Subgroups — #26224187

Order 12032.

886 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689202	C₂	1	1	yes	maximal, characteristic
2	C₄×#13094013	C₂	1	1	yes	maximal, characteristic
3	#20244325	C₂	1	1	yes	maximal, characteristic, complemented
4	#20244181	C₂	1	1	yes	maximal, characteristic
5	#20244000	C₂	1	4	yes	maximal, complemented
6	#20243504	C₂	1	1	yes	maximal, characteristic, complemented
7	#20243171	C₂	1	4	yes	maximal, complemented
8	C₄×#13093934	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₈×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#10625324	C₂²	1	2	yes	—
11	C₂×#13094013	C₄	1	1	yes	characteristic
12	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
13	#15689202	—	2	4	no	—
14	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×(C₂×D₄₇)⋊C₈	C₂²	1	2	yes	—
16	C₂×(C₂×D₄₇)⋊C₈	C₂²	1	4	yes	—
17	C₂×#13093934	C₂²	1	2	yes	—
18	C₂×#13093934	C₂²	1	2	yes	—
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	#15689030	C₄	1	1	yes	characteristic
21	#15689024	C₄	1	1	yes	characteristic
22	#15689013	C₄	1	4	yes	—
23	#15688988	C₄	1	1	yes	characteristic
24	C₄×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
25	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
26	#15688878	C₂²	1	4	yes	—
27	#15688877	C₂²	1	4	yes	—
28	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₈×Q₈×C₄₇	—	2	4	no	—
30	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
31	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₂²×D₁₈₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×D₄₇	C₂³	1	2	yes	—
36	C₂×#10625324	—	2	4	no	—
37	C₂×#10625324	—	2	8	no	—
38	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
39	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
40	#13094013	—	2	4	no	—
41	C₄×D₁₈₈	—	2	2	no	—
42	C₄×D₁₈₈	—	2	2	no	—
43	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×(C₂×D₄₇)⋊C₄	C₂×C₄	1	4	yes	—
45	C₂×(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
46	C₂×(C₂×D₄₇)⋊C₄	C₂×C₄	1	4	yes	—
47	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
51	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
52	(C₂×D₄₇)⋊C₈	—	2	4	no	—
53	(C₂×D₄₇)⋊C₈	—	2	8	no	—
54	#13093934	—	2	4	no	—
55	#13093934	—	2	4	no	—
56	C₄×C₄₇⋊C₈	—	2	2	no	—
57	C₄₇⋊#95	C₂×C₄	1	4	yes	—
58	C₄₇⋊#95	C₂×C₄	1	4	yes	—
59	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
61	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
63	C₄×Q₈×C₄₇	—	2	2	no	—
64	C₄×Q₈×C₄₇	—	2	2	no	—
65	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
67	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
68	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₄₇	—	2	4	no	—
70	C₄⋊C₈×C₄₇	—	2	2	no	—
71	C₄×C₈×C₄₇	—	2	4	no	—
72	C₄×C₈×C₄₇	D₄	1	4	yes	—
73	C₂³×D₄₇	C₂²×C₄	1	2	yes	—
74	C₂×Q₈×C₄₇	—	2	2	no	—
75	C₂×Q₈×C₄₇	—	2	2	no	—
76	C₂×Q₈×C₄₇	M₁₆	1	4	yes	complemented
77	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
82	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
83	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
84	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
85	C₂×D₁₈₈	—	2	2	no	—
86	C₂×D₁₈₈	—	2	2	no	—
87	C₂×D₁₈₈	—	2	2	no	—
88	C₂×C₄×D₄₇	—	2	2	no	—
89	C₂×C₄×D₄₇	—	2	4	no	—
90	C₂×C₄×D₄₇	—	2	2	no	—
91	C₂×C₄×D₄₇	—	2	4	no	—
92	C₂×C₈×C₄₇	—	2	2	no	—
93	C₂×C₈×C₄₇	—	2	2	no	—
94	C₂×C₈×C₄₇	—	2	4	no	—
95	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₄₇	#41	1	4	yes	—
98	C₄⋊C₄×C₄₇	—	2	4	no	—
99	C₄⋊C₄×C₄₇	—	2	2	no	—
100	C₄²×C₄₇	—	2	2	no	—
101	C₄²×C₄₇	—	2	2	no	—
102	C₄²×C₄₇	C₂×D₄	1	2	yes	—
103	C₄²×C₄₇	#41	1	2	yes	—
104	(C₂×D₄₇)⋊C₄	—	2	4	no	—
105	(C₂×D₄₇)⋊C₄	—	2	8	no	—
106	(C₂×D₄₇)⋊C₄	—	4	4	no	—
107	C₄⋊Dic₄₇	—	2	2	no	—
108	C₄×Dic₄₇	—	2	2	no	—
109	C₄×Dic₄₇	—	2	2	no	—
110	C₄×Dic₄₇	—	2	2	no	—
111	C₄×Dic₄₇	—	2	2	no	—
112	C₄×Dic₄₇	—	4	2	no	—
113	C₂×C₄₇⋊C₈	—	2	2	no	—
114	C₂×C₄₇⋊C₈	—	2	2	no	—
115	C₂×C₄₇⋊C₈	—	2	2	no	—
116	C₂×C₄₇⋊C₈	—	2	4	no	—
117	C₂×C₄₇⋊C₈	#41	1	4	yes	—
118	#10625324	—	4	8	no	—
119	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
120	C₂²×D₄₇	—	2	2	no	—
121	C₂²×D₄₇	—	2	4	no	—
122	C₂²×D₄₇	—	2	2	no	—
123	C₂²×D₄₇	—	2	2	no	—
124	C₂²×D₄₇	—	2	4	no	—
125	Q₈×C₄₇	—	2	8	no	—
126	C₂×C₄×C₄₇	—	2	2	no	—
127	C₂×C₄×C₄₇	—	2	2	no	—
128	C₂×C₄×C₄₇	—	2	2	no	—
129	C₂×C₄×C₄₇	—	2	2	no	—
130	C₂×C₄×C₄₇	—	2	4	no	—
131	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
134	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
135	C₂×C₄×C₄₇	#131	1	2	yes	—
136	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
138	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
140	C₂×Dic₄₇	—	2	2	no	—
141	C₂×Dic₄₇	—	2	2	no	—
142	C₂×Dic₄₇	—	2	2	no	—
143	C₂×Dic₄₇	—	2	4	no	—
144	C₂×Dic₄₇	—	2	2	no	—
145	C₂×Dic₄₇	—	2	2	no	—
146	C₂×Dic₄₇	—	2	2	no	—
147	C₂×Dic₄₇	—	2	2	no	—
148	D₁₈₈	—	4	4	no	—
149	C₄×D₄₇	—	4	2	no	—
150	C₄×D₄₇	—	4	4	no	—
151	C₄×D₄₇	—	4	2	no	—
152	C₈×C₄₇	—	2	2	no	—
153	C₈×C₄₇	—	4	2	no	—
154	C₈×C₄₇	#124	1	4	yes	complemented
155	C₄₇⋊C₈	—	2	4	no	—
156	C₄₇⋊C₈	—	4	2	no	—
157	#21634	—	47	1	no	maximal, Sylow
158	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
159	C₂²×C₄₇	C₄×#41	1	2	yes	—
160	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
161	C₂²×C₄₇	C₂×#131	1	1	yes	characteristic
162	C₂²×C₄₇	#568	1	2	yes	—
163	C₂×D₄₇	—	4	2	no	—
164	C₂×D₄₇	—	4	2	no	—
165	C₂×D₄₇	—	4	4	no	—
166	C₂×D₄₇	—	4	4	no	—
167	C₂×D₄₇	—	4	2	no	—
168	C₄×C₄₇	—	2	4	no	—
169	C₄×C₄₇	—	2	2	no	—
170	C₄×C₄₇	—	4	2	no	—
171	C₄×C₄₇	#435	1	4	yes	—
172	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
173	C₄×C₄₇	#529	1	2	yes	—
174	C₄×C₄₇	#532	1	1	yes	characteristic
175	Dic₄₇	—	4	2	no	—
176	Dic₄₇	—	4	2	no	—
177	Dic₄₇	—	4	2	no	—
178	C₂×C₈×Q₈	D₄₇	1	1	yes	characteristic, complemented
179	C₂×#436	—	47	1	no	—
180	C₄×#124	—	47	1	no	—
181	#1984	—	47	1	no	—
182	#1954	—	47	1	no	—
183	#1924	—	47	4	no	—
184	#1831	—	47	1	no	—
185	#1753	—	47	4	no	—
186	C₄×#97	—	47	1	no	—
187	C₂×C₄₇	C₄×#124	1	1	yes	characteristic
188	C₂×C₄₇	C₂×#435	1	1	yes	characteristic
189	C₂×C₄₇	#2932	1	1	yes	characteristic
190	C₂×C₄₇	#2969	1	2	yes	—
191	C₂×C₄₇	#2971	1	2	yes	—
192	D₄₇	—	8	2	no	—
193	C₂²×M₁₆	—	47	2	no	—
194	C₂×#124	—	47	1	no	—
195	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
196	C₂×C₄×D₄	—	47	1	no	—
197	C₈×Q₈	—	2	4	no	—
198	#436	—	94	4	no	—
199	C₂×C₄⋊C₈	—	47	1	no	—
200	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
201	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
202	C₂×C₂²⋊C₈	—	47	2	no	—
203	C₂×C₂²⋊C₈	—	47	4	no	—
204	C₂×#97	—	47	2	no	—
205	C₂×#97	—	47	2	no	—
206	C₂×C₄×C₈	—	47	1	no	—
207	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
208	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
209	#391	—	47	1	no	—
210	#390	—	47	1	no	—
211	#388	—	47	4	no	—
212	#382	—	47	1	no	—
213	C₄×C₂²⋊C₄	—	47	4	no	—
214	C₄³	—	47	1	no	—
215	#336	—	47	4	no	—
216	#336	—	47	4	no	—
217	C₄₇	#21634	1	1	yes	characteristic, complemented, Sylow
218	C₂²×Q₈	C₄×D₄₇	1	1	yes	characteristic
219	C₂²×D₄	—	47	1	no	—
220	C₂³×C₄	—	47	2	no	—
221	C₂×M₁₆	—	94	8	no	—
222	C₂×M₁₆	—	94	4	no	—
223	C₂²×C₈	—	47	2	no	—
224	C₂²×C₈	—	47	2	no	—
225	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
226	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
227	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
228	#124	—	94	4	no	—
229	C₄×Q₈	—	2	2	no	—
230	C₄×Q₈	—	2	2	no	—
231	C₄×D₄	—	94	2	no	—
232	C₄×D₄	—	94	2	no	—
233	C₂×C₄⋊C₄	—	47	1	no	—
234	C₂×C₄⋊C₄	C₄×D₄₇	1	1	yes	characteristic
235	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
236	C₂×C₂²⋊C₄	—	47	2	no	—
237	C₂×C₂²⋊C₄	—	47	4	no	—
238	C₂×C₂²⋊C₄	—	47	4	no	—
239	C₂×C₄²	—	47	1	no	—
240	C₂×C₄²	—	47	1	no	—
241	C₂×C₄²	—	47	2	no	—
242	C₂×C₄²	—	47	2	no	—
243	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
244	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
245	C₄⋊C₈	—	2	4	no	—
246	C₄⋊C₈	—	2	2	no	—
247	C₄⋊C₈	—	94	2	no	—
248	C₂²⋊C₈	—	94	8	no	—
249	C₂²⋊C₈	—	94	4	no	—
250	#97	—	94	4	no	—
251	#97	—	94	4	no	—
252	C₄×C₈	—	2	4	no	—
253	C₄×C₈	—	94	2	no	—
254	C₄×C₈	C₂²⋊D₄₇	1	4	yes	—
255	#95	—	47	4	no	—
256	#95	—	47	4	no	—
257	C₂⁴	—	47	2	no	—
258	C₂×Q₈	—	2	2	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	#10625324	1	4	yes	complemented
261	C₂×D₄	—	94	2	no	—
262	C₂×D₄	—	94	2	no	—
263	C₂×D₄	—	94	2	no	—
264	C₂²×C₄	—	47	2	no	—
265	C₂²×C₄	—	47	2	no	—
266	C₂²×C₄	—	47	2	no	—
267	C₂²×C₄	—	94	2	no	—
268	C₂²×C₄	—	94	4	no	—
269	C₂²×C₄	—	94	2	no	—
270	C₂²×C₄	—	94	4	no	—
271	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
272	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
274	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
275	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
276	M₁₆	—	188	8	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	—	47	4	no	—
281	C₂×C₈	—	94	2	no	—
282	C₂×C₈	—	94	2	no	—
283	C₂×C₈	—	94	2	no	—
284	C₂×C₈	—	94	4	no	—
285	C₂×C₈	#10625354	1	4	yes	—
286	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
287	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
288	C₄⋊C₄	—	2	4	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	94	2	no	—
291	C₂²⋊C₄	—	94	4	no	—
292	C₂²⋊C₄	—	94	8	no	—
293	C₂²⋊C₄	—	188	4	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	94	2	no	—
297	C₄²	—	94	2	no	—
298	C₄²	—	94	2	no	—
299	C₄²	—	94	2	no	—
300	C₄²	—	188	2	no	—
301	C₄²	#10625350	1	2	yes	—
302	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
303	C₂³	—	94	2	no	—
304	C₂³	—	94	4	no	—
305	C₂³	—	94	2	no	—
306	C₂³	—	94	2	no	—
307	C₂³	—	94	4	no	—
308	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
309	Q₈	—	2	8	no	—
310	D₄	—	188	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	94	2	no	—
317	C₂×C₄	—	94	2	no	—
318	C₂×C₄	—	94	2	no	—
319	C₂×C₄	—	94	4	no	—
320	C₂×C₄	—	94	2	no	—
321	C₂×C₄	—	94	2	no	—
322	C₂×C₄	—	94	2	no	—
323	C₂×C₄	—	94	2	no	—
324	C₂×C₄	—	188	2	no	—
325	C₂×C₄	—	188	4	no	—
326	C₂×C₄	—	188	2	no	—
327	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
328	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
329	C₂×C₄	C₂×#10625324	1	4	yes	—
330	C₂×C₄	C₂×#10625324	1	2	yes	—
331	C₂×C₄	#13094030	1	2	yes	—
332	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
333	C₂×C₄	#41×D₄₇	1	2	yes	—
334	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
335	C₂×C₄	C₂×#10625354	1	2	yes	—
336	C₈	—	2	2	no	—
337	C₈	—	4	2	no	—
338	C₈	—	94	4	no	—
339	C₈	—	188	2	no	—
340	C₈	#13094013	1	4	yes	complemented
341	C₂²	—	188	2	no	—
342	C₂²	—	188	2	no	—
343	C₂²	—	188	2	no	—
344	C₂²	—	188	4	no	—
345	C₂²	—	188	4	no	—
346	C₂²	C₂×C₄×C₂²⋊D₄₇	1	1	yes	characteristic
347	C₂²	C₄×#10625354	1	2	yes	—
348	C₂²	C₂²×#10625324	1	1	yes	characteristic
349	C₂²	C₂×#13094030	1	1	yes	characteristic
350	C₂²	#15689888	1	2	yes	—
351	C₄	—	2	4	no	—
352	C₄	—	2	2	no	—
353	C₄	—	4	2	no	—
354	C₄	—	188	2	no	—
355	C₄	—	188	2	no	—
356	C₄	—	188	2	no	—
357	C₄	#15689194	1	4	yes	—
358	C₄	C₂×#13094013	1	1	yes	characteristic
359	C₄	#15689660	1	2	yes	—
360	C₄	#15689669	1	1	yes	characteristic
361	C₂	—	376	2	no	—
362	C₂	C₄×#13094013	1	1	yes	characteristic
363	C₂	C₂×#15689194	1	1	yes	characteristic
364	C₂	#20252104	1	1	yes	characteristic
365	C₂	#20252391	1	2	yes	—
366	C₂	#20252407	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₈×Q₈×C₄₇	C₂
prime radical	C₂²×D₁₈₈	C₂×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₈×Q₈	D₄₇
hypocenter	C₄₇	#21634
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. #20244325	C₂	semidirect
5. #20244000	C₂	semidirect
6. #20243504	C₂	semidirect
7. #20243171	C₂	semidirect
8. C₄×#13093934	C₂	semidirect
9. C₂×C₈×Q₈×C₄₇	C₂	semidirect
76. C₂×Q₈×C₄₇	M₁₆	semidirect
154. C₈×C₄₇	#124	semidirect
178. C₂×C₈×Q₈	D₄₇	semidirect
217. C₄₇	#21634	semidirect
260. C₂×Q₈	#10625324	semidirect
340. C₈	#13094013	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689202	2
C₄×#13094013	2
#20244325	2
#20244181	2
#20244000	2
#20243504	2
#20243171	2
C₄×#13093934	2
C₂×C₈×Q₈×C₄₇	2
#21634	47