Subgroups — #26214473

Order 12032.

958 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	C₂×#15689180	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689176	C₂	1	1	yes	maximal, characteristic
3	#20243241	C₂	1	2	yes	maximal, complemented
4	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal, complemented
5	#20243154	C₂	1	4	yes	maximal
6	#20243140	C₂	1	2	yes	maximal, complemented
7	#20243075	C₂	1	2	yes	maximal, complemented
8	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
9	#15689180	C₄	1	8	yes	complemented
10	#15689176	C₄	1	8	yes	complemented
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×#13093935	C₂²	1	4	yes	—
17	C₂×#13093935	C₂²	1	2	yes	—
18	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
20	#15688878	C₂²	1	4	yes	—
21	#15688878	C₂²	1	4	yes	—
22	C₂×#117×C₄₇	C₂²	1	2	yes	—
23	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
26	C₄×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
27	C₄×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
28	C₄₇×#376	C₂²	1	1	yes	characteristic
29	C₄₇×#376	C₂²	1	1	yes	characteristic
30	C₄³×C₄₇	C₂²	1	2	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
32	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
33	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
34	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
35	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
36	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
37	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
38	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
39	#13093935	—	2	8	no	—
40	#13093935	C₂×C₄	1	8	yes	—
41	C₄×C₄₇⋊C₈	—	2	8	no	—
42	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
43	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
46	#117×C₄₇	—	2	4	no	—
47	#117×C₄₇	—	2	4	no	—
48	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
49	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₄₇	C₂×C₄	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	2	yes	—
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
60	#95×C₄₇	C₂×C₄	1	2	yes	—
61	#95×C₄₇	C₂×C₄	1	2	yes	—
62	#95×C₄₇	C₂×C₄	1	2	yes	—
63	#95×C₄₇	C₂×C₄	1	2	yes	—
64	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	—	2	4	no	—
66	C₂²×C₄×C₄₇	—	2	2	no	—
67	C₂²×C₄×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	—	4	2	no	—
69	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
70	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₄⋊C₄×C₄₇	—	2	8	no	—
82	C₂²⋊C₄×C₄₇	—	2	8	no	—
83	C₄²×C₄₇	—	2	8	no	—
84	C₄²×C₄₇	—	2	2	no	—
85	C₄²×C₄₇	—	2	2	no	—
86	C₄²×C₄₇	—	4	4	no	—
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	C₄²×C₄₇	#41	1	4	yes	—
94	C₄²×C₄₇	#41	1	4	yes	—
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂×C₄₇⋊C₈	—	2	4	no	—
97	C₂×C₄₇⋊C₈	—	2	4	no	—
98	C₂×C₄₇⋊C₈	—	2	4	no	—
99	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
100	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
101	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
102	C₂³×C₄₇	—	2	2	no	—
103	C₂³×C₄₇	—	2	2	no	—
104	C₂³×C₄₇	—	2	2	no	—
105	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
106	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	8	no	—
117	C₂×C₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	—	2	4	no	—
120	C₂×C₄×C₄₇	—	4	4	no	—
121	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
126	C₂×C₄×C₄₇	#117	1	8	yes	—
127	C₂×C₄×C₄₇	#131	1	4	yes	—
128	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
131	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
132	C₄₇⋊C₈	—	2	4	no	—
133	C₄₇⋊C₈	—	2	8	no	—
134	C₄₇⋊C₈	—	2	4	no	—
135	#20470	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	—	4	4	no	—
140	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
141	C₂²×C₄₇	C₂×#117	1	2	yes	—
142	C₂²×C₄₇	C₂×#131	1	2	yes	—
143	C₂²×C₄₇	#568	1	2	yes	—
144	C₄×C₄₇	—	2	2	no	—
145	C₄×C₄₇	—	2	2	no	—
146	C₄×C₄₇	—	2	2	no	—
147	C₄×C₄₇	—	2	8	no	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	4	4	no	—
150	C₄×C₄₇	#405	1	4	yes	—
151	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
152	C₄×C₄₇	C₄×#41	1	2	yes	—
153	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
154	C₂×#432	—	47	1	no	—
155	C₂×#431	—	47	1	no	—
156	C₄×#117	D₄₇	1	1	yes	characteristic
157	#1767	—	47	2	no	—
158	C₄×C₄⋊C₈	—	47	2	no	—
159	#1750	—	47	4	no	—
160	#1746	—	47	2	no	—
161	#1726	—	47	2	no	—
162	C₂×C₄₇	—	4	2	no	—
163	C₂×C₄₇	C₄×#117	1	1	yes	characteristic
164	C₂×C₄₇	C₂×#405	1	1	yes	characteristic
165	C₂×C₄₇	#2873	1	1	yes	characteristic
166	C₂×C₄₇	#2925	1	4	yes	—
167	C₂×#117	C₂×D₄₇	1	2	yes	—
168	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
169	#432	—	47	8	no	—
170	#431	—	47	8	no	—
171	C₂×C₄⋊C₈	—	47	1	no	—
172	C₂×C₄⋊C₈	—	47	1	no	—
173	C₂×C₄⋊C₈	—	47	2	no	—
174	C₂×C₂²⋊C₈	—	47	2	no	—
175	C₂×C₂²⋊C₈	—	47	2	no	—
176	C₂×#97	—	47	2	no	—
177	C₂×#97	—	47	4	no	—
178	C₂×C₄×C₈	—	47	2	no	—
179	C₂×C₄×C₈	—	47	4	no	—
180	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
181	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
182	C₄×C₂²⋊C₄	Dic₄₇	1	2	yes	—
183	C₄×C₂²⋊C₄	Dic₄₇	1	2	yes	—
184	#376	C₂×D₄₇	1	1	yes	characteristic
185	#376	C₂×D₄₇	1	1	yes	characteristic
186	C₄³	C₂×D₄₇	1	2	yes	—
187	#336	—	47	4	no	—
188	#336	—	47	4	no	—
189	C₄₇	#20470	1	1	yes	characteristic, complemented, Sylow
190	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
191	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
192	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
193	C₂²×C₈	—	47	2	no	—
194	C₂²×C₈	—	47	2	no	—
195	C₂²×C₈	—	47	4	no	—
196	#117	—	2	4	no	—
197	#117	—	2	4	no	—
198	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
199	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
200	C₂×C₄²	C₄×D₄₇	1	4	yes	—
201	C₂×C₄²	C₄×D₄₇	1	4	yes	—
202	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
203	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
204	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
205	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
206	C₂×C₄²	C₂²×D₄₇	1	2	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	2	yes	—
210	C₄⋊C₈	—	47	4	no	—
211	C₄⋊C₈	—	47	4	no	—
212	C₄⋊C₈	—	47	8	no	—
213	C₂²⋊C₈	—	47	8	no	—
214	C₂²⋊C₈	—	47	8	no	—
215	#97	—	47	8	no	—
216	#97	—	94	8	no	—
217	C₄×C₈	—	47	8	no	—
218	C₄×C₈	—	94	8	no	—
219	#95	C₂×Dic₄₇	1	2	yes	—
220	#95	C₂×Dic₄₇	1	2	yes	—
221	#95	C₂×Dic₄₇	1	2	yes	—
222	#95	C₂×Dic₄₇	1	2	yes	—
223	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
229	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
230	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
231	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
232	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
233	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
234	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
235	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
236	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
237	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
238	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
240	C₂×C₈	—	47	4	no	—
241	C₂×C₈	—	47	4	no	—
242	C₂×C₈	—	47	8	no	—
243	C₂×C₈	—	94	4	no	—
244	C₂×C₈	—	94	4	no	—
245	C₂×C₈	—	94	4	no	—
246	C₂×C₈	—	94	4	no	—
247	C₄⋊C₄	—	2	8	no	—
248	C₂²⋊C₄	—	2	8	no	—
249	C₄²	—	2	8	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	4	4	no	—
253	C₄²	C₄×Dic₄₇	1	4	yes	—
254	C₄²	C₄×Dic₄₇	1	4	yes	—
255	C₄²	C₂×C₄×D₄₇	1	2	yes	—
256	C₄²	C₂×C₄×D₄₇	1	2	yes	—
257	C₄²	C₂×C₄×D₄₇	1	2	yes	—
258	C₄²	C₂×C₄×D₄₇	1	2	yes	—
259	C₄²	#10625350	1	4	yes	—
260	C₄²	#10625350	1	4	yes	—
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	2	2	no	—
264	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
265	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
281	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
282	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
283	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
284	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
285	C₂×C₄	#13093983	1	8	yes	—
286	C₂×C₄	#13094031	1	4	yes	—
287	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
288	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
289	C₂×C₄	C₂×#10625350	1	2	yes	—
290	C₂×C₄	C₂×#10625350	1	2	yes	—
291	C₈	—	94	4	no	—
292	C₈	—	94	4	no	—
293	C₈	—	94	8	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	4	4	no	—
298	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
299	C₂²	C₂×#13093983	1	2	yes	—
300	C₂²	C₂×#13094031	1	2	yes	—
301	C₂²	#15689890	1	2	yes	—
302	C₄	—	2	8	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	4	no	—
308	C₄	#15689079	1	4	yes	—
309	C₄	C₄×#10625350	1	1	yes	characteristic
310	C₄	C₄×#10625350	1	1	yes	characteristic
311	C₄	C₄×#10625350	1	2	yes	—
312	C₂	—	4	2	no	—
313	C₂	C₄×#13093983	1	1	yes	characteristic
314	C₂	C₂×#15689079	1	1	yes	characteristic
315	C₂	#20251807	1	1	yes	characteristic
316	C₂	#20252054	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄×#117×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#117	D₄₇
hypocenter	C₄₇	#20470
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. #20243241	C₂	semidirect
4. C₄×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
6. #20243140	C₂	semidirect
7. #20243075	C₂	semidirect
9. #15689180	C₄	semidirect
10. #15689176	C₄	semidirect
189. C₄₇	#20470	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×#15689176	2
#20243241	2
C₄×C₄⋊(C₄₇⋊C₈)	2
#20243154	2
#20243140	2
#20243075	2
C₄×#117×C₄₇	2
#20470	47