Subgroups — #26214231

Order 12032.

1166 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689151	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689078	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	#20245870	C₂	1	1	yes	maximal, characteristic
5	#20243264	C₂	1	4	yes	maximal, complemented
6	C₈×Dic₄₇⋊C₄	C₂	1	4	yes	maximal, complemented
7	#20243085	C₂	1	1	yes	maximal, characteristic
8	#20243081	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄₇×#405	C₂	1	1	yes	maximal, characteristic
10	C₂²×#10625329	C₂²	1	1	yes	characteristic
11	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×#13093983	C₂²	1	1	yes	characteristic
13	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
14	#15689151	C₄	1	8	yes	complemented
15	C₂×Dic₄₇⋊C₈	C₂²	1	4	yes	—
16	#15689078	C₄	1	8	yes	complemented
17	C₂×#13093934	C₂²	1	2	yes	—
18	C₂×C₈×Dic₄₇	—	2	4	no	—
19	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
20	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
21	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
22	#15689019	C₄	1	4	yes	—
23	#15688995	C₄	1	4	yes	—
24	C₄×Dic₄₇⋊C₄	C₂²	1	4	yes	—
25	#15688878	C₂²	1	4	yes	—
26	#15688877	C₂²	1	4	yes	—
27	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
28	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
30	C₄₇×#405	—	2	4	no	—
31	C₂×#97×C₄₇	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
33	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
34	C₂×#10625329	—	2	2	no	—
35	C₂×#10625329	—	2	2	no	—
36	C₂×#10625329	C₂×C₄	1	4	yes	—
37	C₂²×C₄₇⋊C₈	—	2	2	no	—
38	C₂²×C₄₇⋊C₈	—	2	2	no	—
39	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
40	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
41	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
42	#13093983	C₂×C₄	1	4	yes	—
43	#13093983	C₂×C₄	1	4	yes	—
44	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
45	C₂×Dic₄₇⋊C₄	C₂×C₄	1	8	yes	—
46	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₄₇	—	2	2	no	—
48	C₂×C₄×Dic₄₇	—	2	2	no	—
49	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
51	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
52	Dic₄₇⋊C₈	C₂×C₄	1	16	yes	complemented
53	#13093934	C₂×C₄	1	8	yes	complemented
54	C₈×Dic₄₇	—	2	4	no	—
55	C₈×Dic₄₇	—	2	4	no	—
56	C₈×Dic₄₇	C₂×C₄	1	8	yes	complemented
57	C₄₇⋊#95	C₂×C₄	1	8	yes	—
58	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×M₁₆×C₄₇	—	2	2	no	—
60	C₂×M₁₆×C₄₇	—	2	2	no	—
61	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
62	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
63	C₂²×C₈×C₄₇	D₄	1	4	yes	—
64	C₂²×C₈×C₄₇	Q₈	1	4	yes	—
65	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
66	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	#117×C₄₇	—	2	2	no	—
69	#117×C₄₇	—	2	2	no	—
70	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
71	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
72	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
73	#97×C₄₇	—	2	4	no	—
74	C₄×C₈×C₄₇	—	2	4	no	—
75	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₄₇	#41	1	2	yes	—
83	C₂²×C₄×C₄₇	#41	1	2	yes	—
84	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
85	C₂²×Dic₄₇	—	2	4	no	—
86	C₂²×Dic₄₇	—	2	8	no	—
87	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
88	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
89	M₁₆×C₄₇	—	2	8	no	—
90	C₂×C₈×C₄₇	—	2	2	no	—
91	C₂×C₈×C₄₇	—	2	2	no	—
92	C₂×C₈×C₄₇	C₄⋊C₄	1	16	yes	complemented
93	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
99	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
100	C₄⋊C₄×C₄₇	—	2	4	no	—
101	C₂²⋊C₄×C₄₇	—	2	4	no	—
102	C₄²×C₄₇	—	2	2	no	—
103	C₄²×C₄₇	—	2	2	no	—
104	C₂²⋊Dic₄₇	C₄²	1	8	yes	—
105	C₄⋊Dic₄₇	C₄²	1	8	yes	—
106	Dic₄₇⋊C₄	—	2	16	no	—
107	C₄×Dic₄₇	—	2	4	no	—
108	C₄×Dic₄₇	—	2	2	no	—
109	C₄×Dic₄₇	—	2	2	no	—
110	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
111	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
112	#10625329	—	2	8	no	—
113	C₂×C₄₇⋊C₈	—	2	8	no	—
114	C₂×C₄₇⋊C₈	—	2	2	no	—
115	C₂×C₄₇⋊C₈	—	2	2	no	—
116	C₂×C₄₇⋊C₈	—	2	2	no	—
117	C₂×C₄₇⋊C₈	—	2	2	no	—
118	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
119	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
120	C₂³×C₄₇	—	2	4	no	—
121	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
122	C₂³×C₄₇	C₄×D₄	1	2	yes	—
123	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
124	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	—	2	4	no	—
126	C₂×C₄×C₄₇	—	2	8	no	—
127	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
128	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
129	C₂×C₄×C₄₇	#117	1	4	yes	—
130	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
131	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
133	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
134	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
140	C₂×Dic₄₇	—	2	16	no	—
141	C₂×Dic₄₇	—	2	4	no	—
142	C₂×Dic₄₇	—	2	8	no	—
143	C₂×Dic₄₇	—	2	4	no	—
144	C₂×Dic₄₇	C₂×C₄²	1	8	yes	—
145	C₂×Dic₄₇	#131	1	8	yes	—
146	C₈×C₄₇	—	2	4	no	—
147	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
148	C₈×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
149	C₄₇⋊C₈	—	2	4	no	—
150	C₄₇⋊C₈	—	2	4	no	—
151	#20435	—	47	1	no	maximal, Sylow
152	C₂²×C₄₇	—	2	4	no	—
153	C₂²×C₄₇	—	2	4	no	—
154	C₂²×C₄₇	—	2	4	no	—
155	C₂²×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
156	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
157	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
158	C₂²×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
159	C₂²×C₄₇	C₂×#131	1	4	yes	—
160	C₄×C₄₇	—	4	4	no	—
161	C₄×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
162	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
163	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
164	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
165	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
166	Dic₄₇	—	2	8	no	—
167	Dic₄₇	—	2	8	no	—
168	C₂×#424	—	47	1	no	—
169	C₂×#405	—	47	1	no	—
170	C₂×#405	D₄₇	1	1	yes	characteristic
171	C₂²×C₄×C₈	—	47	1	no	—
172	#2296	—	47	1	no	—
173	#1771	—	47	4	no	—
174	C₈×C₄⋊C₄	—	47	4	no	—
175	#1729	—	47	1	no	—
176	#1728	—	47	1	no	—
177	C₂×C₄₇	—	2	4	no	—
178	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₄₇	C₄×#131	1	2	yes	—
180	C₂×C₄₇	#2929	1	2	yes	—
181	C₂×C₄₇	#2960	1	2	yes	—
182	C₂²×M₁₆	—	47	1	no	—
183	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
184	C₂³×C₈	—	47	1	no	—
185	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
186	C₂×#117	—	47	1	no	—
187	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
188	C₂²×C₄²	—	47	1	no	—
189	#424	—	47	8	no	—
190	C₂×C₄⋊C₈	—	47	4	no	—
191	#405	—	2	4	no	—
192	#405	—	47	8	no	—
193	C₂×#97	—	47	2	no	—
194	C₂×#97	C₂×D₄₇	1	2	yes	—
195	C₂×C₄×C₈	—	47	2	no	—
196	C₂×C₄×C₈	—	47	2	no	—
197	C₂×C₄×C₈	—	47	2	no	—
198	C₂×C₄×C₈	—	94	4	no	—
199	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
200	#389	—	47	4	no	—
201	#385	—	47	4	no	—
202	C₄×C₄⋊C₄	—	47	4	no	—
203	#336	—	47	4	no	—
204	#336	—	47	4	no	—
205	C₄₇	#20435	1	1	yes	characteristic, complemented, Sylow
206	C₂³×C₄	—	47	2	no	—
207	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
208	C₂×M₁₆	—	2	2	no	—
209	C₂×M₁₆	—	2	2	no	—
210	C₂×M₁₆	—	47	4	no	—
211	C₂×M₁₆	—	94	2	no	—
212	C₂×M₁₆	—	94	2	no	—
213	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
214	C₂²×C₈	—	47	4	no	—
215	C₂²×C₈	—	47	2	no	—
216	C₂²×C₈	—	47	2	no	—
217	C₂²×C₈	—	94	2	no	—
218	C₂²×C₈	—	94	2	no	—
219	C₂²×C₈	Dic₉₄	1	4	yes	—
220	C₂²×C₈	C₄×D₄₇	1	4	yes	—
221	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
222	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
223	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
224	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
225	#117	—	2	2	no	—
226	#117	—	2	2	no	—
227	#117	—	47	4	no	—
228	#117	—	47	4	no	—
229	C₂×C₄⋊C₄	—	47	2	no	—
230	C₂×C₄⋊C₄	—	47	8	no	—
231	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
232	C₂×C₂²⋊C₄	—	47	2	no	—
233	C₂×C₂²⋊C₄	C₄×D₄₇	1	2	yes	—
234	C₂×C₄²	—	47	2	no	—
235	C₂×C₄²	—	47	2	no	—
236	C₂×C₄²	—	47	2	no	—
237	C₂×C₄²	—	94	2	no	—
238	C₂×C₄²	—	94	2	no	—
239	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
240	C₄⋊C₈	—	47	16	no	—
241	#97	—	2	4	no	—
242	#97	—	47	8	no	—
243	C₄×C₈	—	2	4	no	—
244	C₄×C₈	—	47	8	no	—
245	C₄×C₈	—	94	4	no	—
246	C₄×C₈	—	94	4	no	—
247	#95	—	47	8	no	—
248	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
249	C₂²×C₄	—	47	4	no	—
250	C₂²×C₄	—	47	4	no	—
251	C₂²×C₄	—	94	8	no	—
252	C₂²×C₄	—	94	4	no	—
253	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
254	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
255	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
257	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
258	C₂²×C₄	#10625350	1	2	yes	—
259	C₂²×C₄	#10625350	1	2	yes	—
260	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
261	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
262	M₁₆	—	2	8	no	—
263	M₁₆	—	94	8	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	47	4	no	—
267	C₂×C₈	—	47	4	no	—
268	C₂×C₈	—	94	8	no	—
269	C₂×C₈	—	94	2	no	—
270	C₂×C₈	—	94	2	no	—
271	C₂×C₈	—	94	2	no	—
272	C₂×C₈	—	94	2	no	—
273	C₂×C₈	Dic₄₇⋊C₄	1	16	yes	complemented
274	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
275	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
276	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
277	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
278	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
279	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
280	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
281	C₄⋊C₄	—	2	4	no	—
282	C₄⋊C₄	—	47	8	no	—
283	C₄⋊C₄	—	94	16	no	—
284	C₂²⋊C₄	—	2	4	no	—
285	C₂²⋊C₄	—	47	8	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	47	4	no	—
289	C₄²	—	47	4	no	—
290	C₄²	—	94	4	no	—
291	C₄²	—	94	2	no	—
292	C₄²	—	94	2	no	—
293	C₂³	—	2	4	no	—
294	C₂³	C₄²×D₄₇	1	2	yes	—
295	C₂³	C₄×C₂²⋊D₄₇	1	2	yes	—
296	C₂³	C₄×Dic₉₄	1	2	yes	—
297	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	8	no	—
300	C₂×C₄	—	47	8	no	—
301	C₂×C₄	—	47	8	no	—
302	C₂×C₄	—	94	16	no	—
303	C₂×C₄	—	94	4	no	—
304	C₂×C₄	—	94	8	no	—
305	C₂×C₄	—	94	4	no	—
306	C₂×C₄	C₄²×D₄₇	1	2	yes	—
307	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
308	C₂×C₄	#13093981	1	4	yes	—
309	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
310	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
311	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
312	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
313	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
315	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
316	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
317	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
318	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
319	C₈	—	2	4	no	—
320	C₈	—	94	4	no	—
321	C₈	—	94	4	no	—
322	C₈	C₂×Dic₄₇⋊C₄	1	4	yes	—
323	C₈	C₂×Dic₄₇⋊C₄	1	4	yes	—
324	C₂²	—	2	4	no	—
325	C₂²	—	2	4	no	—
326	C₂²	—	2	4	no	—
327	C₂²	C₄×Dic₄₇⋊C₄	1	4	yes	—
328	C₂²	C₂×C₄²×D₄₇	1	1	yes	characteristic
329	C₂²	C₂×C₄×C₂²⋊D₄₇	1	1	yes	characteristic
330	C₂²	C₂×C₄×Dic₉₄	1	1	yes	characteristic
331	C₂²	#131×D₄₇	1	4	yes	—
332	C₄	—	4	4	no	—
333	C₄	—	94	8	no	—
334	C₄	—	94	8	no	—
335	C₄	C₄×Dic₄₇⋊C₄	1	4	yes	—
336	C₄	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
337	C₄	C₂×#13093981	1	1	yes	characteristic
338	C₄	C₂×C₄×C₂²⋊D₄₇	1	1	yes	characteristic
339	C₄	C₂×C₄×Dic₉₄	1	1	yes	characteristic
340	C₂	—	2	4	no	—
341	C₂	C₂×C₄×Dic₄₇⋊C₄	1	1	yes	characteristic
342	C₂	#20251810	1	2	yes	—
343	C₂	#20252080	1	2	yes	—
344	C₂	#20252320	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#405	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#405	D₄₇
hypocenter	C₄₇	#20435
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
5. #20243264	C₂	semidirect
6. C₈×Dic₄₇⋊C₄	C₂	semidirect
14. #15689151	C₄	semidirect
16. #15689078	C₄	semidirect
52. Dic₄₇⋊C₈	C₂×C₄	semidirect
53. #13093934	C₂×C₄	semidirect
56. C₈×Dic₄₇	C₂×C₄	semidirect
92. C₂×C₈×C₄₇	C₄⋊C₄	semidirect
205. C₄₇	#20435	semidirect
273. C₂×C₈	Dic₄₇⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689151	2
C₂×#15689078	2
C₂²×C₈×Dic₄₇	2
#20245870	2
#20243264	2
C₈×Dic₄₇⋊C₄	2
#20243085	2
#20243081	2
C₂×C₄₇×#405	2
#20435	47