Subgroups — #26226648

Order 12032.

906 conjugacy classes of subgroups.

The subgroup lattice has 318 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	#20245795	C₂	1	1	yes	maximal, characteristic
3	#20244151	C₂	1	4	yes	maximal, complemented
4	#20244149	C₂	1	4	yes	maximal, complemented
5	#20243411	C₂	1	4	yes	maximal
6	C₄₇×#2285	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
8	C₂×#13093983	—	2	1	no	—
9	C₂×#13093955	C₂²	1	8	yes	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₄	1	8	yes	complemented
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	#15688993	C₂²	1	2	yes	—
14	#15688989	—	2	1	no	—
15	#15688983	—	2	1	no	—
16	#15688983	C₂²	1	2	yes	—
17	C₄×C₄⋊Dic₄₇	—	2	1	no	—
18	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
19	#15688885	C₂²	1	8	yes	—
20	C₂×#117×C₄₇	—	2	1	no	—
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#384	C₂²	1	2	yes	—
23	C₄₇×#382	—	2	1	no	—
24	C₄₇×#380	—	2	1	no	—
25	C₄₇×#380	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₄₇	—	2	1	no	—
27	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
28	C₂²×C₄₇⋊C₈	—	4	4	no	—
29	C₂²×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
30	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
31	#13093983	—	4	4	no	—
32	C₂×C₄⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄⋊Dic₄₇	—	2	2	no	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
35	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
36	C₂×C₄×Dic₄₇	—	2	2	no	—
37	#13093955	—	4	8	no	—
38	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
39	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
40	C₄₇⋊#95	—	2	2	no	—
41	C₄₇⋊#95	—	2	2	no	—
42	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
45	#117×C₄₇	—	4	4	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
49	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
50	C₂×C₄²×C₄₇	—	2	2	no	—
51	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
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	#95×C₄₇	—	2	2	no	—
58	#95×C₄₇	—	2	2	no	—
59	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	—	2	4	no	—
63	C₂²×C₄×C₄₇	—	4	2	no	—
64	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
66	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
67	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₄₇	—	2	4	no	—
77	C₄⋊C₄×C₄₇	—	2	4	no	—
78	C₄⋊C₄×C₄₇	—	2	4	no	—
79	C₄⋊C₄×C₄₇	—	4	4	no	—
80	C₄⋊C₄×C₄₇	—	4	4	no	—
81	C₂²⋊C₄×C₄₇	—	4	4	no	—
82	C₄²×C₄₇	—	4	4	no	—
83	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
84	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
85	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
86	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
87	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
88	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
89	C₂²⋊Dic₄₇	—	4	4	no	—
90	C₄⋊Dic₄₇	—	2	4	no	—
91	C₄⋊Dic₄₇	—	2	4	no	—
92	C₄⋊Dic₄₇	—	4	4	no	—
93	C₄⋊Dic₄₇	—	4	4	no	—
94	C₄×Dic₄₇	—	4	4	no	—
95	C₂×C₄₇⋊C₈	—	2	8	no	—
96	C₂×C₄₇⋊C₈	—	4	8	no	—
97	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
98	C₂³×C₄₇	—	2	2	no	—
99	C₂³×C₄₇	—	2	2	no	—
100	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
101	C₂³×C₄₇	C₂×SD₁₆	1	2	yes	—
102	C₂³×C₄₇	C₂×SD₁₆	1	2	yes	—
103	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	—	4	4	no	—
112	C₂×C₄×C₄₇	—	4	4	no	—
113	C₂×C₄×C₄₇	—	4	4	no	—
114	C₂×C₄×C₄₇	#106	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	C₂×SD₁₆	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
123	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
124	C₂×C₄×C₄₇	C₂³×C₄	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	—	4	4	no	—
129	C₂×Dic₄₇	—	4	4	no	—
130	C₂×Dic₄₇	—	4	4	no	—
131	C₄₇⋊C₈	—	2	8	no	—
132	#21959	—	47	1	no	maximal, Sylow
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	2	4	no	—
136	C₂²×C₄₇	—	2	4	no	—
137	C₂²×C₄₇	C₂×#106	1	4	yes	—
138	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
139	C₂²×C₄₇	#537	1	2	yes	—
140	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
141	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
142	C₂²×C₄₇	#575	1	2	yes	—
143	C₄×C₄₇	—	2	2	no	—
144	C₄×C₄₇	—	2	2	no	—
145	C₄×C₄₇	—	2	2	no	—
146	C₄×C₄₇	—	8	4	no	—
147	C₄×C₄₇	C₂×#106	1	4	yes	—
148	C₄×C₄₇	C₂×#106	1	4	yes	—
149	C₄×C₄₇	#518	1	1	yes	characteristic
150	C₄×C₄₇	#519	1	1	yes	characteristic
151	C₄×C₄₇	#544	1	2	yes	—
152	Dic₄₇	—	8	4	no	—
153	C₂²×C₄⋊C₈	—	47	1	no	—
154	#2285	—	47	1	no	—
155	#2285	D₄₇	1	1	yes	characteristic
156	#1948	—	47	4	no	—
157	#1948	—	47	4	no	—
158	#1807	—	47	4	no	—
159	C₂×C₄₇	—	2	4	no	—
160	C₂×C₄₇	#2316	1	1	yes	characteristic
161	C₂×C₄₇	C₂²×#106	1	1	yes	characteristic
162	C₂×C₄₇	#2911	1	1	yes	characteristic
163	C₂×C₄₇	#3246	1	2	yes	—
164	C₂×C₄₇	#3246	1	2	yes	—
165	C₂³×C₈	—	47	2	no	—
166	C₂×#117	—	2	1	no	—
167	C₂×#117	—	94	1	no	—
168	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
169	C₂×#106	—	47	8	no	—
170	C₂×C₄⋊C₈	—	47	8	no	—
171	C₂×C₄⋊C₈	—	47	2	no	—
172	C₂×C₄⋊C₈	—	47	2	no	—
173	#384	—	47	2	no	—
174	#384	C₂×D₄₇	1	2	yes	—
175	#382	—	2	1	no	—
176	#382	—	94	1	no	—
177	#380	—	2	1	no	—
178	#380	—	47	2	no	—
179	#380	—	94	1	no	—
180	#380	C₂×D₄₇	1	2	yes	—
181	C₄×C₄⋊C₄	—	2	1	no	—
182	C₄×C₄⋊C₄	—	47	2	no	—
183	C₄×C₄⋊C₄	—	94	1	no	—
184	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
185	#340	—	47	8	no	—
186	C₄₇	#21959	1	1	yes	characteristic, complemented, Sylow
187	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
188	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
189	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
190	C₂²×C₈	—	47	4	no	—
191	C₂²×C₈	—	47	8	no	—
192	C₂²×C₈	—	188	4	no	—
193	#117	—	4	4	no	—
194	#117	—	188	4	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	47	4	no	—
198	C₂×C₄⋊C₄	—	94	2	no	—
199	C₂×C₄⋊C₄	—	94	2	no	—
200	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
201	C₂×C₂²⋊C₄	—	2	2	no	—
202	C₂×C₂²⋊C₄	—	94	2	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	—	94	2	no	—
205	C₂×C₄²	Dic₉₄	1	4	yes	—
206	C₂×C₄²	C₄×D₄₇	1	4	yes	—
207	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
208	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
209	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
210	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
211	#106	—	188	8	no	—
212	C₄⋊C₈	—	47	8	no	—
213	C₄⋊C₈	—	47	8	no	—
214	#95	—	2	2	no	—
215	#95	—	2	2	no	—
216	#95	—	94	2	no	—
217	#95	—	94	2	no	—
218	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	94	4	no	—
224	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
225	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
226	C₂²×C₄	#10625335	1	4	yes	—
227	C₂²×C₄	#10625336	1	4	yes	—
228	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
229	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
230	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
235	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
236	C₂×C₈	—	47	8	no	—
237	C₂×C₈	—	94	8	no	—
238	C₂×C₈	—	188	8	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	4	4	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	94	4	no	—
244	C₄⋊C₄	—	94	4	no	—
245	C₄⋊C₄	—	188	4	no	—
246	C₄⋊C₄	—	188	4	no	—
247	C₂²⋊C₄	—	4	4	no	—
248	C₂²⋊C₄	—	188	4	no	—
249	C₄²	—	4	4	no	—
250	C₄²	—	188	4	no	—
251	C₄²	Dic₄₇⋊C₄	1	4	yes	—
252	C₄²	Dic₄₇⋊C₄	1	4	yes	—
253	C₄²	C₂×Dic₉₄	1	2	yes	—
254	C₄²	C₂×Dic₉₄	1	2	yes	—
255	C₄²	C₂×C₄×D₄₇	1	2	yes	—
256	C₄²	C₂×C₄×D₄₇	1	2	yes	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
260	C₂³	C₂×#10625336	1	2	yes	—
261	C₂³	C₂×#10625335	1	2	yes	—
262	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	188	4	no	—
274	C₂×C₄	—	188	4	no	—
275	C₂×C₄	—	188	4	no	—
276	C₂×C₄	#13093955	1	16	yes	complemented
277	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
278	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
279	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
280	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
281	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
282	C₂×C₄	C₂×#10625336	1	2	yes	—
283	C₂×C₄	C₂×#10625336	1	2	yes	—
284	C₂×C₄	C₂×#10625335	1	2	yes	—
285	C₂×C₄	C₂×#10625335	1	2	yes	—
286	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
287	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
288	C₂×C₄	#13094070	1	2	yes	—
289	C₂×C₄	#13094071	1	2	yes	—
290	C₈	—	94	8	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	2	2	no	—
295	C₂²	C₂×#13093955	1	4	yes	—
296	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
297	C₂²	#15689701	1	2	yes	—
298	C₂²	C₂²×#10625336	1	1	yes	characteristic
299	C₂²	C₂²×#10625335	1	1	yes	characteristic
300	C₂²	#15689921	1	2	yes	—
301	C₄	—	2	2	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	8	4	no	—
305	C₄	—	376	4	no	—
306	C₄	C₂×#13093955	1	4	yes	—
307	C₄	C₂×#13093955	1	4	yes	—
308	C₄	#15689597	1	1	yes	characteristic
309	C₄	#15689600	1	1	yes	characteristic
310	C₄	#15689751	1	2	yes	—
311	C₂	—	2	4	no	—
312	C₂	#20246004	1	1	yes	characteristic
313	C₂	C₂²×#13093955	1	1	yes	characteristic
314	C₂	#20251977	1	1	yes	characteristic
315	C₂	#20254700	1	2	yes	—
316	C₂	#20254704	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2285	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2285	D₄₇
hypocenter	C₄₇	#21959
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
3. #20244151	C₂	semidirect
4. #20244149	C₂	semidirect
10. C₂×C₄⋊(C₄₇⋊C₈)	C₄	semidirect
114. C₂×C₄×C₄₇	#106	semidirect
186. C₄₇	#21959	semidirect
276. C₂×C₄	#13093955	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
#20245795	2
#20244151	2
#20244149	2
#20243411	2
C₄₇×#2285	2
#21959	47