Subgroups — #26226457

Order 12032.

750 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20244003	C₂	1	4	yes	maximal
2	#20243954	C₂	1	4	yes	maximal, complemented
3	#20243540	C₂	1	1	yes	maximal, characteristic, complemented
4	#20243505	C₂	1	1	yes	maximal, characteristic, complemented
5	#20243368	C₂	1	2	yes	maximal
6	#20243241	C₂	1	2	yes	maximal, complemented
7	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
11	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
12	C₂×#13093935	C₂²	1	4	yes	—
13	#15688878	C₂²	1	4	yes	—
14	#15688878	C₂²	1	4	yes	—
15	#15688878	C₂²	1	4	yes	—
16	C₂×#117×C₄₇	C₄	1	1	yes	characteristic
17	C₂×#117×C₄₇	C₄	1	1	yes	characteristic
18	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
20	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₄×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
22	C₄×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
23	C₄₇×#376	C₄	1	1	yes	characteristic
24	C₄₇×#376	C₄	1	1	yes	characteristic
25	C₄³×C₄₇	C₂²	1	1	yes	characteristic
26	C₄³×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
28	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
29	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
30	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
31	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
32	C₂²⋊(C₄₇⋊C₈)	—	4	4	no	—
33	#13093935	—	4	4	no	—
34	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	#117×C₄₇	—	2	4	no	—
38	#117×C₄₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₇	—	2	2	no	—
44	C₂×C₄²×C₄₇	—	2	2	no	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	#95×C₄₇	C₂×C₄	1	2	yes	—
58	#95×C₄₇	C₂×C₄	1	2	yes	—
59	#95×C₄₇	C₂×C₄	1	2	yes	—
60	#95×C₄₇	C₂×C₄	1	2	yes	—
61	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	2	1	no	—
66	C₂²×C₄×C₄₇	—	2	1	no	—
67	C₂²×C₄×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	—	4	2	no	—
70	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₄⋊C₄×C₄₇	—	2	4	no	—
82	C₄⋊C₄×C₄₇	—	2	4	no	—
83	C₂²⋊C₄×C₄₇	—	2	4	no	—
84	C₂²⋊C₄×C₄₇	—	2	4	no	—
85	C₄²×C₄₇	—	2	2	no	—
86	C₄²×C₄₇	—	2	2	no	—
87	C₄²×C₄₇	—	2	2	no	—
88	C₄²×C₄₇	—	2	2	no	—
89	C₄²×C₄₇	—	2	2	no	—
90	C₄²×C₄₇	—	2	1	no	—
91	C₄²×C₄₇	—	2	1	no	—
92	C₄²×C₄₇	—	2	2	no	—
93	C₄²×C₄₇	—	2	2	no	—
94	C₄²×C₄₇	—	2	2	no	—
95	C₄²×C₄₇	—	2	2	no	—
96	C₄²×C₄₇	—	4	2	no	—
97	C₄²×C₄₇	—	4	2	no	—
98	C₄²×C₄₇	M₁₆	1	4	yes	—
99	C₄²×C₄₇	#41	1	2	yes	—
100	C₄²×C₄₇	#41	1	2	yes	—
101	C₂×C₄₇⋊C₈	—	2	4	no	—
102	C₂×C₄₇⋊C₈	—	2	4	no	—
103	C₂×C₄₇⋊C₈	—	2	4	no	—
104	C₂×C₄₇⋊C₈	—	2	4	no	—
105	C₂×C₄₇⋊C₈	—	2	4	no	—
106	C₂×C₄₇⋊C₈	—	2	4	no	—
107	C₂³×C₄₇	—	2	1	no	—
108	C₂³×C₄₇	—	2	2	no	—
109	C₂³×C₄₇	—	2	1	no	—
110	C₂³×C₄₇	—	2	2	no	—
111	C₂³×C₄₇	C₂×M₁₆	1	2	yes	—
112	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	8	no	—
119	C₂×C₄×C₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	—	2	1	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	2	2	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	2	2	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	2	no	—
131	C₂×C₄×C₄₇	—	2	2	no	—
132	C₂×C₄×C₄₇	—	4	4	no	—
133	C₂×C₄×C₄₇	—	4	1	no	—
134	C₂×C₄×C₄₇	—	4	1	no	—
135	C₂×C₄×C₄₇	—	4	2	no	—
136	C₂×C₄×C₄₇	#117	1	2	yes	—
137	C₂×C₄×C₄₇	#117	1	2	yes	—
138	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
139	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
140	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
141	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
142	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
145	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
146	C₄₇⋊C₈	—	4	4	no	—
147	C₄₇⋊C₈	—	4	4	no	—
148	#21926	—	47	1	no	maximal, Sylow
149	C₂²×C₄₇	—	2	2	no	—
150	C₂²×C₄₇	—	2	4	no	—
151	C₂²×C₄₇	—	4	1	no	—
152	C₂²×C₄₇	—	4	1	no	—
153	C₂²×C₄₇	—	4	2	no	—
154	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
155	C₂²×C₄₇	#520	1	2	yes	—
156	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
157	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
158	C₂²×C₄₇	#568	1	2	yes	—
159	C₄×C₄₇	—	2	2	no	—
160	C₄×C₄₇	—	2	2	no	—
161	C₄×C₄₇	—	2	4	no	—
162	C₄×C₄₇	—	4	1	no	—
163	C₄×C₄₇	—	4	1	no	—
164	C₄×C₄₇	—	4	2	no	—
165	C₄×C₄₇	—	4	2	no	—
166	C₄×C₄₇	—	4	2	no	—
167	C₄×C₄₇	#432	1	4	yes	—
168	C₄×C₄₇	#540	1	1	yes	characteristic
169	C₄×C₄₇	#541	1	1	yes	characteristic
170	C₄×C₄₇	#542	1	2	yes	—
171	C₄×#117	D₄₇	1	1	yes	characteristic
172	#1924	—	47	4	no	—
173	#1917	—	47	4	no	—
174	#1842	—	47	1	no	—
175	#1831	—	47	1	no	—
176	#1797	—	47	2	no	—
177	#1767	—	47	2	no	—
178	C₂×C₄₇	—	4	2	no	—
179	C₂×C₄₇	#2314	1	1	yes	characteristic
180	C₂×C₄₇	C₂×#432	1	1	yes	characteristic
181	C₂×C₄₇	#2921	1	1	yes	characteristic
182	C₂×C₄₇	#2989	1	2	yes	—
183	C₂×C₄₇	#2989	1	2	yes	—
184	C₂×#117	Dic₄₇	1	1	yes	characteristic
185	C₂×#117	Dic₄₇	1	1	yes	characteristic
186	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
187	C₂×C₄⋊C₈	—	47	2	no	—
188	C₂×C₄⋊C₈	—	47	2	no	—
189	C₂×C₄⋊C₈	—	47	4	no	—
190	C₂×C₂²⋊C₈	—	47	4	no	—
191	C₂×#97	—	47	4	no	—
192	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
193	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
194	C₄×C₂²⋊C₄	Dic₄₇	1	2	yes	—
195	C₄×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
196	#376	Dic₄₇	1	1	yes	characteristic
197	#376	Dic₄₇	1	1	yes	characteristic
198	C₄³	C₂×D₄₇	1	1	yes	characteristic
199	C₄³	C₂×D₄₇	1	1	yes	characteristic
200	#336	—	47	4	no	—
201	#336	—	47	4	no	—
202	#336	—	47	4	no	—
203	C₄₇	#21926	1	1	yes	characteristic, complemented, Sylow
204	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
205	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
206	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
207	C₂²×C₈	—	47	4	no	—
208	C₂²×C₈	—	47	4	no	—
209	#117	—	2	4	no	—
210	#117	—	2	4	no	—
211	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
212	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
213	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
214	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	—	2	2	no	—
217	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
218	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
219	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
220	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
221	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
222	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
223	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
224	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
225	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
226	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
227	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
228	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
229	C₄⋊C₈	—	94	4	no	—
230	C₄⋊C₈	—	94	4	no	—
231	C₄⋊C₈	—	188	4	no	—
232	C₂²⋊C₈	—	188	4	no	—
233	#97	—	188	4	no	—
234	#95	C₂×Dic₄₇	1	2	yes	—
235	#95	C₂×Dic₄₇	1	2	yes	—
236	#95	C₂×Dic₄₇	1	2	yes	—
237	#95	C₂×Dic₄₇	1	2	yes	—
238	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	#10625329	1	4	yes	—
248	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
249	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
251	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
252	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
253	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
255	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
256	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
257	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
258	C₂×C₈	—	94	4	no	—
259	C₂×C₈	—	94	4	no	—
260	C₂×C₈	—	94	4	no	—
261	C₂×C₈	—	94	4	no	—
262	C₂×C₈	—	94	4	no	—
263	C₂×C₈	—	94	4	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₂²⋊C₄	—	2	4	no	—
267	C₂²⋊C₄	—	2	4	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	4	2	no	—
280	C₄²	—	4	2	no	—
281	C₄²	#10625329	1	4	yes	—
282	C₄²	#10625350	1	2	yes	—
283	C₄²	#10625350	1	2	yes	—
284	C₂³	—	2	1	no	—
285	C₂³	—	2	2	no	—
286	C₂³	—	2	1	no	—
287	C₂³	—	2	2	no	—
288	C₂³	C₂×#10625329	1	2	yes	—
289	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	8	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	#13093983	1	2	yes	—
314	C₂×C₄	#13093983	1	2	yes	—
315	C₂×C₄	C₂×#10625329	1	4	yes	—
316	C₂×C₄	C₂×#10625329	1	2	yes	—
317	C₂×C₄	C₂×#10625329	1	2	yes	—
318	C₂×C₄	C₂×#10625329	1	2	yes	—
319	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
320	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
321	C₂×C₄	#13094070	1	2	yes	—
322	C₂×C₄	#13094071	1	2	yes	—
323	C₈	—	188	4	no	—
324	C₈	—	188	4	no	—
325	C₂²	—	2	4	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	4	2	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	C₂×#13093983	1	1	yes	characteristic
331	C₂²	#15689606	1	2	yes	—
332	C₂²	C₂²×#10625329	1	1	yes	characteristic
333	C₂²	C₂²×#10625329	1	1	yes	characteristic
334	C₂²	#15689890	1	2	yes	—
335	C₄	—	2	4	no	—
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	4	2	no	—
339	C₄	—	4	2	no	—
340	C₄	—	4	2	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	#15689180	1	4	yes	—
344	C₄	#15689728	1	1	yes	characteristic
345	C₄	#15689735	1	1	yes	characteristic
346	C₄	#15689744	1	2	yes	—
347	C₂	—	4	2	no	—
348	C₂	#20245997	1	1	yes	characteristic
349	C₂	C₂×#15689180	1	1	yes	characteristic
350	C₂	#20252026	1	1	yes	characteristic
351	C₂	#20252505	1	2	yes	—
352	C₂	#20252505	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄×#117×C₄₇	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#117	D₄₇
hypocenter	C₄₇	#21926
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
2. #20243954	C₂	semidirect
3. #20243540	C₂	semidirect
4. #20243505	C₂	semidirect
6. #20243241	C₂	semidirect
203. C₄₇	#21926	semidirect

Maximal subgroups

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

subgroup	index
#20244003	2
#20243954	2
#20243540	2
#20243505	2
#20243368	2
#20243241	2
C₄×#117×C₄₇	2
#21926	47