Finite Group Databasebeta

← #18010595

Subgroups — #18010595

Order 4736.

334 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 68 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
1#14558602C211yesmaximal, characteristic
2D37⋊#104C211yesmaximal, characteristic
3#14558526C211yesmaximal, characteristic, complemented
4(C2×#3866)⋊Q8C211yesmaximal, characteristic, complemented
5#14558412C211yesmaximal, characteristic, complemented
6#14558405C211yesmaximal, characteristic, complemented
7#137×D37C211yesmaximal, characteristic
8D37⋊M16C2211yescharacteristic
9Q8×#3866—21no—
10D4×#3866—22no—
11D4×#3866—21no—
12#11924615—21no—
13#11924615C2211yescharacteristic
14C2×C4⋊#3866—21no—
15C2×C4⋊#3866C2211yescharacteristic
16C2×C4×#3866C2211yescharacteristic
17#11924601—21no—
18#11924600—21no—
19#11924599—21no—
20#11924599—21no—
21(C2×#3866)⋊C4—22no—
22#41×D37C2211yescharacteristic
23C2×Q8×D37C2211yescharacteristic
24#11924569C411yescharacteristic, complemented
25#11924568—21no—
26#11924567C411yescharacteristic, complemented
27#11924566C411yescharacteristic, complemented
28#11924565—21no—
29Q16×D37—21no—
30SD16×D37—21no—
31M16×D37C2211yescharacteristic
32#137×C37C411yescharacteristic, complemented
33C22×#3866—21no—
34C22×#3866—22no—
35C22×#3866D412yes—
36#41×C37C2×C411yescharacteristic
37C2×Q8×C37C2×C411yescharacteristic, complemented
38#10596966D411yescharacteristic
39Q8×D37—21no—
40Q8×D37—21no—
41Q8×D37—21no—
42Q8×D37—21no—
43Q8×D37D411yescharacteristic
44#10596964—21no—
45#10596964D411yescharacteristic
46D4×D37—21no—
47D4×D37D411yescharacteristic
48#10596962—21no—
49#10596962C2×C411yescharacteristic
50C2×C4×D37—21no—
51C2×C4×D37—21no—
52C2×C4×D37C2311yescharacteristic
53C2×Dic74—21no—
54C2×Dic74C2×C411yescharacteristic, complemented
55C22⋊#3866—22no—
56C22⋊#3866—21no—
57#10596957—21no—
58C4⋊#3866—22no—
59C4⋊#3866—21no—
60C4⋊#3866—21no—
61C4⋊#3866D411yescharacteristic
62C4⋊#3866D411yescharacteristic
63C4×#3866—22no—
64C4×#3866—21no—
65C4×#3866—21no—
66C4×#3866—21no—
67#10596953—21no—
68D37⋊C8—21no—
69Q16×C37—21no—
70SD16×C37—21no—
71M16×C37C2×C411yescharacteristic, complemented
72#10596942—21no—
73#10596942—21no—
74#10596941—21no—
75#10596940—21no—
76#10596934C2×C411yescharacteristic, complemented
77Dic148—21no—
78#10596930—21no—
79#10596929—21no—
80C8×D37—21no—
81C22×D37—21no—
82C22×D37C2×D411yescharacteristic
83C2×#3866—22no—
84C2×#3866—21no—
85C2×#3866—22no—
86C2×#3866—21no—
87C2×#3866—41no—
88C2×#3866—41no—
89C2×#3866—42no—
90C2×#3866—82no—
91Q8×C37—21no—
92Q8×C37—21no—
93Q8×C37C22⋊C411yescharacteristic
94D4×C37—21no—
95D4×C37C22⋊C411yescharacteristic
96C2×C4×C37—21no—
97C2×C4×C37—21no—
98C2×C4×C37C22×C411yescharacteristic
99C22⋊D37—21no—
100C22⋊D37—21no—
101C22⋊D37—21no—
102C2×Dic37—21no—
103C2×Dic37—21no—
104C2×Dic37#4111yescharacteristic
105D148—21no—
106D148C22⋊C411yescharacteristic
107C4×D37—21no—
108C4×D37—21no—
109C4×D37—21no—
110C4×D37—41no—
111C4×D37—41no—
112C4×D37C2×D411yescharacteristic
113C4×D37C2×D411yescharacteristic
114C4×D37C2×D411yescharacteristic
115C4×D37#4111yescharacteristic
116Dic74—21no—
117Dic74—21no—
118Dic74—21no—
119Dic74—41no—
120Dic74—41no—
121Dic74C22⋊C411yescharacteristic
122#64438—41no—
123C8×C37—21no—
124#64436—21no—
125C22×C37—21no—
126C22×C37C4×D411yescharacteristic
127C2×D37—21no—
128C2×D37—21no—
129C2×D37—21no—
130C2×D37—41no—
131C2×D37—41no—
132C2×D37#12011yescharacteristic
133C2×D37#12111yescharacteristic
134#3866—42no—
135#3866—42no—
136#3866—41no—
137#3866—41no—
138C4×C37—21no—
139C4×C37—41no—
140C4×C37C2×C22⋊C411yescharacteristic
141C4×C37C4×D411yescharacteristic
142Dic37—21no—
143Dic37—41no—
144Dic37#12111yescharacteristic
145Dic37#12311yescharacteristic
146#1882—371nomaximal, Sylow
147C2×C37—21no—
148C2×C37—41no—
149C2×C37#38611yescharacteristic
150D37—21no—
151D37—41no—
152D37#45511yescharacteristic
153D37#45611yescharacteristic
154C2×#137—371no—
155#520—371no—
156C2×#104—371no—
157#419—371no—
158#391—371no—
159#343—371no—
160#339—371no—
161C37#188211yescharacteristic, complemented, Sylow
162C2×#41—371no—
163C22×Q8—371no—
164#137—371no—
165#137—371no—
166#137—371no—
167#137—741no—
168#137—741no—
169#137#386611yescharacteristic, complemented
170C2×Q16—741no—
171C2×SD16—741no—
172C2×M16—371no—
173C2×M16—371no—
174C4×Q8—741no—
175C4×D4—742no—
176C4×D4—741no—
177#117—371no—
178#117—741no—
179C2×C4⋊C4—371no—
180C2×C4⋊C4—741no—
181C2×C42—371no—
182#104—741no—
183#104—741no—
184#103—741no—
185#103—741no—
186#95—742no—
187#41—371no—
188#41—371no—
189#41—371no—
190#41—741no—
191#41—741no—
192#41C2×#386611yescharacteristic
193C2×Q8—371no—
194C2×Q8—371no—
195C2×Q8—741no—
196C2×Q8—741no—
197C2×Q8—741no—
198C2×Q8—741no—
199C2×Q8—741no—
200C2×Q8C2×#386611yescharacteristic, complemented
201C2×D4—371no—
202C2×D4—741no—
203C22×C4—371no—
204C22×C4—372no—
205C22×C4—741no—
206C22×C4—741no—
207C22×C4—742no—
208C22×C4—741no—
209Q16—21no—
210Q16—741no—
211Q16—741no—
212Q16—741no—
213SD16—21no—
214SD16—741no—
215SD16—741no—
216SD16—741no—
217M16—371no—
218M16—741no—
219M16—741no—
220M16—741no—
221M16C2×#386611yescharacteristic, complemented
222C2×C8—741no—
223C2×C8—741no—
224C4⋊C4—371no—
225C4⋊C4—371no—
226C4⋊C4—741no—
227C4⋊C4—741no—
228C4⋊C4—742no—
229C22⋊C4—741no—
230C22⋊C4—742no—
231C42—741no—
232C42—741no—
233C42—741no—
234C42—742no—
235C23—371no—
236C23—741no—
237Q8—21no—
238Q8—21no—
239Q8—371no—
240Q8—741no—
241Q8—741no—
242Q8—741no—
243Q8—1481no—
244Q8—1481no—
245Q8C22⋊#386611yescharacteristic
246D4—21no—
247D4—371no—
248D4—741no—
249D4—741no—
250D4—741no—
251D4—741no—
252D4C22⋊#386611yescharacteristic
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—371no—
256C2×C4—371no—
257C2×C4—371no—
258C2×C4—371no—
259C2×C4—371no—
260C2×C4—2962no—
261C2×C4—741no—
262C2×C4—741no—
263C2×C4—741no—
264C2×C4—741no—
265C2×C4—741no—
266C2×C4—741no—
267C2×C4—742no—
268C2×C4—742no—
269C2×C4—741no—
270C2×C4—1481no—
271C2×C4—1481no—
272C2×C4—1481no—
273C2×C4—1481no—
274C2×C4—1482no—
275C2×C4C22×#386611yescharacteristic
276C8—21no—
277C8—741no—
278C8—1481no—
279C22—21no—
280C22—371no—
281C22—371no—
282C22—741no—
283C22—741no—
284C22—741no—
285C22—1481no—
286C22—1481no—
287C22D4×#386611yescharacteristic
288C4—21no—
289C4—41no—
290C4—371no—
291C4—371no—
292C4—741no—
293C4—1481no—
294C4—1482no—
295C4—1481no—
296C4—1482no—
297C4—1481no—
298C4C2×C22⋊#386611yescharacteristic
299C4D4×#386611yescharacteristic
300C2—21no—
301C2—41no—
302C2—371no—
303C2—371no—
304C2—741no—
305C2—1481no—
306C2#1455848111yescharacteristic
subgroupG/H
centerC2#14558481
commutatorC2×C4×C37C22×C4
FrattiniC2×C4C22×#3866
Fitting#137×C37C4
socleC2×C37#386
prime radical#41×D37C22
cosocleC2×C4×D37C23
hypercenter#137#3866
hypocenterC37#1882
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
3. #14558526C2semidirect
4. (C2×#3866)⋊Q8C2semidirect
5. #14558412C2semidirect
6. #14558405C2semidirect
24. #11924569C4semidirect
26. #11924567C4semidirect
27. #11924566C4semidirect
32. #137×C37C4semidirect
37. C2×Q8×C37C2×C4semidirect
54. C2×Dic74C2×C4semidirect
71. M16×C37C2×C4semidirect
76. #10596934C2×C4semidirect
161. C37#1882semidirect
169. #137#3866semidirect
200. C2×Q8C2×#3866semidirect
221. M16C2×#3866semidirect

Maximal subgroups

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

subgroupindex
#145586022
D37⋊#1042
#145585262
(C2×#3866)⋊Q82
#145584122
#145584052
#137×D372
#188237