Subgroups — C₄₇×#21958 #26061656

Order 12032.

790 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#433	C₂	1	2	yes	maximal
2	C₄₇×#2296	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1940	C₂	1	4	yes	maximal, complemented
4	C₄₇×#1939	C₂	1	4	yes	maximal, complemented
5	C₄₇×#1798	C₂	1	2	yes	maximal
6	C₄₇×#1797	C₂	1	2	yes	maximal
7	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
8	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
10	C₄₇×#433	—	2	8	no	—
11	C₂×C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
12	C₂×C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₄₇	C₂²	1	4	yes	—
16	C₂×#97×C₄₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
18	C₄₇×#389	C₄	1	2	yes	—
19	C₄₇×#389	C₄	1	2	yes	—
20	C₄₇×#385	C₄	1	2	yes	—
21	C₄₇×#385	C₄	1	2	yes	—
22	C₄×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
23	C₄₇×#336	C₂²	1	2	yes	—
24	C₄₇×#336	C₂²	1	2	yes	—
25	C₄₇×#336	C₂²	1	4	yes	—
26	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
30	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
31	#117×C₄₇	—	4	2	no	—
32	#117×C₄₇	D₄	1	4	yes	—
33	#117×C₄₇	Q₈	1	4	yes	—
34	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₄₇	—	4	2	no	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
46	C₄⋊C₈×C₄₇	—	2	8	no	—
47	C₄⋊C₈×C₄₇	—	4	4	no	—
48	C₄⋊C₈×C₄₇	—	4	2	no	—
49	C₄⋊C₈×C₄₇	—	4	2	no	—
50	C₂²⋊C₈×C₄₇	—	2	8	no	—
51	#97×C₄₇	—	2	4	no	—
52	C₄×C₈×C₄₇	—	2	4	no	—
53	#95×C₄₇	C₂×C₄	1	4	yes	—
54	#95×C₄₇	C₂×C₄	1	4	yes	—
55	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	4	1	no	—
60	C₂²×C₄×C₄₇	—	4	1	no	—
61	C₂²×C₄×C₄₇	—	4	2	no	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂×C₈×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	—	2	4	no	—
74	C₂×C₈×C₄₇	—	2	4	no	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	—	2	4	no	—
77	C₄⋊C₄×C₄₇	—	2	8	no	—
78	C₄⋊C₄×C₄₇	—	4	4	no	—
79	C₄⋊C₄×C₄₇	—	4	1	no	—
80	C₄⋊C₄×C₄₇	—	4	1	no	—
81	C₄⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
82	C₂²⋊C₄×C₄₇	—	4	2	no	—
83	C₂²⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
84	C₄²×C₄₇	—	2	4	no	—
85	C₄²×C₄₇	—	2	2	no	—
86	C₄²×C₄₇	—	2	2	no	—
87	C₄²×C₄₇	—	4	2	no	—
88	C₄²×C₄₇	C₂×D₄	1	4	yes	—
89	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
90	C₂³×C₄₇	—	2	1	no	—
91	C₂³×C₄₇	—	2	1	no	—
92	C₂³×C₄₇	—	2	2	no	—
93	C₂³×C₄₇	—	2	2	no	—
94	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	1	no	—
98	C₂×C₄×C₄₇	—	2	1	no	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	4	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	2	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	4	4	no	—
112	C₂×C₄×C₄₇	—	4	4	no	—
113	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
114	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₄₇	#131	1	4	yes	—
116	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
119	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
120	C₈×C₄₇	—	4	4	no	—
121	C₈×C₄₇	—	4	4	no	—
122	#21958	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₄₇	—	2	2	no	—
124	C₂²×C₄₇	—	2	4	no	—
125	C₂²×C₄₇	—	4	1	no	—
126	C₂²×C₄₇	—	4	1	no	—
127	C₂²×C₄₇	—	4	2	no	—
128	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₄₇	#518	1	1	yes	characteristic
130	C₂²×C₄₇	#519	1	1	yes	characteristic
131	C₂²×C₄₇	C₂×#131	1	2	yes	—
132	C₂²×C₄₇	#568	1	2	yes	—
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	2	8	no	—
135	C₄×C₄₇	—	2	4	no	—
136	C₄×C₄₇	—	4	2	no	—
137	C₄×C₄₇	—	4	2	no	—
138	C₄×C₄₇	—	4	2	no	—
139	C₄×C₄₇	#537	1	2	yes	—
140	C₄×C₄₇	#544	1	2	yes	—
141	C₂×#433	C₂×C₄₇	1	2	yes	—
142	#2296	C₂×C₄₇	1	1	yes	characteristic
143	#1940	C₂×C₄₇	1	4	yes	complemented
144	#1939	C₂×C₄₇	1	4	yes	complemented
145	#1798	C₂×C₄₇	1	2	yes	—
146	#1797	C₂×C₄₇	1	2	yes	—
147	C₂×C₄₇	—	4	2	no	—
148	C₂×C₄₇	#2316	1	1	yes	characteristic
149	C₂×C₄₇	#2906	1	2	yes	—
150	C₂×C₄₇	#2977	1	1	yes	characteristic
151	C₂×C₄₇	#2978	1	1	yes	characteristic
152	C₂×C₄₇	#2985	1	2	yes	—
153	C₂×#117	C₂²×C₄₇	1	1	yes	characteristic
154	C₂×#117	C₂²×C₄₇	1	1	yes	characteristic
155	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
156	#433	—	2	8	no	—
157	C₂×C₄⋊C₈	C₂²×C₄₇	1	4	yes	—
158	C₂×C₄⋊C₈	C₂²×C₄₇	1	4	yes	—
159	C₂×C₄⋊C₈	C₂²×C₄₇	1	2	yes	—
160	C₂×C₄⋊C₈	C₂²×C₄₇	1	2	yes	—
161	C₂×C₂²⋊C₈	C₂²×C₄₇	1	4	yes	—
162	C₂×#97	C₂²×C₄₇	1	2	yes	—
163	C₂×C₄×C₈	C₂²×C₄₇	1	2	yes	—
164	#389	C₄×C₄₇	1	2	yes	—
165	#389	C₄×C₄₇	1	2	yes	—
166	#385	C₄×C₄₇	1	2	yes	—
167	#385	C₄×C₄₇	1	2	yes	—
168	C₄×C₄⋊C₄	C₂²×C₄₇	1	4	yes	—
169	#336	C₂²×C₄₇	1	2	yes	—
170	#336	C₂²×C₄₇	1	2	yes	—
171	#336	C₂²×C₄₇	1	4	yes	—
172	C₄₇	#21958	1	1	yes	characteristic, direct product factor, Sylow
173	C₂³×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
174	C₂³×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
175	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
176	C₂²×C₈	C₂³×C₄₇	1	4	yes	—
177	C₂²×C₈	C₂³×C₄₇	1	4	yes	—
178	#117	—	4	2	no	—
179	#117	D₄×C₄₇	1	4	yes	—
180	#117	Q₈×C₄₇	1	4	yes	—
181	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	4	yes	—
182	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
183	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	4	yes	—
185	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
186	C₂×C₂²⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
187	C₂×C₂²⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
188	C₂×C₄²	—	4	2	no	—
189	C₂×C₄²	C₂×C₄×C₄₇	1	2	yes	—
190	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
191	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
192	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
193	C₄⋊C₈	—	2	8	no	—
194	C₄⋊C₈	—	4	4	no	—
195	C₄⋊C₈	—	4	2	no	—
196	C₄⋊C₈	—	4	2	no	—
197	C₂²⋊C₈	—	2	8	no	—
198	#97	—	2	4	no	—
199	C₄×C₈	—	2	4	no	—
200	#95	C₂×C₄×C₄₇	1	4	yes	—
201	#95	C₂×C₄×C₄₇	1	4	yes	—
202	C₂⁴	C₂²×C₄×C₄₇	1	1	yes	characteristic
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	4	1	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
210	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
211	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
212	C₂²×C₄	C₂²×C₄×C₄₇	1	4	yes	—
213	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
214	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
216	C₂²×C₄	C₂×Q₈×C₄₇	1	2	yes	—
217	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₄⋊C₄	—	2	8	no	—
225	C₄⋊C₄	—	4	4	no	—
226	C₄⋊C₄	—	4	1	no	—
227	C₄⋊C₄	—	4	1	no	—
228	C₄⋊C₄	C₄⋊C₄×C₄₇	1	8	yes	—
229	C₂²⋊C₄	—	4	2	no	—
230	C₂²⋊C₄	C₄⋊C₄×C₄₇	1	8	yes	—
231	C₄²	—	2	4	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	4	2	no	—
235	C₄²	C₂×D₄×C₄₇	1	4	yes	—
236	C₄²	C₂×Q₈×C₄₇	1	4	yes	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	C₂×C₄⋊C₄×C₄₇	1	2	yes	—
242	C₂³	C₂³×C₄×C₄₇	1	1	yes	characteristic
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	C₂×C₄⋊C₄×C₄₇	1	8	yes	—
261	C₂×C₄	C₂×C₄⋊C₄×C₄₇	1	2	yes	—
262	C₂×C₄	#131×C₄₇	1	4	yes	—
263	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
264	C₂×C₄	C₂²×Q₈×C₄₇	1	1	yes	characteristic
265	C₂×C₄	H(5,𝔽₂)×C₄₇	1	2	yes	—
266	C₂×C₄	2₋¹⁺⁴×C₄₇	1	2	yes	—
267	C₈	—	4	4	no	—
268	C₈	—	4	4	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	2	no	—
274	C₂²	C₂²×C₄⋊C₄×C₄₇	1	1	yes	characteristic
275	C₂²	C₄₇×#518	1	1	yes	characteristic
276	C₂²	C₄₇×#519	1	1	yes	characteristic
277	C₂²	C₂×#131×C₄₇	1	2	yes	—
278	C₂²	C₄₇×#568	1	2	yes	—
279	C₄	—	2	2	no	—
280	C₄	—	2	8	no	—
281	C₄	—	2	4	no	—
282	C₄	—	4	2	no	—
283	C₄	—	4	2	no	—
284	C₄	—	4	2	no	—
285	C₄	C₄₇×#537	1	2	yes	—
286	C₄	C₄₇×#544	1	2	yes	—
287	C₂	—	4	2	no	—
288	C₂	C₄₇×#2316	1	1	yes	characteristic
289	C₂	C₄₇×#2906	1	2	yes	—
290	C₂	C₄₇×#2977	1	1	yes	characteristic
291	C₂	C₄₇×#2978	1	1	yes	characteristic
292	C₂	C₄₇×#2985	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₄₇×#1940	C₂	semidirect
4. C₄₇×#1939	C₂	semidirect
122. #21958	C₄₇	direct (N × H)
143. #1940	C₂×C₄₇	semidirect
144. #1939	C₂×C₄₇	semidirect
172. C₄₇	#21958	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₇×#433	2
C₄₇×#2296	2
C₄₇×#1940	2
C₄₇×#1939	2
C₄₇×#1798	2
C₄₇×#1797	2
#21958	47

Subgroups — C47×#21958 #26061656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#21958 #26061656