Subgroups — #26226750

Order 12032.

910 conjugacy classes of subgroups.

The subgroup lattice has 310 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	1	yes	maximal, characteristic
2	C₂²×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#20244112	C₂	1	4	yes	maximal, complemented
4	#20243568	C₂	1	4	yes	maximal
5	#20243540	C₂	1	2	yes	maximal, complemented
6	#20243505	C₂	1	2	yes	maximal, complemented
7	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#15689180	—	2	4	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
16	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
17	C₂×#13093935	C₂²	1	2	yes	—
18	#15688878	C₂²	1	8	yes	—
19	C₂×#117×C₄₇	C₄	1	2	yes	—
20	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
22	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
23	C₄×C₂²⋊C₄×C₄₇	C₄	1	1	yes	characteristic
24	C₄×C₂²⋊C₄×C₄₇	C₄	1	1	yes	characteristic
25	C₄×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
26	C₄₇×#376	C₄	1	2	yes	—
27	C₄³×C₄₇	C₂²	1	2	yes	—
28	C₂²×C₄₇⋊C₈	—	2	4	no	—
29	C₂²×C₄₇⋊C₈	—	4	4	no	—
30	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
32	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
33	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
34	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
35	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
36	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
37	C₄⋊(C₄₇⋊C₈)	—	4	8	no	—
38	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
39	#13093935	—	2	4	no	—
40	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	#117×C₄₇	—	2	8	no	—
44	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
45	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
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	4	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	#95×C₄₇	C₂×C₄	1	2	yes	—
58	#95×C₄₇	C₂×C₄	1	2	yes	—
59	#95×C₄₇	C₂×C₄	1	4	yes	—
60	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	—	2	4	no	—
62	C₂²×C₄×C₄₇	—	2	4	no	—
63	C₂²×C₄×C₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
66	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
77	C₄⋊C₄×C₄₇	—	2	8	no	—
78	C₂²⋊C₄×C₄₇	—	2	4	no	—
79	C₂²⋊C₄×C₄₇	M₁₆	1	8	yes	—
80	C₄²×C₄₇	—	2	8	no	—
81	C₄²×C₄₇	—	2	4	no	—
82	C₄²×C₄₇	—	2	4	no	—
83	C₄²×C₄₇	—	2	4	no	—
84	C₄²×C₄₇	—	2	4	no	—
85	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
86	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
87	C₄²×C₄₇	C₂×D₄	1	2	yes	—
88	C₄²×C₄₇	C₂×D₄	1	2	yes	—
89	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
90	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
91	C₂×C₄₇⋊C₈	—	2	2	no	—
92	C₂×C₄₇⋊C₈	—	2	2	no	—
93	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	—	2	8	no	—
96	C₂×C₄₇⋊C₈	—	2	4	no	—
97	C₂×C₄₇⋊C₈	—	4	8	no	—
98	C₂³×C₄₇	—	2	2	no	—
99	C₂³×C₄₇	—	2	2	no	—
100	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
101	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
102	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	8	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	2	8	no	—
114	C₂×C₄×C₄₇	—	2	4	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
123	C₂×C₄×C₄₇	#131	1	4	yes	—
124	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
127	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
128	C₄₇⋊C₈	—	4	2	no	—
129	C₄₇⋊C₈	—	4	4	no	—
130	C₄₇⋊C₈	—	4	2	no	—
131	#21976	—	47	1	no	maximal, Sylow
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	8	no	—
135	C₂²×C₄₇	#423	1	4	yes	—
136	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₄₇	#520	1	2	yes	—
138	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
139	C₂²×C₄₇	C₂×#131	1	2	yes	—
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	—	2	2	no	—
143	C₄×C₄₇	—	2	2	no	—
144	C₄×C₄₇	—	2	8	no	—
145	C₄×C₄₇	—	4	4	no	—
146	C₄×C₄₇	#424	1	4	yes	—
147	C₄×C₄₇	#535	1	1	yes	characteristic
148	C₄×C₄₇	#536	1	1	yes	characteristic
149	C₄×C₄₇	#544	1	2	yes	—
150	C₂×#432	—	47	1	no	—
151	C₂²×C₄⋊C₈	—	47	1	no	—
152	C₄×#117	D₄₇	1	1	yes	characteristic
153	#1940	—	47	4	no	—
154	#1850	—	47	4	no	—
155	#1842	—	47	2	no	—
156	#1831	—	47	2	no	—
157	C₂×C₄₇	—	2	4	no	—
158	C₂×C₄₇	#2317	1	1	yes	characteristic
159	C₂×C₄₇	C₂×#423	1	1	yes	characteristic
160	C₂×C₄₇	C₂×#424	1	1	yes	characteristic
161	C₂×C₄₇	#2990	1	4	yes	—
162	C₂³×C₈	—	47	2	no	—
163	C₂×#117	Dic₄₇	1	2	yes	—
164	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
165	#432	—	94	4	no	—
166	C₂×C₄⋊C₈	—	47	1	no	—
167	C₂×C₄⋊C₈	—	47	2	no	—
168	C₂×C₄⋊C₈	—	47	1	no	—
169	C₂×C₄⋊C₈	—	47	2	no	—
170	C₂×C₄⋊C₈	—	47	8	no	—
171	C₂×C₄⋊C₈	—	94	4	no	—
172	C₂×C₂²⋊C₈	—	47	2	no	—
173	C₂×#97	—	47	2	no	—
174	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
175	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
176	C₄×C₂²⋊C₄	Dic₄₇	1	1	yes	characteristic
177	C₄×C₂²⋊C₄	Dic₄₇	1	1	yes	characteristic
178	C₄×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
179	#376	Dic₄₇	1	2	yes	—
180	C₄³	C₂×D₄₇	1	2	yes	—
181	#336	—	47	8	no	—
182	C₄₇	#21976	1	1	yes	characteristic, complemented, Sylow
183	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
184	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
185	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
186	C₂²×C₈	—	47	4	no	—
187	C₂²×C₈	—	47	2	no	—
188	C₂²×C₈	—	47	2	no	—
189	C₂²×C₈	—	94	4	no	—
190	C₂²×C₈	—	188	4	no	—
191	#117	—	2	8	no	—
192	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
193	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
194	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
195	C₂×C₄²	Dic₉₄	1	4	yes	—
196	C₂×C₄²	D₁₈₈	1	4	yes	—
197	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
198	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
199	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
200	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
201	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
202	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
203	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
204	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
205	C₄⋊C₈	—	94	4	no	—
206	C₄⋊C₈	—	94	4	no	—
207	C₄⋊C₈	—	94	2	no	—
208	C₄⋊C₈	—	94	2	no	—
209	C₄⋊C₈	—	188	8	no	—
210	C₂²⋊C₈	—	94	4	no	—
211	#97	—	94	4	no	—
212	#95	C₂×Dic₄₇	1	2	yes	—
213	#95	C₂×Dic₄₇	1	2	yes	—
214	#95	C₂×Dic₄₇	1	4	yes	—
215	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
221	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
222	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
223	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
224	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
225	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
226	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
227	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
229	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
230	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
231	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
232	C₂×C₈	—	94	2	no	—
233	C₂×C₈	—	94	2	no	—
234	C₂×C₈	—	94	4	no	—
235	C₂×C₈	—	94	4	no	—
236	C₂×C₈	—	94	8	no	—
237	C₂×C₈	—	94	4	no	—
238	C₂×C₈	—	188	8	no	—
239	C₄⋊C₄	—	2	8	no	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₂²⋊C₄	#10625329	1	8	yes	—
242	C₄²	—	2	8	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	2	4	no	—
247	C₄²	C₄⋊Dic₄₇	1	4	yes	—
248	C₄²	C₄⋊Dic₄₇	1	4	yes	—
249	C₄²	C₂×Dic₉₄	1	2	yes	—
250	C₄²	C₂×Dic₉₄	1	2	yes	—
251	C₄²	C₂×D₁₈₈	1	2	yes	—
252	C₄²	C₂×D₁₈₈	1	2	yes	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	2	no	—
255	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
256	C₂³	C₂×#10625329	1	4	yes	—
257	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
273	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
274	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
275	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
276	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
277	C₂×C₄	C₂×#10625329	1	8	yes	—
278	C₂×C₄	#13094031	1	4	yes	—
279	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
280	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
281	C₂×C₄	#13094070	1	2	yes	—
282	C₂×C₄	#13094071	1	2	yes	—
283	C₈	—	188	2	no	—
284	C₈	—	188	4	no	—
285	C₈	—	188	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	8	no	—
288	C₂²	—	2	2	no	—
289	C₂²	#15689147	1	4	yes	—
290	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
291	C₂²	#15689606	1	2	yes	—
292	C₂²	C₂²×#10625329	1	2	yes	—
293	C₂²	C₂×#13094031	1	2	yes	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	8	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	4	no	—
300	C₄	#15689150	1	4	yes	—
301	C₄	#15689692	1	1	yes	characteristic
302	C₄	#15689699	1	1	yes	characteristic
303	C₄	#15689751	1	2	yes	—
304	C₂	—	2	4	no	—
305	C₂	#20246011	1	1	yes	characteristic
306	C₂	C₂×#15689147	1	1	yes	characteristic
307	C₂	C₂×#15689150	1	1	yes	characteristic
308	C₂	#20252513	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄×#117×C₄₇	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#117	D₄₇
hypocenter	C₄₇	#21976
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. #20244112	C₂	semidirect
5. #20243540	C₂	semidirect
6. #20243505	C₂	semidirect
182. C₄₇	#21976	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
C₂²×C₄⋊(C₄₇⋊C₈)	2
#20244112	2
#20243568	2
#20243540	2
#20243505	2
C₄×#117×C₄₇	2
#21976	47