Subgroups — #26105868

Order 12032.

758 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688969	C₂	1	1	yes	maximal, characteristic
3	#20241230	C₂	1	4	yes	maximal
4	C₂×C₄₇×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄×Dic₄₇	C₄	1	2	yes	—
6	#15689176	—	2	2	no	—
7	#15689176	C₄	1	4	yes	complemented
8	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
9	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
11	#15688969	—	2	1	no	—
12	#15688969	—	2	1	no	—
13	#15688969	—	2	1	no	—
14	#15688969	—	2	1	no	—
15	C₂×C₄₇⋊#95	C₄	1	2	yes	—
16	C₂×C₄₇⋊#95	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#431	—	2	2	no	—
19	C₄₇×#431	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
21	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₄₇	—	2	1	no	—
23	C₂³×Dic₄₇	C₂×C₄	1	4	yes	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₂×C₄×Dic₄₇	—	2	2	no	—
26	C₂×C₄×Dic₄₇	—	2	2	no	—
27	C₂×C₄×Dic₄₇	—	2	4	no	—
28	C₂×C₄×Dic₄₇	C₈	1	8	yes	complemented
29	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
30	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
31	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
32	C₄×C₄₇⋊C₈	—	2	4	no	—
33	C₄₇⋊#95	—	2	2	no	—
34	C₄₇⋊#95	—	2	2	no	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₄₇⋊#95	—	2	2	no	—
37	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	—	2	2	no	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₇	D₄	1	2	yes	—
44	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
49	C₄⋊C₈×C₄₇	—	2	4	no	—
50	C₂²⋊C₈×C₄₇	—	2	2	no	—
51	C₂²⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
52	C₄×C₈×C₄₇	—	2	4	no	—
53	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	—	4	2	no	—
55	C₂²×C₄×C₄₇	—	4	2	no	—
56	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
57	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
65	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
72	C₂²×Dic₄₇	—	2	4	no	—
73	C₂²×Dic₄₇	—	2	4	no	—
74	C₂²×Dic₄₇	—	4	4	no	—
75	C₂²×Dic₄₇	—	4	4	no	—
76	C₂²×Dic₄₇	C₂×C₈	1	8	yes	—
77	C₂×C₈×C₄₇	—	2	4	no	—
78	C₂×C₈×C₄₇	—	4	4	no	—
79	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
84	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
85	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
86	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
87	C₄×Dic₄₇	—	2	8	no	—
88	C₄×Dic₄₇	—	2	8	no	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	4	4	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	—	2	2	no	—
93	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
94	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
97	C₂³×C₄₇	C₂×M₁₆	1	2	yes	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	4	2	no	—
103	C₂×C₄×C₄₇	—	4	2	no	—
104	C₂×C₄×C₄₇	—	4	1	no	—
105	C₂×C₄×C₄₇	—	4	1	no	—
106	C₂×C₄×C₄₇	—	4	1	no	—
107	C₂×C₄×C₄₇	—	4	1	no	—
108	C₂×C₄×C₄₇	#95	1	2	yes	—
109	C₂×C₄×C₄₇	#95	1	2	yes	—
110	C₂×C₄×C₄₇	#95	1	2	yes	—
111	C₂×C₄×C₄₇	#95	1	2	yes	—
112	C₂×C₄×C₄₇	C₄×C₈	1	4	yes	—
113	C₂×C₄×C₄₇	#97	1	4	yes	—
114	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
115	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
116	C₂×C₄×C₄₇	C₄⋊C₈	1	8	yes	—
117	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×Dic₄₇	—	2	8	no	—
122	C₂×Dic₄₇	—	4	8	no	—
123	C₂×Dic₄₇	—	4	4	no	—
124	C₂×Dic₄₇	—	4	4	no	—
125	C₂×Dic₄₇	—	4	8	no	—
126	C₈×C₄₇	—	4	4	no	—
127	C₄₇⋊C₈	—	4	4	no	—
128	#7906	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	—	2	4	no	—
130	C₂²×C₄₇	—	2	4	no	—
131	C₂²×C₄₇	—	2	1	no	—
132	C₂²×C₄₇	—	2	1	no	—
133	C₂²×C₄₇	—	2	1	no	—
134	C₂²×C₄₇	—	2	1	no	—
135	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
136	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
137	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
139	C₂²×C₄₇	#410	1	2	yes	—
140	C₂²×C₄₇	#413	1	2	yes	—
141	C₂²×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
142	C₄×C₄₇	—	2	1	no	—
143	C₄×C₄₇	—	2	1	no	—
144	C₄×C₄₇	—	2	1	no	—
145	C₄×C₄₇	—	2	1	no	—
146	C₄×C₄₇	—	2	2	no	—
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	#336	1	4	yes	—
149	C₄×C₄₇	#336	1	4	yes	—
150	Dic₄₇	—	4	8	no	—
151	C₂×#431	—	47	1	no	—
152	C₂×#431	D₄₇	1	1	yes	characteristic
153	C₂×#376	—	47	1	no	—
154	#1280	—	47	4	no	—
155	C₂×C₄₇	—	2	1	no	—
156	C₂×C₄₇	—	2	1	no	—
157	C₂×C₄₇	—	2	2	no	—
158	C₂×C₄₇	#1458	1	4	yes	—
159	C₂×C₄₇	C₂×#336	1	1	yes	characteristic
160	C₂×C₄₇	#1740	1	1	yes	characteristic
161	C₂×C₄₇	#1742	1	1	yes	characteristic
162	C₂²×C₄²	—	47	2	no	—
163	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
164	#431	—	2	2	no	—
165	#431	—	47	4	no	—
166	#431	—	94	2	no	—
167	#431	Dic₄₇	1	4	yes	complemented
168	C₂×C₄⋊C₈	—	2	1	no	—
169	C₂×C₄⋊C₈	—	94	1	no	—
170	C₂×C₂²⋊C₈	—	47	2	no	—
171	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
172	C₂×C₄×C₈	—	2	1	no	—
173	C₂×C₄×C₈	—	94	1	no	—
174	#376	—	94	1	no	—
175	#376	—	94	1	no	—
176	#376	—	94	1	no	—
177	#376	—	94	1	no	—
178	C₂×#95	—	47	2	no	—
179	C₂×#95	—	47	2	no	—
180	C₄₇	#7906	1	1	yes	characteristic, complemented, Sylow
181	C₂³×C₄	—	47	4	no	—
182	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
183	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
184	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	94	2	no	—
187	C₂×C₄²	—	47	8	no	—
188	C₂×C₄²	—	94	4	no	—
189	C₂×C₄²	—	94	2	no	—
190	C₂×C₄²	—	94	2	no	—
191	C₂×C₄²	Dic₉₄	1	2	yes	—
192	C₂×C₄²	C₄×D₄₇	1	2	yes	—
193	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
194	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
195	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
196	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
197	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
199	C₄⋊C₈	—	2	4	no	—
200	C₄⋊C₈	—	94	4	no	—
201	C₂²⋊C₈	—	2	2	no	—
202	C₂²⋊C₈	—	47	4	no	—
203	C₂²⋊C₈	—	94	2	no	—
204	C₂²⋊C₈	C₂×Dic₄₇	1	4	yes	—
205	C₄×C₈	—	2	4	no	—
206	C₄×C₈	—	94	4	no	—
207	#95	—	94	2	no	—
208	#95	—	94	2	no	—
209	#95	—	94	2	no	—
210	#95	—	94	2	no	—
211	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	—	47	8	no	—
215	C₂²×C₄	—	94	4	no	—
216	C₂²×C₄	—	94	4	no	—
217	C₂²×C₄	—	188	4	no	—
218	C₂²×C₄	—	188	4	no	—
219	C₂²×C₄	C₈×D₄₇	1	4	yes	—
220	C₂²×C₄	#10625324	1	4	yes	—
221	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
222	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
223	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
224	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
225	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
226	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
227	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
228	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
229	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
232	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	4	4	no	—
237	C₂×C₈	—	94	4	no	—
238	C₂×C₈	—	188	4	no	—
239	C₄²	—	94	8	no	—
240	C₄²	—	94	8	no	—
241	C₄²	Dic₄₇⋊C₄	1	2	yes	—
242	C₄²	Dic₄₇⋊C₄	1	2	yes	—
243	C₄²	C₄⋊Dic₄₇	1	2	yes	—
244	C₄²	C₄⋊Dic₄₇	1	2	yes	—
245	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
246	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
247	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
248	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
252	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
253	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
254	C₂³	C₂×C₈×D₄₇	1	2	yes	—
255	C₂³	C₂×#10625324	1	2	yes	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	94	8	no	—
267	C₂×C₄	—	188	8	no	—
268	C₂×C₄	—	188	4	no	—
269	C₂×C₄	—	188	4	no	—
270	C₂×C₄	—	188	8	no	—
271	C₂×C₄	C₄₇⋊#95	1	2	yes	—
272	C₂×C₄	C₄₇⋊#95	1	2	yes	—
273	C₂×C₄	C₄₇⋊#95	1	2	yes	—
274	C₂×C₄	C₄₇⋊#95	1	2	yes	—
275	C₂×C₄	C₈×Dic₄₇	1	4	yes	—
276	C₂×C₄	#13093934	1	4	yes	—
277	C₂×C₄	(C₂×D₄₇)⋊C₈	1	4	yes	—
278	C₂×C₄	(C₂×D₄₇)⋊C₈	1	4	yes	—
279	C₂×C₄	Dic₄₇⋊C₈	1	8	yes	—
280	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
281	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
282	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
283	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
284	C₈	—	4	4	no	—
285	C₈	—	188	4	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
293	C₂²	C₂×C₈×Dic₄₇	1	1	yes	characteristic
294	C₂²	C₂×#13093934	1	1	yes	characteristic
295	C₂²	C₂×(C₂×D₄₇)⋊C₈	1	2	yes	—
296	C₂²	#15689096	1	2	yes	—
297	C₂²	#15689105	1	2	yes	—
298	C₂²	C₂×Dic₄₇⋊C₈	1	2	yes	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	188	8	no	—
306	C₄	#15688877	1	4	yes	—
307	C₄	#15688877	1	4	yes	—
308	C₂	—	2	1	no	—
309	C₂	—	2	1	no	—
310	C₂	—	2	2	no	—
311	C₂	#20241700	1	4	yes	—
312	C₂	C₂×#15688877	1	1	yes	characteristic
313	C₂	#20243120	1	1	yes	characteristic
314	C₂	#20243126	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄₇	C₂×C₄×C₈
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#431	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#431	D₄₇
hypocenter	C₄₇	#7906
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
7. #15689176	C₄	semidirect
19. C₄₇×#431	C₄	semidirect
28. C₂×C₄×Dic₄₇	C₈	semidirect
167. #431	Dic₄₇	semidirect
180. C₄₇	#7906	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
C₂×#15688969	2
#20241230	2
C₂×C₄₇×#431	2
#7906	47