Finite Group Databasebeta

← #18010605

Subgroups — #18010605

Order 4736.

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

normal subgroups only

This group has 578 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#14558600C211yesmaximal, characteristic, complemented
2D37⋊#99C211yesmaximal, characteristic, complemented
3#14558474C212yesmaximal, complemented
4#14558410C212yesmaximal, complemented
5#120×D37C211yesmaximal, characteristic
6D4×#3866—24no—
7#11924615—21no—
8C2×C22⋊#3866—21no—
9C2×C22⋊#3866—24no—
10C2×C22⋊#3866C2212yes—
11C2×C22⋊#3866C2212yes—
12#11924595—21no—
13#11924594—21no—
14C24×D37C2211yescharacteristic
15C2×D4×D37—21no—
16C2×D4×D37C2211yescharacteristic
17#11924506—21no—
18#11924506C411yescharacteristic, complemented
19#11924505C411yescharacteristic, complemented
20#11924504—21no—
21#11924504C411yescharacteristic, complemented
22C22⋊C4×D37—21no—
23C22⋊C4×D37C2211yescharacteristic
24#120×C37C411yescharacteristic, complemented
25C24×C37C2×C411yescharacteristic, complemented
26C23×D37—21no—
27C23×D37—22no—
28C23×D37—21no—
29C23×D37—22no—
30C23×D37—44no—
31C23×D37C2×C411yescharacteristic, complemented
32C23×D37D411yescharacteristic
33C23×D37C2311yescharacteristic
34C22×#3866—22no—
35C22×#3866—22no—
36C22×#3866D414yescomplemented
37C2×D4×C37—21no—
38C2×D4×C37C2×C411yescharacteristic
39C2×C22⋊D37—21no—
40C2×C22⋊D37—21no—
41C2×C22⋊D37C2×C411yescharacteristic
42C2×C22⋊D37D411yescharacteristic
43D4×D37—22no—
44D4×D37—22no—
45D4×D37—44no—
46C2×D148—21no—
47C2×D148D411yescharacteristic
48C2×C4×D37—21no—
49C2×C4×D37D411yescharacteristic
50C22⋊#3866—22no—
51C22⋊#3866—22no—
52C22⋊#3866—22no—
53C22⋊#3866—22no—
54C22⋊#3866—22no—
55C22⋊#3866—48no—
56C4⋊#3866—22no—
57C4×#3866—22no—
58C22⋊C4×C37—21no—
59C22⋊C4×C37C2×C411yescharacteristic, complemented
60C22⋊Dic37—21no—
61C22⋊Dic37C2×C411yescharacteristic, complemented
62#10596938—21no—
63#10596938—21no—
64#10596938—21no—
65C23×C37—21no—
66C23×C37—22no—
67C23×C37—42no—
68C23×C37C22⋊C411yescharacteristic, complemented
69C23×C37C22×C411yescharacteristic
70C22×D37—28no—
71C22×D37—24no—
72C22×D37—21no—
73C22×D37—22no—
74C22×D37—21no—
75C22×D37—21no—
76C22×D37—42no—
77C22×D37—42no—
78C22×D37—42no—
79C22×D37—44no—
80C22×D37—42no—
81C22×D37—44no—
82C22×D37—42no—
83C22×D37—42no—
84C22×D37—44no—
85C22×D37—41no—
86C22×D37—42no—
87C22×D37—41no—
88C22×D37C22⋊C411yescharacteristic, complemented
89C22×D37C2×D412yes—
90C22×D37C2×D411yescharacteristic
91C22×D37C2×D411yescharacteristic
92C22×D37#4112yes—
93C2×#3866—24no—
94C2×#3866—44no—
95C2×#3866—44no—
96C2×#3866—42no—
97C2×#3866—82no—
98D4×C37—22no—
99D4×C37—42no—
100C2×C4×C37—21no—
101C2×C4×C37C22⋊C411yescharacteristic, complemented
102C22⋊D37—22no—
103C22⋊D37—22no—
104C22⋊D37—42no—
105C22⋊D37—42no—
106C2×Dic37—21no—
107C2×Dic37C22⋊C411yescharacteristic, complemented
108D148—22no—
109D148—42no—
110C4×D37—21no—
111C4×D37—21no—
112C4×D37—42no—
113C22×C37—22no—
114C22×C37—24no—
115C22×C37—21no—
116C22×C37—41no—
117C22×C37—41no—
118C22×C37—41no—
119C22×C37—42no—
120C22×C37—41no—
121C22×C37C2×C22⋊C411yescharacteristic
122C22×C37C4×D412yes—
123C2×D37—22no—
124C2×D37—24no—
125C2×D37—22no—
126C2×D37—24no—
127C2×D37—21no—
128C2×D37—41no—
129C2×D37—42no—
130C2×D37—48no—
131C2×D37—44no—
132C2×D37—41no—
133C2×D37—42no—
134C2×D37—41no—
135C2×D37—42no—
136C2×D37—44no—
137C2×D37—41no—
138C2×D37—81no—
139C2×D37#12011yescharacteristic
140C2×D37#12112yes—
141C2×D37#12311yescharacteristic
142#3866—44no—
143#3866—82no—
144C4×C37—21no—
145C4×C37—41no—
146Dic37—21no—
147Dic37—41no—
148#1889—371nomaximal, Sylow
149C2×C37—21no—
150C2×C37—22no—
151C2×C37—42no—
152C2×C37—41no—
153C2×C37#38611yescharacteristic
154D37—22no—
155D37—21no—
156D37—42no—
157D37—41no—
158D37B(4,𝔽2)12yes—
159C2×#120—371no—
160#518—371no—
161C2×#99—371no—
162#379—372no—
163#342—372no—
164C37#188911yescharacteristic, complemented, Sylow
165C25—371no—
166C22×D4—371no—
167C22×D4—741no—
168#120—371no—
169#120—371no—
170#120—371no—
171#120—741no—
172#120—741no—
173#120#386611yescharacteristic, complemented
174C4×D4—744no—
175#117—741no—
176C2×C22⋊C4—371no—
177C2×C22⋊C4—372no—
178C2×C22⋊C4—372no—
179C2×C22⋊C4—741no—
180C2×C22⋊C4—741no—
181C2×C22⋊C4—744no—
182#99—741no—
183#99—741no—
184C24—371no—
185C24—371no—
186C24—371no—
187C24—742no—
188C24—741no—
189C24—741no—
190C24—742no—
191C24—1484no—
192C24C2×#386611yescharacteristic, complemented
193C2×D4—21no—
194C2×D4—371no—
195C2×D4—371no—
196C2×D4—371no—
197C2×D4—741no—
198C2×D4—742no—
199C2×D4—742no—
200C2×D4—741no—
201C2×D4—741no—
202C2×D4—1484no—
203C2×D4C2×#386611yescharacteristic
204C22×C4—371no—
205C22×C4—374no—
206C22×C4—741no—
207C22×C4—742no—
208C22×C4—742no—
209C4⋊C4—742no—
210C22⋊C4—21no—
211C22⋊C4—371no—
212C22⋊C4—741no—
213C22⋊C4—741no—
214C22⋊C4—741no—
215C22⋊C4—742no—
216C22⋊C4—742no—
217C22⋊C4—742no—
218C22⋊C4—741no—
219C22⋊C4—742no—
220C22⋊C4—742no—
221C22⋊C4—1488no—
222C22⋊C4C2×#386611yescharacteristic, complemented
223C42—742no—
224C23—21no—
225C23—22no—
226C23—42no—
227C23—371no—
228C23—372no—
229C23—371no—
230C23—372no—
231C23—371no—
232C23—741no—
233C23—741no—
234C23—748no—
235C23—744no—
236C23—741no—
237C23—742no—
238C23—1482no—
239C23—1484no—
240C23—1481no—
241C23—1481no—
242C23—1482no—
243C23—1482no—
244C23—1482no—
245C23—1482no—
246C23—1484no—
247C23—1482no—
248C23—1482no—
249C23—1484no—
250C23C22⋊#386611yescharacteristic, complemented
251C23C22×#386611yescharacteristic
252D4—22no—
253D4—42no—
254D4—742no—
255D4—742no—
256D4—742no—
257D4—1482no—
258D4—1482no—
259D4—1482no—
260C2×C4—21no—
261C2×C4—371no—
262C2×C4—2962no—
263C2×C4—741no—
264C2×C4—741no—
265C2×C4—744no—
266C2×C4—741no—
267C2×C4—1482no—
268C2×C4—1482no—
269C2×C4—1484no—
270C2×C4—1484no—
271C2×C4C22⋊#386611yescharacteristic, complemented
272C22—22no—
273C22—24no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—41no—
280C22—371no—
281C22—371no—
282C22—372no—
283C22—2961no—
284C22—744no—
285C22—742no—
286C22—742no—
287C22—744no—
288C22—741no—
289C22—1481no—
290C22—1482no—
291C22—1481no—
292C22—1481no—
293C22—1482no—
294C22—1482no—
295C22—1484no—
296C22—1488no—
297C22—1484no—
298C22—1481no—
299C22C2×C22⋊#386611yescharacteristic
300C22D4×#386612yes—
301C4—21no—
302C4—41no—
303C4—2962no—
304C4—741no—
305C4—1481no—
306C4—1484no—
307C2—21no—
308C2—22no—
309C2—42no—
310C2—41no—
311C2—372no—
312C2—742no—
313C2—741no—
314C2—1481no—
315C2—1482no—
316C2#1455848111yescharacteristic
subgroupG/H
centerC2#14558481
commutatorC23×C37C22×C4
FrattiniC23C22×#3866
Fitting#120×C37C4
socleC2×C37#386
prime radical#120×D37C2
cosocleC23×D37C23
hypercenter#120#3866
hypocenterC37#1889
LCSG ▷ C23×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C23 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #14558600C2semidirect
2. D37⋊#99C2semidirect
3. #14558474C2semidirect
4. #14558410C2semidirect
18. #11924506C4semidirect
19. #11924505C4semidirect
21. #11924504C4semidirect
24. #120×C37C4semidirect
25. C24×C37C2×C4semidirect
31. C23×D37C2×C4semidirect
36. C22×#3866D4semidirect
59. C22⋊C4×C37C2×C4semidirect
61. C22⋊Dic37C2×C4semidirect
68. C23×C37C22⋊C4semidirect
88. C22×D37C22⋊C4semidirect
101. C2×C4×C37C22⋊C4semidirect
107. C2×Dic37C22⋊C4semidirect
164. C37#1889semidirect
173. #120#3866semidirect
192. C24C2×#3866semidirect
222. C22⋊C4C2×#3866semidirect
250. C23C22⋊#3866semidirect
271. C2×C4C22⋊#3866semidirect

Maximal subgroups

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

subgroupindex
#145586002
D37⋊#992
#145584742
#145584102
#120×D372
#188937