Subgroups — #26242866

Order 12032.

664 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13094022	C₂	1	1	yes	maximal, characteristic
2	#20244325	C₂	1	1	yes	maximal, characteristic
3	#20244322	C₂	1	2	yes	maximal
4	#20243948	C₂	1	2	yes	maximal
5	#20243945	C₂	1	2	yes	maximal
6	#20243943	C₂	1	2	yes	maximal
7	#20243942	C₂	1	2	yes	maximal
8	#20243530	C₂	1	2	yes	maximal
9	C₄₇×#1984	C₂	1	1	yes	maximal, characteristic
10	C₂×#13094022	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
16	#15689030	C₄	1	1	yes	characteristic
17	#15689028	C₄	1	1	yes	characteristic
18	#15689018	C₄	1	2	yes	—
19	#15689018	C₄	1	2	yes	—
20	#15688992	C₄	1	1	yes	characteristic
21	C₄×Dic₄₇⋊C₄	C₂²	1	2	yes	—
22	C₄×Dic₄₇⋊C₄	C₂²	1	2	yes	—
23	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
24	#15688878	C₂²	1	2	yes	—
25	#15688878	C₂²	1	2	yes	—
26	#15688877	C₂²	1	2	yes	—
27	#15688877	C₂²	1	2	yes	—
28	#15688877	C₂²	1	2	yes	—
29	#15688877	C₂²	1	2	yes	—
30	#15688877	C₂²	1	2	yes	—
31	#15688877	C₂²	1	2	yes	—
32	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
35	C₄₇×#336	C₂²	1	2	yes	—
36	C₄₇×#336	C₂²	1	2	yes	—
37	C₂²×Dic₉₄	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
39	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
40	#13094022	C₈	1	8	yes	complemented
41	C₄×Dic₉₄	—	4	1	no	—
42	C₄×Dic₉₄	—	4	1	no	—
43	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
45	C₂×Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
46	C₂×Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
47	C₂×Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
48	C₂×Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
53	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
54	C₄⋊(C₄₇⋊C₈)	—	4	1	no	—
55	Dic₄₇⋊C₈	—	4	2	no	—
56	C₄×C₄₇⋊C₈	—	4	1	no	—
57	C₈×Dic₄₇	—	4	2	no	—
58	C₄₇⋊#95	C₂×C₄	1	2	yes	—
59	C₄₇⋊#95	C₂×C₄	1	2	yes	—
60	C₄₇⋊#95	C₂×C₄	1	2	yes	—
61	C₄₇⋊#95	C₂×C₄	1	2	yes	—
62	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
64	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
65	C₄×Q₈×C₄₇	—	4	1	no	—
66	C₄×Q₈×C₄₇	—	4	1	no	—
67	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₄₇	—	4	1	no	—
74	C₄×C₈×C₄₇	—	4	1	no	—
75	C₂×Q₈×C₄₇	—	4	1	no	—
76	C₂×Q₈×C₄₇	—	4	1	no	—
77	C₂×Q₈×C₄₇	C₂×C₈	1	4	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
85	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
86	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
87	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
88	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
89	C₂×Dic₉₄	—	4	1	no	—
90	C₂×Dic₉₄	—	4	1	no	—
91	C₂×Dic₉₄	C₂×C₈	1	4	yes	—
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	—	2	2	no	—
96	C₂×C₈×C₄₇	—	2	2	no	—
97	C₄⋊C₄×C₄₇	—	4	1	no	—
98	C₄⋊C₄×C₄₇	—	4	1	no	—
99	C₄⋊C₄×C₄₇	—	4	1	no	—
100	C₄²×C₄₇	—	2	1	no	—
101	C₄²×C₄₇	—	2	1	no	—
102	C₄²×C₄₇	—	2	1	no	—
103	C₄²×C₄₇	—	2	1	no	—
104	C₄²×C₄₇	—	2	1	no	—
105	C₄²×C₄₇	—	2	1	no	—
106	C₄⋊Dic₄₇	—	4	1	no	—
107	Dic₄₇⋊C₄	—	4	2	no	—
108	Dic₄₇⋊C₄	—	4	2	no	—
109	Dic₄₇⋊C₄	—	4	2	no	—
110	Dic₄₇⋊C₄	C₂×C₈	1	8	yes	—
111	Dic₄₇⋊C₄	C₂×C₈	1	8	yes	—
112	C₄×Dic₄₇	—	2	2	no	—
113	C₄×Dic₄₇	—	2	2	no	—
114	C₄×Dic₄₇	—	2	2	no	—
115	C₄×Dic₄₇	—	2	2	no	—
116	C₄×Dic₄₇	—	4	2	no	—
117	C₄×Dic₄₇	—	4	1	no	—
118	C₄×Dic₄₇	C₂×C₈	1	4	yes	—
119	C₂×C₄₇⋊C₈	—	2	2	no	—
120	C₂×C₄₇⋊C₈	—	2	2	no	—
121	C₂×C₄₇⋊C₈	—	2	2	no	—
122	C₂×C₄₇⋊C₈	—	2	4	no	—
123	C₂×C₄₇⋊C₈	—	2	2	no	—
124	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
125	Q₈×C₄₇	—	4	4	no	—
126	C₂×C₄×C₄₇	—	2	1	no	—
127	C₂×C₄×C₄₇	—	2	1	no	—
128	C₂×C₄×C₄₇	—	2	1	no	—
129	C₂×C₄×C₄₇	—	2	1	no	—
130	C₂×C₄×C₄₇	—	2	1	no	—
131	C₂×C₄×C₄₇	—	2	1	no	—
132	C₂×C₄×C₄₇	—	2	1	no	—
133	C₂×C₄×C₄₇	—	2	1	no	—
134	C₂×C₄×C₄₇	—	2	1	no	—
135	C₂×C₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	—	2	2	no	—
138	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
139	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
140	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
141	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
144	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
145	C₂×Dic₄₇	—	2	2	no	—
146	C₂×Dic₄₇	—	2	2	no	—
147	C₂×Dic₄₇	—	2	2	no	—
148	C₂×Dic₄₇	—	2	2	no	—
149	C₂×Dic₄₇	—	2	2	no	—
150	C₂×Dic₄₇	—	2	2	no	—
151	C₂×Dic₄₇	—	2	2	no	—
152	C₂×Dic₄₇	—	2	2	no	—
153	C₂×Dic₄₇	—	2	2	no	—
154	C₂×Dic₄₇	—	2	2	no	—
155	C₂×Dic₄₇	C₂²×C₈	1	4	yes	—
156	C₂×Dic₄₇	C₂²×C₈	1	4	yes	—
157	Dic₉₄	—	4	4	no	—
158	C₈×C₄₇	—	4	2	no	—
159	C₈×C₄₇	—	4	2	no	—
160	C₄₇⋊C₈	—	4	2	no	—
161	C₄₇⋊C₈	—	4	2	no	—
162	#23363	—	47	1	no	maximal, Sylow
163	C₂²×C₄₇	#518	1	1	yes	characteristic
164	C₂²×C₄₇	#519	1	2	yes	—
165	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
166	C₂²×C₄₇	#568	1	2	yes	—
167	C₂²×C₄₇	#568	1	1	yes	characteristic
168	C₄×C₄₇	—	2	2	no	—
169	C₄×C₄₇	—	2	2	no	—
170	C₄×C₄₇	—	2	2	no	—
171	C₄×C₄₇	—	4	1	no	—
172	C₄×C₄₇	—	4	1	no	—
173	C₄×C₄₇	—	4	1	no	—
174	C₄×C₄₇	#563	1	1	yes	characteristic
175	C₄×C₄₇	#563	1	1	yes	characteristic
176	C₄×C₄₇	#563	1	2	yes	—
177	Dic₄₇	—	2	4	no	—
178	Dic₄₇	—	2	4	no	—
179	Dic₄₇	—	4	2	no	—
180	Dic₄₇	—	4	2	no	—
181	C₄×C₄⋊Q₈	—	47	1	no	—
182	#1984	—	47	2	no	—
183	#1984	—	47	1	no	—
184	#1984	D₄₇	1	1	yes	characteristic
185	#1916	—	47	2	no	—
186	#1916	—	47	2	no	—
187	#1916	—	47	2	no	—
188	#1916	—	47	2	no	—
189	#1839	—	47	2	no	—
190	C₂×C₄₇	#2381	1	1	yes	characteristic
191	C₂×C₄₇	#2972	1	1	yes	characteristic
192	C₂×C₄₇	#2980	1	1	yes	characteristic
193	C₂×C₄₇	#2982	1	2	yes	—
194	C₂×C₄₇	#2987	1	2	yes	—
195	C₂×C₄⋊Q₈	—	47	1	no	—
196	C₂×C₄×Q₈	—	47	1	no	—
197	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
198	C₂×C₄⋊C₈	—	47	2	no	—
199	C₂×C₄⋊C₈	—	47	1	no	—
200	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
201	C₂×C₄×C₈	—	47	2	no	—
202	C₂×C₄×C₈	—	47	1	no	—
203	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
204	#391	—	47	1	no	—
205	#391	—	47	1	no	—
206	#389	—	47	2	no	—
207	#389	—	47	2	no	—
208	#384	—	47	1	no	—
209	C₄×C₄⋊C₄	—	47	2	no	—
210	C₄×C₄⋊C₄	—	47	2	no	—
211	C₄³	—	47	1	no	—
212	#336	—	47	2	no	—
213	#336	—	47	2	no	—
214	#336	—	47	2	no	—
215	#336	—	47	2	no	—
216	#336	—	47	2	no	—
217	#336	—	47	2	no	—
218	#336	—	47	2	no	—
219	#336	—	47	2	no	—
220	#336	C₂×D₄₇	1	2	yes	—
221	#336	C₂×D₄₇	1	2	yes	—
222	C₄₇	#23363	1	1	yes	characteristic, complemented, Sylow
223	C₂²×Q₈	—	47	1	no	—
224	C₂²×Q₈	C₄×D₄₇	1	1	yes	characteristic
225	C₂²×C₈	—	47	2	no	—
226	C₂²×C₈	—	47	2	no	—
227	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
228	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
229	C₄⋊Q₈	—	47	8	no	—
230	C₄×Q₈	—	4	1	no	—
231	C₄×Q₈	—	4	1	no	—
232	C₄×Q₈	—	188	1	no	—
233	C₄×Q₈	—	188	1	no	—
234	C₂×C₄⋊C₄	—	47	2	no	—
235	C₂×C₄⋊C₄	—	47	2	no	—
236	C₂×C₄⋊C₄	—	47	2	no	—
237	C₂×C₄⋊C₄	—	47	1	no	—
238	C₂×C₄⋊C₄	—	47	2	no	—
239	C₂×C₄⋊C₄	—	47	2	no	—
240	C₂×C₄⋊C₄	C₄×D₄₇	1	1	yes	characteristic
241	C₂×C₄⋊C₄	C₄×D₄₇	1	1	yes	characteristic
242	C₂×C₄⋊C₄	C₄×D₄₇	1	1	yes	characteristic
243	C₂×C₄²	—	47	1	no	—
244	C₂×C₄²	—	47	2	no	—
245	C₂×C₄²	—	47	2	no	—
246	C₂×C₄²	—	47	2	no	—
247	C₂×C₄²	—	47	1	no	—
248	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
249	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
250	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
251	C₄⋊C₈	—	4	1	no	—
252	C₄⋊C₈	—	188	1	no	—
253	C₄⋊C₈	—	188	2	no	—
254	C₄×C₈	—	4	1	no	—
255	C₄×C₈	—	188	1	no	—
256	C₄×C₈	—	188	2	no	—
257	#95	—	47	2	no	—
258	#95	—	47	2	no	—
259	#95	—	47	2	no	—
260	#95	—	47	2	no	—
261	C₂×Q₈	—	4	1	no	—
262	C₂×Q₈	—	4	1	no	—
263	C₂×Q₈	—	47	4	no	—
264	C₂×Q₈	—	188	1	no	—
265	C₂×Q₈	—	188	1	no	—
266	C₂×Q₈	C₈×D₄₇	1	4	yes	—
267	C₂²×C₄	—	47	2	no	—
268	C₂²×C₄	—	47	2	no	—
269	C₂²×C₄	—	47	2	no	—
270	C₂²×C₄	—	47	2	no	—
271	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
272	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
276	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
277	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	94	2	no	—
284	C₂×C₈	—	94	2	no	—
285	C₂×C₈	—	94	2	no	—
286	C₂×C₈	—	94	4	no	—
287	C₂×C₈	—	94	2	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	47	8	no	—
292	C₄⋊C₄	—	47	8	no	—
293	C₄⋊C₄	—	188	2	no	—
294	C₄⋊C₄	—	188	2	no	—
295	C₄⋊C₄	—	188	2	no	—
296	C₄⋊C₄	—	188	1	no	—
297	C₄²	—	2	1	no	—
298	C₄²	—	2	1	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	2	1	no	—
301	C₄²	—	2	1	no	—
302	C₄²	—	2	1	no	—
303	C₄²	—	47	4	no	—
304	C₄²	—	94	2	no	—
305	C₄²	—	94	2	no	—
306	C₄²	—	94	2	no	—
307	C₄²	—	94	2	no	—
308	C₄²	—	188	1	no	—
309	C₄²	—	188	2	no	—
310	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
311	Q₈	—	4	4	no	—
312	Q₈	—	188	4	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	47	4	no	—
326	C₂×C₄	—	47	4	no	—
327	C₂×C₄	—	94	2	no	—
328	C₂×C₄	—	94	2	no	—
329	C₂×C₄	—	94	2	no	—
330	C₂×C₄	—	94	2	no	—
331	C₂×C₄	—	94	2	no	—
332	C₂×C₄	—	94	2	no	—
333	C₂×C₄	—	94	2	no	—
334	C₂×C₄	—	94	2	no	—
335	C₂×C₄	—	94	2	no	—
336	C₂×C₄	—	94	2	no	—
337	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
338	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
339	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
340	C₂×C₄	#13094069	1	1	yes	characteristic
341	C₂×C₄	#13094069	1	1	yes	characteristic
342	C₂×C₄	#13094071	1	2	yes	—
343	C₂×C₄	#13094072	1	2	yes	—
344	C₈	—	4	2	no	—
345	C₈	—	4	2	no	—
346	C₈	—	188	2	no	—
347	C₈	—	188	2	no	—
348	C₂²	#15689596	1	1	yes	characteristic
349	C₂²	#15689599	1	2	yes	—
350	C₂²	C₂²×C₈×D₄₇	1	1	yes	characteristic
351	C₂²	#15689888	1	2	yes	—
352	C₂²	#15689889	1	1	yes	characteristic
353	C₄	—	2	2	no	—
354	C₄	—	2	2	no	—
355	C₄	—	2	2	no	—
356	C₄	—	4	1	no	—
357	C₄	—	4	1	no	—
358	C₄	—	4	1	no	—
359	C₄	—	94	4	no	—
360	C₄	—	94	4	no	—
361	C₄	—	188	2	no	—
362	C₄	—	188	2	no	—
363	C₄	#15689872	1	1	yes	characteristic
364	C₄	#15689873	1	1	yes	characteristic
365	C₄	#15689874	1	2	yes	—
366	C₂	#20246567	1	1	yes	characteristic
367	C₂	#20252410	1	1	yes	characteristic
368	C₂	#20252457	1	1	yes	characteristic
369	C₂	#20252467	1	2	yes	—
370	C₂	#20252491	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₈
Fitting	C₄₇×#1984	C₂
socle, prime radical	C₂³×C₄₇	C₂³×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1984	D₄₇
hypocenter	C₄₇	#23363
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1984

Complements

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

N	H ≅ G/N	type
40. #13094022	C₈	semidirect
222. C₄₇	#23363	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13094022	2
#20244325	2
#20244322	2
#20243948	2
#20243945	2
#20243943	2
#20243942	2
#20243530	2
C₄₇×#1984	2
#23363	47