Subgroups — #26214491

Order 12032.

1050 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×#15689176	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
3	#20243263	C₂	1	4	yes	maximal, complemented
4	#20243172	C₂	1	4	yes	maximal
5	#20243134	C₂	1	1	yes	maximal, characteristic, complemented
6	#20243134	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#13093935	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄²×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
10	C₂²×#10625329	C₂²	1	2	yes	—
11	#15689176	C₄	1	8	yes	complemented
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
14	C₄×#10625329	C₄	1	8	yes	complemented
15	C₂×#13093935	C₂²	1	4	yes	—
16	C₂×#13093935	C₂²	1	2	yes	—
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
18	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
22	#15688878	C₂²	1	4	yes	—
23	#15688878	C₂²	1	4	yes	—
24	C₂×#117×C₄₇	C₄	1	2	yes	—
25	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
27	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
28	C₄×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
29	C₄×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
30	C₄₇×#376	C₄	1	2	yes	—
31	C₄³×C₄₇	C₂²	1	1	yes	characteristic
32	C₄³×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×#10625329	—	2	8	no	—
34	C₂×#10625329	C₂×C₄	1	8	yes	—
35	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
36	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
37	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
38	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
39	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
40	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
41	#13093935	—	2	8	no	—
42	#13093935	C₂×C₄	1	8	yes	—
43	C₄×C₄₇⋊C₈	—	2	8	no	—
44	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
45	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
46	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
47	C₄×C₄₇⋊C₈	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	#117×C₄₇	—	2	8	no	—
52	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
53	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₄₇	C₂×C₄	1	2	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	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
63	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
64	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
66	#95×C₄₇	C₂×C₄	1	4	yes	—
67	#95×C₄₇	C₂×C₄	1	4	yes	—
68	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	—	2	4	no	—
70	C₂²×C₄×C₄₇	—	2	4	no	—
71	C₂²×C₄×C₄₇	—	2	4	no	—
72	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
73	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
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	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
84	C₄⋊C₄×C₄₇	—	2	8	no	—
85	C₂²⋊C₄×C₄₇	—	2	8	no	—
86	C₄²×C₄₇	—	2	4	no	—
87	C₄²×C₄₇	—	2	4	no	—
88	C₄²×C₄₇	—	2	4	no	—
89	C₄²×C₄₇	—	2	4	no	—
90	C₄²×C₄₇	—	2	4	no	—
91	C₄²×C₄₇	C₄²	1	4	yes	—
92	C₄²×C₄₇	C₄²	1	4	yes	—
93	C₄²×C₄₇	M₁₆	1	8	yes	complemented
94	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
95	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
96	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
97	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
98	#10625329	—	2	16	no	—
99	C₂×C₄₇⋊C₈	—	2	4	no	—
100	C₂×C₄₇⋊C₈	—	2	4	no	—
101	C₂×C₄₇⋊C₈	—	2	4	no	—
102	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
103	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
104	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
105	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
106	C₂×C₄₇⋊C₈	#41	1	8	yes	—
107	C₂³×C₄₇	—	2	2	no	—
108	C₂³×C₄₇	—	2	2	no	—
109	C₂³×C₄₇	—	2	2	no	—
110	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
111	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	—	2	4	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	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	—	2	4	no	—
122	C₂×C₄×C₄₇	—	2	4	no	—
123	C₂×C₄×C₄₇	—	2	4	no	—
124	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
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×M₁₆	1	8	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	C₂×#41	1	4	yes	—
135	C₄₇⋊C₈	—	2	4	no	—
136	C₄₇⋊C₈	—	2	4	no	—
137	C₄₇⋊C₈	—	2	4	no	—
138	C₄₇⋊C₈	#117	1	8	yes	complemented
139	#20472	—	47	1	no	maximal, Sylow
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	4	4	no	—
144	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
145	C₂²×C₄₇	C₄×#41	1	2	yes	—
146	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
147	C₂²×C₄₇	C₂×#131	1	2	yes	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	2	2	no	—
150	C₄×C₄₇	—	2	8	no	—
151	C₄×C₄₇	—	2	2	no	—
152	C₄×C₄₇	—	4	4	no	—
153	C₄×C₄₇	C₄×M₁₆	1	4	yes	—
154	C₄×C₄₇	#405	1	4	yes	—
155	C₄×C₄₇	C₂×#117	1	2	yes	—
156	C₄×C₄₇	C₄×#41	1	2	yes	—
157	C₂×#431	—	47	1	no	—
158	C₂×C₄×M₁₆	—	47	1	no	—
159	C₄×#117	D₄₇	1	1	yes	characteristic
160	#1771	—	47	4	no	—
161	#1753	—	47	4	no	—
162	#1744	—	47	1	no	—
163	#1744	—	47	1	no	—
164	C₄×#97	—	47	1	no	—
165	C₄²×C₈	—	47	1	no	—
166	C₂×C₄₇	—	4	2	no	—
167	C₂×C₄₇	C₄×#117	1	1	yes	characteristic
168	C₂×C₄₇	C₂×C₄×M₁₆	1	1	yes	characteristic
169	C₂×C₄₇	C₂×#405	1	1	yes	characteristic
170	C₂×C₄₇	#2965	1	4	yes	—
171	C₂²×M₁₆	—	47	2	no	—
172	C₂×#117	Dic₄₇	1	2	yes	—
173	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
174	#431	—	47	8	no	—
175	C₂×C₄⋊C₈	—	47	2	no	—
176	C₂×C₂²⋊C₈	—	47	2	no	—
177	C₄×M₁₆	—	47	8	no	—
178	C₂×#97	—	47	2	no	—
179	C₂×#97	—	47	4	no	—
180	C₂×C₄×C₈	—	47	1	no	—
181	C₂×C₄×C₈	—	47	1	no	—
182	C₂×C₄×C₈	—	47	1	no	—
183	C₂×C₄×C₈	—	47	1	no	—
184	C₂×C₄×C₈	—	47	4	no	—
185	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
186	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
187	C₄×C₂²⋊C₄	Dic₄₇	1	2	yes	—
188	C₄×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
189	#376	Dic₄₇	1	2	yes	—
190	C₄³	C₂×D₄₇	1	1	yes	characteristic
191	C₄³	C₂×D₄₇	1	1	yes	characteristic
192	#336	—	47	4	no	—
193	#336	—	47	4	no	—
194	C₄₇	#20472	1	1	yes	characteristic, complemented, Sylow
195	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
196	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
197	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
198	C₂×M₁₆	—	47	8	no	—
199	C₂×M₁₆	—	94	8	no	—
200	C₂²×C₈	—	47	2	no	—
201	C₂²×C₈	—	47	2	no	—
202	C₂²×C₈	—	47	2	no	—
203	C₂²×C₈	—	47	2	no	—
204	#117	—	2	8	no	—
205	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
206	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
207	C₂×C₄²	C₄×D₄₇	1	4	yes	—
208	C₂×C₄²	C₄×D₄₇	1	4	yes	—
209	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
210	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
211	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
212	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
213	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
214	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
216	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
217	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
218	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
219	C₄⋊C₈	—	47	8	no	—
220	C₂²⋊C₈	—	47	8	no	—
221	#97	—	47	8	no	—
222	#97	—	94	8	no	—
223	C₄×C₈	—	47	4	no	—
224	C₄×C₈	—	47	4	no	—
225	C₄×C₈	—	47	4	no	—
226	C₄×C₈	—	47	4	no	—
227	C₄×C₈	—	94	8	no	—
228	#95	C₂×Dic₄₇	1	4	yes	—
229	#95	C₂×Dic₄₇	1	4	yes	—
230	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
235	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
236	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
237	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
238	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
239	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
240	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
241	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
242	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
243	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
244	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
245	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
246	M₁₆	—	94	16	no	—
247	C₂×C₈	—	47	4	no	—
248	C₂×C₈	—	47	8	no	—
249	C₂×C₈	—	47	4	no	—
250	C₂×C₈	—	47	4	no	—
251	C₂×C₈	—	47	4	no	—
252	C₂×C₈	—	94	4	no	—
253	C₂×C₈	—	94	4	no	—
254	C₂×C₈	—	94	4	no	—
255	C₄⋊C₄	—	2	8	no	—
256	C₂²⋊C₄	—	2	8	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	2	4	no	—
260	C₄²	—	2	4	no	—
261	C₄²	—	2	4	no	—
262	C₄²	#10625329	1	8	yes	complemented
263	C₄²	C₄×Dic₄₇	1	4	yes	—
264	C₄²	C₄×Dic₄₇	1	4	yes	—
265	C₄²	C₂×C₄×D₄₇	1	2	yes	—
266	C₄²	C₂×C₄×D₄₇	1	2	yes	—
267	C₄²	C₂×C₄×D₄₇	1	2	yes	—
268	C₄²	C₂×C₄×D₄₇	1	2	yes	—
269	C₂³	—	2	2	no	—
270	C₂³	—	2	2	no	—
271	C₂³	—	2	2	no	—
272	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
273	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
287	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
288	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
289	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
290	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
291	C₂×C₄	C₂×#10625329	1	8	yes	—
292	C₂×C₄	C₂×#10625329	1	4	yes	—
293	C₂×C₄	#13094031	1	4	yes	—
294	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
295	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
296	C₂×C₄	#41×D₄₇	1	4	yes	—
297	C₈	—	47	8	no	—
298	C₈	—	94	4	no	—
299	C₈	—	94	4	no	—
300	C₈	—	94	4	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	4	no	—
305	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
306	C₂²	#41×Dic₄₇	1	2	yes	—
307	C₂²	C₂²×#10625329	1	2	yes	—
308	C₂²	C₂×#13094031	1	2	yes	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	8	no	—
312	C₄	—	2	2	no	—
313	C₄	—	4	4	no	—
314	C₄	C₄×#10625329	1	4	yes	—
315	C₄	#15689079	1	4	yes	—
316	C₄	#117×D₄₇	1	2	yes	—
317	C₄	#15689591	1	2	yes	—
318	C₂	—	4	2	no	—
319	C₂	#117×Dic₄₇	1	1	yes	characteristic
320	C₂	C₂×C₄×#10625329	1	1	yes	characteristic
321	C₂	C₂×#15689079	1	1	yes	characteristic
322	C₂	#20252355	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
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	#20472
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. #20243263	C₂	semidirect
5. #20243134	C₂	semidirect
6. #20243134	C₂	semidirect
7. C₄×#13093935	C₂	semidirect
8. C₄²×C₄₇⋊C₈	C₂	semidirect
11. #15689176	C₄	semidirect
14. C₄×#10625329	C₄	semidirect
93. C₄²×C₄₇	M₁₆	semidirect
138. C₄₇⋊C₈	#117	semidirect
194. C₄₇	#20472	semidirect
262. C₄²	#10625329	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
C₂×C₄×#10625329	2
#20243263	2
#20243172	2
#20243134	2
C₄×#13093935	2
C₄²×C₄₇⋊C₈	2
C₄×#117×C₄₇	2
#20472	47