Subgroups — #26234548

Order 12032.

862 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252123	C₂	1	2	yes	maximal
2	C₂×#15689202	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689170	C₂	1	2	yes	maximal, complemented
4	#20244161	C₂	1	2	yes	maximal, complemented
5	#20243591	C₂	1	4	yes	maximal, complemented
6	#20243427	C₂	1	2	yes	maximal, complemented
7	C₄×#13093960	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#1848	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#10625324	C₂²	1	2	yes	—
10	C₂×M₁₆×D₄₇	C₂²	1	4	yes	—
11	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
12	#15689202	C₄	1	4	yes	—
13	#15689202	C₄	1	4	yes	—
14	#15689200	—	2	2	no	—
15	#15689193	—	2	4	no	—
16	#15689188	—	2	2	no	—
17	#15689170	—	2	8	no	—
18	C₂×#13093960	C₂²	1	1	yes	characteristic
19	C₂×#13093960	C₂²	1	1	yes	characteristic
20	C₂×#13093959	C₂²	1	4	yes	—
21	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
22	#15689086	C₂²	1	4	yes	—
23	C₂×(C₂×D₄₇)⋊C₈	C₂²	1	2	yes	—
24	C₄×#10625329	C₂²	1	2	yes	—
25	#15688888	C₂²	1	2	yes	—
26	#15688887	C₂²	1	4	yes	—
27	C₂×#108×C₄₇	C₂²	1	2	yes	—
28	C₂×#108×C₄₇	C₂²	1	2	yes	—
29	C₄₇×#423	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₂²×D₁₈₈	Q₈	1	1	yes	characteristic
32	C₂²×C₄×D₄₇	C₂³	1	2	yes	—
33	C₂×#10625329	C₂³	1	2	yes	—
34	C₂×#10625329	C₂³	1	2	yes	—
35	C₂×#10625324	—	2	4	no	—
36	C₂×#10625324	—	2	4	no	—
37	C₂×#10625324	—	2	4	no	—
38	C₂×#10625324	C₂×C₄	1	8	yes	—
39	M₁₆×D₄₇	—	2	8	no	—
40	M₁₆×D₄₇	—	4	4	no	—
41	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
42	C₂×C₈×D₄₇	—	2	4	no	—
43	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
44	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
45	C₄×D₁₈₈	D₄	1	2	yes	—
46	C₄×D₁₈₈	D₄	1	2	yes	—
47	C₂×C₄⋊Dic₄₇	Q₈	1	1	yes	characteristic
48	C₂×(C₂×D₄₇)⋊C₄	Q₈	1	2	yes	—
49	#13093960	—	2	2	no	—
50	#13093960	D₄	1	4	yes	complemented
51	#13093959	—	2	8	no	—
52	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
53	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
54	(C₂×D₄₇)⋊C₈	—	2	4	no	—
55	(C₂×D₄₇)⋊C₈	—	2	4	no	—
56	(C₂×D₄₇)⋊C₈	C₂×C₄	1	8	yes	—
57	#13093935	—	2	2	no	—
58	C₄×C₄₇⋊C₈	—	2	2	no	—
59	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
60	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
63	#108×C₄₇	—	2	4	no	—
64	#108×C₄₇	—	4	2	no	—
65	C₄⋊C₈×C₄₇	—	2	2	no	—
66	C₄⋊C₈×C₄₇	—	2	2	no	—
67	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
69	C₂³×D₄₇	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
74	C₂×D₁₈₈	—	4	2	no	—
75	C₂×D₁₈₈	C₄⋊C₄	1	2	yes	—
76	C₂×D₁₈₈	C₄⋊C₄	1	2	yes	—
77	C₂×C₄×D₄₇	—	2	2	no	—
78	C₂×C₄×D₄₇	—	2	2	no	—
79	C₂×C₄×D₄₇	—	2	2	no	—
80	C₂×C₄×D₄₇	—	2	2	no	—
81	C₂×C₄×D₄₇	C₂²×C₄	1	4	yes	—
82	C₂×C₄×D₄₇	C₂×D₄	1	4	yes	—
83	M₁₆×C₄₇	—	2	4	no	—
84	M₁₆×C₄₇	—	2	4	no	—
85	C₂×C₈×C₄₇	—	2	1	no	—
86	C₂×C₈×C₄₇	—	2	1	no	—
87	C₂×C₈×C₄₇	—	2	4	no	—
88	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₄₇	#41	1	2	yes	—
92	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
96	(C₂×D₄₇)⋊C₄	C₄⋊C₄	1	4	yes	—
97	(C₂×D₄₇)⋊C₄	C₄⋊C₄	1	4	yes	—
98	C₄⋊Dic₄₇	C₄⋊C₄	1	2	yes	—
99	C₄⋊Dic₄₇	C₄⋊C₄	1	2	yes	—
100	#10625329	—	2	2	no	—
101	#10625329	—	2	2	no	—
102	#10625329	—	2	2	no	—
103	#10625329	C₂×D₄	1	4	yes	—
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	2	no	—
108	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
109	#10625324	—	2	16	no	—
110	#10625324	—	4	8	no	—
111	C₈×D₄₇	—	4	8	no	—
112	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
113	C₂²×D₄₇	—	2	2	no	—
114	C₂²×D₄₇	—	2	2	no	—
115	C₂²×D₄₇	—	4	2	no	—
116	C₂²×D₄₇	—	4	2	no	—
117	C₂²×D₄₇	C₂×C₄⋊C₄	1	4	yes	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	4	1	no	—
120	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
130	C₂×Dic₄₇	—	2	2	no	—
131	C₂×Dic₄₇	—	2	2	no	—
132	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
133	D₁₈₈	—	2	4	no	—
134	D₁₈₈	—	4	2	no	—
135	C₄×D₄₇	—	2	4	no	—
136	C₄×D₄₇	—	2	4	no	—
137	C₄×D₄₇	—	2	4	no	—
138	C₄×D₄₇	—	2	4	no	—
139	C₈×C₄₇	—	2	2	no	—
140	C₈×C₄₇	—	4	4	no	—
141	C₈×C₄₇	C₄×D₄	1	4	yes	—
142	C₄₇⋊C₈	—	2	4	no	—
143	C₄₇⋊C₈	—	2	4	no	—
144	C₄₇⋊C₈	—	4	2	no	—
145	#22669	—	47	1	no	maximal, Sylow
146	C₂²×C₄₇	—	2	1	no	—
147	C₂²×C₄₇	—	2	1	no	—
148	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₄₇	D₄×Q₈	1	1	yes	characteristic
150	C₂²×C₄₇	#554	1	1	yes	characteristic
151	C₂×D₄₇	—	2	4	no	—
152	C₂×D₄₇	—	2	4	no	—
153	C₂×D₄₇	—	2	4	no	—
154	C₂×D₄₇	—	8	2	no	—
155	C₂×D₄₇	—	8	2	no	—
156	C₄×C₄₇	—	2	1	no	—
157	C₄×C₄₇	—	2	1	no	—
158	C₄×C₄₇	—	2	1	no	—
159	C₄×C₄₇	#430	1	2	yes	complemented
160	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
161	C₄×C₄₇	#520	1	1	yes	characteristic
162	C₄×C₄₇	D₄²	1	1	yes	characteristic
163	C₄×C₄₇	#547	1	1	yes	characteristic
164	Dic₄₇	—	2	4	no	—
165	D₄×M₁₆	—	47	2	no	—
166	C₂×#436	—	47	1	no	—
167	C₂×#430	—	47	2	no	—
168	#1950	—	47	2	no	—
169	#1855	—	47	4	no	—
170	#1848	D₄₇	1	1	yes	characteristic, complemented
171	#1811	—	47	2	no	—
172	C₄×#108	—	47	1	no	—
173	C₂×C₄₇	—	2	1	no	—
174	C₂×C₄₇	—	2	1	no	—
175	C₂×C₄₇	D₄×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₄₇	C₂×#430	1	1	yes	characteristic
177	C₂×C₄₇	#2916	1	1	yes	characteristic
178	D₄₇	—	4	4	no	—
179	C₂²×M₁₆	—	47	2	no	—
180	C₂²×M₁₆	—	47	4	no	—
181	C₂×C₄×D₄	—	47	1	no	—
182	#436	—	47	4	no	—
183	#436	—	47	4	no	—
184	#436	—	94	2	no	—
185	#435	—	94	4	no	—
186	C₈×D₄	—	94	2	no	—
187	#430	—	94	8	no	—
188	C₂×#108	—	47	1	no	—
189	C₂×#108	—	47	4	no	—
190	C₂×#108	—	47	1	no	—
191	C₂×#108	C₂×D₄₇	1	2	yes	—
192	C₂×#108	C₂×D₄₇	1	2	yes	—
193	#423	C₂×D₄₇	1	2	yes	—
194	C₂×C₄⋊C₈	—	47	1	no	—
195	#407	—	47	4	no	—
196	C₂×C₂²⋊C₈	—	47	2	no	—
197	C₄×M₁₆	—	47	2	no	—
198	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
199	#341	—	47	4	no	—
200	#341	—	47	2	no	—
201	C₄₇	#22669	1	1	yes	characteristic, complemented, Sylow
202	C₂²×D₄	—	47	1	no	—
203	C₂³×C₄	—	47	2	no	—
204	C₂×M₁₆	—	47	8	no	—
205	C₂×M₁₆	—	47	2	no	—
206	C₂×M₁₆	—	47	2	no	—
207	C₂×M₁₆	—	94	4	no	—
208	C₂×M₁₆	—	94	4	no	—
209	C₂×M₁₆	—	94	4	no	—
210	C₂×M₁₆	—	94	8	no	—
211	C₂×M₁₆	—	188	4	no	—
212	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
213	C₂²×C₈	—	47	2	no	—
214	C₂²×C₈	—	94	4	no	—
215	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
216	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
217	C₄×D₄	—	47	2	no	—
218	C₄×D₄	—	47	2	no	—
219	C₄×D₄	—	47	2	no	—
220	C₄×D₄	—	47	2	no	—
221	C₂×C₄⋊C₄	—	47	1	no	—
222	C₂×C₂²⋊C₄	—	47	2	no	—
223	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
224	#108	—	2	4	no	—
225	#108	—	4	2	no	—
226	#108	—	47	4	no	—
227	#108	—	94	8	no	—
228	#108	—	94	2	no	—
229	C₄⋊C₈	—	2	2	no	—
230	C₄⋊C₈	—	2	2	no	—
231	C₄⋊C₈	—	47	2	no	—
232	C₄⋊C₈	—	47	2	no	—
233	C₂²⋊C₈	—	47	8	no	—
234	C₂²⋊C₈	—	94	4	no	—
235	C₂²⋊C₈	—	94	4	no	—
236	#97	—	94	2	no	—
237	C₄×C₈	—	94	2	no	—
238	C₄×C₈	C₄×D₄₇	1	2	yes	—
239	C₄×C₈	C₄×D₄₇	1	2	yes	—
240	C₂⁴	—	47	2	no	—
241	C₂×D₄	—	47	2	no	—
242	C₂×D₄	—	47	2	no	—
243	C₂×D₄	—	188	2	no	—
244	C₂²×C₄	—	47	2	no	—
245	C₂²×C₄	—	47	4	no	—
246	C₂²×C₄	—	47	4	no	—
247	C₂²×C₄	—	94	2	no	—
248	C₂²×C₄	—	94	2	no	—
249	C₂²×C₄	—	94	2	no	—
250	C₂²×C₄	—	94	2	no	—
251	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
252	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
253	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
254	M₁₆	—	2	4	no	—
255	M₁₆	—	2	4	no	—
256	M₁₆	—	47	4	no	—
257	M₁₆	—	94	2	no	—
258	M₁₆	—	94	16	no	—
259	M₁₆	—	94	2	no	—
260	M₁₆	—	94	2	no	—
261	M₁₆	—	188	8	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	47	4	no	—
266	C₂×C₈	—	94	2	no	—
267	C₂×C₈	—	94	2	no	—
268	C₂×C₈	—	94	2	no	—
269	C₂×C₈	—	94	2	no	—
270	C₂×C₈	—	188	8	no	—
271	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
272	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
273	C₂×C₈	D₄×D₄₇	1	2	yes	—
274	C₂×C₈	#10625354	1	2	yes	—
275	C₄⋊C₄	—	47	2	no	—
276	C₄⋊C₄	—	47	2	no	—
277	C₂²⋊C₄	—	47	4	no	—
278	C₂²⋊C₄	—	47	4	no	—
279	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
280	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
281	C₄²	D₄×D₄₇	1	1	yes	characteristic
282	C₄²	D₄×D₄₇	1	1	yes	characteristic
283	C₂³	—	47	4	no	—
284	C₂³	—	94	2	no	—
285	C₂³	—	94	2	no	—
286	C₂³	—	188	2	no	—
287	C₂³	—	188	2	no	—
288	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
289	D₄	—	94	4	no	—
290	D₄	—	188	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	47	4	no	—
294	C₂×C₄	—	94	2	no	—
295	C₂×C₄	—	94	2	no	—
296	C₂×C₄	—	94	4	no	—
297	C₂×C₄	—	94	4	no	—
298	C₂×C₄	—	94	4	no	—
299	C₂×C₄	—	94	4	no	—
300	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
301	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
302	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
303	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
306	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
307	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
308	C₂×C₄	#13094069	1	1	yes	characteristic
309	C₂×C₄	#13094072	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	4	4	no	—
312	C₈	—	94	4	no	—
313	C₈	—	94	4	no	—
314	C₈	—	188	2	no	—
315	C₈	#13093986	1	4	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	94	4	no	—
319	C₂²	—	94	4	no	—
320	C₂²	—	94	4	no	—
321	C₂²	—	376	2	no	—
322	C₂²	—	376	2	no	—
323	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
324	C₂²	#15689788	1	1	yes	characteristic
325	C₂²	#15689834	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	94	4	no	—
330	C₄	#15689170	1	2	yes	complemented
331	C₄	C₂×#13093986	1	1	yes	characteristic
332	C₄	#15689604	1	1	yes	characteristic
333	C₄	#15689758	1	1	yes	characteristic
334	C₄	#15689776	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	—	188	4	no	—
338	C₂	#20246288	1	1	yes	characteristic
339	C₂	C₂×#15689170	1	1	yes	characteristic
340	C₂	#20251996	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1848	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂²×D₁₈₈	Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1848	D₄₇
hypocenter	C₄₇	#22669
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
3. C₂×#15689170	C₂	semidirect
4. #20244161	C₂	semidirect
5. #20243591	C₂	semidirect
6. #20243427	C₂	semidirect
7. C₄×#13093960	C₂	semidirect
8. C₄₇×#1848	C₂	semidirect
50. #13093960	D₄	semidirect
159. C₄×C₄₇	#430	semidirect
170. #1848	D₄₇	semidirect
201. C₄₇	#22669	semidirect
330. C₄	#15689170	semidirect

Maximal subgroups

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

subgroup	index
#20252123	2
C₂×#15689202	2
C₂×#15689170	2
#20244161	2
#20243591	2
#20243427	2
C₄×#13093960	2
C₄₇×#1848	2
#22669	47