Subgroups — #26226872

Order 12032.

762 conjugacy classes of subgroups.

The subgroup lattice has 378 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	#20246011	C₂	1	1	yes	maximal, characteristic
3	#20244156	C₂	1	2	yes	maximal, complemented
4	#20244146	C₂	1	2	yes	maximal, complemented
5	#20243585	C₂	1	2	yes	maximal
6	#20243575	C₂	1	2	yes	maximal
7	#20243554	C₂	1	1	yes	maximal, characteristic, complemented
8	#20243547	C₂	1	1	yes	maximal, characteristic, complemented
9	#20243521	C₂	1	1	yes	maximal, characteristic, complemented
10	#20243518	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2314	C₂	1	1	yes	maximal, characteristic
12	C₂×#13093983	C₂²	1	1	yes	characteristic
13	C₂²×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
14	#15689184	C₄	1	8	yes	complemented
15	C₂×#13093957	C₂²	1	2	yes	—
16	C₂×#13093957	C₂²	1	2	yes	—
17	C₂×#13093957	C₂²	1	2	yes	—
18	C₂×#13093955	C₂²	1	2	yes	—
19	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
20	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
21	C₂×#13093935	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
23	#15689022	—	2	2	no	—
24	#15688998	—	2	2	no	—
25	#15688993	C₂²	1	1	yes	characteristic
26	#15688993	C₂²	1	1	yes	characteristic
27	#15688989	C₂²	1	1	yes	characteristic
28	#15688989	C₂²	1	1	yes	characteristic
29	#15688885	C₂²	1	2	yes	—
30	#15688885	C₂²	1	2	yes	—
31	#15688885	C₂²	1	2	yes	—
32	#15688885	C₂²	1	2	yes	—
33	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
34	C₄₇×#388	—	2	2	no	—
35	C₄₇×#387	—	2	2	no	—
36	C₄₇×#384	C₂²	1	1	yes	characteristic
37	C₄₇×#382	C₂²	1	1	yes	characteristic
38	C₄₇×#380	C₂²	1	2	yes	—
39	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
40	C₂³×Dic₄₇	—	2	1	no	—
41	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
42	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
43	#13093983	—	4	2	no	—
44	C₂×C₄⋊Dic₄₇	—	2	1	no	—
45	C₂×C₄⋊Dic₄₇	—	2	2	no	—
46	C₂×C₄⋊Dic₄₇	—	2	2	no	—
47	C₂×C₄⋊Dic₄₇	—	4	2	no	—
48	C₂×C₄⋊Dic₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₄⋊Dic₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊Dic₄₇	C₂³	1	1	yes	characteristic
51	C₂×C₄⋊Dic₄₇	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
53	C₂×C₄×Dic₄₇	—	2	1	no	—
54	#13093957	—	4	2	no	—
55	#13093957	—	4	2	no	—
56	#13093957	—	4	2	no	—
57	#13093955	—	4	2	no	—
58	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	complemented
59	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
60	#13093935	C₂×C₄	1	4	yes	complemented
61	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	complemented
62	C₄₇⋊#95	—	2	2	no	—
63	C₄₇⋊#95	—	2	2	no	—
64	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
65	#117×C₄₇	C₂×C₄	1	4	yes	—
66	#117×C₄₇	Q₈	1	4	yes	—
67	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
68	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
69	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
71	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
73	C₂×C₄²×C₄₇	—	2	2	no	—
74	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
76	#95×C₄₇	—	2	2	no	—
77	#95×C₄₇	—	2	2	no	—
78	#95×C₄₇	—	2	2	no	—
79	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	—	2	2	no	—
81	C₂²×C₄×C₄₇	—	2	2	no	—
82	C₂²×C₄×C₄₇	—	2	2	no	—
83	C₂²×C₄×C₄₇	—	2	2	no	—
84	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
89	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
90	C₂²×Dic₄₇	—	2	1	no	—
91	C₂²×Dic₄₇	—	2	1	no	—
92	C₂²×Dic₄₇	—	2	1	no	—
93	C₂²×Dic₄₇	—	2	1	no	—
94	C₂²×Dic₄₇	—	4	2	no	—
95	C₂²×Dic₄₇	—	8	2	no	—
96	C₄⋊C₄×C₄₇	—	2	2	no	—
97	C₄⋊C₄×C₄₇	—	2	2	no	—
98	C₄⋊C₄×C₄₇	—	2	4	no	—
99	C₄⋊C₄×C₄₇	—	8	2	no	—
100	C₄⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
101	C₂²⋊C₄×C₄₇	—	8	2	no	—
102	C₂²⋊C₄×C₄₇	C₄⋊C₄	1	8	yes	—
103	C₄²×C₄₇	—	8	2	no	—
104	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
105	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
106	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
107	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
108	C₂²⋊Dic₄₇	—	4	2	no	—
109	C₄⋊Dic₄₇	—	2	2	no	—
110	C₄⋊Dic₄₇	—	2	2	no	—
111	C₄⋊Dic₄₇	—	2	2	no	—
112	C₄⋊Dic₄₇	—	2	2	no	—
113	C₄⋊Dic₄₇	—	4	2	no	—
114	C₄⋊Dic₄₇	—	8	2	no	—
115	C₄⋊Dic₄₇	—	8	2	no	—
116	C₄×Dic₄₇	—	4	2	no	—
117	C₂×C₄₇⋊C₈	—	2	2	no	—
118	C₂×C₄₇⋊C₈	—	2	2	no	—
119	C₂×C₄₇⋊C₈	—	2	2	no	—
120	C₂×C₄₇⋊C₈	—	2	2	no	—
121	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
122	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
123	C₂³×C₄₇	—	2	1	no	—
124	C₂³×C₄₇	—	2	1	no	—
125	C₂³×C₄₇	—	2	2	no	—
126	C₂³×C₄₇	—	2	2	no	—
127	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
128	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	—	2	1	no	—
130	C₂×C₄×C₄₇	—	2	1	no	—
131	C₂×C₄×C₄₇	—	2	2	no	—
132	C₂×C₄×C₄₇	—	2	2	no	—
133	C₂×C₄×C₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	—	2	4	no	—
136	C₂×C₄×C₄₇	—	2	4	no	—
137	C₂×C₄×C₄₇	—	4	2	no	—
138	C₂×C₄×C₄₇	—	4	2	no	—
139	C₂×C₄×C₄₇	—	4	2	no	—
140	C₂×C₄×C₄₇	—	4	2	no	—
141	C₂×C₄×C₄₇	—	4	2	no	—
142	C₂×C₄×C₄₇	—	4	2	no	—
143	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
144	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
145	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
146	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
148	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
149	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
150	C₂×Dic₄₇	—	4	1	no	—
151	C₂×Dic₄₇	—	4	1	no	—
152	C₂×Dic₄₇	—	4	1	no	—
153	C₂×Dic₄₇	—	4	1	no	—
154	C₂×Dic₄₇	—	4	2	no	—
155	C₂×Dic₄₇	—	4	2	no	—
156	C₂×Dic₄₇	—	4	2	no	—
157	C₂×Dic₄₇	—	4	2	no	—
158	C₂×Dic₄₇	—	8	4	no	—
159	C₄₇⋊C₈	—	2	4	no	—
160	C₄₇⋊C₈	—	2	4	no	—
161	#21991	—	47	1	no	maximal, Sylow
162	C₂²×C₄₇	—	2	2	no	—
163	C₂²×C₄₇	—	2	4	no	—
164	C₂²×C₄₇	—	4	1	no	—
165	C₂²×C₄₇	—	4	1	no	—
166	C₂²×C₄₇	—	4	2	no	—
167	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
168	C₂²×C₄₇	#535	1	1	yes	characteristic
169	C₂²×C₄₇	#536	1	1	yes	characteristic
170	C₂²×C₄₇	#576	1	2	yes	—
171	C₂²×C₄₇	#578	1	2	yes	—
172	C₄×C₄₇	—	2	2	no	—
173	C₄×C₄₇	—	2	4	no	—
174	C₄×C₄₇	—	2	4	no	—
175	C₄×C₄₇	—	8	2	no	—
176	C₄×C₄₇	—	8	2	no	—
177	C₄×C₄₇	#520	1	2	yes	—
178	C₄×C₄₇	#544	1	2	yes	—
179	Dic₄₇	—	8	1	no	—
180	Dic₄₇	—	8	1	no	—
181	Dic₄₇	—	8	1	no	—
182	Dic₄₇	—	8	1	no	—
183	C₂×#433	—	47	1	no	—
184	#2317	—	47	1	no	—
185	#2314	D₄₇	1	1	yes	characteristic
186	#1949	—	47	2	no	—
187	#1947	—	47	2	no	—
188	#1854	—	47	2	no	—
189	#1852	—	47	2	no	—
190	#1847	—	47	1	no	—
191	#1845	—	47	1	no	—
192	#1836	—	47	1	no	—
193	#1835	—	47	1	no	—
194	C₂×C₄₇	—	4	2	no	—
195	C₂×C₄₇	#2317	1	1	yes	characteristic
196	C₂×C₄₇	#2913	1	2	yes	—
197	C₂×C₄₇	#3220	1	1	yes	characteristic
198	C₂×C₄₇	#3223	1	1	yes	characteristic
199	C₂×C₄₇	#3243	1	2	yes	—
200	C₂×#117	—	47	1	no	—
201	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
202	C₂²×C₄⋊C₄	—	47	1	no	—
203	#433	—	47	8	no	—
204	C₂×#107	—	47	2	no	—
205	C₂×#107	—	47	2	no	—
206	C₂×#107	—	47	2	no	—
207	C₂×#106	—	47	2	no	—
208	C₂×C₄⋊C₈	—	47	2	no	—
209	C₂×C₂²⋊C₈	—	47	2	no	—
210	C₂×#97	—	47	1	no	—
211	C₂×C₄×C₈	—	47	1	no	—
212	#389	—	94	2	no	—
213	#388	—	2	2	no	—
214	#387	—	2	2	no	—
215	#385	—	94	2	no	—
216	#384	—	47	1	no	—
217	#384	—	47	1	no	—
218	#384	C₂×D₄₇	1	1	yes	characteristic
219	#382	—	47	1	no	—
220	#382	—	47	1	no	—
221	#382	C₂×D₄₇	1	1	yes	characteristic
222	#380	C₂×D₄₇	1	2	yes	—
223	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
224	#340	—	47	2	no	—
225	#340	—	47	2	no	—
226	#340	—	47	2	no	—
227	#340	—	47	2	no	—
228	C₄₇	#21991	1	1	yes	characteristic, complemented, Sylow
229	C₂³×C₄	—	94	1	no	—
230	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
231	C₂²×C₈	—	47	2	no	—
232	C₂²×C₈	—	47	2	no	—
233	#117	—	188	2	no	—
234	#117	Dic₉₄	1	4	yes	—
235	#117	C₄×D₄₇	1	4	yes	—
236	C₂×C₄⋊C₄	—	2	2	no	—
237	C₂×C₄⋊C₄	—	47	1	no	—
238	C₂×C₄⋊C₄	—	47	1	no	—
239	C₂×C₄⋊C₄	—	47	1	no	—
240	C₂×C₄⋊C₄	—	47	1	no	—
241	C₂×C₄⋊C₄	—	94	2	no	—
242	C₂×C₄⋊C₄	—	94	2	no	—
243	C₂×C₄⋊C₄	—	94	1	no	—
244	C₂×C₄⋊C₄	—	188	2	no	—
245	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
246	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
247	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
248	C₂×C₂²⋊C₄	—	2	2	no	—
249	C₂×C₂²⋊C₄	—	94	1	no	—
250	C₂×C₂²⋊C₄	C₂²⋊D₄₇	1	2	yes	—
251	C₂×C₄²	—	2	2	no	—
252	C₂×C₄²	—	94	1	no	—
253	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
254	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
255	#107	—	188	2	no	—
256	#107	—	188	2	no	—
257	#107	—	188	2	no	—
258	#106	—	188	2	no	—
259	C₄⋊C₈	—	47	8	no	—
260	C₂²⋊C₈	—	47	8	no	—
261	#97	—	47	4	no	—
262	C₄×C₈	—	47	4	no	—
263	#95	—	2	2	no	—
264	#95	—	2	2	no	—
265	#95	—	2	2	no	—
266	#95	—	94	2	no	—
267	#95	—	94	2	no	—
268	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	94	1	no	—
274	C₂²×C₄	—	94	1	no	—
275	C₂²×C₄	—	94	1	no	—
276	C₂²×C₄	—	94	1	no	—
277	C₂²×C₄	—	376	2	no	—
278	C₂²×C₄	—	188	2	no	—
279	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
280	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
281	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
282	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
283	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
284	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
285	C₂×C₈	—	47	4	no	—
286	C₂×C₈	—	47	4	no	—
287	C₂×C₈	—	94	2	no	—
288	C₂×C₈	—	94	2	no	—
289	C₂×C₈	—	94	2	no	—
290	C₂×C₈	—	94	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	4	no	—
294	C₄⋊C₄	—	8	2	no	—
295	C₄⋊C₄	—	94	2	no	—
296	C₄⋊C₄	—	94	2	no	—
297	C₄⋊C₄	—	94	2	no	—
298	C₄⋊C₄	—	94	2	no	—
299	C₄⋊C₄	—	376	2	no	—
300	C₄⋊C₄	—	376	2	no	—
301	C₄⋊C₄	—	188	2	no	—
302	C₄⋊C₄	Dic₄₇⋊C₄	1	8	yes	—
303	C₂²⋊C₄	—	8	2	no	—
304	C₂²⋊C₄	—	188	2	no	—
305	C₂²⋊C₄	Dic₄₇⋊C₄	1	8	yes	—
306	C₄²	—	8	2	no	—
307	C₄²	—	188	2	no	—
308	C₄²	C₂×Dic₉₄	1	2	yes	—
309	C₄²	C₂×Dic₉₄	1	2	yes	—
310	C₄²	C₂×C₄×D₄₇	1	2	yes	—
311	C₄²	C₂×C₄×D₄₇	1	2	yes	—
312	C₂³	—	2	2	no	—
313	C₂³	—	2	2	no	—
314	C₂³	—	2	1	no	—
315	C₂³	—	2	1	no	—
316	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
317	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	4	no	—
325	C₂×C₄	—	2	4	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	376	4	no	—
333	C₂×C₄	—	188	2	no	—
334	C₂×C₄	—	188	2	no	—
335	C₂×C₄	—	188	2	no	—
336	C₂×C₄	—	188	2	no	—
337	C₂×C₄	—	188	1	no	—
338	C₂×C₄	—	188	1	no	—
339	C₂×C₄	—	188	1	no	—
340	C₂×C₄	—	188	1	no	—
341	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
342	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
343	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
344	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
345	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
346	C₂×C₄	#13094070	1	2	yes	—
347	C₂×C₄	#13094071	1	2	yes	—
348	C₈	—	94	4	no	—
349	C₈	—	94	4	no	—
350	C₂²	—	2	4	no	—
351	C₂²	—	2	2	no	—
352	C₂²	—	4	1	no	—
353	C₂²	—	4	2	no	—
354	C₂²	—	4	1	no	—
355	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
356	C₂²	#15689687	1	1	yes	characteristic
357	C₂²	#15689694	1	1	yes	characteristic
358	C₂²	#15689926	1	2	yes	—
359	C₂²	#15689938	1	2	yes	—
360	C₄	—	2	4	no	—
361	C₄	—	2	4	no	—
362	C₄	—	2	2	no	—
363	C₄	—	8	2	no	—
364	C₄	—	8	2	no	—
365	C₄	—	376	1	no	—
366	C₄	—	376	1	no	—
367	C₄	—	376	1	no	—
368	C₄	—	376	1	no	—
369	C₄	#15689603	1	2	yes	—
370	C₄	#15689751	1	2	yes	—
371	C₂	—	4	2	no	—
372	C₂	#20246008	1	1	yes	characteristic
373	C₂	#20251986	1	2	yes	—
374	C₂	#20254432	1	1	yes	characteristic
375	C₂	#20254465	1	1	yes	characteristic
376	C₂	#20254665	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2314	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2314	D₄₇
hypocenter	C₄₇	#21991
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ #2314

Complements

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

N	H ≅ G/N	type
3. #20244156	C₂	semidirect
4. #20244146	C₂	semidirect
7. #20243554	C₂	semidirect
8. #20243547	C₂	semidirect
9. #20243521	C₂	semidirect
10. #20243518	C₂	semidirect
14. #15689184	C₄	semidirect
58. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
60. #13093935	C₂×C₄	semidirect
61. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
228. C₄₇	#21991	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
#20246011	2
#20244156	2
#20244146	2
#20243585	2
#20243575	2
#20243554	2
#20243547	2
#20243521	2
#20243518	2
C₄₇×#2314	2
#21991	47