Finite Group Databasebeta

← C37×#11754 #22633007

Subgroups — C37×#11754 #22633007

Order 9472.

500 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2598C211yesmaximal, characteristic
2C37×#1740C211yesmaximal, characteristic, complemented
3C37×#1464C211yesmaximal, characteristic, complemented
4C37×#1352C212yesmaximal, complemented
5C37×#1343C212yesmaximal
6C2×C4×D4×C37C411yescharacteristic, complemented
7C2×#117×C37—21no—
8C2×#117×C37C2211yescharacteristic
9C37×#432—21no—
10C37×#400C411yescharacteristic, complemented
11C37×#399C2211yescharacteristic
12C37×#394C411yescharacteristic, complemented
13C37×#392C2211yescharacteristic
14C37×#388—21no—
15C37×#386—21no—
16C37×#386—21no—
17C37×#385—21no—
18C37×#383C411yescharacteristic, complemented
19C37×#342—21no—
20C37×#342C2212yes—
21C37×#324—21no—
22C37×#323C2212yes—
23C22×D4×C37—21no—
24C22×D4×C37C2×C411yescharacteristic
25C23×C4×C37—21no—
26C23×C4×C37D411yescharacteristic
27C4×D4×C37—41no—
28C4×D4×C37—41no—
29#117×C37—24no—
30#117×C37—41no—
31#117×C37—41no—
32#117×C37—42no—
33C2×C4⋊C4×C37—21no—
34C2×C4⋊C4×C37—41no—
35C2×C4⋊C4×C37C2×C411yescharacteristic
36C2×C4⋊C4×C37C2×C411yescharacteristic
37C2×C4⋊C4×C37D411yescharacteristic
38C2×C22⋊C4×C37—22no—
39C2×C22⋊C4×C37—21no—
40C2×C22⋊C4×C37—21no—
41C2×C22⋊C4×C37—21no—
42C2×C22⋊C4×C37—21no—
43C2×C22⋊C4×C37—21no—
44C2×C22⋊C4×C37D411yescharacteristic
45C2×C22⋊C4×C37C2311yescharacteristic
46C2×C42×C37—41no—
47C2×C42×C37C2×C411yescharacteristic
48C2×C42×C37D411yescharacteristic
49C4⋊C8×C37—41no—
50C22⋊C8×C37—22no—
51#97×C37—41no—
52#95×C37—21no—
53#95×C37—21no—
54#95×C37—21no—
55#95×C37—21no—
56#95×C37C2×C411yescharacteristic
57#95×C37C2×C411yescharacteristic
58C24×C37—21no—
59C24×C37C2×D411yescharacteristic
60C2×D4×C37—41no—
61C2×D4×C37—42no—
62C2×D4×C37—42no—
63C2×D4×C37—41no—
64C2×D4×C37—41no—
65C2×D4×C37—81no—
66C22×C4×C37—21no—
67C22×C4×C37—21no—
68C22×C4×C37—21no—
69C22×C4×C37—21no—
70C22×C4×C37—22no—
71C22×C4×C37—21no—
72C22×C4×C37—21no—
73C22×C4×C37—21no—
74C22×C4×C37—42no—
75C22×C4×C37—42no—
76C22×C4×C37—81no—
77C22×C4×C37—81no—
78C22×C4×C37C22⋊C411yescharacteristic
79C22×C4×C37C22⋊C411yescharacteristic
80C22×C4×C37C22⋊C411yescharacteristic
81C22×C4×C37C22⋊C411yescharacteristic
82C22×C4×C37C22×C411yescharacteristic
83C22×C4×C37C2×D411yescharacteristic
84C2×C8×C37—42no—
85C4⋊C4×C37—41no—
86C4⋊C4×C37—41no—
87C4⋊C4×C37—41no—
88C4⋊C4×C37—44no—
89C4⋊C4×C37—81no—
90C22⋊C4×C37—24no—
91C22⋊C4×C37—41no—
92C22⋊C4×C37—41no—
93C22⋊C4×C37—42no—
94C22⋊C4×C37—42no—
95C22⋊C4×C37—41no—
96C22⋊C4×C37—41no—
97C22⋊C4×C37—42no—
98C22⋊C4×C37—81no—
99C22⋊C4×C37—81no—
100C42×C37—21no—
101C42×C37—21no—
102C42×C37—21no—
103C42×C37—21no—
104C42×C37—44no—
105C23×C37—21no—
106C23×C37—21no—
107C23×C37—22no—
108C23×C37—41no—
109C23×C37—41no—
110C23×C37—41no—
111C23×C37—41no—
112C23×C37—41no—
113C23×C37—41no—
114C23×C37—41no—
115C23×C37—41no—
116C23×C37C2×C22⋊C411yescharacteristic
117C23×C37AGL(1,ℤ/8)11yescharacteristic
118C23×C37#13711yescharacteristic
119D4×C37—42no—
120D4×C37—42no—
121D4×C37—82no—
122D4×C37—82no—
123C2×C4×C37—22no—
124C2×C4×C37—22no—
125C2×C4×C37—22no—
126C2×C4×C37—22no—
127C2×C4×C37—21no—
128C2×C4×C37—21no—
129C2×C4×C37—21no—
130C2×C4×C37—21no—
131C2×C4×C37—21no—
132C2×C4×C37—21no—
133C2×C4×C37—21no—
134C2×C4×C37—21no—
135C2×C4×C37—41no—
136C2×C4×C37—41no—
137C2×C4×C37—41no—
138C2×C4×C37—41no—
139C2×C4×C37—44no—
140C2×C4×C37—41no—
141C2×C4×C37—41no—
142C2×C4×C37—41no—
143C2×C4×C37—41no—
144C2×C4×C37—41no—
145C2×C4×C37—41no—
146C2×C4×C37—41no—
147C2×C4×C37—41no—
148C2×C4×C37—41no—
149C2×C4×C37—84no—
150C2×C4×C37—82no—
151C2×C4×C37—82no—
152C2×C4×C37#9911yescharacteristic
153C2×C4×C37#9911yescharacteristic
154C2×C4×C37#10412yes—
155C8×C37—82no—
156#11754C3711yesmaximal, characteristic, direct product factor, Sylow
157C22×C37—21no—
158C22×C37—21no—
159C22×C37—21no—
160C22×C37—21no—
161C22×C37—41no—
162C22×C37—41no—
163C22×C37—42no—
164C22×C37—42no—
165C22×C37—41no—
166C22×C37—41no—
167C22×C37—42no—
168C22×C37—81no—
169C22×C37—81no—
170C22×C37—81no—
171C22×C37—81no—
172C22×C37—81no—
173C22×C37C2×#9911yescharacteristic
174C22×C37#41711yescharacteristic
175C22×C37C2×#10411yescharacteristic
176C4×C37—21no—
177C4×C37—21no—
178C4×C37—22no—
179C4×C37—41no—
180C4×C37—41no—
181C4×C37—41no—
182C4×C37—41no—
183C4×C37—81no—
184C4×C37—81no—
185C4×C37—84no—
186C4×C37—81no—
187#2598C2×C3711yescharacteristic
188#1740C2×C3711yescharacteristic, complemented
189#1464C2×C3711yescharacteristic, complemented
190#1352C2×C3712yescomplemented
191#1343C2×C3712yes—
192C2×C37—21no—
193C2×C37—21no—
194C2×C37—41no—
195C2×C37—41no—
196C2×C37—82no—
197C2×C37#150411yescharacteristic
198C2×C37#212011yescharacteristic
199C2×C37#212811yescharacteristic
200C2×C4×D4C4×C3711yescharacteristic, complemented
201C2×#117—21no—
202C2×#117C22×C3711yescharacteristic
203#432—21no—
204#400C4×C3711yescharacteristic, complemented
205#399C22×C3711yescharacteristic
206#394C4×C3711yescharacteristic, complemented
207#392C22×C3711yescharacteristic
208#388—21no—
209#386—21no—
210#386—21no—
211#385—21no—
212#383C4×C3711yescharacteristic, complemented
213#342—21no—
214#342C22×C3712yes—
215#324—21no—
216#323C22×C3712yes—
217C37#1175411yescharacteristic, direct product factor, Sylow
218C22×D4—21no—
219C22×D4C2×C4×C3711yescharacteristic
220C23×C4—21no—
221C23×C4D4×C3711yescharacteristic
222C4×D4—41no—
223C4×D4—41no—
224#117—24no—
225#117—41no—
226#117—41no—
227#117—42no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4C2×C4×C3711yescharacteristic
231C2×C4⋊C4C2×C4×C3711yescharacteristic
232C2×C4⋊C4D4×C3711yescharacteristic
233C2×C22⋊C4—22no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4D4×C3711yescharacteristic
240C2×C22⋊C4C23×C3711yescharacteristic
241C2×C42—41no—
242C2×C42C2×C4×C3711yescharacteristic
243C2×C42D4×C3711yescharacteristic
244C4⋊C8—41no—
245C22⋊C8—22no—
246#97—41no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95C2×C4×C3711yescharacteristic
252#95C2×C4×C3711yescharacteristic
253C24—21no—
254C24C2×D4×C3711yescharacteristic
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—81no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—81no—
272C22×C4—81no—
273C22×C4C22⋊C4×C3711yescharacteristic
274C22×C4C22⋊C4×C3711yescharacteristic
275C22×C4C22⋊C4×C3711yescharacteristic
276C22×C4C22⋊C4×C3711yescharacteristic
277C22×C4C22×C4×C3711yescharacteristic
278C22×C4C2×D4×C3711yescharacteristic
279C2×C8—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—44no—
284C4⋊C4—81no—
285C22⋊C4—24no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—42no—
293C22⋊C4—81no—
294C22⋊C4—81no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—44no—
300C23—21no—
301C23—21no—
302C23—22no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23C2×C22⋊C4×C3711yescharacteristic
312C23AGL(1,ℤ/8)×C3711yescharacteristic
313C23#137×C3711yescharacteristic
314D4—42no—
315D4—42no—
316D4—82no—
317D4—82no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—44no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—84no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4#99×C3711yescharacteristic
348C2×C4#99×C3711yescharacteristic
349C2×C4#104×C3712yes—
350C8—82no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—42no—
358C22—42no—
359C22—41no—
360C22—41no—
361C22—42no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22C2×#99×C3711yescharacteristic
368C22C37×#41711yescharacteristic
369C22C2×#104×C3711yescharacteristic
370C4—21no—
371C4—21no—
372C4—22no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—81no—
378C4—81no—
379C4—84no—
380C4—81no—
381C2—21no—
382C2—21no—
383C2—41no—
384C2—41no—
385C2—82no—
386C2C37×#150411yescharacteristic
387C2C37×#212011yescharacteristic
388C2C37×#212811yescharacteristic
subgroupG/H
center, socleC22×C37C2×#99
commutatorC22×C4C22×C4×C37
Frattini, cosocleC2×C22⋊C4C23×C37
prime radicalC37×#392C22
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C37 ◁ C22×C4×C37 ◁ C2×#117×C37 ◁ G

Complements

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

NH ≅ G/Ntype
2. C37×#1740C2semidirect
3. C37×#1464C2semidirect
4. C37×#1352C2semidirect
6. C2×C4×D4×C37C4semidirect
10. C37×#400C4semidirect
12. C37×#394C4semidirect
18. C37×#383C4semidirect
156. #11754C37direct (N × H)
188. #1740C2×C37semidirect
189. #1464C2×C37semidirect
190. #1352C2×C37semidirect
200. C2×C4×D4C4×C37semidirect
204. #400C4×C37semidirect
206. #394C4×C37semidirect
212. #383C4×C37semidirect
217. C37#11754direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#25982
C37×#17402
C37×#14642
C37×#13522
C37×#13432
#1175437