Subgroups — C₂×C₄₇×#2026 #26063148

Order 12032.

1302 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#103×C₄₇	C₂	1	2	yes	maximal
3	C₂×C₄₇×#391	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#2026	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄₇×#340	C₂	1	2	yes	maximal
6	C₂×C₄₇×#336	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
9	C₂×#125×C₄₇	—	2	8	no	—
10	C₂×#125×C₄₇	—	2	2	no	—
11	C₂×#125×C₄₇	C₂²	1	4	yes	—
12	C₂²×C₄⋊C₄×C₄₇	—	2	2	no	—
13	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×#103×C₄₇	—	2	8	no	—
17	C₂×#103×C₄₇	C₂²	1	8	yes	—
18	C₄₇×#391	—	2	2	no	—
19	C₄₇×#391	C₂²	1	4	yes	—
20	C₂×#95×C₄₇	—	2	2	no	—
21	C₄₇×#340	C₂²	1	8	yes	—
22	C₄₇×#336	C₂²	1	4	yes	—
23	C₂²×Q₈×C₄₇	—	2	1	no	—
24	C₂²×Q₈×C₄₇	—	2	2	no	—
25	C₂²×Q₈×C₄₇	—	2	4	no	—
26	C₂²×Q₈×C₄₇	—	2	4	no	—
27	C₂²×Q₈×C₄₇	Q₈	1	4	yes	—
28	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
29	C₂³×C₄×C₄₇	—	2	2	no	—
30	C₂³×C₄×C₄₇	—	2	1	no	—
31	C₂³×C₄×C₄₇	—	2	1	no	—
32	C₂³×C₄×C₄₇	D₄	1	2	yes	—
33	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₄₇	—	2	4	no	—
35	C₂²×C₈×C₄₇	—	2	4	no	—
36	C₂²×C₈×C₄₇	—	2	4	no	—
37	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
38	#125×C₄₇	—	2	16	no	—
39	#125×C₄₇	—	2	16	no	—
40	C₂×C₄⋊C₄×C₄₇	—	2	16	no	—
41	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
42	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
44	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
46	C₂×C₄⋊C₄×C₄₇	D₄	1	8	yes	—
47	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
49	C₂×C₄²×C₄₇	—	2	4	no	—
50	C₂×C₄²×C₄₇	D₄	1	2	yes	—
51	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
52	#103×C₄₇	—	2	16	no	—
53	#95×C₄₇	—	2	8	no	—
54	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂×Q₈×C₄₇	—	2	4	no	—
56	C₂×Q₈×C₄₇	—	2	2	no	—
57	C₂×Q₈×C₄₇	—	2	16	no	—
58	C₂×Q₈×C₄₇	—	2	4	no	—
59	C₂×Q₈×C₄₇	—	2	4	no	—
60	C₂×Q₈×C₄₇	—	4	2	no	—
61	C₂×Q₈×C₄₇	—	4	2	no	—
62	C₂×Q₈×C₄₇	—	4	8	no	—
63	C₂×Q₈×C₄₇	C₂×Q₈	1	4	yes	—
64	C₂²×C₄×C₄₇	—	2	4	no	—
65	C₂²×C₄×C₄₇	—	2	8	no	—
66	C₂²×C₄×C₄₇	—	2	4	no	—
67	C₂²×C₄×C₄₇	—	2	4	no	—
68	C₂²×C₄×C₄₇	—	2	4	no	—
69	C₂²×C₄×C₄₇	—	2	4	no	—
70	C₂²×C₄×C₄₇	—	2	2	no	—
71	C₂²×C₄×C₄₇	—	2	2	no	—
72	C₂²×C₄×C₄₇	—	4	2	no	—
73	C₂²×C₄×C₄₇	—	4	2	no	—
74	C₂²×C₄×C₄₇	—	4	4	no	—
75	C₂²×C₄×C₄₇	—	4	4	no	—
76	C₂²×C₄×C₄₇	—	4	2	no	—
77	C₂²×C₄×C₄₇	—	4	4	no	—
78	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
79	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₄₇	#41	1	4	yes	—
83	C₂²×C₄×C₄₇	#41	1	2	yes	—
84	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₄₇	—	2	4	no	—
86	C₂×C₈×C₄₇	—	2	4	no	—
87	C₂×C₈×C₄₇	—	2	4	no	—
88	C₂×C₈×C₄₇	—	4	8	no	—
89	C₄⋊C₄×C₄₇	—	2	32	no	—
90	C₄⋊C₄×C₄₇	—	2	8	no	—
91	C₄⋊C₄×C₄₇	—	2	4	no	—
92	C₄⋊C₄×C₄₇	—	2	4	no	—
93	C₄⋊C₄×C₄₇	—	4	2	no	—
94	C₄⋊C₄×C₄₇	C₂×D₄	1	8	yes	—
95	C₄²×C₄₇	—	2	4	no	—
96	C₄²×C₄₇	—	2	4	no	—
97	C₂³×C₄₇	#120	1	2	yes	—
98	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
99	C₂³×C₄₇	#124	1	2	yes	—
100	C₂³×C₄₇	#135	1	4	yes	—
101	C₂³×C₄₇	#137	1	4	yes	—
102	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
103	Q₈×C₄₇	—	2	8	no	—
104	Q₈×C₄₇	—	4	4	no	—
105	Q₈×C₄₇	—	4	8	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	8	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	—	4	2	no	—
112	C₂×C₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	4	4	no	—
114	C₂×C₄×C₄₇	—	4	4	no	—
115	C₂×C₄×C₄₇	—	4	8	no	—
116	C₂×C₄×C₄₇	—	4	8	no	—
117	C₂×C₄×C₄₇	—	4	16	no	—
118	C₂×C₄×C₄₇	—	4	2	no	—
119	C₂×C₄×C₄₇	—	4	4	no	—
120	C₂×C₄×C₄₇	—	8	4	no	—
121	C₂×C₄×C₄₇	#120	1	2	yes	—
122	C₂×C₄×C₄₇	C₂²⋊Q₈	1	8	yes	—
123	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₄₇	#124	1	8	yes	—
125	C₂×C₄×C₄₇	#124	1	2	yes	—
126	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
127	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
129	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
130	C₈×C₄₇	—	4	4	no	—
131	C₂×#2026	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂²×C₄₇	#393	1	4	yes	—
133	C₂²×C₄₇	#452	1	8	yes	—
134	C₂²×C₄₇	#479	1	8	yes	—
135	C₂²×C₄₇	#488	1	4	yes	—
136	C₂²×C₄₇	#491	1	4	yes	—
137	C₂²×C₄₇	C₂×#120	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
139	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
140	C₂²×C₄₇	C₂×#135	1	2	yes	—
141	C₂²×C₄₇	C₂×#137	1	2	yes	—
142	C₄×C₄₇	—	4	4	no	—
143	C₄×C₄₇	—	4	4	no	—
144	C₄×C₄₇	—	4	8	no	—
145	C₄×C₄₇	—	8	2	no	—
146	C₄×C₄₇	#393	1	4	yes	—
147	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
148	C₄×C₄₇	C₂×#124	1	2	yes	—
149	C₂²×#125	C₂×C₄₇	1	1	yes	characteristic
150	C₂²×#103	C₂×C₄₇	1	2	yes	—
151	C₂×#391	C₂×C₄₇	1	1	yes	characteristic
152	#2026	C₂×C₄₇	1	8	yes	direct product factor
153	C₂×#340	C₂×C₄₇	1	2	yes	—
154	C₂×#336	C₂×C₄₇	1	1	yes	characteristic
155	C₂×C₄₇	#2026	1	8	yes	direct product factor
156	C₂×C₄₇	C₂×#393	1	1	yes	characteristic
157	C₂×C₄₇	C₂×#452	1	2	yes	—
158	C₂×C₄₇	C₂×#479	1	2	yes	—
159	C₂×C₄₇	C₂×#488	1	1	yes	characteristic
160	C₂×C₄₇	C₂×#491	1	1	yes	characteristic
161	C₂³×Q₈	C₂²×C₄₇	1	1	yes	characteristic
162	C₂³×C₈	C₂²×C₄₇	1	2	yes	—
163	C₂×#125	—	2	8	no	—
164	C₂×#125	—	2	2	no	—
165	C₂×#125	C₂²×C₄₇	1	4	yes	—
166	C₂²×C₄⋊C₄	—	2	2	no	—
167	C₂²×C₄⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
168	C₂²×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
169	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
170	C₂×#103	—	2	8	no	—
171	C₂×#103	C₂²×C₄₇	1	8	yes	—
172	#391	—	2	2	no	—
173	#391	C₂²×C₄₇	1	4	yes	—
174	C₂×#95	—	2	2	no	—
175	#340	C₂²×C₄₇	1	8	yes	—
176	#336	C₂²×C₄₇	1	4	yes	—
177	C₄₇	C₂×#2026	1	1	yes	characteristic, direct product factor, Sylow
178	C₂²×Q₈	—	2	1	no	—
179	C₂²×Q₈	—	2	2	no	—
180	C₂²×Q₈	—	2	4	no	—
181	C₂²×Q₈	—	2	4	no	—
182	C₂²×Q₈	Q₈×C₄₇	1	4	yes	—
183	C₂²×Q₈	C₂³×C₄₇	1	2	yes	—
184	C₂³×C₄	—	2	2	no	—
185	C₂³×C₄	—	2	1	no	—
186	C₂³×C₄	—	2	1	no	—
187	C₂³×C₄	D₄×C₄₇	1	2	yes	—
188	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
189	C₂²×C₈	—	2	4	no	—
190	C₂²×C₈	—	2	4	no	—
191	C₂²×C₈	—	2	4	no	—
192	C₂²×C₈	C₂³×C₄₇	1	4	yes	—
193	#125	—	2	16	no	—
194	#125	—	2	16	no	—
195	C₂×C₄⋊C₄	—	2	16	no	—
196	C₂×C₄⋊C₄	—	2	8	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	4	no	—
199	C₂×C₄⋊C₄	—	2	4	no	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	D₄×C₄₇	1	8	yes	—
202	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
203	C₂×C₄⋊C₄	C₂³×C₄₇	1	4	yes	—
204	C₂×C₄²	—	2	4	no	—
205	C₂×C₄²	D₄×C₄₇	1	2	yes	—
206	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
207	#103	—	2	16	no	—
208	#95	—	2	8	no	—
209	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
210	C₂×Q₈	—	2	4	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	2	16	no	—
213	C₂×Q₈	—	2	4	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	4	2	no	—
216	C₂×Q₈	—	4	2	no	—
217	C₂×Q₈	—	4	8	no	—
218	C₂×Q₈	C₂×Q₈×C₄₇	1	4	yes	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	—	4	4	no	—
230	C₂²×C₄	—	4	4	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	—	4	4	no	—
233	C₂²×C₄	C₂×D₄×C₄₇	1	4	yes	—
234	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₄₇	1	4	yes	—
236	C₂²×C₄	C₂×Q₈×C₄₇	1	2	yes	—
237	C₂²×C₄	#41×C₄₇	1	4	yes	—
238	C₂²×C₄	#41×C₄₇	1	2	yes	—
239	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	4	8	no	—
244	C₄⋊C₄	—	2	32	no	—
245	C₄⋊C₄	—	2	8	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	C₂×D₄×C₄₇	1	8	yes	—
250	C₄²	—	2	4	no	—
251	C₄²	—	2	4	no	—
252	C₂³	#120×C₄₇	1	2	yes	—
253	C₂³	C₂²⋊Q₈×C₄₇	1	2	yes	—
254	C₂³	#124×C₄₇	1	2	yes	—
255	C₂³	#135×C₄₇	1	4	yes	—
256	C₂³	#137×C₄₇	1	4	yes	—
257	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
258	Q₈	—	2	8	no	—
259	Q₈	—	4	4	no	—
260	Q₈	—	4	8	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	4	8	no	—
267	C₂×C₄	—	4	16	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	8	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	8	4	no	—
276	C₂×C₄	#120×C₄₇	1	2	yes	—
277	C₂×C₄	C₂²⋊Q₈×C₄₇	1	8	yes	—
278	C₂×C₄	C₂²⋊Q₈×C₄₇	1	2	yes	—
279	C₂×C₄	#124×C₄₇	1	8	yes	—
280	C₂×C₄	#124×C₄₇	1	2	yes	—
281	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
282	C₂×C₄	C₂²×Q₈×C₄₇	1	1	yes	characteristic
283	C₂×C₄	C₂×#41×C₄₇	1	2	yes	—
284	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
285	C₈	—	4	4	no	—
286	C₂²	C₄₇×#393	1	4	yes	—
287	C₂²	C₄₇×#452	1	8	yes	—
288	C₂²	C₄₇×#479	1	8	yes	—
289	C₂²	C₄₇×#488	1	4	yes	—
290	C₂²	C₄₇×#491	1	4	yes	—
291	C₂²	C₂×#120×C₄₇	1	1	yes	characteristic
292	C₂²	C₂×C₂²⋊Q₈×C₄₇	1	1	yes	characteristic
293	C₂²	C₂×#124×C₄₇	1	1	yes	characteristic
294	C₂²	C₂×#135×C₄₇	1	2	yes	—
295	C₂²	C₂×#137×C₄₇	1	2	yes	—
296	C₄	—	4	4	no	—
297	C₄	—	4	4	no	—
298	C₄	—	4	8	no	—
299	C₄	—	8	2	no	—
300	C₄	C₄₇×#393	1	4	yes	—
301	C₄	C₂×C₂²⋊Q₈×C₄₇	1	2	yes	—
302	C₄	C₂×#124×C₄₇	1	2	yes	—
303	C₂	C₄₇×#2026	1	8	yes	direct product factor
304	C₂	C₂×C₄₇×#393	1	1	yes	characteristic
305	C₂	C₂×C₄₇×#452	1	2	yes	—
306	C₂	C₂×C₄₇×#479	1	2	yes	—
307	C₂	C₂×C₄₇×#488	1	1	yes	characteristic
308	C₂	C₂×C₄₇×#491	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₄₇	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×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	direct (N × H)
131. C₂×#2026	C₄₇	direct (N × H)
152. #2026	C₂×C₄₇	direct (N × H)
155. C₂×C₄₇	#2026	direct (N × H)
177. C₄₇	C₂×#2026	direct (N × H)
303. C₂	C₄₇×#2026	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#125×C₄₇	2
C₂²×#103×C₄₇	2
C₂×C₄₇×#391	2
C₄₇×#2026	2
C₂×C₄₇×#340	2
C₂×C₄₇×#336	2
C₂×#2026	47

Subgroups — C2×C47×#2026 #26063148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₇×#2026 #26063148