Finite Group Databasebeta

← C37×#13317 #22634570

Subgroups — C37×#13317 #22634570

Order 9472.

552 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C37×#510C211yesmaximal, characteristic, complemented
2C37×#3054C211yesmaximal, characteristic
3C2×C37×#470C211yesmaximal, characteristic, complemented
4C2×C37×#360C211yesmaximal, characteristic, complemented
5C37×#2141C211yesmaximal, characteristic, complemented
6C2×C37×#349C211yesmaximal, characteristic, complemented
7C37×#1386C211yesmaximal, characteristic, complemented
8C2×#135×C37C2211yescharacteristic
9C22×Q16×C37C2211yescharacteristic
10C22×M16×C37C2211yescharacteristic
11C37×#525—21no—
12C2×C22⋊Q8×C37—21no—
13C37×#510—22no—
14C37×#510—22no—
15C2×Q32×C37—21no—
16C2×SD32×C37—21no—
17C2×M32×C37C2211yescharacteristic
18C2×M32×C37C2211yescharacteristic
19C37×#470—21no—
20C37×#470—22no—
21C37×#470—22no—
22C37×#469—21no—
23C37×#468—22no—
24C37×#427C2211yescharacteristic
25C2×#106×C37C2211yescharacteristic
26C37×#416—21no—
27C2×#103×C37—21no—
28C37×#360—22no—
29C37×#358—22no—
30C37×#349—22no—
31C2×#41×C37—21no—
32C22×Q8×C37—21no—
33C23×C4×C37D411yescharacteristic
34#135×C37—21no—
35#135×C37—21no—
36#135×C37—21no—
37#135×C37—21no—
38C2×Q16×C37—21no—
39C2×Q16×C37—21no—
40C2×Q16×C37—21no—
41C2×Q16×C37—21no—
42C2×Q16×C37—21no—
43C2×Q16×C37D411yescharacteristic
44C2×Q16×C37D412yescomplemented
45C2×SD16×C37—21no—
46C2×D8×C37D411yescharacteristic
47C2×M16×C37—21no—
48C2×M16×C37—21no—
49C2×M16×C37—21no—
50C2×M16×C37—21no—
51C2×M16×C37—21no—
52C2×M16×C37—21no—
53C22×C8×C37D411yescharacteristic
54C22×C8×C37C2311yescharacteristic
55#123×C37—41no—
56C22⋊Q8×C37—24no—
57C22⋊Q8×C37—22no—
58C22⋊Q8×C37—42no—
59#121×C37—22no—
60#120×C37—41no—
61C4×D4×C37—42no—
62#117×C37—21no—
63C2×C4⋊C4×C37—21no—
64C2×C4⋊C4×C37—41no—
65C2×C22⋊C4×C37—41no—
66Q32×C37—24no—
67SD32×C37—24no—
68M32×C37—21no—
69M32×C37—21no—
70M32×C37—21no—
71M32×C37D412yescomplemented
72C2×C16×C37—21no—
73C2×C16×C37—21no—
74#107×C37D412yes—
75#106×C37—21no—
76#106×C37—21no—
77#106×C37D412yescomplemented
78#103×C37—24no—
79#103×C37—22no—
80#102×C37—22no—
81C24×C37D811yescharacteristic
82#41×C37—41no—
83#41×C37—41no—
84#41×C37—41no—
85#41×C37—41no—
86C2×Q8×C37—21no—
87C2×Q8×C37—22no—
88C2×Q8×C37—41no—
89C2×Q8×C37—41no—
90C2×Q8×C37—41no—
91C2×Q8×C37—41no—
92C2×Q8×C37—41no—
93C2×D4×C37—21no—
94C2×D4×C37—42no—
95C2×D4×C37—41no—
96C22×C4×C37—21no—
97C22×C4×C37—21no—
98C22×C4×C37—21no—
99C22×C4×C37—21no—
100C22×C4×C37—21no—
101C22×C4×C37—21no—
102C22×C4×C37—41no—
103C22×C4×C37—41no—
104C22×C4×C37—41no—
105C22×C4×C37D811yescharacteristic
106C22×C4×C37C2×D411yescharacteristic
107Q16×C37—22no—
108Q16×C37—21no—
109Q16×C37—22no—
110Q16×C37—21no—
111Q16×C37—41no—
112Q16×C37—41no—
113SD16×C37—41no—
114SD16×C37—41no—
115D8×C37—21no—
116D8×C37—21no—
117M16×C37—41no—
118M16×C37—41no—
119M16×C37—41no—
120M16×C37—41no—
121C2×C8×C37—21no—
122C2×C8×C37—21no—
123C2×C8×C37—21no—
124C2×C8×C37C2×D411yescharacteristic
125C2×C8×C37C2×D411yescharacteristic
126C2×C8×C37C2×D411yescharacteristic
127C2×C8×C37C2×D411yescharacteristic
128C2×C8×C37C2×D411yescharacteristic
129C2×C8×C37#4111yescharacteristic
130C4⋊C4×C37—22no—
131C4⋊C4×C37—22no—
132C4⋊C4×C37—44no—
133C4⋊C4×C37—42no—
134C4⋊C4×C37—41no—
135C22⋊C4×C37—42no—
136C22⋊C4×C37—44no—
137C22⋊C4×C37—42no—
138C22⋊C4×C37—41no—
139C42×C37—41no—
140C16×C37—22no—
141C16×C37—41no—
142C23×C37—21no—
143C23×C37—21no—
144C23×C37—21no—
145C23×C37—42no—
146C23×C37—41no—
147C23×C37C2×D811yescharacteristic
148Q8×C37—41no—
149Q8×C37—42no—
150Q8×C37—42no—
151Q8×C37—41no—
152Q8×C37—81no—
153Q8×C37—81no—
154D4×C37—41no—
155D4×C37—41no—
156D4×C37—84no—
157D4×C37—81no—
158D4×C37—81no—
159C2×C4×C37—21no—
160C2×C4×C37—21no—
161C2×C4×C37—21no—
162C2×C4×C37—41no—
163C2×C4×C37—41no—
164C2×C4×C37—41no—
165C2×C4×C37—41no—
166C2×C4×C37—42no—
167C2×C4×C37—42no—
168C2×C4×C37—42no—
169C2×C4×C37—41no—
170C2×C4×C37—82no—
171C2×C4×C37—81no—
172C2×C4×C37—81no—
173C2×C4×C37—81no—
174C2×C4×C37—81no—
175C2×C4×C37#12011yescharacteristic
176C2×C4×C37#12011yescharacteristic
177C2×C4×C37#12111yescharacteristic
178C2×C4×C37#13511yescharacteristic
179C2×C4×C37AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C37AGL(1,ℤ/8)11yescharacteristic
181C8×C37—41no—
182C8×C37#12011yescharacteristic
183C8×C37#12111yescharacteristic
184C8×C37#12111yescharacteristic
185C8×C37#12711yescharacteristic
186#13317C3711yesmaximal, characteristic, direct product factor, Sylow
187C22×C37—22no—
188C22×C37—42no—
189C22×C37—42no—
190C22×C37—41no—
191C22×C37—81no—
192C22×C37—82no—
193C22×C37—81no—
194C22×C37#44711yescharacteristic
195C22×C37#44711yescharacteristic
196C22×C37#46611yescharacteristic
197C4×C37—41no—
198C4×C37—82no—
199C4×C37—82no—
200C4×C37—81no—
201C4×C37—82no—
202C4×C37#39211yescharacteristic
203C4×C37#44911yescharacteristic
204C4×C37#44911yescharacteristic
205C4×C37#46911yescharacteristic
206C2×#510C2×C3711yescharacteristic, complemented
207#3054C2×C3711yescharacteristic
208C2×#470C2×C3711yescharacteristic, complemented
209C2×#360C2×C3711yescharacteristic, complemented
210#2141C2×C3711yescharacteristic, complemented
211C2×#349C2×C3711yescharacteristic, complemented
212#1386C2×C3711yescharacteristic, complemented
213C2×C37—21no—
214C2×C37—21no—
215C2×C37—81no—
216C2×C37—161no—
217C2×C37#199911yescharacteristic
218C2×C37#219112yes—
219C2×#135C22×C3711yescharacteristic
220C22×Q16C22×C3711yescharacteristic
221C22×M16C22×C3711yescharacteristic
222#525—21no—
223C2×C22⋊Q8—21no—
224#510—22no—
225#510—22no—
226C2×Q32—21no—
227C2×SD32—21no—
228C2×M32C22×C3711yescharacteristic
229C2×M32C22×C3711yescharacteristic
230#470—21no—
231#470—22no—
232#470—22no—
233#469—21no—
234#468—22no—
235#427C22×C3711yescharacteristic
236C2×#106C22×C3711yescharacteristic
237#416—21no—
238C2×#103—21no—
239#360—22no—
240#358—22no—
241#349—22no—
242C37#1331711yescharacteristic, direct product factor, Sylow
243C2×#41—21no—
244C22×Q8—21no—
245C23×C4D4×C3711yescharacteristic
246#135—21no—
247#135—21no—
248#135—21no—
249#135—21no—
250C2×Q16—21no—
251C2×Q16—21no—
252C2×Q16—21no—
253C2×Q16—21no—
254C2×Q16—21no—
255C2×Q16D4×C3711yescharacteristic
256C2×Q16D4×C3712yescomplemented
257C2×SD16—21no—
258C2×D8D4×C3711yescharacteristic
259C2×M16—21no—
260C2×M16—21no—
261C2×M16—21no—
262C2×M16—21no—
263C2×M16—21no—
264C2×M16—21no—
265C22×C8D4×C3711yescharacteristic
266C22×C8C23×C3711yescharacteristic
267#123—41no—
268C22⋊Q8—24no—
269C22⋊Q8—22no—
270C22⋊Q8—42no—
271#121—22no—
272#120—41no—
273C4×D4—42no—
274#117—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—41no—
277C2×C22⋊C4—41no—
278Q32—24no—
279SD32—24no—
280M32—21no—
281M32—21no—
282M32—21no—
283M32D4×C3712yescomplemented
284C2×C16—21no—
285C2×C16—21no—
286#107D4×C3712yes—
287#106—21no—
288#106—21no—
289#106D4×C3712yescomplemented
290#103—24no—
291#103—22no—
292#102—22no—
293C24D8×C3711yescharacteristic
294#41—41no—
295#41—41no—
296#41—41no—
297#41—41no—
298C2×Q8—21no—
299C2×Q8—22no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×D4—21no—
306C2×D4—42no—
307C2×D4—41no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4D8×C3711yescharacteristic
318C22×C4C2×D4×C3711yescharacteristic
319Q16—22no—
320Q16—21no—
321Q16—22no—
322Q16—21no—
323Q16—41no—
324Q16—41no—
325SD16—41no—
326SD16—41no—
327D8—21no—
328D8—21no—
329M16—41no—
330M16—41no—
331M16—41no—
332M16—41no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8C2×D4×C3711yescharacteristic
337C2×C8C2×D4×C3711yescharacteristic
338C2×C8C2×D4×C3711yescharacteristic
339C2×C8C2×D4×C3711yescharacteristic
340C2×C8C2×D4×C3711yescharacteristic
341C2×C8#41×C3711yescharacteristic
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C22⋊C4—42no—
348C22⋊C4—44no—
349C22⋊C4—42no—
350C22⋊C4—41no—
351C42—41no—
352C16—22no—
353C16—41no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—42no—
358C23—41no—
359C23C2×D8×C3711yescharacteristic
360Q8—41no—
361Q8—42no—
362Q8—42no—
363Q8—41no—
364Q8—81no—
365Q8—81no—
366D4—41no—
367D4—41no—
368D4—84no—
369D4—81no—
370D4—81no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4#120×C3711yescharacteristic
388C2×C4#120×C3711yescharacteristic
389C2×C4#121×C3711yescharacteristic
390C2×C4#135×C3711yescharacteristic
391C2×C4AGL(1,ℤ/8)×C3711yescharacteristic
392C2×C4AGL(1,ℤ/8)×C3711yescharacteristic
393C8—41no—
394C8#120×C3711yescharacteristic
395C8#121×C3711yescharacteristic
396C8#121×C3711yescharacteristic
397C8#127×C3711yescharacteristic
398C22—22no—
399C22—42no—
400C22—42no—
401C22—41no—
402C22—81no—
403C22—82no—
404C22—81no—
405C22C37×#44711yescharacteristic
406C22C37×#44711yescharacteristic
407C22C37×#46611yescharacteristic
408C4—41no—
409C4—82no—
410C4—82no—
411C4—81no—
412C4—82no—
413C4C37×#39211yescharacteristic
414C4C37×#44911yescharacteristic
415C4C37×#44911yescharacteristic
416C4C37×#46911yescharacteristic
417C2—21no—
418C2—21no—
419C2—81no—
420C2—161no—
421C2C37×#199911yescharacteristic
422C2C37×#219112yes—
subgroupG/H
center, socleC22×C37#447
commutator, Frattini, cosocleC22×C8C23×C37
prime radicalC37×#3054C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C37 ◁ C22×C4×C37 ◁ C22×M16×C37 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C37×#510C2semidirect
3. C2×C37×#470C2semidirect
4. C2×C37×#360C2semidirect
5. C37×#2141C2semidirect
6. C2×C37×#349C2semidirect
7. C37×#1386C2semidirect
44. C2×Q16×C37D4semidirect
71. M32×C37D4semidirect
77. #106×C37D4semidirect
186. #13317C37direct (N × H)
206. C2×#510C2×C37semidirect
208. C2×#470C2×C37semidirect
209. C2×#360C2×C37semidirect
210. #2141C2×C37semidirect
211. C2×#349C2×C37semidirect
212. #1386C2×C37semidirect
242. C37#13317direct (N × H)
256. C2×Q16D4×C37semidirect
283. M32D4×C37semidirect
289. #106D4×C37semidirect

Maximal subgroups

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

subgroupindex
C2×C37×#5102
C37×#30542
C2×C37×#4702
C2×C37×#3602
C37×#21412
C2×C37×#3492
C37×#13862
#1331737