Subgroups — C₄₇×#21214 #26060912

Order 12032.

1598 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#537	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₇×#410	C₂	1	4	yes	maximal, complemented
3	C₄₇×#2316	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#2274	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1964	C₂	1	8	yes	maximal, complemented
6	C₂²×#41×C₄₇	C₂²	1	2	yes	—
7	C₄₇×#537	—	2	4	no	—
8	C₄₇×#537	—	2	4	no	—
9	C₂×C₂²⋊Q₈×C₄₇	C₄	1	4	yes	complemented
10	C₂×#121×C₄₇	C₄	1	4	yes	complemented
11	C₂×#121×C₄₇	C₂²	1	4	yes	—
12	C₂×#117×C₄₇	C₂²	1	2	yes	—
13	C₂×#117×C₄₇	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#410	—	2	16	no	—
16	C₂×#99×C₄₇	C₂²	1	16	yes	—
17	C₄₇×#389	—	2	4	no	—
18	C₄₇×#380	C₂²	1	4	yes	—
19	C₄×C₄⋊C₄×C₄₇	—	2	4	no	—
20	C₄×C₂²⋊C₄×C₄₇	C₂²	1	4	yes	—
21	C₂×#41×C₄₇	—	2	4	no	—
22	C₂×#41×C₄₇	—	2	4	no	—
23	C₂×#41×C₄₇	—	2	8	no	—
24	C₂×#41×C₄₇	—	2	8	no	—
25	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
26	C₂²×D₄×C₄₇	C₂×C₄	1	2	yes	—
27	C₂²×D₄×C₄₇	C₂×C₄	1	4	yes	—
28	C₂²×D₄×C₄₇	C₂³	1	4	yes	—
29	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
30	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
31	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂²⋊Q₈×C₄₇	—	2	16	no	—
33	#121×C₄₇	—	2	8	no	—
34	#121×C₄₇	—	2	8	no	—
35	#121×C₄₇	—	2	16	no	—
36	#117×C₄₇	—	2	4	no	—
37	#117×C₄₇	—	2	8	no	—
38	#117×C₄₇	D₄	1	8	yes	complemented
39	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
43	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
44	C₂×C₄⋊C₄×C₄₇	D₄	1	4	yes	—
45	C₂×C₄⋊C₄×C₄₇	D₄	1	4	yes	—
46	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
47	C₂×C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
48	C₂×C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
49	C₂×C₄²×C₄₇	—	2	2	no	—
50	C₂×C₄²×C₄₇	—	2	2	no	—
51	C₂×C₄²×C₄₇	—	2	4	no	—
52	#99×C₄₇	—	2	32	no	—
53	#95×C₄₇	—	2	4	no	—
54	#95×C₄₇	—	2	4	no	—
55	C₂⁴×C₄₇	C₂²×C₄	1	4	yes	—
56	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
57	#41×C₄₇	—	4	8	no	—
58	#41×C₄₇	—	4	8	no	—
59	#41×C₄₇	—	4	8	no	—
60	#41×C₄₇	—	4	8	no	—
61	C₂×Q₈×C₄₇	—	2	4	no	—
62	C₂×Q₈×C₄₇	—	2	8	no	—
63	C₂×D₄×C₄₇	—	2	4	no	—
64	C₂×D₄×C₄₇	—	2	4	no	—
65	C₂×D₄×C₄₇	—	2	8	no	—
66	C₂×D₄×C₄₇	—	2	8	no	—
67	C₂×D₄×C₄₇	—	2	4	no	—
68	C₂×D₄×C₄₇	—	2	8	no	—
69	C₂×D₄×C₄₇	—	2	8	no	—
70	C₂×D₄×C₄₇	—	8	4	no	—
71	C₂²×C₄×C₄₇	—	2	4	no	—
72	C₂²×C₄×C₄₇	—	2	8	no	—
73	C₂²×C₄×C₄₇	—	2	16	no	—
74	C₂²×C₄×C₄₇	—	2	4	no	—
75	C₂²×C₄×C₄₇	—	8	2	no	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
84	C₂²×C₄×C₄₇	#41	1	4	yes	—
85	C₄⋊C₄×C₄₇	—	2	8	no	—
86	C₄⋊C₄×C₄₇	—	2	8	no	—
87	C₄⋊C₄×C₄₇	—	4	4	no	—
88	C₄⋊C₄×C₄₇	—	8	4	no	—
89	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	16	yes	—
90	C₂²⋊C₄×C₄₇	—	2	4	no	—
91	C₂²⋊C₄×C₄₇	—	2	16	no	—
92	C₂²⋊C₄×C₄₇	—	2	8	no	—
93	C₂²⋊C₄×C₄₇	C₂×D₄	1	8	yes	—
94	C₄²×C₄₇	—	2	8	no	—
95	C₄²×C₄₇	—	4	4	no	—
96	C₄²×C₄₇	—	8	4	no	—
97	C₂³×C₄₇	—	2	4	no	—
98	C₂³×C₄₇	—	2	8	no	—
99	C₂³×C₄₇	—	4	1	no	—
100	C₂³×C₄₇	—	4	1	no	—
101	C₂³×C₄₇	—	4	4	no	—
102	C₂³×C₄₇	—	4	4	no	—
103	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
104	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
105	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
106	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
107	C₂³×C₄₇	C₂×#41	1	2	yes	—
108	Q₈×C₄₇	—	4	8	no	—
109	D₄×C₄₇	—	4	8	no	—
110	D₄×C₄₇	—	4	4	no	—
111	D₄×C₄₇	—	4	4	no	—
112	D₄×C₄₇	—	4	8	no	—
113	D₄×C₄₇	—	8	8	no	—
114	C₂×C₄×C₄₇	—	2	4	no	—
115	C₂×C₄×C₄₇	—	2	8	no	—
116	C₂×C₄×C₄₇	—	2	8	no	—
117	C₂×C₄×C₄₇	—	4	4	no	—
118	C₂×C₄×C₄₇	—	4	4	no	—
119	C₂×C₄×C₄₇	—	4	4	no	—
120	C₂×C₄×C₄₇	—	4	4	no	—
121	C₂×C₄×C₄₇	—	4	4	no	—
122	C₂×C₄×C₄₇	—	4	16	no	—
123	C₂×C₄×C₄₇	—	4	4	no	—
124	C₂×C₄×C₄₇	—	4	4	no	—
125	C₂×C₄×C₄₇	—	4	4	no	—
126	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
127	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
128	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
129	C₂×C₄×C₄₇	#121	1	4	yes	—
130	C₂×C₄×C₄₇	#121	1	4	yes	—
131	C₂×C₄×C₄₇	#124	1	4	yes	—
132	C₂×C₄×C₄₇	#127	1	4	yes	—
133	#21214	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂²×C₄₇	—	2	1	no	—
135	C₂²×C₄₇	—	2	1	no	—
136	C₂²×C₄₇	—	2	1	no	—
137	C₂²×C₄₇	—	2	1	no	—
138	C₂²×C₄₇	—	2	1	no	—
139	C₂²×C₄₇	—	2	1	no	—
140	C₂²×C₄₇	—	4	2	no	—
141	C₂²×C₄₇	—	4	2	no	—
142	C₂²×C₄₇	—	4	4	no	—
143	C₂²×C₄₇	—	4	4	no	—
144	C₂²×C₄₇	—	4	4	no	—
145	C₂²×C₄₇	—	8	8	no	—
146	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×C₄×D₄	1	2	yes	—
148	C₂²×C₄₇	C₂×#121	1	2	yes	—
149	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
150	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
151	C₄×C₄₇	—	2	8	no	—
152	C₄×C₄₇	—	4	8	no	—
153	C₄×C₄₇	—	4	4	no	—
154	C₄×C₄₇	—	8	4	no	—
155	C₄×C₄₇	#390	1	8	yes	—
156	C₂×#537	C₂×C₄₇	1	1	yes	characteristic
157	C₂×#410	C₂×C₄₇	1	4	yes	complemented
158	#2316	C₂×C₄₇	1	1	yes	characteristic, complemented
159	#2274	C₂×C₄₇	1	1	yes	characteristic, complemented
160	#1964	C₂×C₄₇	1	8	yes	complemented
161	C₂×C₄₇	—	2	2	no	—
162	C₂×C₄₇	—	2	1	no	—
163	C₂×C₄₇	—	2	2	no	—
164	C₂×C₄₇	—	2	1	no	—
165	C₂×C₄₇	—	8	4	no	—
166	C₂×C₄₇	C₂×#390	1	1	yes	characteristic
167	C₂×C₄₇	#2883	1	1	yes	characteristic
168	C₂×C₄₇	#2884	1	1	yes	characteristic
169	C₂²×#41	C₂²×C₄₇	1	2	yes	—
170	#537	—	2	4	no	—
171	#537	—	2	4	no	—
172	C₂×C₂²⋊Q₈	C₄×C₄₇	1	4	yes	complemented
173	C₂×#121	C₄×C₄₇	1	4	yes	complemented
174	C₂×#121	C₂²×C₄₇	1	4	yes	—
175	C₂×#117	C₂²×C₄₇	1	2	yes	—
176	C₂×#117	C₂²×C₄₇	1	2	yes	—
177	C₂²×C₄⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
178	#410	—	2	16	no	—
179	C₂×#99	C₂²×C₄₇	1	16	yes	—
180	#389	—	2	4	no	—
181	#380	C₂²×C₄₇	1	4	yes	—
182	C₄×C₄⋊C₄	—	2	4	no	—
183	C₄×C₂²⋊C₄	C₂²×C₄₇	1	4	yes	—
184	C₄₇	#21214	1	1	yes	characteristic, direct product factor, Sylow
185	C₂×#41	—	2	4	no	—
186	C₂×#41	—	2	4	no	—
187	C₂×#41	—	2	8	no	—
188	C₂×#41	—	2	8	no	—
189	C₂²×Q₈	C₂×C₄×C₄₇	1	2	yes	—
190	C₂²×D₄	C₂×C₄×C₄₇	1	2	yes	—
191	C₂²×D₄	C₂×C₄×C₄₇	1	4	yes	—
192	C₂²×D₄	C₂³×C₄₇	1	4	yes	—
193	C₂³×C₄	C₂×C₄×C₄₇	1	4	yes	—
194	C₂³×C₄	C₂³×C₄₇	1	2	yes	—
195	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
196	C₂²⋊Q₈	—	2	16	no	—
197	#121	—	2	8	no	—
198	#121	—	2	8	no	—
199	#121	—	2	16	no	—
200	#117	—	2	4	no	—
201	#117	—	2	8	no	—
202	#117	D₄×C₄₇	1	8	yes	complemented
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	4	yes	—
207	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	4	yes	—
208	C₂×C₄⋊C₄	D₄×C₄₇	1	4	yes	—
209	C₂×C₄⋊C₄	D₄×C₄₇	1	4	yes	—
210	C₂×C₂²⋊C₄	C₂×C₄×C₄₇	1	8	yes	—
211	C₂×C₂²⋊C₄	C₂³×C₄₇	1	4	yes	—
212	C₂×C₂²⋊C₄	C₂³×C₄₇	1	4	yes	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	—	2	4	no	—
216	#99	—	2	32	no	—
217	#95	—	2	4	no	—
218	#95	—	2	4	no	—
219	C₂⁴	C₂²×C₄×C₄₇	1	4	yes	—
220	C₂⁴	C₂⁴×C₄₇	1	1	yes	characteristic
221	#41	—	4	8	no	—
222	#41	—	4	8	no	—
223	#41	—	4	8	no	—
224	#41	—	4	8	no	—
225	C₂×Q₈	—	2	4	no	—
226	C₂×Q₈	—	2	8	no	—
227	C₂×D₄	—	2	8	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	8	no	—
231	C₂×D₄	—	2	4	no	—
232	C₂×D₄	—	2	8	no	—
233	C₂×D₄	—	2	8	no	—
234	C₂×D₄	—	8	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	8	no	—
237	C₂²×C₄	—	2	16	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	8	2	no	—
240	C₂²×C₄	C₂²×C₄×C₄₇	1	4	yes	—
241	C₂²×C₄	C₂²×C₄×C₄₇	1	4	yes	—
242	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
243	C₂²×C₄	C₂×D₄×C₄₇	1	4	yes	—
244	C₂²×C₄	C₂×D₄×C₄₇	1	4	yes	—
245	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
247	C₂²×C₄	C₂×D₄×C₄₇	1	4	yes	—
248	C₂²×C₄	#41×C₄₇	1	4	yes	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄⋊C₄	—	2	8	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₄⋊C₄	—	8	4	no	—
253	C₄⋊C₄	C₂²⋊C₄×C₄₇	1	16	yes	—
254	C₂²⋊C₄	—	2	4	no	—
255	C₂²⋊C₄	—	2	16	no	—
256	C₂²⋊C₄	—	2	8	no	—
257	C₂²⋊C₄	C₂×D₄×C₄₇	1	8	yes	—
258	C₄²	—	2	8	no	—
259	C₄²	—	4	4	no	—
260	C₄²	—	8	4	no	—
261	C₂³	—	2	4	no	—
262	C₂³	—	2	8	no	—
263	C₂³	—	4	1	no	—
264	C₂³	—	4	1	no	—
265	C₂³	—	4	4	no	—
266	C₂³	—	4	4	no	—
267	C₂³	C₂³×C₄×C₄₇	1	1	yes	characteristic
268	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
269	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
270	C₂³	C₂²×D₄×C₄₇	1	2	yes	—
271	C₂³	C₂×#41×C₄₇	1	2	yes	—
272	Q₈	—	4	8	no	—
273	D₄	—	4	8	no	—
274	D₄	—	4	4	no	—
275	D₄	—	4	4	no	—
276	D₄	—	4	8	no	—
277	D₄	—	8	8	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	16	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	8	yes	—
291	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	4	yes	—
292	C₂×C₄	C₄×D₄×C₄₇	1	8	yes	—
293	C₂×C₄	#121×C₄₇	1	4	yes	—
294	C₂×C₄	#121×C₄₇	1	4	yes	—
295	C₂×C₄	#124×C₄₇	1	4	yes	—
296	C₂×C₄	#127×C₄₇	1	4	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	4	no	—
306	C₂²	—	4	4	no	—
307	C₂²	—	4	4	no	—
308	C₂²	—	8	8	no	—
309	C₂²	C₂²×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
310	C₂²	C₂×C₄×D₄×C₄₇	1	2	yes	—
311	C₂²	C₂×#121×C₄₇	1	2	yes	—
312	C₂²	C₂×#124×C₄₇	1	1	yes	characteristic
313	C₂²	C₂×#127×C₄₇	1	1	yes	characteristic
314	C₄	—	2	8	no	—
315	C₄	—	4	8	no	—
316	C₄	—	4	4	no	—
317	C₄	—	8	4	no	—
318	C₄	C₄₇×#390	1	8	yes	—
319	C₂	—	2	2	no	—
320	C₂	—	2	1	no	—
321	C₂	—	2	2	no	—
322	C₂	—	2	1	no	—
323	C₂	—	8	4	no	—
324	C₂	C₂×C₄₇×#390	1	1	yes	characteristic
325	C₂	C₄₇×#2883	1	1	yes	characteristic
326	C₂	C₄₇×#2884	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₄₇
Frattini, cosocle	C₂⁴	C₂⁴×C₄₇
prime radical	C₂×C₄₇×#537	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂²×C₄⋊C₄×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₇×#410	C₂	semidirect
3. C₄₇×#2316	C₂	semidirect
4. C₄₇×#2274	C₂	semidirect
5. C₄₇×#1964	C₂	semidirect
9. C₂×C₂²⋊Q₈×C₄₇	C₄	semidirect
10. C₂×#121×C₄₇	C₄	semidirect
38. #117×C₄₇	D₄	semidirect
133. #21214	C₄₇	direct (N × H)
157. C₂×#410	C₂×C₄₇	semidirect
158. #2316	C₂×C₄₇	semidirect
159. #2274	C₂×C₄₇	semidirect
160. #1964	C₂×C₄₇	semidirect
172. C₂×C₂²⋊Q₈	C₄×C₄₇	semidirect
173. C₂×#121	C₄×C₄₇	semidirect
184. C₄₇	#21214	direct (N × H)
202. #117	D₄×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×#537	2
C₂×C₄₇×#410	2
C₄₇×#2316	2
C₄₇×#2274	2
C₄₇×#1964	2
#21214	47

Subgroups — C47×#21214 #26060912

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#21214 #26060912