Finite Group Databasebeta

← C37×#19517 #22640770

Subgroups — C37×#19517 #22640770

Order 9472.

580 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C37×#3254C211yesmaximal, characteristic
2C37×#3246C211yesmaximal, characteristic
3C37×#3231C211yesmaximal, characteristic, complemented
4C37×#2981C211yesmaximal, characteristic
5C37×#1703C212yesmaximal, complemented
6C37×#1693C212yesmaximal, complemented
7C37×#1679C212yesmaximal, complemented
8C37×#1671C212yesmaximal, complemented
9C37×#1570C211yesmaximal, characteristic, complemented
10C37×#1559C211yesmaximal, characteristic, complemented
11C37×#1552C211yesmaximal, characteristic, complemented
12C37×#533C2211yescharacteristic
13C37×#533C2211yescharacteristic
14C37×#532C2211yescharacteristic
15C37×#500C2211yescharacteristic
16C37×#498C2211yescharacteristic
17C37×#487C2211yescharacteristic
18C37×#487C2211yescharacteristic
19C37×#485—22no—
20C37×#479—22no—
21C37×#477—21no—
22C37×#476C2212yes—
23C37×#475—21no—
24C37×#474C2212yes—
25C37×#468—22no—
26C37×#465—22no—
27C37×#461C2212yes—
28C37×#461C2212yes—
29C37×#460—21no—
30C37×#460—21no—
31C37×#436C2212yes—
32C37×#435—22no—
33C8×D4×C37C2212yes—
34C37×#432C2211yescharacteristic
35C37×#431C2211yescharacteristic
36C37×#423C2211yescharacteristic
37C2×C4⋊C8×C37C2211yescharacteristic
38C37×#408—22no—
39C37×#334C2212yes—
40C37×#334C2212yes—
41C37×#333C2212yes—
42C37×#332C2212yes—
43C37×#332C2212yes—
44C37×#332C2212yes—
45C2×#41×C37—21no—
46C2×SD16×C37—22no—
47C2×SD16×C37—22no—
48C2×M16×C37—21no—
49C2×M16×C37—22no—
50C22×C8×C37—21no—
51C22×C8×C37—22no—
52C4⋊Q8×C37D411yescharacteristic
53C4⋊Q8×C37C2311yescharacteristic
54C4⋊Q8×C37C2311yescharacteristic
55C4⋊Q8×C37C2311yescharacteristic
56C4⋊Q8×C37C2311yescharacteristic
57#127×C37D411yescharacteristic
58#125×C37—21no—
59#125×C37—21no—
60#124×C37—21no—
61C22⋊Q8×C37—21no—
62C22⋊Q8×C37—21no—
63C22⋊Q8×C37—21no—
64C22⋊Q8×C37—21no—
65#121×C37—21no—
66#121×C37D412yes—
67C4×Q8×C37—21no—
68C4×Q8×C37—21no—
69C4×Q8×C37—21no—
70C4×Q8×C37—21no—
71C4×D4×C37—21no—
72C4×D4×C37C2312yes—
73#117×C37—21no—
74#117×C37—21no—
75C2×C42×C37C2311yescharacteristic
76#107×C37—22no—
77#107×C37—22no—
78#106×C37—22no—
79#106×C37—22no—
80C4⋊C8×C37—21no—
81C4⋊C8×C37—21no—
82C4⋊C8×C37D412yes—
83C4⋊C8×C37C2312yes—
84C4⋊C8×C37C2312yes—
85C4⋊C8×C37C2312yes—
86#103×C37—22no—
87#103×C37—22no—
88#103×C37—22no—
89#103×C37—22no—
90#102×C37—22no—
91#102×C37—22no—
92C22⋊C8×C37—21no—
93C22⋊C8×C37—22no—
94C22⋊C8×C37—22no—
95C22⋊C8×C37D412yes—
96#97×C37—21no—
97C4×C8×C37C2311yescharacteristic
98C4×C8×C37C2311yescharacteristic
99#41×C37—82no—
100C2×Q8×C37—21no—
101C2×Q8×C37—21no—
102C2×Q8×C37—21no—
103C2×Q8×C37—21no—
104C2×Q8×C37—21no—
105C2×D4×C37—22no—
106C2×D4×C37—21no—
107C2×D4×C37—21no—
108C2×D4×C37C2×D412yes—
109C22×C4×C37—22no—
110C22×C4×C37C2×D411yescharacteristic
111C22×C4×C37C2×D411yescharacteristic
112C22×C4×C37#4111yescharacteristic
113SD16×C37—82no—
114SD16×C37—82no—
115M16×C37—82no—
116M16×C37—81no—
117C2×C8×C37—22no—
118C2×C8×C37—22no—
119C2×C8×C37—22no—
120C2×C8×C37—82no—
121C2×C8×C37—81no—
122C2×C8×C37C2×D412yes—
123C2×C8×C37C2×D412yes—
124C4⋊C4×C37—21no—
125C4⋊C4×C37—21no—
126C4⋊C4×C37—21no—
127C4⋊C4×C37—21no—
128C4⋊C4×C37—21no—
129C4⋊C4×C37—21no—
130C4⋊C4×C37—21no—
131C4⋊C4×C37—21no—
132C4⋊C4×C37—21no—
133C4⋊C4×C37—41no—
134C4⋊C4×C37—41no—
135C4⋊C4×C37—41no—
136C4⋊C4×C37—41no—
137C4⋊C4×C37C2×D412yes—
138C22⋊C4×C37—22no—
139C22⋊C4×C37—22no—
140C22⋊C4×C37—41no—
141C22⋊C4×C37—41no—
142C42×C37—41no—
143C42×C37—41no—
144C42×C37C2×D411yescharacteristic
145C42×C37C2×D411yescharacteristic
146C42×C37#4111yescharacteristic
147C42×C37C2411yescharacteristic
148C23×C37—22no—
149C23×C37#12111yescharacteristic
150Q8×C37—41no—
151Q8×C37—41no—
152Q8×C37—42no—
153Q8×C37—41no—
154Q8×C37—41no—
155Q8×C37—81no—
156Q8×C37—81no—
157D4×C37—22no—
158D4×C37—22no—
159D4×C37—81no—
160D4×C37—81no—
161D4×C37—82no—
162C2×C4×C37—21no—
163C2×C4×C37—21no—
164C2×C4×C37—22no—
165C2×C4×C37—22no—
166C2×C4×C37—41no—
167C2×C4×C37—42no—
168C2×C4×C37—42no—
169C2×C4×C37—41no—
170C2×C4×C37—41no—
171C2×C4×C37—41no—
172C2×C4×C37—41no—
173C2×C4×C37#12111yescharacteristic
174C2×C4×C37#12111yescharacteristic
175C2×C4×C37#12111yescharacteristic
176C2×C4×C37C22×D411yescharacteristic
177C2×C4×C37C22×D411yescharacteristic
178C2×C4×C37C2×#4111yescharacteristic
179C8×C37—22no—
180C8×C37—22no—
181C8×C37—42no—
182C8×C37—42no—
183C8×C37—42no—
184#19517C3711yesmaximal, characteristic, direct product factor, Sylow
185C22×C37—21no—
186C22×C37—21no—
187C22×C37—21no—
188C22×C37—42no—
189C22×C37—42no—
190C22×C37—42no—
191C22×C37C2×#12111yescharacteristic
192C4×C37—21no—
193C4×C37—21no—
194C4×C37—21no—
195C4×C37—21no—
196C4×C37—42no—
197C4×C37—81no—
198C4×C37—81no—
199C4×C37—81no—
200C4×C37—81no—
201C4×C37#57712yes—
202C4×C37#57711yescharacteristic
203C4×C37#57711yescharacteristic
204#3254C2×C3711yescharacteristic
205#3246C2×C3711yescharacteristic
206#3231C2×C3711yescharacteristic, complemented
207#2981C2×C3711yescharacteristic
208#1703C2×C3712yescomplemented
209#1693C2×C3712yescomplemented
210#1679C2×C3712yescomplemented
211#1671C2×C3712yescomplemented
212#1570C2×C3711yescharacteristic, complemented
213#1559C2×C3711yescharacteristic, complemented
214#1552C2×C3711yescharacteristic, complemented
215C2×C37—41no—
216C2×C37—82no—
217C2×C37#304211yescharacteristic
218C2×C37#306011yescharacteristic
219C2×C37#306911yescharacteristic
220#533C22×C3711yescharacteristic
221#533C22×C3711yescharacteristic
222#532C22×C3711yescharacteristic
223#500C22×C3711yescharacteristic
224#498C22×C3711yescharacteristic
225#487C22×C3711yescharacteristic
226#487C22×C3711yescharacteristic
227#485—22no—
228#479—22no—
229#477—21no—
230#476C22×C3712yes—
231#475—21no—
232#474C22×C3712yes—
233#468—22no—
234#465—22no—
235#461C22×C3712yes—
236#461C22×C3712yes—
237#460—21no—
238#460—21no—
239#436C22×C3712yes—
240#435—22no—
241C8×D4C22×C3712yes—
242#432C22×C3711yescharacteristic
243#431C22×C3711yescharacteristic
244#423C22×C3711yescharacteristic
245C2×C4⋊C8C22×C3711yescharacteristic
246#408—22no—
247#334C22×C3712yes—
248#334C22×C3712yes—
249#333C22×C3712yes—
250#332C22×C3712yes—
251#332C22×C3712yes—
252#332C22×C3712yes—
253C37#1951711yescharacteristic, direct product factor, Sylow
254C2×#41—21no—
255C2×SD16—22no—
256C2×SD16—22no—
257C2×M16—21no—
258C2×M16—22no—
259C22×C8—21no—
260C22×C8—22no—
261C4⋊Q8D4×C3711yescharacteristic
262C4⋊Q8C23×C3711yescharacteristic
263C4⋊Q8C23×C3711yescharacteristic
264C4⋊Q8C23×C3711yescharacteristic
265C4⋊Q8C23×C3711yescharacteristic
266#127D4×C3711yescharacteristic
267#125—21no—
268#125—21no—
269#124—21no—
270C22⋊Q8—21no—
271C22⋊Q8—21no—
272C22⋊Q8—21no—
273C22⋊Q8—21no—
274#121—21no—
275#121D4×C3712yes—
276C4×Q8—21no—
277C4×Q8—21no—
278C4×Q8—21no—
279C4×Q8—21no—
280C4×D4—21no—
281C4×D4C23×C3712yes—
282#117—21no—
283#117—21no—
284C2×C42C23×C3711yescharacteristic
285#107—22no—
286#107—22no—
287#106—22no—
288#106—22no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8D4×C3712yes—
292C4⋊C8C23×C3712yes—
293C4⋊C8C23×C3712yes—
294C4⋊C8C23×C3712yes—
295#103—22no—
296#103—22no—
297#103—22no—
298#103—22no—
299#102—22no—
300#102—22no—
301C22⋊C8—21no—
302C22⋊C8—22no—
303C22⋊C8—22no—
304C22⋊C8D4×C3712yes—
305#97—21no—
306C4×C8C23×C3711yescharacteristic
307C4×C8C23×C3711yescharacteristic
308#41—82no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×D4—22no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4C2×D4×C3712yes—
318C22×C4—22no—
319C22×C4C2×D4×C3711yescharacteristic
320C22×C4C2×D4×C3711yescharacteristic
321C22×C4#41×C3711yescharacteristic
322SD16—82no—
323SD16—82no—
324M16—82no—
325M16—81no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—82no—
330C2×C8—81no—
331C2×C8C2×D4×C3712yes—
332C2×C8C2×D4×C3712yes—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4C2×D4×C3712yes—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C42—41no—
352C42—41no—
353C42C2×D4×C3711yescharacteristic
354C42C2×D4×C3711yescharacteristic
355C42#41×C3711yescharacteristic
356C42C24×C3711yescharacteristic
357C23—22no—
358C23#121×C3711yescharacteristic
359Q8—41no—
360Q8—41no—
361Q8—42no—
362Q8—41no—
363Q8—41no—
364Q8—81no—
365Q8—81no—
366D4—22no—
367D4—22no—
368D4—81no—
369D4—81no—
370D4—82no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4#121×C3711yescharacteristic
383C2×C4#121×C3711yescharacteristic
384C2×C4#121×C3711yescharacteristic
385C2×C4C22×D4×C3711yescharacteristic
386C2×C4C22×D4×C3711yescharacteristic
387C2×C4C2×#41×C3711yescharacteristic
388C8—22no—
389C8—22no—
390C8—42no—
391C8—42no—
392C8—42no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22C2×#121×C3711yescharacteristic
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—42no—
405C4—81no—
406C4—81no—
407C4—81no—
408C4—81no—
409C4C37×#57712yes—
410C4C37×#57711yescharacteristic
411C4C37×#57711yescharacteristic
412C2—41no—
413C2—82no—
414C2C37×#304211yescharacteristic
415C2C37×#306011yescharacteristic
416C2C37×#306911yescharacteristic
subgroupG/H
center, socleC22×C37C2×#121
commutator, Frattini, cosocleC42C24×C37
prime radicalC37×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C37 ◁ C2×C42×C37 ◁ G

Complements

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

NH ≅ G/Ntype
3. C37×#3231C2semidirect
5. C37×#1703C2semidirect
6. C37×#1693C2semidirect
7. C37×#1679C2semidirect
8. C37×#1671C2semidirect
9. C37×#1570C2semidirect
10. C37×#1559C2semidirect
11. C37×#1552C2semidirect
184. #19517C37direct (N × H)
206. #3231C2×C37semidirect
208. #1703C2×C37semidirect
209. #1693C2×C37semidirect
210. #1679C2×C37semidirect
211. #1671C2×C37semidirect
212. #1570C2×C37semidirect
213. #1559C2×C37semidirect
214. #1552C2×C37semidirect
253. C37#19517direct (N × H)

Maximal subgroups

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

subgroupindex
C37×#32542
C37×#32462
C37×#32312
C37×#29812
C37×#17032
C37×#16932
C37×#16792
C37×#16712
C37×#15702
C37×#15592
C37×#15522
#1951737