Subgroups — #26226719

Order 12032.

946 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	C₂²×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
3	#20244106	C₂	1	4	yes	maximal, complemented
4	#20243565	C₂	1	4	yes	maximal
5	#20243539	C₂	1	2	yes	maximal, complemented
6	#20243504	C₂	1	2	yes	maximal, complemented
7	C₂×C₄₇×#431	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
9	C₂×#13093983	C₄	1	1	yes	characteristic
10	C₂×#13093983	C₄	1	1	yes	characteristic
11	C₂×C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×Dic₄₇⋊C₈	C₂²	1	8	yes	—
15	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
16	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
17	C₄×C₂²⋊Dic₄₇	C₄	1	2	yes	—
18	C₄×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
19	#15688969	C₄	1	2	yes	—
20	C₄²×Dic₄₇	C₂²	1	2	yes	—
21	#15688877	C₂²	1	8	yes	—
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#431	—	2	4	no	—
24	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
28	C₂²×C₄₇⋊C₈	—	2	4	no	—
29	C₂²×C₄₇⋊C₈	—	4	4	no	—
30	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
31	#13093983	—	2	4	no	—
32	#13093983	C₈	1	8	yes	complemented
33	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
41	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
42	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
43	Dic₄₇⋊C₈	—	4	8	no	—
44	C₄₇⋊#95	C₂×C₄	1	4	yes	—
45	C₄₇⋊#95	C₂×C₄	1	4	yes	—
46	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	D₄	1	4	yes	—
54	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₄₇	—	2	2	no	—
58	C₄⋊C₈×C₄₇	—	2	2	no	—
59	C₂²⋊C₈×C₄₇	—	2	4	no	—
60	C₄×C₈×C₄₇	—	2	4	no	—
61	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	—	4	2	no	—
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
66	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
68	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
77	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
78	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₄₇	—	2	2	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	2	4	no	—
82	C₂×C₈×C₄₇	—	2	4	no	—
83	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
84	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
85	C₄²×C₄₇	C₂×D₄	1	2	yes	—
86	C₄²×C₄₇	C₂×D₄	1	2	yes	—
87	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
88	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
89	C₂²⋊Dic₄₇	—	2	4	no	—
90	C₂²⋊Dic₄₇	C₂×C₈	1	8	yes	—
91	C₄⋊Dic₄₇	—	2	4	no	—
92	C₄⋊Dic₄₇	C₂×C₈	1	8	yes	complemented
93	C₄×Dic₄₇	—	2	4	no	—
94	C₄×Dic₄₇	—	2	4	no	—
95	C₄×Dic₄₇	—	2	4	no	—
96	C₄×Dic₄₇	—	4	4	no	—
97	C₄×Dic₄₇	C₂×C₈	1	8	yes	complemented
98	C₂×C₄₇⋊C₈	—	2	4	no	—
99	C₂×C₄₇⋊C₈	—	2	8	no	—
100	C₂×C₄₇⋊C₈	—	4	8	no	—
101	C₂³×C₄₇	—	2	2	no	—
102	C₂³×C₄₇	—	2	2	no	—
103	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
105	C₂³×C₄₇	C₂×M₁₆	1	2	yes	—
106	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	4	4	no	—
114	C₂×C₄×C₄₇	C₄⋊C₈	1	16	yes	complemented
115	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
121	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
123	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
124	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
127	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
128	C₂×Dic₄₇	—	2	4	no	—
129	C₂×Dic₄₇	—	2	4	no	—
130	C₂×Dic₄₇	—	2	4	no	—
131	C₂×Dic₄₇	—	2	4	no	—
132	C₂×Dic₄₇	—	2	4	no	—
133	C₂×Dic₄₇	C₂²×C₈	1	8	yes	—
134	C₈×C₄₇	—	4	2	no	—
135	C₈×C₄₇	—	4	2	no	—
136	C₄₇⋊C₈	—	4	4	no	—
137	#21972	—	47	1	no	maximal, Sylow
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	—	2	2	no	—
140	C₂²×C₄₇	—	2	4	no	—
141	C₂²×C₄₇	—	2	4	no	—
142	C₂²×C₄₇	C₂×C₄⋊C₈	1	4	yes	—
143	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×C₄₇	#520	1	2	yes	—
145	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
146	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
147	C₂²×C₄₇	#568	1	2	yes	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	2	2	no	—
150	C₄×C₄₇	—	2	2	no	—
151	C₄×C₄₇	C₂×C₄⋊C₈	1	4	yes	—
152	C₄×C₄₇	C₂×C₄⋊C₈	1	4	yes	—
153	C₄×C₄₇	#535	1	1	yes	characteristic
154	C₄×C₄₇	#536	1	1	yes	characteristic
155	C₄×C₄₇	#544	1	2	yes	—
156	Dic₄₇	—	2	8	no	—
157	Dic₄₇	—	4	4	no	—
158	C₂×#431	D₄₇	1	1	yes	characteristic
159	C₂²×C₄⋊C₈	—	47	1	no	—
160	C₄×#117	—	47	1	no	—
161	#1939	—	47	4	no	—
162	#1850	—	47	4	no	—
163	#1842	—	47	2	no	—
164	#1831	—	47	2	no	—
165	C₂×C₄₇	—	2	4	no	—
166	C₂×C₄₇	#2317	1	1	yes	characteristic
167	C₂×C₄₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
168	C₂×C₄₇	#2905	1	1	yes	characteristic
169	C₂×C₄₇	#2988	1	2	yes	—
170	C₂×C₄₇	#2989	1	2	yes	—
171	C₂³×C₈	—	47	2	no	—
172	C₂×#117	—	47	1	no	—
173	C₂×#117	—	47	1	no	—
174	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
175	#431	—	2	4	no	—
176	C₂×C₄⋊C₈	—	47	2	no	—
177	C₂×C₄⋊C₈	—	47	2	no	—
178	C₂×C₄⋊C₈	—	47	8	no	—
179	C₂×C₄⋊C₈	—	94	4	no	—
180	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
181	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
182	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
183	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
184	C₄×C₄⋊C₄	—	47	2	no	—
185	C₄×C₄⋊C₄	—	47	2	no	—
186	C₄×C₂²⋊C₄	—	47	2	no	—
187	C₄×C₂²⋊C₄	—	47	2	no	—
188	#376	—	47	2	no	—
189	C₄³	—	47	2	no	—
190	#336	—	47	8	no	—
191	C₄₇	#21972	1	1	yes	characteristic, complemented, Sylow
192	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
193	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
194	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
195	C₂²×C₈	—	47	4	no	—
196	C₂²×C₈	—	94	4	no	—
197	C₂²×C₈	—	188	4	no	—
198	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
199	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
200	#117	—	47	8	no	—
201	#117	—	94	4	no	—
202	C₂×C₄⋊C₄	—	47	2	no	—
203	C₂×C₄⋊C₄	—	47	2	no	—
204	C₂×C₂²⋊C₄	—	47	2	no	—
205	C₂×C₂²⋊C₄	—	47	2	no	—
206	C₂×C₄²	—	47	2	no	—
207	C₂×C₄²	—	47	2	no	—
208	C₂×C₄²	—	47	4	no	—
209	C₂×C₄²	—	47	4	no	—
210	C₂×C₄²	Dic₉₄	1	4	yes	—
211	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
212	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
213	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
214	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
216	C₄⋊C₈	—	2	2	no	—
217	C₄⋊C₈	—	2	2	no	—
218	C₄⋊C₈	—	94	4	no	—
219	C₄⋊C₈	—	94	4	no	—
220	C₄⋊C₈	—	188	8	no	—
221	C₂²⋊C₈	—	2	4	no	—
222	C₄×C₈	—	2	4	no	—
223	#95	—	47	4	no	—
224	#95	—	47	4	no	—
225	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	—	47	4	no	—
230	C₂²×C₄	—	47	4	no	—
231	C₂²×C₄	C₈×D₄₇	1	4	yes	—
232	C₂²×C₄	#10625324	1	4	yes	—
233	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
234	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
235	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
236	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
242	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	94	4	no	—
248	C₂×C₈	—	94	8	no	—
249	C₂×C₈	—	188	8	no	—
250	C₄⋊C₄	—	47	8	no	—
251	C₄⋊C₄	—	94	4	no	—
252	C₂²⋊C₄	—	47	8	no	—
253	C₂²⋊C₄	—	94	4	no	—
254	C₄²	—	47	8	no	—
255	C₄²	—	94	4	no	—
256	C₄²	—	94	4	no	—
257	C₄²	—	94	4	no	—
258	C₄²	—	188	4	no	—
259	C₄²	Dic₄₇⋊C₄	1	4	yes	—
260	C₄²	Dic₄₇⋊C₄	1	4	yes	—
261	C₄²	C₂×Dic₉₄	1	2	yes	—
262	C₄²	C₂×Dic₉₄	1	2	yes	—
263	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
264	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
268	C₂³	C₂×C₈×D₄₇	1	2	yes	—
269	C₂³	C₂×#10625324	1	2	yes	—
270	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	47	8	no	—
279	C₂×C₄	—	94	4	no	—
280	C₂×C₄	—	94	4	no	—
281	C₂×C₄	—	94	4	no	—
282	C₂×C₄	—	94	4	no	—
283	C₂×C₄	—	94	4	no	—
284	C₂×C₄	Dic₄₇⋊C₈	1	16	yes	complemented
285	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
286	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
287	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
288	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
289	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
290	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
291	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
292	C₂×C₄	C₂×#10625324	1	2	yes	—
293	C₂×C₄	C₂×#10625324	1	2	yes	—
294	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
295	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
296	C₂×C₄	#13094070	1	2	yes	—
297	C₂×C₄	#13094071	1	2	yes	—
298	C₈	—	4	2	no	—
299	C₈	—	4	2	no	—
300	C₈	—	188	4	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	2	2	no	—
305	C₂²	C₂×Dic₄₇⋊C₈	1	4	yes	—
306	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
307	C₂²	#15689603	1	2	yes	—
308	C₂²	C₂²×C₈×D₄₇	1	1	yes	characteristic
309	C₂²	C₂²×#10625324	1	1	yes	characteristic
310	C₂²	#15689889	1	2	yes	—
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	94	8	no	—
315	C₄	—	188	4	no	—
316	C₄	C₂×Dic₄₇⋊C₈	1	4	yes	—
317	C₄	C₂×Dic₄₇⋊C₈	1	4	yes	—
318	C₄	#15689687	1	1	yes	characteristic
319	C₄	#15689694	1	1	yes	characteristic
320	C₄	#15689751	1	2	yes	—
321	C₂	—	2	4	no	—
322	C₂	#20246008	1	1	yes	characteristic
323	C₂	C₂²×Dic₄₇⋊C₈	1	1	yes	characteristic
324	C₂	#20251951	1	1	yes	characteristic
325	C₂	#20252496	1	2	yes	—
326	C₂	#20252501	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₈
Fitting	C₂×C₄₇×#431	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#431	D₄₇
hypocenter	C₄₇	#21972
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20244106	C₂	semidirect
5. #20243539	C₂	semidirect
6. #20243504	C₂	semidirect
32. #13093983	C₈	semidirect
92. C₄⋊Dic₄₇	C₂×C₈	semidirect
97. C₄×Dic₄₇	C₂×C₈	semidirect
114. C₂×C₄×C₄₇	C₄⋊C₈	semidirect
191. C₄₇	#21972	semidirect
284. C₂×C₄	Dic₄₇⋊C₈	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
C₄×#13093983	2
#20244106	2
#20243565	2
#20243539	2
#20243504	2
C₂×C₄₇×#431	2
#21972	47