Subgroups — C₂×C₄₇×#1728 #26060031

Order 12032.

2782 conjugacy classes of subgroups.

The subgroup lattice has 292 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	2	yes	maximal
2	C₂³×C₄²×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1728	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄₇×#336	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₄₇	C₄	1	2	yes	—
6	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₄₇	—	2	8	no	—
8	C₂²×M₁₆×C₄₇	—	2	8	no	—
9	C₂²×M₁₆×C₄₇	C₄	1	16	yes	complemented
10	C₂²×M₁₆×C₄₇	C₂²	1	8	yes	—
11	C₂³×C₈×C₄₇	C₂²	1	4	yes	—
12	C₂²×C₄²×C₄₇	—	2	8	no	—
13	C₂²×C₄²×C₄₇	C₄	1	4	yes	—
14	C₂²×C₄²×C₄₇	C₄	1	2	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	4	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
17	C₄₇×#336	C₂²	1	16	yes	complemented
18	C₂⁵×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₄₇	—	2	8	no	—
20	C₂³×C₄×C₄₇	—	2	16	no	—
21	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
22	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
23	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
26	C₂³×C₄×C₄₇	D₄	1	4	yes	—
27	C₂³×C₄×C₄₇	D₄	1	8	yes	—
28	C₂³×C₄×C₄₇	Q₈	1	4	yes	—
29	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
30	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₄₇	—	2	64	no	—
32	C₂×M₁₆×C₄₇	—	2	16	no	—
33	C₂×M₁₆×C₄₇	—	2	16	no	—
34	C₂×M₁₆×C₄₇	C₂×C₄	1	32	yes	complemented
35	C₂²×C₈×C₄₇	—	2	8	no	—
36	C₂²×C₈×C₄₇	—	2	8	no	—
37	C₂²×C₈×C₄₇	C₂×C₄	1	16	yes	complemented
38	C₂²×C₈×C₄₇	C₂³	1	8	yes	—
39	C₂×C₄²×C₄₇	—	2	16	no	—
40	C₂×C₄²×C₄₇	—	2	8	no	—
41	C₂×C₄²×C₄₇	—	2	8	no	—
42	C₂×C₄²×C₄₇	—	2	16	no	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
46	C₂⁴×C₄₇	—	2	8	no	—
47	C₂⁴×C₄₇	C₄²	1	4	yes	—
48	C₂⁴×C₄₇	C₂²⋊C₄	1	4	yes	—
49	C₂⁴×C₄₇	C₄⋊C₄	1	4	yes	—
50	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
51	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	2	64	no	—
53	C₂²×C₄×C₄₇	—	2	16	no	—
54	C₂²×C₄×C₄₇	—	2	8	no	—
55	C₂²×C₄×C₄₇	—	2	8	no	—
56	C₂²×C₄×C₄₇	—	2	32	no	—
57	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
58	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	32	yes	—
62	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
63	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
64	C₂²×C₄×C₄₇	C₄⋊C₄	1	32	yes	—
65	C₂²×C₄×C₄₇	M₁₆	1	16	yes	complemented
66	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
75	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
78	M₁₆×C₄₇	—	2	64	no	—
79	C₂×C₈×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	complemented
83	C₄²×C₄₇	—	2	16	no	—
84	C₄²×C₄₇	—	2	8	no	—
85	C₄²×C₄₇	—	2	8	no	—
86	C₂³×C₄₇	—	2	8	no	—
87	C₂³×C₄₇	—	2	16	no	—
88	C₂³×C₄₇	—	2	16	no	—
89	C₂³×C₄₇	—	2	16	no	—
90	C₂³×C₄₇	#95	1	16	yes	—
91	C₂³×C₄₇	C₂×C₄²	1	4	yes	—
92	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
93	C₂³×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
94	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
95	C₂³×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
96	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
97	C₂³×C₄₇	C₂×M₁₆	1	8	yes	—
98	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	—	2	64	no	—
100	C₂×C₄×C₄₇	—	2	8	no	—
101	C₂×C₄×C₄₇	—	2	8	no	—
102	C₂×C₄×C₄₇	—	2	32	no	—
103	C₂×C₄×C₄₇	#95	1	32	yes	—
104	C₂×C₄×C₄₇	#95	1	16	yes	—
105	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
106	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
108	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	16	yes	—
109	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
110	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	16	yes	—
112	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
113	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
114	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₄₇	C₂×M₁₆	1	16	yes	complemented
116	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
120	C₈×C₄₇	—	2	16	no	—
121	C₂×#1728	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂²×C₄₇	—	2	8	no	—
123	C₂²×C₄₇	—	2	8	no	—
124	C₂²×C₄₇	—	2	8	no	—
125	C₂²×C₄₇	—	2	32	no	—
126	C₂²×C₄₇	C₂×#95	1	8	yes	—
127	C₂²×C₄₇	C₂×#95	1	4	yes	—
128	C₂²×C₄₇	C₄×M₁₆	1	8	yes	—
129	C₂²×C₄₇	#407	1	8	yes	—
130	C₂²×C₄₇	#423	1	8	yes	—
131	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
132	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₄₇	C₂²×M₁₆	1	4	yes	—
135	C₄×C₄₇	—	2	16	no	—
136	C₄×C₄₇	C₂×#95	1	4	yes	—
137	C₄×C₄₇	C₂×#95	1	8	yes	—
138	C₄×C₄₇	C₂²×C₂²⋊C₄	1	2	yes	—
139	C₄×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
140	C₂³×M₁₆	C₂×C₄₇	1	2	yes	—
141	C₂³×C₄²	C₂×C₄₇	1	1	yes	characteristic
142	#1728	C₂×C₄₇	1	8	yes	direct product factor
143	C₂×#336	C₂×C₄₇	1	4	yes	complemented
144	C₂×C₄₇	—	2	8	no	—
145	C₂×C₄₇	#1728	1	8	yes	direct product factor
146	C₂×C₄₇	C₂²×#95	1	1	yes	characteristic
147	C₂×C₄₇	C₂×C₄×M₁₆	1	2	yes	—
148	C₂×C₄₇	C₂×#407	1	2	yes	—
149	C₂×C₄₇	C₂×#423	1	2	yes	—
150	C₂⁴×C₄	C₄×C₄₇	1	2	yes	—
151	C₂⁴×C₄	C₂²×C₄₇	1	1	yes	characteristic
152	C₂²×M₁₆	—	2	8	no	—
153	C₂²×M₁₆	—	2	8	no	—
154	C₂²×M₁₆	C₄×C₄₇	1	16	yes	complemented
155	C₂²×M₁₆	C₂²×C₄₇	1	8	yes	—
156	C₂³×C₈	C₂²×C₄₇	1	4	yes	—
157	C₂²×C₄²	—	2	8	no	—
158	C₂²×C₄²	C₄×C₄₇	1	4	yes	—
159	C₂²×C₄²	C₄×C₄₇	1	2	yes	—
160	C₂²×C₄²	C₂²×C₄₇	1	2	yes	—
161	C₂²×C₄²	C₂²×C₄₇	1	4	yes	—
162	#336	C₂²×C₄₇	1	16	yes	complemented
163	C₄₇	C₂×#1728	1	1	yes	characteristic, direct product factor, Sylow
164	C₂⁵	C₂×C₄×C₄₇	1	1	yes	characteristic
165	C₂³×C₄	—	2	8	no	—
166	C₂³×C₄	—	2	16	no	—
167	C₂³×C₄	C₂×C₄×C₄₇	1	8	yes	—
168	C₂³×C₄	C₂×C₄×C₄₇	1	4	yes	—
169	C₂³×C₄	C₂×C₄×C₄₇	1	2	yes	—
170	C₂³×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
171	C₂³×C₄	C₂×C₄×C₄₇	1	8	yes	—
172	C₂³×C₄	D₄×C₄₇	1	4	yes	—
173	C₂³×C₄	D₄×C₄₇	1	8	yes	—
174	C₂³×C₄	Q₈×C₄₇	1	4	yes	—
175	C₂³×C₄	C₂³×C₄₇	1	2	yes	—
176	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
177	C₂×M₁₆	—	2	64	no	—
178	C₂×M₁₆	—	2	16	no	—
179	C₂×M₁₆	—	2	16	no	—
180	C₂×M₁₆	C₂×C₄×C₄₇	1	32	yes	complemented
181	C₂²×C₈	—	2	8	no	—
182	C₂²×C₈	—	2	8	no	—
183	C₂²×C₈	C₂×C₄×C₄₇	1	16	yes	complemented
184	C₂²×C₈	C₂³×C₄₇	1	8	yes	—
185	C₂×C₄²	—	2	16	no	—
186	C₂×C₄²	—	2	8	no	—
187	C₂×C₄²	—	2	8	no	—
188	C₂×C₄²	—	2	16	no	—
189	C₂×C₄²	C₂×C₄×C₄₇	1	4	yes	—
190	C₂×C₄²	C₂×C₄×C₄₇	1	8	yes	—
191	C₂×C₄²	C₂³×C₄₇	1	4	yes	—
192	C₂⁴	—	2	8	no	—
193	C₂⁴	C₄²×C₄₇	1	4	yes	—
194	C₂⁴	C₂²⋊C₄×C₄₇	1	4	yes	—
195	C₂⁴	C₄⋊C₄×C₄₇	1	4	yes	—
196	C₂⁴	C₂²×C₄×C₄₇	1	2	yes	—
197	C₂⁴	C₂²×C₄×C₄₇	1	1	yes	characteristic
198	C₂²×C₄	—	2	64	no	—
199	C₂²×C₄	—	2	16	no	—
200	C₂²×C₄	—	2	8	no	—
201	C₂²×C₄	—	2	8	no	—
202	C₂²×C₄	—	2	32	no	—
203	C₂²×C₄	C₄²×C₄₇	1	8	yes	—
204	C₂²×C₄	C₄²×C₄₇	1	4	yes	—
205	C₂²×C₄	C₂²⋊C₄×C₄₇	1	32	yes	—
206	C₂²×C₄	C₂²⋊C₄×C₄₇	1	8	yes	—
207	C₂²×C₄	C₂²⋊C₄×C₄₇	1	4	yes	—
208	C₂²×C₄	C₄⋊C₄×C₄₇	1	32	yes	—
209	C₂²×C₄	C₄⋊C₄×C₄₇	1	8	yes	—
210	C₂²×C₄	C₄⋊C₄×C₄₇	1	4	yes	—
211	C₂²×C₄	M₁₆×C₄₇	1	16	yes	complemented
212	C₂²×C₄	C₂²×C₄×C₄₇	1	8	yes	—
213	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
214	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
215	C₂²×C₄	C₂²×C₄×C₄₇	1	4	yes	—
216	C₂²×C₄	C₂²×C₄×C₄₇	1	8	yes	—
217	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
218	C₂²×C₄	C₂×D₄×C₄₇	1	4	yes	—
219	C₂²×C₄	C₂×D₄×C₄₇	1	4	yes	—
220	C₂²×C₄	C₂×D₄×C₄₇	1	8	yes	—
221	C₂²×C₄	C₂×Q₈×C₄₇	1	4	yes	—
222	C₂²×C₄	C₂×Q₈×C₄₇	1	2	yes	—
223	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
224	M₁₆	—	2	64	no	—
225	C₂×C₈	—	2	32	no	—
226	C₂×C₈	—	2	8	no	—
227	C₂×C₈	—	2	8	no	—
228	C₂×C₈	C₂²×C₄×C₄₇	1	16	yes	complemented
229	C₄²	—	2	8	no	—
230	C₄²	—	2	16	no	—
231	C₄²	—	2	8	no	—
232	C₂³	—	2	8	no	—
233	C₂³	—	2	16	no	—
234	C₂³	—	2	16	no	—
235	C₂³	—	2	16	no	—
236	C₂³	#95×C₄₇	1	16	yes	—
237	C₂³	C₂×C₄²×C₄₇	1	4	yes	—
238	C₂³	C₂×C₄²×C₄₇	1	2	yes	—
239	C₂³	C₂×C₂²⋊C₄×C₄₇	1	4	yes	—
240	C₂³	C₂×C₂²⋊C₄×C₄₇	1	2	yes	—
241	C₂³	C₂×C₄⋊C₄×C₄₇	1	4	yes	—
242	C₂³	C₂×C₄⋊C₄×C₄₇	1	2	yes	—
243	C₂³	C₂×M₁₆×C₄₇	1	8	yes	—
244	C₂³	C₂³×C₄×C₄₇	1	1	yes	characteristic
245	C₂×C₄	—	2	64	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	32	no	—
249	C₂×C₄	#95×C₄₇	1	32	yes	—
250	C₂×C₄	#95×C₄₇	1	16	yes	—
251	C₂×C₄	C₂×C₄²×C₄₇	1	2	yes	—
252	C₂×C₄	C₂×C₄²×C₄₇	1	4	yes	—
253	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	8	yes	—
254	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	16	yes	—
255	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	2	yes	—
256	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	4	yes	—
257	C₂×C₄	C₂×C₄⋊C₄×C₄₇	1	16	yes	—
258	C₂×C₄	C₂×C₄⋊C₄×C₄₇	1	8	yes	—
259	C₂×C₄	C₂×C₄⋊C₄×C₄₇	1	2	yes	—
260	C₂×C₄	C₂×C₄⋊C₄×C₄₇	1	4	yes	—
261	C₂×C₄	C₂×M₁₆×C₄₇	1	16	yes	complemented
262	C₂×C₄	C₂³×C₄×C₄₇	1	2	yes	—
263	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
265	C₂×C₄	C₂²×Q₈×C₄₇	1	1	yes	characteristic
266	C₈	—	2	16	no	—
267	C₂²	—	2	8	no	—
268	C₂²	—	2	8	no	—
269	C₂²	—	2	8	no	—
270	C₂²	—	2	32	no	—
271	C₂²	C₂×#95×C₄₇	1	8	yes	—
272	C₂²	C₂×#95×C₄₇	1	4	yes	—
273	C₂²	C₄×M₁₆×C₄₇	1	8	yes	—
274	C₂²	C₄₇×#407	1	8	yes	—
275	C₂²	C₄₇×#423	1	8	yes	—
276	C₂²	C₂²×C₄²×C₄₇	1	1	yes	characteristic
277	C₂²	C₂²×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
278	C₂²	C₂²×C₄⋊C₄×C₄₇	1	1	yes	characteristic
279	C₂²	C₂²×M₁₆×C₄₇	1	4	yes	—
280	C₄	—	2	16	no	—
281	C₄	C₂×#95×C₄₇	1	4	yes	—
282	C₄	C₂×#95×C₄₇	1	8	yes	—
283	C₄	C₂²×C₂²⋊C₄×C₄₇	1	2	yes	—
284	C₄	C₂²×C₄⋊C₄×C₄₇	1	2	yes	—
285	C₂	—	2	8	no	—
286	C₂	C₄₇×#1728	1	8	yes	direct product factor
287	C₂	C₂²×#95×C₄₇	1	1	yes	characteristic
288	C₂	C₂×C₄×M₁₆×C₄₇	1	2	yes	—
289	C₂	C₂×C₄₇×#407	1	2	yes	—
290	C₂	C₂×C₄₇×#423	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₄₇×#1728	C₂	direct (N × H)
4. C₂×C₄₇×#336	C₂	semidirect
9. C₂²×M₁₆×C₄₇	C₄	semidirect
17. C₄₇×#336	C₂²	semidirect
34. C₂×M₁₆×C₄₇	C₂×C₄	semidirect
37. C₂²×C₈×C₄₇	C₂×C₄	semidirect
65. C₂²×C₄×C₄₇	M₁₆	semidirect
82. C₂×C₈×C₄₇	C₂²×C₄	semidirect
115. C₂×C₄×C₄₇	C₂×M₁₆	semidirect
121. C₂×#1728	C₄₇	direct (N × H)
142. #1728	C₂×C₄₇	direct (N × H)
143. C₂×#336	C₂×C₄₇	semidirect
145. C₂×C₄₇	#1728	direct (N × H)
154. C₂²×M₁₆	C₄×C₄₇	semidirect
162. #336	C₂²×C₄₇	semidirect
163. C₄₇	C₂×#1728	direct (N × H)
180. C₂×M₁₆	C₂×C₄×C₄₇	semidirect
183. C₂²×C₈	C₂×C₄×C₄₇	semidirect
211. C₂²×C₄	M₁₆×C₄₇	semidirect
228. C₂×C₈	C₂²×C₄×C₄₇	semidirect
261. C₂×C₄	C₂×M₁₆×C₄₇	semidirect
286. C₂	C₄₇×#1728	direct (N × H)

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
C₂³×C₄²×C₄₇	2
C₄₇×#1728	2
C₂×C₄₇×#336	2
C₂×#1728	47

Subgroups — C2×C47×#1728 #26060031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₇×#1728 #26060031