Subgroups — C₂×C₄₇×#1784 #26060287

Order 12032.

3798 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#101×C₄₇	C₂	1	4	yes	maximal, complemented
3	C₂×C₄₇×#407	C₂	1	2	yes	maximal
4	C₄₇×#1784	C₂	1	8	yes	maximal, direct product factor
5	C₂³×Q₈×C₄₇	—	2	16	no	—
6	C₂³×Q₈×C₄₇	—	2	4	no	—
7	C₂³×Q₈×C₄₇	—	2	4	no	—
8	C₂³×Q₈×C₄₇	C₄	1	4	yes	—
9	C₂³×Q₈×C₄₇	C₄	1	2	yes	—
10	C₂³×Q₈×C₄₇	C₂²	1	4	yes	—
11	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
12	C₂⁴×C₄×C₄₇	C₄	1	2	yes	—
13	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₄₇	C₂²	1	4	yes	—
15	C₂×#101×C₄₇	—	2	16	no	—
16	C₂×#101×C₄₇	C₂²	1	16	yes	complemented
17	C₄₇×#407	—	2	4	no	—
18	C₄₇×#407	C₂²	1	8	yes	—
19	C₂×C₂²⋊C₈×C₄₇	—	2	4	no	—
20	C₂⁵×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×Q₈×C₄₇	—	2	64	no	—
22	C₂²×Q₈×C₄₇	—	2	64	no	—
23	C₂²×Q₈×C₄₇	—	2	32	no	—
24	C₂²×Q₈×C₄₇	—	2	32	no	—
25	C₂²×Q₈×C₄₇	—	2	16	no	—
26	C₂²×Q₈×C₄₇	—	2	16	no	—
27	C₂²×Q₈×C₄₇	—	2	4	no	—
28	C₂²×Q₈×C₄₇	—	2	4	no	—
29	C₂²×Q₈×C₄₇	—	2	16	no	—
30	C₂²×Q₈×C₄₇	—	2	8	no	—
31	C₂²×Q₈×C₄₇	—	2	4	no	—
32	C₂²×Q₈×C₄₇	—	2	4	no	—
33	C₂²×Q₈×C₄₇	C₂×C₄	1	8	yes	—
34	C₂²×Q₈×C₄₇	C₂×C₄	1	4	yes	—
35	C₂²×Q₈×C₄₇	D₄	1	16	yes	—
36	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
37	C₂³×C₄×C₄₇	—	2	4	no	—
38	C₂³×C₄×C₄₇	—	2	4	no	—
39	C₂³×C₄×C₄₇	—	2	16	no	—
40	C₂³×C₄×C₄₇	—	2	8	no	—
41	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
42	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
43	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
45	C₂³×C₄×C₄₇	D₄	1	4	yes	—
46	C₂³×C₄×C₄₇	D₄	1	4	yes	—
47	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
48	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₄₇	—	2	4	no	—
50	C₂×M₁₆×C₄₇	—	2	16	no	—
51	C₂×M₁₆×C₄₇	C₂³	1	8	yes	—
52	C₂²×C₈×C₄₇	—	2	4	no	—
53	#101×C₄₇	—	2	32	no	—
54	C₂²⋊C₈×C₄₇	—	2	16	no	—
55	C₂⁴×C₄₇	—	2	8	no	—
56	C₂⁴×C₄₇	—	2	4	no	—
57	C₂⁴×C₄₇	C₂²⋊C₄	1	2	yes	—
58	C₂⁴×C₄₇	C₂²⋊C₄	1	2	yes	—
59	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×Q₈×C₄₇	—	2	128	no	—
62	C₂×Q₈×C₄₇	—	2	128	no	—
63	C₂×Q₈×C₄₇	—	2	32	no	—
64	C₂×Q₈×C₄₇	—	2	32	no	—
65	C₂×Q₈×C₄₇	—	2	32	no	—
66	C₂×Q₈×C₄₇	—	2	16	no	—
67	C₂×Q₈×C₄₇	—	2	8	no	—
68	C₂×Q₈×C₄₇	—	2	8	no	—
69	C₂×Q₈×C₄₇	—	2	32	no	—
70	C₂×Q₈×C₄₇	—	2	32	no	—
71	C₂×Q₈×C₄₇	—	2	8	no	—
72	C₂×Q₈×C₄₇	—	2	8	no	—
73	C₂×Q₈×C₄₇	C₂²⋊C₄	1	16	yes	—
74	C₂×Q₈×C₄₇	C₂×D₄	1	16	yes	—
75	C₂²×C₄×C₄₇	—	2	32	no	—
76	C₂²×C₄×C₄₇	—	2	32	no	—
77	C₂²×C₄×C₄₇	—	2	16	no	—
78	C₂²×C₄×C₄₇	—	2	4	no	—
79	C₂²×C₄×C₄₇	—	2	8	no	—
80	C₂²×C₄×C₄₇	—	2	4	no	—
81	C₂²×C₄×C₄₇	—	2	4	no	—
82	C₂²×C₄×C₄₇	—	2	32	no	—
83	C₂²×C₄×C₄₇	—	2	8	no	—
84	C₂²×C₄×C₄₇	—	2	16	no	—
85	C₂²×C₄×C₄₇	—	2	16	no	—
86	C₂²×C₄×C₄₇	—	2	8	no	—
87	C₂²×C₄×C₄₇	C₂²⋊C₄	1	16	yes	—
88	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
89	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
90	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
91	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
92	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
95	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
96	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
99	M₁₆×C₄₇	—	2	16	no	—
100	M₁₆×C₄₇	—	4	8	no	—
101	C₂×C₈×C₄₇	—	2	8	no	—
102	C₂×C₈×C₄₇	—	4	8	no	—
103	C₂³×C₄₇	—	2	32	no	—
104	C₂³×C₄₇	—	2	4	no	—
105	C₂³×C₄₇	—	2	4	no	—
106	C₂³×C₄₇	—	2	4	no	—
107	C₂³×C₄₇	—	2	4	no	—
108	C₂³×C₄₇	—	2	4	no	—
109	C₂³×C₄₇	—	2	8	no	—
110	C₂³×C₄₇	—	2	8	no	—
111	C₂³×C₄₇	#101	1	8	yes	complemented
112	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂³×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
114	C₂³×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
115	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
117	Q₈×C₄₇	—	2	64	no	—
118	Q₈×C₄₇	—	2	32	no	—
119	Q₈×C₄₇	—	2	32	no	—
120	C₂×C₄×C₄₇	—	2	64	no	—
121	C₂×C₄×C₄₇	—	2	8	no	—
122	C₂×C₄×C₄₇	—	2	4	no	—
123	C₂×C₄×C₄₇	—	2	8	no	—
124	C₂×C₄×C₄₇	—	2	16	no	—
125	C₂×C₄×C₄₇	—	2	4	no	—
126	C₂×C₄×C₄₇	—	2	16	no	—
127	C₂×C₄×C₄₇	—	2	4	no	—
128	C₂×C₄×C₄₇	—	2	16	no	—
129	C₂×C₄×C₄₇	—	2	16	no	—
130	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	16	yes	—
131	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
132	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
133	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
134	C₂×C₄×C₄₇	#120	1	8	yes	—
135	C₂×C₄×C₄₇	#120	1	8	yes	—
136	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
137	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
138	C₈×C₄₇	—	4	8	no	—
139	C₂×#1784	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂²×C₄₇	—	2	16	no	—
141	C₂²×C₄₇	—	2	16	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	4	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	2	4	no	—
146	C₂²×C₄₇	—	2	16	no	—
147	C₂²×C₄₇	—	2	8	no	—
148	C₂²×C₄₇	#379	1	2	yes	—
149	C₂²×C₄₇	#379	1	2	yes	—
150	C₂²×C₄₇	C₂×#101	1	4	yes	—
151	C₂²×C₄₇	C₂×#101	1	8	yes	complemented
152	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	2	yes	—
153	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
154	C₄×C₄₇	—	2	16	no	—
155	C₄×C₄₇	—	2	4	no	—
156	C₄×C₄₇	#379	1	4	yes	—
157	C₄×C₄₇	C₂×#120	1	4	yes	—
158	C₂⁴×Q₈	C₂×C₄₇	1	1	yes	characteristic
159	C₂²×#101	C₂×C₄₇	1	4	yes	complemented
160	C₂×#407	C₂×C₄₇	1	2	yes	—
161	#1784	C₂×C₄₇	1	8	yes	direct product factor
162	C₂×C₄₇	—	2	2	no	—
163	C₂×C₄₇	—	2	2	no	—
164	C₂×C₄₇	—	2	8	no	—
165	C₂×C₄₇	#1784	1	4	yes	direct product factor
166	C₂×C₄₇	C₂×#379	1	1	yes	characteristic
167	C₂×C₄₇	C₂²×#101	1	2	yes	—
168	C₂³×Q₈	—	2	16	no	—
169	C₂³×Q₈	—	2	4	no	—
170	C₂³×Q₈	—	2	4	no	—
171	C₂³×Q₈	C₄×C₄₇	1	4	yes	—
172	C₂³×Q₈	C₄×C₄₇	1	2	yes	—
173	C₂³×Q₈	C₂²×C₄₇	1	4	yes	—
174	C₂³×Q₈	C₂²×C₄₇	1	2	yes	—
175	C₂⁴×C₄	C₄×C₄₇	1	2	yes	—
176	C₂⁴×C₄	C₂²×C₄₇	1	1	yes	characteristic
177	C₂²×M₁₆	C₂²×C₄₇	1	4	yes	—
178	C₂×#101	—	2	16	no	—
179	C₂×#101	C₂²×C₄₇	1	16	yes	complemented
180	#407	—	2	4	no	—
181	#407	C₂²×C₄₇	1	8	yes	—
182	C₂×C₂²⋊C₈	—	2	4	no	—
183	C₄₇	C₂×#1784	1	1	yes	characteristic, direct product factor, Sylow
184	C₂⁵	C₂×C₄×C₄₇	1	1	yes	characteristic
185	C₂²×Q₈	—	2	64	no	—
186	C₂²×Q₈	—	2	64	no	—
187	C₂²×Q₈	—	2	32	no	—
188	C₂²×Q₈	—	2	32	no	—
189	C₂²×Q₈	—	2	16	no	—
190	C₂²×Q₈	—	2	16	no	—
191	C₂²×Q₈	—	2	4	no	—
192	C₂²×Q₈	—	2	4	no	—
193	C₂²×Q₈	—	2	16	no	—
194	C₂²×Q₈	—	2	4	no	—
195	C₂²×Q₈	—	2	8	no	—
196	C₂²×Q₈	—	2	4	no	—
197	C₂²×Q₈	C₂×C₄×C₄₇	1	8	yes	—
198	C₂²×Q₈	C₂×C₄×C₄₇	1	4	yes	—
199	C₂²×Q₈	D₄×C₄₇	1	16	yes	—
200	C₂²×Q₈	C₂³×C₄₇	1	4	yes	—
201	C₂³×C₄	—	2	4	no	—
202	C₂³×C₄	—	2	4	no	—
203	C₂³×C₄	—	2	16	no	—
204	C₂³×C₄	—	2	8	no	—
205	C₂³×C₄	C₂×C₄×C₄₇	1	8	yes	—
206	C₂³×C₄	C₂×C₄×C₄₇	1	4	yes	—
207	C₂³×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
208	C₂³×C₄	C₂×C₄×C₄₇	1	2	yes	—
209	C₂³×C₄	D₄×C₄₇	1	4	yes	—
210	C₂³×C₄	D₄×C₄₇	1	4	yes	—
211	C₂³×C₄	C₂³×C₄₇	1	2	yes	—
212	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
213	C₂×M₁₆	—	2	4	no	—
214	C₂×M₁₆	—	2	16	no	—
215	C₂×M₁₆	C₂³×C₄₇	1	8	yes	—
216	C₂²×C₈	—	2	4	no	—
217	#101	—	2	32	no	—
218	C₂²⋊C₈	—	2	16	no	—
219	C₂⁴	—	2	4	no	—
220	C₂⁴	—	2	8	no	—
221	C₂⁴	C₂²⋊C₄×C₄₇	1	2	yes	—
222	C₂⁴	C₂²⋊C₄×C₄₇	1	2	yes	—
223	C₂⁴	C₂²×C₄×C₄₇	1	2	yes	—
224	C₂⁴	C₂²×C₄×C₄₇	1	1	yes	characteristic
225	C₂×Q₈	—	2	128	no	—
226	C₂×Q₈	—	2	128	no	—
227	C₂×Q₈	—	2	32	no	—
228	C₂×Q₈	—	2	32	no	—
229	C₂×Q₈	—	2	32	no	—
230	C₂×Q₈	—	2	16	no	—
231	C₂×Q₈	—	2	8	no	—
232	C₂×Q₈	—	2	8	no	—
233	C₂×Q₈	—	2	32	no	—
234	C₂×Q₈	—	2	32	no	—
235	C₂×Q₈	—	2	8	no	—
236	C₂×Q₈	—	2	8	no	—
237	C₂×Q₈	C₂²⋊C₄×C₄₇	1	16	yes	—
238	C₂×Q₈	C₂×D₄×C₄₇	1	16	yes	—
239	C₂²×C₄	—	2	32	no	—
240	C₂²×C₄	—	2	32	no	—
241	C₂²×C₄	—	2	16	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	2	8	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	32	no	—
247	C₂²×C₄	—	2	8	no	—
248	C₂²×C₄	—	2	16	no	—
249	C₂²×C₄	—	2	16	no	—
250	C₂²×C₄	—	2	8	no	—
251	C₂²×C₄	C₂²⋊C₄×C₄₇	1	16	yes	—
252	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
253	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
254	C₂²×C₄	C₂²⋊C₄×C₄₇	1	8	yes	—
255	C₂²×C₄	C₂²×C₄×C₄₇	1	8	yes	—
256	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
257	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₄×C₄₇	1	8	yes	—
259	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
260	C₂²×C₄	C₂×D₄×C₄₇	1	8	yes	—
261	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
262	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
263	M₁₆	—	2	16	no	—
264	M₁₆	—	4	8	no	—
265	C₂×C₈	—	2	8	no	—
266	C₂×C₈	—	4	8	no	—
267	C₂³	—	2	32	no	—
268	C₂³	—	2	4	no	—
269	C₂³	—	2	4	no	—
270	C₂³	—	2	4	no	—
271	C₂³	—	2	4	no	—
272	C₂³	—	2	4	no	—
273	C₂³	—	2	8	no	—
274	C₂³	—	2	8	no	—
275	C₂³	#101×C₄₇	1	8	yes	complemented
276	C₂³	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
277	C₂³	C₂×C₂²⋊C₄×C₄₇	1	4	yes	—
278	C₂³	C₂×C₂²⋊C₄×C₄₇	1	4	yes	—
279	C₂³	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
280	C₂³	C₂³×C₄×C₄₇	1	1	yes	characteristic
281	Q₈	—	2	64	no	—
282	Q₈	—	2	32	no	—
283	Q₈	—	2	32	no	—
284	C₂×C₄	—	2	64	no	—
285	C₂×C₄	—	2	8	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	8	no	—
288	C₂×C₄	—	2	16	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	16	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	16	no	—
293	C₂×C₄	—	2	16	no	—
294	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	16	yes	—
295	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	4	yes	—
296	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	4	yes	—
297	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	2	yes	—
298	C₂×C₄	#120×C₄₇	1	8	yes	—
299	C₂×C₄	#120×C₄₇	1	8	yes	—
300	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
301	C₂×C₄	C₂²×D₄×C₄₇	1	4	yes	—
302	C₈	—	4	8	no	—
303	C₂²	—	2	16	no	—
304	C₂²	—	2	16	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	8	no	—
307	C₂²	—	2	16	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	2	2	no	—
311	C₂²	C₄₇×#379	1	2	yes	—
312	C₂²	C₄₇×#379	1	2	yes	—
313	C₂²	C₂×#101×C₄₇	1	4	yes	—
314	C₂²	C₂×#101×C₄₇	1	8	yes	complemented
315	C₂²	C₂²×C₂²⋊C₄×C₄₇	1	2	yes	—
316	C₂²	C₂²×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
317	C₄	—	2	16	no	—
318	C₄	—	2	4	no	—
319	C₄	C₄₇×#379	1	4	yes	—
320	C₄	C₂×#120×C₄₇	1	4	yes	—
321	C₂	—	2	2	no	—
322	C₂	—	2	2	no	—
323	C₂	—	2	8	no	—
324	C₂	C₄₇×#1784	1	4	yes	direct product factor
325	C₂	C₂×C₄₇×#379	1	1	yes	characteristic
326	C₂	C₂²×#101×C₄₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂²×#101×C₄₇	C₂	semidirect
4. C₄₇×#1784	C₂	direct (N × H)
16. C₂×#101×C₄₇	C₂²	semidirect
111. C₂³×C₄₇	#101	semidirect
139. C₂×#1784	C₄₇	direct (N × H)
151. C₂²×C₄₇	C₂×#101	semidirect
159. C₂²×#101	C₂×C₄₇	semidirect
161. #1784	C₂×C₄₇	direct (N × H)
165. C₂×C₄₇	#1784	direct (N × H)
179. C₂×#101	C₂²×C₄₇	semidirect
183. C₄₇	C₂×#1784	direct (N × H)
275. C₂³	#101×C₄₇	semidirect
314. C₂²	C₂×#101×C₄₇	semidirect
324. C₂	C₄₇×#1784	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₄₇	2
C₂²×#101×C₄₇	2
C₂×C₄₇×#407	2
C₄₇×#1784	2
C₂×#1784	47

Subgroups — C2×C47×#1784 #26060287

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₇×#1784 #26060287