Subgroups — #26215903

Order 12032.

1438 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689172	C₂	1	2	yes	maximal, complemented
2	C₂×#15689087	C₂	1	2	yes	maximal
3	#20243430	C₂	1	4	yes	maximal, complemented
4	#20243426	C₂	1	4	yes	maximal, complemented
5	#20243115	C₂	1	2	yes	maximal, complemented
6	C₂³×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	4	yes	—
8	#15689172	—	2	8	no	—
9	C₂×#13093960	C₂²	1	8	yes	complemented
10	#15689087	—	2	4	no	—
11	#15689087	C₂²	1	4	yes	—
12	#15689087	C₂²	1	4	yes	—
13	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
14	#15688888	C₂²	1	8	yes	complemented
15	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₄₇	—	2	8	no	—
17	C₂²×M₁₆×C₄₇	C₄	1	8	yes	—
18	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
20	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
21	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
22	C₂×#10625329	—	2	4	no	—
23	C₂×#10625329	—	2	8	no	—
24	C₂×#10625329	—	2	8	no	—
25	C₂×#10625329	C₂³	1	8	yes	—
26	C₂²×C₄₇⋊C₈	—	2	4	no	—
27	#13093960	—	2	16	no	—
28	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
29	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
30	C₂⁵×C₄₇	Q₈	1	1	yes	characteristic
31	C₂³×C₄×C₄₇	—	2	4	no	—
32	C₂³×C₄×C₄₇	—	2	4	no	—
33	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
34	C₂³×C₄×C₄₇	D₄	1	4	yes	—
35	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
37	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₄₇	—	2	4	no	—
42	C₂×M₁₆×C₄₇	—	2	32	no	—
43	C₂×M₁₆×C₄₇	—	2	8	no	—
44	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
45	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
46	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
47	C₂×M₁₆×C₄₇	D₄	1	8	yes	—
48	C₂²×C₈×C₄₇	—	2	8	no	—
49	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
50	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
52	C₂⁴×C₄₇	—	2	4	no	—
53	C₂⁴×C₄₇	—	4	4	no	—
54	C₂⁴×C₄₇	C₄⋊C₄	1	4	yes	—
55	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
56	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
57	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	2	8	no	—
60	C₂²×C₄×C₄₇	—	2	8	no	—
61	C₂²×C₄×C₄₇	—	2	16	no	—
62	C₂²×C₄×C₄₇	—	2	4	no	—
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	—	2	2	no	—
67	C₂²×C₄×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
70	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	#41	1	4	yes	—
82	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
83	M₁₆×C₄₇	—	2	8	no	—
84	M₁₆×C₄₇	—	2	16	no	—
85	M₁₆×C₄₇	C₂²⋊C₄	1	16	yes	—
86	C₂×C₈×C₄₇	—	2	4	no	—
87	C₂×C₈×C₄₇	—	2	16	no	—
88	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₄₇	C₂×D₄	1	8	yes	—
91	#10625329	—	2	8	no	—
92	#10625329	—	2	8	no	—
93	#10625329	—	4	8	no	—
94	C₂×C₄₇⋊C₈	—	2	8	no	—
95	C₂×C₄₇⋊C₈	—	4	8	no	—
96	C₂³×C₄₇	—	2	2	no	—
97	C₂³×C₄₇	—	2	4	no	—
98	C₂³×C₄₇	—	2	2	no	—
99	C₂³×C₄₇	—	2	8	no	—
100	C₂³×C₄₇	—	2	8	no	—
101	C₂³×C₄₇	—	4	4	no	—
102	C₂³×C₄₇	—	4	2	no	—
103	C₂³×C₄₇	—	4	2	no	—
104	C₂³×C₄₇	—	4	16	no	—
105	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
106	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
107	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
108	C₂³×C₄₇	C₂²⋊Q₈	1	4	yes	—
109	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	8	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	—	2	8	no	—
120	C₂×C₄×C₄₇	—	2	8	no	—
121	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
122	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₄₇	#117	1	4	yes	—
126	C₂×C₄×C₄₇	#121	1	8	yes	—
127	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
128	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
130	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
132	C₈×C₄₇	—	2	4	no	—
133	C₈×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
134	C₄₇⋊C₈	—	4	8	no	—
135	#20667	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	—	2	8	no	—
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	2	4	no	—
139	C₂²×C₄₇	—	2	2	no	—
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	—	2	4	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	4	4	no	—
144	C₂²×C₄₇	—	4	16	no	—
145	C₂²×C₄₇	—	4	4	no	—
146	C₂²×C₄₇	#380	1	4	yes	—
147	C₂²×C₄₇	#430	1	4	yes	complemented
148	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
150	C₄×C₄₇	—	2	2	no	—
151	C₄×C₄₇	—	2	2	no	—
152	C₄×C₄₇	#380	1	4	yes	—
153	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
154	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
155	C₄×C₄₇	C₂×#121	1	2	yes	—
156	C₂³×M₁₆	D₄₇	1	1	yes	characteristic
157	C₂×#430	—	47	2	no	—
158	C₂×#407	—	47	2	no	—
159	#1811	—	47	4	no	—
160	#1810	—	47	4	no	—
161	#1738	—	47	2	no	—
162	C₂×C₄₇	—	2	2	no	—
163	C₂×C₄₇	—	2	4	no	—
164	C₂×C₄₇	—	2	4	no	—
165	C₂×C₄₇	—	4	2	no	—
166	C₂×C₄₇	C₂×#380	1	1	yes	characteristic
167	C₂×C₄₇	C₂×#430	1	2	yes	—
168	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
169	C₂²×M₁₆	—	2	8	no	—
170	C₂²×M₁₆	—	47	4	no	—
171	C₂²×M₁₆	Dic₄₇	1	8	yes	—
172	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
173	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
174	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
175	C₂³×C₈	C₂×D₄₇	1	2	yes	—
176	#430	—	94	8	no	—
177	C₂×#108	—	47	8	no	—
178	#407	—	47	4	no	—
179	#407	—	47	4	no	—
180	#407	—	94	4	no	—
181	C₂×C₂²⋊C₈	—	94	4	no	—
182	#341	—	47	8	no	—
183	C₄₇	#20667	1	1	yes	characteristic, complemented, Sylow
184	C₂⁵	Dic₉₄	1	1	yes	characteristic
185	C₂³×C₄	—	2	4	no	—
186	C₂³×C₄	—	2	4	no	—
187	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
188	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
189	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
190	C₂³×C₄	D₁₈₈	1	4	yes	—
191	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
192	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
193	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
194	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
195	C₂×M₁₆	—	2	4	no	—
196	C₂×M₁₆	—	2	32	no	—
197	C₂×M₁₆	—	2	8	no	—
198	C₂×M₁₆	—	47	8	no	—
199	C₂×M₁₆	—	94	4	no	—
200	C₂×M₁₆	—	94	8	no	—
201	C₂×M₁₆	—	94	8	no	—
202	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
203	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
204	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
205	C₂×M₁₆	C₂²⋊D₄₇	1	8	yes	—
206	C₂²×C₈	—	2	8	no	—
207	C₂²×C₈	—	94	4	no	—
208	C₂²×C₈	C₂×Dic₄₇	1	8	yes	—
209	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
210	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
211	#108	—	94	16	no	—
212	C₂²⋊C₈	—	94	8	no	—
213	C₂²⋊C₈	—	94	8	no	—
214	C₂⁴	—	2	4	no	—
215	C₂⁴	—	4	4	no	—
216	C₂⁴	C₄⋊Dic₄₇	1	4	yes	—
217	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
218	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
219	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	8	no	—
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	—	2	16	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
231	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
232	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
233	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
234	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
235	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
237	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
238	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
239	C₂²×C₄	#10625350	1	4	yes	—
240	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
241	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
242	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
243	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
244	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
245	M₁₆	—	2	8	no	—
246	M₁₆	—	2	16	no	—
247	M₁₆	—	94	8	no	—
248	M₁₆	—	94	8	no	—
249	M₁₆	—	188	8	no	—
250	M₁₆	C₂²⋊Dic₄₇	1	16	yes	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	16	no	—
253	C₂×C₈	—	94	8	no	—
254	C₂×C₈	—	188	8	no	—
255	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
256	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
257	C₂×C₈	C₂×C₂²⋊D₄₇	1	8	yes	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	4	no	—
260	C₂³	—	2	8	no	—
261	C₂³	—	2	2	no	—
262	C₂³	—	2	8	no	—
263	C₂³	—	4	4	no	—
264	C₂³	—	4	2	no	—
265	C₂³	—	4	2	no	—
266	C₂³	—	4	16	no	—
267	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
268	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
269	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
270	C₂³	#13094003	1	4	yes	—
271	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
284	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
285	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
286	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
287	C₂×C₄	#13093983	1	4	yes	—
288	C₂×C₄	#13093998	1	8	yes	—
289	C₂×C₄	#13094003	1	4	yes	—
290	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
291	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
292	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
293	C₂×C₄	C₂×#10625350	1	2	yes	—
294	C₈	—	2	4	no	—
295	C₈	—	188	8	no	—
296	C₈	C₂×C₂²⋊Dic₄₇	1	8	yes	—
297	C₂²	—	2	8	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	4	4	no	—
305	C₂²	—	4	16	no	—
306	C₂²	—	4	4	no	—
307	C₂²	#15688983	1	4	yes	—
308	C₂²	#15689172	1	4	yes	complemented
309	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
310	C₂²	C₂×#13094003	1	2	yes	—
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	#15688983	1	4	yes	—
314	C₄	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
315	C₄	C₂×#13093983	1	1	yes	characteristic
316	C₄	C₂×#13093998	1	2	yes	—
317	C₂	—	2	2	no	—
318	C₂	—	2	4	no	—
319	C₂	—	2	4	no	—
320	C₂	—	4	2	no	—
321	C₂	C₂×#15688983	1	1	yes	characteristic
322	C₂	C₂×#15689172	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂³×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂⁵×C₄₇	Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂³×M₁₆	D₄₇
hypocenter	C₄₇	#20667
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#15689172	C₂	semidirect
3. #20243430	C₂	semidirect
4. #20243426	C₂	semidirect
5. #20243115	C₂	semidirect
9. C₂×#13093960	C₂²	semidirect
14. #15688888	C₂²	semidirect
147. C₂²×C₄₇	#430	semidirect
183. C₄₇	#20667	semidirect
308. C₂²	#15689172	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689172	2
C₂×#15689087	2
#20243430	2
#20243426	2
#20243115	2
C₂³×M₁₆×C₄₇	2
#20667	47