Subgroups — #26121436

Order 12032.

1078 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093934	C₂	1	2	yes	maximal
2	C₂²×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688877	C₂	1	2	yes	maximal
4	C₄×C₄₇⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₇×#336	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
9	C₂×#13093934	—	2	4	no	—
10	C₂×#13093934	—	2	4	no	—
11	C₂×#13093934	C₄	1	8	yes	complemented
12	C₂×C₄×C₄₇⋊C₈	—	2	4	no	—
13	C₂×C₄×C₄₇⋊C₈	C₄	1	4	yes	complemented
14	C₂×C₄₇⋊#95	C₄	1	4	yes	—
15	#15688877	C₄	1	8	yes	complemented
16	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#336	C₄	1	4	yes	complemented
19	C₂³×Dic₄₇	C₂×C₄	1	4	yes	—
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	1	no	—
22	C₂²×C₄₇⋊C₈	—	2	1	no	—
23	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
27	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
28	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
29	C₂²×C₄₇⋊C₈	Q₈	1	2	yes	—
30	C₂×C₄×Dic₄₇	—	2	2	no	—
31	C₂×C₄×Dic₄₇	—	2	2	no	—
32	C₂×C₄×Dic₄₇	—	2	2	no	—
33	C₂×C₄×Dic₄₇	—	2	2	no	—
34	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
35	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
36	#13093934	—	2	16	no	—
37	C₄×C₄₇⋊C₈	—	2	8	no	—
38	C₄₇⋊#95	—	2	8	no	—
39	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
40	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	—	2	2	no	—
42	C₂²×C₈×C₄₇	—	2	2	no	—
43	C₂²×C₈×C₄₇	—	2	2	no	—
44	C₂²×C₈×C₄₇	—	2	2	no	—
45	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₄₇	—	2	2	no	—
49	C₂×C₄²×C₄₇	—	2	2	no	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
52	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	2	4	no	—
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
56	C₂²×C₄×C₄₇	M₁₆	1	8	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	#41	1	2	yes	—
66	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
67	C₂²×Dic₄₇	—	2	4	no	—
68	C₂²×Dic₄₇	—	2	4	no	—
69	C₂²×Dic₄₇	—	2	8	no	—
70	C₂²×Dic₄₇	—	2	8	no	—
71	C₂²×Dic₄₇	C₄²	1	8	yes	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	—	2	4	no	—
74	C₂×C₈×C₄₇	—	2	4	no	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	—	2	4	no	—
77	C₂×C₈×C₄₇	—	2	4	no	—
78	C₂×C₈×C₄₇	C₄²	1	8	yes	—
79	C₄²×C₄₇	—	2	2	no	—
80	C₄²×C₄₇	—	2	2	no	—
81	C₄²×C₄₇	—	2	4	no	—
82	C₄×Dic₄₇	—	2	4	no	—
83	C₄×Dic₄₇	—	2	4	no	—
84	C₄×Dic₄₇	—	2	4	no	—
85	C₄×Dic₄₇	—	2	4	no	—
86	C₂×C₄₇⋊C₈	—	2	2	no	—
87	C₂×C₄₇⋊C₈	—	2	2	no	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	C₂×C₄₇⋊C₈	C₄²	1	4	yes	—
91	C₂×C₄₇⋊C₈	C₄²	1	4	yes	—
92	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	4	yes	—
93	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	8	yes	—
94	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
95	C₂×C₄₇⋊C₈	C₄⋊C₄	1	8	yes	—
96	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
97	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
98	C₂³×C₄₇	C₄×D₄	1	2	yes	—
99	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
100	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
101	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
102	C₂³×C₄₇	#131	1	4	yes	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	2	16	no	—
106	C₂×C₄×C₄₇	#97	1	8	yes	complemented
107	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
110	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₄₇	#117	1	2	yes	—
115	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
117	C₂×C₄×C₄₇	#117	1	2	yes	—
118	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
124	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
125	C₂×Dic₄₇	—	2	8	no	—
126	C₂×Dic₄₇	—	2	8	no	—
127	C₂×Dic₄₇	—	2	8	no	—
128	C₂×Dic₄₇	—	4	8	no	—
129	C₂×Dic₄₇	—	4	8	no	—
130	C₈×C₄₇	—	2	8	no	—
131	C₄₇⋊C₈	—	2	4	no	—
132	C₄₇⋊C₈	#95	1	8	yes	complemented
133	#10696	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	C₄³	1	1	yes	characteristic
135	C₂²×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
136	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
138	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
139	C₂²×C₄₇	C₂×#97	1	2	yes	—
140	C₂²×C₄₇	C₄×M₁₆	1	4	yes	—
141	C₂²×C₄₇	#405	1	4	yes	—
142	C₂²×C₄₇	#405	1	2	yes	—
143	C₂²×C₄₇	#435	1	8	yes	—
144	C₂²×C₄₇	#436	1	4	yes	—
145	C₂²×C₄₇	#446	1	4	yes	—
146	C₄×C₄₇	—	2	8	no	—
147	C₄×C₄₇	C₂×#95	1	1	yes	characteristic
148	C₄×C₄₇	#376	1	1	yes	characteristic
149	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
150	C₄×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
151	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
153	Dic₄₇	—	4	8	no	—
154	C₂²×#97	—	47	2	no	—
155	C₂²×C₄×C₈	—	47	1	no	—
156	C₂×#336	—	47	2	no	—
157	C₂×#336	D₄₇	1	1	yes	characteristic
158	C₄×#95	—	47	1	no	—
159	C₂×C₄₇	C₄×#95	1	1	yes	characteristic
160	C₂×C₄₇	C₄×#97	1	1	yes	characteristic
161	C₂×C₄₇	#1726	1	1	yes	characteristic
162	C₂×C₄₇	#1752	1	2	yes	—
163	C₂×C₄₇	#1753	1	4	yes	—
164	C₂×C₄₇	#1770	1	2	yes	—
165	C₂×C₄₇	#1771	1	4	yes	—
166	C₂³×C₈	—	47	1	no	—
167	C₂³×C₈	—	47	1	no	—
168	C₂³×C₈	C₂×D₄₇	1	2	yes	—
169	C₂²×C₄²	—	47	2	no	—
170	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
171	C₂×#97	—	47	8	no	—
172	C₂×#97	—	94	4	no	—
173	C₂×#97	—	94	4	no	—
174	C₂×C₄×C₈	—	47	4	no	—
175	C₂×C₄×C₈	—	94	4	no	—
176	C₂×#95	—	47	4	no	—
177	#336	—	47	8	no	—
178	#336	Dic₄₇	1	4	yes	complemented
179	C₄₇	#10696	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₄	—	47	4	no	—
181	C₂³×C₄	C₄×D₄₇	1	2	yes	—
182	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	47	2	no	—
188	C₂²×C₈	—	47	4	no	—
189	C₂²×C₈	—	47	4	no	—
190	C₂²×C₈	—	47	2	no	—
191	C₂²×C₈	—	47	2	no	—
192	C₂²×C₈	—	47	4	no	—
193	C₂²×C₈	—	47	2	no	—
194	C₂²×C₈	—	94	2	no	—
195	C₂²×C₈	—	94	1	no	—
196	C₂²×C₈	—	94	1	no	—
197	C₂²×C₈	C₄×D₄₇	1	4	yes	—
198	C₂²×C₈	C₄×D₄₇	1	4	yes	—
199	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	—	47	4	no	—
203	C₂×C₄²	—	47	4	no	—
204	C₂×C₄²	—	94	2	no	—
205	C₂×C₄²	—	94	2	no	—
206	C₂×C₄²	—	94	2	no	—
207	C₂×C₄²	—	94	2	no	—
208	C₂×C₄²	C₄×D₄₇	1	2	yes	—
209	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
210	#97	—	94	16	no	—
211	C₄×C₈	—	94	8	no	—
212	#95	—	94	8	no	—
213	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	47	8	no	—
217	C₂²×C₄	—	94	4	no	—
218	C₂²×C₄	—	94	4	no	—
219	C₂²×C₄	—	94	8	no	—
220	C₂²×C₄	—	94	8	no	—
221	C₂²×C₄	#10625324	1	8	yes	—
222	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
223	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
225	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
226	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
227	C₂²×C₄	D₄×D₄₇	1	2	yes	—
228	C₂²×C₄	#10625352	1	1	yes	characteristic
229	C₂²×C₄	#10625352	1	2	yes	—
230	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
231	C₂²×C₄	#10625354	1	1	yes	characteristic
232	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	47	4	no	—
240	C₂×C₈	—	47	4	no	—
241	C₂×C₈	—	47	8	no	—
242	C₂×C₈	—	47	4	no	—
243	C₂×C₈	—	47	8	no	—
244	C₂×C₈	—	47	4	no	—
245	C₂×C₈	—	94	2	no	—
246	C₂×C₈	—	94	2	no	—
247	C₂×C₈	—	94	4	no	—
248	C₂×C₈	—	94	4	no	—
249	C₂×C₈	C₄×Dic₄₇	1	8	yes	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	94	4	no	—
254	C₄²	—	94	4	no	—
255	C₄²	—	94	4	no	—
256	C₄²	—	94	4	no	—
257	C₂³	C₄²×D₄₇	1	2	yes	—
258	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
259	C₂³	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
260	C₂³	#13093986	1	1	yes	characteristic
261	C₂³	#13093989	1	1	yes	characteristic
262	C₂³	C₂×#10625324	1	4	yes	—
263	C₂³	#13094032	1	4	yes	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	16	no	—
267	C₂×C₄	—	94	8	no	—
268	C₂×C₄	—	94	8	no	—
269	C₂×C₄	—	94	8	no	—
270	C₂×C₄	—	188	8	no	—
271	C₂×C₄	—	188	8	no	—
272	C₂×C₄	#13093934	1	8	yes	complemented
273	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
274	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
275	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
276	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
277	C₂×C₄	C₂²⋊C₄×D₄₇	1	2	yes	—
278	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
279	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
280	C₂×C₄	#13093980	1	2	yes	—
281	C₂×C₄	#13093980	1	1	yes	characteristic
282	C₂×C₄	#13093982	1	1	yes	characteristic
283	C₂×C₄	#13093982	1	2	yes	—
284	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
285	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
286	C₂×C₄	#13093986	1	2	yes	—
287	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
288	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
289	C₂×C₄	#13093989	1	2	yes	—
290	C₂×C₄	Q₈×Dic₄₇	1	1	yes	characteristic
291	C₈	—	2	8	no	—
292	C₈	—	47	8	no	—
293	C₈	—	94	4	no	—
294	C₂²	C₄²×Dic₄₇	1	1	yes	characteristic
295	C₂²	#15688970	1	2	yes	—
296	C₂²	C₂²⋊C₄×Dic₄₇	1	1	yes	characteristic
297	C₂²	#15688974	1	2	yes	—
298	C₂²	C₄⋊C₄×Dic₄₇	1	1	yes	characteristic
299	C₂²	C₂×#13093934	1	2	yes	—
300	C₂²	C₄×#10625324	1	4	yes	—
301	C₂²	#15689080	1	4	yes	—
302	C₂²	#15689081	1	2	yes	—
303	C₂²	#15689191	1	8	yes	—
304	C₂²	#15689200	1	4	yes	—
305	C₂²	#15689250	1	4	yes	—
306	C₄	—	2	8	no	—
307	C₄	—	188	8	no	—
308	C₄	#95×D₄₇	1	1	yes	characteristic
309	C₄	#15688968	1	1	yes	characteristic
310	C₄	#15688970	1	1	yes	characteristic
311	C₄	C₂²⋊C₄×Dic₄₇	1	2	yes	—
312	C₄	#15688974	1	1	yes	characteristic
313	C₄	C₄⋊C₄×Dic₄₇	1	2	yes	—
314	C₂	#95×Dic₄₇	1	1	yes	characteristic
315	C₂	C₄×#13093934	1	1	yes	characteristic
316	C₂	#20243076	1	1	yes	characteristic
317	C₂	#20243167	1	2	yes	—
318	C₂	#20243168	1	4	yes	—
319	C₂	#20243258	1	2	yes	—
320	C₂	#20243259	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₄³
Fitting	C₂×C₄₇×#336	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#336	D₄₇
hypocenter	C₄₇	#10696
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
11. C₂×#13093934	C₄	semidirect
13. C₂×C₄×C₄₇⋊C₈	C₄	semidirect
15. #15688877	C₄	semidirect
18. C₄₇×#336	C₄	semidirect
106. C₂×C₄×C₄₇	#97	semidirect
132. C₄₇⋊C₈	#95	semidirect
178. #336	Dic₄₇	semidirect
179. C₄₇	#10696	semidirect
272. C₂×C₄	#13093934	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093934	2
C₂²×C₄×C₄₇⋊C₈	2
C₂×#15688877	2
C₄×C₄₇⋊#95	2
C₂×C₄₇×#336	2
#10696	47