Subgroups — C₂×#347×D₄₇ #26368930

Order 12032.

782 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₄₇	C₂	1	2	yes	maximal
2	C₂×#97×D₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688901	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15688900	C₂	1	2	yes	maximal, complemented
5	#347×D₄₇	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄₇×#347	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₄₇	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
9	C₂×C₄²×D₄₇	C₄	1	1	yes	characteristic
10	C₂×#13093964	C₂²	1	2	yes	—
11	C₂×#13093963	C₄	1	4	yes	complemented
12	M₃₂×D₄₇	C₄	1	8	yes	complemented
13	M₃₂×D₄₇	C₂²	1	8	yes	—
14	C₂×C₁₆×D₄₇	C₄	1	4	yes	complemented
15	C₂×#13093935	C₄	1	1	yes	characteristic
16	C₂×#13093934	C₄	1	1	yes	characteristic
17	C₂×#13093934	C₂²	1	1	yes	characteristic
18	#97×D₄₇	C₄	1	4	yes	—
19	#97×D₄₇	C₂²	1	4	yes	—
20	#15688901	C₂²	1	4	yes	complemented
21	#15688900	C₂²	1	8	yes	complemented
22	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
23	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#347	C₂²	1	4	yes	complemented
25	C₂²×C₄×D₄₇	—	2	1	no	—
26	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×D₄₇	—	2	1	no	—
30	C₂×C₈×D₄₇	—	2	2	no	—
31	C₂×C₈×D₄₇	—	2	1	no	—
32	C₂×C₈×D₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₄₇	C₂×C₄	1	4	yes	—
36	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₄₇	C₂³	1	2	yes	—
38	C₂×C₄×Dic₄₇	—	2	1	no	—
39	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
40	C₄²×D₄₇	—	2	2	no	—
41	C₄²×D₄₇	C₂×C₄	1	4	yes	—
42	#13093964	C₂×C₄	1	4	yes	complemented
43	#13093964	C₂³	1	4	yes	—
44	#13093963	C₂×C₄	1	16	yes	complemented
45	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
46	C₁₆×D₄₇	C₂×C₄	1	16	yes	complemented
47	#13093935	C₂×C₄	1	4	yes	—
48	#13093934	C₂×C₄	1	2	yes	—
49	#13093934	C₂×C₄	1	4	yes	—
50	#13093934	C₂³	1	2	yes	—
51	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
54	M₃₂×C₄₇	C₂×C₄	1	4	yes	complemented
55	M₃₂×C₄₇	C₂³	1	4	yes	—
56	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	complemented
57	#97×C₄₇	C₂×C₄	1	2	yes	—
58	#97×C₄₇	C₂³	1	2	yes	—
59	C₂³×D₄₇	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	—	2	1	no	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×Dic₄₇	—	2	1	no	—
63	C₂²×Dic₄₇	M₁₆	1	1	yes	characteristic
64	C₂×C₄×D₄₇	—	2	4	no	—
65	C₂×C₄×D₄₇	—	4	4	no	—
66	C₂×C₄×D₄₇	C₄²	1	2	yes	—
67	C₂×C₄×D₄₇	C₄²	1	1	yes	characteristic
68	C₂×C₄×D₄₇	C₄²	1	1	yes	characteristic
69	C₂×C₄×D₄₇	M₁₆	1	2	yes	—
70	C₂×C₄×D₄₇	M₁₆	1	1	yes	characteristic
71	C₂×C₄×D₄₇	M₁₆	1	2	yes	—
72	C₂×C₄×D₄₇	M₁₆	1	1	yes	characteristic
73	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₄₇	—	2	1	no	—
75	C₂×C₈×C₄₇	—	2	1	no	—
76	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₄²×C₄₇	—	2	1	no	—
82	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
83	C₄×Dic₄₇	—	2	1	no	—
84	C₄×Dic₄₇	—	2	4	no	—
85	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
86	C₂×C₄₇⋊C₈	—	2	1	no	—
87	C₂×C₄₇⋊C₈	—	2	1	no	—
88	C₂×C₄₇⋊C₈	C₄²	1	2	yes	—
89	C₂×C₄₇⋊C₈	C₄²	1	2	yes	—
90	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
91	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
92	C₈×D₄₇	—	2	4	no	—
93	C₈×D₄₇	—	2	4	no	—
94	C₈×D₄₇	C₄²	1	4	yes	—
95	C₈×D₄₇	C₄²	1	4	yes	—
96	C₈×D₄₇	C₂²×C₄	1	4	yes	—
97	C₈×D₄₇	C₂²×C₄	1	4	yes	—
98	C₁₆×C₄₇	C₂²×C₄	1	8	yes	complemented
99	C₄₇⋊C₁₆	C₂²×C₄	1	8	yes	complemented
100	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
101	C₂²×D₄₇	—	2	4	no	—
102	C₂²×D₄₇	#97	1	1	yes	characteristic
103	C₂²×D₄₇	#97	1	2	yes	—
104	C₂²×D₄₇	#97	1	1	yes	characteristic
105	C₂²×D₄₇	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	4	2	no	—
108	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×Dic₄₇	—	2	2	no	—
115	C₂×Dic₄₇	—	4	2	no	—
116	C₂×Dic₄₇	#97	1	2	yes	—
117	C₂×Dic₄₇	#97	1	1	yes	characteristic
118	C₂×Dic₄₇	#97	1	1	yes	characteristic
119	C₂×Dic₄₇	C₂×M₁₆	1	2	yes	—
120	C₄×D₄₇	—	4	8	no	—
121	C₄×D₄₇	#97	1	2	yes	—
122	C₄×D₄₇	#97	1	2	yes	—
123	C₄×D₄₇	#97	1	2	yes	—
124	C₄×D₄₇	#97	1	2	yes	—
125	C₄×D₄₇	C₂×C₄²	1	2	yes	—
126	C₄×D₄₇	C₂×C₄²	1	2	yes	—
127	C₄×D₄₇	C₂×M₁₆	1	2	yes	—
128	C₄×D₄₇	C₂×M₁₆	1	2	yes	—
129	C₈×C₄₇	—	2	2	no	—
130	C₈×C₄₇	C₂×C₄²	1	2	yes	—
131	C₈×C₄₇	C₂³×C₄	1	2	yes	—
132	C₄₇⋊C₈	—	2	2	no	—
133	C₄₇⋊C₈	C₂×C₄²	1	4	yes	—
134	C₂²×#347	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
137	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
138	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
139	C₂×D₄₇	—	2	2	no	—
140	C₂×D₄₇	—	2	4	no	—
141	C₂×D₄₇	—	2	4	no	—
142	C₂×D₄₇	#347	1	4	yes	direct product factor
143	C₂×D₄₇	C₂×#97	1	2	yes	—
144	C₂×D₄₇	C₂×#97	1	2	yes	—
145	C₄×C₄₇	—	4	2	no	—
146	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
147	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
148	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
149	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
150	Dic₄₇	—	4	2	no	—
151	Dic₄₇	C₂×#97	1	2	yes	—
152	Dic₄₇	C₂×#97	1	2	yes	—
153	C₂²×M₃₂	—	47	2	no	—
154	C₂²×#97	—	47	1	no	—
155	C₂×#347	—	47	2	no	—
156	C₂×#347	—	47	1	no	—
157	C₂×#347	—	47	8	no	—
158	C₂×#347	D₄₇	1	1	yes	characteristic, direct product factor
159	C₂×C₄₇	—	2	1	no	—
160	C₂×C₄₇	—	2	1	no	—
161	C₂×C₄₇	C₂×#347	1	2	yes	complemented
162	C₂×C₄₇	C₂²×#97	1	1	yes	characteristic
163	D₄₇	—	2	2	no	—
164	D₄₇	C₂×#347	1	4	yes	direct product factor
165	C₂³×C₈	—	47	1	no	—
166	C₂³×C₈	—	47	1	no	—
167	C₂²×C₄²	—	47	1	no	—
168	C₂×M₃₂	—	47	2	no	—
169	C₂×M₃₂	—	47	8	no	—
170	C₂×M₃₂	—	47	8	no	—
171	C₂×M₃₂	—	47	4	no	—
172	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
173	C₂²×C₁₆	—	47	4	no	—
174	C₂×#97	—	47	1	no	—
175	C₂×#97	—	47	1	no	—
176	C₂×#97	—	47	1	no	—
177	C₂×#97	—	47	4	no	—
178	C₂×#97	—	47	4	no	—
179	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
180	#347	—	47	8	no	—
181	#347	—	47	4	no	—
182	#347	C₂×D₄₇	1	4	yes	direct product factor
183	C₄₇	C₂²×#347	1	1	yes	characteristic, complemented, Sylow
184	C₂³×C₄	—	47	1	no	—
185	C₂³×C₄	—	94	1	no	—
186	C₂²×C₈	—	47	1	no	—
187	C₂²×C₈	—	47	2	no	—
188	C₂²×C₈	—	47	2	no	—
189	C₂²×C₈	—	47	4	no	—
190	C₂²×C₈	—	47	2	no	—
191	C₂²×C₈	—	47	2	no	—
192	C₂²×C₈	—	47	1	no	—
193	C₂²×C₈	—	47	4	no	—
194	C₂²×C₈	—	94	1	no	—
195	C₂²×C₈	—	94	1	no	—
196	C₂²×C₈	—	94	2	no	—
197	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
199	C₂×C₄²	—	47	1	no	—
200	C₂×C₄²	—	47	4	no	—
201	C₂×C₄²	—	94	2	no	—
202	C₂×C₄²	—	94	1	no	—
203	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
204	M₃₂	—	47	4	no	—
205	M₃₂	—	47	4	no	—
206	M₃₂	—	47	16	no	—
207	M₃₂	C₄×D₄₇	1	4	yes	complemented
208	M₃₂	C₂²×D₄₇	1	4	yes	—
209	C₂×C₁₆	—	47	4	no	—
210	C₂×C₁₆	—	47	16	no	—
211	C₂×C₁₆	C₄×D₄₇	1	4	yes	complemented
212	#97	—	47	4	no	—
213	#97	—	47	2	no	—
214	#97	—	47	2	no	—
215	#97	—	47	4	no	—
216	#97	C₄×D₄₇	1	2	yes	—
217	#97	C₂²×D₄₇	1	2	yes	—
218	C₂⁴	—	47	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	47	1	no	—
221	C₂²×C₄	—	47	2	no	—
222	C₂²×C₄	—	47	1	no	—
223	C₂²×C₄	—	47	1	no	—
224	C₂²×C₄	—	47	2	no	—
225	C₂²×C₄	—	47	1	no	—
226	C₂²×C₄	—	47	2	no	—
227	C₂²×C₄	—	47	1	no	—
228	C₂²×C₄	—	47	2	no	—
229	C₂²×C₄	—	94	4	no	—
230	C₂²×C₄	—	94	1	no	—
231	C₂²×C₄	—	188	4	no	—
232	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	47	2	no	—
236	C₂×C₈	—	47	2	no	—
237	C₂×C₈	—	47	2	no	—
238	C₂×C₈	—	47	4	no	—
239	C₂×C₈	—	47	4	no	—
240	C₂×C₈	—	47	4	no	—
241	C₂×C₈	—	47	4	no	—
242	C₂×C₈	—	47	2	no	—
243	C₂×C₈	—	94	4	no	—
244	C₂×C₈	—	94	4	no	—
245	C₂×C₈	—	94	1	no	—
246	C₂×C₈	—	94	1	no	—
247	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
248	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
249	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
250	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
251	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
252	C₄²	—	2	1	no	—
253	C₄²	—	47	2	no	—
254	C₄²	—	94	4	no	—
255	C₄²	—	94	1	no	—
256	C₄²	C₂×C₄×D₄₇	1	2	yes	—
257	C₁₆	—	47	8	no	—
258	C₁₆	C₂×C₄×D₄₇	1	8	yes	complemented
259	C₂³	—	47	1	no	—
260	C₂³	—	47	2	no	—
261	C₂³	—	47	1	no	—
262	C₂³	—	47	2	no	—
263	C₂³	—	94	4	no	—
264	C₂³	M₁₆×D₄₇	1	1	yes	characteristic
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	47	2	no	—
268	C₂×C₄	—	47	1	no	—
269	C₂×C₄	—	47	1	no	—
270	C₂×C₄	—	47	2	no	—
271	C₂×C₄	—	47	2	no	—
272	C₂×C₄	—	47	2	no	—
273	C₂×C₄	—	47	2	no	—
274	C₂×C₄	—	47	2	no	—
275	C₂×C₄	—	47	2	no	—
276	C₂×C₄	—	47	2	no	—
277	C₂×C₄	—	47	2	no	—
278	C₂×C₄	—	47	2	no	—
279	C₂×C₄	—	94	2	no	—
280	C₂×C₄	—	188	8	no	—
281	C₂×C₄	—	188	2	no	—
282	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
283	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
284	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
285	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
286	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
287	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
288	C₈	—	2	2	no	—
289	C₈	—	47	4	no	—
290	C₈	—	94	2	no	—
291	C₈	C₄²×D₄₇	1	2	yes	—
292	C₈	C₂²×C₄×D₄₇	1	2	yes	—
293	C₂²	—	2	2	no	—
294	C₂²	—	47	2	no	—
295	C₂²	—	47	4	no	—
296	C₂²	—	47	2	no	—
297	C₂²	—	94	4	no	—
298	C₂²	—	94	4	no	—
299	C₂²	—	94	2	no	—
300	C₂²	#97×D₄₇	1	1	yes	characteristic
301	C₂²	#97×D₄₇	1	1	yes	characteristic
302	C₂²	C₂×M₁₆×D₄₇	1	1	yes	characteristic
303	C₄	—	4	2	no	—
304	C₄	—	47	2	no	—
305	C₄	—	47	2	no	—
306	C₄	—	188	2	no	—
307	C₄	#97×D₄₇	1	1	yes	characteristic
308	C₄	#97×D₄₇	1	1	yes	characteristic
309	C₄	C₂×C₄²×D₄₇	1	1	yes	characteristic
310	C₄	C₂×M₁₆×D₄₇	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	1	no	—
313	C₂	—	47	4	no	—
314	C₂	—	94	2	no	—
315	C₂	#347×D₄₇	1	2	yes	direct product factor
316	C₂	C₂×#97×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄²×D₄₇
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#347	C₂
socle	C₂²×C₄₇	C₂×#97
prime radical	C₂³×D₄₇	M₁₆
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂×#347	D₄₇
hypocenter	C₄₇	C₂²×#347
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
3. C₂×#15688901	C₂	semidirect
4. C₂×#15688900	C₂	semidirect
5. #347×D₄₇	C₂	direct (N × H)
6. C₂×C₄₇×#347	C₂	semidirect
11. C₂×#13093963	C₄	semidirect
12. M₃₂×D₄₇	C₄	semidirect
14. C₂×C₁₆×D₄₇	C₄	semidirect
20. #15688901	C₂²	semidirect
21. #15688900	C₂²	semidirect
24. C₄₇×#347	C₂²	semidirect
42. #13093964	C₂×C₄	semidirect
44. #13093963	C₂×C₄	semidirect
45. C₂×C₄₇⋊C₁₆	C₂×C₄	semidirect
46. C₁₆×D₄₇	C₂×C₄	semidirect
54. M₃₂×C₄₇	C₂×C₄	semidirect
56. C₂×C₁₆×C₄₇	C₂×C₄	semidirect
98. C₁₆×C₄₇	C₂²×C₄	semidirect
99. C₄₇⋊C₁₆	C₂²×C₄	semidirect
142. C₂×D₄₇	#347	direct (N × H)
158. C₂×#347	D₄₇	direct (N × H)
161. C₂×C₄₇	C₂×#347	semidirect
164. D₄₇	C₂×#347	direct (N × H)
182. #347	C₂×D₄₇	direct (N × H)
183. C₄₇	C₂²×#347	semidirect
207. M₃₂	C₄×D₄₇	semidirect
211. C₂×C₁₆	C₄×D₄₇	semidirect
258. C₁₆	C₂×C₄×D₄₇	semidirect
315. C₂	#347×D₄₇	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×M₃₂×D₄₇	2
C₂×#97×D₄₇	2
C₂×#15688901	2
C₂×#15688900	2
#347×D₄₇	2
C₂×C₄₇×#347	2
C₂²×#347	47

Subgroups — C2×#347×D47 #26368930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#347×D₄₇ #26368930