Subgroups — C₄₇×#20377 #26060075

Order 12032.

1670 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×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₇×#407	C₂	1	2	yes	maximal
3	C₂×C₄×M₁₆×C₄₇	C₂	1	2	yes	maximal, complemented
4	C₄₇×#1750	C₂	1	8	yes	maximal, complemented
5	C₄₇×#1728	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₄₇	C₂²	1	4	yes	—
8	C₂×#117×C₄₇	—	2	8	no	—
9	C₂×#117×C₄₇	C₄	1	4	yes	—
10	C₂×#117×C₄₇	C₂²	1	4	yes	—
11	C₂²×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
12	C₂²×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
13	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#407	C₄	1	16	yes	complemented
16	C₂×C₂²⋊C₈×C₄₇	C₂²	1	8	yes	—
17	C₄×M₁₆×C₄₇	—	2	8	no	—
18	C₂×#97×C₄₇	C₂²	1	4	yes	complemented
19	C₂×C₄×C₈×C₄₇	C₂²	1	4	yes	complemented
20	C₄₇×#336	C₂²	1	8	yes	complemented
21	C₂⁵×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₄₇	—	2	4	no	—
23	C₂³×C₄×C₄₇	—	2	4	no	—
24	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
27	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
30	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₄₇	—	2	8	no	—
32	C₂×M₁₆×C₄₇	—	2	8	no	—
33	C₂×M₁₆×C₄₇	C₂×C₄	1	16	yes	—
34	C₂²×C₈×C₄₇	C₂³	1	8	yes	—
35	#117×C₄₇	—	2	16	no	—
36	#117×C₄₇	—	2	8	no	—
37	#117×C₄₇	—	2	8	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
39	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
40	C₂×C₂²⋊C₄×C₄₇	—	2	8	no	—
41	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
42	C₂×C₄²×C₄₇	—	2	4	no	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₇	D₄	1	8	yes	—
46	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
47	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
48	C₂²⋊C₈×C₄₇	C₂×C₄	1	32	yes	complemented
49	#97×C₄₇	—	2	8	no	—
50	C₄×C₈×C₄₇	—	2	8	no	—
51	C₂⁴×C₄₇	—	2	2	no	—
52	C₂⁴×C₄₇	—	2	2	no	—
53	C₂⁴×C₄₇	—	4	4	no	—
54	C₂⁴×C₄₇	C₄²	1	4	yes	—
55	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	2	4	no	—
60	C₂²×C₄×C₄₇	—	2	4	no	—
61	C₂²×C₄×C₄₇	—	2	32	no	—
62	C₂²×C₄×C₄₇	—	2	4	no	—
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	—	2	4	no	—
65	C₂²×C₄×C₄₇	—	4	4	no	—
66	C₂²×C₄×C₄₇	—	4	4	no	—
67	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
68	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
69	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₄₇	#41	1	4	yes	—
78	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
79	M₁₆×C₄₇	—	2	32	no	—
80	C₂×C₈×C₄₇	—	2	8	no	—
81	C₂×C₈×C₄₇	—	2	8	no	—
82	C₂×C₈×C₄₇	C₂²×C₄	1	16	yes	—
83	C₄⋊C₄×C₄₇	—	2	16	no	—
84	C₂²⋊C₄×C₄₇	—	2	16	no	—
85	C₄²×C₄₇	—	2	4	no	—
86	C₄²×C₄₇	—	2	4	no	—
87	C₄²×C₄₇	C₂²⋊C₄	1	8	yes	—
88	C₄²×C₄₇	C₂×D₄	1	4	yes	—
89	C₄²×C₄₇	C₂×D₄	1	4	yes	—
90	C₂³×C₄₇	—	2	4	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	—	2	2	no	—
93	C₂³×C₄₇	—	2	8	no	—
94	C₂³×C₄₇	—	2	8	no	—
95	C₂³×C₄₇	—	4	8	no	—
96	C₂³×C₄₇	—	4	8	no	—
97	C₂³×C₄₇	—	4	8	no	—
98	C₂³×C₄₇	C₂×C₄²	1	4	yes	—
99	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
100	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
101	C₂³×C₄₇	#117	1	2	yes	—
102	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	8	no	—
106	C₂×C₄×C₄₇	—	2	8	no	—
107	C₂×C₄×C₄₇	—	2	16	no	—
108	C₂×C₄×C₄₇	—	2	16	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	8	no	—
111	C₂×C₄×C₄₇	—	4	8	no	—
112	C₂×C₄×C₄₇	—	4	8	no	—
113	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
114	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
116	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
117	C₂×C₄×C₄₇	#117	1	2	yes	—
118	C₂×C₄×C₄₇	C₄×D₄	1	16	yes	—
119	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
122	C₈×C₄₇	—	2	16	no	—
123	#20377	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂²×C₄₇	—	2	4	no	—
125	C₂²×C₄₇	—	2	4	no	—
126	C₂²×C₄₇	—	2	4	no	—
127	C₂²×C₄₇	—	2	8	no	—
128	C₂²×C₄₇	—	2	4	no	—
129	C₂²×C₄₇	—	4	16	no	—
130	C₂²×C₄₇	—	4	4	no	—
131	C₂²×C₄₇	—	4	4	no	—
132	C₂²×C₄₇	C₄×C₂²⋊C₄	1	4	yes	—
133	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
134	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
136	C₂²×C₄₇	#568	1	4	yes	—
137	C₄×C₄₇	—	2	8	no	—
138	C₄×C₄₇	—	2	4	no	—
139	C₄×C₄₇	—	4	4	no	—
140	C₄×C₄₇	C₄×C₂²⋊C₄	1	4	yes	—
141	C₄×C₄₇	C₂×C₄×D₄	1	4	yes	—
142	C₂²×#117	C₂×C₄₇	1	1	yes	characteristic
143	C₂×#407	C₂×C₄₇	1	2	yes	—
144	C₂×C₄×M₁₆	C₂×C₄₇	1	2	yes	complemented
145	#1750	C₂×C₄₇	1	8	yes	complemented
146	#1728	C₂×C₄₇	1	2	yes	complemented
147	C₂×C₄₇	—	2	4	no	—
148	C₂×C₄₇	—	4	4	no	—
149	C₂×C₄₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₄₇	#2873	1	2	yes	—
151	C₂×C₄₇	#2879	1	2	yes	—
152	C₂×C₄₇	#2921	1	2	yes	—
153	C₂⁴×C₄	C₂²×C₄₇	1	1	yes	characteristic
154	C₂²×M₁₆	C₂²×C₄₇	1	4	yes	—
155	C₂×#117	—	2	8	no	—
156	C₂×#117	C₄×C₄₇	1	4	yes	—
157	C₂×#117	C₂²×C₄₇	1	4	yes	—
158	C₂²×C₄⋊C₄	C₄×C₄₇	1	2	yes	—
159	C₂²×C₂²⋊C₄	C₄×C₄₇	1	2	yes	—
160	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
161	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
162	#407	C₄×C₄₇	1	16	yes	complemented
163	C₂×C₂²⋊C₈	C₂²×C₄₇	1	8	yes	—
164	C₄×M₁₆	—	2	8	no	—
165	C₂×#97	C₂²×C₄₇	1	4	yes	complemented
166	C₂×C₄×C₈	C₂²×C₄₇	1	4	yes	complemented
167	#336	C₂²×C₄₇	1	8	yes	complemented
168	C₄₇	#20377	1	1	yes	characteristic, direct product factor, Sylow
169	C₂⁵	C₂×C₄×C₄₇	1	1	yes	characteristic
170	C₂³×C₄	—	2	4	no	—
171	C₂³×C₄	—	2	4	no	—
172	C₂³×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
173	C₂³×C₄	C₂×C₄×C₄₇	1	2	yes	—
174	C₂³×C₄	C₂×C₄×C₄₇	1	8	yes	—
175	C₂³×C₄	C₂×C₄×C₄₇	1	2	yes	—
176	C₂³×C₄	C₂×C₄×C₄₇	1	2	yes	—
177	C₂³×C₄	C₂³×C₄₇	1	2	yes	—
178	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
179	C₂×M₁₆	—	2	8	no	—
180	C₂×M₁₆	—	2	8	no	—
181	C₂×M₁₆	C₂×C₄×C₄₇	1	16	yes	—
182	C₂²×C₈	C₂³×C₄₇	1	8	yes	—
183	#117	—	2	16	no	—
184	#117	—	2	8	no	—
185	#117	—	2	8	no	—
186	C₂×C₄⋊C₄	—	2	8	no	—
187	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	8	yes	—
188	C₂×C₂²⋊C₄	—	2	8	no	—
189	C₂×C₂²⋊C₄	C₂×C₄×C₄₇	1	8	yes	—
190	C₂×C₄²	—	2	4	no	—
191	C₂×C₄²	C₂×C₄×C₄₇	1	2	yes	—
192	C₂×C₄²	C₂×C₄×C₄₇	1	2	yes	—
193	C₂×C₄²	D₄×C₄₇	1	8	yes	—
194	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
195	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
196	C₂²⋊C₈	C₂×C₄×C₄₇	1	32	yes	complemented
197	#97	—	2	8	no	—
198	C₄×C₈	—	2	8	no	—
199	C₂⁴	—	2	2	no	—
200	C₂⁴	—	2	2	no	—
201	C₂⁴	—	4	4	no	—
202	C₂⁴	C₄²×C₄₇	1	4	yes	—
203	C₂⁴	C₂²×C₄×C₄₇	1	2	yes	—
204	C₂⁴	C₂²×C₄×C₄₇	1	1	yes	characteristic
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	32	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	4	4	no	—
214	C₂²×C₄	—	4	4	no	—
215	C₂²×C₄	C₄²×C₄₇	1	4	yes	—
216	C₂²×C₄	C₄²×C₄₇	1	8	yes	—
217	C₂²×C₄	C₂²⋊C₄×C₄₇	1	8	yes	—
218	C₂²×C₄	C₂²×C₄×C₄₇	1	4	yes	—
219	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
220	C₂²×C₄	C₂²×C₄×C₄₇	1	8	yes	—
221	C₂²×C₄	C₂²×C₄×C₄₇	1	4	yes	—
222	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
223	C₂²×C₄	C₂²×C₄×C₄₇	1	4	yes	—
224	C₂²×C₄	C₂×D₄×C₄₇	1	4	yes	—
225	C₂²×C₄	#41×C₄₇	1	4	yes	—
226	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
227	M₁₆	—	2	32	no	—
228	C₂×C₈	—	2	8	no	—
229	C₂×C₈	—	2	8	no	—
230	C₂×C₈	C₂²×C₄×C₄₇	1	16	yes	—
231	C₄⋊C₄	—	2	16	no	—
232	C₂²⋊C₄	—	2	16	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	C₂²⋊C₄×C₄₇	1	8	yes	—
236	C₄²	C₂×D₄×C₄₇	1	4	yes	—
237	C₄²	C₂×D₄×C₄₇	1	4	yes	—
238	C₂³	—	2	4	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	8	no	—
242	C₂³	—	2	8	no	—
243	C₂³	—	4	8	no	—
244	C₂³	—	4	8	no	—
245	C₂³	—	4	8	no	—
246	C₂³	C₂×C₄²×C₄₇	1	4	yes	—
247	C₂³	C₂×C₄²×C₄₇	1	2	yes	—
248	C₂³	C₂×C₂²⋊C₄×C₄₇	1	2	yes	—
249	C₂³	#117×C₄₇	1	2	yes	—
250	C₂³	C₂³×C₄×C₄₇	1	1	yes	characteristic
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	16	no	—
256	C₂×C₄	—	2	16	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	4	8	no	—
260	C₂×C₄	—	4	8	no	—
261	C₂×C₄	C₂×C₄²×C₄₇	1	4	yes	—
262	C₂×C₄	C₂×C₄²×C₄₇	1	2	yes	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	8	yes	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	2	yes	—
265	C₂×C₄	#117×C₄₇	1	2	yes	—
266	C₂×C₄	C₄×D₄×C₄₇	1	16	yes	—
267	C₂×C₄	C₂³×C₄×C₄₇	1	2	yes	—
268	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
269	C₂×C₄	C₂×#41×C₄₇	1	2	yes	—
270	C₈	—	2	16	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	8	no	—
276	C₂²	—	4	16	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	4	4	no	—
279	C₂²	C₄×C₂²⋊C₄×C₄₇	1	4	yes	—
280	C₂²	C₂²×C₄²×C₄₇	1	1	yes	characteristic
281	C₂²	C₂²×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
282	C₂²	C₂×#117×C₄₇	1	1	yes	characteristic
283	C₂²	C₄₇×#568	1	4	yes	—
284	C₄	—	2	4	no	—
285	C₄	—	2	8	no	—
286	C₄	—	4	4	no	—
287	C₄	C₄×C₂²⋊C₄×C₄₇	1	4	yes	—
288	C₄	C₂×C₄×D₄×C₄₇	1	4	yes	—
289	C₂	—	2	4	no	—
290	C₂	—	4	4	no	—
291	C₂	C₂×C₄×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
292	C₂	C₄₇×#2873	1	2	yes	—
293	C₂	C₄₇×#2879	1	2	yes	—
294	C₂	C₄₇×#2921	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₄₇	C₂⁴
commutator	C₂²	C₂²×C₄²×C₄₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁵×C₄₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×M₁₆×C₄₇	C₂	semidirect
4. C₄₇×#1750	C₂	semidirect
5. C₄₇×#1728	C₂	semidirect
15. C₄₇×#407	C₄	semidirect
18. C₂×#97×C₄₇	C₂²	semidirect
19. C₂×C₄×C₈×C₄₇	C₂²	semidirect
20. C₄₇×#336	C₂²	semidirect
48. C₂²⋊C₈×C₄₇	C₂×C₄	semidirect
123. #20377	C₄₇	direct (N × H)
144. C₂×C₄×M₁₆	C₂×C₄₇	semidirect
145. #1750	C₂×C₄₇	semidirect
146. #1728	C₂×C₄₇	semidirect
162. #407	C₄×C₄₇	semidirect
165. C₂×#97	C₂²×C₄₇	semidirect
166. C₂×C₄×C₈	C₂²×C₄₇	semidirect
167. #336	C₂²×C₄₇	semidirect
168. C₄₇	#20377	direct (N × H)
196. C₂²⋊C₈	C₂×C₄×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#117×C₄₇	2
C₂×C₄₇×#407	2
C₂×C₄×M₁₆×C₄₇	2
C₄₇×#1750	2
C₄₇×#1728	2
#20377	47

Subgroups — C47×#20377 #26060075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#20377 #26060075