Subgroups — #26215012

Order 12032.

1010 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689180	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20243265	C₂	1	4	yes	maximal, complemented
4	#20243260	C₂	1	4	yes	maximal, complemented
5	#20243152	C₂	1	2	yes	maximal
6	C₂²⋊C₈×Dic₄₇	C₂	1	2	yes	maximal
7	C₂×C₄₇×#432	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄⋊Dic₄₇	C₄	1	2	yes	—
9	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
11	#15689180	C₄	1	8	yes	complemented
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×#13093935	C₂²	1	2	yes	—
15	C₂×#13093934	C₂²	1	4	yes	—
16	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
17	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
18	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
19	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
20	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
21	C₂×C₄₇⋊#95	C₄	1	2	yes	—
22	#15688877	C₂²	1	4	yes	—
23	#15688877	C₂²	1	4	yes	—
24	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×#432	C₄	1	8	yes	complemented
26	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
28	C₂×#97×C₄₇	C₂²	1	2	yes	—
29	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
30	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
32	C₂×C₄⋊Dic₄₇	—	2	8	no	—
33	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
35	C₂×C₄×Dic₄₇	—	4	2	no	—
36	C₂×C₄×Dic₄₇	—	4	2	no	—
37	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
40	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
41	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	complemented
42	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
43	#13093935	C₂×C₄	1	8	yes	complemented
44	#13093934	—	4	4	no	—
45	C₈×Dic₄₇	—	4	4	no	—
46	C₄₇⋊#95	C₂×C₄	1	4	yes	—
47	C₄₇⋊#95	C₂×C₄	1	4	yes	—
48	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
52	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
55	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	complemented
59	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
60	#97×C₄₇	C₂×C₄	1	8	yes	complemented
61	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	—	2	4	no	—
63	C₂²×C₄×C₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	4	2	no	—
66	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
67	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₄₇	—	2	4	no	—
77	C₂²×Dic₄₇	—	4	2	no	—
78	C₂²×Dic₄₇	—	4	2	no	—
79	C₂²×Dic₄₇	—	4	2	no	—
80	C₂²×Dic₄₇	—	4	2	no	—
81	C₂²×Dic₄₇	M₁₆	1	8	yes	—
82	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
83	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₄₇	—	2	4	no	—
85	C₂×C₈×C₄₇	—	2	4	no	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
87	C₄²×C₄₇	C₄²	1	4	yes	—
88	C₄²×C₄₇	C₄²	1	4	yes	—
89	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
90	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
91	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
92	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
93	C₄⋊Dic₄₇	—	2	8	no	—
94	C₄⋊Dic₄₇	—	2	8	no	—
95	C₄×Dic₄₇	—	2	4	no	—
96	C₄×Dic₄₇	—	2	4	no	—
97	C₄×Dic₄₇	—	4	2	no	—
98	C₄×Dic₄₇	—	4	2	no	—
99	C₂×C₄₇⋊C₈	—	2	4	no	—
100	C₂×C₄₇⋊C₈	—	2	4	no	—
101	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
102	C₂³×C₄₇	—	2	1	no	—
103	C₂³×C₄₇	—	2	1	no	—
104	C₂³×C₄₇	—	2	2	no	—
105	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
106	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
107	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	—	2	1	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	8	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	4	4	no	—
118	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
126	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
127	C₂×Dic₄₇	—	2	4	no	—
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	—	4	8	no	—
133	C₂×Dic₄₇	—	4	8	no	—
134	C₂×Dic₄₇	C₂×M₁₆	1	8	yes	—
135	C₈×C₄₇	—	2	8	no	—
136	C₄₇⋊C₈	—	2	8	no	—
137	#20534	—	47	1	no	maximal, Sylow
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	—	2	4	no	—
140	C₂²×C₄₇	—	2	4	no	—
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	C₄×M₁₆	1	4	yes	—
143	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
144	C₂²×C₄₇	#518	1	1	yes	characteristic
145	C₂²×C₄₇	#519	1	1	yes	characteristic
146	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
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	—	2	2	no	—
152	C₄×C₄₇	—	2	2	no	—
153	C₄×C₄₇	#520	1	2	yes	—
154	C₄×C₄₇	#520	1	2	yes	—
155	Dic₄₇	—	2	8	no	—
156	Dic₄₇	—	4	4	no	—
157	C₂×#432	—	47	1	no	—
158	C₂×#432	D₄₇	1	1	yes	characteristic
159	C₂×C₄×C₄⋊C₄	—	47	1	no	—
160	#1771	—	47	4	no	—
161	#1771	—	47	4	no	—
162	#1749	—	47	2	no	—
163	C₄×C₂²⋊C₈	—	47	2	no	—
164	C₂×C₄₇	—	2	4	no	—
165	C₂×C₄₇	#2274	1	1	yes	characteristic
166	C₂×C₄₇	C₂×C₄×M₁₆	1	1	yes	characteristic
167	C₂×C₄₇	#2873	1	1	yes	characteristic
168	C₂×C₄₇	#2973	1	2	yes	—
169	C₂×C₄₇	#2983	1	2	yes	—
170	C₂²×C₄⋊C₄	—	47	2	no	—
171	C₂²×C₄²	—	47	1	no	—
172	C₂²×C₄²	—	47	1	no	—
173	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
174	#432	—	47	8	no	—
175	#432	Dic₄₇	1	8	yes	complemented
176	C₂×C₄⋊C₈	—	47	2	no	—
177	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
178	C₂×C₂²⋊C₈	—	47	2	no	—
179	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
180	C₂×#97	—	47	2	no	—
181	C₂×#97	—	47	4	no	—
182	C₂×#97	C₂×D₄₇	1	2	yes	—
183	C₂×C₄×C₈	—	47	4	no	—
184	C₄×C₄⋊C₄	—	47	2	no	—
185	C₄×C₄⋊C₄	—	47	2	no	—
186	C₄×C₄⋊C₄	—	47	2	no	—
187	C₄×C₄⋊C₄	—	47	2	no	—
188	C₂×#95	—	47	2	no	—
189	#336	—	47	4	no	—
190	#336	—	47	4	no	—
191	C₄₇	#20534	1	1	yes	characteristic, complemented, Sylow
192	C₂³×C₄	—	47	2	no	—
193	C₂³×C₄	—	47	2	no	—
194	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
195	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
196	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
197	C₂²×C₈	—	47	4	no	—
198	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
199	C₂×C₄⋊C₄	—	47	4	no	—
200	C₂×C₄⋊C₄	—	47	4	no	—
201	C₂×C₄⋊C₄	—	94	8	no	—
202	C₂×C₄²	—	47	2	no	—
203	C₂×C₄²	—	47	2	no	—
204	C₂×C₄²	—	47	2	no	—
205	C₂×C₄²	—	47	2	no	—
206	C₂×C₄²	—	188	2	no	—
207	C₂×C₄²	—	188	2	no	—
208	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
209	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
210	C₂×C₄²	C₄×D₄₇	1	4	yes	—
211	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
212	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
213	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
214	C₄⋊C₈	—	47	8	no	—
215	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	complemented
216	C₂²⋊C₈	—	47	8	no	—
217	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
218	#97	—	47	8	no	—
219	#97	—	188	4	no	—
220	#97	C₂×Dic₄₇	1	8	yes	complemented
221	C₄×C₈	—	188	4	no	—
222	#95	—	47	4	no	—
223	#95	—	47	4	no	—
224	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
225	C₂²×C₄	—	2	4	no	—
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	8	no	—
231	C₂²×C₄	—	47	4	no	—
232	C₂²×C₄	—	94	4	no	—
233	C₂²×C₄	—	188	2	no	—
234	C₂²×C₄	—	188	2	no	—
235	C₂²×C₄	—	188	2	no	—
236	C₂²×C₄	—	188	2	no	—
237	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
238	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
239	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
245	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
246	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	47	8	no	—
250	C₂×C₈	—	94	4	no	—
251	C₂×C₈	—	94	4	no	—
252	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
253	C₄⋊C₄	—	94	8	no	—
254	C₄⋊C₄	—	94	8	no	—
255	C₄²	—	94	4	no	—
256	C₄²	—	94	4	no	—
257	C₄²	—	188	2	no	—
258	C₄²	—	188	2	no	—
259	C₄²	C₄×Dic₄₇	1	4	yes	—
260	C₄²	C₄×Dic₄₇	1	4	yes	—
261	C₄²	C₂×C₄×D₄₇	1	2	yes	—
262	C₄²	C₂×C₄×D₄₇	1	2	yes	—
263	C₄²	C₂²×Dic₄₇	1	2	yes	—
264	C₄²	C₂²×Dic₄₇	1	2	yes	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	2	no	—
268	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
269	C₂³	M₁₆×D₄₇	1	4	yes	—
270	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	47	8	no	—
282	C₂×C₄	—	94	4	no	—
283	C₂×C₄	—	94	4	no	—
284	C₂×C₄	—	94	4	no	—
285	C₂×C₄	—	94	4	no	—
286	C₂×C₄	—	94	4	no	—
287	C₂×C₄	—	188	8	no	—
288	C₂×C₄	—	188	8	no	—
289	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
290	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
291	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
292	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
293	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
294	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
295	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
296	C₂×C₄	#13094070	1	2	yes	—
297	C₂×C₄	#13094071	1	2	yes	—
298	C₈	—	2	8	no	—
299	C₈	—	94	8	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	M₁₆×Dic₄₇	1	4	yes	—
305	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
306	C₂²	#15689597	1	1	yes	characteristic
307	C₂²	#15689600	1	1	yes	characteristic
308	C₂²	C₂×M₁₆×D₄₇	1	2	yes	—
309	C₂²	#15689889	1	2	yes	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	94	8	no	—
316	C₄	—	188	4	no	—
317	C₄	#15689603	1	2	yes	—
318	C₄	#15689606	1	2	yes	—
319	C₂	—	2	4	no	—
320	C₂	#20245744	1	1	yes	characteristic
321	C₂	C₂×M₁₆×Dic₄₇	1	1	yes	characteristic
322	C₂	#20251806	1	1	yes	characteristic
323	C₂	#20252416	1	2	yes	—
324	C₂	#20252472	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₂×C₄₇×#432	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#432	D₄₇
hypocenter	C₄₇	#20534
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #20243265	C₂	semidirect
4. #20243260	C₂	semidirect
11. #15689180	C₄	semidirect
25. C₄₇×#432	C₄	semidirect
41. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
43. #13093935	C₂×C₄	semidirect
58. C₄⋊C₈×C₄₇	C₂×C₄	semidirect
60. #97×C₄₇	C₂×C₄	semidirect
175. #432	Dic₄₇	semidirect
191. C₄₇	#20534	semidirect
215. C₄⋊C₈	C₂×Dic₄₇	semidirect
220. #97	C₂×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689180	2
C₂×C₄×C₄⋊Dic₄₇	2
#20243265	2
#20243260	2
#20243152	2
C₂²⋊C₈×Dic₄₇	2
C₂×C₄₇×#432	2
#20534	47