Subgroups — #26215118

Order 12032.

994 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
3	#20243258	C₂	1	2	yes	maximal, complemented
4	#20243253	C₂	1	2	yes	maximal, complemented
5	#20243243	C₂	1	2	yes	maximal, complemented
6	C₄⋊C₈×Dic₄₇	C₂	1	2	yes	maximal, complemented
7	#20243169	C₂	1	4	yes	maximal
8	C₂×C₄₇×#423	C₂	1	1	yes	maximal, characteristic
9	C₂×#13093983	C₄	1	2	yes	—
10	#15689176	C₄	1	8	yes	complemented
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×#13093934	C₂²	1	4	yes	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
16	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
17	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
18	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
20	C₄×C₂²⋊Dic₄₇	C₄	1	2	yes	—
21	C₄×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
22	#15688969	C₄	1	2	yes	—
23	C₄²×Dic₄₇	C₂²	1	2	yes	—
24	#15688877	C₂²	1	4	yes	—
25	#15688877	C₂²	1	4	yes	—
26	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
27	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#423	C₄	1	8	yes	complemented
29	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
30	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
32	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
33	#13093983	—	2	8	no	—
34	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
38	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
40	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
41	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
42	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
43	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
44	#13093934	—	4	4	no	—
45	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
46	C₈×Dic₄₇	—	4	4	no	—
47	C₄₇⋊#95	C₂×C₄	1	2	yes	—
48	C₄₇⋊#95	C₂×C₄	1	2	yes	—
49	C₄₇⋊#95	C₂×C₄	1	4	yes	—
50	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
52	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₄₇	—	4	2	no	—
54	C₂×M₁₆×C₄₇	—	4	2	no	—
55	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
56	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
57	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
60	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
61	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
64	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
65	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	—	2	4	no	—
67	C₂²×C₄×C₄₇	—	2	4	no	—
68	C₂²×C₄×C₄₇	—	2	1	no	—
69	C₂²×C₄×C₄₇	—	2	1	no	—
70	C₂²×C₄×C₄₇	—	2	1	no	—
71	C₂²×C₄×C₄₇	—	2	1	no	—
72	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
73	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	2	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	C₂⁴	1	1	yes	characteristic
82	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
83	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
84	M₁₆×C₄₇	—	4	8	no	—
85	C₂×C₈×C₄₇	—	2	4	no	—
86	C₂×C₈×C₄₇	—	2	4	no	—
87	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
88	C₄²×C₄₇	C₄²	1	4	yes	—
89	C₄²×C₄₇	C₄²	1	4	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	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
94	C₂²⋊Dic₄₇	—	2	8	no	—
95	C₄⋊Dic₄₇	—	2	4	no	—
96	C₄⋊Dic₄₇	M₁₆	1	8	yes	complemented
97	C₄×Dic₄₇	—	2	8	no	—
98	C₄×Dic₄₇	—	2	2	no	—
99	C₄×Dic₄₇	—	2	2	no	—
100	C₄×Dic₄₇	—	2	2	no	—
101	C₄×Dic₄₇	—	2	2	no	—
102	C₄×Dic₄₇	—	2	8	no	—
103	C₂×C₄₇⋊C₈	—	2	2	no	—
104	C₂×C₄₇⋊C₈	—	2	2	no	—
105	C₂×C₄₇⋊C₈	—	2	2	no	—
106	C₂×C₄₇⋊C₈	—	2	2	no	—
107	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
108	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
109	C₂³×C₄₇	—	2	1	no	—
110	C₂³×C₄₇	—	2	1	no	—
111	C₂³×C₄₇	—	2	2	no	—
112	C₂³×C₄₇	—	2	2	no	—
113	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
114	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	2	1	no	—
117	C₂×C₄×C₄₇	—	2	2	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	2	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	2	8	no	—
124	C₂×C₄×C₄₇	—	2	8	no	—
125	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
126	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
127	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
128	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
130	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
131	C₂×C₄×C₄₇	#131	1	4	yes	—
132	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
136	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
137	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
138	C₂×Dic₄₇	—	2	4	no	—
139	C₂×Dic₄₇	—	2	4	no	—
140	C₂×Dic₄₇	—	2	4	no	—
141	C₂×Dic₄₇	—	2	4	no	—
142	C₂×Dic₄₇	—	2	4	no	—
143	C₂×Dic₄₇	C₂×M₁₆	1	8	yes	—
144	C₈×C₄₇	—	2	8	no	—
145	C₄₇⋊C₈	—	2	4	no	—
146	C₄₇⋊C₈	—	2	4	no	—
147	#20550	—	47	1	no	maximal, Sylow
148	C₂²×C₄₇	—	2	2	no	—
149	C₂²×C₄₇	—	2	2	no	—
150	C₂²×C₄₇	—	2	2	no	—
151	C₂²×C₄₇	—	4	2	no	—
152	C₂²×C₄₇	—	4	2	no	—
153	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
154	C₂²×C₄₇	#520	1	1	yes	characteristic
155	C₂²×C₄₇	#520	1	1	yes	characteristic
156	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
157	C₂²×C₄₇	C₂×#131	1	2	yes	—
158	C₄×C₄₇	—	2	2	no	—
159	C₄×C₄₇	—	2	2	no	—
160	C₄×C₄₇	—	2	2	no	—
161	C₄×C₄₇	C₄×M₁₆	1	4	yes	—
162	C₄×C₄₇	#405	1	4	yes	—
163	C₄×C₄₇	#518	1	1	yes	characteristic
164	C₄×C₄₇	#519	1	1	yes	characteristic
165	C₄×C₄₇	#520	1	1	yes	characteristic
166	C₄×C₄₇	#520	1	1	yes	characteristic
167	Dic₄₇	—	2	8	no	—
168	Dic₄₇	—	4	4	no	—
169	C₂×#431	—	47	1	no	—
170	C₂×#423	D₄₇	1	1	yes	characteristic
171	C₄×#117	—	47	1	no	—
172	#1770	—	47	2	no	—
173	C₈×C₄⋊C₄	—	47	2	no	—
174	#1767	—	47	2	no	—
175	C₄×C₄⋊C₈	—	47	2	no	—
176	#1753	—	47	4	no	—
177	C₂×C₄₇	—	4	2	no	—
178	C₂×C₄₇	#2274	1	1	yes	characteristic
179	C₂×C₄₇	C₂×C₄×M₁₆	1	1	yes	characteristic
180	C₂×C₄₇	C₂×#405	1	1	yes	characteristic
181	C₂×C₄₇	#2990	1	2	yes	—
182	C₂×C₄₇	#2991	1	2	yes	—
183	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
184	C₂×#117	—	47	2	no	—
185	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
186	#431	—	47	8	no	—
187	#423	Dic₄₇	1	8	yes	complemented
188	C₂×C₄⋊C₈	—	47	1	no	—
189	C₂×C₄⋊C₈	—	47	1	no	—
190	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
191	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
192	C₂×C₂²⋊C₈	—	47	2	no	—
193	C₂×#97	—	47	4	no	—
194	C₂×C₄×C₈	—	47	2	no	—
195	C₂×C₄×C₈	—	47	4	no	—
196	C₄×C₄⋊C₄	—	47	2	no	—
197	C₄×C₄⋊C₄	—	47	1	no	—
198	C₄×C₄⋊C₄	—	47	1	no	—
199	C₄×C₂²⋊C₄	—	47	2	no	—
200	C₄×C₂²⋊C₄	—	47	2	no	—
201	#376	—	47	2	no	—
202	C₄³	—	47	2	no	—
203	#336	—	47	4	no	—
204	#336	—	47	4	no	—
205	C₄₇	#20550	1	1	yes	characteristic, complemented, Sylow
206	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
207	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
208	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
209	C₂×M₁₆	—	4	2	no	—
210	C₂×M₁₆	—	4	2	no	—
211	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
212	C₂²×C₈	—	47	2	no	—
213	C₂²×C₈	—	47	2	no	—
214	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
215	#117	—	94	8	no	—
216	C₂×C₄⋊C₄	—	47	2	no	—
217	C₂×C₄⋊C₄	—	47	2	no	—
218	C₂×C₂²⋊C₄	—	47	4	no	—
219	C₂×C₄²	—	47	4	no	—
220	C₂×C₄²	—	47	4	no	—
221	C₂×C₄²	—	47	2	no	—
222	C₂×C₄²	—	47	2	no	—
223	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
224	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
225	C₂×C₄²	C₄×D₄₇	1	4	yes	—
226	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
227	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
228	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
229	C₄⋊C₈	—	47	4	no	—
230	C₄⋊C₈	—	47	4	no	—
231	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
232	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
233	C₂²⋊C₈	—	47	8	no	—
234	#97	—	188	4	no	—
235	C₄×C₈	—	47	8	no	—
236	C₄×C₈	—	188	4	no	—
237	#95	—	47	2	no	—
238	#95	—	47	4	no	—
239	#95	—	47	2	no	—
240	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	47	4	no	—
248	C₂²×C₄	—	47	4	no	—
249	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
250	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
251	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
255	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
257	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
258	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
259	M₁₆	—	4	8	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	47	4	no	—
263	C₂×C₈	—	47	4	no	—
264	C₂×C₈	—	94	2	no	—
265	C₂×C₈	—	94	2	no	—
266	C₂×C₈	—	94	2	no	—
267	C₂×C₈	—	94	2	no	—
268	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
269	C₄⋊C₄	—	47	8	no	—
270	C₄⋊C₄	—	94	4	no	—
271	C₂²⋊C₄	—	94	8	no	—
272	C₄²	—	94	8	no	—
273	C₄²	—	94	8	no	—
274	C₄²	—	94	2	no	—
275	C₄²	—	94	2	no	—
276	C₄²	—	94	2	no	—
277	C₄²	—	94	2	no	—
278	C₄²	C₄×Dic₄₇	1	4	yes	—
279	C₄²	C₄×Dic₄₇	1	4	yes	—
280	C₄²	C₂×C₄×D₄₇	1	2	yes	—
281	C₄²	C₂×C₄×D₄₇	1	2	yes	—
282	C₄²	C₂²×Dic₄₇	1	2	yes	—
283	C₄²	C₂²×Dic₄₇	1	2	yes	—
284	C₂³	—	2	1	no	—
285	C₂³	—	2	1	no	—
286	C₂³	—	2	2	no	—
287	C₂³	—	2	2	no	—
288	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
289	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	8	no	—
295	C₂×C₄	—	2	8	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	47	8	no	—
301	C₂×C₄	—	94	4	no	—
302	C₂×C₄	—	94	4	no	—
303	C₂×C₄	—	94	4	no	—
304	C₂×C₄	—	94	4	no	—
305	C₂×C₄	—	94	4	no	—
306	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
307	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
308	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
309	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
310	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
311	C₂×C₄	M₁₆×D₄₇	1	4	yes	—
312	C₂×C₄	#13094032	1	4	yes	—
313	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
314	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
315	C₂×C₄	#13094069	1	1	yes	characteristic
316	C₂×C₄	#13094069	1	1	yes	characteristic
317	C₂×C₄	#13094072	1	1	yes	characteristic
318	C₂×C₄	#13094072	1	1	yes	characteristic
319	C₈	—	2	8	no	—
320	C₈	—	94	4	no	—
321	C₈	—	94	4	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
328	C₂²	#15689604	1	1	yes	characteristic
329	C₂²	#15689604	1	1	yes	characteristic
330	C₂²	C₂×M₁₆×D₄₇	1	2	yes	—
331	C₂²	C₂×#13094032	1	2	yes	—
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	2	no	—
335	C₄	—	94	8	no	—
336	C₄	—	188	4	no	—
337	C₄	M₁₆×Dic₄₇	1	4	yes	—
338	C₄	#15689081	1	4	yes	—
339	C₄	#15689598	1	1	yes	characteristic
340	C₄	#15689601	1	1	yes	characteristic
341	C₄	#15689604	1	1	yes	characteristic
342	C₄	#15689604	1	1	yes	characteristic
343	C₂	—	4	2	no	—
344	C₂	#20245745	1	1	yes	characteristic
345	C₂	C₂×M₁₆×Dic₄₇	1	1	yes	characteristic
346	C₂	C₂×#15689081	1	1	yes	characteristic
347	C₂	#20252514	1	2	yes	—
348	C₂	#20252518	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₂×C₄₇×#423	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#423	D₄₇
hypocenter	C₄₇	#20550
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #20243258	C₂	semidirect
4. #20243253	C₂	semidirect
5. #20243243	C₂	semidirect
6. C₄⋊C₈×Dic₄₇	C₂	semidirect
10. #15689176	C₄	semidirect
28. C₄₇×#423	C₄	semidirect
96. C₄⋊Dic₄₇	M₁₆	semidirect
187. #423	Dic₄₇	semidirect
205. C₄₇	#20550	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
C₄×#13093983	2
#20243258	2
#20243253	2
#20243243	2
C₄⋊C₈×Dic₄₇	2
#20243169	2
C₂×C₄₇×#423	2
#20550	47