Finite Group Databasebeta

← #18003958

Subgroups — #18003958

Order 4736.

698 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 145 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×#11924583C211yesmaximal, characteristic, complemented
2#14558331C211yesmaximal, characteristic, complemented
3#14558104C211yesmaximal, characteristic, complemented
4#14557714C214yesmaximal, complemented
5#14557704C214yesmaximal, complemented
6#14557550C211yesmaximal, characteristic, complemented
7#14557549C211yesmaximal, characteristic, complemented
8C2×#11924454C211yesmaximal, characteristic, complemented
9C37×#575C211yesmaximal, characteristic, complemented
10#11924583—22no—
11#11924583—22no—
12#11924583—22no—
13#11924583C2214yes—
14C2×#10596962—21no—
15C2×#10596962C2211yescharacteristic
16C2×#10596964—21no—
17C2×#10596964—22no—
18C22×Dic74—21no—
19C22×Dic74C2211yescharacteristic
20C2×Q8×D37—21no—
21#11924567C2212yescomplemented
22#11924566—21no—
23#11924562C2212yescomplemented
24#11924561—21no—
25#11924556—22no—
26C2×#10596934C2211yescharacteristic
27#11924534C2212yescomplemented
28#11924523C2212yescomplemented
29#11924517—22no—
30#11924515—22no—
31#11924495C2211yescharacteristic
32#11924463C2214yescomplemented
33#11924462C2214yescomplemented
34#11924454C2214yescomplemented
35C2×#41×C37C2211yescharacteristic
36C2×#41×C37C2211yescharacteristic
37#137×C37—21no—
38#137×C37C2212yescomplemented
39AGL(1,ℤ/8)×C37—21no—
40AGL(1,ℤ/8)×C37C2212yescomplemented
41#135×C37—22no—
42C2×M16×C37C2211yescharacteristic
43#41×C37—21no—
44#41×C37—21no—
45#41×C37—21no—
46#41×C37—21no—
47#41×C37—21no—
48#41×C37—21no—
49#41×C37D412yes—
50#41×C37C2312yes—
51C2×Q8×C37D411yescharacteristic
52C2×Q8×C37C2311yescharacteristic
53C2×D4×C37—21no—
54C2×D4×C37—21no—
55C2×D4×C37D411yescharacteristic
56C2×D4×C37C2311yescharacteristic
57C22×C4×C37—21no—
58C22×C4×C37—21no—
59C22×C4×C37C2311yescharacteristic
60C2×C22⋊D37—22no—
61C2×C22⋊D37—21no—
62C22×Dic37—22no—
63C22×Dic37D412yes—
64Q8×D37—24no—
65Q8×D37—24no—
66#10596964—28no—
67#10596964—24no—
68#10596964—24no—
69#10596964—28no—
70#10596962—24no—
71#10596962—21no—
72#10596962—21no—
73#10596962—24no—
74#10596962D412yes—
75#10596962C2312yes—
76C2×D148D411yescharacteristic
77C2×C4×D37—22no—
78C2×C4×D37—21no—
79C2×Dic74—22no—
80C2×Dic74—24no—
81C2×Dic74—21no—
82C2×Dic74—21no—
83C2×Dic74—21no—
84C2×Dic74—21no—
85C2×Dic74—24no—
86C2×Dic74D412yes—
87C2×Dic74D411yescharacteristic
88C2×Dic74C2312yes—
89Q16×C37—22no—
90SD16×C37—22no—
91SD16×C37—22no—
92D8×C37—22no—
93M16×C37—21no—
94M16×C37C2312yes—
95C2×C8×C37—21no—
96C22⋊Dic37—21no—
97C22⋊Dic37—24no—
98#10596942—22no—
99#10596941—22no—
100#10596940—22no—
101#10596939—22no—
102C4⋊Dic37—21no—
103#10596936—24no—
104C4×Dic37C2312yes—
105#10596934—21no—
106#10596934C2312yes—
107C2×#64436—21no—
108C23×C37—21no—
109C23×C37—21no—
110C23×C37C2×D411yescharacteristic
111C22×D37—21no—
112Q8×C37—21no—
113Q8×C37—21no—
114Q8×C37—21no—
115Q8×C37C2×D412yes—
116D4×C37—22no—
117D4×C37—22no—
118D4×C37—21no—
119D4×C37—21no—
120D4×C37—21no—
121D4×C37—41no—
122D4×C37—41no—
123D4×C37C2×D412yes—
124C2×C4×C37—21no—
125C2×C4×C37—22no—
126C2×C4×C37—22no—
127C2×C4×C37—21no—
128C2×C4×C37—41no—
129C2×C4×C37—41no—
130C2×C4×C37C2×D411yescharacteristic
131C2×C4×C37C2×D411yescharacteristic
132C2×C4×C37C2×D411yescharacteristic
133C2×C4×C37C2×D411yescharacteristic
134C2×C4×C37C2×D411yescharacteristic
135C2×C4×C37C2411yescharacteristic
136C22⋊D37—28no—
137C22⋊D37—22no—
138C22⋊D37—22no—
139C2×Dic37—28no—
140C2×Dic37—22no—
141C2×Dic37—21no—
142C2×Dic37—22no—
143C2×Dic37—22no—
144C2×Dic37C2×D414yes—
145D148—21no—
146D148C2×D412yes—
147C4×D37—28no—
148C4×D37—22no—
149C4×D37—22no—
150Dic74—28no—
151Dic74—22no—
152Dic74—21no—
153Dic74—22no—
154Dic74—22no—
155Dic74—22no—
156Dic74C2×D412yes—
157C8×C37—22no—
158#64436—22no—
159C22×C37—22no—
160C22×C37—41no—
161C22×C37—42no—
162C22×C37—42no—
163C22×C37—41no—
164C22×C37#12011yescharacteristic
165C22×C37#12011yescharacteristic
166C22×C37C22×D411yescharacteristic
167C2×D37—22no—
168C2×D37—42no—
169C4×C37—22no—
170C4×C37—41no—
171C4×C37#12011yescharacteristic
172C4×C37#12011yescharacteristic
173C4×C37C22×D411yescharacteristic
174C4×C37C22×D411yescharacteristic
175Dic37—24no—
176Dic37—22no—
177Dic37—42no—
178#3020—371nomaximal, Sylow
179C2×C37—21no—
180C2×C37—21no—
181C2×C37—21no—
182C2×C37—42no—
183C2×C37—81no—
184C2×C37C2×#12011yescharacteristic
185D37—42no—
186C2×2−1+4—371no—
187#575—371no—
188#575D3711yescharacteristic, complemented
189#536—371no—
190#456—374no—
191#454—374no—
192#421—371no—
193#421—371no—
194C2×#101—371no—
195C37#302011yescharacteristic, complemented, Sylow
1962−1+4—374no—
1972−1+4—742no—
1982−1+4—742no—
1992−1+4—742no—
200C2×#41—371no—
201C2×#41—741no—
202C2×#41—742no—
203C2×#41—741no—
204C2×#41C2×D3711yescharacteristic
205C2×#41C2×D3711yescharacteristic
206C22×Q8—371no—
207C22×Q8—741no—
208C22×Q8—741no—
209#137—21no—
210#137—372no—
211#137—741no—
212#137C2×D3712yescomplemented
213AGL(1,ℤ/8)—21no—
214AGL(1,ℤ/8)—372no—
215AGL(1,ℤ/8)—741no—
216AGL(1,ℤ/8)C2×D3712yescomplemented
217#135—22no—
218#135—742no—
219C2×M16—371no—
220C2×M16C2×D3711yescharacteristic
221C4⋊Q8—372no—
222#124—372no—
223#123—742no—
224C22⋊Q8—742no—
225#117—371no—
226#104—374no—
227#104—374no—
228#101—374no—
229#41—21no—
230#41—21no—
231#41—21no—
232#41—21no—
233#41—21no—
234#41—21no—
235#41—372no—
236#41—372no—
237#41—744no—
238#41—744no—
239#41—748no—
240#41—748no—
241#41—744no—
242#41—744no—
243#41—741no—
244#41—741no—
245#41C22⋊D3712yes—
246#41C22×D3712yes—
247C2×Q8—372no—
248C2×Q8—372no—
249C2×Q8—371no—
250C2×Q8—744no—
251C2×Q8—742no—
252C2×Q8—744no—
253C2×Q8—744no—
254C2×Q8—744no—
255C2×Q8—741no—
256C2×Q8—741no—
257C2×Q8—741no—
258C2×Q8—741no—
259C2×Q8C22⋊D3711yescharacteristic
260C2×Q8C22×D3711yescharacteristic
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—371no—
264C2×D4—742no—
265C2×D4—741no—
266C2×D4C22⋊D3711yescharacteristic
267C2×D4C22×D3711yescharacteristic
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—372no—
271C22×C4—742no—
272C22×C4—742no—
273C22×C4—741no—
274C22×C4C22×D3711yescharacteristic
275Q16—22no—
276Q16—742no—
277SD16—22no—
278SD16—22no—
279SD16—742no—
280SD16—742no—
281D8—22no—
282D8—742no—
283M16—21no—
284M16—372no—
285M16—741no—
286M16C22×D3712yes—
287C2×C8—21no—
288C2×C8—741no—
289C4⋊C4—744no—
290C4⋊C4—741no—
291C22⋊C4—744no—
292C22⋊C4—741no—
293C42—372no—
294C23—21no—
295C23—21no—
296C23—741no—
297C23D4×D3711yescharacteristic
298Q8—21no—
299Q8—21no—
300Q8—21no—
301Q8—372no—
302Q8—748no—
303Q8—742no—
304Q8—742no—
305Q8—741no—
306Q8—742no—
307Q8—742no—
308Q8C2×C22⋊D3712yes—
309D4—22no—
310D4—22no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—41no—
315D4—41no—
316D4—372no—
317D4—748no—
318D4—742no—
319D4—742no—
320D4—741no—
321D4C2×C22⋊D3712yes—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—374no—
329C2×C4—748no—
330C2×C4—748no—
331C2×C4—742no—
332C2×C4—742no—
333C2×C4—741no—
334C2×C4—742no—
335C2×C4—742no—
336C2×C4—742no—
337C2×C4D4×D3711yescharacteristic
338C2×C4D4×D3711yescharacteristic
339C2×C4D4×D3711yescharacteristic
340C2×C4C2×C22⋊D3711yescharacteristic
341C2×C4C2×C22⋊D3711yescharacteristic
342C2×C4C23×D3711yescharacteristic
343C8—22no—
344C8—742no—
345C22—22no—
346C22—41no—
347C22—42no—
348C22—42no—
349C22—41no—
350C22—742no—
351C22—1482no—
352C22#1192450411yescharacteristic
353C22#1192450411yescharacteristic
354C22C2×D4×D3711yescharacteristic
355C4—22no—
356C4—41no—
357C4—744no—
358C4—742no—
359C4—1482no—
360C4#1192450411yescharacteristic
361C4#1192450411yescharacteristic
362C4C2×D4×D3711yescharacteristic
363C4C22×C22⋊D3711yescharacteristic
364C2—21no—
365C2—21no—
366C2—21no—
367C2—42no—
368C2—81no—
369C2—1482no—
370C2C2×#1192450411yescharacteristic
subgroupG/H
centerC2C2×#11924504
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#575C2
socleC2×C37C2×#120
hypercenter#575D37
hypocenterC37#3020
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#11924583C2semidirect
2. #14558331C2semidirect
3. #14558104C2semidirect
4. #14557714C2semidirect
5. #14557704C2semidirect
6. #14557550C2semidirect
7. #14557549C2semidirect
8. C2×#11924454C2semidirect
9. C37×#575C2semidirect
21. #11924567C22semidirect
23. #11924562C22semidirect
27. #11924534C22semidirect
28. #11924523C22semidirect
32. #11924463C22semidirect
33. #11924462C22semidirect
34. #11924454C22semidirect
38. #137×C37C22semidirect
40. AGL(1,ℤ/8)×C37C22semidirect
188. #575D37semidirect
195. C37#3020semidirect
212. #137C2×D37semidirect
216. AGL(1,ℤ/8)C2×D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245832
#145583312
#145581042
#145577142
#145577042
#145575502
#145575492
C2×#119244542
C37×#5752
#302037