Subgroups — #26369356

Order 12032.

682 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	#20245188	C₂	1	4	yes	maximal, complemented
3	#20245184	C₂	1	4	yes	maximal, complemented
4	#20245143	C₂	1	2	yes	maximal, complemented
5	#20244999	C₂	1	1	yes	maximal, characteristic
6	M₃₂×Dic₄₇	C₂	1	2	yes	maximal, complemented
7	C₄₇×#2112	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄₇⋊C₈	C₄	1	1	yes	characteristic
9	C₂²×C₄×Dic₄₇	C₄	1	1	yes	characteristic
10	C₂×#13093964	C₂²	1	2	yes	—
11	C₂×C₈×Dic₄₇	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
13	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
14	C₂×C₈×Dic₄₇	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
17	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
18	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
19	Dic₄₇⋊C₁₆	C₂²	1	8	yes	complemented
20	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	4	yes	—
21	#15688898	C₂²	1	4	yes	complemented
22	C₁₆×Dic₄₇	C₂²	1	4	yes	complemented
23	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
25	C₄₇×C₂²⋊C₁₆	C₂²	1	4	yes	—
26	C₂³×Dic₄₇	C₈	1	2	yes	—
27	C₂²×C₄₇⋊C₈	—	2	4	no	—
28	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₄₇	C₈	1	2	yes	—
33	C₂×C₄×Dic₄₇	C₈	1	2	yes	—
34	C₂×C₄×Dic₄₇	C₈	1	2	yes	—
35	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
39	#13093964	—	2	4	no	—
40	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
41	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
42	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
43	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
44	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
45	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
46	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
47	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
48	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
49	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
50	C₈×Dic₄₇	C₂³	1	2	yes	—
51	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	—	2	4	no	—
53	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
55	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
56	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
57	M₃₂×C₄₇	—	2	4	no	—
58	C₂×C₁₆×C₄₇	C₂³	1	4	yes	—
59	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×Dic₄₇	—	2	8	no	—
67	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
68	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
69	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
70	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
71	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
72	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
73	C₂×C₈×C₄₇	—	2	2	no	—
74	C₂×C₈×C₄₇	—	2	4	no	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₇	#41	1	4	yes	—
79	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
80	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
81	C₄×Dic₄₇	C₂×C₈	1	1	yes	characteristic
82	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
83	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
84	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
85	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
86	C₄×Dic₄₇	C₂×C₈	1	1	yes	characteristic
87	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
88	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
89	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄₇⋊C₈	—	2	2	no	—
91	C₂×C₄₇⋊C₈	—	2	4	no	—
92	C₂×C₄₇⋊C₈	—	2	4	no	—
93	C₂×C₄₇⋊C₈	M₁₆	1	4	yes	—
94	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
97	C₁₆×C₄₇	—	2	4	no	—
98	C₄₇⋊C₁₆	—	2	4	no	—
99	C₂³×C₄₇	—	2	2	no	—
100	C₂³×C₄₇	—	2	4	no	—
101	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
102	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	#117	1	2	yes	—
108	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×Dic₄₇	—	2	4	no	—
113	C₂×Dic₄₇	—	2	4	no	—
114	C₂×Dic₄₇	—	2	4	no	—
115	C₂×Dic₄₇	—	2	4	no	—
116	C₂×Dic₄₇	—	2	4	no	—
117	C₂×Dic₄₇	M₃₂	1	8	yes	complemented
118	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
119	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
120	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
121	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
122	C₈×C₄₇	—	2	2	no	—
123	C₈×C₄₇	#117	1	2	yes	—
124	C₈×C₄₇	C₂×#41	1	2	yes	—
125	C₄₇⋊C₈	—	2	2	no	—
126	C₄₇⋊C₈	C₂×M₁₆	1	2	yes	—
127	C₄₇⋊C₈	C₂×M₁₆	1	2	yes	—
128	#33418	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	—	2	8	no	—
130	C₂²×C₄₇	—	2	2	no	—
131	C₂²×C₄₇	—	2	2	no	—
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	#431	1	2	yes	—
134	C₂²×C₄₇	C₂×M₃₂	1	4	yes	—
135	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
136	C₄×C₄₇	—	2	2	no	—
137	C₄×C₄₇	#431	1	2	yes	—
138	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
139	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
140	Dic₄₇	—	2	2	no	—
141	Dic₄₇	—	2	2	no	—
142	Dic₄₇	C₂×M₃₂	1	4	yes	—
143	Dic₄₇	C₂×M₃₂	1	4	yes	—
144	C₂²×C₄×C₈	—	47	1	no	—
145	#2163	—	47	4	no	—
146	#2162	—	47	4	no	—
147	#2150	—	47	2	no	—
148	#2112	—	47	1	no	—
149	#2112	D₄₇	1	1	yes	characteristic
150	C₄×M₃₂	—	47	2	no	—
151	C₂×C₄₇	—	2	2	no	—
152	C₂×C₄₇	—	2	4	no	—
153	C₂×C₄₇	C₂×#431	1	1	yes	characteristic
154	C₂×C₄₇	C₂²×M₃₂	1	2	yes	—
155	C₂³×C₈	—	47	1	no	—
156	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
157	C₂²×C₄²	—	47	1	no	—
158	C₂×M₃₂	—	47	2	no	—
159	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
160	C₂×C₄×C₈	—	47	1	no	—
161	C₂×C₄×C₈	—	47	2	no	—
162	C₂×C₄×C₈	—	47	2	no	—
163	C₂×C₄×C₈	—	47	1	no	—
164	C₂×C₄×C₈	—	47	1	no	—
165	C₂×C₄×C₈	—	47	1	no	—
166	C₂×C₄×C₈	—	47	2	no	—
167	C₂×C₄×C₈	—	47	2	no	—
168	C₄⋊C₁₆	—	47	8	no	—
169	C₂²⋊C₁₆	—	47	4	no	—
170	C₂²⋊C₁₆	C₂×D₄₇	1	4	yes	—
171	#346	—	47	4	no	—
172	C₄×C₁₆	—	47	4	no	—
173	C₄₇	#33418	1	1	yes	characteristic, complemented, Sylow
174	C₂³×C₄	—	47	2	no	—
175	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
176	C₂²×C₈	—	2	4	no	—
177	C₂²×C₈	—	47	1	no	—
178	C₂²×C₈	—	47	2	no	—
179	C₂²×C₈	—	47	2	no	—
180	C₂²×C₈	—	47	1	no	—
181	C₂²×C₈	—	94	4	no	—
182	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
183	C₂²×C₈	C₄×D₄₇	1	2	yes	—
184	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
185	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
186	C₂×C₄²	—	47	2	no	—
187	C₂×C₄²	—	47	2	no	—
188	C₂×C₄²	—	47	2	no	—
189	C₂×C₄²	—	47	2	no	—
190	C₂×C₄²	—	47	2	no	—
191	C₂×C₄²	—	47	1	no	—
192	C₂×C₄²	—	47	1	no	—
193	M₃₂	—	2	4	no	—
194	M₃₂	—	94	4	no	—
195	C₂×C₁₆	—	47	4	no	—
196	C₂×C₁₆	C₂²×D₄₇	1	4	yes	—
197	C₄×C₈	—	47	2	no	—
198	C₄×C₈	—	47	2	no	—
199	C₄×C₈	—	47	1	no	—
200	C₄×C₈	—	47	1	no	—
201	C₄×C₈	—	47	2	no	—
202	C₄×C₈	—	47	2	no	—
203	C₄×C₈	—	47	2	no	—
204	C₄×C₈	—	47	1	no	—
205	C₄×C₈	—	47	1	no	—
206	C₄×C₈	—	47	2	no	—
207	C₂⁴	C₈×D₄₇	1	1	yes	characteristic
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	47	2	no	—
211	C₂²×C₄	—	47	2	no	—
212	C₂²×C₄	—	47	2	no	—
213	C₂²×C₄	—	47	2	no	—
214	C₂²×C₄	—	47	2	no	—
215	C₂²×C₄	—	47	2	no	—
216	C₂²×C₄	—	94	8	no	—
217	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
218	C₂²×C₄	C₈×D₄₇	1	2	yes	—
219	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
220	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	47	4	no	—
225	C₂×C₈	—	47	1	no	—
226	C₂×C₈	—	47	1	no	—
227	C₂×C₈	—	47	2	no	—
228	C₂×C₈	—	94	4	no	—
229	C₂×C₈	—	94	4	no	—
230	C₂×C₈	—	94	2	no	—
231	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
232	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
233	C₂×C₈	#10625352	1	4	yes	—
234	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
235	C₄²	—	47	2	no	—
236	C₄²	—	47	1	no	—
237	C₄²	—	47	2	no	—
238	C₄²	—	47	2	no	—
239	C₄²	—	47	2	no	—
240	C₄²	—	47	2	no	—
241	C₄²	—	47	1	no	—
242	C₄²	—	47	1	no	—
243	C₄²	—	47	2	no	—
244	C₄²	—	47	1	no	—
245	C₁₆	—	2	4	no	—
246	C₁₆	—	94	4	no	—
247	C₂³	—	2	4	no	—
248	C₂³	—	2	2	no	—
249	C₂³	C₂×C₈×D₄₇	1	2	yes	—
250	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	47	8	no	—
256	C₂×C₄	—	47	2	no	—
257	C₂×C₄	—	47	2	no	—
258	C₂×C₄	—	47	2	no	—
259	C₂×C₄	—	47	2	no	—
260	C₂×C₄	—	94	4	no	—
261	C₂×C₄	—	94	4	no	—
262	C₂×C₄	—	94	4	no	—
263	C₂×C₄	—	94	4	no	—
264	C₂×C₄	—	94	4	no	—
265	C₂×C₄	#13093980	1	2	yes	—
266	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
267	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
268	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
269	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	47	2	no	—
272	C₈	—	47	2	no	—
273	C₈	—	94	2	no	—
274	C₈	#13093980	1	2	yes	—
275	C₈	C₂×#10625352	1	2	yes	—
276	C₂²	—	2	8	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	#15689173	1	2	yes	—
281	C₂²	M₃₂×D₄₇	1	4	yes	—
282	C₂²	C₂²×C₈×D₄₇	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	47	4	no	—
285	C₄	—	47	4	no	—
286	C₄	—	94	2	no	—
287	C₄	—	94	2	no	—
288	C₄	#15689173	1	2	yes	—
289	C₄	C₂×#13093980	1	1	yes	characteristic
290	C₄	C₂×M₁₆×D₄₇	1	1	yes	characteristic
291	C₂	—	2	4	no	—
292	C₂	—	2	2	no	—
293	C₂	C₂×#15689173	1	1	yes	characteristic
294	C₂	C₂×M₃₂×D₄₇	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₈
Fitting	C₄₇×#2112	C₂
prime radical	C₂⁴×C₄₇	C₂×C₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2112	D₄₇
hypocenter	C₄₇	#33418
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ #2112

Complements

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

N	H ≅ G/N	type
2. #20245188	C₂	semidirect
3. #20245184	C₂	semidirect
4. #20245143	C₂	semidirect
6. M₃₂×Dic₄₇	C₂	semidirect
19. Dic₄₇⋊C₁₆	C₂²	semidirect
21. #15688898	C₂²	semidirect
22. C₁₆×Dic₄₇	C₂²	semidirect
117. C₂×Dic₄₇	M₃₂	semidirect
173. C₄₇	#33418	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×Dic₄₇	2
#20245188	2
#20245184	2
#20245143	2
#20244999	2
M₃₂×Dic₄₇	2
C₄₇×#2112	2
#33418	47