Subgroups — C₄₇×#12681 #26052379

Order 12032.

1038 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₇×#349	C₂	1	4	yes	maximal, complemented
3	C₄₇×#2112	C₂	1	2	yes	maximal, complemented
4	C₂⁴×C₄×C₄₇	C₄	1	1	yes	characteristic
5	C₂²×M₁₆×C₄₇	—	2	8	no	—
6	C₂²×M₁₆×C₄₇	—	2	4	no	—
7	C₂²×M₁₆×C₄₇	C₄	1	2	yes	—
8	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
9	C₂³×C₈×C₄₇	C₄	1	1	yes	characteristic
10	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×M₃₂×C₄₇	C₂²	1	4	yes	complemented
12	C₄₇×#349	—	2	8	no	—
13	C₄₇×C₂²⋊C₁₆	—	2	4	no	—
14	C₂⁵×C₄₇	C₈	1	1	yes	characteristic
15	C₂³×C₄×C₄₇	—	2	4	no	—
16	C₂³×C₄×C₄₇	—	2	2	no	—
17	C₂³×C₄×C₄₇	—	2	2	no	—
18	C₂³×C₄×C₄₇	—	2	4	no	—
19	C₂³×C₄×C₄₇	C₈	1	1	yes	characteristic
20	C₂³×C₄×C₄₇	C₈	1	2	yes	—
21	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
23	C₂×M₁₆×C₄₇	—	2	32	no	—
24	C₂×M₁₆×C₄₇	—	2	4	no	—
25	C₂×M₁₆×C₄₇	—	2	4	no	—
26	C₂×M₁₆×C₄₇	—	2	4	no	—
27	C₂×M₁₆×C₄₇	—	2	4	no	—
28	C₂×M₁₆×C₄₇	—	2	4	no	—
29	C₂×M₁₆×C₄₇	D₄	1	8	yes	—
30	C₂²×C₈×C₄₇	—	2	2	no	—
31	C₂²×C₈×C₄₇	—	2	4	no	—
32	C₂²×C₈×C₄₇	—	2	4	no	—
33	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₄₇	D₄	1	4	yes	—
36	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
37	M₃₂×C₄₇	—	2	8	no	—
38	C₂×C₁₆×C₄₇	—	2	4	no	—
39	C₂⁴×C₄₇	—	2	4	no	—
40	C₂⁴×C₄₇	—	2	2	no	—
41	C₂⁴×C₄₇	—	4	4	no	—
42	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
43	C₂⁴×C₄₇	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₄₇	—	2	8	no	—
45	C₂²×C₄×C₄₇	—	2	16	no	—
46	C₂²×C₄×C₄₇	—	2	8	no	—
47	C₂²×C₄×C₄₇	—	2	2	no	—
48	C₂²×C₄×C₄₇	—	2	4	no	—
49	C₂²×C₄×C₄₇	—	2	4	no	—
50	C₂²×C₄×C₄₇	—	2	4	no	—
51	C₂²×C₄×C₄₇	—	2	4	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
63	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	M₁₆×C₄₇	—	2	16	no	—
66	M₁₆×C₄₇	—	2	16	no	—
67	C₂×C₈×C₄₇	—	2	8	no	—
68	C₂×C₈×C₄₇	—	2	2	no	—
69	C₂×C₈×C₄₇	—	2	2	no	—
70	C₂×C₈×C₄₇	—	2	2	no	—
71	C₂×C₈×C₄₇	—	2	8	no	—
72	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
73	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
75	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
76	C₁₆×C₄₇	—	4	4	no	—
77	C₂³×C₄₇	—	2	4	no	—
78	C₂³×C₄₇	—	2	2	no	—
79	C₂³×C₄₇	—	2	8	no	—
80	C₂³×C₄₇	—	2	2	no	—
81	C₂³×C₄₇	—	2	2	no	—
82	C₂³×C₄₇	—	4	4	no	—
83	C₂³×C₄₇	—	4	16	no	—
84	C₂³×C₄₇	—	4	4	no	—
85	C₂³×C₄₇	—	4	4	no	—
86	C₂³×C₄₇	C₂²⋊C₈	1	2	yes	—
87	C₂³×C₄₇	C₂²⋊C₈	1	4	yes	—
88	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₄₇	—	2	8	no	—
90	C₂×C₄×C₄₇	—	2	16	no	—
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
102	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
107	C₈×C₄₇	—	2	4	no	—
108	C₈×C₄₇	—	2	2	no	—
109	C₈×C₄₇	#120	1	4	yes	—
110	#12681	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂²×C₄₇	—	2	8	no	—
112	C₂²×C₄₇	—	2	2	no	—
113	C₂²×C₄₇	—	2	2	no	—
114	C₂²×C₄₇	—	2	2	no	—
115	C₂²×C₄₇	—	2	4	no	—
116	C₂²×C₄₇	—	4	4	no	—
117	C₂²×C₄₇	—	4	16	no	—
118	C₂²×C₄₇	—	4	4	no	—
119	C₂²×C₄₇	—	4	4	no	—
120	C₂²×C₄₇	#349	1	4	yes	complemented
121	C₂²×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
122	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
123	C₄×C₄₇	—	2	2	no	—
124	C₄×C₄₇	—	2	4	no	—
125	C₄×C₄₇	#379	1	1	yes	characteristic
126	C₄×C₄₇	#407	1	2	yes	—
127	C₄×C₄₇	#407	1	1	yes	characteristic
128	C₂³×M₁₆	C₂×C₄₇	1	1	yes	characteristic
129	C₂×#349	C₂×C₄₇	1	4	yes	complemented
130	#2112	C₂×C₄₇	1	2	yes	complemented
131	C₂×C₄₇	—	2	4	no	—
132	C₂×C₄₇	—	2	2	no	—
133	C₂×C₄₇	—	4	4	no	—
134	C₂×C₄₇	#1779	1	1	yes	characteristic
135	C₂×C₄₇	C₂×#349	1	2	yes	—
136	C₂⁴×C₄	C₄×C₄₇	1	1	yes	characteristic
137	C₂²×M₁₆	—	2	8	no	—
138	C₂²×M₁₆	—	2	4	no	—
139	C₂²×M₁₆	C₄×C₄₇	1	2	yes	—
140	C₂²×M₁₆	C₂²×C₄₇	1	2	yes	—
141	C₂³×C₈	C₄×C₄₇	1	1	yes	characteristic
142	C₂³×C₈	C₂²×C₄₇	1	1	yes	characteristic
143	C₂×M₃₂	C₂²×C₄₇	1	4	yes	complemented
144	#349	—	2	8	no	—
145	C₂²⋊C₁₆	—	2	4	no	—
146	C₄₇	#12681	1	1	yes	characteristic, direct product factor, Sylow
147	C₂⁵	C₈×C₄₇	1	1	yes	characteristic
148	C₂³×C₄	—	2	4	no	—
149	C₂³×C₄	—	2	2	no	—
150	C₂³×C₄	—	2	2	no	—
151	C₂³×C₄	—	2	4	no	—
152	C₂³×C₄	C₈×C₄₇	1	1	yes	characteristic
153	C₂³×C₄	C₈×C₄₇	1	2	yes	—
154	C₂³×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
155	C₂³×C₄	C₂×C₄×C₄₇	1	2	yes	—
156	C₂×M₁₆	—	2	32	no	—
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	—	2	4	no	—
159	C₂×M₁₆	—	2	4	no	—
160	C₂×M₁₆	—	2	4	no	—
161	C₂×M₁₆	—	2	4	no	—
162	C₂×M₁₆	D₄×C₄₇	1	8	yes	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	2	4	no	—
165	C₂²×C₈	—	2	4	no	—
166	C₂²×C₈	C₂×C₄×C₄₇	1	2	yes	—
167	C₂²×C₈	C₂×C₄×C₄₇	1	1	yes	characteristic
168	C₂²×C₈	D₄×C₄₇	1	4	yes	—
169	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
170	M₃₂	—	2	8	no	—
171	C₂×C₁₆	—	2	4	no	—
172	C₂⁴	—	2	4	no	—
173	C₂⁴	—	2	2	no	—
174	C₂⁴	—	4	4	no	—
175	C₂⁴	C₂×C₈×C₄₇	1	1	yes	characteristic
176	C₂⁴	C₂×C₈×C₄₇	1	2	yes	—
177	C₂²×C₄	—	2	8	no	—
178	C₂²×C₄	—	2	8	no	—
179	C₂²×C₄	—	2	16	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
192	C₂²×C₄	C₂²⋊C₄×C₄₇	1	4	yes	—
193	C₂²×C₄	C₂×C₈×C₄₇	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₈×C₄₇	1	2	yes	—
195	C₂²×C₄	M₁₆×C₄₇	1	2	yes	—
196	C₂²×C₄	M₁₆×C₄₇	1	4	yes	—
197	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
198	M₁₆	—	2	16	no	—
199	M₁₆	—	2	16	no	—
200	C₂×C₈	—	2	8	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	8	no	—
205	C₂×C₈	C₂²⋊C₄×C₄₇	1	4	yes	—
206	C₂×C₈	C₂²⋊C₄×C₄₇	1	2	yes	—
207	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
208	C₂×C₈	C₂×D₄×C₄₇	1	4	yes	—
209	C₁₆	—	4	4	no	—
210	C₂³	—	2	4	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	8	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	4	4	no	—
216	C₂³	—	4	16	no	—
217	C₂³	—	4	4	no	—
218	C₂³	—	4	4	no	—
219	C₂³	C₂²⋊C₈×C₄₇	1	2	yes	—
220	C₂³	C₂²⋊C₈×C₄₇	1	4	yes	—
221	C₂³	C₂²×C₈×C₄₇	1	1	yes	characteristic
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	16	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	C₂²⋊C₈×C₄₇	1	4	yes	—
235	C₂×C₄	C₂²⋊C₈×C₄₇	1	2	yes	—
236	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	2	yes	—
237	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
238	C₂×C₄	C₂×M₁₆×C₄₇	1	2	yes	—
239	C₂×C₄	C₂×M₁₆×C₄₇	1	1	yes	characteristic
240	C₈	—	2	4	no	—
241	C₈	—	2	2	no	—
242	C₈	#120×C₄₇	1	4	yes	—
243	C₂²	—	2	8	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	4	no	—
248	C₂²	—	4	4	no	—
249	C₂²	—	4	16	no	—
250	C₂²	—	4	4	no	—
251	C₂²	—	4	4	no	—
252	C₂²	C₄₇×#349	1	4	yes	complemented
253	C₂²	C₂×C₂²⋊C₈×C₄₇	1	2	yes	—
254	C₂²	C₂×C₂²⋊C₈×C₄₇	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	4	no	—
257	C₄	C₄₇×#379	1	1	yes	characteristic
258	C₄	C₄₇×#407	1	2	yes	—
259	C₄	C₄₇×#407	1	1	yes	characteristic
260	C₂	—	2	4	no	—
261	C₂	—	2	2	no	—
262	C₂	—	4	4	no	—
263	C₂	C₄₇×#1779	1	1	yes	characteristic
264	C₂	C₂×C₄₇×#349	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₄₇	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₈×C₄₇
Frattini, cosocle	C₂²×C₈	C₂³×C₄₇
socle	C₂²×C₄₇	C₂×C₂²⋊C₈
prime radical	C₂⁵×C₄₇	C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₇ ◁ C₂³×C₈×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₇×#349	C₂	semidirect
3. C₄₇×#2112	C₂	semidirect
11. C₂×M₃₂×C₄₇	C₂²	semidirect
110. #12681	C₄₇	direct (N × H)
120. C₂²×C₄₇	#349	semidirect
129. C₂×#349	C₂×C₄₇	semidirect
130. #2112	C₂×C₄₇	semidirect
143. C₂×M₃₂	C₂²×C₄₇	semidirect
146. C₄₇	#12681	direct (N × H)
252. C₂²	C₄₇×#349	semidirect

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₄₇	2
C₂×C₄₇×#349	2
C₄₇×#2112	2
#12681	47

Subgroups — C47×#12681 #26052379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#12681 #26052379