Subgroups — #26226736

Order 12032.

898 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
3	#20244107	C₂	1	4	yes	maximal, complemented
4	#20243573	C₂	1	4	yes	maximal
5	#20243540	C₂	1	2	yes	maximal, complemented
6	#20243505	C₂	1	2	yes	maximal, complemented
7	C₄×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	#15689176	—	2	4	no	—
10	#15689147	—	2	4	no	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	8	yes	—
16	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
18	#15688878	C₂²	1	8	yes	—
19	C₂×#117×C₄₇	C₄	1	2	yes	—
20	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
24	C₄×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
25	C₄×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
26	C₄₇×#376	C₄	1	2	yes	—
27	C₄³×C₄₇	C₂²	1	2	yes	—
28	C₂×#10625329	—	2	4	no	—
29	C₂×#10625329	—	4	4	no	—
30	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
32	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
33	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
34	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
35	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
36	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
37	C₄⋊(C₄₇⋊C₈)	—	4	8	no	—
38	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
39	C₄×C₄₇⋊C₈	—	2	4	no	—
40	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	#117×C₄₇	—	2	8	no	—
44	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	D₄	1	4	yes	—
53	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	#95×C₄₇	C₂×C₄	1	4	yes	—
58	#95×C₄₇	C₂×C₄	1	2	yes	—
59	#95×C₄₇	C₂×C₄	1	2	yes	—
60	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	—	2	4	no	—
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	2	no	—
66	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₄₇	—	2	4	no	—
79	C₄⋊C₄×C₄₇	M₁₆	1	8	yes	complemented
80	C₂²⋊C₄×C₄₇	—	2	8	no	—
81	C₄²×C₄₇	—	2	8	no	—
82	C₄²×C₄₇	—	2	4	no	—
83	C₄²×C₄₇	—	2	4	no	—
84	C₄²×C₄₇	—	2	4	no	—
85	C₄²×C₄₇	—	2	4	no	—
86	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
87	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
88	C₄²×C₄₇	C₂×D₄	1	2	yes	—
89	C₄²×C₄₇	C₂×D₄	1	2	yes	—
90	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
91	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
92	#10625329	—	4	8	no	—
93	C₂×C₄₇⋊C₈	—	2	2	no	—
94	C₂×C₄₇⋊C₈	—	2	2	no	—
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂×C₄₇⋊C₈	—	2	4	no	—
97	C₂×C₄₇⋊C₈	—	2	8	no	—
98	C₂×C₄₇⋊C₈	—	2	4	no	—
99	C₂³×C₄₇	—	2	2	no	—
100	C₂³×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	—	2	1	no	—
102	C₂³×C₄₇	—	2	1	no	—
103	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	2	8	no	—
115	C₂×C₄×C₄₇	—	2	8	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
125	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
126	C₂×C₄×C₄₇	#131	1	2	yes	—
127	C₂×C₄×C₄₇	#131	1	2	yes	—
128	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
131	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
132	C₄₇⋊C₈	—	4	2	no	—
133	C₄₇⋊C₈	—	4	4	no	—
134	C₄₇⋊C₈	—	4	2	no	—
135	#21974	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	—	4	2	no	—
140	C₂²×C₄₇	—	4	2	no	—
141	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
142	C₂²×C₄₇	#520	1	2	yes	—
143	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
144	C₂²×C₄₇	C₂×#131	1	1	yes	characteristic
145	C₂²×C₄₇	C₂×#131	1	1	yes	characteristic
146	C₄×C₄₇	—	2	2	no	—
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	2	8	no	—
150	C₄×C₄₇	—	4	4	no	—
151	C₄×C₄₇	#423	1	4	yes	—
152	C₄×C₄₇	#424	1	4	yes	—
153	C₄×C₄₇	#535	1	1	yes	characteristic
154	C₄×C₄₇	#536	1	1	yes	characteristic
155	C₄×C₄₇	#544	1	2	yes	—
156	C₂×#431	—	47	1	no	—
157	C₂×#423	—	47	1	no	—
158	C₄×#117	D₄₇	1	1	yes	characteristic
159	#1939	—	47	4	no	—
160	#1851	—	47	4	no	—
161	#1842	—	47	2	no	—
162	#1831	—	47	2	no	—
163	C₂×C₄₇	—	4	2	no	—
164	C₂×C₄₇	#2317	1	1	yes	characteristic
165	C₂×C₄₇	C₂×#423	1	1	yes	characteristic
166	C₂×C₄₇	C₂×#424	1	1	yes	characteristic
167	C₂×C₄₇	#2990	1	2	yes	—
168	C₂×C₄₇	#2991	1	2	yes	—
169	C₂²×M₁₆	—	47	2	no	—
170	C₂×#117	Dic₄₇	1	2	yes	—
171	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
172	#431	—	94	4	no	—
173	#423	—	94	4	no	—
174	C₂×C₄⋊C₈	—	47	1	no	—
175	C₂×C₄⋊C₈	—	47	2	no	—
176	C₂×C₄⋊C₈	—	47	1	no	—
177	C₂×C₄⋊C₈	—	47	2	no	—
178	C₂×C₄⋊C₈	—	47	8	no	—
179	C₂×C₂²⋊C₈	—	47	2	no	—
180	C₂×C₄×C₈	—	47	2	no	—
181	C₄×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
182	C₄×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
183	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
184	C₄×C₂²⋊C₄	Dic₄₇	1	2	yes	—
185	C₄×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
186	#376	Dic₄₇	1	2	yes	—
187	C₄³	C₂×D₄₇	1	2	yes	—
188	#336	—	47	8	no	—
189	C₄₇	#21974	1	1	yes	characteristic, complemented, Sylow
190	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
191	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
192	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
193	C₂×M₁₆	—	94	4	no	—
194	C₂×M₁₆	—	188	4	no	—
195	C₂²×C₈	—	47	4	no	—
196	C₂²×C₈	—	47	2	no	—
197	C₂²×C₈	—	47	2	no	—
198	#117	—	2	8	no	—
199	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
200	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
201	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
202	C₂×C₄²	Dic₉₄	1	4	yes	—
203	C₂×C₄²	D₁₈₈	1	4	yes	—
204	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
205	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
206	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
207	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
208	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
209	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
210	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
211	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
212	C₄⋊C₈	—	94	4	no	—
213	C₄⋊C₈	—	94	4	no	—
214	C₄⋊C₈	—	94	2	no	—
215	C₄⋊C₈	—	94	2	no	—
216	C₄⋊C₈	—	188	8	no	—
217	C₂²⋊C₈	—	94	4	no	—
218	C₄×C₈	—	94	4	no	—
219	#95	C₂×Dic₄₇	1	4	yes	—
220	#95	C₂×Dic₄₇	1	2	yes	—
221	#95	C₂×Dic₄₇	1	2	yes	—
222	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
229	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
230	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
231	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
232	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
233	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
234	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
235	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
236	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
237	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
238	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
240	M₁₆	—	188	8	no	—
241	C₂×C₈	—	94	2	no	—
242	C₂×C₈	—	94	2	no	—
243	C₂×C₈	—	94	4	no	—
244	C₂×C₈	—	94	4	no	—
245	C₂×C₈	—	94	8	no	—
246	C₂×C₈	—	94	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	#10625329	1	8	yes	complemented
249	C₂²⋊C₄	—	2	8	no	—
250	C₄²	—	2	8	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	4	no	—
255	C₄²	C₄⋊Dic₄₇	1	4	yes	—
256	C₄²	C₄⋊Dic₄₇	1	4	yes	—
257	C₄²	C₂×Dic₉₄	1	2	yes	—
258	C₄²	C₂×Dic₉₄	1	2	yes	—
259	C₄²	C₂×D₁₈₈	1	2	yes	—
260	C₄²	C₂×D₁₈₈	1	2	yes	—
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	2	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
266	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
282	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
283	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
284	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
285	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
286	C₂×C₄	C₂×#10625329	1	8	yes	—
287	C₂×C₄	C₂×#10625329	1	4	yes	—
288	C₂×C₄	#13094031	1	2	yes	—
289	C₂×C₄	#13094031	1	2	yes	—
290	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
291	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
292	C₂×C₄	#13094070	1	2	yes	—
293	C₂×C₄	#13094071	1	2	yes	—
294	C₈	—	188	2	no	—
295	C₈	—	188	4	no	—
296	C₈	—	188	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
303	C₂²	#15689606	1	2	yes	—
304	C₂²	C₂²×#10625329	1	2	yes	—
305	C₂²	C₂×#13094031	1	1	yes	characteristic
306	C₂²	C₂×#13094031	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	8	no	—
310	C₄	—	2	2	no	—
311	C₄	—	4	4	no	—
312	C₄	#15689147	1	4	yes	—
313	C₄	#15689150	1	4	yes	—
314	C₄	#15689692	1	1	yes	characteristic
315	C₄	#15689699	1	1	yes	characteristic
316	C₄	#15689751	1	2	yes	—
317	C₂	—	4	2	no	—
318	C₂	#20246011	1	1	yes	characteristic
319	C₂	C₂×#15689147	1	1	yes	characteristic
320	C₂	C₂×#15689150	1	1	yes	characteristic
321	C₂	#20252513	1	2	yes	—
322	C₂	#20252520	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	#21974
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
3. #20244107	C₂	semidirect
5. #20243540	C₂	semidirect
6. #20243505	C₂	semidirect
79. C₄⋊C₄×C₄₇	M₁₆	semidirect
189. C₄₇	#21974	semidirect
248. C₄⋊C₄	#10625329	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
C₂×#15689147	2
#20244107	2
#20243573	2
#20243540	2
#20243505	2
C₄×#117×C₄₇	2
#21974	47