Subgroups — C₂×C₄₇×#1995 #26061026

Order 12032.

1206 conjugacy classes of subgroups.

The subgroup lattice has 316 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×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₇×#376	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1995	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄₇×#339	C₂	1	4	yes	maximal
6	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
7	C₄₇×#533	—	2	2	no	—
8	C₄₇×#533	—	2	2	no	—
9	C₄₇×#533	C₄	1	4	yes	complemented
10	C₄₇×#533	C₂²	1	4	yes	—
11	C₂×C₄⋊Q₈×C₄₇	C₄	1	2	yes	complemented
12	C₂×#125×C₄₇	C₄	1	2	yes	complemented
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×C₄²×C₄₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#423	—	2	2	no	—
19	C₄₇×#423	C₂²	1	4	yes	—
20	C₂×C₄⋊C₈×C₄₇	—	2	2	no	—
21	C₄₇×#376	—	2	2	no	—
22	C₄₇×#376	C₂²	1	4	yes	—
23	C₂×#95×C₄₇	—	2	2	no	—
24	C₄₇×#339	C₂²	1	16	yes	—
25	C₂²×Q₈×C₄₇	—	2	2	no	—
26	C₂³×C₄×C₄₇	—	2	2	no	—
27	C₂³×C₄×C₄₇	D₄	1	2	yes	—
28	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₄₇	—	2	2	no	—
30	C₂×M₁₆×C₄₇	—	2	8	no	—
31	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
32	C₂²×C₈×C₄₇	—	2	2	no	—
33	C₄⋊Q₈×C₄₇	—	2	4	no	—
34	C₄⋊Q₈×C₄₇	C₂×C₄	1	8	yes	complemented
35	#125×C₄₇	—	2	4	no	—
36	#125×C₄₇	C₂×C₄	1	8	yes	complemented
37	C₂²⋊Q₈×C₄₇	—	2	8	no	—
38	C₂²⋊Q₈×C₄₇	—	2	8	no	—
39	C₄×Q₈×C₄₇	—	2	8	no	—
40	C₄×Q₈×C₄₇	—	2	8	no	—
41	#117×C₄₇	—	2	2	no	—
42	#117×C₄₇	—	2	2	no	—
43	#117×C₄₇	C₂×C₄	1	4	yes	—
44	#117×C₄₇	C₂³	1	4	yes	—
45	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
48	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
49	C₂×C₄²×C₄₇	—	2	2	no	—
50	C₂×C₄²×C₄₇	—	2	2	no	—
51	C₂×C₄²×C₄₇	—	2	4	no	—
52	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₄₇	D₄	1	2	yes	—
56	C₂×C₄²×C₄₇	D₄	1	2	yes	—
57	C₂×C₄²×C₄₇	D₄	1	2	yes	—
58	C₂×C₄²×C₄₇	Q₈	1	8	yes	complemented
59	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
60	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
61	C₄⋊C₈×C₄₇	—	2	8	no	—
62	#95×C₄₇	—	2	8	no	—
63	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂×Q₈×C₄₇	—	2	8	no	—
65	C₂×Q₈×C₄₇	—	4	8	no	—
66	C₂²×C₄×C₄₇	—	2	4	no	—
67	C₂²×C₄×C₄₇	—	2	8	no	—
68	C₂²×C₄×C₄₇	—	2	4	no	—
69	C₂²×C₄×C₄₇	—	4	4	no	—
70	C₂²×C₄×C₄₇	—	4	8	no	—
71	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
81	C₂²×C₄×C₄₇	#41	1	4	yes	—
82	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
83	M₁₆×C₄₇	—	2	8	no	—
84	M₁₆×C₄₇	—	4	4	no	—
85	C₂×C₈×C₄₇	—	2	4	no	—
86	C₂×C₈×C₄₇	—	4	4	no	—
87	C₄⋊C₄×C₄₇	—	2	16	no	—
88	C₄⋊C₄×C₄₇	—	2	8	no	—
89	C₄⋊C₄×C₄₇	—	2	4	no	—
90	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
91	C₂²⋊C₄×C₄₇	—	2	4	no	—
92	C₂²⋊C₄×C₄₇	—	2	4	no	—
93	C₄²×C₄₇	—	2	4	no	—
94	C₄²×C₄₇	—	2	8	no	—
95	C₄²×C₄₇	—	2	4	no	—
96	C₄²×C₄₇	—	2	4	no	—
97	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
98	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
99	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
100	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
101	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
102	C₄²×C₄₇	C₂×D₄	1	4	yes	—
103	C₄²×C₄₇	C₂×Q₈	1	8	yes	complemented
104	C₂³×C₄₇	—	2	4	no	—
105	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂³×C₄₇	C₂²⋊Q₈	1	4	yes	—
108	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
109	Q₈×C₄₇	—	4	16	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	2	8	no	—
113	C₂×C₄×C₄₇	—	2	8	no	—
114	C₂×C₄×C₄₇	—	4	4	no	—
115	C₂×C₄×C₄₇	—	4	4	no	—
116	C₂×C₄×C₄₇	—	4	8	no	—
117	C₂×C₄×C₄₇	—	4	16	no	—
118	C₂×C₄×C₄₇	—	4	8	no	—
119	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
120	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
121	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
124	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
125	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
126	C₂×C₄×C₄₇	#124	1	2	yes	—
127	C₂×C₄×C₄₇	#124	1	2	yes	—
128	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
129	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
130	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
133	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
134	C₈×C₄₇	—	4	4	no	—
135	C₂×#1995	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	2	8	no	—
139	C₂²×C₄₇	#391	1	2	yes	—
140	C₂²×C₄₇	#391	1	2	yes	—
141	C₂²×C₄₇	#421	1	4	yes	—
142	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
144	C₄×C₄₇	—	2	4	no	—
145	C₄×C₄₇	—	2	4	no	—
146	C₄×C₄₇	—	4	8	no	—
147	C₄×C₄₇	—	4	8	no	—
148	C₄×C₄₇	#391	1	2	yes	—
149	C₄×C₄₇	#391	1	2	yes	—
150	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
151	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
152	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
153	C₂×#533	C₂×C₄₇	1	1	yes	characteristic
154	C₂×#423	C₂×C₄₇	1	1	yes	characteristic
155	C₂×#376	C₂×C₄₇	1	1	yes	characteristic
156	#1995	C₂×C₄₇	1	8	yes	direct product factor
157	C₂×#339	C₂×C₄₇	1	4	yes	—
158	C₂×C₄₇	—	2	2	no	—
159	C₂×C₄₇	—	2	2	no	—
160	C₂×C₄₇	#1995	1	4	yes	direct product factor
161	C₂×C₄₇	C₂×#391	1	1	yes	characteristic
162	C₂×C₄₇	C₂×#421	1	2	yes	—
163	C₂²×M₁₆	C₂²×C₄₇	1	2	yes	—
164	#533	—	2	2	no	—
165	#533	—	2	2	no	—
166	#533	C₄×C₄₇	1	4	yes	complemented
167	#533	C₂²×C₄₇	1	4	yes	—
168	C₂×C₄⋊Q₈	C₄×C₄₇	1	2	yes	complemented
169	C₂×#125	C₄×C₄₇	1	2	yes	complemented
170	C₂×C₂²⋊Q₈	—	2	2	no	—
171	C₂×C₄×Q₈	—	2	2	no	—
172	C₂×#117	C₂²×C₄₇	1	2	yes	—
173	C₂²×C₄²	C₂²×C₄₇	1	2	yes	—
174	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
175	#423	—	2	2	no	—
176	#423	C₂²×C₄₇	1	4	yes	—
177	C₂×C₄⋊C₈	—	2	2	no	—
178	#376	—	2	2	no	—
179	#376	C₂²×C₄₇	1	4	yes	—
180	C₂×#95	—	2	2	no	—
181	#339	C₂²×C₄₇	1	16	yes	—
182	C₄₇	C₂×#1995	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²×Q₈	—	2	2	no	—
184	C₂³×C₄	—	2	2	no	—
185	C₂³×C₄	D₄×C₄₇	1	2	yes	—
186	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	8	no	—
189	C₂×M₁₆	C₂³×C₄₇	1	4	yes	—
190	C₂²×C₈	—	2	2	no	—
191	C₄⋊Q₈	—	2	4	no	—
192	C₄⋊Q₈	C₂×C₄×C₄₇	1	8	yes	complemented
193	#125	—	2	4	no	—
194	#125	C₂×C₄×C₄₇	1	8	yes	complemented
195	C₂²⋊Q₈	—	2	8	no	—
196	C₂²⋊Q₈	—	2	8	no	—
197	C₄×Q₈	—	2	8	no	—
198	C₄×Q₈	—	2	8	no	—
199	#117	—	2	2	no	—
200	#117	—	2	2	no	—
201	#117	C₂×C₄×C₄₇	1	4	yes	—
202	#117	C₂³×C₄₇	1	4	yes	—
203	C₂×C₄⋊C₄	—	2	4	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	4	yes	—
206	C₂×C₂²⋊C₄	—	2	2	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	—	2	2	no	—
209	C₂×C₄²	—	2	4	no	—
210	C₂×C₄²	C₂×C₄×C₄₇	1	1	yes	characteristic
211	C₂×C₄²	C₂×C₄×C₄₇	1	1	yes	characteristic
212	C₂×C₄²	C₂×C₄×C₄₇	1	2	yes	—
213	C₂×C₄²	D₄×C₄₇	1	2	yes	—
214	C₂×C₄²	D₄×C₄₇	1	2	yes	—
215	C₂×C₄²	D₄×C₄₇	1	2	yes	—
216	C₂×C₄²	Q₈×C₄₇	1	8	yes	complemented
217	C₂×C₄²	C₂³×C₄₇	1	4	yes	—
218	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
219	C₄⋊C₈	—	2	8	no	—
220	#95	—	2	8	no	—
221	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
222	C₂×Q₈	—	2	8	no	—
223	C₂×Q₈	—	4	8	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	8	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	4	4	no	—
228	C₂²×C₄	—	4	8	no	—
229	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
230	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
231	C₂²×C₄	C₂²⋊C₄×C₄₇	1	4	yes	—
232	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
233	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₄₇	1	4	yes	—
235	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
238	C₂²×C₄	C₂×Q₈×C₄₇	1	4	yes	—
239	C₂²×C₄	#41×C₄₇	1	4	yes	—
240	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
241	M₁₆	—	2	8	no	—
242	M₁₆	—	4	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	4	no	—
245	C₄⋊C₄	—	2	16	no	—
246	C₄⋊C₄	—	2	8	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	C₂²×C₄×C₄₇	1	8	yes	—
249	C₂²⋊C₄	—	2	4	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	8	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	4	no	—
255	C₄²	C₂²⋊C₄×C₄₇	1	4	yes	—
256	C₄²	C₂²⋊C₄×C₄₇	1	2	yes	—
257	C₄²	C₂²⋊C₄×C₄₇	1	2	yes	—
258	C₄²	C₂²×C₄×C₄₇	1	2	yes	—
259	C₄²	C₂²×C₄×C₄₇	1	2	yes	—
260	C₄²	C₂×D₄×C₄₇	1	4	yes	—
261	C₄²	C₂×Q₈×C₄₇	1	8	yes	complemented
262	C₂³	—	2	4	no	—
263	C₂³	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
264	C₂³	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
265	C₂³	C₂²⋊Q₈×C₄₇	1	4	yes	—
266	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
267	Q₈	—	4	16	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	8	no	—
275	C₂×C₄	—	4	16	no	—
276	C₂×C₄	—	4	8	no	—
277	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	4	yes	—
278	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	4	yes	—
279	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
280	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
281	C₂×C₄	C₄×Q₈×C₄₇	1	4	yes	—
282	C₂×C₄	C₄×Q₈×C₄₇	1	4	yes	—
283	C₂×C₄	C₂²⋊Q₈×C₄₇	1	4	yes	—
284	C₂×C₄	#124×C₄₇	1	2	yes	—
285	C₂×C₄	#124×C₄₇	1	2	yes	—
286	C₂×C₄	C₄⋊Q₈×C₄₇	1	2	yes	—
287	C₂×C₄	C₄⋊Q₈×C₄₇	1	2	yes	—
288	C₂×C₄	C₂³×C₄×C₄₇	1	1	yes	characteristic
289	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
290	C₂×C₄	C₂²×Q₈×C₄₇	1	2	yes	—
291	C₂×C₄	C₂×#41×C₄₇	1	2	yes	—
292	C₈	—	4	4	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	8	no	—
296	C₂²	C₄₇×#391	1	2	yes	—
297	C₂²	C₄₇×#391	1	2	yes	—
298	C₂²	C₄₇×#421	1	4	yes	—
299	C₂²	C₂²×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
300	C₂²	C₂×C₂²⋊Q₈×C₄₇	1	2	yes	—
301	C₄	—	2	4	no	—
302	C₄	—	2	4	no	—
303	C₄	—	4	8	no	—
304	C₄	—	4	8	no	—
305	C₄	C₄₇×#391	1	2	yes	—
306	C₄	C₄₇×#391	1	2	yes	—
307	C₄	C₂×C₄×Q₈×C₄₇	1	2	yes	—
308	C₄	C₂×#124×C₄₇	1	1	yes	characteristic
309	C₄	C₂×C₄⋊Q₈×C₄₇	1	1	yes	characteristic
310	C₂	—	2	2	no	—
311	C₂	—	2	2	no	—
312	C₂	C₄₇×#1995	1	4	yes	direct product factor
313	C₂	C₂×C₄₇×#391	1	1	yes	characteristic
314	C₂	C₂×C₄₇×#421	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₄₇×#1995	C₂	direct (N × H)
9. C₄₇×#533	C₄	semidirect
11. C₂×C₄⋊Q₈×C₄₇	C₄	semidirect
12. C₂×#125×C₄₇	C₄	semidirect
34. C₄⋊Q₈×C₄₇	C₂×C₄	semidirect
36. #125×C₄₇	C₂×C₄	semidirect
58. C₂×C₄²×C₄₇	Q₈	semidirect
103. C₄²×C₄₇	C₂×Q₈	semidirect
135. C₂×#1995	C₄₇	direct (N × H)
156. #1995	C₂×C₄₇	direct (N × H)
160. C₂×C₄₇	#1995	direct (N × H)
166. #533	C₄×C₄₇	semidirect
168. C₂×C₄⋊Q₈	C₄×C₄₇	semidirect
169. C₂×#125	C₄×C₄₇	semidirect
182. C₄₇	C₂×#1995	direct (N × H)
192. C₄⋊Q₈	C₂×C₄×C₄₇	semidirect
194. #125	C₂×C₄×C₄₇	semidirect
216. C₂×C₄²	Q₈×C₄₇	semidirect
261. C₄²	C₂×Q₈×C₄₇	semidirect
312. C₂	C₄₇×#1995	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×#533	2
C₂×C₄₇×#423	2
C₂×C₄₇×#376	2
C₄₇×#1995	2
C₂×C₄₇×#339	2
C₂×#1995	47

Subgroups — C2×C47×#1995 #26061026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₇×#1995 #26061026