Subgroups — #26106435

Order 12032.

544 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252021	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689176	C₂	1	1	yes	maximal, characteristic
3	#20241276	C₂	1	4	yes	maximal
4	C₂×C₄₇×#432	C₂	1	1	yes	maximal, characteristic
5	#15689180	—	2	1	no	—
6	#15689180	—	2	1	no	—
7	#15689176	—	2	1	no	—
8	#15689176	—	2	1	no	—
9	#15689176	—	2	1	no	—
10	#15689176	—	2	1	no	—
11	#15689176	C₄	1	2	yes	—
12	#15689176	C₄	1	2	yes	—
13	#15689147	C₄	1	2	yes	—
14	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
15	#15689087	C₂²	1	2	yes	—
16	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
17	C₄×#10625329	C₄	1	2	yes	—
18	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₄₇×#432	—	2	1	no	—
21	C₄₇×#432	—	2	1	no	—
22	C₄₇×#432	C₄	1	2	yes	—
23	C₄₇×#432	C₄	1	2	yes	—
24	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
25	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
26	C₂×#97×C₄₇	—	2	1	no	—
27	C₂×#10625329	C₂×C₄	1	4	yes	—
28	C₂²×C₄₇⋊C₈	—	2	2	no	—
29	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
30	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
31	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
32	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
33	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
34	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
35	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
36	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
37	#13093935	—	2	2	no	—
38	C₄×C₄₇⋊C₈	—	2	2	no	—
39	C₄×C₄₇⋊C₈	—	2	2	no	—
40	C₄×C₄₇⋊C₈	—	2	2	no	—
41	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
42	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
43	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	—	2	2	no	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
57	C₄⋊C₈×C₄₇	—	2	2	no	—
58	C₄⋊C₈×C₄₇	—	2	2	no	—
59	C₂²⋊C₈×C₄₇	—	2	2	no	—
60	C₂²⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
61	#97×C₄₇	—	2	2	no	—
62	#97×C₄₇	—	2	2	no	—
63	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	—	2	4	no	—
65	C₂²×C₄×C₄₇	—	4	2	no	—
66	C₂²×C₄×C₄₇	—	4	2	no	—
67	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₄₇	—	2	4	no	—
86	C₂×C₈×C₄₇	—	4	4	no	—
87	C₄²×C₄₇	C₄²	1	1	yes	characteristic
88	C₄²×C₄₇	C₄²	1	1	yes	characteristic
89	C₄²×C₄₇	C₄²	1	1	yes	characteristic
90	C₄²×C₄₇	C₄²	1	1	yes	characteristic
91	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
93	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
94	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
95	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
96	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
99	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
100	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
101	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
102	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
103	#10625329	—	4	4	no	—
104	C₂×C₄₇⋊C₈	—	2	4	no	—
105	C₂×C₄₇⋊C₈	—	2	4	no	—
106	C₂×C₄₇⋊C₈	—	4	4	no	—
107	C₂³×C₄₇	—	2	2	no	—
108	C₂³×C₄₇	—	2	2	no	—
109	C₂³×C₄₇	—	4	1	no	—
110	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×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	2	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	4	1	no	—
124	C₂×C₄×C₄₇	—	4	1	no	—
125	C₂×C₄×C₄₇	—	4	1	no	—
126	C₂×C₄×C₄₇	—	4	1	no	—
127	C₂×C₄×C₄₇	—	4	1	no	—
128	C₂×C₄×C₄₇	—	4	1	no	—
129	C₂×C₄×C₄₇	—	4	1	no	—
130	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
131	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
132	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
133	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
134	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
135	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
136	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
137	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
138	C₂×C₄×C₄₇	#108	1	4	yes	—
139	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₈×C₄₇	—	4	4	no	—
144	C₄₇⋊C₈	—	4	4	no	—
145	C₄₇⋊C₈	—	4	4	no	—
146	#8058	—	47	1	no	maximal, Sylow
147	C₂²×C₄₇	—	2	1	no	—
148	C₂²×C₄₇	—	2	1	no	—
149	C₂²×C₄₇	—	2	1	no	—
150	C₂²×C₄₇	—	2	1	no	—
151	C₂²×C₄₇	—	2	1	no	—
152	C₂²×C₄₇	—	2	1	no	—
153	C₂²×C₄₇	—	2	1	no	—
154	C₂²×C₄₇	—	4	4	no	—
155	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
156	C₂²×C₄₇	C₂×#108	1	2	yes	—
157	C₂²×C₄₇	#430	1	2	yes	—
158	C₄×C₄₇	—	2	1	no	—
159	C₄×C₄₇	—	2	1	no	—
160	C₄×C₄₇	—	2	1	no	—
161	C₄×C₄₇	—	2	1	no	—
162	C₄×C₄₇	—	2	1	no	—
163	C₄×C₄₇	—	2	1	no	—
164	C₄×C₄₇	—	2	1	no	—
165	C₄×C₄₇	—	2	1	no	—
166	C₄×C₄₇	—	2	1	no	—
167	C₄×C₄₇	#341	1	2	yes	—
168	C₄×C₄₇	#341	1	2	yes	—
169	C₄×C₄₇	#410	1	2	yes	—
170	#2920	—	47	1	no	—
171	C₂×#432	D₄₇	1	1	yes	characteristic
172	C₂×#431	—	47	1	no	—
173	#1298	—	47	4	no	—
174	C₂×C₄₇	—	2	2	no	—
175	C₂×C₄₇	—	2	2	no	—
176	C₂×C₄₇	—	4	1	no	—
177	C₂×C₄₇	C₂×#341	1	1	yes	characteristic
178	C₂×C₄₇	#1738	1	1	yes	characteristic
179	C₂×C₄₇	#1740	1	1	yes	characteristic
180	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
181	#432	—	2	1	no	—
182	#432	—	2	1	no	—
183	#432	—	94	1	no	—
184	#432	—	94	1	no	—
185	#432	Dic₄₇	1	2	yes	—
186	#432	Dic₄₇	1	2	yes	—
187	#431	—	47	2	no	—
188	#431	—	47	2	no	—
189	#431	—	94	1	no	—
190	#431	—	94	1	no	—
191	#431	—	94	1	no	—
192	#431	—	94	1	no	—
193	#423	—	47	2	no	—
194	C₂×C₄⋊C₈	—	2	1	no	—
195	C₂×C₄⋊C₈	—	94	1	no	—
196	#407	—	47	2	no	—
197	C₂×C₂²⋊C₈	—	47	2	no	—
198	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
199	C₄×M₁₆	—	47	2	no	—
200	C₂×#97	—	2	1	no	—
201	C₂×C₄×C₈	—	94	1	no	—
202	C₄₇	#8058	1	1	yes	characteristic, complemented, Sylow
203	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
204	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
205	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
206	C₂×M₁₆	—	47	4	no	—
207	C₂²×C₈	—	2	2	no	—
208	C₂²×C₈	—	94	2	no	—
209	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
210	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
211	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
212	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
213	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
214	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
215	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
216	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
217	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
218	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
219	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
220	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
221	C₄⋊C₈	—	2	2	no	—
222	C₄⋊C₈	—	2	2	no	—
223	C₄⋊C₈	—	94	2	no	—
224	C₄⋊C₈	—	94	2	no	—
225	C₄⋊C₈	—	94	2	no	—
226	C₄⋊C₈	—	94	2	no	—
227	C₂²⋊C₈	—	2	2	no	—
228	C₂²⋊C₈	—	47	4	no	—
229	C₂²⋊C₈	—	94	2	no	—
230	C₂²⋊C₈	—	94	2	no	—
231	C₂²⋊C₈	—	94	2	no	—
232	C₂²⋊C₈	C₂×Dic₄₇	1	4	yes	—
233	#97	—	2	2	no	—
234	#97	—	2	2	no	—
235	#97	—	94	2	no	—
236	C₄×C₈	—	94	2	no	—
237	C₄×C₈	—	94	2	no	—
238	C₄×C₈	—	94	2	no	—
239	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
244	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
245	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
246	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
247	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
248	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
249	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
250	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
251	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
252	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
253	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
254	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
258	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
260	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
261	M₁₆	—	188	4	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	4	4	no	—
264	C₂×C₈	—	94	4	no	—
265	C₂×C₈	—	94	4	no	—
266	C₂×C₈	—	188	4	no	—
267	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
268	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
269	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
270	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
271	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
272	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
273	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
274	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
275	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
276	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
277	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
278	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
279	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
280	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
281	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
282	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
283	C₂³	—	2	2	no	—
284	C₂³	—	2	2	no	—
285	C₂³	—	4	1	no	—
286	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
287	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
288	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	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	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
307	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
308	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
309	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
310	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
311	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
312	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
313	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
314	C₂×C₄	#13093959	1	4	yes	—
315	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
316	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
317	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
318	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
319	C₈	—	4	4	no	—
320	C₈	—	188	4	no	—
321	C₈	—	188	4	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	4	4	no	—
330	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
331	C₂²	C₂×#13093959	1	2	yes	—
332	C₂²	#15689172	1	2	yes	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	#15688887	1	2	yes	—
343	C₄	#15688887	1	2	yes	—
344	C₄	#15689096	1	2	yes	—
345	C₂	—	2	2	no	—
346	C₂	—	2	2	no	—
347	C₂	—	4	1	no	—
348	C₂	C₂×#15688887	1	1	yes	characteristic
349	C₂	#20243115	1	1	yes	characteristic
350	C₂	#20243120	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#432	C₂
socle	C₂²×C₄₇	C₂×#95
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#432	D₄₇
hypocenter	C₄₇	#8058
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
202. C₄₇	#8058	semidirect

Maximal subgroups

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

subgroup	index
#20252021	2
C₂×#15689176	2
#20241276	2
C₂×C₄₇×#432	2
#8058	47