Subgroups — C₂×#15689872 #20257517

Order 6016.

1024 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689872	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13094022	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13094019	C₂	1	2	yes	maximal, complemented
4	C₂×#13094018	C₂	1	2	yes	maximal, complemented
5	C₂×#13094014	C₂	1	2	yes	maximal, complemented
6	C₂×#13094013	C₂	1	1	yes	maximal, characteristic
7	C₂×#13094009	C₂	1	2	yes	maximal
8	C₂×#13094002	C₂	1	2	yes	maximal, complemented
9	C₂×#13094000	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
12	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₄₇	C₂²	1	2	yes	—
14	#13094022	C₂²	1	8	yes	complemented
15	#13094019	C₂²	1	16	yes	complemented
16	#13094018	C₂²	1	16	yes	complemented
17	#13094014	C₂²	1	16	yes	complemented
18	#13094013	C₂²	1	8	yes	—
19	#13094009	C₂²	1	16	yes	—
20	#13094002	C₂²	1	16	yes	complemented
21	#13094000	C₂²	1	16	yes	complemented
22	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
23	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
24	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
25	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
26	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
27	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
28	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
29	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	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×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
34	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
35	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
37	C₄⋊Q₈×C₄₇	C₂²	1	8	yes	complemented
38	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
39	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
40	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
41	C₂³×D₄₇	C₂³	1	2	yes	—
42	C₂×Q₈×C₄₇	—	4	2	no	—
43	C₂×Q₈×C₄₇	—	4	2	no	—
44	C₂×Q₈×C₄₇	C₂³	1	4	yes	—
45	C₂×Q₈×C₄₇	C₂³	1	4	yes	—
46	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
51	C₂²×Dic₄₇	C₂³	1	2	yes	—
52	C₂²×Dic₄₇	C₂³	1	2	yes	—
53	C₂²×Dic₄₇	C₂³	1	2	yes	—
54	C₂×D₁₈₈	—	4	2	no	—
55	C₂×D₁₈₈	C₂³	1	4	yes	—
56	C₂×C₄×D₄₇	—	4	4	no	—
57	C₂×C₄×D₄₇	C₂³	1	8	yes	—
58	C₂×Dic₉₄	—	4	2	no	—
59	C₂×Dic₉₄	C₂³	1	4	yes	—
60	C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
61	C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
62	C₄²×C₄₇	C₂³	1	4	yes	—
63	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
64	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
65	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
66	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
67	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
68	C₄⋊Dic₄₇	C₂³	1	8	yes	—
69	Dic₄₇⋊C₄	C₂³	1	8	yes	—
70	Dic₄₇⋊C₄	C₂³	1	8	yes	—
71	Dic₄₇⋊C₄	C₂³	1	8	yes	—
72	Dic₄₇⋊C₄	C₂³	1	8	yes	—
73	Dic₄₇⋊C₄	C₂³	1	8	yes	—
74	C₄×Dic₄₇	C₂³	1	4	yes	—
75	C₄×Dic₄₇	C₂³	1	4	yes	—
76	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
77	C₂²×D₄₇	—	2	4	no	—
78	C₂²×D₄₇	—	2	8	no	—
79	C₂²×D₄₇	C₂⁴	1	4	yes	—
80	Q₈×C₄₇	—	4	4	no	—
81	Q₈×C₄₇	—	4	4	no	—
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	—	2	4	no	—
84	C₂×C₄×C₄₇	—	2	2	no	—
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	—	2	4	no	—
87	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
88	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
89	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
90	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
91	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
92	C₂×Dic₄₇	—	2	4	no	—
93	C₂×Dic₄₇	—	2	4	no	—
94	C₂×Dic₄₇	—	2	4	no	—
95	C₂×Dic₄₇	C₂⁴	1	4	yes	—
96	C₂×Dic₄₇	C₂⁴	1	4	yes	—
97	C₂×Dic₄₇	C₂⁴	1	4	yes	—
98	D₁₈₈	—	4	4	no	—
99	C₄×D₄₇	—	4	8	no	—
100	Dic₉₄	—	4	4	no	—
101	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
102	C₂²×C₄₇	2₋¹⁺⁴	1	4	yes	—
103	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
104	C₂×D₄₇	—	2	4	no	—
105	C₂×D₄₇	—	2	8	no	—
106	C₂×D₄₇	—	4	8	no	—
107	C₄×C₄₇	—	2	2	no	—
108	C₄×C₄₇	—	2	4	no	—
109	C₄×C₄₇	—	2	2	no	—
110	C₄×C₄₇	—	2	2	no	—
111	C₄×C₄₇	—	2	4	no	—
112	Dic₄₇	—	2	4	no	—
113	Dic₄₇	—	2	4	no	—
114	Dic₄₇	—	2	4	no	—
115	C₂×#563	—	47	1	no	maximal, Sylow
116	C₂×C₄₇	#563	1	4	yes	complemented
117	C₂×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×C₄₇	C₂×2₋¹⁺⁴	1	2	yes	—
119	D₄₇	—	4	4	no	—
120	#563	—	47	16	no	—
121	C₂×C₄⋊Q₈	—	47	1	no	—
122	C₂×C₄⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
123	C₂×#126	—	47	2	no	—
124	C₂×#126	—	47	2	no	—
125	C₂×#125	—	47	2	no	—
126	C₂×#124	—	47	1	no	—
127	C₂×#123	—	47	2	no	—
128	C₂×C₂²⋊Q₈	—	47	2	no	—
129	C₂×C₂²⋊Q₈	—	47	2	no	—
130	C₄₇	C₂×#563	1	1	yes	characteristic, complemented, Sylow
131	C₂²×Q₈	—	47	1	no	—
132	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
133	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
134	C₂²×D₄	—	47	1	no	—
135	C₂³×C₄	—	47	2	no	—
136	C₄⋊Q₈	—	47	8	no	—
137	C₄⋊Q₈	C₂×D₄₇	1	8	yes	complemented
138	#126	—	47	16	no	—
139	#126	—	47	16	no	—
140	#125	—	47	16	no	—
141	#124	—	47	8	no	—
142	#123	—	47	16	no	—
143	C₂²⋊Q₈	—	47	16	no	—
144	C₂²⋊Q₈	—	47	16	no	—
145	C₂×C₄⋊C₄	—	47	2	no	—
146	C₂×C₄⋊C₄	—	47	2	no	—
147	C₂×C₄⋊C₄	—	47	2	no	—
148	C₂×C₄⋊C₄	—	47	2	no	—
149	C₂×C₄⋊C₄	—	47	2	no	—
150	C₂×C₄⋊C₄	—	47	2	no	—
151	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
152	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
153	C₂×C₂²⋊C₄	—	47	2	no	—
154	C₂×C₂²⋊C₄	—	47	2	no	—
155	C₂×C₂²⋊C₄	—	47	2	no	—
156	C₂×C₂²⋊C₄	—	47	2	no	—
157	C₂×C₂²⋊C₄	—	47	2	no	—
158	C₂×C₄²	—	47	1	no	—
159	C₂×C₄²	—	47	1	no	—
160	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
161	C₂⁴	—	47	2	no	—
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	47	4	no	—
165	C₂×Q₈	—	188	2	no	—
166	C₂×Q₈	C₂²×D₄₇	1	4	yes	—
167	C₂×Q₈	C₂²×D₄₇	1	4	yes	—
168	C₂×D₄	—	47	4	no	—
169	C₂×D₄	—	188	2	no	—
170	C₂²×C₄	—	47	2	no	—
171	C₂²×C₄	—	47	2	no	—
172	C₂²×C₄	—	47	2	no	—
173	C₂²×C₄	—	47	8	no	—
174	C₂²×C₄	—	188	4	no	—
175	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
176	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
177	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
178	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
180	C₄⋊C₄	—	47	8	no	—
181	C₄⋊C₄	—	47	8	no	—
182	C₄⋊C₄	—	47	8	no	—
183	C₄⋊C₄	—	47	8	no	—
184	C₄⋊C₄	—	47	8	no	—
185	C₄⋊C₄	—	47	8	no	—
186	C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
187	C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
188	C₂²⋊C₄	—	47	8	no	—
189	C₂²⋊C₄	—	47	8	no	—
190	C₂²⋊C₄	—	47	8	no	—
191	C₂²⋊C₄	—	47	8	no	—
192	C₂²⋊C₄	—	47	8	no	—
193	C₄²	—	47	4	no	—
194	C₄²	—	47	4	no	—
195	C₄²	C₂²×D₄₇	1	4	yes	—
196	C₂³	—	47	4	no	—
197	C₂³	—	94	4	no	—
198	C₂³	—	94	8	no	—
199	C₂³	C₂³×D₄₇	1	1	yes	characteristic
200	Q₈	—	4	4	no	—
201	Q₈	—	4	4	no	—
202	Q₈	—	188	4	no	—
203	D₄	—	188	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	47	4	no	—
210	C₂×C₄	—	47	4	no	—
211	C₂×C₄	—	47	4	no	—
212	C₂×C₄	—	94	4	no	—
213	C₂×C₄	—	94	4	no	—
214	C₂×C₄	—	94	4	no	—
215	C₂×C₄	—	188	8	no	—
216	C₂×C₄	C₂³×D₄₇	1	2	yes	—
217	C₂×C₄	C₂³×D₄₇	1	2	yes	—
218	C₂×C₄	C₂³×D₄₇	1	4	yes	—
219	C₂×C₄	C₂³×D₄₇	1	2	yes	—
220	C₂×C₄	C₂³×D₄₇	1	4	yes	—
221	C₂²	—	94	4	no	—
222	C₂²	—	94	8	no	—
223	C₂²	—	188	8	no	—
224	C₂²	#13094069	1	2	yes	—
225	C₂²	#13094072	1	4	yes	—
226	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
227	C₄	—	2	4	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	4	no	—
232	C₄	—	94	4	no	—
233	C₄	—	94	4	no	—
234	C₄	—	94	4	no	—
235	C₂	—	188	4	no	—
236	C₂	#15689872	1	4	yes	direct product factor
237	C₂	C₂×#13094069	1	1	yes	characteristic
238	C₂	C₂×#13094072	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁵
Frattini	C₂²	C₂⁴×D₄₇
Fitting	C₂×C₄⋊Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	C₂×#563
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #15689872	C₂	direct (N × H)
2. C₂×#13094022	C₂	semidirect
3. C₂×#13094019	C₂	semidirect
4. C₂×#13094018	C₂	semidirect
5. C₂×#13094014	C₂	semidirect
8. C₂×#13094002	C₂	semidirect
9. C₂×#13094000	C₂	semidirect
10. C₂×C₄⋊Q₈×C₄₇	C₂	semidirect
14. #13094022	C₂²	semidirect
15. #13094019	C₂²	semidirect
16. #13094018	C₂²	semidirect
17. #13094014	C₂²	semidirect
20. #13094002	C₂²	semidirect
21. #13094000	C₂²	semidirect
37. C₄⋊Q₈×C₄₇	C₂²	semidirect
116. C₂×C₄₇	#563	semidirect
122. C₂×C₄⋊Q₈	D₄₇	semidirect
130. C₄₇	C₂×#563	semidirect
137. C₄⋊Q₈	C₂×D₄₇	semidirect
236. C₂	#15689872	direct (N × H)

Subgroups — C2×#15689872 #20257517

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689872 #20257517