Subgroups — C₄₇×#23455 #26063153

Order 12032.

1266 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₇×#419	C₂	1	2	yes	maximal
3	C₂×C₄₇×#391	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#2026	C₂	1	4	yes	maximal, complemented
5	C₄₇×#2024	C₂	1	4	yes	maximal, complemented
6	C₄₇×#1736	C₂	1	2	yes	maximal
7	C₄₇×#1728	C₂	1	1	yes	maximal, characteristic
8	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
10	C₄₇×#533	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
12	C₂×#125×C₄₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊Q₈×C₄₇	—	2	2	no	—
14	C₂×C₄×Q₈×C₄₇	—	2	2	no	—
15	C₂×#117×C₄₇	C₂²	1	2	yes	—
16	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#419	—	2	8	no	—
19	C₂×#103×C₄₇	C₂²	1	8	yes	—
20	C₄₇×#391	—	2	2	no	—
21	C₄₇×#391	C₂²	1	2	yes	—
22	C₄₇×#391	C₂²	1	2	yes	—
23	C₂×#95×C₄₇	—	2	2	no	—
24	C₄₇×#340	C₂²	1	8	yes	—
25	C₄₇×#336	C₂²	1	2	yes	—
26	C₄₇×#336	C₂²	1	2	yes	—
27	C₂²×Q₈×C₄₇	—	2	2	no	—
28	C₂²×Q₈×C₄₇	—	2	1	no	—
29	C₂²×Q₈×C₄₇	—	2	2	no	—
30	C₂²×Q₈×C₄₇	—	2	4	no	—
31	C₂²×Q₈×C₄₇	—	2	4	no	—
32	C₂²×Q₈×C₄₇	Q₈	1	4	yes	complemented
33	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
34	C₂³×C₄×C₄₇	—	2	1	no	—
35	C₂³×C₄×C₄₇	—	2	1	no	—
36	C₂³×C₄×C₄₇	D₄	1	2	yes	—
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₄₇	—	2	4	no	—
39	C₂×M₁₆×C₄₇	—	2	4	no	—
40	C₂×M₁₆×C₄₇	—	2	4	no	—
41	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
42	C₄⋊Q₈×C₄₇	—	2	8	no	—
43	#125×C₄₇	—	2	8	no	—
44	C₂²⋊Q₈×C₄₇	—	2	16	no	—
45	C₄×Q₈×C₄₇	—	2	16	no	—
46	#117×C₄₇	—	2	4	no	—
47	#117×C₄₇	D₄	1	8	yes	—
48	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
49	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
51	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
52	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
54	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
55	C₂×C₄²×C₄₇	—	2	4	no	—
56	C₂×C₄²×C₄₇	—	2	2	no	—
57	C₂×C₄²×C₄₇	D₄	1	2	yes	—
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	#103×C₄₇	—	2	16	no	—
61	#95×C₄₇	—	2	4	no	—
62	#95×C₄₇	—	2	4	no	—
63	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂×Q₈×C₄₇	—	2	8	no	—
65	C₂×Q₈×C₄₇	—	2	4	no	—
66	C₂×Q₈×C₄₇	—	2	2	no	—
67	C₂×Q₈×C₄₇	—	2	16	no	—
68	C₂×Q₈×C₄₇	—	2	4	no	—
69	C₂×Q₈×C₄₇	—	2	4	no	—
70	C₂×Q₈×C₄₇	—	4	4	no	—
71	C₂×Q₈×C₄₇	—	4	2	no	—
72	C₂×Q₈×C₄₇	—	4	2	no	—
73	C₂×Q₈×C₄₇	—	4	8	no	—
74	C₂×Q₈×C₄₇	—	4	4	no	—
75	C₂×Q₈×C₄₇	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₄₇	—	2	4	no	—
77	C₂²×C₄×C₄₇	—	2	4	no	—
78	C₂²×C₄×C₄₇	—	2	4	no	—
79	C₂²×C₄×C₄₇	—	2	2	no	—
80	C₂²×C₄×C₄₇	—	2	4	no	—
81	C₂²×C₄×C₄₇	—	2	2	no	—
82	C₂²×C₄×C₄₇	—	4	2	no	—
83	C₂²×C₄×C₄₇	—	4	2	no	—
84	C₂²×C₄×C₄₇	—	4	2	no	—
85	C₂²×C₄×C₄₇	—	4	4	no	—
86	C₂²×C₄×C₄₇	—	4	2	no	—
87	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
91	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
92	C₂²×C₄×C₄₇	#41	1	4	yes	—
93	C₂²×C₄×C₄₇	#41	1	2	yes	—
94	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
95	M₁₆×C₄₇	—	4	8	no	—
96	C₂×C₈×C₄₇	—	2	4	no	—
97	C₂×C₈×C₄₇	—	2	4	no	—
98	C₂×C₈×C₄₇	—	2	4	no	—
99	C₄⋊C₄×C₄₇	—	2	16	no	—
100	C₄⋊C₄×C₄₇	—	2	8	no	—
101	C₄⋊C₄×C₄₇	—	2	4	no	—
102	C₄⋊C₄×C₄₇	—	2	4	no	—
103	C₄⋊C₄×C₄₇	—	4	2	no	—
104	C₄⋊C₄×C₄₇	C₂×D₄	1	8	yes	—
105	C₂²⋊C₄×C₄₇	—	2	8	no	—
106	C₄²×C₄₇	—	2	4	no	—
107	C₄²×C₄₇	—	2	8	no	—
108	C₄²×C₄₇	—	2	2	no	—
109	C₄²×C₄₇	—	2	2	no	—
110	C₂³×C₄₇	—	2	2	no	—
111	C₂³×C₄₇	—	2	2	no	—
112	C₂³×C₄₇	#120	1	2	yes	—
113	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
114	C₂³×C₄₇	#124	1	2	yes	—
115	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
116	Q₈×C₄₇	—	2	8	no	—
117	Q₈×C₄₇	—	4	16	no	—
118	Q₈×C₄₇	—	4	4	no	—
119	Q₈×C₄₇	—	4	8	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	—	2	8	no	—
124	C₂×C₄×C₄₇	—	2	4	no	—
125	C₂×C₄×C₄₇	—	2	2	no	—
126	C₂×C₄×C₄₇	—	2	2	no	—
127	C₂×C₄×C₄₇	—	4	8	no	—
128	C₂×C₄×C₄₇	—	4	2	no	—
129	C₂×C₄×C₄₇	—	4	2	no	—
130	C₂×C₄×C₄₇	—	4	4	no	—
131	C₂×C₄×C₄₇	—	4	4	no	—
132	C₂×C₄×C₄₇	—	4	8	no	—
133	C₂×C₄×C₄₇	—	4	2	no	—
134	C₂×C₄×C₄₇	—	4	4	no	—
135	C₂×C₄×C₄₇	—	8	4	no	—
136	C₂×C₄×C₄₇	#120	1	2	yes	—
137	C₂×C₄×C₄₇	C₂²⋊Q₈	1	8	yes	—
138	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
139	C₂×C₄×C₄₇	#124	1	8	yes	—
140	C₂×C₄×C₄₇	#124	1	2	yes	—
141	C₂×C₄×C₄₇	#137	1	4	yes	—
142	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
143	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
145	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
146	C₈×C₄₇	—	4	4	no	—
147	#23455	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂²×C₄₇	—	2	4	no	—
149	C₂²×C₄₇	—	2	2	no	—
150	C₂²×C₄₇	—	2	2	no	—
151	C₂²×C₄₇	—	2	4	no	—
152	C₂²×C₄₇	#393	1	4	yes	—
153	C₂²×C₄₇	C₂×#120	1	1	yes	characteristic
154	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
155	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
156	C₂²×C₄₇	C₂×#137	1	2	yes	—
157	C₂²×C₄₇	#575	1	2	yes	—
158	C₄×C₄₇	—	2	4	no	—
159	C₄×C₄₇	—	4	4	no	—
160	C₄×C₄₇	—	4	2	no	—
161	C₄×C₄₇	—	4	8	no	—
162	C₄×C₄₇	—	8	2	no	—
163	C₄×C₄₇	#393	1	4	yes	—
164	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
165	C₄×C₄₇	C₂×#124	1	2	yes	—
166	C₂×#533	C₂×C₄₇	1	1	yes	characteristic
167	C₂×#419	C₂×C₄₇	1	2	yes	—
168	C₂×#391	C₂×C₄₇	1	1	yes	characteristic
169	#2026	C₂×C₄₇	1	4	yes	complemented
170	#2024	C₂×C₄₇	1	4	yes	complemented
171	#1736	C₂×C₄₇	1	2	yes	—
172	#1728	C₂×C₄₇	1	1	yes	characteristic
173	C₂×C₄₇	—	2	4	no	—
174	C₂×C₄₇	C₂×#393	1	1	yes	characteristic
175	C₂×C₄₇	#3007	1	2	yes	—
176	C₂×C₄₇	#3078	1	2	yes	—
177	C₂×C₄₇	#3139	1	1	yes	characteristic
178	C₂×C₄₇	#3140	1	1	yes	characteristic
179	C₂³×Q₈	C₂²×C₄₇	1	1	yes	characteristic
180	C₂²×M₁₆	C₂²×C₄₇	1	2	yes	—
181	#533	—	2	8	no	—
182	C₂×C₄⋊Q₈	C₂²×C₄₇	1	2	yes	—
183	C₂×#125	C₂²×C₄₇	1	2	yes	—
184	C₂×C₂²⋊Q₈	—	2	2	no	—
185	C₂×C₄×Q₈	—	2	2	no	—
186	C₂×#117	C₂²×C₄₇	1	2	yes	—
187	C₂²×C₄⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
188	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
189	#419	—	2	8	no	—
190	C₂×#103	C₂²×C₄₇	1	8	yes	—
191	#391	—	2	2	no	—
192	#391	C₂²×C₄₇	1	2	yes	—
193	#391	C₂²×C₄₇	1	2	yes	—
194	C₂×#95	—	2	2	no	—
195	#340	C₂²×C₄₇	1	8	yes	—
196	#336	C₂²×C₄₇	1	2	yes	—
197	#336	C₂²×C₄₇	1	2	yes	—
198	C₄₇	#23455	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²×Q₈	—	2	2	no	—
200	C₂²×Q₈	—	2	1	no	—
201	C₂²×Q₈	—	2	2	no	—
202	C₂²×Q₈	—	2	4	no	—
203	C₂²×Q₈	—	2	4	no	—
204	C₂²×Q₈	Q₈×C₄₇	1	4	yes	complemented
205	C₂²×Q₈	C₂³×C₄₇	1	2	yes	—
206	C₂³×C₄	—	2	1	no	—
207	C₂³×C₄	—	2	1	no	—
208	C₂³×C₄	D₄×C₄₇	1	2	yes	—
209	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
210	C₂×M₁₆	—	2	4	no	—
211	C₂×M₁₆	—	2	4	no	—
212	C₂×M₁₆	—	2	4	no	—
213	C₂²×C₈	C₂³×C₄₇	1	4	yes	—
214	C₄⋊Q₈	—	2	8	no	—
215	#125	—	2	8	no	—
216	C₂²⋊Q₈	—	2	16	no	—
217	C₄×Q₈	—	2	16	no	—
218	#117	—	2	4	no	—
219	#117	D₄×C₄₇	1	8	yes	—
220	C₂×C₄⋊C₄	—	2	4	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	4	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
225	C₂×C₄⋊C₄	C₂³×C₄₇	1	4	yes	—
226	C₂×C₂²⋊C₄	—	2	2	no	—
227	C₂×C₄²	—	2	2	no	—
228	C₂×C₄²	—	2	4	no	—
229	C₂×C₄²	D₄×C₄₇	1	2	yes	—
230	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
231	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
232	#103	—	2	16	no	—
233	#95	—	2	4	no	—
234	#95	—	2	4	no	—
235	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
236	C₂×Q₈	—	2	8	no	—
237	C₂×Q₈	—	2	4	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	16	no	—
240	C₂×Q₈	—	2	4	no	—
241	C₂×Q₈	—	2	4	no	—
242	C₂×Q₈	—	4	4	no	—
243	C₂×Q₈	—	4	2	no	—
244	C₂×Q₈	—	4	2	no	—
245	C₂×Q₈	—	4	8	no	—
246	C₂×Q₈	—	4	4	no	—
247	C₂×Q₈	C₂×Q₈×C₄₇	1	4	yes	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	4	no	—
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	4	2	no	—
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	—	4	4	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
260	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
261	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₄₇	1	4	yes	—
263	C₂²×C₄	C₂×Q₈×C₄₇	1	2	yes	—
264	C₂²×C₄	#41×C₄₇	1	4	yes	—
265	C₂²×C₄	#41×C₄₇	1	2	yes	—
266	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
267	M₁₆	—	4	8	no	—
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	2	4	no	—
270	C₂×C₈	—	2	4	no	—
271	C₄⋊C₄	—	2	16	no	—
272	C₄⋊C₄	—	2	8	no	—
273	C₄⋊C₄	—	2	4	no	—
274	C₄⋊C₄	—	2	4	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	C₂×D₄×C₄₇	1	8	yes	—
277	C₂²⋊C₄	—	2	8	no	—
278	C₄²	—	2	4	no	—
279	C₄²	—	2	8	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₂³	—	2	2	no	—
283	C₂³	—	2	2	no	—
284	C₂³	#120×C₄₇	1	2	yes	—
285	C₂³	C₂²⋊Q₈×C₄₇	1	2	yes	—
286	C₂³	#124×C₄₇	1	2	yes	—
287	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
288	Q₈	—	2	8	no	—
289	Q₈	—	4	16	no	—
290	Q₈	—	4	4	no	—
291	Q₈	—	4	8	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	8	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	4	8	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	8	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	8	4	no	—
308	C₂×C₄	#120×C₄₇	1	2	yes	—
309	C₂×C₄	C₂²⋊Q₈×C₄₇	1	8	yes	—
310	C₂×C₄	C₂²⋊Q₈×C₄₇	1	2	yes	—
311	C₂×C₄	#124×C₄₇	1	8	yes	—
312	C₂×C₄	#124×C₄₇	1	2	yes	—
313	C₂×C₄	#137×C₄₇	1	4	yes	—
314	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
315	C₂×C₄	C₂²×Q₈×C₄₇	1	1	yes	characteristic
316	C₂×C₄	C₂×#41×C₄₇	1	2	yes	—
317	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
318	C₈	—	4	4	no	—
319	C₂²	—	2	4	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	4	no	—
323	C₂²	C₄₇×#393	1	4	yes	—
324	C₂²	C₂×#120×C₄₇	1	1	yes	characteristic
325	C₂²	C₂×C₂²⋊Q₈×C₄₇	1	1	yes	characteristic
326	C₂²	C₂×#124×C₄₇	1	1	yes	characteristic
327	C₂²	C₂×#137×C₄₇	1	2	yes	—
328	C₂²	C₄₇×#575	1	2	yes	—
329	C₄	—	2	4	no	—
330	C₄	—	4	4	no	—
331	C₄	—	4	2	no	—
332	C₄	—	4	8	no	—
333	C₄	—	8	2	no	—
334	C₄	C₄₇×#393	1	4	yes	—
335	C₄	C₂×C₂²⋊Q₈×C₄₇	1	2	yes	—
336	C₄	C₂×#124×C₄₇	1	2	yes	—
337	C₂	—	2	4	no	—
338	C₂	C₂×C₄₇×#393	1	1	yes	characteristic
339	C₂	C₄₇×#3007	1	2	yes	—
340	C₂	C₄₇×#3078	1	2	yes	—
341	C₂	C₄₇×#3139	1	1	yes	characteristic
342	C₂	C₄₇×#3140	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
prime radical	C₂⁴×C₄₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ C₂²×C₄²×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄₇×#2026	C₂	semidirect
5. C₄₇×#2024	C₂	semidirect
32. C₂²×Q₈×C₄₇	Q₈	semidirect
147. #23455	C₄₇	direct (N × H)
169. #2026	C₂×C₄₇	semidirect
170. #2024	C₂×C₄₇	semidirect
198. C₄₇	#23455	direct (N × H)
204. C₂²×Q₈	Q₈×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₇×#533	2
C₂×C₄₇×#419	2
C₂×C₄₇×#391	2
C₄₇×#2026	2
C₄₇×#2024	2
C₄₇×#1736	2
C₄₇×#1728	2
#23455	47

Subgroups — C47×#23455 #26063153

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#23455 #26063153