Finite Group Databasebeta

← C2×C37×#2223 #22654869

Subgroups — C2×C37×#2223 #22654869

Order 9472.

658 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×SD32×C37C211yesmaximal, characteristic, complemented
2C2×C37×#504C211yesmaximal, characteristic, complemented
3C2×C37×#473C211yesmaximal, characteristic, complemented
4C2×C37×#472C211yesmaximal, characteristic
5C37×#2223C218yesmaximal, direct product factor
6C2×C37×#368C211yesmaximal, characteristic, complemented
7C2×C37×#362C211yesmaximal, characteristic, complemented
8C2×C37×#361C211yesmaximal, characteristic, complemented
9C2×#137×C37—21no—
10C2×AGL(1,ℤ/8)×C37—21no—
11C22×Q16×C37C2211yescharacteristic
12C22×D8×C37C2211yescharacteristic
13C2×#131×C37C2211yescharacteristic
14C2×SD32×C37—21no—
15C2×SD32×C37—21no—
16C2×SD32×C37—22no—
17C2×SD32×C37C2214yescomplemented
18C37×#504—22no—
19C37×#504C2214yescomplemented
20C2×M32×C37—21no—
21C2×M32×C37C2211yescharacteristic
22C22×C16×C37—21no—
23C22×C16×C37C2211yescharacteristic
24C37×#473—22no—
25C37×#473C2214yescomplemented
26C37×#472—22no—
27C37×#472C2214yes—
28C2×#108×C37C2211yescharacteristic
29C2×#108×C37C2211yescharacteristic
30C2×#101×C37—21no—
31C2×#100×C37—21no—
32C37×#368C2214yescomplemented
33C37×#362C2214yescomplemented
34C37×#361C2214yescomplemented
35C2×#41×C37D411yescharacteristic
36C22×Q8×C37—21no—
37C22×D4×C37—21no—
38#137×C37—24no—
39#137×C37—42no—
40AGL(1,ℤ/8)×C37—24no—
41AGL(1,ℤ/8)×C37—42no—
42C2×Q16×C37—21no—
43C2×Q16×C37—21no—
44C2×Q16×C37—21no—
45C2×Q16×C37—21no—
46C2×Q16×C37—21no—
47C2×Q16×C37—41no—
48C2×Q16×C37C2312yes—
49C2×SD16×C37—41no—
50C2×SD16×C37—41no—
51C2×D8×C37—21no—
52C2×D8×C37—21no—
53C2×D8×C37—21no—
54C2×D8×C37—21no—
55C2×D8×C37—21no—
56C2×D8×C37—41no—
57C2×D8×C37C2312yes—
58#131×C37—21no—
59#131×C37—21no—
60#131×C37D412yes—
61#131×C37C2312yes—
62C2×M16×C37—21no—
63C2×M16×C37—21no—
64C2×M16×C37—21no—
65C2×M16×C37D411yescharacteristic
66C22×C8×C37—21no—
67C22×C8×C37C2311yescharacteristic
68SD32×C37—24no—
69SD32×C37—24no—
70M32×C37—21no—
71M32×C37—24no—
72M32×C37C2312yes—
73C2×C16×C37—21no—
74C2×C16×C37—24no—
75C2×C16×C37D411yescharacteristic
76C2×C16×C37D411yescharacteristic
77C2×C16×C37D412yes—
78C2×C16×C37C2312yes—
79#108×C37—21no—
80#108×C37—21no—
81#108×C37C2312yes—
82#108×C37C2312yes—
83#101×C37—24no—
84#100×C37—24no—
85C24×C37—41no—
86#41×C37—21no—
87#41×C37—21no—
88#41×C37D812yes—
89#41×C37C2×D412yes—
90C2×Q8×C37—22no—
91C2×Q8×C37—41no—
92C2×Q8×C37—41no—
93C2×Q8×C37—41no—
94C2×Q8×C37—41no—
95C2×Q8×C37—41no—
96C2×Q8×C37D811yescharacteristic
97C2×D4×C37—21no—
98C2×D4×C37—22no—
99C2×D4×C37—41no—
100C2×D4×C37—41no—
101C2×D4×C37—41no—
102C2×D4×C37—41no—
103C2×D4×C37—41no—
104C2×D4×C37D811yescharacteristic
105C22×C4×C37—21no—
106C22×C4×C37—41no—
107C22×C4×C37C2×D411yescharacteristic
108Q16×C37—22no—
109Q16×C37—22no—
110Q16×C37—44no—
111Q16×C37—41no—
112Q16×C37—41no—
113SD16×C37—44no—
114SD16×C37—44no—
115D8×C37—22no—
116D8×C37—22no—
117D8×C37—44no—
118D8×C37—41no—
119D8×C37—41no—
120M16×C37—21no—
121M16×C37—22no—
122M16×C37—22no—
123M16×C37—22no—
124M16×C37—22no—
125M16×C37—41no—
126M16×C37—41no—
127M16×C37C2×D412yes—
128C2×C8×C37—21no—
129C2×C8×C37—22no—
130C2×C8×C37—22no—
131C2×C8×C37—41no—
132C2×C8×C37—41no—
133C2×C8×C37C2×D411yescharacteristic
134C2×C8×C37C2×D411yescharacteristic
135C2×C8×C37C2×D411yescharacteristic
136C2×C8×C37#4111yescharacteristic
137C2×C8×C37#4111yescharacteristic
138C2×C8×C37C2411yescharacteristic
139C16×C37—22no—
140C16×C37C2×D412yes—
141C16×C37C2×D412yes—
142C23×C37—21no—
143C23×C37—42no—
144C23×C37—81no—
145C23×C37—81no—
146C23×C37—162no—
147C23×C37#13511yescharacteristic
148Q8×C37—21no—
149Q8×C37—42no—
150Q8×C37—42no—
151Q8×C37—81no—
152Q8×C37—81no—
153Q8×C37C2×D812yes—
154D4×C37—22no—
155D4×C37—22no—
156D4×C37—21no—
157D4×C37—42no—
158D4×C37—42no—
159D4×C37—81no—
160D4×C37—81no—
161D4×C37C2×D812yes—
162C2×C4×C37—21no—
163C2×C4×C37—22no—
164C2×C4×C37—22no—
165C2×C4×C37—42no—
166C2×C4×C37—81no—
167C2×C4×C37—81no—
168C2×C4×C37#12111yescharacteristic
169C2×C4×C37#12111yescharacteristic
170C2×C4×C37C2×D811yescharacteristic
171C2×C4×C37C2×D811yescharacteristic
172C2×C4×C37#13511yescharacteristic
173C2×C4×C37C22×D411yescharacteristic
174C8×C37—22no—
175C8×C37—42no—
176C8×C37—42no—
177C8×C37#12111yescharacteristic
178C8×C37#12111yescharacteristic
179C8×C37C22×D411yescharacteristic
180C8×C37C2×#4111yescharacteristic
181C2×#2223C3711yesmaximal, characteristic, direct product factor, Sylow
182C22×C37—22no—
183C22×C37—22no—
184C22×C37—42no—
185C22×C37—82no—
186C22×C37—162no—
187C22×C37—162no—
188C22×C37—162no—
189C22×C37#46611yescharacteristic
190C22×C37#46611yescharacteristic
191C22×C37C2×#13511yescharacteristic
192C4×C37—22no—
193C4×C37—82no—
194C4×C37#46611yescharacteristic
195C4×C37#46611yescharacteristic
196C4×C37C2×#12111yescharacteristic
197C4×C37C22×D811yescharacteristic
198C22×SD32C2×C3711yescharacteristic, complemented
199C2×#504C2×C3711yescharacteristic, complemented
200C2×#473C2×C3711yescharacteristic, complemented
201C2×#472C2×C3711yescharacteristic
202#2223C2×C3718yesdirect product factor
203C2×#368C2×C3711yescharacteristic, complemented
204C2×#362C2×C3711yescharacteristic, complemented
205C2×#361C2×C3711yescharacteristic, complemented
206C2×C37—21no—
207C2×C37—21no—
208C2×C37—42no—
209C2×C37—162no—
210C2×C37#222312yesdirect product factor
211C2×C37C2×#46611yescharacteristic
212C2×#137—21no—
213C2×AGL(1,ℤ/8)—21no—
214C22×Q16C22×C3711yescharacteristic
215C22×D8C22×C3711yescharacteristic
216C2×#131C22×C3711yescharacteristic
217C2×SD32—21no—
218C2×SD32—21no—
219C2×SD32—22no—
220C2×SD32C22×C3714yescomplemented
221#504—22no—
222#504C22×C3714yescomplemented
223C2×M32—21no—
224C2×M32C22×C3711yescharacteristic
225C22×C16—21no—
226C22×C16C22×C3711yescharacteristic
227#473—22no—
228#473C22×C3714yescomplemented
229#472—22no—
230#472C22×C3714yes—
231C2×#108C22×C3711yescharacteristic
232C2×#108C22×C3711yescharacteristic
233C2×#101—21no—
234C2×#100—21no—
235#368C22×C3714yescomplemented
236#362C22×C3714yescomplemented
237#361C22×C3714yescomplemented
238C37C2×#222311yescharacteristic, direct product factor, Sylow
239C2×#41D4×C3711yescharacteristic
240C22×Q8—21no—
241C22×D4—21no—
242#137—24no—
243#137—42no—
244AGL(1,ℤ/8)—24no—
245AGL(1,ℤ/8)—42no—
246C2×Q16—21no—
247C2×Q16—21no—
248C2×Q16—21no—
249C2×Q16—21no—
250C2×Q16—21no—
251C2×Q16—41no—
252C2×Q16C23×C3712yes—
253C2×SD16—41no—
254C2×SD16—41no—
255C2×D8—21no—
256C2×D8—21no—
257C2×D8—21no—
258C2×D8—21no—
259C2×D8—21no—
260C2×D8—41no—
261C2×D8C23×C3712yes—
262#131—21no—
263#131—21no—
264#131D4×C3712yes—
265#131C23×C3712yes—
266C2×M16—21no—
267C2×M16—21no—
268C2×M16—21no—
269C2×M16D4×C3711yescharacteristic
270C22×C8—21no—
271C22×C8C23×C3711yescharacteristic
272SD32—24no—
273SD32—24no—
274M32—21no—
275M32—24no—
276M32C23×C3712yes—
277C2×C16—21no—
278C2×C16—24no—
279C2×C16D4×C3711yescharacteristic
280C2×C16D4×C3711yescharacteristic
281C2×C16D4×C3712yes—
282C2×C16C23×C3712yes—
283#108—21no—
284#108—21no—
285#108C23×C3712yes—
286#108C23×C3712yes—
287#101—24no—
288#100—24no—
289C24—41no—
290#41—21no—
291#41—21no—
292#41D8×C3712yes—
293#41C2×D4×C3712yes—
294C2×Q8—22no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8D8×C3711yescharacteristic
301C2×D4—21no—
302C2×D4—22no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4D8×C3711yescharacteristic
309C22×C4—21no—
310C22×C4—41no—
311C22×C4C2×D4×C3711yescharacteristic
312Q16—22no—
313Q16—22no—
314Q16—44no—
315Q16—41no—
316Q16—41no—
317SD16—44no—
318SD16—44no—
319D8—22no—
320D8—22no—
321D8—44no—
322D8—41no—
323D8—41no—
324M16—21no—
325M16—22no—
326M16—22no—
327M16—22no—
328M16—22no—
329M16—41no—
330M16—41no—
331M16C2×D4×C3712yes—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8C2×D4×C3711yescharacteristic
338C2×C8C2×D4×C3711yescharacteristic
339C2×C8C2×D4×C3711yescharacteristic
340C2×C8#41×C3711yescharacteristic
341C2×C8#41×C3711yescharacteristic
342C2×C8C24×C3711yescharacteristic
343C16—22no—
344C16C2×D4×C3712yes—
345C16C2×D4×C3712yes—
346C23—21no—
347C23—42no—
348C23—81no—
349C23—81no—
350C23—162no—
351C23#135×C3711yescharacteristic
352Q8—21no—
353Q8—42no—
354Q8—42no—
355Q8—81no—
356Q8—81no—
357Q8C2×D8×C3712yes—
358D4—22no—
359D4—22no—
360D4—21no—
361D4—42no—
362D4—42no—
363D4—81no—
364D4—81no—
365D4C2×D8×C3712yes—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—42no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4#121×C3711yescharacteristic
373C2×C4#121×C3711yescharacteristic
374C2×C4C2×D8×C3711yescharacteristic
375C2×C4C2×D8×C3711yescharacteristic
376C2×C4#135×C3711yescharacteristic
377C2×C4C22×D4×C3711yescharacteristic
378C8—22no—
379C8—42no—
380C8—42no—
381C8#121×C3711yescharacteristic
382C8#121×C3711yescharacteristic
383C8C22×D4×C3711yescharacteristic
384C8C2×#41×C3711yescharacteristic
385C22—22no—
386C22—22no—
387C22—42no—
388C22—82no—
389C22—162no—
390C22—162no—
391C22—162no—
392C22C37×#46611yescharacteristic
393C22C37×#46611yescharacteristic
394C22C2×#135×C3711yescharacteristic
395C4—22no—
396C4—82no—
397C4C37×#46611yescharacteristic
398C4C37×#46611yescharacteristic
399C4C2×#121×C3711yescharacteristic
400C4C22×D8×C3711yescharacteristic
401C2—21no—
402C2—21no—
403C2—42no—
404C2—162no—
405C2C37×#222312yesdirect product factor
406C2C2×C37×#46611yescharacteristic
subgroupG/H
center, socleC22×C37#466
commutator, Frattini, cosocleC2×C8C24×C37
prime radicalC2×C37×#472C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C37 ◁ C22×C4×C37 ◁ C2×#131×C37 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×SD32×C37C2semidirect
2. C2×C37×#504C2semidirect
3. C2×C37×#473C2semidirect
5. C37×#2223C2direct (N × H)
6. C2×C37×#368C2semidirect
7. C2×C37×#362C2semidirect
8. C2×C37×#361C2semidirect
17. C2×SD32×C37C22semidirect
19. C37×#504C22semidirect
25. C37×#473C22semidirect
32. C37×#368C22semidirect
33. C37×#362C22semidirect
34. C37×#361C22semidirect
181. C2×#2223C37direct (N × H)
198. C22×SD32C2×C37semidirect
199. C2×#504C2×C37semidirect
200. C2×#473C2×C37semidirect
202. #2223C2×C37direct (N × H)
203. C2×#368C2×C37semidirect
204. C2×#362C2×C37semidirect
205. C2×#361C2×C37semidirect
210. C2×C37#2223direct (N × H)
220. C2×SD32C22×C37semidirect
222. #504C22×C37semidirect
228. #473C22×C37semidirect
235. #368C22×C37semidirect
236. #362C22×C37semidirect
237. #361C22×C37semidirect
238. C37C2×#2223direct (N × H)
405. C2C37×#2223direct (N × H)

Maximal subgroups

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

subgroupindex
C22×SD32×C372
C2×C37×#5042
C2×C37×#4732
C2×C37×#4722
C37×#22232
C2×C37×#3682
C2×C37×#3622
C2×C37×#3612
C2×#222337