Subgroups — #20247182

Order 6016.

512 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689059	C₂	1	2	yes	maximal, complemented
3	#15689048	C₂	1	2	yes	maximal, complemented
4	#15689030	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689026	C₂	1	1	yes	maximal, characteristic
6	#15689026	C₂	1	1	yes	maximal, characteristic
7	#15689021	C₂	1	1	yes	maximal, characteristic, complemented
8	#15689011	C₂	1	2	yes	maximal, complemented
9	#15689011	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₄₇⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#124×C₄₇	C₂	1	1	yes	maximal, characteristic
12	Q₈×Dic₄₇	—	2	4	no	—
13	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
16	C₂×Dic₄₇⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₇⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
19	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
20	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
23	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
26	C₄₇⋊#95	C₂²	1	1	yes	characteristic
27	C₄₇⋊#95	C₂²	1	1	yes	characteristic
28	C₄₇⋊#95	C₂²	1	2	yes	—
29	C₄₇⋊#95	C₂²	1	2	yes	—
30	C₄₇⋊#95	C₂²	1	2	yes	—
31	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
33	#124×C₄₇	—	2	4	no	—
34	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
35	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
36	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
37	C₂⁴×C₄₇	C₂³	1	2	yes	—
38	C₂×Q₈×C₄₇	—	2	2	no	—
39	C₂×Q₈×C₄₇	—	2	2	no	—
40	C₂×Q₈×C₄₇	D₄	1	4	yes	—
41	C₂×D₄×C₄₇	—	2	2	no	—
42	C₂×D₄×C₄₇	—	2	2	no	—
43	C₂×D₄×C₄₇	—	2	2	no	—
44	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
48	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
49	C₂²×Dic₄₇	C₂³	1	2	yes	—
50	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
51	C₂²×Dic₄₇	C₂³	1	2	yes	—
52	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
53	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
54	C₂²⋊C₄×C₄₇	—	2	4	no	—
55	C₂²⋊C₄×C₄₇	—	2	4	no	—
56	C₄²×C₄₇	—	2	2	no	—
57	C₂²⋊Dic₄₇	—	2	4	no	—
58	C₂²⋊Dic₄₇	—	2	4	no	—
59	C₂²⋊Dic₄₇	—	4	2	no	—
60	C₂²⋊Dic₄₇	—	4	2	no	—
61	C₄⋊Dic₄₇	—	2	4	no	—
62	C₄⋊Dic₄₇	—	2	2	no	—
63	C₄⋊Dic₄₇	—	2	2	no	—
64	Dic₄₇⋊C₄	—	2	2	no	—
65	Dic₄₇⋊C₄	—	2	2	no	—
66	Dic₄₇⋊C₄	—	4	2	no	—
67	C₄×Dic₄₇	—	2	4	no	—
68	C₄×Dic₄₇	—	2	2	no	—
69	C₄×Dic₄₇	—	2	2	no	—
70	C₂³×C₄₇	—	2	2	no	—
71	C₂³×C₄₇	—	2	2	no	—
72	C₂³×C₄₇	—	2	2	no	—
73	C₂³×C₄₇	—	2	2	no	—
74	C₂³×C₄₇	—	2	2	no	—
75	C₂³×C₄₇	—	2	2	no	—
76	C₂³×C₄₇	—	2	2	no	—
77	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
78	Q₈×C₄₇	—	2	8	no	—
79	D₄×C₄₇	—	4	4	no	—
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	2	2	no	—
82	C₂×C₄×C₄₇	—	2	1	no	—
83	C₂×C₄×C₄₇	—	2	1	no	—
84	C₂×C₄×C₄₇	—	2	1	no	—
85	C₂×C₄×C₄₇	—	2	1	no	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	2	1	no	—
88	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
89	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₄₇	#41	1	2	yes	—
91	C₂×C₄×C₄₇	#41	1	2	yes	—
92	C₂×Dic₄₇	—	2	1	no	—
93	C₂×Dic₄₇	—	2	1	no	—
94	C₂×Dic₄₇	—	2	1	no	—
95	C₂×Dic₄₇	—	2	1	no	—
96	C₂×Dic₄₇	—	2	1	no	—
97	C₂×Dic₄₇	—	2	2	no	—
98	C₂×Dic₄₇	—	2	1	no	—
99	C₂×Dic₄₇	—	2	1	no	—
100	C₂×Dic₄₇	—	2	2	no	—
101	C₂×Dic₄₇	—	2	2	no	—
102	C₂×Dic₄₇	—	2	2	no	—
103	C₂×Dic₄₇	—	2	2	no	—
104	C₂×Dic₄₇	—	2	2	no	—
105	C₂×Dic₄₇	—	2	1	no	—
106	C₂×Dic₄₇	—	2	1	no	—
107	C₂×Dic₄₇	—	2	1	no	—
108	C₂×Dic₄₇	#41	1	2	yes	—
109	C₂×Dic₄₇	#41	1	2	yes	—
110	C₂²×C₄₇	—	4	2	no	—
111	C₂²×C₄₇	—	4	2	no	—
112	C₂²×C₄₇	—	4	2	no	—
113	C₂²×C₄₇	—	4	2	no	—
114	C₂²×C₄₇	—	4	2	no	—
115	C₂²×C₄₇	—	4	2	no	—
116	C₂²×C₄₇	—	4	2	no	—
117	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
119	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
120	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
121	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
122	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
124	C₄×C₄₇	—	2	4	no	—
125	C₄×C₄₇	—	4	1	no	—
126	C₄×C₄₇	—	4	1	no	—
127	C₄×C₄₇	#123	1	4	yes	—
128	Dic₄₇	—	2	2	no	—
129	Dic₄₇	—	2	2	no	—
130	Dic₄₇	—	4	1	no	—
131	Dic₄₇	—	4	1	no	—
132	Dic₄₇	—	4	2	no	—
133	Dic₄₇	—	4	2	no	—
134	#2448	—	47	1	no	maximal, Sylow
135	C₂×C₄₇	—	8	2	no	—
136	C₂×C₄₇	C₂×#123	1	1	yes	characteristic
137	C₂×C₄₇	#529	1	1	yes	characteristic
138	C₂×C₄₇	#542	1	1	yes	characteristic
139	C₂×C₄₇	#548	1	1	yes	characteristic
140	C₂×C₄₇	#550	1	1	yes	characteristic
141	C₂×C₄₇	#556	1	1	yes	characteristic
142	C₂×C₄₇	#559	1	1	yes	characteristic
143	C₂×#124	D₄₇	1	1	yes	characteristic
144	C₂×C₄×Q₈	—	47	1	no	—
145	#399	—	47	2	no	—
146	#397	—	47	2	no	—
147	#391	—	47	1	no	—
148	#390	—	47	1	no	—
149	#390	—	47	1	no	—
150	#389	—	47	1	no	—
151	#388	—	47	2	no	—
152	#388	—	47	2	no	—
153	C₄×C₄⋊C₄	—	47	1	no	—
154	C₄₇	#2448	1	1	yes	characteristic, complemented, Sylow
155	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
156	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
157	#124	—	2	4	no	—
158	C₄×Q₈	—	94	4	no	—
159	C₂×C₄⋊C₄	—	47	1	no	—
160	C₂×C₄⋊C₄	—	47	2	no	—
161	C₂×C₄⋊C₄	—	47	1	no	—
162	C₂×C₄⋊C₄	—	47	1	no	—
163	C₂×C₄⋊C₄	—	47	1	no	—
164	C₂×C₄⋊C₄	—	47	2	no	—
165	C₂×C₂²⋊C₄	—	47	2	no	—
166	C₂×C₂²⋊C₄	—	47	2	no	—
167	C₂×C₂²⋊C₄	—	47	2	no	—
168	C₂×C₂²⋊C₄	—	47	2	no	—
169	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
170	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
171	C₂×C₄²	—	47	1	no	—
172	C₂×C₄²	—	47	2	no	—
173	C₂×C₄²	—	47	1	no	—
174	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
175	#95	—	47	1	no	—
176	#95	—	47	1	no	—
177	#95	—	47	2	no	—
178	#95	—	47	2	no	—
179	#95	—	47	2	no	—
180	C₂⁴	C₂²×D₄₇	1	2	yes	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	C₂²⋊D₄₇	1	4	yes	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂²×C₄	—	47	1	no	—
188	C₂²×C₄	—	47	1	no	—
189	C₂²×C₄	—	47	1	no	—
190	C₂²×C₄	—	47	2	no	—
191	C₂²×C₄	—	47	2	no	—
192	C₂²×C₄	—	47	1	no	—
193	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
194	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
195	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
196	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
197	C₄⋊C₄	—	94	2	no	—
198	C₄⋊C₄	—	94	2	no	—
199	C₄⋊C₄	—	94	2	no	—
200	C₄⋊C₄	—	94	4	no	—
201	C₄⋊C₄	—	94	2	no	—
202	C₄⋊C₄	—	188	2	no	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	94	4	no	—
206	C₂²⋊C₄	—	94	4	no	—
207	C₂²⋊C₄	—	188	2	no	—
208	C₂²⋊C₄	—	188	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	94	2	no	—
211	C₄²	—	94	4	no	—
212	C₄²	—	94	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	C₂³×D₄₇	1	1	yes	characteristic
221	Q₈	—	2	8	no	—
222	D₄	—	4	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	47	2	no	—
232	C₂×C₄	—	47	2	no	—
233	C₂×C₄	—	94	1	no	—
234	C₂×C₄	—	94	1	no	—
235	C₂×C₄	—	94	1	no	—
236	C₂×C₄	—	94	1	no	—
237	C₂×C₄	—	94	2	no	—
238	C₂×C₄	—	94	1	no	—
239	C₂×C₄	—	94	2	no	—
240	C₂×C₄	—	94	2	no	—
241	C₂×C₄	—	94	1	no	—
242	C₂×C₄	—	94	1	no	—
243	C₂×C₄	—	94	2	no	—
244	C₂×C₄	—	94	1	no	—
245	C₂×C₄	—	94	1	no	—
246	C₂×C₄	—	94	2	no	—
247	C₂×C₄	—	94	2	no	—
248	C₂×C₄	—	94	1	no	—
249	C₂×C₄	#10625352	1	2	yes	—
250	C₂×C₄	#10625352	1	2	yes	—
251	C₂×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
252	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
261	C₂²	#41×D₄₇	1	1	yes	characteristic
262	C₂²	#41×D₄₇	1	1	yes	characteristic
263	C₂²	C₂×#10625352	1	1	yes	characteristic
264	C₂²	C₂×#10625352	1	1	yes	characteristic
265	C₂²	#13094070	1	1	yes	characteristic
266	C₂²	#13094071	1	1	yes	characteristic
267	C₄	—	2	4	no	—
268	C₄	—	4	1	no	—
269	C₄	—	4	1	no	—
270	C₄	—	94	2	no	—
271	C₄	—	94	2	no	—
272	C₄	—	188	1	no	—
273	C₄	—	188	2	no	—
274	C₄	—	188	1	no	—
275	C₄	—	188	2	no	—
276	C₄	#13094005	1	4	yes	—
277	C₂	—	8	2	no	—
278	C₂	C₂×#13094005	1	1	yes	characteristic
279	C₂	#15689652	1	1	yes	characteristic
280	C₂	#15689742	1	1	yes	characteristic
281	C₂	#15689780	1	1	yes	characteristic
282	C₂	#15689792	1	1	yes	characteristic
283	C₂	#15689842	1	1	yes	characteristic
284	C₂	#15689855	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×#124×C₄₇	C₂
hypercenter	C₂×#124	D₄₇
hypocenter	C₄₇	#2448
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#124

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×Dic₄₇	C₂	semidirect
2. #15689059	C₂	semidirect
3. #15689048	C₂	semidirect
4. #15689030	C₂	semidirect
7. #15689021	C₂	semidirect
8. #15689011	C₂	semidirect
9. #15689011	C₂	semidirect
10. C₄×Dic₄₇⋊C₄	C₂	semidirect
154. C₄₇	#2448	semidirect