Subgroups — C₂×#15689596 #20256573

Order 6016.

1920 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂²⋊D₄₇	C₂	1	1	yes	maximal, characteristic
2	#15689596	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×C₂²⋊D₄₇	C₂	1	4	yes	maximal, complemented
4	C₂×(C₂²⋊D₄₇)⋊C₄	C₂	1	4	yes	maximal, complemented
5	C₂×#13093980	C₂	1	2	yes	maximal, complemented
6	C₂²×C₂²⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂²⋊C₄×D₄₇	C₂	1	2	yes	maximal, complemented
8	C₂²×C₂²⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×D₄₇	C₂²	1	1	yes	characteristic
10	C₂²×C₂²⋊D₄₇	C₄	1	16	yes	complemented
11	C₂²×C₂²⋊D₄₇	C₂²	1	2	yes	—
12	C₂²×C₂²⋊D₄₇	C₂²	1	2	yes	—
13	C₂²×C₂²⋊D₄₇	C₂²	1	8	yes	—
14	C₂³×Dic₄₇	C₂²	1	1	yes	characteristic
15	C₂³×Dic₄₇	C₂²	1	2	yes	complemented
16	C₂²×C₄×D₄₇	C₂²	1	4	yes	complemented
17	C₄×C₂²⋊D₄₇	C₂²	1	32	yes	complemented
18	(C₂²⋊D₄₇)⋊C₄	C₂²	1	32	yes	complemented
19	#13093980	C₂²	1	16	yes	complemented
20	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	complemented
21	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₄₇	C₂²	1	8	yes	complemented
23	C₂×C₂²⋊Dic₄₇	C₂²	1	2	yes	complemented
24	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	complemented
25	C₂²⋊C₄×D₄₇	C₂²	1	16	yes	complemented
26	C₂×C₄×Dic₄₇	C₂²	1	4	yes	complemented
27	C₂⁵×C₄₇	C₂²	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	C₂²	1	2	yes	complemented
29	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₄×C₄₇	C₂²	1	8	yes	complemented
31	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	complemented
32	C₂⁴×C₄₇	—	4	4	no	—
33	C₂⁴×C₄₇	C₂×C₄	1	8	yes	complemented
34	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
35	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
36	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
37	C₂⁴×C₄₇	C₂³	1	4	yes	—
38	C₂³×D₄₇	—	4	4	no	—
39	C₂³×D₄₇	C₂×C₄	1	8	yes	complemented
40	C₂³×D₄₇	C₂³	1	2	yes	—
41	C₂³×D₄₇	C₂³	1	4	yes	—
42	C₂²×C₄×C₄₇	—	4	4	no	—
43	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
44	C₂²×C₄×C₄₇	C₂³	1	8	yes	complemented
45	C₂×C₂²⋊D₄₇	C₂×C₄	1	16	yes	complemented
46	C₂×C₂²⋊D₄₇	C₂×C₄	1	16	yes	complemented
47	C₂×C₂²⋊D₄₇	C₂×C₄	1	64	yes	complemented
48	C₂×C₂²⋊D₄₇	C₂³	1	8	yes	—
49	C₂×C₂²⋊D₄₇	C₂³	1	8	yes	—
50	C₂²×Dic₄₇	—	4	4	no	—
51	C₂²×Dic₄₇	C₂×C₄	1	8	yes	complemented
52	C₂²×Dic₄₇	C₂³	1	4	yes	—
53	C₂²×Dic₄₇	C₂³	1	2	yes	—
54	C₂²×Dic₄₇	C₂³	1	4	yes	—
55	C₂²×Dic₄₇	C₂³	1	8	yes	complemented
56	C₂×C₄×D₄₇	—	4	8	no	—
57	C₂×C₄×D₄₇	C₂³	1	16	yes	complemented
58	C₂²⋊C₄×C₄₇	C₂³	1	8	yes	—
59	C₂²⋊C₄×C₄₇	C₂³	1	8	yes	complemented
60	C₂²⋊Dic₄₇	C₂³	1	8	yes	—
61	C₂²⋊Dic₄₇	C₂³	1	8	yes	complemented
62	(C₂×D₄₇)⋊C₄	C₂³	1	16	yes	complemented
63	Dic₄₇⋊C₄	C₂³	1	16	yes	complemented
64	C₄×Dic₄₇	C₂³	1	16	yes	complemented
65	C₂³×C₄₇	—	2	4	no	—
66	C₂³×C₄₇	—	2	4	no	—
67	C₂³×C₄₇	—	2	4	no	—
68	C₂³×C₄₇	—	4	16	no	—
69	C₂³×C₄₇	—	4	8	no	—
70	C₂³×C₄₇	C₂²×C₄	1	4	yes	—
71	C₂³×C₄₇	C₂²×C₄	1	4	yes	—
72	C₂³×C₄₇	C₂²×C₄	1	4	yes	—
73	C₂³×C₄₇	C₂²×C₄	1	16	yes	complemented
74	C₂³×C₄₇	C₂⁴	1	2	yes	—
75	C₂³×C₄₇	C₂⁴	1	2	yes	—
76	C₂³×C₄₇	C₂⁴	1	2	yes	—
77	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
78	C₂²×D₄₇	—	2	8	no	—
79	C₂²×D₄₇	—	4	16	no	—
80	C₂²×D₄₇	—	4	8	no	—
81	C₂²×D₄₇	C₂²×C₄	1	8	yes	—
82	C₂²×D₄₇	C₂²×C₄	1	16	yes	complemented
83	C₂²×D₄₇	C₂⁴	1	4	yes	—
84	C₂×C₄×C₄₇	—	2	8	no	—
85	C₂×C₄×C₄₇	—	4	8	no	—
86	C₂×C₄×C₄₇	C₂⁴	1	8	yes	—
87	C₂²⋊D₄₇	C₂²×C₄	1	32	yes	complemented
88	C₂²⋊D₄₇	C₂²×C₄	1	32	yes	complemented
89	C₂×Dic₄₇	—	2	8	no	—
90	C₂×Dic₄₇	—	4	8	no	—
91	C₂×Dic₄₇	C₂²×C₄	1	8	yes	—
92	C₂×Dic₄₇	C₂²×C₄	1	16	yes	complemented
93	C₂×Dic₄₇	C₂⁴	1	8	yes	—
94	C₂×Dic₄₇	C₂⁴	1	4	yes	—
95	C₄×D₄₇	—	4	16	no	—
96	C₂²×C₄₇	—	2	8	no	—
97	C₂²×C₄₇	—	2	8	no	—
98	C₂²×C₄₇	—	2	8	no	—
99	C₂²×C₄₇	—	2	4	no	—
100	C₂²×C₄₇	—	2	4	no	—
101	C₂²×C₄₇	—	2	4	no	—
102	C₂²×C₄₇	—	4	16	no	—
103	C₂²×C₄₇	C₂³×C₄	1	4	yes	—
104	C₂²×C₄₇	C₂³×C₄	1	4	yes	—
105	C₂²×C₄₇	C₂³×C₄	1	4	yes	—
106	C₂²×C₄₇	C₂³×C₄	1	2	yes	—
107	C₂²×C₄₇	H(5,𝔽₂)	1	4	yes	—
108	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
109	C₂×D₄₇	—	2	16	no	—
110	C₂×D₄₇	—	2	8	no	—
111	C₂×D₄₇	—	4	16	no	—
112	C₂×D₄₇	C₂³×C₄	1	8	yes	—
113	C₄×C₄₇	—	2	8	no	—
114	Dic₄₇	—	2	8	no	—
115	Dic₄₇	C₂³×C₄	1	8	yes	—
116	C₂×#518	—	47	1	no	maximal, Sylow
117	C₂×C₄₇	—	2	4	no	—
118	C₂×C₄₇	—	2	4	no	—
119	C₂×C₄₇	—	2	4	no	—
120	C₂×C₄₇	#518	1	4	yes	complemented
121	C₂×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
122	C₂×C₄₇	C₂×H(5,𝔽₂)	1	2	yes	—
123	D₄₇	—	2	8	no	—
124	C₂³×D₄	—	47	1	no	—
125	#518	—	47	16	no	—
126	C₂×C₄×D₄	—	47	4	no	—
127	C₂×C₄×D₄	—	47	4	no	—
128	C₂×#117	—	47	2	no	—
129	C₂²×C₂²⋊C₄	—	47	1	no	—
130	C₂²×C₂²⋊C₄	—	47	2	no	—
131	C₂²×C₂²⋊C₄	D₄₇	1	1	yes	characteristic, complemented
132	C₄₇	C₂×#518	1	1	yes	characteristic, complemented, Sylow
133	C₂⁵	—	47	1	no	—
134	C₂⁵	C₂×D₄₇	1	1	yes	characteristic
135	C₂²×D₄	—	47	16	no	—
136	C₂²×D₄	—	47	2	no	—
137	C₂²×D₄	—	47	2	no	—
138	C₂²×D₄	—	47	8	no	—
139	C₂³×C₄	—	47	1	no	—
140	C₂³×C₄	—	47	2	no	—
141	C₂³×C₄	—	47	4	no	—
142	C₂³×C₄	C₂×D₄₇	1	2	yes	complemented
143	C₄×D₄	—	47	32	no	—
144	C₄×D₄	—	47	32	no	—
145	#117	—	47	16	no	—
146	C₂×C₄⋊C₄	—	47	4	no	—
147	C₂×C₂²⋊C₄	—	47	2	no	—
148	C₂×C₂²⋊C₄	—	47	2	no	—
149	C₂×C₂²⋊C₄	—	47	8	no	—
150	C₂×C₂²⋊C₄	—	47	4	no	—
151	C₂×C₂²⋊C₄	—	47	16	no	—
152	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
153	C₂×C₂²⋊C₄	C₂×D₄₇	1	8	yes	complemented
154	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	complemented
155	C₂×C₄²	—	47	4	no	—
156	C₂⁴	—	4	4	no	—
157	C₂⁴	—	47	2	no	—
158	C₂⁴	—	47	8	no	—
159	C₂⁴	—	47	4	no	—
160	C₂⁴	—	188	4	no	—
161	C₂⁴	C₄×D₄₇	1	8	yes	complemented
162	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
163	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
164	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
165	C₂⁴	C₂²×D₄₇	1	4	yes	—
166	C₂×D₄	—	47	16	no	—
167	C₂×D₄	—	47	16	no	—
168	C₂×D₄	—	47	64	no	—
169	C₂×D₄	—	47	8	no	—
170	C₂×D₄	—	47	8	no	—
171	C₂²×C₄	—	4	4	no	—
172	C₂²×C₄	—	47	8	no	—
173	C₂²×C₄	—	47	4	no	—
174	C₂²×C₄	—	47	2	no	—
175	C₂²×C₄	—	47	4	no	—
176	C₂²×C₄	—	47	8	no	—
177	C₂²×C₄	—	47	16	no	—
178	C₂²×C₄	—	188	8	no	—
179	C₂²×C₄	—	188	4	no	—
180	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
181	C₂²×C₄	C₂²×D₄₇	1	8	yes	complemented
182	C₄⋊C₄	—	47	16	no	—
183	C₂²⋊C₄	—	47	8	no	—
184	C₂²⋊C₄	—	47	8	no	—
185	C₂²⋊C₄	—	47	16	no	—
186	C₂²⋊C₄	C₂²×D₄₇	1	8	yes	—
187	C₂²⋊C₄	C₂²×D₄₇	1	8	yes	complemented
188	C₄²	—	47	16	no	—
189	C₂³	—	2	4	no	—
190	C₂³	—	2	4	no	—
191	C₂³	—	2	4	no	—
192	C₂³	—	4	16	no	—
193	C₂³	—	4	8	no	—
194	C₂³	—	47	8	no	—
195	C₂³	—	47	4	no	—
196	C₂³	—	47	16	no	—
197	C₂³	—	94	8	no	—
198	C₂³	—	188	16	no	—
199	C₂³	—	188	8	no	—
200	C₂³	C₂×C₄×D₄₇	1	4	yes	—
201	C₂³	C₂×C₄×D₄₇	1	4	yes	—
202	C₂³	C₂×C₄×D₄₇	1	4	yes	—
203	C₂³	C₂×C₄×D₄₇	1	16	yes	complemented
204	C₂³	C₂³×D₄₇	1	2	yes	—
205	C₂³	C₂³×D₄₇	1	2	yes	—
206	C₂³	C₂³×D₄₇	1	2	yes	—
207	C₂³	C₂³×D₄₇	1	1	yes	characteristic
208	D₄	—	47	32	no	—
209	D₄	—	47	32	no	—
210	C₂×C₄	—	2	8	no	—
211	C₂×C₄	—	4	8	no	—
212	C₂×C₄	—	47	8	no	—
213	C₂×C₄	—	47	16	no	—
214	C₂×C₄	—	47	4	no	—
215	C₂×C₄	—	47	8	no	—
216	C₂×C₄	—	94	8	no	—
217	C₂×C₄	—	188	16	no	—
218	C₂×C₄	—	188	8	no	—
219	C₂×C₄	C₂³×D₄₇	1	8	yes	—
220	C₂²	—	2	8	no	—
221	C₂²	—	2	8	no	—
222	C₂²	—	2	8	no	—
223	C₂²	—	2	4	no	—
224	C₂²	—	2	4	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	4	16	no	—
227	C₂²	—	47	8	no	—
228	C₂²	—	94	16	no	—
229	C₂²	—	94	8	no	—
230	C₂²	—	188	16	no	—
231	C₂²	C₂²×C₄×D₄₇	1	4	yes	—
232	C₂²	C₂²×C₄×D₄₇	1	4	yes	—
233	C₂²	C₂²×C₄×D₄₇	1	4	yes	—
234	C₂²	C₂²×C₄×D₄₇	1	2	yes	—
235	C₂²	#13094069	1	4	yes	—
236	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
237	C₄	—	2	8	no	—
238	C₄	—	47	8	no	—
239	C₄	—	94	8	no	—
240	C₂	—	2	4	no	—
241	C₂	—	2	4	no	—
242	C₂	—	2	4	no	—
243	C₂	—	94	8	no	—
244	C₂	#15689596	1	4	yes	direct product factor
245	C₂	C₂³×C₄×D₄₇	1	1	yes	characteristic
246	C₂	C₂×#13094069	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₄₇
commutator	C₂×C₄₇	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₄₇
Fitting	C₂²×C₂²⋊C₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂²×C₂²⋊C₄	D₄₇
hypocenter	C₄₇	C₂×#518
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. #15689596	C₂	direct (N × H)
3. C₂×C₄×C₂²⋊D₄₇	C₂	semidirect
4. C₂×(C₂²⋊D₄₇)⋊C₄	C₂	semidirect
5. C₂×#13093980	C₂	semidirect
6. C₂²×C₂²⋊Dic₄₇	C₂	semidirect
7. C₂×C₂²⋊C₄×D₄₇	C₂	semidirect
8. C₂²×C₂²⋊C₄×C₄₇	C₂	semidirect
10. C₂²×C₂²⋊D₄₇	C₄	semidirect
15. C₂³×Dic₄₇	C₂²	semidirect
16. C₂²×C₄×D₄₇	C₂²	semidirect
17. C₄×C₂²⋊D₄₇	C₂²	semidirect
18. (C₂²⋊D₄₇)⋊C₄	C₂²	semidirect
19. #13093980	C₂²	semidirect
20. C₂×Dic₄₇⋊C₄	C₂²	semidirect
22. C₂×C₂²⋊Dic₄₇	C₂²	semidirect
23. C₂×C₂²⋊Dic₄₇	C₂²	semidirect
24. C₂×(C₂×D₄₇)⋊C₄	C₂²	semidirect
25. C₂²⋊C₄×D₄₇	C₂²	semidirect
26. C₂×C₄×Dic₄₇	C₂²	semidirect
28. C₂³×C₄×C₄₇	C₂²	semidirect
30. C₂×C₂²⋊C₄×C₄₇	C₂²	semidirect
31. C₂×C₂²⋊C₄×C₄₇	C₂²	semidirect
33. C₂⁴×C₄₇	C₂×C₄	semidirect
39. C₂³×D₄₇	C₂×C₄	semidirect
44. C₂²×C₄×C₄₇	C₂³	semidirect
45. C₂×C₂²⋊D₄₇	C₂×C₄	semidirect
46. C₂×C₂²⋊D₄₇	C₂×C₄	semidirect
47. C₂×C₂²⋊D₄₇	C₂×C₄	semidirect
51. C₂²×Dic₄₇	C₂×C₄	semidirect
55. C₂²×Dic₄₇	C₂³	semidirect
57. C₂×C₄×D₄₇	C₂³	semidirect
59. C₂²⋊C₄×C₄₇	C₂³	semidirect
61. C₂²⋊Dic₄₇	C₂³	semidirect
62. (C₂×D₄₇)⋊C₄	C₂³	semidirect
63. Dic₄₇⋊C₄	C₂³	semidirect
64. C₄×Dic₄₇	C₂³	semidirect
73. C₂³×C₄₇	C₂²×C₄	semidirect
82. C₂²×D₄₇	C₂²×C₄	semidirect
87. C₂²⋊D₄₇	C₂²×C₄	semidirect
88. C₂²⋊D₄₇	C₂²×C₄	semidirect
92. C₂×Dic₄₇	C₂²×C₄	semidirect
120. C₂×C₄₇	#518	semidirect
131. C₂²×C₂²⋊C₄	D₄₇	semidirect
132. C₄₇	C₂×#518	semidirect
142. C₂³×C₄	C₂×D₄₇	semidirect
153. C₂×C₂²⋊C₄	C₂×D₄₇	semidirect
154. C₂×C₂²⋊C₄	C₂×D₄₇	semidirect
161. C₂⁴	C₄×D₄₇	semidirect
181. C₂²×C₄	C₂²×D₄₇	semidirect
187. C₂²⋊C₄	C₂²×D₄₇	semidirect
203. C₂³	C₂×C₄×D₄₇	semidirect
244. C₂	#15689596	direct (N × H)

Subgroups — C2×#15689596 #20256573

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689596 #20256573