Subgroups — C₂×#15689889 #20258097

Order 6016.

1460 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10625350	C₂	1	1	yes	maximal, characteristic
2	#15689889	C₂	1	16	yes	maximal, direct product factor
3	C₂×#13094032	C₂	1	4	yes	maximal, complemented
4	C₂×#13094030	C₂	1	4	yes	maximal, complemented
5	C₂²×#10625329	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×M₁₆×D₄₇	C₂	1	4	yes	maximal, complemented
7	C₂²×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625350	C₄	1	2	yes	—
9	C₂×#10625350	C₄	1	2	yes	—
10	C₂×#10625350	C₄	1	8	yes	—
11	C₂×#10625350	C₂²	1	2	yes	—
12	C₂×#10625350	C₂²	1	2	yes	—
13	C₂×#10625350	C₂²	1	8	yes	—
14	C₂²×Dic₉₄	C₄	1	1	yes	characteristic
15	C₂²×D₁₈₈	C₄	1	1	yes	characteristic
16	C₂²×C₂²⋊D₄₇	C₄	1	2	yes	—
17	C₂²×C₄×D₄₇	C₂²	1	2	yes	—
18	#13094032	C₂²	1	32	yes	complemented
19	#13094030	C₂²	1	32	yes	complemented
20	C₂×#10625329	C₂²	1	2	yes	—
21	C₂×#10625329	C₂²	1	2	yes	complemented
22	C₂×#10625329	C₂²	1	8	yes	complemented
23	C₂×#10625324	C₂²	1	8	yes	complemented
24	M₁₆×D₄₇	C₂²	1	32	yes	complemented
25	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	complemented
26	C₂×C₈×D₄₇	C₂²	1	8	yes	complemented
27	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
29	C₂×M₁₆×C₄₇	C₂²	1	2	yes	complemented
30	C₂×M₁₆×C₄₇	C₂²	1	8	yes	complemented
31	C₂²×C₈×C₄₇	C₂²	1	2	yes	complemented
32	C₂⁴×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂³×D₄₇	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
38	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
42	C₂×C₂²⋊D₄₇	C₂×C₄	1	4	yes	—
43	C₂×C₂²⋊D₄₇	C₂×C₄	1	4	yes	—
44	C₂×C₂²⋊D₄₇	C₂×C₄	1	16	yes	—
45	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
46	#10625350	C₂×C₄	1	8	yes	—
47	#10625350	C₂×C₄	1	8	yes	—
48	#10625350	C₂×C₄	1	16	yes	—
49	#10625350	C₂×C₄	1	16	yes	—
50	#10625350	C₂³	1	8	yes	—
51	#10625350	C₂³	1	8	yes	—
52	C₂×D₁₈₈	C₂×C₄	1	2	yes	—
53	C₂×D₁₈₈	C₂×C₄	1	2	yes	—
54	C₂×D₁₈₈	C₂×C₄	1	8	yes	—
55	C₂×C₄×D₄₇	C₂×C₄	1	4	yes	—
56	C₂×C₄×D₄₇	C₂×C₄	1	8	yes	—
57	C₂×C₄×D₄₇	C₂³	1	4	yes	—
58	C₂×C₄×D₄₇	C₂³	1	8	yes	—
59	C₂×Dic₉₄	C₂×C₄	1	2	yes	—
60	C₂×Dic₉₄	C₂×C₄	1	2	yes	—
61	C₂×Dic₉₄	C₂×C₄	1	8	yes	—
62	M₁₆×C₄₇	C₂³	1	8	yes	—
63	M₁₆×C₄₇	C₂³	1	8	yes	complemented
64	C₂×C₈×C₄₇	C₂³	1	4	yes	—
65	C₂×C₈×C₄₇	C₂³	1	8	yes	complemented
66	#10625329	C₂³	1	8	yes	—
67	#10625329	C₂³	1	8	yes	complemented
68	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
69	C₂×C₄₇⋊C₈	C₂³	1	8	yes	complemented
70	#10625324	C₂³	1	32	yes	complemented
71	C₈×D₄₇	C₂³	1	32	yes	complemented
72	C₂³×C₄₇	—	4	2	no	—
73	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂³×C₄₇	C₂²×C₄	1	4	yes	—
77	C₂²×D₄₇	—	4	4	no	—
78	C₂²×D₄₇	C₂²×C₄	1	4	yes	—
79	C₂²×D₄₇	C₂²×C₄	1	8	yes	—
80	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
81	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
88	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
89	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
90	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
91	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
92	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
93	C₂²⋊D₄₇	C₂²×C₄	1	16	yes	—
94	C₂²⋊D₄₇	C₂²×C₄	1	16	yes	—
95	C₂×Dic₄₇	C₂²×C₄	1	8	yes	—
96	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
97	D₁₈₈	C₂²×C₄	1	8	yes	—
98	D₁₈₈	C₂²×C₄	1	8	yes	—
99	C₄×D₄₇	C₂²×C₄	1	8	yes	—
100	C₄×D₄₇	C₂²×C₄	1	16	yes	—
101	C₄×D₄₇	C₂⁴	1	8	yes	—
102	Dic₉₄	C₂²×C₄	1	8	yes	—
103	Dic₉₄	C₂²×C₄	1	8	yes	—
104	C₈×C₄₇	C₂⁴	1	8	yes	—
105	C₄₇⋊C₈	C₂⁴	1	8	yes	—
106	C₂²×C₄₇	—	2	2	no	—
107	C₂²×C₄₇	—	2	2	no	—
108	C₂²×C₄₇	—	2	2	no	—
109	C₂²×C₄₇	—	4	4	no	—
110	C₂²×C₄₇	C₂³×C₄	1	2	yes	—
111	C₂²×C₄₇	C₂³×C₄	1	2	yes	—
112	C₂²×C₄₇	C₂³×C₄	1	2	yes	—
113	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×D₄₇	—	2	8	no	—
115	C₂×D₄₇	—	4	8	no	—
116	C₂×D₄₇	C₂³×C₄	1	8	yes	—
117	C₄×C₄₇	C₂³×C₄	1	2	yes	—
118	C₄×C₄₇	C₂³×C₄	1	2	yes	—
119	C₄×C₄₇	C₂³×C₄	1	2	yes	—
120	C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
121	C₄×C₄₇	C₂⁵	1	1	yes	characteristic
122	Dic₄₇	C₂³×C₄	1	8	yes	—
123	C₂×#568	—	47	1	no	maximal, Sylow
124	C₂×C₄₇	—	2	2	no	—
125	C₂×C₄₇	—	2	2	no	—
126	C₂×C₄₇	—	2	2	no	—
127	C₂×C₄₇	#568	1	2	yes	complemented
128	C₂×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
129	D₄₇	—	2	8	no	—
130	C₂²×#41	—	47	1	no	—
131	#568	—	47	16	no	—
132	C₂×#131	—	47	4	no	—
133	C₂×#131	—	47	4	no	—
134	C₂²×M₁₆	—	47	1	no	—
135	C₂²×M₁₆	—	47	4	no	—
136	C₂²×M₁₆	D₄₇	1	1	yes	characteristic, complemented
137	C₄₇	C₂×#568	1	1	yes	characteristic, complemented, Sylow
138	C₂×#41	—	47	2	no	—
139	C₂×#41	—	47	2	no	—
140	C₂×#41	—	47	8	no	—
141	C₂×#41	—	47	2	no	—
142	C₂×#41	—	47	2	no	—
143	C₂×#41	—	47	8	no	—
144	C₂²×Q₈	—	47	1	no	—
145	C₂²×D₄	—	47	2	no	—
146	C₂²×D₄	—	47	1	no	—
147	C₂³×C₄	—	47	2	no	—
148	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
149	#131	—	47	32	no	—
150	#131	—	47	32	no	—
151	C₂×M₁₆	—	47	2	no	—
152	C₂×M₁₆	—	47	2	no	—
153	C₂×M₁₆	—	47	8	no	—
154	C₂×M₁₆	—	47	32	no	—
155	C₂×M₁₆	—	47	8	no	—
156	C₂×M₁₆	C₂×D₄₇	1	2	yes	—
157	C₂×M₁₆	C₂×D₄₇	1	2	yes	complemented
158	C₂×M₁₆	C₂×D₄₇	1	8	yes	complemented
159	C₂²×C₈	—	47	2	no	—
160	C₂²×C₈	—	47	8	no	—
161	C₂²×C₈	C₂×D₄₇	1	2	yes	complemented
162	C₂⁴	—	47	2	no	—
163	C₂⁴	C₄×D₄₇	1	1	yes	characteristic
164	#41	—	47	8	no	—
165	#41	—	47	8	no	—
166	#41	—	47	16	no	—
167	#41	—	47	16	no	—
168	#41	—	47	8	no	—
169	#41	—	47	8	no	—
170	C₂×Q₈	—	47	2	no	—
171	C₂×Q₈	—	47	2	no	—
172	C₂×Q₈	—	47	8	no	—
173	C₂×D₄	—	47	4	no	—
174	C₂×D₄	—	47	4	no	—
175	C₂×D₄	—	47	16	no	—
176	C₂×D₄	—	47	2	no	—
177	C₂×D₄	—	47	2	no	—
178	C₂×D₄	—	47	8	no	—
179	C₂²×C₄	—	47	2	no	—
180	C₂²×C₄	—	47	4	no	—
181	C₂²×C₄	—	47	8	no	—
182	C₂²×C₄	—	47	8	no	—
183	C₂²×C₄	—	47	4	no	—
184	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
185	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
186	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
187	C₂²×C₄	C₄×D₄₇	1	4	yes	—
188	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
189	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
191	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
192	M₁₆	—	47	8	no	—
193	M₁₆	—	47	8	no	—
194	M₁₆	—	47	32	no	—
195	M₁₆	C₂²×D₄₇	1	8	yes	—
196	M₁₆	C₂²×D₄₇	1	8	yes	complemented
197	C₂×C₈	—	47	8	no	—
198	C₂×C₈	—	47	4	no	—
199	C₂×C₈	—	47	32	no	—
200	C₂×C₈	C₂²×D₄₇	1	4	yes	—
201	C₂×C₈	C₂²×D₄₇	1	8	yes	complemented
202	C₂³	—	4	2	no	—
203	C₂³	—	47	4	no	—
204	C₂³	—	47	8	no	—
205	C₂³	—	188	4	no	—
206	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
207	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
208	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
209	C₂³	C₂×C₄×D₄₇	1	4	yes	—
210	Q₈	—	47	8	no	—
211	Q₈	—	47	8	no	—
212	D₄	—	47	16	no	—
213	D₄	—	47	16	no	—
214	D₄	—	47	8	no	—
215	D₄	—	47	8	no	—
216	C₂×C₄	—	47	8	no	—
217	C₂×C₄	—	47	4	no	—
218	C₂×C₄	—	47	16	no	—
219	C₂×C₄	—	47	8	no	—
220	C₂×C₄	—	47	8	no	—
221	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
222	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
223	C₂×C₄	C₂×C₄×D₄₇	1	4	yes	—
224	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
225	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄×D₄₇	1	4	yes	—
227	C₂×C₄	C₂×C₄×D₄₇	1	4	yes	—
228	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
229	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
230	C₂×C₄	C₂³×D₄₇	1	2	yes	—
231	C₂×C₄	C₂³×D₄₇	1	2	yes	—
232	C₂×C₄	C₂³×D₄₇	1	2	yes	—
233	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
234	C₈	—	47	8	no	—
235	C₈	C₂³×D₄₇	1	8	yes	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	4	4	no	—
240	C₂²	—	47	8	no	—
241	C₂²	—	94	8	no	—
242	C₂²	—	188	8	no	—
243	C₂²	C₂²×C₄×D₄₇	1	2	yes	—
244	C₂²	C₂²×C₄×D₄₇	1	2	yes	—
245	C₂²	C₂²×C₄×D₄₇	1	2	yes	—
246	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
247	C₄	—	47	8	no	—
248	C₄	C₂²×C₄×D₄₇	1	2	yes	—
249	C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
250	C₄	C₂²×C₄×D₄₇	1	2	yes	—
251	C₄	C₂²×C₄×D₄₇	1	2	yes	—
252	C₄	C₂⁴×D₄₇	1	1	yes	characteristic
253	C₂	—	2	2	no	—
254	C₂	—	2	2	no	—
255	C₂	—	2	2	no	—
256	C₂	—	94	8	no	—
257	C₂	#15689889	1	2	yes	direct product factor
258	C₂	C₂³×C₄×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₄₇
commutator	C₂×C₄₇	C₂⁴×C₄
Frattini	C₄	C₂⁴×D₄₇
Fitting	C₂²×M₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂³×C₄
hypercenter	C₂²×M₁₆	D₄₇
hypocenter	C₄₇	C₂×#568
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
2. #15689889	C₂	direct (N × H)
3. C₂×#13094032	C₂	semidirect
4. C₂×#13094030	C₂	semidirect
5. C₂²×#10625329	C₂	semidirect
6. C₂×M₁₆×D₄₇	C₂	semidirect
7. C₂²×M₁₆×C₄₇	C₂	semidirect
18. #13094032	C₂²	semidirect
19. #13094030	C₂²	semidirect
21. C₂×#10625329	C₂²	semidirect
22. C₂×#10625329	C₂²	semidirect
23. C₂×#10625324	C₂²	semidirect
24. M₁₆×D₄₇	C₂²	semidirect
25. C₂²×C₄₇⋊C₈	C₂²	semidirect
26. C₂×C₈×D₄₇	C₂²	semidirect
29. C₂×M₁₆×C₄₇	C₂²	semidirect
30. C₂×M₁₆×C₄₇	C₂²	semidirect
31. C₂²×C₈×C₄₇	C₂²	semidirect
63. M₁₆×C₄₇	C₂³	semidirect
65. C₂×C₈×C₄₇	C₂³	semidirect
67. #10625329	C₂³	semidirect
69. C₂×C₄₇⋊C₈	C₂³	semidirect
70. #10625324	C₂³	semidirect
71. C₈×D₄₇	C₂³	semidirect
127. C₂×C₄₇	#568	semidirect
136. C₂²×M₁₆	D₄₇	semidirect
137. C₄₇	C₂×#568	semidirect
157. C₂×M₁₆	C₂×D₄₇	semidirect
158. C₂×M₁₆	C₂×D₄₇	semidirect
161. C₂²×C₈	C₂×D₄₇	semidirect
196. M₁₆	C₂²×D₄₇	semidirect
201. C₂×C₈	C₂²×D₄₇	semidirect
257. C₂	#15689889	direct (N × H)

Subgroups — C2×#15689889 #20258097

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689889 #20258097