Subgroups — #26214826

Order 12032.

1206 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×Dic₄₇	C₂	1	2	yes	maximal
2	#20245749	C₂	1	2	yes	maximal, complemented
3	#107×Dic₄₇	C₂	1	2	yes	maximal, complemented
4	C₄×#13093958	C₂	1	1	yes	maximal, characteristic, complemented
5	#20243215	C₂	1	4	yes	maximal, complemented
6	#20243213	C₂	1	2	yes	maximal, complemented
7	#20243074	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×D₄×Dic₄₇	C₂²	1	4	yes	—
10	D₈×Dic₄₇	C₄	1	16	yes	complemented
11	C₂×#13093958	C₂²	1	2	yes	—
12	C₂×#13093945	C₂²	1	4	yes	—
13	C₂×#13093935	C₂²	1	1	yes	characteristic
14	C₂×#13093934	C₂²	1	2	yes	—
15	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
16	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
17	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
18	C₂²⋊C₄×Dic₄₇	—	2	4	no	—
19	C₄×C₂²⋊Dic₄₇	—	2	2	no	—
20	#15688969	C₂²	1	1	yes	characteristic
21	C₂×C₄₇⋊#95	—	2	2	no	—
22	#15688886	C₂²	1	2	yes	—
23	#15688885	C₂²	1	4	yes	—
24	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₄₇	C₂²	1	2	yes	—
26	C₄×D₈×C₄₇	C₄	1	8	yes	complemented
27	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×#102×C₄₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂³×Dic₄₇	—	2	4	no	—
31	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
32	D₄×Dic₄₇	—	2	8	no	—
33	D₄×Dic₄₇	C₂×C₄	1	16	yes	complemented
34	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
35	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
36	C₂×C₄×Dic₄₇	—	2	2	no	—
37	C₂×C₄×Dic₄₇	—	2	4	no	—
38	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
39	#13093958	C₂×C₄	1	8	yes	complemented
40	#13093945	C₂×C₄	1	16	yes	complemented
41	#13093935	—	2	2	no	—
42	#13093934	—	2	4	no	—
43	C₈×Dic₄₇	C₂×C₄	1	8	yes	complemented
44	C₄₇⋊#95	—	2	2	no	—
45	C₄₇⋊#95	—	2	4	no	—
46	C₄₇⋊#95	D₄	1	1	yes	characteristic
47	C₄₇⋊#95	D₄	1	1	yes	characteristic
48	C₄₇⋊#95	D₄	1	2	yes	—
49	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
50	C₂³×C₄×C₄₇	—	2	2	no	—
51	C₂×D₈×C₄₇	C₂×C₄	1	8	yes	—
52	C₂×D₈×C₄₇	C₂×C₄	1	4	yes	—
53	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
55	C₄×D₄×C₄₇	—	2	4	no	—
56	C₄×D₄×C₄₇	C₂×C₄	1	8	yes	complemented
57	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
58	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	#107×C₄₇	C₂×C₄	1	4	yes	complemented
61	#102×C₄₇	C₂×C₄	1	8	yes	complemented
62	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
63	C₂⁴×C₄₇	—	2	2	no	—
64	C₂×D₄×C₄₇	—	2	2	no	—
65	C₂×D₄×C₄₇	—	2	4	no	—
66	C₂×D₄×C₄₇	C₂²×C₄	1	8	yes	—
67	C₂×D₄×C₄₇	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	—	2	8	no	—
70	C₂²×C₄×C₄₇	—	8	4	no	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₄₇	—	2	4	no	—
75	C₂²×Dic₄₇	—	2	16	no	—
76	C₂²×Dic₄₇	—	8	4	no	—
77	C₂²×Dic₄₇	—	8	4	no	—
78	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
79	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
80	D₈×C₄₇	C₄²	1	16	yes	complemented
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
85	C₄⋊C₄×C₄₇	—	2	2	no	—
86	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
87	C₂²⋊C₄×C₄₇	—	2	8	no	—
88	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
89	C₄²×C₄₇	#41	1	2	yes	—
90	C₂²⋊Dic₄₇	—	2	16	no	—
91	C₄⋊Dic₄₇	—	2	4	no	—
92	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
93	C₄×Dic₄₇	—	8	4	no	—
94	C₄×Dic₄₇	—	8	2	no	—
95	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
96	C₄×Dic₄₇	#41	1	4	yes	—
97	C₂×C₄₇⋊C₈	—	2	2	no	—
98	C₂×C₄₇⋊C₈	—	2	2	no	—
99	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
100	C₂³×C₄₇	—	2	8	no	—
101	C₂³×C₄₇	—	2	4	no	—
102	C₂³×C₄₇	—	4	2	no	—
103	C₂³×C₄₇	—	4	4	no	—
104	C₂³×C₄₇	—	4	2	no	—
105	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
106	D₄×C₄₇	—	2	8	no	—
107	D₄×C₄₇	—	2	4	no	—
108	D₄×C₄₇	C₂×C₄²	1	8	yes	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	4	4	no	—
114	C₂×C₄×C₄₇	—	8	8	no	—
115	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
120	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
121	C₂×Dic₄₇	—	2	2	no	—
122	C₂×Dic₄₇	—	2	2	no	—
123	C₂×Dic₄₇	—	2	2	no	—
124	C₂×Dic₄₇	—	2	2	no	—
125	C₂×Dic₄₇	—	2	8	no	—
126	C₂×Dic₄₇	—	4	4	no	—
127	C₂×Dic₄₇	—	4	4	no	—
128	C₂×Dic₄₇	—	8	16	no	—
129	C₂×Dic₄₇	C₄×D₄	1	4	yes	—
130	C₂×Dic₄₇	C₄×D₄	1	4	yes	—
131	C₈×C₄₇	—	2	2	no	—
132	C₈×C₄₇	C₂×C₄²	1	4	yes	—
133	C₄₇⋊C₈	—	2	4	no	—
134	#20508	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	—	2	8	no	—
136	C₂²×C₄₇	—	4	8	no	—
137	C₂²×C₄₇	—	4	4	no	—
138	C₂²×C₄₇	—	4	8	no	—
139	C₂²×C₄₇	—	4	4	no	—
140	C₂²×C₄₇	C₂×C₄×D₄	1	2	yes	—
141	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂²×C₄₇	#576	1	2	yes	—
143	C₂²×C₄₇	#576	1	1	yes	characteristic
144	C₂²×C₄₇	#578	1	1	yes	characteristic
145	C₄×C₄₇	—	2	2	no	—
146	C₄×C₄₇	—	2	2	no	—
147	C₄×C₄₇	—	4	4	no	—
148	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
149	C₄×C₄₇	C₄×#41	1	2	yes	—
150	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
151	Dic₄₇	—	2	4	no	—
152	Dic₄₇	—	2	4	no	—
153	Dic₄₇	—	4	8	no	—
154	C₂×C₄×D₈	—	47	2	no	—
155	C₂×C₄×D₈	D₄₇	1	1	yes	characteristic
156	#2275	—	47	2	no	—
157	C₄×#107	—	47	2	no	—
158	C₄×#107	—	47	1	no	—
159	#1762	—	47	4	no	—
160	#1762	—	47	2	no	—
161	#1726	—	47	1	no	—
162	C₂×C₄₇	—	4	8	no	—
163	C₂×C₄₇	C₄²×D₄	1	1	yes	characteristic
164	C₂×C₄₇	#2948	1	2	yes	—
165	C₂×C₄₇	#2948	1	1	yes	characteristic
166	C₂×C₄₇	#2951	1	2	yes	—
167	C₂×C₄₇	#2951	1	1	yes	characteristic
168	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
169	C₂×C₄×D₄	—	47	4	no	—
170	C₂×C₄×D₄	C₂×D₄₇	1	2	yes	—
171	C₄×D₈	—	47	16	no	—
172	C₄×D₈	Dic₄₇	1	8	yes	complemented
173	C₂×#107	—	47	2	no	—
174	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
175	C₂×#102	—	47	4	no	—
176	C₂×#102	C₂×D₄₇	1	2	yes	—
177	C₂×#97	—	47	2	no	—
178	C₂×#97	—	47	1	no	—
179	C₂×C₄×C₈	—	47	2	no	—
180	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
181	C₄×C₄⋊C₄	—	47	4	no	—
182	C₄×C₄⋊C₄	—	47	2	no	—
183	C₄×C₂²⋊C₄	—	94	4	no	—
184	C₄×C₂²⋊C₄	—	94	2	no	—
185	#376	—	47	1	no	—
186	C₂×#95	—	94	2	no	—
187	#340	—	47	2	no	—
188	#340	—	47	4	no	—
189	C₄₇	#20508	1	1	yes	characteristic, complemented, Sylow
190	C₂²×D₄	C₂²×D₄₇	1	2	yes	—
191	C₂³×C₄	—	2	2	no	—
192	C₂³×C₄	—	94	4	no	—
193	C₂×D₈	C₄×D₄₇	1	8	yes	—
194	C₂×D₈	C₂×Dic₄₇	1	4	yes	—
195	C₂²×C₈	—	47	2	no	—
196	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
197	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
198	C₄×D₄	—	2	4	no	—
199	C₄×D₄	—	47	16	no	—
200	C₄×D₄	—	94	8	no	—
201	C₄×D₄	C₂×Dic₄₇	1	8	yes	complemented
202	C₂×C₄⋊C₄	—	47	4	no	—
203	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₂²⋊C₄	—	94	4	no	—
206	C₂×C₄²	—	47	2	no	—
207	C₂×C₄²	—	94	4	no	—
208	C₂×C₄²	—	94	2	no	—
209	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
210	#107	—	47	8	no	—
211	#107	C₂×Dic₄₇	1	4	yes	complemented
212	#102	—	47	16	no	—
213	#102	C₂×Dic₄₇	1	8	yes	complemented
214	#97	—	94	4	no	—
215	#97	—	94	2	no	—
216	C₄×C₈	—	47	8	no	—
217	C₄×C₈	C₂×Dic₄₇	1	4	yes	complemented
218	#95	—	47	1	no	—
219	#95	—	47	2	no	—
220	#95	—	47	1	no	—
221	#95	—	94	4	no	—
222	#95	—	94	2	no	—
223	C₂⁴	—	2	2	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	C₂×C₄×D₄₇	1	8	yes	—
227	C₂×D₄	C₂²×Dic₄₇	1	4	yes	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	8	no	—
230	C₂²×C₄	—	8	4	no	—
231	C₂²×C₄	—	47	2	no	—
232	C₂²×C₄	—	47	2	no	—
233	C₂²×C₄	—	94	16	no	—
234	C₂²×C₄	—	94	4	no	—
235	C₂²×C₄	—	376	4	no	—
236	C₂²×C₄	—	376	4	no	—
237	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
238	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
240	D₈	C₄×Dic₄₇	1	16	yes	complemented
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	47	4	no	—
243	C₂×C₈	—	94	2	no	—
244	C₂×C₈	—	94	2	no	—
245	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
246	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
247	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	47	8	no	—
250	C₄⋊C₄	—	94	4	no	—
251	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
252	C₂²⋊C₄	—	2	8	no	—
253	C₂²⋊C₄	—	94	16	no	—
254	C₄²	—	47	4	no	—
255	C₄²	—	47	4	no	—
256	C₄²	—	376	2	no	—
257	C₄²	—	376	4	no	—
258	C₄²	#10625352	1	2	yes	—
259	C₄²	C₂²×Dic₄₇	1	2	yes	—
260	C₂³	—	2	8	no	—
261	C₂³	—	2	4	no	—
262	C₂³	—	4	4	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	4	2	no	—
265	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
266	D₄	—	2	8	no	—
267	D₄	—	2	4	no	—
268	D₄	C₂×C₄×Dic₄₇	1	8	yes	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	8	8	no	—
275	C₂×C₄	—	47	4	no	—
276	C₂×C₄	—	47	4	no	—
277	C₂×C₄	—	94	2	no	—
278	C₂×C₄	—	94	2	no	—
279	C₂×C₄	—	94	2	no	—
280	C₂×C₄	—	94	2	no	—
281	C₂×C₄	—	94	8	no	—
282	C₂×C₄	—	376	16	no	—
283	C₂×C₄	—	188	4	no	—
284	C₂×C₄	—	188	4	no	—
285	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
286	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
287	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
288	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
289	C₂×C₄	#41×D₄₇	1	2	yes	—
290	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	94	4	no	—
293	C₈	C₂×C₄×Dic₄₇	1	4	yes	—
294	C₂²	—	2	8	no	—
295	C₂²	—	4	8	no	—
296	C₂²	—	4	4	no	—
297	C₂²	—	4	8	no	—
298	C₂²	—	4	4	no	—
299	C₂²	C₄×D₄×D₄₇	1	2	yes	—
300	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
301	C₂²	#15689924	1	2	yes	—
302	C₂²	#15689925	1	1	yes	characteristic
303	C₂²	#15689937	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	4	4	no	—
307	C₄	—	94	4	no	—
308	C₄	—	94	4	no	—
309	C₄	—	188	8	no	—
310	C₄	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
311	C₄	C₄×#10625352	1	2	yes	—
312	C₄	#41×Dic₄₇	1	1	yes	characteristic
313	C₂	—	4	8	no	—
314	C₂	C₄×D₄×Dic₄₇	1	1	yes	characteristic
315	C₂	#20252235	1	2	yes	—
316	C₂	#20252240	1	1	yes	characteristic
317	C₂	#20252258	1	2	yes	—
318	C₂	#20252264	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄×D₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₈×C₄₇	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×D₈	D₄₇
hypocenter	C₄₇	#20508
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. #20245749	C₂	semidirect
3. #107×Dic₄₇	C₂	semidirect
4. C₄×#13093958	C₂	semidirect
5. #20243215	C₂	semidirect
6. #20243213	C₂	semidirect
7. #20243074	C₂	semidirect
10. D₈×Dic₄₇	C₄	semidirect
26. C₄×D₈×C₄₇	C₄	semidirect
33. D₄×Dic₄₇	C₂×C₄	semidirect
39. #13093958	C₂×C₄	semidirect
40. #13093945	C₂×C₄	semidirect
43. C₈×Dic₄₇	C₂×C₄	semidirect
56. C₄×D₄×C₄₇	C₂×C₄	semidirect
60. #107×C₄₇	C₂×C₄	semidirect
61. #102×C₄₇	C₂×C₄	semidirect
62. C₄×C₈×C₄₇	C₂×C₄	semidirect
80. D₈×C₄₇	C₄²	semidirect
172. C₄×D₈	Dic₄₇	semidirect
189. C₄₇	#20508	semidirect
201. C₄×D₄	C₂×Dic₄₇	semidirect
211. #107	C₂×Dic₄₇	semidirect
213. #102	C₂×Dic₄₇	semidirect
217. C₄×C₈	C₂×Dic₄₇	semidirect
240. D₈	C₄×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₈×Dic₄₇	2
#20245749	2
#107×Dic₄₇	2
C₄×#13093958	2
#20243215	2
#20243213	2
#20243074	2
C₂×C₄×D₈×C₄₇	2
#20508	47