Subgroups — #26214565

Order 12032.

1022 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689180	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
3	#20243255	C₂	1	2	yes	maximal, complemented
4	C₈×C₄⋊Dic₄₇	C₂	1	2	yes	maximal, complemented
5	#20243169	C₂	1	4	yes	maximal
6	#20243136	C₂	1	2	yes	maximal, complemented
7	#97×Dic₄₇	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#13093983	C₄	1	2	yes	—
10	#15689180	C₄	1	8	yes	complemented
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×#13093935	C₂²	1	2	yes	—
14	C₂×#13093934	C₂²	1	4	yes	—
15	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
16	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
17	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
19	C₄×C₂²⋊Dic₄₇	C₄	1	2	yes	—
20	C₄×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
21	#15688969	C₄	1	2	yes	—
22	C₄²×Dic₄₇	C₂²	1	2	yes	—
23	#15688877	C₂²	1	4	yes	—
24	#15688877	C₂²	1	4	yes	—
25	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
26	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₄₇	C₄	1	8	yes	complemented
28	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
32	#13093983	—	2	8	no	—
33	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
38	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
39	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
40	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
41	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
42	#13093935	C₂×C₄	1	8	yes	—
43	#13093934	—	2	8	no	—
44	C₈×Dic₄₇	—	2	8	no	—
45	C₄₇⋊#95	C₂×C₄	1	2	yes	—
46	C₄₇⋊#95	C₂×C₄	1	2	yes	—
47	C₄₇⋊#95	C₂×C₄	1	4	yes	—
48	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₄₇	—	2	4	no	—
52	C₂×M₁₆×C₄₇	—	2	4	no	—
53	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
54	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
55	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
56	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
59	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
60	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	#97×C₄₇	C₂×C₄	1	4	yes	—
63	#97×C₄₇	C₂×C₄	1	4	yes	—
64	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	—
65	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
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	—	4	2	no	—
70	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
71	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
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	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
80	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
81	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
82	M₁₆×C₄₇	—	2	16	no	—
83	C₂×C₈×C₄₇	—	2	2	no	—
84	C₂×C₈×C₄₇	—	2	2	no	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₄₇	#41	1	4	yes	—
88	C₂×C₈×C₄₇	#41	1	4	yes	—
89	C₄²×C₄₇	C₄²	1	4	yes	—
90	C₄²×C₄₇	C₄²	1	4	yes	—
91	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
92	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
93	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
94	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
95	C₂²⋊Dic₄₇	—	2	8	no	—
96	C₄⋊Dic₄₇	—	2	4	no	—
97	C₄⋊Dic₄₇	M₁₆	1	8	yes	complemented
98	C₄×Dic₄₇	—	2	4	no	—
99	C₄×Dic₄₇	—	2	4	no	—
100	C₄×Dic₄₇	—	2	8	no	—
101	C₄×Dic₄₇	—	4	4	no	—
102	C₂×C₄₇⋊C₈	—	2	4	no	—
103	C₂×C₄₇⋊C₈	—	2	4	no	—
104	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
105	C₂³×C₄₇	—	2	1	no	—
106	C₂³×C₄₇	—	2	1	no	—
107	C₂³×C₄₇	—	2	2	no	—
108	C₂³×C₄₇	—	2	2	no	—
109	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
110	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	—	2	2	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	4	no	—
118	C₂×C₄×C₄₇	—	2	8	no	—
119	C₂×C₄×C₄₇	—	4	4	no	—
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	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
126	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
129	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
130	C₂×Dic₄₇	—	2	4	no	—
131	C₂×Dic₄₇	—	2	4	no	—
132	C₂×Dic₄₇	—	2	4	no	—
133	C₂×Dic₄₇	—	2	4	no	—
134	C₂×Dic₄₇	—	2	4	no	—
135	C₂×Dic₄₇	C₂×M₁₆	1	8	yes	—
136	C₈×C₄₇	—	2	4	no	—
137	C₈×C₄₇	#117	1	8	yes	complemented
138	C₄₇⋊C₈	—	2	8	no	—
139	#20480	—	47	1	no	maximal, Sylow
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	4	2	no	—
144	C₂²×C₄₇	—	4	2	no	—
145	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
146	C₂²×C₄₇	C₄×#41	1	1	yes	characteristic
147	C₂²×C₄₇	C₄×#41	1	1	yes	characteristic
148	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
149	C₂²×C₄₇	#568	1	2	yes	—
150	C₄×C₄₇	—	2	2	no	—
151	C₄×C₄₇	—	2	2	no	—
152	C₄×C₄₇	—	2	2	no	—
153	C₄×C₄₇	—	2	2	no	—
154	C₄×C₄₇	C₄×M₁₆	1	4	yes	—
155	C₄×C₄₇	C₂×#117	1	2	yes	—
156	C₄×C₄₇	C₄×#41	1	2	yes	—
157	Dic₄₇	—	2	8	no	—
158	Dic₄₇	—	4	4	no	—
159	C₂×#432	—	47	1	no	—
160	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
161	C₄×#117	—	47	1	no	—
162	#1770	—	47	2	no	—
163	C₈×C₄⋊C₄	—	47	2	no	—
164	#1753	—	47	4	no	—
165	#1745	—	47	2	no	—
166	C₄×#97	—	47	2	no	—
167	C₂×C₄₇	—	4	2	no	—
168	C₂×C₄₇	C₄×#117	1	1	yes	characteristic
169	C₂×C₄₇	C₂×C₄×M₁₆	1	1	yes	characteristic
170	C₂×C₄₇	#2873	1	1	yes	characteristic
171	C₂×C₄₇	#2970	1	2	yes	—
172	C₂×C₄₇	#2971	1	2	yes	—
173	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
174	C₂×#117	—	47	2	no	—
175	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
176	#432	—	47	8	no	—
177	C₂×C₄⋊C₈	—	47	2	no	—
178	C₂×C₂²⋊C₈	—	47	2	no	—
179	C₄×M₁₆	Dic₄₇	1	8	yes	complemented
180	C₂×#97	—	47	2	no	—
181	C₂×#97	—	47	4	no	—
182	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
183	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
184	C₂×C₄×C₈	—	47	4	no	—
185	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
186	C₄×C₄⋊C₄	—	47	2	no	—
187	C₄×C₄⋊C₄	—	47	1	no	—
188	C₄×C₄⋊C₄	—	47	1	no	—
189	C₄×C₂²⋊C₄	—	47	2	no	—
190	C₄×C₂²⋊C₄	—	47	2	no	—
191	#376	—	47	2	no	—
192	C₄³	—	47	2	no	—
193	#336	—	47	4	no	—
194	#336	—	47	4	no	—
195	C₄₇	#20480	1	1	yes	characteristic, complemented, Sylow
196	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
197	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
198	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
199	C₂×M₁₆	—	2	4	no	—
200	C₂×M₁₆	—	2	4	no	—
201	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
202	C₂²×C₈	—	47	4	no	—
203	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
204	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
205	#117	—	94	8	no	—
206	C₂×C₄⋊C₄	—	47	2	no	—
207	C₂×C₄⋊C₄	—	47	2	no	—
208	C₂×C₂²⋊C₄	—	47	4	no	—
209	C₂×C₄²	—	47	4	no	—
210	C₂×C₄²	—	47	2	no	—
211	C₂×C₄²	—	47	2	no	—
212	C₂×C₄²	—	47	4	no	—
213	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
214	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
215	C₂×C₄²	C₄×D₄₇	1	4	yes	—
216	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
217	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
218	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
219	C₄⋊C₈	—	47	8	no	—
220	C₂²⋊C₈	—	47	8	no	—
221	#97	—	47	8	no	—
222	#97	—	94	8	no	—
223	#97	C₂×Dic₄₇	1	4	yes	—
224	#97	C₂×Dic₄₇	1	4	yes	—
225	C₄×C₈	—	94	8	no	—
226	C₄×C₈	C₂×Dic₄₇	1	8	yes	—
227	#95	—	47	2	no	—
228	#95	—	47	4	no	—
229	#95	—	47	2	no	—
230	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	47	4	no	—
236	C₂²×C₄	—	47	4	no	—
237	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
238	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
239	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
244	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
245	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
246	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
247	M₁₆	—	2	16	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	47	8	no	—
251	C₂×C₈	—	94	4	no	—
252	C₂×C₈	—	94	4	no	—
253	C₂×C₈	#10625350	1	4	yes	—
254	C₂×C₈	#10625350	1	4	yes	—
255	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
256	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
257	C₄⋊C₄	—	47	8	no	—
258	C₄⋊C₄	—	94	4	no	—
259	C₂²⋊C₄	—	94	8	no	—
260	C₄²	—	94	8	no	—
261	C₄²	—	94	4	no	—
262	C₄²	—	94	4	no	—
263	C₄²	—	188	4	no	—
264	C₄²	C₄×Dic₄₇	1	4	yes	—
265	C₄²	C₄×Dic₄₇	1	4	yes	—
266	C₄²	C₂×C₄×D₄₇	1	2	yes	—
267	C₄²	C₂×C₄×D₄₇	1	2	yes	—
268	C₄²	C₂²×Dic₄₇	1	2	yes	—
269	C₄²	C₂²×Dic₄₇	1	2	yes	—
270	C₂³	—	2	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	2	no	—
273	C₂³	—	2	2	no	—
274	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
275	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	47	8	no	—
286	C₂×C₄	—	94	4	no	—
287	C₂×C₄	—	94	4	no	—
288	C₂×C₄	—	94	4	no	—
289	C₂×C₄	—	94	4	no	—
290	C₂×C₄	—	94	4	no	—
291	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
292	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
293	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
294	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
295	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
296	C₂×C₄	M₁₆×D₄₇	1	4	yes	—
297	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
298	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
299	C₂×C₄	C₂×#10625350	1	2	yes	—
300	C₂×C₄	C₂×#10625350	1	2	yes	—
301	C₈	—	2	4	no	—
302	C₈	—	94	8	no	—
303	C₈	#13093983	1	8	yes	complemented
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	2	no	—
309	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
310	C₂²	C₄×#10625350	1	1	yes	characteristic
311	C₂²	C₄×#10625350	1	1	yes	characteristic
312	C₂²	C₂×M₁₆×D₄₇	1	2	yes	—
313	C₂²	#15689889	1	2	yes	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	94	8	no	—
319	C₄	—	188	4	no	—
320	C₄	M₁₆×Dic₄₇	1	4	yes	—
321	C₄	C₂×#13093983	1	2	yes	—
322	C₄	C₄×#10625350	1	2	yes	—
323	C₂	—	4	2	no	—
324	C₂	C₄×#13093983	1	1	yes	characteristic
325	C₂	C₂×M₁₆×Dic₄₇	1	1	yes	characteristic
326	C₂	#20251806	1	1	yes	characteristic
327	C₂	#20252403	1	2	yes	—
328	C₂	#20252406	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₂×C₄×M₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₄₇
hypocenter	C₄₇	#20480
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20243255	C₂	semidirect
4. C₈×C₄⋊Dic₄₇	C₂	semidirect
6. #20243136	C₂	semidirect
7. #97×Dic₄₇	C₂	semidirect
10. #15689180	C₄	semidirect
27. C₄×M₁₆×C₄₇	C₄	semidirect
97. C₄⋊Dic₄₇	M₁₆	semidirect
137. C₈×C₄₇	#117	semidirect
179. C₄×M₁₆	Dic₄₇	semidirect
195. C₄₇	#20480	semidirect
303. C₈	#13093983	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689180	2
C₄×#13093983	2
#20243255	2
C₈×C₄⋊Dic₄₇	2
#20243169	2
#20243136	2
#97×Dic₄₇	2
C₂×C₄×M₁₆×C₄₇	2
#20480	47