Subgroups — #26216767

Order 12032.

1110 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×#15689180	C₂	1	2	yes	maximal
2	#20243540	C₂	1	2	yes	maximal, complemented
3	#20243505	C₂	1	2	yes	maximal, complemented
4	#20243368	C₂	1	4	yes	maximal
5	#20243241	C₂	1	4	yes	maximal, complemented
6	C₂×C₄³×C₄₇	C₂	1	1	yes	maximal, characteristic
7	#15689180	—	2	8	no	—
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
10	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
11	C₂×#13093935	C₂²	1	4	yes	—
12	#15688878	C₂²	1	8	yes	—
13	C₂²×C₄²×C₄₇	C₄	1	2	yes	—
14	C₂²×C₄²×C₄₇	C₄	1	1	yes	characteristic
15	C₂²×C₄²×C₄₇	C₄	1	1	yes	characteristic
16	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄³×C₄₇	C₄	1	1	yes	characteristic
19	C₄³×C₄₇	C₄	1	2	yes	—
20	C₄³×C₄₇	C₄	1	1	yes	characteristic
21	C₄³×C₄₇	C₂²	1	2	yes	—
22	C₄³×C₄₇	C₂²	1	1	yes	characteristic
23	C₄³×C₄₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄₇⋊C₈	C₂³	1	8	yes	—
25	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
26	C₄⋊(C₄₇⋊C₈)	—	2	16	no	—
27	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
28	#13093935	—	2	8	no	—
29	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	—	2	8	no	—
35	C₂×C₄²×C₄₇	—	2	4	no	—
36	C₂×C₄²×C₄₇	—	2	4	no	—
37	C₂×C₄²×C₄₇	—	2	8	no	—
38	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	D₄	1	4	yes	—
53	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
58	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	—	2	4	no	—
60	C₂²×C₄×C₄₇	—	2	4	no	—
61	C₂²×C₄×C₄₇	—	2	4	no	—
62	C₂²×C₄×C₄₇	—	2	4	no	—
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	—	2	4	no	—
67	C₂²×C₄×C₄₇	—	2	4	no	—
68	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
70	C₂²×C₄×C₄₇	M₁₆	1	8	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	2	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
83	C₄²×C₄₇	—	2	4	no	—
84	C₄²×C₄₇	—	2	4	no	—
85	C₄²×C₄₇	—	2	4	no	—
86	C₄²×C₄₇	—	2	4	no	—
87	C₄²×C₄₇	—	2	4	no	—
88	C₄²×C₄₇	—	2	2	no	—
89	C₄²×C₄₇	—	2	2	no	—
90	C₄²×C₄₇	—	2	4	no	—
91	C₄²×C₄₇	—	2	4	no	—
92	C₄²×C₄₇	—	2	4	no	—
93	C₄²×C₄₇	—	2	4	no	—
94	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
95	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
96	C₄²×C₄₇	M₁₆	1	4	yes	—
97	C₄²×C₄₇	M₁₆	1	4	yes	—
98	C₄²×C₄₇	C₂×D₄	1	2	yes	—
99	C₄²×C₄₇	C₂×D₄	1	2	yes	—
100	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
101	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
102	C₄²×C₄₇	#41	1	8	yes	—
103	C₂×C₄₇⋊C₈	—	2	8	no	—
104	C₂×C₄₇⋊C₈	—	2	16	no	—
105	C₂³×C₄₇	—	2	2	no	—
106	C₂³×C₄₇	—	2	1	no	—
107	C₂³×C₄₇	—	2	1	no	—
108	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
109	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
110	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	—	2	16	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	2	8	no	—
124	C₂×C₄×C₄₇	—	2	4	no	—
125	C₂×C₄×C₄₇	—	2	4	no	—
126	C₂×C₄×C₄₇	—	2	8	no	—
127	C₂×C₄×C₄₇	—	2	16	no	—
128	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×C₄×C₄₇	#117	1	4	yes	—
134	C₂×C₄×C₄₇	#117	1	4	yes	—
135	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
136	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
137	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
138	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
139	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
140	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
141	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
144	C₄₇⋊C₈	—	4	8	no	—
145	#20810	—	47	1	no	maximal, Sylow
146	C₂²×C₄₇	—	2	2	no	—
147	C₂²×C₄₇	—	2	4	no	—
148	C₂²×C₄₇	—	2	4	no	—
149	C₂²×C₄₇	—	2	2	no	—
150	C₂²×C₄₇	#423	1	4	yes	—
151	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₂²×C₄₇	C₂×#117	1	2	yes	—
153	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
154	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
155	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	2	2	no	—
158	C₄×C₄₇	—	2	2	no	—
159	C₄×C₄₇	—	2	4	no	—
160	C₄×C₄₇	—	2	4	no	—
161	C₄×C₄₇	—	2	4	no	—
162	C₄×C₄₇	—	2	2	no	—
163	C₄×C₄₇	#423	1	4	yes	—
164	C₄×C₄₇	#432	1	8	yes	complemented
165	C₄×C₄₇	#532	1	2	yes	—
166	C₄×C₄₇	#533	1	2	yes	—
167	C₂×#432	—	47	2	no	—
168	C₂×C₄³	D₄₇	1	1	yes	characteristic
169	#1842	—	47	2	no	—
170	#1831	—	47	2	no	—
171	#1797	—	47	4	no	—
172	#1767	—	47	4	no	—
173	C₂×C₄₇	—	2	4	no	—
174	C₂×C₄₇	#2285	1	1	yes	characteristic
175	C₂×C₄₇	C₂×#423	1	1	yes	characteristic
176	C₂×C₄₇	C₂×#423	1	1	yes	characteristic
177	C₂×C₄₇	C₂×#432	1	2	yes	—
178	C₂×C₄₇	#2920	1	2	yes	—
179	C₂²×C₄²	Dic₄₇	1	2	yes	—
180	C₂²×C₄²	Dic₄₇	1	1	yes	characteristic
181	C₂²×C₄²	Dic₄₇	1	1	yes	characteristic
182	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
183	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
184	#432	—	94	8	no	—
185	C₂×C₄⋊C₈	—	47	8	no	—
186	C₂×C₄⋊C₈	—	47	4	no	—
187	C₂×C₂²⋊C₈	—	47	4	no	—
188	C₂×#97	—	47	4	no	—
189	C₄³	Dic₄₇	1	1	yes	characteristic
190	C₄³	Dic₄₇	1	2	yes	—
191	C₄³	Dic₄₇	1	1	yes	characteristic
192	C₄³	C₂×D₄₇	1	1	yes	characteristic
193	C₄³	C₂×D₄₇	1	1	yes	characteristic
194	C₄³	C₂×D₄₇	1	2	yes	—
195	#336	—	47	8	no	—
196	C₄₇	#20810	1	1	yes	characteristic, complemented, Sylow
197	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
198	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
199	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
200	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
201	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
202	C₂²×C₈	—	47	8	no	—
203	C₂×C₄²	—	2	8	no	—
204	C₂×C₄²	—	2	4	no	—
205	C₂×C₄²	—	2	4	no	—
206	C₂×C₄²	—	2	8	no	—
207	C₂×C₄²	Dic₉₄	1	4	yes	—
208	C₂×C₄²	D₁₈₈	1	4	yes	—
209	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
210	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
211	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
212	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
213	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
214	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
215	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
216	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
217	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
218	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
219	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
220	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
221	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
222	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
223	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
224	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
225	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
226	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
227	C₄⋊C₈	—	94	8	no	—
228	C₄⋊C₈	—	94	16	no	—
229	C₂²⋊C₈	—	94	8	no	—
230	#97	—	94	8	no	—
231	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
232	C₂²×C₄	—	2	4	no	—
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	—	2	2	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	#10625329	1	8	yes	—
242	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
243	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
244	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
245	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
246	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
247	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
248	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
249	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
251	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
252	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
253	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
255	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
256	C₂×C₈	—	94	8	no	—
257	C₂×C₈	—	94	16	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	2	4	no	—
260	C₄²	—	2	4	no	—
261	C₄²	—	2	4	no	—
262	C₄²	—	2	4	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	2	4	no	—
267	C₄²	—	2	4	no	—
268	C₄²	—	2	4	no	—
269	C₄²	#10625329	1	4	yes	—
270	C₄²	#10625329	1	4	yes	—
271	C₄²	C₄⋊Dic₄₇	1	4	yes	—
272	C₄²	C₄⋊Dic₄₇	1	4	yes	—
273	C₄²	C₂×Dic₉₄	1	2	yes	—
274	C₄²	C₂×Dic₉₄	1	2	yes	—
275	C₄²	C₂×D₁₈₈	1	2	yes	—
276	C₄²	C₂×D₁₈₈	1	2	yes	—
277	C₄²	#10625350	1	8	yes	—
278	C₂³	—	2	2	no	—
279	C₂³	—	2	1	no	—
280	C₂³	—	2	1	no	—
281	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
282	C₂³	C₂×#10625329	1	4	yes	—
283	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
284	C₂×C₄	—	2	16	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	2	8	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	16	no	—
301	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
302	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
303	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
304	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
305	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
306	C₂×C₄	#13093983	1	4	yes	—
307	C₂×C₄	#13093983	1	4	yes	—
308	C₂×C₄	C₂×#10625329	1	4	yes	—
309	C₂×C₄	C₂×#10625329	1	4	yes	—
310	C₂×C₄	C₂×#10625329	1	2	yes	—
311	C₂×C₄	C₂×#10625329	1	2	yes	—
312	C₂×C₄	C₂×#10625329	1	4	yes	—
313	C₂×C₄	C₂×#10625329	1	4	yes	—
314	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
315	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
316	C₂×C₄	C₂×#10625350	1	4	yes	—
317	C₈	—	188	8	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	4	no	—
321	C₂²	—	2	4	no	—
322	C₂²	#15689147	1	4	yes	—
323	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
324	C₂²	C₂×#13093983	1	2	yes	—
325	C₂²	C₂²×#10625329	1	1	yes	characteristic
326	C₂²	C₂²×#10625329	1	2	yes	—
327	C₂²	C₂²×#10625329	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	4	no	—
333	C₄	—	2	4	no	—
334	C₄	—	2	4	no	—
335	C₄	#15689147	1	4	yes	—
336	C₄	#15689180	1	8	yes	complemented
337	C₄	#15689673	1	2	yes	—
338	C₄	#15689680	1	2	yes	—
339	C₂	—	2	4	no	—
340	C₂	#20245795	1	1	yes	characteristic
341	C₂	C₂×#15689147	1	1	yes	characteristic
342	C₂	C₂×#15689147	1	1	yes	characteristic
343	C₂	C₂×#15689180	1	2	yes	—
344	C₂	#20252021	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₄³×C₄₇	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄³	D₄₇
hypocenter	C₄₇	#20810
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #20243540	C₂	semidirect
3. #20243505	C₂	semidirect
5. #20243241	C₂	semidirect
164. C₄×C₄₇	#432	semidirect
196. C₄₇	#20810	semidirect
336. C₄	#15689180	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
#20243540	2
#20243505	2
#20243368	2
#20243241	2
C₂×C₄³×C₄₇	2
#20810	47