Subgroups — #26369963

Order 12032.

536 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245297	C₂	1	2	yes	maximal, complemented
2	#20245214	C₂	1	4	yes	maximal, complemented
3	D₄×C₄₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	#20245191	C₂	1	2	yes	maximal, complemented
5	#20245187	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245144	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	2	yes	maximal, complemented
8	C₈×#41×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#13093964	C₂²	1	2	yes	—
10	C₂²×C₄₇⋊C₁₆	C₂²	1	4	yes	—
11	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	4	yes	—
12	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
13	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
15	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
16	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
17	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	4	yes	—
18	#15688899	C₂²	1	4	yes	—
19	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
20	C₄×#41×C₄₇	C₄	1	1	yes	characteristic
21	C₈×Q₈×C₄₇	C₄	1	1	yes	characteristic
22	C₈×Q₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₈×D₄×C₄₇	C₄	1	2	yes	—
24	C₈×D₄×C₄₇	C₄	1	1	yes	characteristic
25	C₈×D₄×C₄₇	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₄₇	C₂²	1	2	yes	—
27	C₄₇×#431	C₄	1	1	yes	characteristic
28	C₄₇×#431	C₄	1	2	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	#13093964	—	4	2	no	—
32	C₂×C₄₇⋊C₁₆	—	4	4	no	—
33	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
34	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
35	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
36	C₂×#41×C₄₇	C₈	1	1	yes	characteristic
37	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
40	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
41	C₄×Q₈×C₄₇	C₈	1	1	yes	characteristic
42	C₄×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₄×D₄×C₄₇	C₈	1	2	yes	—
44	C₄×D₄×C₄₇	C₈	1	1	yes	characteristic
45	C₄×D₄×C₄₇	C₂×C₄	1	2	yes	—
46	C₄×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
47	#117×C₄₇	C₈	1	1	yes	characteristic
48	#117×C₄₇	C₈	1	2	yes	—
49	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
52	C₄⋊C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₄⋊C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
54	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
55	C₂²⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
56	C₂²⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
57	C₂²⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₄₇	C₂³	1	2	yes	—
64	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
65	#41×C₄₇	—	2	4	no	—
66	C₂×Q₈×C₄₇	C₂×C₈	1	1	yes	characteristic
67	C₂×D₄×C₄₇	C₂×C₈	1	2	yes	—
68	C₂×D₄×C₄₇	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	—	2	1	no	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₂×C₈×C₄₇	—	2	1	no	—
79	C₂×C₈×C₄₇	—	4	2	no	—
80	C₂×C₈×C₄₇	—	4	1	no	—
81	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
88	C₄⋊C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
89	C₄⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
90	C₄⋊C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
91	C₂²⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
92	C₂²⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
93	C₂²⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
94	C₄²×C₄₇	C₂×C₈	1	2	yes	—
95	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
96	C₄²×C₄₇	C₂×C₈	1	2	yes	—
97	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
98	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
99	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
101	C₄₇⋊C₁₆	—	2	2	no	—
102	C₄₇⋊C₁₆	—	2	4	no	—
103	C₄₇⋊C₁₆	—	2	2	no	—
104	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
105	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
106	Q₈×C₄₇	—	2	2	no	—
107	D₄×C₄₇	—	2	4	no	—
108	D₄×C₄₇	M₃₂	1	4	yes	complemented
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
121	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
126	C₈×C₄₇	—	2	2	no	—
127	C₈×C₄₇	—	2	1	no	—
128	C₈×C₄₇	—	2	1	no	—
129	C₈×C₄₇	—	2	1	no	—
130	C₈×C₄₇	—	2	2	no	—
131	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
132	C₈×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
133	#33504	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	2	1	no	—
137	C₂²×C₄₇	—	2	1	no	—
138	C₂²×C₄₇	—	2	1	no	—
139	C₂²×C₄₇	C₂×M₃₂	1	4	yes	—
140	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
141	C₄×C₄₇	—	2	1	no	—
142	C₄×C₄₇	—	2	2	no	—
143	C₄×C₄₇	—	2	2	no	—
144	C₄×C₄₇	—	2	1	no	—
145	C₄×C₄₇	—	2	2	no	—
146	C₄×C₄₇	—	2	1	no	—
147	C₄×C₄₇	C₂×M₃₂	1	2	yes	—
148	C₄×C₄₇	#504	1	2	yes	—
149	C₄×C₄₇	#520	1	1	yes	characteristic
150	C₄×C₄₇	#568	1	1	yes	characteristic
151	C₈×#41	D₄₇	1	1	yes	characteristic
152	#2183	—	47	2	no	—
153	#2168	—	47	4	no	—
154	D₄×C₁₆	—	47	2	no	—
155	#2163	—	47	2	no	—
156	#2162	—	47	1	no	—
157	#2150	—	47	1	no	—
158	C₂×C₄⋊C₁₆	—	47	2	no	—
159	C₂×C₄₇	—	2	4	no	—
160	C₂×C₄₇	—	4	1	no	—
161	C₂×C₄₇	#2988	1	1	yes	characteristic
162	C₂×C₄₇	C₂²×M₃₂	1	1	yes	characteristic
163	C₂×C₄₇	C₂×#504	1	1	yes	characteristic
164	C₄×#41	Dic₄₇	1	1	yes	characteristic
165	C₂×M₃₂	—	47	2	no	—
166	C₂²×C₁₆	—	47	4	no	—
167	C₈×Q₈	Dic₄₇	1	1	yes	characteristic
168	C₈×Q₈	C₂×D₄₇	1	1	yes	characteristic
169	C₈×D₄	Dic₄₇	1	2	yes	—
170	C₈×D₄	Dic₄₇	1	1	yes	characteristic
171	C₈×D₄	C₂×D₄₇	1	1	yes	characteristic
172	C₈×D₄	C₂×D₄₇	1	2	yes	—
173	#431	Dic₄₇	1	1	yes	characteristic
174	#431	Dic₄₇	1	2	yes	—
175	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
176	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
177	C₄⋊C₁₆	—	47	1	no	—
178	C₄⋊C₁₆	—	47	2	no	—
179	C₄⋊C₁₆	—	47	4	no	—
180	C₄⋊C₁₆	—	47	2	no	—
181	C₄⋊C₁₆	—	47	1	no	—
182	C₂²⋊C₁₆	—	47	2	no	—
183	C₂²⋊C₁₆	—	47	4	no	—
184	#346	—	47	4	no	—
185	C₄×C₁₆	—	47	2	no	—
186	C₄₇	#33504	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	C₄₇⋊C₈	1	1	yes	characteristic
188	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
189	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
190	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
191	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
192	C₄×Q₈	C₄₇⋊C₈	1	1	yes	characteristic
193	C₄×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
194	C₄×D₄	C₄₇⋊C₈	1	2	yes	—
195	C₄×D₄	C₄₇⋊C₈	1	1	yes	characteristic
196	C₄×D₄	C₂×Dic₄₇	1	2	yes	—
197	C₄×D₄	C₂×Dic₄₇	1	1	yes	characteristic
198	#117	C₄₇⋊C₈	1	1	yes	characteristic
199	#117	C₄₇⋊C₈	1	2	yes	—
200	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
201	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
202	M₃₂	—	188	2	no	—
203	C₂×C₁₆	—	47	2	no	—
204	C₂×C₁₆	—	47	2	no	—
205	C₂×C₁₆	—	47	4	no	—
206	C₂×C₁₆	—	188	4	no	—
207	C₄⋊C₈	C₂×Dic₄₇	1	2	yes	—
208	C₄⋊C₈	C₂×Dic₄₇	1	1	yes	characteristic
209	C₄⋊C₈	C₂×Dic₄₇	1	1	yes	characteristic
210	C₄⋊C₈	C₂×Dic₄₇	1	2	yes	—
211	C₂²⋊C₈	C₂×Dic₄₇	1	2	yes	—
212	C₂²⋊C₈	C₂×Dic₄₇	1	2	yes	—
213	C₂²⋊C₈	C₂×Dic₄₇	1	2	yes	—
214	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
215	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
216	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
217	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₄₇	1	2	yes	—
220	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
221	#41	—	2	4	no	—
222	C₂×Q₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
223	C₂×D₄	C₂×C₄₇⋊C₈	1	2	yes	—
224	C₂×D₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
226	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
229	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
230	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
238	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
239	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
240	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
241	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
242	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
243	C₄⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
244	C₄⋊C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
245	C₄⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
246	C₄⋊C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
247	C₂²⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
248	C₂²⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
249	C₂²⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
250	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
251	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
252	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
253	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
254	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
255	C₄²	C₂²×Dic₄₇	1	2	yes	—
256	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
257	C₁₆	—	94	2	no	—
258	C₁₆	—	94	4	no	—
259	C₁₆	—	94	2	no	—
260	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
261	C₂³	C₂²×C₄₇⋊C₈	1	2	yes	—
262	Q₈	—	2	2	no	—
263	D₄	—	2	4	no	—
264	D₄	#13093964	1	4	yes	complemented
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
275	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
277	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
278	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
279	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
281	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	2	no	—
287	C₈	#13094070	1	1	yes	characteristic
288	C₈	#13094071	1	1	yes	characteristic
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	C₂×#13093964	1	4	yes	—
295	C₂²	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	1	no	—
302	C₄	C₂×#13093964	1	2	yes	—
303	C₄	#15689540	1	2	yes	—
304	C₄	#15689606	1	1	yes	characteristic
305	C₄	#15689890	1	1	yes	characteristic
306	C₂	—	2	4	no	—
307	C₂	—	4	1	no	—
308	C₂	#20252499	1	1	yes	characteristic
309	C₂	C₂²×#13093964	1	1	yes	characteristic
310	C₂	C₂×#15689540	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₈
Fitting	C₈×#41×C₄₇	C₂
prime radical	C₂×#41×C₄₇	C₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₈×#41	D₄₇
hypocenter	C₄₇	#33504
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. #20245297	C₂	semidirect
2. #20245214	C₂	semidirect
3. D₄×C₄₇⋊C₁₆	C₂	semidirect
4. #20245191	C₂	semidirect
5. #20245187	C₂	semidirect
6. #20245144	C₂	semidirect
7. C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	semidirect
108. D₄×C₄₇	M₃₂	semidirect
186. C₄₇	#33504	semidirect
264. D₄	#13093964	semidirect

Maximal subgroups

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

subgroup	index
#20245297	2
#20245214	2
D₄×C₄₇⋊C₁₆	2
#20245191	2
#20245187	2
#20245144	2
C₂×C₄⋊(C₄₇⋊C₁₆)	2
C₈×#41×C₄₇	2
#33504	47