Subgroups — #26187010

Order 12032.

606 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252364	C₂	1	2	yes	maximal, complemented
2	#20242239	C₂	1	2	yes	maximal, complemented
3	#20242232	C₂	1	2	yes	maximal, complemented
4	#20242225	C₂	1	2	yes	maximal, complemented
5	#20242218	C₂	1	2	yes	maximal, complemented
6	#20242151	C₂	1	1	yes	maximal, characteristic, complemented
7	#20241848	C₂	1	2	yes	maximal, complemented
8	C₂×#15688845	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#3168	C₂	1	1	yes	maximal, characteristic
10	#15689254	C₂²	1	2	yes	—
11	Q₈×C₄₇⋊C₈	C₂²	1	2	yes	—
12	#15689201	C₂²	1	2	yes	—
13	#15689195	—	2	2	no	—
14	D₄×C₄₇⋊C₈	—	2	2	no	—
15	D₄×C₄₇⋊C₈	C₂²	1	2	yes	—
16	#15689184	—	2	2	no	—
17	#15689180	C₂²	1	2	yes	—
18	#15689079	—	2	2	no	—
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
21	#15688876	C₂²	1	2	yes	—
22	#15688874	C₂²	1	1	yes	characteristic
23	#15688874	C₂²	1	1	yes	characteristic
24	Q₈⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
25	Q₈⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
26	D₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
27	D₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
28	#15688845	C₂²	1	2	yes	—
29	#15688845	C₂²	1	2	yes	—
30	C₄₇×#529	C₂²	1	2	yes	—
31	C₄₇×#499	C₄	1	1	yes	characteristic
32	C₄₇×#495	C₄	1	1	yes	characteristic
33	C₄₇×#488	C₄	1	2	yes	—
34	C₄₇×#467	C₄	1	1	yes	characteristic
35	C₄₇×#466	C₄	1	1	yes	characteristic
36	C₄₇×#465	C₄	1	2	yes	—
37	C₄×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
38	C₄×SD₁₆×C₄₇	C₂²	1	2	yes	—
39	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
41	C₂×#10625329	—	2	2	no	—
42	C₂²×C₄₇⋊C₈	—	2	1	no	—
43	C₂²×C₄₇⋊C₈	—	2	2	no	—
44	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
45	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
46	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
47	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
48	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
49	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
50	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
51	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
52	#13093935	—	2	2	no	—
53	#13093935	—	2	2	no	—
54	C₄×C₄₇⋊C₈	—	2	1	no	—
55	C₄×C₄₇⋊C₈	—	2	2	no	—
56	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
57	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
58	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
59	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
60	C₂×Q₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
61	C₂×SD₁₆×C₄₇	C₂×C₄	1	2	yes	—
62	C₂×D₈×C₄₇	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
65	#126×C₄₇	—	2	2	no	—
66	#125×C₄₇	C₂×C₄	1	2	yes	—
67	#124×C₄₇	C₂×C₄	1	2	yes	—
68	#123×C₄₇	—	2	2	no	—
69	C₂²⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
70	#121×C₄₇	C₂×C₄	1	2	yes	—
71	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
72	C₄×D₄×C₄₇	—	2	2	no	—
73	C₄×D₄×C₄₇	C₂³	1	2	yes	—
74	#117×C₄₇	—	2	2	no	—
75	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
76	#107×C₄₇	C₂×C₄	1	1	yes	characteristic
77	#107×C₄₇	C₂×C₄	1	1	yes	characteristic
78	#106×C₄₇	C₂×C₄	1	2	yes	—
79	#103×C₄₇	C₂×C₄	1	2	yes	—
80	#103×C₄₇	C₂×C₄	1	2	yes	—
81	#102×C₄₇	C₂×C₄	1	2	yes	—
82	#102×C₄₇	C₂×C₄	1	2	yes	—
83	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
84	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
85	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
87	C₂×Q₈×C₄₇	C₂²×C₄	1	2	yes	—
88	C₂×D₄×C₄₇	—	2	2	no	—
89	C₂×D₄×C₄₇	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₄₇	—	2	2	no	—
91	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
94	Q₁₆×C₄₇	—	4	1	no	—
95	SD₁₆×C₄₇	—	4	2	no	—
96	D₈×C₄₇	—	4	1	no	—
97	C₂×C₈×C₄₇	—	4	1	no	—
98	C₂×C₈×C₄₇	—	4	1	no	—
99	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
103	C₄⋊C₄×C₄₇	—	2	2	no	—
104	C₄⋊C₄×C₄₇	—	2	2	no	—
105	C₄⋊C₄×C₄₇	—	2	2	no	—
106	C₄⋊C₄×C₄₇	—	2	2	no	—
107	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
108	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
109	C₂²⋊C₄×C₄₇	—	2	2	no	—
110	C₂²⋊C₄×C₄₇	—	2	2	no	—
111	C₂²⋊C₄×C₄₇	—	2	2	no	—
112	C₂²⋊C₄×C₄₇	—	2	2	no	—
113	C₂²⋊C₄×C₄₇	—	2	2	no	—
114	C₄²×C₄₇	—	2	2	no	—
115	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₄₇	#41	1	1	yes	characteristic
118	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
119	#10625329	—	4	4	no	—
120	C₂×C₄₇⋊C₈	—	2	4	no	—
121	C₂×C₄₇⋊C₈	—	2	2	no	—
122	C₂×C₄₇⋊C₈	—	2	2	no	—
123	C₂×C₄₇⋊C₈	—	2	2	no	—
124	C₂×C₄₇⋊C₈	—	4	2	no	—
125	C₂×C₄₇⋊C₈	—	4	4	no	—
126	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
127	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
128	C₂³×C₄₇	—	2	2	no	—
129	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
130	Q₈×C₄₇	—	2	2	no	—
131	Q₈×C₄₇	—	2	2	no	—
132	D₄×C₄₇	—	2	2	no	—
133	D₄×C₄₇	—	2	2	no	—
134	D₄×C₄₇	—	4	4	no	—
135	C₂×C₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	2	1	no	—
137	C₂×C₄×C₄₇	—	2	1	no	—
138	C₂×C₄×C₄₇	—	2	2	no	—
139	C₂×C₄×C₄₇	—	2	2	no	—
140	C₂×C₄×C₄₇	—	2	2	no	—
141	C₂×C₄×C₄₇	—	2	2	no	—
142	C₂×C₄×C₄₇	—	4	2	no	—
143	C₂×C₄×C₄₇	—	4	2	no	—
144	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
150	C₈×C₄₇	—	2	1	no	—
151	C₈×C₄₇	—	2	1	no	—
152	C₈×C₄₇	—	2	1	no	—
153	C₈×C₄₇	—	2	1	no	—
154	C₄₇⋊C₈	—	2	2	no	—
155	C₄₇⋊C₈	—	4	2	no	—
156	C₄₇⋊C₈	—	4	2	no	—
157	C₄₇⋊C₈	—	4	2	no	—
158	C₄₇⋊C₈	#135	1	4	yes	—
159	#17978	—	47	1	no	maximal, Sylow
160	C₂²×C₄₇	—	2	1	no	—
161	C₂²×C₄₇	—	2	1	no	—
162	C₂²×C₄₇	—	4	2	no	—
163	C₂²×C₄₇	—	4	2	no	—
164	C₂²×C₄₇	—	4	2	no	—
165	C₂²×C₄₇	#444	1	2	yes	—
166	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
167	C₄×C₄₇	—	2	1	no	—
168	C₄×C₄₇	—	2	1	no	—
169	C₄×C₄₇	—	2	1	no	—
170	C₄×C₄₇	—	2	1	no	—
171	C₄×C₄₇	—	4	2	no	—
172	C₄×C₄₇	—	4	2	no	—
173	C₄×C₄₇	—	4	2	no	—
174	C₄×C₄₇	#568	1	1	yes	characteristic
175	C₄×C₄₇	#568	1	1	yes	characteristic
176	C₄×C₄₇	C₂×#135	1	2	yes	—
177	#3168	D₄₇	1	1	yes	characteristic
178	#2966	—	47	2	no	—
179	#1584	—	47	2	no	—
180	#1583	—	47	2	no	—
181	#1582	—	47	2	no	—
182	#1581	—	47	2	no	—
183	#1568	—	47	1	no	—
184	#1494	—	47	2	no	—
185	C₂×#322	—	47	1	no	—
186	C₂×C₄₇	—	2	2	no	—
187	C₂×C₄₇	—	8	2	no	—
188	C₂×C₄₇	#2932	1	1	yes	characteristic
189	C₂×C₄₇	C₄×#135	1	1	yes	characteristic
190	C₂×C₄₇	C₂×#444	1	1	yes	characteristic
191	#529	C₂×D₄₇	1	2	yes	—
192	#499	Dic₄₇	1	1	yes	characteristic
193	#495	Dic₄₇	1	1	yes	characteristic
194	#488	Dic₄₇	1	2	yes	—
195	#467	Dic₄₇	1	1	yes	characteristic
196	#466	Dic₄₇	1	1	yes	characteristic
197	#465	Dic₄₇	1	2	yes	—
198	#446	—	47	2	no	—
199	C₈×Q₈	—	47	2	no	—
200	C₄×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
201	C₄×SD₁₆	C₂×D₄₇	1	2	yes	—
202	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
203	#436	—	47	2	no	—
204	#435	—	94	2	no	—
205	C₈×D₄	—	47	2	no	—
206	C₈×D₄	—	94	2	no	—
207	#433	—	94	2	no	—
208	#432	—	47	2	no	—
209	#405	—	94	2	no	—
210	C₂×C₄×C₈	—	47	1	no	—
211	C₂×C₄×C₈	—	47	1	no	—
212	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
213	#335	—	47	2	no	—
214	#334	—	47	1	no	—
215	#334	—	47	1	no	—
216	Q₈⋊C₈	—	47	2	no	—
217	Q₈⋊C₈	—	47	2	no	—
218	D₄⋊C₈	—	47	2	no	—
219	D₄⋊C₈	—	47	2	no	—
220	#322	—	47	2	no	—
221	#322	—	47	2	no	—
222	C₄₇	#17978	1	1	yes	characteristic, complemented, Sylow
223	C₂×Q₁₆	C₂×Dic₄₇	1	1	yes	characteristic
224	C₂×SD₁₆	C₂×Dic₄₇	1	2	yes	—
225	C₂×D₈	C₂×Dic₄₇	1	1	yes	characteristic
226	C₂×M₁₆	—	94	2	no	—
227	C₂²×C₈	—	47	2	no	—
228	C₂²×C₈	—	94	2	no	—
229	C₂²×C₈	—	94	1	no	—
230	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
231	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
232	#126	—	2	2	no	—
233	#125	C₂×Dic₄₇	1	2	yes	—
234	#124	C₂×Dic₄₇	1	2	yes	—
235	#123	—	2	2	no	—
236	C₂²⋊Q₈	C₂×Dic₄₇	1	2	yes	—
237	#121	C₂×Dic₄₇	1	2	yes	—
238	C₄×Q₈	C₂²×D₄₇	1	2	yes	—
239	C₄×D₄	—	2	2	no	—
240	C₄×D₄	C₂²×D₄₇	1	2	yes	—
241	#117	—	2	2	no	—
242	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
243	#107	C₂×Dic₄₇	1	1	yes	characteristic
244	#107	C₂×Dic₄₇	1	1	yes	characteristic
245	#106	C₂×Dic₄₇	1	2	yes	—
246	C₄⋊C₈	—	47	2	no	—
247	C₄⋊C₈	—	47	2	no	—
248	C₄⋊C₈	—	94	2	no	—
249	C₄⋊C₈	—	94	2	no	—
250	#103	C₂×Dic₄₇	1	2	yes	—
251	#103	C₂×Dic₄₇	1	2	yes	—
252	#102	C₂×Dic₄₇	1	2	yes	—
253	#102	C₂×Dic₄₇	1	2	yes	—
254	C₂²⋊C₈	—	94	2	no	—
255	C₂²⋊C₈	—	94	2	no	—
256	C₂²⋊C₈	—	94	2	no	—
257	#97	—	94	2	no	—
258	#97	—	94	2	no	—
259	C₄×C₈	—	47	1	no	—
260	C₄×C₈	—	47	1	no	—
261	C₄×C₈	—	47	2	no	—
262	C₄×C₈	—	47	2	no	—
263	C₄×C₈	—	94	2	no	—
264	C₄×C₈	—	94	1	no	—
265	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
266	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
268	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
269	C₂×Q₈	C₂²×Dic₄₇	1	2	yes	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	C₂²×Dic₄₇	1	2	yes	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
274	C₂²×C₄	#10625352	1	1	yes	characteristic
275	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
276	Q₁₆	—	4	1	no	—
277	SD₁₆	—	4	2	no	—
278	D₈	—	4	1	no	—
279	M₁₆	—	188	4	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	47	2	no	—
283	C₂×C₈	—	47	2	no	—
284	C₂×C₈	—	94	4	no	—
285	C₂×C₈	—	94	2	no	—
286	C₂×C₈	—	94	2	no	—
287	C₂×C₈	—	94	2	no	—
288	C₂×C₈	—	188	4	no	—
289	C₂×C₈	—	188	2	no	—
290	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
291	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
292	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
293	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	C₂²×Dic₄₇	1	2	yes	—
299	C₄⋊C₄	C₂²×Dic₄₇	1	2	yes	—
300	C₂²⋊C₄	—	2	2	no	—
301	C₂²⋊C₄	—	2	2	no	—
302	C₂²⋊C₄	—	2	2	no	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	2	2	no	—
305	C₄²	—	2	2	no	—
306	C₄²	D₄×D₄₇	1	1	yes	characteristic
307	C₄²	#10625352	1	1	yes	characteristic
308	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
309	C₄²	C₂³×D₄₇	1	1	yes	characteristic
310	C₂³	—	2	2	no	—
311	C₂³	D₄×Dic₄₇	1	1	yes	characteristic
312	Q₈	—	2	2	no	—
313	Q₈	—	2	2	no	—
314	D₄	—	2	2	no	—
315	D₄	—	2	2	no	—
316	D₄	—	4	4	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
327	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
328	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
329	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
330	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
331	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	47	4	no	—
337	C₈	—	94	2	no	—
338	C₈	—	188	2	no	—
339	C₈	—	188	2	no	—
340	C₈	—	188	2	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	4	2	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	#15689245	1	2	yes	—
347	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	4	2	no	—
353	C₄	—	4	2	no	—
354	C₄	—	4	2	no	—
355	C₄	#15689890	1	1	yes	characteristic
356	C₄	#15689890	1	1	yes	characteristic
357	C₄	#135×D₄₇	1	2	yes	—
358	C₂	—	2	2	no	—
359	C₂	—	8	2	no	—
360	C₂	#20252106	1	1	yes	characteristic
361	C₂	#135×Dic₄₇	1	1	yes	characteristic
362	C₂	C₂×#15689245	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#3168	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₄₇×#466	C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#3168	D₄₇
hypocenter	C₄₇	#17978
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3168

Complements

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

N	H ≅ G/N	type
1. #20252364	C₂	semidirect
2. #20242239	C₂	semidirect
3. #20242232	C₂	semidirect
4. #20242225	C₂	semidirect
5. #20242218	C₂	semidirect
6. #20242151	C₂	semidirect
7. #20241848	C₂	semidirect
8. C₂×#15688845	C₂	semidirect
222. C₄₇	#17978	semidirect

Maximal subgroups

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

subgroup	index
#20252364	2
#20242239	2
#20242232	2
#20242225	2
#20242218	2
#20242151	2
#20241848	2
C₂×#15688845	2
C₄₇×#3168	2
#17978	47