Subgroups — #26369393

Order 12032.

590 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245195	C₂	1	4	yes	maximal, complemented
2	#20245191	C₂	1	2	yes	maximal, complemented
3	#20245187	C₂	1	2	yes	maximal, complemented
4	#20245144	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245002	C₂	1	2	yes	maximal
7	#20244985	C₂	1	2	yes	maximal, complemented
8	C₄×#131×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#13093964	C₂²	1	2	yes	—
10	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
11	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
14	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
15	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
16	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	4	yes	—
17	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	4	yes	—
18	#15688899	C₂²	1	4	yes	—
19	C₄×C₄₇⋊C₁₆	C₂²	1	4	yes	—
20	C₂×#131×C₄₇	C₄	1	1	yes	characteristic
21	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
22	C₄×#41×C₄₇	C₄	1	1	yes	characteristic
23	C₄₇×#405	C₄	1	2	yes	—
24	C₄₇×#405	C₄	1	1	yes	characteristic
25	C₄₇×#405	C₂²	1	2	yes	—
26	C₄₇×#405	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₄₇	C₄	1	2	yes	—
28	C₄×M₁₆×C₄₇	C₄	1	1	yes	characteristic
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
31	#13093964	—	2	4	no	—
32	C₂×C₄₇⋊C₁₆	—	2	4	no	—
33	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
34	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
35	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
36	C₂×#41×C₄₇	C₂×C₄	1	1	yes	characteristic
37	#131×C₄₇	—	2	4	no	—
38	#131×C₄₇	—	2	2	no	—
39	#131×C₄₇	—	2	2	no	—
40	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
48	C₄×Q₈×C₄₇	C₈	1	1	yes	characteristic
49	C₄×Q₈×C₄₇	C₈	1	1	yes	characteristic
50	C₄×D₄×C₄₇	C₈	1	2	yes	—
51	C₄×D₄×C₄₇	C₈	1	1	yes	characteristic
52	C₄×D₄×C₄₇	C₈	1	2	yes	—
53	C₄×D₄×C₄₇	C₈	1	1	yes	characteristic
54	#117×C₄₇	C₂×C₄	1	2	yes	—
55	#117×C₄₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
58	#97×C₄₇	C₂×C₄	1	2	yes	—
59	#97×C₄₇	C₂×C₄	1	2	yes	—
60	#97×C₄₇	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₄₇	C₂³	1	2	yes	—
66	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	#41×C₄₇	—	2	2	no	—
69	#41×C₄₇	—	2	2	no	—
70	C₂×Q₈×C₄₇	C₂×C₈	1	1	yes	characteristic
71	C₂×D₄×C₄₇	C₂×C₈	1	2	yes	—
72	C₂×D₄×C₄₇	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	M₁₆×C₄₇	—	2	2	no	—
80	M₁₆×C₄₇	—	2	4	no	—
81	M₁₆×C₄₇	—	2	4	no	—
82	M₁₆×C₄₇	M₁₆	1	4	yes	—
83	C₂×C₈×C₄₇	—	2	2	no	—
84	C₂×C₈×C₄₇	—	2	4	no	—
85	C₂×C₈×C₄₇	—	2	2	no	—
86	C₂×C₈×C₄₇	—	2	2	no	—
87	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₄₇	#41	1	2	yes	—
93	C₂×C₈×C₄₇	#41	1	2	yes	—
94	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
96	C₄⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
97	C₄⋊C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
98	C₄⋊C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
99	C₂²⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
100	C₂²⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
101	C₂²⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
102	C₄²×C₄₇	C₂×C₈	1	2	yes	—
103	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
104	C₄²×C₄₇	C₂×C₈	1	2	yes	—
105	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
106	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
108	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
109	C₄₇⋊C₁₆	—	2	2	no	—
110	C₄₇⋊C₁₆	—	2	2	no	—
111	C₄₇⋊C₁₆	—	2	4	no	—
112	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
113	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
114	Q₈×C₄₇	—	2	2	no	—
115	D₄×C₄₇	—	2	4	no	—
116	D₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	4	no	—
121	C₂×C₄×C₄₇	—	4	1	no	—
122	C₂×C₄×C₄₇	—	4	2	no	—
123	C₂×C₄×C₄₇	#117	1	2	yes	—
124	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
129	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
133	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
134	C₈×C₄₇	—	2	2	no	—
135	C₈×C₄₇	—	2	1	no	—
136	C₈×C₄₇	—	2	1	no	—
137	C₈×C₄₇	#117	1	2	yes	—
138	C₈×C₄₇	C₂×M₁₆	1	4	yes	—
139	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
140	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
141	#33423	—	47	1	no	maximal, Sylow
142	C₂²×C₄₇	—	2	1	no	—
143	C₂²×C₄₇	—	2	1	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	2	2	no	—
146	C₂²×C₄₇	—	2	2	no	—
147	C₂²×C₄₇	#431	1	2	yes	—
148	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
149	C₄×C₄₇	—	2	1	no	—
150	C₄×C₄₇	—	2	2	no	—
151	C₄×C₄₇	—	2	1	no	—
152	C₄×C₄₇	—	2	1	no	—
153	C₄×C₄₇	—	2	2	no	—
154	C₄×C₄₇	—	2	2	no	—
155	C₄×C₄₇	#431	1	2	yes	—
156	C₄×C₄₇	#504	1	2	yes	—
157	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
158	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
159	C₄×#131	D₄₇	1	1	yes	characteristic
160	#2164	—	47	4	no	—
161	#2163	—	47	2	no	—
162	#2162	—	47	2	no	—
163	#2150	—	47	1	no	—
164	C₂×C₄⋊C₁₆	—	47	1	no	—
165	#2113	—	47	2	no	—
166	#2108	—	47	2	no	—
167	C₂×C₄₇	—	2	2	no	—
168	C₂×C₄₇	—	4	2	no	—
169	C₂×C₄₇	C₂×#431	1	1	yes	characteristic
170	C₂×C₄₇	C₂×#504	1	1	yes	characteristic
171	C₂×C₄₇	#3407	1	1	yes	characteristic
172	C₂×#131	Dic₄₇	1	1	yes	characteristic
173	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
174	C₄×#41	Dic₄₇	1	1	yes	characteristic
175	C₂×M₃₂	—	47	2	no	—
176	C₂²×C₁₆	—	47	2	no	—
177	#405	Dic₄₇	1	1	yes	characteristic
178	#405	Dic₄₇	1	2	yes	—
179	#405	C₂×D₄₇	1	1	yes	characteristic
180	#405	C₂×D₄₇	1	2	yes	—
181	C₄×M₁₆	Dic₄₇	1	2	yes	—
182	C₄×M₁₆	Dic₄₇	1	1	yes	characteristic
183	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
184	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
185	C₄⋊C₁₆	—	47	1	no	—
186	C₄⋊C₁₆	—	47	1	no	—
187	C₄⋊C₁₆	—	47	2	no	—
188	C₄⋊C₁₆	—	47	2	no	—
189	C₄⋊C₁₆	—	47	2	no	—
190	C₂²⋊C₁₆	—	47	4	no	—
191	C₂²⋊C₁₆	—	47	4	no	—
192	#346	—	47	4	no	—
193	C₄×C₁₆	—	47	4	no	—
194	C₄₇	#33423	1	1	yes	characteristic, complemented, Sylow
195	C₂×#41	C₂×Dic₄₇	1	1	yes	characteristic
196	#131	—	2	4	no	—
197	#131	—	2	2	no	—
198	#131	—	2	2	no	—
199	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
200	C₂×M₁₆	C₂×Dic₄₇	1	1	yes	characteristic
201	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
202	C₂×M₁₆	C₂×Dic₄₇	1	1	yes	characteristic
203	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
204	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
205	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
206	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
207	C₄×Q₈	C₄₇⋊C₈	1	1	yes	characteristic
208	C₄×Q₈	C₄₇⋊C₈	1	1	yes	characteristic
209	C₄×D₄	C₄₇⋊C₈	1	2	yes	—
210	C₄×D₄	C₄₇⋊C₈	1	1	yes	characteristic
211	C₄×D₄	C₄₇⋊C₈	1	2	yes	—
212	C₄×D₄	C₄₇⋊C₈	1	1	yes	characteristic
213	#117	C₂×Dic₄₇	1	2	yes	—
214	#117	C₂×Dic₄₇	1	1	yes	characteristic
215	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
216	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
217	M₃₂	—	94	4	no	—
218	C₂×C₁₆	—	47	2	no	—
219	C₂×C₁₆	—	47	4	no	—
220	C₂×C₁₆	—	47	2	no	—
221	C₂×C₁₆	—	94	4	no	—
222	#97	C₂×Dic₄₇	1	2	yes	—
223	#97	C₂×Dic₄₇	1	2	yes	—
224	#97	C₂×Dic₄₇	1	2	yes	—
225	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
226	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
227	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
228	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₄₇	1	2	yes	—
230	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
232	#41	—	2	2	no	—
233	#41	—	2	2	no	—
234	C₂×Q₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
235	C₂×D₄	C₂×C₄₇⋊C₈	1	2	yes	—
236	C₂×D₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
238	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
240	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
241	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
242	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
243	M₁₆	—	2	2	no	—
244	M₁₆	—	2	4	no	—
245	M₁₆	—	2	4	no	—
246	M₁₆	#10625329	1	4	yes	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	#10625350	1	2	yes	—
252	C₂×C₈	#10625350	1	2	yes	—
253	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
254	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
255	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
256	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
257	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
258	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
259	C₄⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
260	C₄⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
261	C₄⋊C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
262	C₄⋊C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
263	C₂²⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
264	C₂²⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
265	C₂²⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
266	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
267	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
268	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
269	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
270	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
271	C₄²	C₂²×Dic₄₇	1	2	yes	—
272	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
273	C₁₆	—	94	2	no	—
274	C₁₆	—	94	2	no	—
275	C₁₆	—	94	4	no	—
276	C₂³	C₂²×C₄₇⋊C₈	1	2	yes	—
277	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
278	Q₈	—	2	2	no	—
279	D₄	—	2	4	no	—
280	D₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	#13093983	1	2	yes	—
288	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
290	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
291	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
293	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
294	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
295	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
296	C₂×C₄	C₂×#10625329	1	2	yes	—
297	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	#13093983	1	2	yes	—
302	C₈	C₂×#10625329	1	4	yes	—
303	C₈	C₂×#10625350	1	1	yes	characteristic
304	C₈	C₂×#10625350	1	1	yes	characteristic
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	#15689176	1	2	yes	—
311	C₂²	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	2	no	—
318	C₄	#15689176	1	2	yes	—
319	C₄	#15689540	1	2	yes	—
320	C₄	C₂×#13093983	1	1	yes	characteristic
321	C₄	C₂²×#10625329	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	4	2	no	—
324	C₂	C₂×#15689176	1	1	yes	characteristic
325	C₂	C₂×#15689540	1	1	yes	characteristic
326	C₂	#20256491	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₈
Fitting	C₄×#131×C₄₇	C₂
prime radical	C₂×#41×C₄₇	C₂×C₄
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₄×#131	D₄₇
hypocenter	C₄₇	#33423
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. #20245195	C₂	semidirect
2. #20245191	C₂	semidirect
3. #20245187	C₂	semidirect
4. #20245144	C₂	semidirect
5. C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	semidirect
7. #20244985	C₂	semidirect
194. C₄₇	#33423	semidirect

Maximal subgroups

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

subgroup	index
#20245195	2
#20245191	2
#20245187	2
#20245144	2
C₂×C₄⋊(C₄₇⋊C₁₆)	2
#20245002	2
#20244985	2
C₄×#131×C₄₇	2
#33423	47