Subgroups — #26370008

Order 12032.

484 conjugacy classes of subgroups.

The subgroup lattice has 326 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	#20245220	C₂	1	2	yes	maximal, complemented
3	#20245214	C₂	1	2	yes	maximal, complemented
4	#20245214	C₂	1	2	yes	maximal, complemented
5	#20245195	C₂	1	2	yes	maximal, complemented
6	#20245191	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245147	C₂	1	2	yes	maximal, complemented
8	#20245144	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#2966	C₂	1	1	yes	maximal, characteristic
10	C₂×#13093964	C₂²	1	1	yes	characteristic
11	C₂×#13093964	C₂²	1	1	yes	characteristic
12	C₂×#13093964	C₂²	1	2	yes	—
13	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
14	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
15	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
16	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
17	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
18	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
19	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
20	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
21	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
22	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
23	#15688899	C₂²	1	2	yes	—
24	#15688899	C₂²	1	2	yes	—
25	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
26	C₄₇×#529	C₄	1	1	yes	characteristic
27	C₄₇×#446	C₄	1	1	yes	characteristic
28	C₈×Q₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#436	C₄	1	1	yes	characteristic
30	C₄₇×#435	C₄	1	2	yes	—
31	C₈×D₄×C₄₇	C₂²	1	1	yes	characteristic
32	C₈×D₄×C₄₇	C₂²	1	2	yes	—
33	C₄₇×#433	C₄	1	2	yes	—
34	C₄₇×#432	C₄	1	1	yes	characteristic
35	C₄₇×#405	C₂²	1	2	yes	—
36	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
37	#13093964	—	4	1	no	—
38	#13093964	—	4	2	no	—
39	#13093964	—	4	1	no	—
40	C₂×C₄₇⋊C₁₆	—	4	2	no	—
41	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
42	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
43	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
44	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
45	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
49	#126×C₄₇	C₈	1	2	yes	—
50	#125×C₄₇	C₈	1	1	yes	characteristic
51	#124×C₄₇	C₈	1	1	yes	characteristic
52	#123×C₄₇	C₈	1	2	yes	—
53	C₂²⋊Q₈×C₄₇	C₈	1	1	yes	characteristic
54	#121×C₄₇	C₈	1	1	yes	characteristic
55	C₄×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
56	C₄×D₄×C₄₇	C₂×C₄	1	2	yes	—
57	C₄×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
58	#117×C₄₇	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
60	C₄⋊C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
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	C₂²⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
65	C₂²⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
66	C₂²⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
67	#97×C₄₇	C₂×C₄	1	2	yes	—
68	#97×C₄₇	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₄₇	C₂³	1	2	yes	—
72	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
74	C₂×Q₈×C₄₇	C₂×C₈	1	1	yes	characteristic
75	C₂×D₄×C₄₇	C₂×C₈	1	2	yes	—
76	C₂×D₄×C₄₇	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
81	M₁₆×C₄₇	—	4	2	no	—
82	C₂×C₈×C₄₇	—	2	2	no	—
83	C₂×C₈×C₄₇	—	2	1	no	—
84	C₂×C₈×C₄₇	—	2	2	no	—
85	C₂×C₈×C₄₇	—	2	1	no	—
86	C₂×C₈×C₄₇	—	4	1	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	2	yes	—
91	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
94	C₄⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
95	C₄⋊C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
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	2	yes	—
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	C₂²⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
103	C₂²⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
104	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
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	2	no	—
112	C₄₇⋊C₁₆	—	2	2	no	—
113	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
114	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
115	Q₈×C₄₇	—	4	1	no	—
116	D₄×C₄₇	—	4	1	no	—
117	D₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	—	2	1	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	4	1	no	—
123	C₂×C₄×C₄₇	—	4	1	no	—
124	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
132	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
133	C₈×C₄₇	—	2	2	no	—
134	C₈×C₄₇	—	2	1	no	—
135	C₈×C₄₇	—	2	1	no	—
136	C₈×C₄₇	—	2	1	no	—
137	C₈×C₄₇	—	2	2	no	—
138	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
139	C₈×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
140	#33509	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	—	2	1	no	—
142	C₂²×C₄₇	—	2	1	no	—
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	2	1	no	—
146	C₂²×C₄₇	—	2	2	no	—
147	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
148	C₄×C₄₇	—	2	1	no	—
149	C₄×C₄₇	—	2	1	no	—
150	C₄×C₄₇	—	2	1	no	—
151	C₄×C₄₇	—	2	1	no	—
152	C₄×C₄₇	—	2	2	no	—
153	C₄×C₄₇	—	2	2	no	—
154	C₄×C₄₇	—	2	1	no	—
155	C₄×C₄₇	—	2	2	no	—
156	C₄×C₄₇	#520	1	1	yes	characteristic
157	C₄×C₄₇	#568	1	1	yes	characteristic
158	#2966	D₄₇	1	1	yes	characteristic
159	#2183	—	47	2	no	—
160	#2169	—	47	2	no	—
161	#2168	—	47	2	no	—
162	#2168	—	47	2	no	—
163	#2164	—	47	2	no	—
164	#2163	—	47	1	no	—
165	#2151	—	47	2	no	—
166	#2150	—	47	1	no	—
167	C₂×C₄₇	—	4	1	no	—
168	C₂×C₄₇	—	4	2	no	—
169	C₂×C₄₇	#2988	1	1	yes	characteristic
170	C₂×C₄₇	#3407	1	1	yes	characteristic
171	C₂×C₄₇	#3407	1	1	yes	characteristic
172	#529	Dic₄₇	1	1	yes	characteristic
173	C₂×M₃₂	—	47	1	no	—
174	C₂×M₃₂	—	47	1	no	—
175	C₂×M₃₂	—	47	2	no	—
176	C₂²×C₁₆	—	47	2	no	—
177	#446	Dic₄₇	1	1	yes	characteristic
178	C₈×Q₈	C₂×D₄₇	1	1	yes	characteristic
179	#436	Dic₄₇	1	1	yes	characteristic
180	#435	Dic₄₇	1	2	yes	—
181	C₈×D₄	C₂×D₄₇	1	1	yes	characteristic
182	C₈×D₄	C₂×D₄₇	1	2	yes	—
183	#433	Dic₄₇	1	2	yes	—
184	#432	Dic₄₇	1	1	yes	characteristic
185	#405	C₂×D₄₇	1	2	yes	—
186	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
187	C₄⋊C₁₆	—	47	2	no	—
188	C₄⋊C₁₆	—	47	2	no	—
189	C₄⋊C₁₆	—	47	1	no	—
190	C₄⋊C₁₆	—	47	2	no	—
191	C₄⋊C₁₆	—	47	2	no	—
192	C₄⋊C₁₆	—	47	1	no	—
193	C₂²⋊C₁₆	—	47	2	no	—
194	C₂²⋊C₁₆	—	47	2	no	—
195	C₂²⋊C₁₆	—	47	2	no	—
196	#346	—	47	2	no	—
197	#346	—	47	2	no	—
198	C₄×C₁₆	—	47	2	no	—
199	C₄₇	#33509	1	1	yes	characteristic, complemented, Sylow
200	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
201	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
204	#126	C₄₇⋊C₈	1	2	yes	—
205	#125	C₄₇⋊C₈	1	1	yes	characteristic
206	#124	C₄₇⋊C₈	1	1	yes	characteristic
207	#123	C₄₇⋊C₈	1	2	yes	—
208	C₂²⋊Q₈	C₄₇⋊C₈	1	1	yes	characteristic
209	#121	C₄₇⋊C₈	1	1	yes	characteristic
210	C₄×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
211	C₄×D₄	C₂×Dic₄₇	1	2	yes	—
212	C₄×D₄	C₂×Dic₄₇	1	1	yes	characteristic
213	#117	C₂×Dic₄₇	1	2	yes	—
214	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
215	M₃₂	—	188	1	no	—
216	M₃₂	—	188	2	no	—
217	M₃₂	—	188	1	no	—
218	C₂×C₁₆	—	47	2	no	—
219	C₂×C₁₆	—	47	2	no	—
220	C₂×C₁₆	—	47	2	no	—
221	C₂×C₁₆	—	47	2	no	—
222	C₂×C₁₆	—	188	2	no	—
223	C₄⋊C₈	C₂×Dic₄₇	1	1	yes	characteristic
224	C₄⋊C₈	C₂×Dic₄₇	1	1	yes	characteristic
225	C₄⋊C₈	C₂×Dic₄₇	1	2	yes	—
226	C₄⋊C₈	C₂×Dic₄₇	1	2	yes	—
227	C₂²⋊C₈	C₂×Dic₄₇	1	2	yes	—
228	C₂²⋊C₈	C₂×Dic₄₇	1	2	yes	—
229	C₂²⋊C₈	C₂×Dic₄₇	1	2	yes	—
230	#97	C₂×Dic₄₇	1	2	yes	—
231	#97	C₂×Dic₄₇	1	2	yes	—
232	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
233	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
234	C₄×C₈	C₂²×D₄₇	1	2	yes	—
235	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
237	C₂×Q₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
238	C₂×D₄	C₂×C₄₇⋊C₈	1	2	yes	—
239	C₂×D₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
242	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
243	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
244	M₁₆	—	4	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
251	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
252	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
253	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
254	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
255	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
256	C₄⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
257	C₄⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
258	C₄⋊C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
259	C₄⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
260	C₄⋊C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
261	C₄⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
262	C₂²⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
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	C₂²⋊C₄	C₂×C₄₇⋊C₈	1	2	yes	—
267	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
268	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
269	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
270	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
271	C₄²	C₂²×Dic₄₇	1	2	yes	—
272	C₁₆	—	94	2	no	—
273	C₁₆	—	94	2	no	—
274	C₁₆	—	94	2	no	—
275	C₁₆	—	94	2	no	—
276	C₂³	C₂²×C₄₇⋊C₈	1	2	yes	—
277	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
278	Q₈	—	4	1	no	—
279	D₄	—	4	1	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
288	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
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	2	yes	—
294	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
295	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
296	C₈	—	2	2	no	—
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	2	no	—
301	C₈	#13094070	1	1	yes	characteristic
302	C₈	#13094071	1	1	yes	characteristic
303	C₂²	—	2	1	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	#15689606	1	1	yes	characteristic
319	C₄	#15689890	1	1	yes	characteristic
320	C₂	—	4	1	no	—
321	C₂	—	4	2	no	—
322	C₂	#20252499	1	1	yes	characteristic
323	C₂	#20256491	1	1	yes	characteristic
324	C₂	#20256491	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₈
Fitting	C₄₇×#2966	C₂
prime radical	#121×C₄₇	C₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2966	D₄₇
hypocenter	C₄₇	#33509
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ #2966

Complements

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

N	H ≅ G/N	type
1. #20245297	C₂	semidirect
2. #20245220	C₂	semidirect
3. #20245214	C₂	semidirect
4. #20245214	C₂	semidirect
5. #20245195	C₂	semidirect
6. #20245191	C₂	semidirect
7. #20245147	C₂	semidirect
8. #20245144	C₂	semidirect
199. C₄₇	#33509	semidirect

Maximal subgroups

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

subgroup	index
#20245297	2
#20245220	2
#20245214	2
#20245195	2
#20245191	2
#20245147	2
#20245144	2
C₄₇×#2966	2
#33509	47