Subgroups — C₂×#15689748 #20256782

Order 6016.

1296 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689748	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13094004	C₂	1	2	yes	maximal, complemented
3	C₂×#13094003	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13094002	C₂	1	2	yes	maximal, complemented
5	C₂×#13094002	C₂	1	2	yes	maximal, complemented
6	C₂×#13094001	C₂	1	2	yes	maximal, complemented
7	C₂×#13094000	C₂	1	2	yes	maximal, complemented
8	C₂²×(C₂×D₄₇)⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂×C₂²⋊C₄×D₄₇	C₂	1	2	yes	maximal
10	C₂×C₂²⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×D₄₇	C₂²	1	1	yes	characteristic
12	C₂²×Dic₉₄	C₂²	1	2	yes	complemented
13	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic, complemented
14	C₂³×Dic₄₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₄₇	C₂²	1	2	yes	—
16	C₂²×C₄×D₄₇	C₂²	1	2	yes	—
17	#13094004	C₂²	1	16	yes	complemented
18	#13094003	C₂²	1	8	yes	complemented
19	#13094002	C₂²	1	16	yes	complemented
20	#13094002	C₂²	1	16	yes	complemented
21	#13094001	C₂²	1	16	yes	complemented
22	#13094000	C₂²	1	16	yes	complemented
23	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	complemented
24	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic, complemented
25	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	complemented
26	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	complemented
27	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	complemented
28	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
29	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	8	yes	—
30	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
31	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
32	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
33	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
34	C₂²⋊C₄×D₄₇	C₂²	1	16	yes	—
35	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic, complemented
36	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
37	C₂²⋊Q₈×C₄₇	C₂²	1	8	yes	complemented
38	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic, complemented
39	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	complemented
40	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
41	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
42	C₂³×D₄₇	—	4	2	no	—
43	C₂³×D₄₇	—	4	2	no	—
44	C₂³×D₄₇	Q₈	1	8	yes	complemented
45	C₂³×D₄₇	C₂³	1	2	yes	—
46	C₂³×D₄₇	C₂³	1	4	yes	—
47	C₂×Q₈×C₄₇	—	4	2	no	—
48	C₂×Q₈×C₄₇	C₂³	1	4	yes	complemented
49	C₂²×C₄×C₄₇	—	4	2	no	—
50	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
51	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
55	C₂²×Dic₄₇	—	4	2	no	—
56	C₂²×Dic₄₇	C₂³	1	2	yes	—
57	C₂²×Dic₄₇	C₂³	1	2	yes	—
58	C₂²×Dic₄₇	C₂³	1	2	yes	—
59	C₂²×Dic₄₇	C₂³	1	4	yes	—
60	C₂×C₄×D₄₇	—	4	4	no	—
61	C₂×C₄×D₄₇	—	4	4	no	—
62	C₂×C₄×D₄₇	C₂³	1	8	yes	—
63	C₂×C₄×D₄₇	C₂³	1	8	yes	—
64	C₂×Dic₉₄	—	4	2	no	—
65	C₂×Dic₉₄	—	4	4	no	—
66	C₂×Dic₉₄	C₂³	1	8	yes	complemented
67	C₂×Dic₉₄	C₂³	1	4	yes	complemented
68	C₄⋊C₄×C₄₇	C₂³	1	4	yes	complemented
69	C₄⋊C₄×C₄₇	C₂³	1	8	yes	complemented
70	C₂²⋊C₄×C₄₇	C₂³	1	8	yes	—
71	C₂²⋊Dic₄₇	C₂³	1	8	yes	—
72	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
73	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
74	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
75	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
76	C₄⋊Dic₄₇	C₂³	1	8	yes	complemented
77	C₄⋊Dic₄₇	C₂³	1	4	yes	complemented
78	Dic₄₇⋊C₄	C₂³	1	8	yes	complemented
79	Dic₄₇⋊C₄	C₂³	1	8	yes	complemented
80	Dic₄₇⋊C₄	C₂³	1	8	yes	complemented
81	C₂³×C₄₇	—	2	2	no	—
82	C₂³×C₄₇	—	2	2	no	—
83	C₂³×C₄₇	C₂×Q₈	1	4	yes	—
84	C₂³×C₄₇	C₂⁴	1	2	yes	—
85	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
86	C₂²×D₄₇	—	2	4	no	—
87	C₂²×D₄₇	—	2	4	no	—
88	C₂²×D₄₇	—	4	16	no	—
89	C₂²×D₄₇	—	4	4	no	—
90	C₂²×D₄₇	—	4	4	no	—
91	C₂²×D₄₇	C₂×Q₈	1	8	yes	—
92	C₂²×D₄₇	C₂×Q₈	1	16	yes	complemented
93	C₂²×D₄₇	C₂⁴	1	4	yes	—
94	Q₈×C₄₇	—	4	4	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	4	no	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	4	4	no	—
100	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
101	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
102	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
103	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
104	C₂×Dic₄₇	—	2	4	no	—
105	C₂×Dic₄₇	—	2	4	no	—
106	C₂×Dic₄₇	—	2	4	no	—
107	C₂×Dic₄₇	—	4	4	no	—
108	C₂×Dic₄₇	C₂⁴	1	4	yes	—
109	C₂×Dic₄₇	C₂⁴	1	4	yes	—
110	C₂×Dic₄₇	C₂⁴	1	4	yes	—
111	C₄×D₄₇	—	4	8	no	—
112	C₄×D₄₇	—	4	8	no	—
113	Dic₉₄	—	4	4	no	—
114	Dic₉₄	—	4	8	no	—
115	C₂²×C₄₇	—	2	8	no	—
116	C₂²×C₄₇	—	2	2	no	—
117	C₂²×C₄₇	—	2	2	no	—
118	C₂²×C₄₇	C₂²×Q₈	1	4	yes	—
119	C₂²×C₄₇	C₂²×Q₈	1	2	yes	—
120	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
121	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
122	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
123	C₂×D₄₇	—	2	16	no	—
124	C₂×D₄₇	—	2	4	no	—
125	C₂×D₄₇	—	2	4	no	—
126	C₂×D₄₇	—	4	16	no	—
127	C₂×D₄₇	C₂²×Q₈	1	8	yes	—
128	C₄×C₄₇	—	2	4	no	—
129	C₄×C₄₇	—	2	2	no	—
130	C₄×C₄₇	—	2	4	no	—
131	C₄×C₄₇	—	2	2	no	—
132	Dic₄₇	—	2	4	no	—
133	Dic₄₇	—	2	4	no	—
134	Dic₄₇	—	2	4	no	—
135	C₂×#543	—	47	1	no	maximal, Sylow
136	C₂×C₄₇	—	2	4	no	—
137	C₂×C₄₇	#543	1	4	yes	complemented
138	C₂×C₄₇	C₂³×Q₈	1	1	yes	characteristic
139	C₂×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
140	C₂×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
141	D₄₇	—	2	8	no	—
142	#543	—	47	16	no	—
143	C₂×C₂²⋊Q₈	—	47	2	no	—
144	C₂×C₂²⋊Q₈	—	47	1	no	—
145	C₂×C₂²⋊Q₈	—	47	2	no	—
146	C₂×C₂²⋊Q₈	—	47	2	no	—
147	C₂×C₂²⋊Q₈	—	47	2	no	—
148	C₂×C₂²⋊Q₈	—	47	2	no	—
149	C₂×C₂²⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
150	C₂²×C₂²⋊C₄	—	47	1	no	—
151	C₂²×C₂²⋊C₄	—	47	2	no	—
152	C₄₇	C₂×#543	1	1	yes	characteristic, complemented, Sylow
153	C₂⁵	—	47	1	no	—
154	C₂²×Q₈	—	47	1	no	—
155	C₂²×Q₈	—	47	2	no	—
156	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic, complemented
157	C₂³×C₄	—	47	1	no	—
158	C₂³×C₄	—	47	2	no	—
159	C₂³×C₄	—	47	2	no	—
160	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
161	C₂²⋊Q₈	—	47	16	no	—
162	C₂²⋊Q₈	—	47	8	no	—
163	C₂²⋊Q₈	—	47	16	no	—
164	C₂²⋊Q₈	—	47	16	no	—
165	C₂²⋊Q₈	—	47	16	no	—
166	C₂²⋊Q₈	—	47	16	no	—
167	C₂²⋊Q₈	C₂×D₄₇	1	8	yes	complemented
168	C₂×C₄⋊C₄	—	47	2	no	—
169	C₂×C₄⋊C₄	—	47	2	no	—
170	C₂×C₄⋊C₄	—	47	2	no	—
171	C₂×C₄⋊C₄	—	47	1	no	—
172	C₂×C₄⋊C₄	—	47	2	no	—
173	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic, complemented
174	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	complemented
175	C₂×C₂²⋊C₄	—	47	2	no	—
176	C₂×C₂²⋊C₄	—	47	2	no	—
177	C₂×C₂²⋊C₄	—	47	2	no	—
178	C₂×C₂²⋊C₄	—	47	2	no	—
179	C₂×C₂²⋊C₄	—	47	2	no	—
180	C₂×C₂²⋊C₄	—	47	8	no	—
181	C₂×C₂²⋊C₄	—	47	16	no	—
182	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
183	C₂⁴	—	47	2	no	—
184	C₂⁴	—	47	8	no	—
185	C₂⁴	—	47	4	no	—
186	C₂⁴	—	188	2	no	—
187	C₂⁴	—	188	2	no	—
188	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	—	47	4	no	—
191	C₂×Q₈	—	47	8	no	—
192	C₂×Q₈	—	188	2	no	—
193	C₂×Q₈	—	188	4	no	—
194	C₂×Q₈	C₂²×D₄₇	1	4	yes	complemented
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	47	2	no	—
197	C₂²×C₄	—	47	2	no	—
198	C₂²×C₄	—	47	2	no	—
199	C₂²×C₄	—	47	4	no	—
200	C₂²×C₄	—	47	8	no	—
201	C₂²×C₄	—	47	8	no	—
202	C₂²×C₄	—	188	4	no	—
203	C₂²×C₄	—	188	4	no	—
204	C₂²×C₄	—	188	2	no	—
205	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
206	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
208	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
209	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
210	C₄⋊C₄	—	47	8	no	—
211	C₄⋊C₄	—	47	8	no	—
212	C₄⋊C₄	—	47	8	no	—
213	C₄⋊C₄	—	47	4	no	—
214	C₄⋊C₄	—	47	8	no	—
215	C₄⋊C₄	C₂²×D₄₇	1	4	yes	complemented
216	C₄⋊C₄	C₂²×D₄₇	1	8	yes	complemented
217	C₂²⋊C₄	—	47	8	no	—
218	C₂²⋊C₄	—	47	8	no	—
219	C₂²⋊C₄	—	47	8	no	—
220	C₂²⋊C₄	—	47	8	no	—
221	C₂²⋊C₄	—	47	8	no	—
222	C₂²⋊C₄	C₂²×D₄₇	1	8	yes	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	47	8	no	—
226	C₂³	—	47	4	no	—
227	C₂³	—	47	16	no	—
228	C₂³	—	94	4	no	—
229	C₂³	—	94	4	no	—
230	C₂³	—	188	16	no	—
231	C₂³	—	188	4	no	—
232	C₂³	—	188	4	no	—
233	C₂³	Q₈×D₄₇	1	4	yes	—
234	C₂³	C₂³×D₄₇	1	2	yes	—
235	C₂³	C₂³×D₄₇	1	1	yes	characteristic
236	Q₈	—	4	4	no	—
237	Q₈	—	188	4	no	—
238	Q₈	—	188	8	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	47	4	no	—
245	C₂×C₄	—	47	4	no	—
246	C₂×C₄	—	47	4	no	—
247	C₂×C₄	—	94	4	no	—
248	C₂×C₄	—	94	4	no	—
249	C₂×C₄	—	94	4	no	—
250	C₂×C₄	—	188	8	no	—
251	C₂×C₄	—	188	8	no	—
252	C₂×C₄	—	188	4	no	—
253	C₂×C₄	C₂³×D₄₇	1	2	yes	—
254	C₂×C₄	C₂³×D₄₇	1	2	yes	—
255	C₂×C₄	C₂³×D₄₇	1	4	yes	—
256	C₂×C₄	C₂³×D₄₇	1	4	yes	—
257	C₂²	—	2	8	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	47	8	no	—
261	C₂²	—	94	16	no	—
262	C₂²	—	94	4	no	—
263	C₂²	—	94	4	no	—
264	C₂²	—	188	16	no	—
265	C₂²	C₂×Q₈×D₄₇	1	4	yes	—
266	C₂²	C₂×Q₈×D₄₇	1	2	yes	—
267	C₂²	#13094069	1	2	yes	—
268	C₂²	#13094070	1	2	yes	—
269	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	2	4	no	—
273	C₄	—	2	4	no	—
274	C₄	—	94	4	no	—
275	C₄	—	94	4	no	—
276	C₄	—	94	4	no	—
277	C₂	—	2	4	no	—
278	C₂	—	94	8	no	—
279	C₂	#15689748	1	4	yes	direct product factor
280	C₂	C₂²×Q₈×D₄₇	1	1	yes	characteristic
281	C₂	C₂×#13094069	1	1	yes	characteristic
282	C₂	C₂×#13094070	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁵
Frattini	C₂²	C₂⁴×D₄₇
Fitting	C₂×C₂²⋊Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂×C₂²⋊Q₈	D₄₇
hypocenter	C₄₇	C₂×#543
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #15689748	C₂	direct (N × H)
2. C₂×#13094004	C₂	semidirect
3. C₂×#13094003	C₂	semidirect
4. C₂×#13094002	C₂	semidirect
5. C₂×#13094002	C₂	semidirect
6. C₂×#13094001	C₂	semidirect
7. C₂×#13094000	C₂	semidirect
10. C₂×C₂²⋊Q₈×C₄₇	C₂	semidirect
12. C₂²×Dic₉₄	C₂²	semidirect
13. C₂²×Dic₉₄	C₂²	semidirect
17. #13094004	C₂²	semidirect
18. #13094003	C₂²	semidirect
19. #13094002	C₂²	semidirect
20. #13094002	C₂²	semidirect
21. #13094001	C₂²	semidirect
22. #13094000	C₂²	semidirect
23. C₂×C₄⋊Dic₄₇	C₂²	semidirect
24. C₂×C₄⋊Dic₄₇	C₂²	semidirect
25. C₂×Dic₄₇⋊C₄	C₂²	semidirect
26. C₂×Dic₄₇⋊C₄	C₂²	semidirect
27. C₂×Dic₄₇⋊C₄	C₂²	semidirect
35. C₂²×Q₈×C₄₇	C₂²	semidirect
37. C₂²⋊Q₈×C₄₇	C₂²	semidirect
38. C₂×C₄⋊C₄×C₄₇	C₂²	semidirect
39. C₂×C₄⋊C₄×C₄₇	C₂²	semidirect
44. C₂³×D₄₇	Q₈	semidirect
48. C₂×Q₈×C₄₇	C₂³	semidirect
66. C₂×Dic₉₄	C₂³	semidirect
67. C₂×Dic₉₄	C₂³	semidirect
68. C₄⋊C₄×C₄₇	C₂³	semidirect
69. C₄⋊C₄×C₄₇	C₂³	semidirect
76. C₄⋊Dic₄₇	C₂³	semidirect
77. C₄⋊Dic₄₇	C₂³	semidirect
78. Dic₄₇⋊C₄	C₂³	semidirect
79. Dic₄₇⋊C₄	C₂³	semidirect
80. Dic₄₇⋊C₄	C₂³	semidirect
92. C₂²×D₄₇	C₂×Q₈	semidirect
137. C₂×C₄₇	#543	semidirect
149. C₂×C₂²⋊Q₈	D₄₇	semidirect
152. C₄₇	C₂×#543	semidirect
156. C₂²×Q₈	C₂×D₄₇	semidirect
167. C₂²⋊Q₈	C₂×D₄₇	semidirect
173. C₂×C₄⋊C₄	C₂×D₄₇	semidirect
174. C₂×C₄⋊C₄	C₂×D₄₇	semidirect
194. C₂×Q₈	C₂²×D₄₇	semidirect
215. C₄⋊C₄	C₂²×D₄₇	semidirect
216. C₄⋊C₄	C₂²×D₄₇	semidirect
279. C₂	#15689748	direct (N × H)

Subgroups — C2×#15689748 #20256782

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689748 #20256782