Subgroups — #26121883

Order 12032.

1062 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	C₂×#15688878	C₂	1	2	yes	maximal
2	C₂×#15688878	C₂	1	2	yes	maximal
3	C₂×#15688878	C₂	1	2	yes	maximal
4	C₂×C₄³×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄₇⋊C₈	C₂²	1	4	yes	—
6	#15688878	—	2	4	no	—
7	#15688878	—	2	4	no	—
8	#15688878	C₄	1	8	yes	complemented
9	C₂²×C₄²×C₄₇	C₄	1	2	yes	—
10	C₂²×C₄²×C₄₇	C₄	1	2	yes	—
11	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	C₄³×C₄₇	—	2	4	no	—
15	C₂²×C₄₇⋊C₈	—	2	4	no	—
16	C₂²×C₄₇⋊C₈	—	2	4	no	—
17	C₂²×C₄₇⋊C₈	—	2	4	no	—
18	C₂²×C₄₇⋊C₈	—	2	4	no	—
19	C₂²×C₄₇⋊C₈	—	2	4	no	—
20	C₂²×C₄₇⋊C₈	—	2	4	no	—
21	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
22	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₄₇	—	2	4	no	—
28	C₂×C₄²×C₄₇	—	2	4	no	—
29	C₂×C₄²×C₄₇	—	2	4	no	—
30	C₂×C₄²×C₄₇	—	2	4	no	—
31	C₂×C₄²×C₄₇	—	2	4	no	—
32	C₂×C₄²×C₄₇	—	2	2	no	—
33	C₂×C₄²×C₄₇	—	2	2	no	—
34	C₂×C₄²×C₄₇	—	2	2	no	—
35	C₂×C₄²×C₄₇	—	2	2	no	—
36	C₂×C₄²×C₄₇	—	2	2	no	—
37	C₂×C₄²×C₄₇	—	2	2	no	—
38	C₂×C₄²×C₄₇	C₈	1	8	yes	complemented
39	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₄₇	D₄	1	4	yes	—
41	C₂×C₄²×C₄₇	D₄	1	2	yes	—
42	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
43	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	—	2	1	no	—
45	C₂²×C₄×C₄₇	—	2	1	no	—
46	C₂²×C₄×C₄₇	—	2	2	no	—
47	C₂²×C₄×C₄₇	—	2	1	no	—
48	C₂²×C₄×C₄₇	—	2	1	no	—
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
59	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
61	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
64	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
65	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
66	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
67	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	#41	1	2	yes	—
73	C₂²×C₄×C₄₇	#41	1	2	yes	—
74	C₂²×C₄×C₄₇	#41	1	2	yes	—
75	C₂²×C₄×C₄₇	#41	1	2	yes	—
76	C₄²×C₄₇	—	2	16	no	—
77	C₄²×C₄₇	—	2	16	no	—
78	C₄²×C₄₇	—	2	8	no	—
79	C₄²×C₄₇	—	2	4	no	—
80	C₄²×C₄₇	—	2	4	no	—
81	C₄²×C₄₇	—	2	8	no	—
82	C₂×C₄₇⋊C₈	—	2	8	no	—
83	C₂×C₄₇⋊C₈	—	2	8	no	—
84	C₂×C₄₇⋊C₈	—	2	8	no	—
85	C₂×C₄₇⋊C₈	—	4	4	no	—
86	C₂×C₄₇⋊C₈	—	4	4	no	—
87	C₂×C₄₇⋊C₈	—	4	4	no	—
88	C₂×C₄₇⋊C₈	—	4	4	no	—
89	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
90	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂³×C₄₇	#117	1	1	yes	characteristic
93	C₂³×C₄₇	#117	1	1	yes	characteristic
94	C₂³×C₄₇	#117	1	1	yes	characteristic
95	C₂³×C₄₇	#117	1	1	yes	characteristic
96	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
97	C₂³×C₄₇	C₂×M₁₆	1	2	yes	—
98	C₂³×C₄₇	C₂×M₁₆	1	2	yes	—
99	C₂³×C₄₇	C₂×M₁₆	1	2	yes	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	4	no	—
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	4	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	2	4	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	—	2	4	no	—
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	—	2	4	no	—
120	C₂×C₄×C₄₇	C₄×C₈	1	4	yes	—
121	C₂×C₄×C₄₇	#97	1	4	yes	—
122	C₂×C₄×C₄₇	C₂²⋊C₈	1	8	yes	—
123	C₂×C₄×C₄₇	C₄⋊C₈	1	8	yes	—
124	C₂×C₄×C₄₇	#117	1	4	yes	—
125	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
126	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
128	C₂×C₄×C₄₇	#121	1	2	yes	—
129	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
130	C₂×C₄×C₄₇	#123	1	4	yes	—
131	C₂×C₄×C₄₇	#124	1	2	yes	—
132	C₂×C₄×C₄₇	#125	1	2	yes	—
133	C₂×C₄×C₄₇	#126	1	4	yes	—
134	C₄₇⋊C₈	—	4	8	no	—
135	#10795	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	#376	1	1	yes	characteristic
137	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
139	C₂²×C₄₇	#380	1	1	yes	characteristic
140	C₂²×C₄₇	#381	1	1	yes	characteristic
141	C₂²×C₄₇	#382	1	1	yes	characteristic
142	C₂²×C₄₇	#383	1	1	yes	characteristic
143	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
144	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
145	C₂²×C₄₇	C₄×M₁₆	1	1	yes	characteristic
146	C₂²×C₄₇	C₄×M₁₆	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
148	C₂²×C₄₇	#407	1	2	yes	—
149	C₂²×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
150	C₂²×C₄₇	#423	1	2	yes	—
151	C₂²×C₄₇	#431	1	2	yes	—
152	C₂²×C₄₇	#431	1	2	yes	—
153	C₂²×C₄₇	#431	1	2	yes	—
154	C₂²×C₄₇	#431	1	2	yes	—
155	C₂²×C₄₇	#432	1	2	yes	—
156	C₂²×C₄₇	#432	1	2	yes	—
157	C₂²×C₄₇	#432	1	2	yes	—
158	C₂²×C₄₇	#432	1	2	yes	—
159	C₄×C₄₇	—	2	8	no	—
160	C₄×C₄₇	—	2	8	no	—
161	C₄×C₄₇	—	2	4	no	—
162	C₄×C₄₇	—	2	4	no	—
163	C₄×C₄₇	#387	1	4	yes	—
164	C₄×C₄₇	#388	1	4	yes	—
165	C₂×C₄³	D₄₇	1	1	yes	characteristic
166	C₂×#336	—	47	2	no	—
167	C₂×#336	—	47	2	no	—
168	C₂×#336	—	47	2	no	—
169	C₂×C₄₇	#1434	1	1	yes	characteristic
170	C₂×C₄₇	#1744	1	1	yes	characteristic
171	C₂×C₄₇	#1745	1	1	yes	characteristic
172	C₂×C₄₇	C₄×C₂²⋊C₈	1	1	yes	characteristic
173	C₂×C₄₇	#1749	1	1	yes	characteristic
174	C₂×C₄₇	C₄×C₄⋊C₈	1	1	yes	characteristic
175	C₂×C₄₇	#1767	1	1	yes	characteristic
176	C₂×C₄₇	#1797	1	2	yes	—
177	C₂×C₄₇	#1817	1	2	yes	—
178	C₂×C₄₇	#1831	1	2	yes	—
179	C₂×C₄₇	#1839	1	2	yes	—
180	C₂³×C₈	—	47	4	no	—
181	C₂²×C₄²	Dic₄₇	1	2	yes	—
182	C₂²×C₄²	Dic₄₇	1	2	yes	—
183	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
184	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
185	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
186	C₄³	—	2	4	no	—
187	#336	—	47	8	no	—
188	#336	—	94	4	no	—
189	#336	—	94	4	no	—
190	C₄₇	#10795	1	1	yes	characteristic, complemented, Sylow
191	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
192	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
193	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
194	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
195	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
196	C₂²×C₈	—	47	8	no	—
197	C₂²×C₈	—	94	4	no	—
198	C₂²×C₈	—	94	4	no	—
199	C₂²×C₈	—	94	4	no	—
200	C₂²×C₈	—	94	4	no	—
201	C₂²×C₈	—	94	4	no	—
202	C₂²×C₈	—	94	4	no	—
203	C₂×C₄²	—	2	4	no	—
204	C₂×C₄²	—	2	4	no	—
205	C₂×C₄²	—	2	4	no	—
206	C₂×C₄²	—	2	4	no	—
207	C₂×C₄²	—	2	4	no	—
208	C₂×C₄²	—	2	2	no	—
209	C₂×C₄²	—	2	2	no	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	C₄₇⋊C₈	1	8	yes	complemented
215	C₂×C₄²	Dic₉₄	1	2	yes	—
216	C₂×C₄²	C₄×D₄₇	1	4	yes	—
217	C₂×C₄²	D₁₈₈	1	2	yes	—
218	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
219	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	C₂×C₄₇⋊C₈	1	4	yes	—
235	C₂²×C₄	C₂×C₄₇⋊C₈	1	4	yes	—
236	C₂²×C₄	C₂×C₄₇⋊C₈	1	4	yes	—
237	C₂²×C₄	#10625329	1	4	yes	—
238	C₂²×C₄	#10625329	1	4	yes	—
239	C₂²×C₄	#10625329	1	4	yes	—
240	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
241	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
242	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
243	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
244	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
246	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
247	C₂²×C₄	#10625350	1	2	yes	—
248	C₂²×C₄	#10625350	1	2	yes	—
249	C₂²×C₄	#10625350	1	2	yes	—
250	C₂²×C₄	#10625350	1	2	yes	—
251	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
252	C₂×C₈	—	94	8	no	—
253	C₂×C₈	—	94	8	no	—
254	C₂×C₈	—	94	8	no	—
255	C₂×C₈	—	188	4	no	—
256	C₂×C₈	—	188	4	no	—
257	C₂×C₈	—	188	4	no	—
258	C₂×C₈	—	188	4	no	—
259	C₄²	—	2	16	no	—
260	C₄²	—	2	16	no	—
261	C₄²	—	2	8	no	—
262	C₄²	—	2	4	no	—
263	C₄²	—	2	4	no	—
264	C₄²	—	2	8	no	—
265	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
266	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
267	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
268	C₂³	#13093983	1	1	yes	characteristic
269	C₂³	#13093983	1	1	yes	characteristic
270	C₂³	#13093983	1	1	yes	characteristic
271	C₂³	#13093983	1	1	yes	characteristic
272	C₂³	C₂²×C₄₇⋊C₈	1	2	yes	—
273	C₂³	C₂×#10625329	1	2	yes	—
274	C₂³	C₂×#10625329	1	2	yes	—
275	C₂³	C₂×#10625329	1	2	yes	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	C₄×C₄₇⋊C₈	1	4	yes	—
297	C₂×C₄	#13093935	1	4	yes	—
298	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	8	yes	—
299	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	8	yes	—
300	C₂×C₄	#13093981	1	4	yes	—
301	C₂×C₄	C₄×C₂²⋊D₄₇	1	4	yes	—
302	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
303	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
304	C₂×C₄	#13093998	1	2	yes	—
305	C₂×C₄	#13094003	1	2	yes	—
306	C₂×C₄	#13094007	1	4	yes	—
307	C₂×C₄	#13094012	1	2	yes	—
308	C₂×C₄	#13094016	1	2	yes	—
309	C₂×C₄	#13094018	1	4	yes	—
310	C₈	—	188	8	no	—
311	C₂²	#15688969	1	1	yes	characteristic
312	C₂²	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
313	C₂²	C₄×C₄⋊Dic₄₇	1	1	yes	characteristic
314	C₂²	#15688983	1	1	yes	characteristic
315	C₂²	#15688986	1	1	yes	characteristic
316	C₂²	#15688989	1	1	yes	characteristic
317	C₂²	#15688991	1	1	yes	characteristic
318	C₂²	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
319	C₂²	C₂×#13093935	1	1	yes	characteristic
320	C₂²	C₄×#10625329	1	1	yes	characteristic
321	C₂²	C₄×#10625329	1	1	yes	characteristic
322	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
323	C₂²	#15689087	1	2	yes	—
324	C₂²	C₂×C₄⋊(C₄₇⋊C₈)	1	2	yes	—
325	C₂²	#15689147	1	2	yes	—
326	C₂²	#15689176	1	2	yes	—
327	C₂²	#15689176	1	2	yes	—
328	C₂²	#15689176	1	2	yes	—
329	C₂²	#15689176	1	2	yes	—
330	C₂²	#15689180	1	2	yes	—
331	C₂²	#15689180	1	2	yes	—
332	C₂²	#15689180	1	2	yes	—
333	C₂²	#15689180	1	2	yes	—
334	C₄	—	2	8	no	—
335	C₄	—	2	8	no	—
336	C₄	—	2	4	no	—
337	C₄	—	2	4	no	—
338	C₄	#15689007	1	4	yes	—
339	C₄	#15689014	1	4	yes	—
340	C₂	#20241615	1	1	yes	characteristic
341	C₂	#20243134	1	1	yes	characteristic
342	C₂	#20243137	1	1	yes	characteristic
343	C₂	C₄×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
344	C₂	#20243150	1	1	yes	characteristic
345	C₂	C₄×C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
346	C₂	#20243241	1	1	yes	characteristic
347	C₂	#20243368	1	2	yes	—
348	C₂	#20243456	1	2	yes	—
349	C₂	#20243505	1	2	yes	—
350	C₂	#20243531	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂×C₄×C₈
Fitting	C₂×C₄³×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×C₄³	D₄₇
hypocenter	C₄₇	#10795
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
8. #15688878	C₄	semidirect
38. C₂×C₄²×C₄₇	C₈	semidirect
190. C₄₇	#10795	semidirect
214. C₂×C₄²	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688878	2
C₂×C₄³×C₄₇	2
#10795	47