Subgroups — #26215088

Order 12032.

1006 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
3	#20243260	C₂	1	4	yes	maximal, complemented
4	#20243243	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₈×Dic₄₇	C₂	1	2	yes	maximal, complemented
6	#20243167	C₂	1	2	yes	maximal
7	#20243162	C₂	1	2	yes	maximal
8	C₂×C₄₇×#432	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
10	C₂×#13093983	C₄	1	2	yes	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₄	1	8	yes	complemented
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×#13093934	C₂²	1	4	yes	—
16	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
17	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
18	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
19	C₄×C₂²⋊Dic₄₇	C₄	1	2	yes	—
20	C₄×C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
21	C₄×C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
22	#15688969	C₄	1	2	yes	—
23	C₄²×Dic₄₇	C₂²	1	2	yes	—
24	#15688877	C₂²	1	4	yes	—
25	#15688877	C₂²	1	4	yes	—
26	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
27	C₄₇×#432	C₄	1	8	yes	complemented
28	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
30	C₂×#97×C₄₇	C₂²	1	2	yes	—
31	C₂²×C₄₇⋊C₈	—	4	4	no	—
32	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
33	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
34	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
35	#13093983	—	2	8	no	—
36	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
40	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
41	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
42	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
43	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
44	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
45	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
46	#13093934	—	4	4	no	—
47	C₈×Dic₄₇	—	4	4	no	—
48	C₄₇⋊#95	C₂×C₄	1	4	yes	—
49	C₄₇⋊#95	C₂×C₄	1	2	yes	—
50	C₄₇⋊#95	C₂×C₄	1	2	yes	—
51	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
52	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
55	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
58	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
62	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
63	#97×C₄₇	C₂×C₄	1	8	yes	—
64	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	—	2	4	no	—
66	C₂²×C₄×C₄₇	—	2	4	no	—
67	C₂²×C₄×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
70	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
79	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
80	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₄₇	—	2	4	no	—
82	C₂×C₈×C₄₇	—	2	4	no	—
83	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
84	C₄²×C₄₇	C₄²	1	4	yes	—
85	C₄²×C₄₇	C₄²	1	4	yes	—
86	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
87	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
88	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
89	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
90	C₂²⋊Dic₄₇	—	2	4	no	—
91	C₂²⋊Dic₄₇	M₁₆	1	8	yes	—
92	C₄⋊Dic₄₇	—	2	8	no	—
93	C₄×Dic₄₇	—	2	8	no	—
94	C₄×Dic₄₇	—	2	2	no	—
95	C₄×Dic₄₇	—	2	2	no	—
96	C₄×Dic₄₇	—	2	2	no	—
97	C₄×Dic₄₇	—	2	2	no	—
98	C₄×Dic₄₇	—	2	8	no	—
99	C₂×C₄₇⋊C₈	—	2	4	no	—
100	C₂×C₄₇⋊C₈	—	2	4	no	—
101	C₂×C₄₇⋊C₈	—	4	8	no	—
102	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
103	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
104	C₂³×C₄₇	—	2	2	no	—
105	C₂³×C₄₇	—	2	2	no	—
106	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
107	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
108	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	8	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	2	8	no	—
118	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₄₇	#131	1	4	yes	—
124	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
127	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
128	C₂×Dic₄₇	—	2	4	no	—
129	C₂×Dic₄₇	—	2	4	no	—
130	C₂×Dic₄₇	—	2	4	no	—
131	C₂×Dic₄₇	—	2	4	no	—
132	C₂×Dic₄₇	—	2	4	no	—
133	C₂×Dic₄₇	C₂×M₁₆	1	8	yes	—
134	C₈×C₄₇	—	2	8	no	—
135	C₄₇⋊C₈	—	2	4	no	—
136	C₄₇⋊C₈	—	2	4	no	—
137	#20546	—	47	1	no	maximal, Sylow
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	—	2	8	no	—
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	C₄×M₁₆	1	4	yes	—
142	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
143	C₂²×C₄₇	#520	1	2	yes	—
144	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
145	C₂²×C₄₇	C₂×#131	1	2	yes	—
146	C₄×C₄₇	—	2	2	no	—
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	2	2	no	—
150	C₄×C₄₇	#405	1	4	yes	—
151	C₄×C₄₇	#518	1	1	yes	characteristic
152	C₄×C₄₇	#519	1	1	yes	characteristic
153	C₄×C₄₇	#520	1	2	yes	—
154	Dic₄₇	—	2	8	no	—
155	Dic₄₇	—	4	4	no	—
156	C₂×#432	D₄₇	1	1	yes	characteristic
157	C₂²×C₄⋊C₈	—	47	1	no	—
158	C₄×#117	—	47	1	no	—
159	#1771	—	47	4	no	—
160	#1767	—	47	2	no	—
161	C₄×C₄⋊C₈	—	47	2	no	—
162	#1752	—	47	2	no	—
163	C₈×C₂²⋊C₄	—	47	2	no	—
164	C₂×C₄₇	—	2	4	no	—
165	C₂×C₄₇	#2274	1	1	yes	characteristic
166	C₂×C₄₇	C₂×C₄×M₁₆	1	1	yes	characteristic
167	C₂×C₄₇	C₂×#405	1	1	yes	characteristic
168	C₂×C₄₇	#2990	1	4	yes	—
169	C₂³×C₈	—	47	2	no	—
170	C₂×#117	—	47	2	no	—
171	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
172	#432	Dic₄₇	1	8	yes	complemented
173	C₂×C₄⋊C₈	—	47	8	no	—
174	C₂×C₄⋊C₈	—	47	1	no	—
175	C₂×C₄⋊C₈	—	47	1	no	—
176	C₂×C₄⋊C₈	—	47	2	no	—
177	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
178	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
179	C₂×#97	—	47	4	no	—
180	C₂×#97	C₂×D₄₇	1	2	yes	—
181	C₂×C₄×C₈	—	47	4	no	—
182	C₄×C₄⋊C₄	—	47	2	no	—
183	C₄×C₄⋊C₄	—	47	2	no	—
184	C₄×C₂²⋊C₄	—	47	2	no	—
185	C₄×C₂²⋊C₄	—	47	1	no	—
186	C₄×C₂²⋊C₄	—	47	1	no	—
187	#376	—	47	2	no	—
188	C₄³	—	47	2	no	—
189	#336	—	47	4	no	—
190	#336	—	47	4	no	—
191	C₄₇	#20546	1	1	yes	characteristic, complemented, Sylow
192	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
193	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
194	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
195	C₂²×C₈	—	47	2	no	—
196	C₂²×C₈	—	47	2	no	—
197	C₂²×C₈	—	47	8	no	—
198	C₂²×C₈	—	188	4	no	—
199	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
200	#117	—	94	8	no	—
201	C₂×C₄⋊C₄	—	47	4	no	—
202	C₂×C₂²⋊C₄	—	47	2	no	—
203	C₂×C₂²⋊C₄	—	47	2	no	—
204	C₂×C₄²	—	47	4	no	—
205	C₂×C₄²	—	47	4	no	—
206	C₂×C₄²	—	47	2	no	—
207	C₂×C₄²	—	47	2	no	—
208	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
209	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
210	C₂×C₄²	C₄×D₄₇	1	4	yes	—
211	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
212	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
213	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
214	C₄⋊C₈	—	47	4	no	—
215	C₄⋊C₈	—	47	4	no	—
216	C₄⋊C₈	—	47	8	no	—
217	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
218	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
219	#97	—	188	4	no	—
220	#97	C₂×Dic₄₇	1	8	yes	—
221	C₄×C₈	—	188	4	no	—
222	#95	—	47	2	no	—
223	#95	—	47	4	no	—
224	#95	—	47	2	no	—
225	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	47	4	no	—
231	C₂²×C₄	—	47	4	no	—
232	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
233	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
234	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
239	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
240	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
241	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	47	4	no	—
245	C₂×C₈	—	47	4	no	—
246	C₂×C₈	—	94	4	no	—
247	C₂×C₈	—	94	4	no	—
248	C₂×C₈	—	188	8	no	—
249	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
250	C₄⋊C₄	—	94	8	no	—
251	C₂²⋊C₄	—	47	8	no	—
252	C₂²⋊C₄	—	94	4	no	—
253	C₄²	—	94	8	no	—
254	C₄²	—	94	8	no	—
255	C₄²	—	94	2	no	—
256	C₄²	—	94	2	no	—
257	C₄²	—	94	2	no	—
258	C₄²	—	94	2	no	—
259	C₄²	C₄×Dic₄₇	1	4	yes	—
260	C₄²	C₄×Dic₄₇	1	4	yes	—
261	C₄²	C₂×C₄×D₄₇	1	2	yes	—
262	C₄²	C₂×C₄×D₄₇	1	2	yes	—
263	C₄²	C₂²×Dic₄₇	1	2	yes	—
264	C₄²	C₂²×Dic₄₇	1	2	yes	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
268	C₂³	M₁₆×D₄₇	1	4	yes	—
269	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	47	8	no	—
280	C₂×C₄	—	94	4	no	—
281	C₂×C₄	—	94	4	no	—
282	C₂×C₄	—	94	4	no	—
283	C₂×C₄	—	94	4	no	—
284	C₂×C₄	—	94	4	no	—
285	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
286	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
287	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
288	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
289	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
290	C₂×C₄	#13094032	1	4	yes	—
291	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
292	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
293	C₂×C₄	#13094070	1	2	yes	—
294	C₂×C₄	#13094071	1	2	yes	—
295	C₈	—	2	8	no	—
296	C₈	—	94	4	no	—
297	C₈	—	94	4	no	—
298	C₂²	—	2	8	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	M₁₆×Dic₄₇	1	4	yes	—
302	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
303	C₂²	#15689603	1	2	yes	—
304	C₂²	C₂×M₁₆×D₄₇	1	2	yes	—
305	C₂²	C₂×#13094032	1	2	yes	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	94	8	no	—
311	C₄	—	188	4	no	—
312	C₄	#15689081	1	4	yes	—
313	C₄	#15689597	1	1	yes	characteristic
314	C₄	#15689600	1	1	yes	characteristic
315	C₄	#15689606	1	2	yes	—
316	C₂	—	2	4	no	—
317	C₂	#20245744	1	1	yes	characteristic
318	C₂	C₂×M₁₆×Dic₄₇	1	1	yes	characteristic
319	C₂	C₂×#15689081	1	1	yes	characteristic
320	C₂	#20252515	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₂×C₄₇×#432	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#432	D₄₇
hypocenter	C₄₇	#20546
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #20243260	C₂	semidirect
4. #20243243	C₂	semidirect
5. C₄⋊C₈×Dic₄₇	C₂	semidirect
11. C₂×C₄⋊(C₄₇⋊C₈)	C₄	semidirect
27. C₄₇×#432	C₄	semidirect
172. #432	Dic₄₇	semidirect
191. C₄₇	#20546	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₄₇⋊C₈)	2
C₄×#13093983	2
#20243260	2
#20243243	2
C₄⋊C₈×Dic₄₇	2
#20243167	2
#20243162	2
C₂×C₄₇×#432	2
#20546	47