Subgroups — #26220741

Order 12032.

1170 conjugacy classes of subgroups.

The subgroup lattice has 334 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×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
3	#20244329	C₂	1	4	yes	maximal, complemented
4	#20244325	C₂	1	4	yes	maximal, complemented
5	#20243085	C₂	1	4	yes	maximal
6	C₂×C₄₇×#533	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	2	yes	—
8	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₄×#10625329	—	2	4	no	—
13	C₂×#13093935	C₂²	1	2	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
15	#15688878	C₂²	1	16	yes	—
16	C₄₇×#533	—	2	4	no	—
17	C₄₇×#533	—	2	4	no	—
18	C₂×C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic
19	C₂×C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic
20	C₂×#125×C₄₇	C₄	1	2	yes	—
21	C₂×C₂²⋊Q₈×C₄₇	C₄	1	2	yes	—
22	C₂×C₂²⋊Q₈×C₄₇	C₄	1	2	yes	—
23	C₂×C₄×Q₈×C₄₇	C₂²	1	2	yes	—
24	C₂×C₄×Q₈×C₄₇	C₂²	1	2	yes	—
25	C₂×#117×C₄₇	C₂²	1	2	yes	—
26	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#10625329	—	2	4	no	—
28	C₂×#10625329	—	2	4	no	—
29	C₂×#10625329	Q₈	1	8	yes	complemented
30	C₂²×C₄₇⋊C₈	—	2	4	no	—
31	C₂²×C₄₇⋊C₈	—	2	8	no	—
32	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
33	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
34	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
35	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
36	#13093935	—	2	4	no	—
37	C₄×C₄₇⋊C₈	—	2	4	no	—
38	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
39	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
40	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₄⋊Q₈×C₄₇	—	2	4	no	—
44	C₄⋊Q₈×C₄₇	—	2	4	no	—
45	#125×C₄₇	—	2	8	no	—
46	C₂²⋊Q₈×C₄₇	—	2	8	no	—
47	C₂²⋊Q₈×C₄₇	—	2	8	no	—
48	C₄×Q₈×C₄₇	—	2	4	no	—
49	C₄×Q₈×C₄₇	—	2	4	no	—
50	C₄×Q₈×C₄₇	—	2	4	no	—
51	C₄×Q₈×C₄₇	—	2	4	no	—
52	#117×C₄₇	—	2	4	no	—
53	#117×C₄₇	—	2	4	no	—
54	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
57	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
58	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
60	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
61	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₄₇	D₄	1	4	yes	—
64	C₂×C₄²×C₄₇	D₄	1	4	yes	—
65	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
66	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
68	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×Q₈×C₄₇	—	2	4	no	—
70	C₂×Q₈×C₄₇	—	2	4	no	—
71	C₂×Q₈×C₄₇	—	2	8	no	—
72	C₂×Q₈×C₄₇	—	4	4	no	—
73	C₂²×C₄×C₄₇	—	2	4	no	—
74	C₂²×C₄×C₄₇	—	4	2	no	—
75	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
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	4	yes	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
87	C₂²×C₄×C₄₇	#41	1	4	yes	—
88	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₄₇	—	2	4	no	—
90	C₄⋊C₄×C₄₇	—	2	4	no	—
91	C₄⋊C₄×C₄₇	—	2	8	no	—
92	C₄⋊C₄×C₄₇	—	2	8	no	—
93	C₄⋊C₄×C₄₇	—	2	4	no	—
94	C₄⋊C₄×C₄₇	—	2	4	no	—
95	C₂²⋊C₄×C₄₇	—	2	8	no	—
96	C₄²×C₄₇	—	2	4	no	—
97	C₄²×C₄₇	—	2	4	no	—
98	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
99	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
100	C₄²×C₄₇	C₂×D₄	1	2	yes	—
101	C₄²×C₄₇	C₂×D₄	1	2	yes	—
102	C₄²×C₄₇	C₂×D₄	1	2	yes	—
103	C₄²×C₄₇	C₂×D₄	1	2	yes	—
104	#10625329	—	2	16	no	—
105	C₂×C₄₇⋊C₈	—	2	4	no	—
106	C₂×C₄₇⋊C₈	—	2	4	no	—
107	C₂×C₄₇⋊C₈	—	2	4	no	—
108	C₂×C₄₇⋊C₈	—	2	8	no	—
109	C₂×C₄₇⋊C₈	—	4	8	no	—
110	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
111	C₂³×C₄₇	—	2	2	no	—
112	C₂³×C₄₇	—	2	2	no	—
113	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
114	C₂³×C₄₇	C₄×Q₈	1	4	yes	—
115	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
116	Q₈×C₄₇	—	4	8	no	—
117	Q₈×C₄₇	—	4	8	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	8	no	—
122	C₂×C₄×C₄₇	—	2	4	no	—
123	C₂×C₄×C₄₇	—	2	4	no	—
124	C₂×C₄×C₄₇	—	2	8	no	—
125	C₂×C₄×C₄₇	—	2	8	no	—
126	C₂×C₄×C₄₇	—	4	4	no	—
127	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
128	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
129	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
130	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
131	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
132	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
133	C₂×C₄×C₄₇	C₂²⋊Q₈	1	8	yes	—
134	C₂×C₄×C₄₇	#124	1	4	yes	—
135	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
136	C₂×C₄×C₄₇	#131	1	4	yes	—
137	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
140	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
141	C₄₇⋊C₈	—	2	8	no	—
142	C₄₇⋊C₈	—	4	4	no	—
143	#21302	—	47	1	no	maximal, Sylow
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	2	2	no	—
146	C₂²×C₄₇	—	2	8	no	—
147	C₂²×C₄₇	#391	1	4	yes	—
148	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₄₇	C₂×C₄×Q₈	1	2	yes	—
150	C₂²×C₄₇	C₂×#131	1	2	yes	—
151	C₂²×C₄₇	#568	1	2	yes	—
152	C₄×C₄₇	—	2	2	no	—
153	C₄×C₄₇	—	2	2	no	—
154	C₄×C₄₇	—	2	2	no	—
155	C₄×C₄₇	—	4	4	no	—
156	C₄×C₄₇	—	4	4	no	—
157	C₄×C₄₇	#391	1	4	yes	—
158	C₄×C₄₇	#408	1	4	yes	—
159	C₄×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
160	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
161	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
162	C₂×#533	D₄₇	1	1	yes	characteristic
163	C₂²×C₄⋊C₈	—	47	1	no	—
164	C₂×C₄×M₁₆	—	47	1	no	—
165	#1985	—	47	4	no	—
166	#1984	—	47	4	no	—
167	#1729	—	47	4	no	—
168	C₂×C₄₇	—	2	4	no	—
169	C₂×C₄₇	C₂×#391	1	1	yes	characteristic
170	C₂×C₄₇	C₂×#408	1	1	yes	characteristic
171	C₂×C₄₇	#2879	1	1	yes	characteristic
172	C₂×C₄₇	#2962	1	4	yes	—
173	C₂²×M₁₆	—	47	2	no	—
174	C₂³×C₈	—	47	2	no	—
175	#533	—	2	4	no	—
176	#533	—	2	4	no	—
177	C₂×C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic
178	C₂×C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic
179	C₂×#125	Dic₄₇	1	2	yes	—
180	C₂×C₂²⋊Q₈	Dic₄₇	1	2	yes	—
181	C₂×C₂²⋊Q₈	Dic₄₇	1	2	yes	—
182	C₂×C₄×Q₈	C₂×D₄₇	1	2	yes	—
183	C₂×C₄×Q₈	C₂×D₄₇	1	2	yes	—
184	C₂×#117	C₂×D₄₇	1	2	yes	—
185	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
186	C₂×C₄⋊C₈	—	47	2	no	—
187	C₂×C₄⋊C₈	—	47	2	no	—
188	C₂×C₄⋊C₈	—	94	4	no	—
189	C₄×M₁₆	—	94	4	no	—
190	C₂×#97	—	47	2	no	—
191	C₂×C₄×C₈	—	47	2	no	—
192	#336	—	47	16	no	—
193	C₄₇	#21302	1	1	yes	characteristic, complemented, Sylow
194	C₂²×Q₈	C₂×Dic₄₇	1	2	yes	—
195	C₂²×Q₈	C₂×Dic₄₇	1	2	yes	—
196	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
197	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
198	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
199	C₂×M₁₆	—	47	8	no	—
200	C₂×M₁₆	—	94	4	no	—
201	C₂×M₁₆	—	94	4	no	—
202	C₂²×C₈	—	47	4	no	—
203	C₂²×C₈	—	47	4	no	—
204	C₂²×C₈	—	94	8	no	—
205	C₂²×C₈	—	94	4	no	—
206	C₄⋊Q₈	—	2	4	no	—
207	C₄⋊Q₈	—	2	4	no	—
208	#125	—	2	8	no	—
209	C₂²⋊Q₈	—	2	8	no	—
210	C₂²⋊Q₈	—	2	8	no	—
211	C₄×Q₈	—	2	4	no	—
212	C₄×Q₈	—	2	4	no	—
213	C₄×Q₈	—	2	4	no	—
214	C₄×Q₈	—	2	4	no	—
215	#117	—	2	4	no	—
216	#117	—	2	4	no	—
217	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
218	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
219	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
220	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
221	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
222	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
223	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
224	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
225	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
226	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
227	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
228	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
229	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
230	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
231	C₄⋊C₈	—	94	4	no	—
232	C₄⋊C₈	—	94	4	no	—
233	#97	—	94	4	no	—
234	C₄×C₈	—	94	4	no	—
235	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
236	C₂×Q₈	—	2	4	no	—
237	C₂×Q₈	—	2	4	no	—
238	C₂×Q₈	—	2	8	no	—
239	C₂×Q₈	—	4	4	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
243	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
244	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
245	C₂²×C₄	#10625354	1	4	yes	—
246	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
247	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
248	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
249	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
251	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
252	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
254	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
255	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
256	M₁₆	—	94	16	no	—
257	C₂×C₈	—	47	8	no	—
258	C₂×C₈	—	94	4	no	—
259	C₂×C₈	—	94	4	no	—
260	C₂×C₈	—	94	4	no	—
261	C₂×C₈	—	94	8	no	—
262	C₂×C₈	—	188	8	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	8	no	—
266	C₄⋊C₄	—	2	8	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₂²⋊C₄	—	2	8	no	—
270	C₄²	—	2	4	no	—
271	C₄²	—	2	4	no	—
272	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
273	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
274	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
275	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
276	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
277	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
278	C₂³	—	2	2	no	—
279	C₂³	—	2	2	no	—
280	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
281	C₂³	Q₈×Dic₄₇	1	4	yes	—
282	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
283	Q₈	—	4	8	no	—
284	Q₈	—	4	8	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	8	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	8	no	—
292	C₂×C₄	—	2	8	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
295	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
296	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
297	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
298	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
299	C₂×C₄	Q₈×Dic₄₇	1	4	yes	—
300	C₂×C₄	#13094000	1	8	yes	—
301	C₂×C₄	#13094013	1	4	yes	—
302	C₂×C₄	#13094022	1	4	yes	—
303	C₂×C₄	#13094031	1	4	yes	—
304	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
306	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
307	C₂×C₄	C₂×#10625354	1	2	yes	—
308	C₈	—	94	8	no	—
309	C₈	—	188	4	no	—
310	C₂²	—	2	8	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	#15689030	1	4	yes	—
314	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
315	C₂²	C₂×Q₈×Dic₄₇	1	2	yes	—
316	C₂²	C₂×#13094031	1	2	yes	—
317	C₂²	#15689890	1	2	yes	—
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	4	4	no	—
322	C₄	—	4	4	no	—
323	C₄	#15689030	1	4	yes	—
324	C₄	#15689090	1	4	yes	—
325	C₄	C₂×#13094000	1	2	yes	—
326	C₄	C₂×#13094013	1	1	yes	characteristic
327	C₄	C₂×#13094022	1	1	yes	characteristic
328	C₂	—	2	4	no	—
329	C₂	C₂×#15689030	1	1	yes	characteristic
330	C₂	C₂×#15689090	1	1	yes	characteristic
331	C₂	#20251833	1	1	yes	characteristic
332	C₂	#20252337	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₄₇×#533	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#533	D₄₇
hypocenter	C₄₇	#21302
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
3. #20244329	C₂	semidirect
4. #20244325	C₂	semidirect
29. C₂×#10625329	Q₈	semidirect
193. C₄₇	#21302	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×C₄×#10625329	2
#20244329	2
#20244325	2
#20243085	2
C₂×C₄₇×#533	2
#21302	47