
129Q8×C37—44no—
130Q8×C37—82no—
131D4×C37—28no—
132D4×C37—44no—
133D4×C37—48no—
134C2×C4×C37—24no—
135C2×C4×C37—24no—
136C2×C4×C37—41no—
137C2×C4×C37—42no—
138C2×C4×C37—41no—
139C2×C4×C37—42no—
140C2×C4×C37—42no—
141C2×C4×C37—44no—
142C2×C4×C37—44no—
143C2×C4×C37—42no—
144C2×C4×C37—42no—
145C2×C4×C37—42no—
146C2×C4×C37—42no—
147C2×C4×C37—42no—
148C2×C4×C37—44no—
149C2×C4×C37—82no—
150C2×C4×C37—82no—
151C2×C4×C37—81no—
152C2×C4×C37—81no—
153C2×C4×C37#12012yes—
154C2×C4×C37#12112yes—
155C8×C37—44no—
156#9966C3711yesmaximal, characteristic, direct product factor, Sylow
157C22×C37—22no—
158C22×C37—22no—
159C22×C37—22no—
160C22×C37—22no—
161C22×C37—21no—
162C22×C37—21no—
163C22×C37—21no—
164C22×C37—21no—
165C22×C37—48no—
166C22×C37—44no—
167C22×C37—84no—
168C22×C37—84no—
169C22×C37—84no—
170C22×C37—84no—
171C22×C37—84no—
172C22×C37—82no—
173C22×C37—82no—
174C22×C37#39211yescharacteristic
175C22×C37#44812yes—
176C22×C37#45311yescharacteristic
177C22×C37#45411yescharacteristic
178C22×C37B(4,𝔽2)12yes—
179C22×C37#46112yes—
180C22×C37#49211yescharacteristic
181C22×C37#49711yescharacteristic
182C4×C37—22no—
183C4×C37—22no—
184C4×C37—44no—
185C4×C37—82no—
186C4×C37—82no—
187C4×C37—82no—
188C2×#450C2×C3712yes—
189#2659C2×C3711yescharacteristic, complemented
190C2×#328C2×C3712yescomplemented
191#1438C2×C3711yescharacteristic
192#1280C2×C3711yescharacteristic, complemented
193C2×C37—21no—
194C2×C37—22no—
195C2×C37—21no—
196C2×C37—84no—
197C2×C37#159612yes—
198C2×C37#159912yes—
199C2×C37#200611yescharacteristic
200C2×C37#201311yescharacteristic
201C2×C37#202111yescharacteristic
202C23×D4C22×C3711yescharacteristic
203C22×SD16—22no—
204C2×C22⋊Q8C22×C3712yes—
205C22×C22⋊C4—22no—
206#450—24no—
207#450—24no—
208C2×#102—22no—
209C2×C22⋊C8C22×C3712yes—
210#398—22no—
211#395—22no—
212#380—22no—
213C2×#95C22×C3711yescharacteristic
214C2×#95C22×C3711yescharacteristic
215#328—24no—
216C37#996611yescharacteristic, direct product factor, Sylow
217C25—21no—
218C22×Q8D4×C3712yes—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—21no—
222C22×D4—44no—
223C22×D4D4×C3712yes—
224C23×C4—21no—
225C23×C4D4×C3712yes—
226C23×C4C23×C3711yescharacteristic
227C2×SD16—28no—
228C2×SD16—48no—
229C22×C8—22no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8—22no—
233C22⋊Q8—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4D4×C3712yes—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242#102—28no—
243C22⋊C8—22no—
244C22⋊C8D4×C3714yes—
245#95—24no—
246#95—22no—
247#95—22no—
248C24—24no—
249C24—22no—
250C24—42no—
251C24—42no—
252C24—42no—
253C24—41no—
254C24—41no—
255C24—42no—
256C24—42no—
257C24C2×D4×C3711yescharacteristic
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—44no—
261C2×D4—28no—
262C2×D4—24no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—48no—
266C2×D4—44no—
267C2×D4—42no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4SD16×C3714yescomplemented
281C22×C4C2×D4×C3712yes—
282C22×C4C2×D4×C3711yescharacteristic
283C22×C4C2×D4×C3712yes—
284C22×C4#41×C3711yescharacteristic
285SD16—416no—
286C2×C8—24no—
287C2×C8—44no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C22⋊C4—416no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C23—21no—
298C23—21no—
299C23—22no—
300C23—24no—
301C23—44no—
302C23—44no—
303C23—42no—
304C23—44no—
305C23—44no—
306C23—44no—
307C23—44no—
308C23—42no—
309C23—44no—
310C23—44no—
311C23—82no—
312C23—82no—
313C23—82no—
314C23—82no—
315C23—82no—
316C23—82no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23#120×C3711yescharacteristic
322C23#121×C3711yescharacteristic
323C23#127×C3711yescharacteristic
324C23C2×SD16×C3712yes—
325C23#137×C3712yes—
326Q8—44no—
327Q8—82no—
328D4—28no—
329D4—44no—
330D4—48no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4#120×C3712yes—
351C2×C4#121×C3712yes—
352C8—44no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—48no—
362C22—44no—
363C22—84no—
364C22—84no—
365C22—84no—
366C22—84no—
367C22—84no—
368C22—82no—
369C22—82no—
370C22C37×#39211yescharacteristic
371C22C37×#44812yes—
372C22C37×#45311yescharacteristic
373C22C37×#45411yescharacteristic
374C22C37×B(4,𝔽2)12yes—
375C22C37×#46112yes—
376C22C37×#49211yescharacteristic
377C22C37×#49711yescharacteristic
378C4—22no—
379C4—22no—
380C4—44no—
381C4—82no—
382C4—82no—
383C4—82no—
384C2—21no—
385C2—22no—
386C2—21no—
387C2—84no—
388C2C37×#159612yes—
389C2C37×#159912yes—
390C2C37×#200611yescharacteristic
391C2C37×#201311yescharacteristic
392C2C37×#202111yescharacteristic
subgroupG/H
center, socleC23×C37#120
commutator, Frattini, cosocleC23×C4C23×C37
prime radicalC23×D4×C37C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C37 ◁ C23×C4×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C37×#2659C2semidirect
3. C2×C37×#328C2semidirect
5. C37×#1280C2semidirect
83. C22×C4×C37SD16semidirect
156. #9966C37direct (N × H)
189. #2659C2×C37semidirect
190. C2×#328C2×C37semidirect
192. #1280C2×C37semidirect
216. C37#9966direct (N × H)
280. C22×C4SD16×C37semidirect

Maximal subgroups

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

subgroupindex
C2×C37×#4502
C37×#26592
C2×C37×#3282
C37×#14382
C37×#12802
#996637