Subgroups — C₂×C₄₇×#2073 #26063322

Order 12032.

1278 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#102×C₄₇	C₂	1	2	yes	maximal
2	C₂×C₄₇×#390	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₇×#382	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#2073	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄₇×#340	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₇×#336	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₄₇	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
10	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
11	C₂²×C₂²⋊C₄×C₄₇	—	2	2	no	—
12	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×#102×C₄₇	—	2	8	no	—
14	C₂×#102×C₄₇	C₂²	1	8	yes	—
15	C₄₇×#390	—	2	2	no	—
16	C₄₇×#390	C₂²	1	4	yes	—
17	C₄₇×#382	—	2	2	no	—
18	C₄₇×#382	C₂²	1	4	yes	complemented
19	C₂×#95×C₄₇	—	2	2	no	—
20	C₄₇×#340	C₂²	1	8	yes	complemented
21	C₄₇×#336	C₂²	1	4	yes	complemented
22	C₂⁵×C₄₇	—	2	1	no	—
23	C₂²×D₄×C₄₇	—	2	1	no	—
24	C₂²×D₄×C₄₇	—	2	4	no	—
25	C₂²×D₄×C₄₇	—	2	4	no	—
26	C₂²×D₄×C₄₇	—	2	2	no	—
27	C₂²×D₄×C₄₇	D₄	1	4	yes	—
28	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
29	C₂³×C₄×C₄₇	—	2	1	no	—
30	C₂³×C₄×C₄₇	—	2	2	no	—
31	C₂³×C₄×C₄₇	D₄	1	2	yes	—
32	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	—	2	4	no	—
34	C₂²×C₈×C₄₇	—	2	4	no	—
35	C₂²×C₈×C₄₇	—	2	4	no	—
36	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
37	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
42	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
44	C₂×C₂²⋊C₄×C₄₇	—	2	8	no	—
45	C₂×C₂²⋊C₄×C₄₇	—	4	8	no	—
46	C₂×C₄²×C₄₇	—	2	4	no	—
47	C₂×C₄²×C₄₇	D₄	1	2	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
49	#102×C₄₇	—	2	16	no	—
50	#95×C₄₇	—	2	8	no	—
51	C₂⁴×C₄₇	—	2	4	no	—
52	C₂⁴×C₄₇	—	2	2	no	—
53	C₂⁴×C₄₇	—	4	2	no	—
54	C₂⁴×C₄₇	—	4	2	no	—
55	C₂⁴×C₄₇	—	4	4	no	—
56	C₂⁴×C₄₇	—	4	4	no	—
57	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂×D₄×C₄₇	—	2	4	no	—
59	C₂×D₄×C₄₇	—	2	16	no	—
60	C₂×D₄×C₄₇	—	2	4	no	—
61	C₂×D₄×C₄₇	—	2	4	no	—
62	C₂×D₄×C₄₇	—	2	2	no	—
63	C₂×D₄×C₄₇	—	4	2	no	—
64	C₂×D₄×C₄₇	—	4	2	no	—
65	C₂×D₄×C₄₇	—	4	8	no	—
66	C₂×D₄×C₄₇	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₄₇	—	2	4	no	—
68	C₂²×C₄×C₄₇	—	2	4	no	—
69	C₂²×C₄×C₄₇	—	2	4	no	—
70	C₂²×C₄×C₄₇	—	2	2	no	—
71	C₂²×C₄×C₄₇	—	2	4	no	—
72	C₂²×C₄×C₄₇	—	4	4	no	—
73	C₂²×C₄×C₄₇	—	4	2	no	—
74	C₂²×C₄×C₄₇	—	4	4	no	—
75	C₂²×C₄×C₄₇	—	4	4	no	—
76	C₂²×C₄×C₄₇	—	4	4	no	—
77	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₄₇	#41	1	2	yes	—
81	C₂²×C₄×C₄₇	#41	1	4	yes	—
82	C₂²×C₄×C₄₇	#41	1	4	yes	—
83	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₄₇	—	2	4	no	—
85	C₂×C₈×C₄₇	—	2	4	no	—
86	C₂×C₈×C₄₇	—	2	4	no	—
87	C₂×C₈×C₄₇	—	4	8	no	—
88	C₄⋊C₄×C₄₇	—	2	4	no	—
89	C₄⋊C₄×C₄₇	—	2	4	no	—
90	C₄⋊C₄×C₄₇	—	2	4	no	—
91	C₄⋊C₄×C₄₇	—	4	4	no	—
92	C₄⋊C₄×C₄₇	—	4	2	no	—
93	C₂²⋊C₄×C₄₇	—	4	16	no	—
94	C₄²×C₄₇	—	2	4	no	—
95	C₄²×C₄₇	—	2	4	no	—
96	C₂³×C₄₇	—	2	4	no	—
97	C₂³×C₄₇	—	4	2	no	—
98	C₂³×C₄₇	—	4	2	no	—
99	C₂³×C₄₇	—	4	4	no	—
100	C₂³×C₄₇	—	4	4	no	—
101	C₂³×C₄₇	—	4	8	no	—
102	C₂³×C₄₇	—	4	16	no	—
103	C₂³×C₄₇	—	8	8	no	—
104	C₂³×C₄₇	—	8	4	no	—
105	C₂³×C₄₇	—	8	4	no	—
106	C₂³×C₄₇	#121	1	2	yes	—
107	C₂³×C₄₇	#123	1	2	yes	—
108	C₂³×C₄₇	#124	1	2	yes	—
109	C₂³×C₄₇	#135	1	4	yes	—
110	C₂³×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
111	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
112	D₄×C₄₇	—	2	8	no	—
113	D₄×C₄₇	—	4	4	no	—
114	D₄×C₄₇	—	4	8	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	—	4	8	no	—
119	C₂×C₄×C₄₇	—	4	4	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	4	no	—
122	C₂×C₄×C₄₇	—	4	4	no	—
123	C₂×C₄×C₄₇	—	4	4	no	—
124	C₂×C₄×C₄₇	—	8	8	no	—
125	C₂×C₄×C₄₇	—	8	4	no	—
126	C₂×C₄×C₄₇	#121	1	2	yes	—
127	C₂×C₄×C₄₇	#123	1	8	yes	—
128	C₂×C₄×C₄₇	#123	1	2	yes	—
129	C₂×C₄×C₄₇	#124	1	2	yes	—
130	C₂×C₄×C₄₇	#126	1	8	yes	—
131	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
133	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
134	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
135	C₈×C₄₇	—	4	4	no	—
136	C₂×#2073	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂²×C₄₇	—	4	4	no	—
138	C₂²×C₄₇	—	4	8	no	—
139	C₂²×C₄₇	—	8	4	no	—
140	C₂²×C₄₇	—	8	16	no	—
141	C₂²×C₄₇	—	8	2	no	—
142	C₂²×C₄₇	—	8	2	no	—
143	C₂²×C₄₇	#397	1	4	yes	—
144	C₂²×C₄₇	#463	1	8	yes	—
145	C₂²×C₄₇	#482	1	8	yes	—
146	C₂²×C₄₇	#488	1	4	yes	—
147	C₂²×C₄₇	#490	1	4	yes	—
148	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
149	C₂²×C₄₇	C₂×#123	1	1	yes	characteristic
150	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
151	C₂²×C₄₇	C₂×#135	1	2	yes	—
152	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
153	C₄×C₄₇	—	4	4	no	—
154	C₄×C₄₇	—	8	2	no	—
155	C₄×C₄₇	—	8	4	no	—
156	C₄×C₄₇	#397	1	4	yes	—
157	C₄×C₄₇	C₂×#123	1	2	yes	—
158	C₄×C₄₇	C₂×#126	1	2	yes	—
159	C₂²×#102	C₂×C₄₇	1	2	yes	—
160	C₂×#390	C₂×C₄₇	1	1	yes	characteristic
161	C₂×#382	C₂×C₄₇	1	1	yes	characteristic, complemented
162	#2073	C₂×C₄₇	1	8	yes	direct product factor
163	C₂×#340	C₂×C₄₇	1	2	yes	complemented
164	C₂×#336	C₂×C₄₇	1	1	yes	characteristic, complemented
165	C₂×C₄₇	—	8	4	no	—
166	C₂×C₄₇	#2073	1	8	yes	direct product factor
167	C₂×C₄₇	C₂×#397	1	1	yes	characteristic
168	C₂×C₄₇	C₂×#463	1	2	yes	—
169	C₂×C₄₇	C₂×#482	1	2	yes	—
170	C₂×C₄₇	C₂×#488	1	1	yes	characteristic
171	C₂×C₄₇	C₂×#490	1	1	yes	characteristic
172	C₂³×D₄	C₂²×C₄₇	1	1	yes	characteristic
173	C₂³×C₈	C₂²×C₄₇	1	2	yes	—
174	C₂²×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
175	C₂²×C₄⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
176	C₂²×C₂²⋊C₄	—	2	2	no	—
177	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
178	C₂×#102	—	2	8	no	—
179	C₂×#102	C₂²×C₄₇	1	8	yes	—
180	#390	—	2	2	no	—
181	#390	C₂²×C₄₇	1	4	yes	—
182	#382	—	2	2	no	—
183	#382	C₂²×C₄₇	1	4	yes	complemented
184	C₂×#95	—	2	2	no	—
185	#340	C₂²×C₄₇	1	8	yes	complemented
186	#336	C₂²×C₄₇	1	4	yes	complemented
187	C₄₇	C₂×#2073	1	1	yes	characteristic, direct product factor, Sylow
188	C₂⁵	—	2	1	no	—
189	C₂²×D₄	—	2	1	no	—
190	C₂²×D₄	—	2	4	no	—
191	C₂²×D₄	—	2	4	no	—
192	C₂²×D₄	—	2	2	no	—
193	C₂²×D₄	D₄×C₄₇	1	4	yes	—
194	C₂²×D₄	C₂³×C₄₇	1	2	yes	—
195	C₂³×C₄	—	2	1	no	—
196	C₂³×C₄	—	2	2	no	—
197	C₂³×C₄	D₄×C₄₇	1	2	yes	—
198	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
199	C₂²×C₈	—	2	4	no	—
200	C₂²×C₈	—	2	4	no	—
201	C₂²×C₈	—	2	4	no	—
202	C₂²×C₈	C₂³×C₄₇	1	4	yes	—
203	C₂×C₄⋊C₄	—	2	4	no	—
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
209	C₂×C₄⋊C₄	C₂³×C₄₇	1	4	yes	—
210	C₂×C₂²⋊C₄	—	2	8	no	—
211	C₂×C₂²⋊C₄	—	4	8	no	—
212	C₂×C₄²	—	2	4	no	—
213	C₂×C₄²	D₄×C₄₇	1	2	yes	—
214	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
215	#102	—	2	16	no	—
216	#95	—	2	8	no	—
217	C₂⁴	—	2	4	no	—
218	C₂⁴	—	2	2	no	—
219	C₂⁴	—	4	2	no	—
220	C₂⁴	—	4	2	no	—
221	C₂⁴	—	4	4	no	—
222	C₂⁴	—	4	4	no	—
223	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	2	16	no	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	—	2	4	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂×D₄	—	4	2	no	—
231	C₂×D₄	—	4	8	no	—
232	C₂×D₄	C₂×D₄×C₄₇	1	4	yes	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	4	4	no	—
241	C₂²×C₄	—	4	4	no	—
242	C₂²×C₄	—	4	4	no	—
243	C₂²×C₄	C₂×D₄×C₄₇	1	4	yes	—
244	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
246	C₂²×C₄	#41×C₄₇	1	2	yes	—
247	C₂²×C₄	#41×C₄₇	1	4	yes	—
248	C₂²×C₄	#41×C₄₇	1	4	yes	—
249	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	4	8	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₂²⋊C₄	—	4	16	no	—
260	C₄²	—	2	4	no	—
261	C₄²	—	2	4	no	—
262	C₂³	—	2	4	no	—
263	C₂³	—	4	16	no	—
264	C₂³	—	4	2	no	—
265	C₂³	—	4	2	no	—
266	C₂³	—	4	4	no	—
267	C₂³	—	4	4	no	—
268	C₂³	—	4	8	no	—
269	C₂³	—	8	8	no	—
270	C₂³	—	8	4	no	—
271	C₂³	—	8	4	no	—
272	C₂³	#121×C₄₇	1	2	yes	—
273	C₂³	#123×C₄₇	1	2	yes	—
274	C₂³	#124×C₄₇	1	2	yes	—
275	C₂³	#135×C₄₇	1	4	yes	—
276	C₂³	AGL(1,ℤ/8)×C₄₇	1	4	yes	—
277	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
278	D₄	—	2	8	no	—
279	D₄	—	4	4	no	—
280	D₄	—	4	8	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	8	8	no	—
291	C₂×C₄	—	8	4	no	—
292	C₂×C₄	#121×C₄₇	1	2	yes	—
293	C₂×C₄	#123×C₄₇	1	8	yes	—
294	C₂×C₄	#123×C₄₇	1	2	yes	—
295	C₂×C₄	#124×C₄₇	1	2	yes	—
296	C₂×C₄	#126×C₄₇	1	8	yes	—
297	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
298	C₂×C₄	C₂×#41×C₄₇	1	2	yes	—
299	C₂×C₄	C₂×#41×C₄₇	1	2	yes	—
300	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
301	C₈	—	4	4	no	—
302	C₂²	—	4	8	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	8	4	no	—
305	C₂²	—	8	16	no	—
306	C₂²	—	8	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	C₄₇×#397	1	4	yes	—
309	C₂²	C₄₇×#463	1	8	yes	—
310	C₂²	C₄₇×#482	1	8	yes	—
311	C₂²	C₄₇×#488	1	4	yes	—
312	C₂²	C₄₇×#490	1	4	yes	—
313	C₂²	C₂×#121×C₄₇	1	1	yes	characteristic
314	C₂²	C₂×#123×C₄₇	1	1	yes	characteristic
315	C₂²	C₂×#124×C₄₇	1	1	yes	characteristic
316	C₂²	C₂×#135×C₄₇	1	2	yes	—
317	C₂²	C₂×AGL(1,ℤ/8)×C₄₇	1	2	yes	—
318	C₄	—	4	4	no	—
319	C₄	—	8	4	no	—
320	C₄	—	8	2	no	—
321	C₄	C₄₇×#397	1	4	yes	—
322	C₄	C₂×#123×C₄₇	1	2	yes	—
323	C₄	C₂×#126×C₄₇	1	2	yes	—
324	C₂	—	8	4	no	—
325	C₂	C₄₇×#2073	1	8	yes	direct product factor
326	C₂	C₂×C₄₇×#397	1	1	yes	characteristic
327	C₂	C₂×C₄₇×#463	1	2	yes	—
328	C₂	C₂×C₄₇×#482	1	2	yes	—
329	C₂	C₂×C₄₇×#488	1	1	yes	characteristic
330	C₂	C₂×C₄₇×#490	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₄₇	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
prime radical	C₂³×D₄×C₄₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₇ ◁ C₂²×C₄²×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄₇×#382	C₂	semidirect
4. C₄₇×#2073	C₂	direct (N × H)
5. C₂×C₄₇×#340	C₂	semidirect
6. C₂×C₄₇×#336	C₂	semidirect
18. C₄₇×#382	C₂²	semidirect
20. C₄₇×#340	C₂²	semidirect
21. C₄₇×#336	C₂²	semidirect
136. C₂×#2073	C₄₇	direct (N × H)
161. C₂×#382	C₂×C₄₇	semidirect
162. #2073	C₂×C₄₇	direct (N × H)
163. C₂×#340	C₂×C₄₇	semidirect
164. C₂×#336	C₂×C₄₇	semidirect
166. C₂×C₄₇	#2073	direct (N × H)
183. #382	C₂²×C₄₇	semidirect
185. #340	C₂²×C₄₇	semidirect
186. #336	C₂²×C₄₇	semidirect
187. C₄₇	C₂×#2073	direct (N × H)
325. C₂	C₄₇×#2073	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#102×C₄₇	2
C₂×C₄₇×#390	2
C₂×C₄₇×#382	2
C₄₇×#2073	2
C₂×C₄₇×#340	2
C₂×C₄₇×#336	2
C₂×#2073	47

Subgroups — C2×C47×#2073 #26063322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₇×#2073 #26063322