Subgroups — C₂²×#15688847 #26182064

Order 12032.

2878 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal, complemented
2	C₂×#15688847	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₂²⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄₇⋊C₈	—	2	2	no	—
5	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	6	no	—
6	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	12	yes	complemented
7	#15688847	C₂²	1	16	yes	direct product factor
8	C₂⁶×C₄₇	C₄	1	1	yes	characteristic
9	C₂⁴×C₄×C₄₇	C₄	1	1	yes	characteristic
10	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
11	C₂²×C₂²⋊C₄×C₄₇	—	2	2	no	—
12	C₂²×C₂²⋊C₄×C₄₇	—	2	6	no	—
13	C₂²×C₂²⋊C₄×C₄₇	C₄	1	6	yes	—
14	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	6	yes	—
15	C₂²×C₄₇⋊C₈	—	2	12	no	—
16	C₂²×C₄₇⋊C₈	—	4	8	no	—
17	C₂²⋊(C₄₇⋊C₈)	—	2	12	no	—
18	C₂²⋊(C₄₇⋊C₈)	C₂³	1	8	yes	complemented
19	C₂⁵×C₄₇	—	2	3	no	—
20	C₂⁵×C₄₇	—	2	1	no	—
21	C₂⁵×C₄₇	—	2	4	no	—
22	C₂⁵×C₄₇	—	4	4	no	—
23	C₂⁵×C₄₇	—	4	4	no	—
24	C₂⁵×C₄₇	C₈	1	8	yes	complemented
25	C₂⁵×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂⁵×C₄₇	C₂×C₄	1	6	yes	—
27	C₂³×C₄×C₄₇	—	2	3	no	—
28	C₂³×C₄×C₄₇	—	2	1	no	—
29	C₂³×C₄×C₄₇	—	4	4	no	—
30	C₂³×C₄×C₄₇	—	4	4	no	—
31	C₂³×C₄×C₄₇	C₂×C₄	1	6	yes	—
32	C₂³×C₄×C₄₇	C₂×C₄	1	3	yes	—
33	C₂³×C₄×C₄₇	D₄	1	2	yes	—
34	C₂³×C₄×C₄₇	D₄	1	6	yes	—
35	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
36	C₂×C₂²⋊C₄×C₄₇	—	2	12	no	—
37	C₂×C₂²⋊C₄×C₄₇	—	2	12	no	—
38	C₂×C₂²⋊C₄×C₄₇	—	2	12	no	—
39	C₂×C₂²⋊C₄×C₄₇	—	2	12	no	—
40	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	12	yes	—
41	C₂×C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
42	C₂⁴×C₄₇	—	2	4	no	—
43	C₂⁴×C₄₇	—	2	4	no	—
44	C₂⁴×C₄₇	—	2	24	no	—
45	C₂⁴×C₄₇	—	2	8	no	—
46	C₂⁴×C₄₇	—	2	6	no	—
47	C₂⁴×C₄₇	—	2	6	no	—
48	C₂⁴×C₄₇	—	2	12	no	—
49	C₂⁴×C₄₇	—	4	32	no	—
50	C₂⁴×C₄₇	—	4	12	no	—
51	C₂⁴×C₄₇	—	4	12	no	—
52	C₂⁴×C₄₇	—	4	32	no	—
53	C₂⁴×C₄₇	—	4	12	no	—
54	C₂⁴×C₄₇	—	4	12	no	—
55	C₂⁴×C₄₇	—	4	4	no	—
56	C₂⁴×C₄₇	—	4	4	no	—
57	C₂⁴×C₄₇	C₂²⋊C₄	1	3	yes	—
58	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂⁴×C₄₇	C₂×C₈	1	4	yes	—
60	C₂⁴×C₄₇	C₂×C₈	1	24	yes	complemented
61	C₂⁴×C₄₇	M₁₆	1	4	yes	—
62	C₂⁴×C₄₇	C₂²×C₄	1	3	yes	—
63	C₂⁴×C₄₇	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₄₇	—	2	8	no	—
65	C₂²×C₄×C₄₇	—	2	6	no	—
66	C₂²×C₄×C₄₇	—	2	6	no	—
67	C₂²×C₄×C₄₇	—	4	6	no	—
68	C₂²×C₄×C₄₇	—	4	12	no	—
69	C₂²×C₄×C₄₇	—	4	6	no	—
70	C₂²×C₄×C₄₇	—	4	12	no	—
71	C₂²×C₄×C₄₇	—	4	4	no	—
72	C₂²×C₄×C₄₇	—	4	12	no	—
73	C₂²×C₄×C₄₇	C₂²⋊C₄	1	6	yes	—
74	C₂²×C₄×C₄₇	C₂²⋊C₄	1	6	yes	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
77	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
78	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
79	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
80	C₂²⋊C₄×C₄₇	—	2	24	no	—
81	C₂²⋊C₄×C₄₇	—	2	8	no	—
82	C₂×C₄₇⋊C₈	—	2	8	no	—
83	C₂×C₄₇⋊C₈	—	4	24	no	—
84	C₂³×C₄₇	—	2	16	no	—
85	C₂³×C₄₇	—	2	16	no	—
86	C₂³×C₄₇	—	2	6	no	—
87	C₂³×C₄₇	—	2	6	no	—
88	C₂³×C₄₇	—	2	6	no	—
89	C₂³×C₄₇	—	2	6	no	—
90	C₂³×C₄₇	—	2	4	no	—
91	C₂³×C₄₇	—	2	4	no	—
92	C₂³×C₄₇	—	2	24	no	—
93	C₂³×C₄₇	—	2	24	no	—
94	C₂³×C₄₇	—	4	48	no	—
95	C₂³×C₄₇	—	4	4	no	—
96	C₂³×C₄₇	—	4	4	no	—
97	C₂³×C₄₇	—	4	12	no	—
98	C₂³×C₄₇	—	4	12	no	—
99	C₂³×C₄₇	—	4	96	no	—
100	C₂³×C₄₇	—	4	48	no	—
101	C₂³×C₄₇	—	4	32	no	—
102	C₂³×C₄₇	—	4	12	no	—
103	C₂³×C₄₇	—	4	12	no	—
104	C₂³×C₄₇	C₂²⋊C₈	1	12	yes	—
105	C₂³×C₄₇	C₂²⋊C₈	1	4	yes	—
106	C₂³×C₄₇	#99	1	4	yes	—
107	C₂³×C₄₇	#100	1	4	yes	—
108	C₂³×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
109	C₂³×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
110	C₂³×C₄₇	C₂²×C₈	1	6	yes	—
111	C₂³×C₄₇	C₂²×C₈	1	8	yes	complemented
112	C₂³×C₄₇	C₂×M₁₆	1	6	yes	—
113	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	—	2	12	no	—
115	C₂×C₄×C₄₇	—	2	12	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	4	6	no	—
118	C₂×C₄×C₄₇	—	4	3	no	—
119	C₂×C₄×C₄₇	—	4	6	no	—
120	C₂×C₄×C₄₇	—	4	1	no	—
121	C₂×C₄×C₄₇	—	4	12	no	—
122	C₂×C₄×C₄₇	—	4	12	no	—
123	C₂×C₄×C₄₇	—	4	4	no	—
124	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	6	yes	—
125	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
126	C₄₇⋊C₈	—	4	8	no	—
127	C₂²×#323	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	—	2	3	no	—
129	C₂²×C₄₇	—	2	1	no	—
130	C₂²×C₄₇	—	2	3	no	—
131	C₂²×C₄₇	—	2	1	no	—
132	C₂²×C₄₇	—	2	24	no	—
133	C₂²×C₄₇	—	2	12	no	—
134	C₂²×C₄₇	—	2	12	no	—
135	C₂²×C₄₇	—	2	8	no	—
136	C₂²×C₄₇	—	4	8	no	—
137	C₂²×C₄₇	—	4	8	no	—
138	C₂²×C₄₇	—	4	48	no	—
139	C₂²×C₄₇	—	4	4	no	—
140	C₂²×C₄₇	—	4	4	no	—
141	C₂²×C₄₇	—	4	48	no	—
142	C₂²×C₄₇	#323	1	16	yes	complemented
143	C₂²×C₄₇	C₂×C₂²⋊C₈	1	6	yes	—
144	C₂²×C₄₇	C₂×C₂²⋊C₈	1	6	yes	—
145	C₂²×C₄₇	C₂×#99	1	6	yes	—
146	C₂²×C₄₇	C₂×#100	1	6	yes	—
147	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
149	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
150	C₄×C₄₇	—	2	2	no	—
151	C₄×C₄₇	—	2	6	no	—
152	C₄×C₄₇	—	4	4	no	—
153	C₂³×C₂²⋊C₄	D₄₇	1	1	yes	characteristic
154	C₂²×C₂²⋊C₈	—	47	2	no	—
155	C₂×#323	—	47	12	no	—
156	C₂×C₄₇	—	2	1	no	—
157	C₂×C₄₇	—	2	1	no	—
158	C₂×C₄₇	—	2	6	no	—
159	C₂×C₄₇	—	4	8	no	—
160	C₂×C₄₇	C₂×#323	1	12	yes	complemented
161	C₂×C₄₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
162	C₂×C₄₇	C₂²×#99	1	1	yes	characteristic
163	C₂×C₄₇	C₂²×#100	1	1	yes	characteristic
164	C₂⁶	Dic₄₇	1	1	yes	characteristic
165	C₂⁴×C₄	Dic₄₇	1	1	yes	characteristic
166	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
167	C₂³×C₈	—	94	2	no	—
168	C₂²×C₂²⋊C₄	—	2	6	no	—
169	C₂²×C₂²⋊C₄	—	2	2	no	—
170	C₂²×C₂²⋊C₄	Dic₄₇	1	6	yes	—
171	C₂²×C₂²⋊C₄	C₂×D₄₇	1	6	yes	—
172	C₂×C₂²⋊C₈	—	47	12	no	—
173	C₂×C₂²⋊C₈	—	94	6	no	—
174	#323	—	47	16	no	—
175	C₄₇	C₂²×#323	1	1	yes	characteristic, complemented, Sylow
176	C₂⁵	—	2	4	no	—
177	C₂⁵	—	2	3	no	—
178	C₂⁵	—	2	1	no	—
179	C₂⁵	—	4	4	no	—
180	C₂⁵	—	4	4	no	—
181	C₂⁵	C₄₇⋊C₈	1	8	yes	complemented
182	C₂⁵	C₂×Dic₄₇	1	1	yes	characteristic
183	C₂⁵	C₂×Dic₄₇	1	6	yes	—
184	C₂³×C₄	—	2	3	no	—
185	C₂³×C₄	—	2	1	no	—
186	C₂³×C₄	—	4	4	no	—
187	C₂³×C₄	—	4	4	no	—
188	C₂³×C₄	C₂×Dic₄₇	1	6	yes	—
189	C₂³×C₄	C₂×Dic₄₇	1	3	yes	—
190	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
191	C₂³×C₄	C₂²⋊D₄₇	1	6	yes	—
192	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
193	C₂²×C₈	—	94	12	no	—
194	C₂²×C₈	—	188	8	no	—
195	C₂×C₂²⋊C₄	—	2	12	no	—
196	C₂×C₂²⋊C₄	—	2	12	no	—
197	C₂×C₂²⋊C₄	—	2	12	no	—
198	C₂×C₂²⋊C₄	—	2	12	no	—
199	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	12	yes	—
200	C₂×C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
201	C₂²⋊C₈	—	47	8	no	—
202	C₂²⋊C₈	—	94	12	no	—
203	C₂⁴	—	2	4	no	—
204	C₂⁴	—	2	4	no	—
205	C₂⁴	—	2	24	no	—
206	C₂⁴	—	2	8	no	—
207	C₂⁴	—	2	6	no	—
208	C₂⁴	—	2	6	no	—
209	C₂⁴	—	2	12	no	—
210	C₂⁴	—	4	32	no	—
211	C₂⁴	—	4	12	no	—
212	C₂⁴	—	4	12	no	—
213	C₂⁴	—	4	12	no	—
214	C₂⁴	—	4	12	no	—
215	C₂⁴	—	4	32	no	—
216	C₂⁴	—	4	4	no	—
217	C₂⁴	—	4	4	no	—
218	C₂⁴	C₂×C₄₇⋊C₈	1	4	yes	—
219	C₂⁴	C₂×C₄₇⋊C₈	1	24	yes	complemented
220	C₂⁴	#10625329	1	4	yes	—
221	C₂⁴	C₂²⋊Dic₄₇	1	3	yes	—
222	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
223	C₂⁴	C₂²×Dic₄₇	1	3	yes	—
224	C₂⁴	C₂²×Dic₄₇	1	4	yes	—
225	C₂²×C₄	—	2	8	no	—
226	C₂²×C₄	—	2	6	no	—
227	C₂²×C₄	—	2	6	no	—
228	C₂²×C₄	—	4	6	no	—
229	C₂²×C₄	—	4	12	no	—
230	C₂²×C₄	—	4	6	no	—
231	C₂²×C₄	—	4	12	no	—
232	C₂²×C₄	—	4	4	no	—
233	C₂²×C₄	—	4	12	no	—
234	C₂²×C₄	C₂²⋊Dic₄₇	1	6	yes	—
235	C₂²×C₄	C₂²⋊Dic₄₇	1	6	yes	—
236	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
237	C₂²×C₄	C₂²×Dic₄₇	1	3	yes	—
238	C₂²×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
239	C₂²×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
240	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
241	C₂×C₈	—	94	8	no	—
242	C₂×C₈	—	188	24	no	—
243	C₂²⋊C₄	—	2	24	no	—
244	C₂²⋊C₄	—	2	8	no	—
245	C₂³	—	2	6	no	—
246	C₂³	—	2	6	no	—
247	C₂³	—	2	6	no	—
248	C₂³	—	2	6	no	—
249	C₂³	—	2	16	no	—
250	C₂³	—	2	16	no	—
251	C₂³	—	2	4	no	—
252	C₂³	—	2	24	no	—
253	C₂³	—	2	24	no	—
254	C₂³	—	2	4	no	—
255	C₂³	—	4	48	no	—
256	C₂³	—	4	4	no	—
257	C₂³	—	4	4	no	—
258	C₂³	—	4	96	no	—
259	C₂³	—	4	48	no	—
260	C₂³	—	4	12	no	—
261	C₂³	—	4	12	no	—
262	C₂³	—	4	12	no	—
263	C₂³	—	4	12	no	—
264	C₂³	—	4	32	no	—
265	C₂³	C₂²⋊(C₄₇⋊C₈)	1	12	yes	—
266	C₂³	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
267	C₂³	#13093939	1	4	yes	—
268	C₂³	#13093941	1	4	yes	—
269	C₂³	C₂×C₂²⋊Dic₄₇	1	3	yes	—
270	C₂³	C₂×C₂²⋊Dic₄₇	1	3	yes	—
271	C₂³	C₂²×C₄₇⋊C₈	1	6	yes	—
272	C₂³	C₂²×C₄₇⋊C₈	1	8	yes	complemented
273	C₂³	C₂×#10625329	1	6	yes	—
274	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
275	C₂×C₄	—	2	12	no	—
276	C₂×C₄	—	2	12	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	6	no	—
279	C₂×C₄	—	4	3	no	—
280	C₂×C₄	—	4	6	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	12	no	—
283	C₂×C₄	—	4	12	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	6	yes	—
286	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
287	C₈	—	188	8	no	—
288	C₂²	—	2	24	no	—
289	C₂²	—	2	12	no	—
290	C₂²	—	2	12	no	—
291	C₂²	—	2	3	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	3	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	8	no	—
296	C₂²	—	4	8	no	—
297	C₂²	—	4	48	no	—
298	C₂²	—	4	8	no	—
299	C₂²	—	4	48	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	4	4	no	—
302	C₂²	#15688847	1	16	yes	direct product factor
303	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	6	yes	—
304	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	6	yes	—
305	C₂²	C₂×#13093939	1	6	yes	—
306	C₂²	C₂×#13093941	1	6	yes	—
307	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
308	C₂²	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
309	C₂²	C₂²×#10625329	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	6	no	—
312	C₄	—	4	4	no	—
313	C₂	—	2	6	no	—
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	4	8	no	—
317	C₂	C₂×#15688847	1	12	yes	direct product factor
318	C₂	C₂²×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
319	C₂	C₂²×#13093939	1	1	yes	characteristic
320	C₂	C₂²×#13093941	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²⋊Dic₄₇
commutator	C₂²×C₄₇	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂³×C₂²⋊C₄×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂²⋊C₄
prime radical	C₂⁶×C₄₇	C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂³×C₂²⋊C₄	D₄₇
hypocenter	C₄₇	C₂²×#323
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂²⋊(C₄₇⋊C₈)	C₂	semidirect
2. C₂×#15688847	C₂	direct (N × H)
6. C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	semidirect
7. #15688847	C₂²	direct (N × H)
18. C₂²⋊(C₄₇⋊C₈)	C₂³	semidirect
24. C₂⁵×C₄₇	C₈	semidirect
60. C₂⁴×C₄₇	C₂×C₈	semidirect
111. C₂³×C₄₇	C₂²×C₈	semidirect
142. C₂²×C₄₇	#323	semidirect
160. C₂×C₄₇	C₂×#323	semidirect
175. C₄₇	C₂²×#323	semidirect
181. C₂⁵	C₄₇⋊C₈	semidirect
219. C₂⁴	C₂×C₄₇⋊C₈	semidirect
272. C₂³	C₂²×C₄₇⋊C₈	semidirect
302. C₂²	#15688847	direct (N × H)
317. C₂	C₂×#15688847	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×C₂²⋊(C₄₇⋊C₈)	2
C₂×#15688847	2
C₂³×C₂²⋊C₄×C₄₇	2
C₂²×#323	47

Subgroups — C22×#15688847 #26182064

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15688847 #26182064