Subgroups — #26214519

Order 12032.

906 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689184	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689079	C₂	1	1	yes	maximal, characteristic
3	#20243263	C₂	1	2	yes	maximal, complemented
4	#20243257	C₂	1	2	yes	maximal, complemented
5	#20243172	C₂	1	2	yes	maximal
6	#20243166	C₂	1	2	yes	maximal
7	#20243140	C₂	1	1	yes	maximal, characteristic, complemented
8	#20243137	C₂	1	1	yes	maximal, characteristic, complemented
9	#20243075	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#13093935	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2298	C₂	1	1	yes	maximal, characteristic
12	C₂²×#10625329	C₂²	1	1	yes	characteristic
13	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	#15689184	C₄	1	8	yes	complemented
15	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
17	#15689079	C₄	1	8	yes	complemented
18	C₂×#13093935	C₂²	1	2	yes	—
19	C₂×#13093935	C₂²	1	2	yes	—
20	C₂×#13093935	C₂²	1	2	yes	—
21	C₂×#13093935	C₂²	1	1	yes	characteristic
22	C₂×#13093935	C₂²	1	1	yes	characteristic
23	C₂×#13093935	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
25	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
28	#15688878	C₂²	1	2	yes	—
29	#15688878	C₂²	1	2	yes	—
30	#15688878	C₂²	1	2	yes	—
31	#15688878	C₂²	1	2	yes	—
32	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
33	C₄₇×#388	C₄	1	4	yes	—
34	C₄₇×#387	C₄	1	4	yes	—
35	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
36	C₄×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
37	C₄₇×#376	C₂²	1	1	yes	characteristic
38	C₄³×C₄₇	C₂²	1	1	yes	characteristic
39	C₂×#10625329	—	2	4	no	—
40	C₂×#10625329	C₂×C₄	1	4	yes	—
41	C₂²×C₄₇⋊C₈	—	2	4	no	—
42	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
43	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
44	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
45	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
48	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
49	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
50	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
51	#13093935	—	2	4	no	—
52	#13093935	—	2	4	no	—
53	#13093935	—	2	4	no	—
54	#13093935	C₂×C₄	1	4	yes	—
55	#13093935	C₂×C₄	1	4	yes	—
56	#13093935	C₂×C₄	1	4	yes	—
57	C₄×C₄₇⋊C₈	—	2	4	no	—
58	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
59	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
60	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
61	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
62	#117×C₄₇	C₂×C₄	1	4	yes	—
63	#117×C₄₇	C₂×C₄	1	4	yes	—
64	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
65	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
66	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
67	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
68	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
69	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
70	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
71	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
73	#95×C₄₇	C₂×C₄	1	4	yes	—
74	#95×C₄₇	C₂×C₄	1	4	yes	—
75	#95×C₄₇	C₂×C₄	1	4	yes	—
76	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	—	2	2	no	—
78	C₂²×C₄×C₄₇	—	2	2	no	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₄₇	—	4	4	no	—
88	C₄⋊C₄×C₄₇	C₄²	1	8	yes	—
89	C₂²⋊C₄×C₄₇	—	4	4	no	—
90	C₂²⋊C₄×C₄₇	C₄²	1	8	yes	—
91	C₄²×C₄₇	—	2	2	no	—
92	C₄²×C₄₇	—	2	2	no	—
93	C₄²×C₄₇	—	2	2	no	—
94	C₄²×C₄₇	—	2	2	no	—
95	C₄²×C₄₇	—	4	4	no	—
96	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
97	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
98	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
99	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
100	#10625329	—	2	8	no	—
101	C₂×C₄₇⋊C₈	—	2	2	no	—
102	C₂×C₄₇⋊C₈	—	2	2	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	—	2	8	no	—
108	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
109	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
110	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
111	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
112	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
113	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
114	C₂×C₄₇⋊C₈	#41	1	4	yes	—
115	C₂×C₄₇⋊C₈	#41	1	4	yes	—
116	C₂³×C₄₇	—	2	2	no	—
117	C₂³×C₄₇	—	2	4	no	—
118	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
119	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	4	no	—
122	C₂×C₄×C₄₇	—	2	4	no	—
123	C₂×C₄×C₄₇	—	2	2	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	2	4	no	—
126	C₂×C₄×C₄₇	—	2	4	no	—
127	C₂×C₄×C₄₇	—	2	4	no	—
128	C₂×C₄×C₄₇	—	2	4	no	—
129	C₂×C₄×C₄₇	—	2	4	no	—
130	C₂×C₄×C₄₇	—	2	4	no	—
131	C₂×C₄×C₄₇	—	2	4	no	—
132	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
133	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
134	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
135	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
138	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
139	C₄₇⋊C₈	—	2	2	no	—
140	C₄₇⋊C₈	—	2	2	no	—
141	C₄₇⋊C₈	—	2	4	no	—
142	C₄₇⋊C₈	—	2	4	no	—
143	C₄₇⋊C₈	#117	1	4	yes	—
144	C₄₇⋊C₈	#117	1	4	yes	—
145	#20475	—	47	1	no	maximal, Sylow
146	C₂²×C₄₇	—	2	4	no	—
147	C₂²×C₄₇	—	2	2	no	—
148	C₂²×C₄₇	—	4	4	no	—
149	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
150	C₂²×C₄₇	C₄×#41	1	2	yes	—
151	C₂²×C₄₇	#568	1	4	yes	—
152	C₄×C₄₇	—	2	4	no	—
153	C₄×C₄₇	—	2	4	no	—
154	C₄×C₄₇	—	2	2	no	—
155	C₄×C₄₇	—	4	4	no	—
156	C₄×C₄₇	—	4	4	no	—
157	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
158	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
159	C₄×C₄₇	C₄×#41	1	2	yes	—
160	C₂×#433	—	47	1	no	—
161	C₂×#405	—	47	1	no	—
162	#2298	D₄₇	1	1	yes	characteristic
163	#1771	—	47	2	no	—
164	#1770	—	47	2	no	—
165	#1753	—	47	2	no	—
166	#1752	—	47	2	no	—
167	#1746	—	47	1	no	—
168	#1745	—	47	1	no	—
169	#1726	—	47	1	no	—
170	C₄×#97	—	47	1	no	—
171	C₂×C₄₇	—	4	2	no	—
172	C₂×C₄₇	C₄×#117	1	1	yes	characteristic
173	C₂×C₄₇	#2875	1	2	yes	—
174	C₂×C₄₇	#2967	1	4	yes	—
175	C₂²×M₁₆	—	47	1	no	—
176	C₂³×C₈	—	47	1	no	—
177	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
178	#433	—	47	8	no	—
179	C₂×C₄⋊C₈	—	47	2	no	—
180	C₂×C₂²⋊C₈	—	47	2	no	—
181	#405	—	47	8	no	—
182	C₂×#97	—	47	1	no	—
183	C₂×#97	—	47	1	no	—
184	C₂×#97	—	47	1	no	—
185	C₂×#97	—	47	2	no	—
186	C₂×#97	—	47	2	no	—
187	C₂×#97	—	47	2	no	—
188	C₂×C₄×C₈	—	47	1	no	—
189	C₂×C₄×C₈	—	47	1	no	—
190	C₂×C₄×C₈	—	47	1	no	—
191	C₂×C₄×C₈	—	47	2	no	—
192	#388	Dic₄₇	1	4	yes	—
193	#387	Dic₄₇	1	4	yes	—
194	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
195	C₄×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
196	#376	C₂×D₄₇	1	1	yes	characteristic
197	C₄³	C₂×D₄₇	1	1	yes	characteristic
198	#336	—	47	2	no	—
199	#336	—	47	2	no	—
200	#336	—	47	2	no	—
201	#336	—	47	2	no	—
202	C₄₇	#20475	1	1	yes	characteristic, complemented, Sylow
203	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
204	C₂×M₁₆	—	47	4	no	—
205	C₂×M₁₆	—	94	4	no	—
206	C₂²×C₈	—	47	1	no	—
207	C₂²×C₈	—	47	1	no	—
208	C₂²×C₈	—	47	4	no	—
209	C₂²×C₈	—	47	1	no	—
210	C₂²×C₈	—	47	1	no	—
211	C₂²×C₈	—	47	2	no	—
212	C₂²×C₈	—	47	2	no	—
213	C₂²×C₈	—	94	4	no	—
214	#117	C₄×D₄₇	1	4	yes	—
215	#117	C₄×D₄₇	1	4	yes	—
216	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
217	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
218	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
219	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
220	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
221	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
222	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
223	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
224	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
225	C₄⋊C₈	—	47	8	no	—
226	C₂²⋊C₈	—	47	8	no	—
227	#97	—	47	4	no	—
228	#97	—	47	4	no	—
229	#97	—	47	4	no	—
230	#97	—	94	4	no	—
231	#97	—	94	4	no	—
232	#97	—	94	4	no	—
233	C₄×C₈	—	47	4	no	—
234	C₄×C₈	—	47	4	no	—
235	C₄×C₈	—	47	4	no	—
236	C₄×C₈	—	94	4	no	—
237	#95	C₂×Dic₄₇	1	4	yes	—
238	#95	C₂×Dic₄₇	1	4	yes	—
239	#95	C₂×Dic₄₇	1	4	yes	—
240	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
244	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
245	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
246	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
247	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
248	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
249	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
250	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
251	M₁₆	—	94	8	no	—
252	C₂×C₈	—	47	2	no	—
253	C₂×C₈	—	47	2	no	—
254	C₂×C₈	—	47	4	no	—
255	C₂×C₈	—	47	4	no	—
256	C₂×C₈	—	47	2	no	—
257	C₂×C₈	—	47	4	no	—
258	C₂×C₈	—	47	2	no	—
259	C₂×C₈	—	47	4	no	—
260	C₂×C₈	—	94	8	no	—
261	C₂×C₈	—	94	2	no	—
262	C₂×C₈	—	94	2	no	—
263	C₂×C₈	—	94	2	no	—
264	C₂×C₈	—	94	2	no	—
265	C₂×C₈	—	94	2	no	—
266	C₂×C₈	—	94	2	no	—
267	C₄⋊C₄	—	4	4	no	—
268	C₄⋊C₄	C₄×Dic₄₇	1	8	yes	—
269	C₂²⋊C₄	—	4	4	no	—
270	C₂²⋊C₄	C₄×Dic₄₇	1	8	yes	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	4	4	no	—
276	C₄²	C₂×C₄×D₄₇	1	2	yes	—
277	C₄²	C₂×C₄×D₄₇	1	2	yes	—
278	C₄²	C₂×C₄×D₄₇	1	2	yes	—
279	C₄²	C₂×C₄×D₄₇	1	2	yes	—
280	C₂³	—	2	2	no	—
281	C₂³	—	2	4	no	—
282	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
283	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
297	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
298	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
299	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
300	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
301	C₂×C₄	#41×D₄₇	1	2	yes	—
302	C₂×C₄	#41×D₄₇	1	2	yes	—
303	C₈	—	47	4	no	—
304	C₈	—	47	4	no	—
305	C₈	—	94	2	no	—
306	C₈	—	94	2	no	—
307	C₈	—	94	4	no	—
308	C₈	—	94	4	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	4	4	no	—
312	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
313	C₂²	#41×Dic₄₇	1	2	yes	—
314	C₂²	#15689890	1	4	yes	—
315	C₄	—	2	4	no	—
316	C₄	—	2	4	no	—
317	C₄	—	2	2	no	—
318	C₄	—	4	4	no	—
319	C₄	—	4	4	no	—
320	C₄	#117×D₄₇	1	1	yes	characteristic
321	C₄	#117×D₄₇	1	1	yes	characteristic
322	C₄	#15689591	1	2	yes	—
323	C₂	—	4	2	no	—
324	C₂	#117×Dic₄₇	1	1	yes	characteristic
325	C₂	#20251816	1	2	yes	—
326	C₂	#20252375	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄₇×#2298	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2298	D₄₇
hypocenter	C₄₇	#20475
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2298

Complements

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

N	H ≅ G/N	type
3. #20243263	C₂	semidirect
4. #20243257	C₂	semidirect
7. #20243140	C₂	semidirect
8. #20243137	C₂	semidirect
9. #20243075	C₂	semidirect
10. C₄×#13093935	C₂	semidirect
14. #15689184	C₄	semidirect
17. #15689079	C₄	semidirect
202. C₄₇	#20475	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689184	2
C₂×#15689079	2
#20243263	2
#20243257	2
#20243172	2
#20243166	2
#20243140	2
#20243137	2
#20243075	2
C₄×#13093935	2
C₄₇×#2298	2
#20475	47