Subgroups — #26122067

Order 12032.

950 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688969	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688877	C₂	1	4	yes	maximal
4	C₂²×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
6	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
8	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₄	1	4	yes	complemented
10	#15688969	—	2	2	no	—
11	#15688969	—	2	2	no	—
12	C₂×C₄₇⋊#95	C₄	1	2	yes	—
13	C₂×C₄₇⋊#95	C₄	1	2	yes	—
14	#15688877	—	2	8	no	—
15	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₄₇	—	2	2	no	—
18	C₂×C₄⋊C₈×C₄₇	—	2	2	no	—
19	C₂×C₄⋊C₈×C₄₇	C₄	1	4	yes	complemented
20	C₂³×Dic₄₇	C₂×C₄	1	4	yes	—
21	C₂²×C₄₇⋊C₈	—	2	4	no	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
26	C₂×C₄×Dic₄₇	—	2	2	no	—
27	C₂×C₄×Dic₄₇	—	2	2	no	—
28	C₂×C₄×Dic₄₇	—	2	2	no	—
29	C₂×C₄×Dic₄₇	—	2	2	no	—
30	C₂×C₄×Dic₄₇	D₄	1	4	yes	—
31	C₂×C₄×Dic₄₇	Q₈	1	4	yes	—
32	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
33	C₄₇⋊#95	—	2	4	no	—
34	C₄₇⋊#95	—	2	4	no	—
35	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₄₇	—	2	4	no	—
39	C₂²×C₈×C₄₇	—	2	2	no	—
40	C₂²×C₈×C₄₇	—	2	2	no	—
41	C₂²×C₈×C₄₇	—	2	4	no	—
42	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₇	D₄	1	2	yes	—
46	C₂×C₄²×C₄₇	D₄	1	2	yes	—
47	C₂×C₄²×C₄₇	D₄	1	2	yes	—
48	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
49	C₄⋊C₈×C₄₇	—	2	8	no	—
50	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	—	2	4	no	—
52	C₂²×C₄×C₄₇	—	2	4	no	—
53	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
54	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₄₇	#41	1	2	yes	—
68	C₂²×C₄×C₄₇	#41	1	2	yes	—
69	C₂²×Dic₄₇	—	2	4	no	—
70	C₂²×Dic₄₇	—	2	4	no	—
71	C₂²×Dic₄₇	—	2	8	no	—
72	C₂²×Dic₄₇	—	2	8	no	—
73	C₂²×Dic₄₇	C₄⋊C₄	1	8	yes	—
74	C₂×C₈×C₄₇	—	2	4	no	—
75	C₂×C₈×C₄₇	—	2	8	no	—
76	C₂×C₈×C₄₇	—	4	4	no	—
77	C₂×C₈×C₄₇	—	4	2	no	—
78	C₂×C₈×C₄₇	—	4	2	no	—
79	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
80	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
81	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
82	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
83	C₄×Dic₄₇	—	2	4	no	—
84	C₄×Dic₄₇	—	2	4	no	—
85	C₄×Dic₄₇	—	2	4	no	—
86	C₄×Dic₄₇	—	2	4	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	2	8	no	—
89	C₂×C₄₇⋊C₈	—	4	4	no	—
90	C₂×C₄₇⋊C₈	—	4	2	no	—
91	C₂×C₄₇⋊C₈	—	4	2	no	—
92	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
93	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₄₇	#117	1	2	yes	—
97	C₂³×C₄₇	#131	1	4	yes	—
98	C₂³×C₄₇	#131	1	4	yes	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	#95	1	4	yes	—
106	C₂×C₄×C₄₇	#95	1	4	yes	—
107	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₄₇	#121	1	2	yes	—
116	C₂×C₄×C₄₇	#121	1	2	yes	—
117	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
120	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
121	C₂×C₄×C₄₇	#125	1	2	yes	—
122	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
125	C₂×Dic₄₇	—	2	8	no	—
126	C₂×Dic₄₇	—	2	16	no	—
127	C₂×Dic₄₇	—	4	4	no	—
128	C₂×Dic₄₇	—	4	4	no	—
129	C₂×Dic₄₇	—	4	8	no	—
130	C₈×C₄₇	—	4	4	no	—
131	C₄₇⋊C₈	—	4	4	no	—
132	#10833	—	47	1	no	maximal, Sylow
133	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
134	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
135	C₂²×C₄₇	#380	1	2	yes	—
136	C₂²×C₄₇	#382	1	1	yes	characteristic
137	C₂²×C₄₇	#384	1	1	yes	characteristic
138	C₂²×C₄₇	#405	1	2	yes	—
139	C₂²×C₄₇	#405	1	2	yes	—
140	C₂²×C₄₇	#408	1	2	yes	—
141	C₂²×C₄₇	#408	1	2	yes	—
142	C₂²×C₄₇	#424	1	8	yes	—
143	C₂²×C₄₇	#424	1	2	yes	—
144	C₂²×C₄₇	#424	1	2	yes	—
145	C₂²×C₄₇	#433	1	8	yes	—
146	C₄×C₄₇	—	2	4	no	—
147	C₄×C₄₇	—	2	4	no	—
148	C₄×C₄₇	#389	1	2	yes	—
149	C₄×C₄₇	#389	1	2	yes	—
150	C₄×C₄₇	#390	1	1	yes	characteristic
151	C₄×C₄₇	#390	1	1	yes	characteristic
152	C₄×C₄₇	#391	1	1	yes	characteristic
153	C₄×C₄₇	#391	1	1	yes	characteristic
154	Dic₄₇	—	4	8	no	—
155	C₂²×C₄⋊C₈	—	47	1	no	—
156	C₂²×C₄⋊C₈	D₄₇	1	1	yes	characteristic
157	C₂×#376	—	47	1	no	—
158	C₂×#336	—	47	4	no	—
159	C₂×C₄₇	C₄⋊#95	1	1	yes	characteristic
160	C₂×C₄₇	#1729	1	1	yes	characteristic
161	C₂×C₄₇	#1729	1	1	yes	characteristic
162	C₂×C₄₇	#1768	1	4	yes	—
163	C₂×C₄₇	#1798	1	4	yes	—
164	C₂×C₄₇	#1832	1	2	yes	—
165	C₂×C₄₇	#1843	1	2	yes	—
166	C₂³×C₈	—	47	2	no	—
167	C₂³×C₈	C₂×D₄₇	1	2	yes	—
168	C₂²×C₄²	—	47	2	no	—
169	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
170	C₂×C₄⋊C₈	—	2	2	no	—
171	C₂×C₄⋊C₈	—	2	2	no	—
172	C₂×C₄⋊C₈	—	47	4	no	—
173	C₂×C₄⋊C₈	—	94	2	no	—
174	C₂×C₄⋊C₈	—	94	2	no	—
175	C₂×C₄⋊C₈	Dic₄₇	1	4	yes	complemented
176	#376	—	94	2	no	—
177	#376	—	94	2	no	—
178	C₂×#95	—	47	2	no	—
179	C₂×#95	—	47	2	no	—
180	#336	—	94	8	no	—
181	C₄₇	#10833	1	1	yes	characteristic, complemented, Sylow
182	C₂³×C₄	—	47	4	no	—
183	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
184	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
185	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
186	C₂²×C₈	—	2	4	no	—
187	C₂²×C₈	—	2	4	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	47	4	no	—
191	C₂²×C₈	—	94	2	no	—
192	C₂²×C₈	—	94	4	no	—
193	C₂²×C₈	—	94	4	no	—
194	C₂²×C₈	—	94	2	no	—
195	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
196	C₂×C₄²	—	47	4	no	—
197	C₂×C₄²	—	47	4	no	—
198	C₂×C₄²	—	94	2	no	—
199	C₂×C₄²	—	94	2	no	—
200	C₂×C₄²	—	94	2	no	—
201	C₂×C₄²	—	94	2	no	—
202	C₂×C₄²	Dic₉₄	1	2	yes	—
203	C₂×C₄²	C₄×D₄₇	1	2	yes	—
204	C₂×C₄²	D₁₈₈	1	2	yes	—
205	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
206	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
207	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
208	C₄⋊C₈	—	2	8	no	—
209	C₄⋊C₈	—	94	8	no	—
210	#95	—	94	4	no	—
211	#95	—	94	4	no	—
212	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	47	8	no	—
216	C₂²×C₄	—	94	4	no	—
217	C₂²×C₄	—	94	4	no	—
218	C₂²×C₄	—	94	8	no	—
219	C₂²×C₄	—	94	8	no	—
220	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
221	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
222	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
223	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
224	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
225	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
226	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
229	C₂²×C₄	D₄×D₄₇	1	2	yes	—
230	C₂²×C₄	#10625352	1	2	yes	—
231	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
232	C₂²×C₄	#10625354	1	2	yes	—
233	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	8	no	—
238	C₂×C₈	—	4	4	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	94	4	no	—
242	C₂×C₈	—	94	8	no	—
243	C₂×C₈	—	188	4	no	—
244	C₂×C₈	—	188	2	no	—
245	C₂×C₈	—	188	2	no	—
246	C₄²	—	94	4	no	—
247	C₄²	—	94	4	no	—
248	C₄²	—	94	4	no	—
249	C₄²	—	94	4	no	—
250	C₄²	Dic₄₇⋊C₄	1	4	yes	—
251	C₄²	C₄⋊Dic₄₇	1	4	yes	—
252	C₄²	(C₂×D₄₇)⋊C₄	1	4	yes	—
253	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
254	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
255	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
256	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
257	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
258	C₂³	#13093982	1	2	yes	—
259	C₂³	#13094030	1	4	yes	—
260	C₂³	#13094032	1	4	yes	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	94	8	no	—
268	C₂×C₄	—	94	16	no	—
269	C₂×C₄	—	188	4	no	—
270	C₂×C₄	—	188	4	no	—
271	C₂×C₄	—	188	8	no	—
272	C₂×C₄	C₄₇⋊#95	1	4	yes	—
273	C₂×C₄	C₄₇⋊#95	1	4	yes	—
274	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
275	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
276	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
278	C₂×C₄	#13093986	1	2	yes	—
279	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
280	C₂×C₄	#13093989	1	2	yes	—
281	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
282	C₂×C₄	#13093995	1	2	yes	—
283	C₂×C₄	#13093999	1	2	yes	—
284	C₂×C₄	#13094000	1	2	yes	—
285	C₂×C₄	#13094004	1	2	yes	—
286	C₂×C₄	#13094011	1	1	yes	characteristic
287	C₂×C₄	#13094013	1	1	yes	characteristic
288	C₂×C₄	#13094015	1	2	yes	—
289	C₂×C₄	#13094020	1	1	yes	characteristic
290	C₂×C₄	#13094022	1	1	yes	characteristic
291	C₂×C₄	#13094023	1	2	yes	—
292	C₈	—	4	4	no	—
293	C₈	—	188	4	no	—
294	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
295	C₂²	C₄⋊C₄×Dic₄₇	1	2	yes	—
296	C₂²	#15688980	1	2	yes	—
297	C₂²	#15688988	1	1	yes	characteristic
298	C₂²	#15688992	1	1	yes	characteristic
299	C₂²	#15689078	1	2	yes	—
300	C₂²	#15689081	1	2	yes	—
301	C₂²	#15689089	1	2	yes	—
302	C₂²	#15689091	1	2	yes	—
303	C₂²	#15689148	1	8	yes	—
304	C₂²	#15689149	1	2	yes	—
305	C₂²	#15689151	1	2	yes	—
306	C₂²	#15689183	1	8	yes	—
307	C₄	—	2	4	no	—
308	C₄	—	2	4	no	—
309	C₄	—	188	8	no	—
310	C₄	#15689019	1	2	yes	—
311	C₄	#15689022	1	2	yes	—
312	C₄	#15689024	1	1	yes	characteristic
313	C₄	#15689026	1	1	yes	characteristic
314	C₄	#15689028	1	1	yes	characteristic
315	C₄	#15689030	1	1	yes	characteristic
316	C₂	#20241617	1	1	yes	characteristic
317	C₂	#20243084	1	1	yes	characteristic
318	C₂	#20243086	1	1	yes	characteristic
319	C₂	#20243247	1	4	yes	—
320	C₂	#20243369	1	4	yes	—
321	C₂	#20243507	1	2	yes	—
322	C₂	#20243541	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₄₇
commutator	C₂³×C₄₇	C₂×C₄²
Fitting	C₂²×C₄⋊C₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂²×C₄⋊C₈	D₄₇
hypocenter	C₄₇	#10833
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
9. C₂×C₄⋊(C₄₇⋊C₈)	C₄	semidirect
19. C₂×C₄⋊C₈×C₄₇	C₄	semidirect
175. C₂×C₄⋊C₈	Dic₄₇	semidirect
181. C₄₇	#10833	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₄₇⋊C₈)	2
C₂×#15688969	2
C₂×#15688877	2
C₂²×C₄⋊C₈×C₄₇	2
#10833	47