Subgroups — C₂×#15689938 #20258153

Order 6016.

1408 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13094071	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689938	C₂	1	16	yes	maximal, direct product factor
3	C₂×#13094055	C₂	1	3	yes	maximal, complemented
4	C₂×#13094054	C₂	1	3	yes	maximal, complemented
5	C₂×#13094044	C₂	1	3	yes	maximal, complemented
6	C₂²×#10625337	C₂	1	3	yes	maximal, complemented
7	C₂×#13094031	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×2₋¹⁺⁴×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#13094071	—	2	4	no	—
10	#13094071	C₂²	1	8	yes	complemented
11	C₂×#10625350	C₂²	1	3	yes	—
12	C₂×#10625352	—	2	3	no	—
13	C₂²×Dic₉₄	C₂²	1	3	yes	—
14	C₂×Q₈×D₄₇	—	2	1	no	—
15	#13094055	C₂²	1	24	yes	complemented
16	#13094054	C₂²	1	24	yes	complemented
17	#13094044	C₂²	1	24	yes	complemented
18	C₂×#10625337	C₂²	1	3	yes	—
19	C₂×#10625337	C₂²	1	6	yes	—
20	C₂×#10625337	C₂²	1	24	yes	complemented
21	C₂×#10625335	C₂²	1	3	yes	—
22	C₂×#10625336	C₂²	1	3	yes	—
23	C₂×#10625334	C₂²	1	1	yes	characteristic
24	#13094031	C₂²	1	8	yes	complemented
25	C₂×#10625329	C₂²	1	3	yes	—
26	C₂²×C₄₇⋊C₈	C₂²	1	3	yes	—
27	2₋¹⁺⁴×C₄₇	—	2	4	no	—
28	2₋¹⁺⁴×C₄₇	C₂²	1	8	yes	complemented
29	C₂×#41×C₄₇	—	2	3	no	—
30	C₂×#41×C₄₇	C₂²	1	3	yes	—
31	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
32	C₂²×Q₈×C₄₇	—	2	1	no	—
33	C₂²×Q₈×C₄₇	C₂²	1	3	yes	—
34	#41×C₄₇	—	2	24	no	—
35	#41×C₄₇	—	2	6	no	—
36	#41×C₄₇	D₄	1	4	yes	—
37	#41×C₄₇	C₂³	1	12	yes	—
38	#41×C₄₇	C₂³	1	4	yes	—
39	C₂×Q₈×C₄₇	—	2	8	no	—
40	C₂×Q₈×C₄₇	—	2	3	no	—
41	C₂×Q₈×C₄₇	—	2	6	no	—
42	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
43	C₂×Q₈×C₄₇	C₂³	1	3	yes	—
44	C₂×Q₈×C₄₇	C₂³	1	12	yes	—
45	C₂×D₄×C₄₇	—	2	3	no	—
46	C₂×D₄×C₄₇	D₄	1	3	yes	—
47	C₂×D₄×C₄₇	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	—	2	3	no	—
49	C₂²×C₄×C₄₇	—	2	3	no	—
50	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
51	C₂×C₂²⋊D₄₇	—	2	3	no	—
52	C₂²×Dic₄₇	—	2	3	no	—
53	Q₈×D₄₇	—	2	8	no	—
54	#10625352	—	2	24	no	—
55	#10625350	—	2	6	no	—
56	#10625350	C₂³	1	12	yes	—
57	C₂×D₁₈₈	C₂³	1	1	yes	characteristic
58	C₂×C₄×D₄₇	—	2	3	no	—
59	C₂×Dic₉₄	—	2	3	no	—
60	C₂×Dic₉₄	—	2	6	no	—
61	C₂×Dic₉₄	C₂³	1	3	yes	—
62	C₂×Dic₉₄	C₂³	1	12	yes	—
63	#10625337	C₂³	1	12	yes	—
64	#10625337	C₂³	1	24	yes	—
65	#10625336	C₂³	1	12	yes	—
66	#10625335	C₂³	1	12	yes	—
67	#10625334	C₂³	1	4	yes	—
68	#10625329	C₂³	1	12	yes	—
69	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₄₇⋊C₈	C₂³	1	3	yes	—
71	C₂×C₄₇⋊C₈	C₂³	1	12	yes	—
72	C₂³×C₄₇	—	2	1	no	—
73	C₂³×C₄₇	C₂×D₄	1	3	yes	—
74	C₂²×D₄₇	—	2	1	no	—
75	Q₈×C₄₇	—	2	3	no	—
76	Q₈×C₄₇	—	2	12	no	—
77	Q₈×C₄₇	C₂×D₄	1	4	yes	—
78	Q₈×C₄₇	C₂⁴	1	6	yes	—
79	D₄×C₄₇	—	2	12	no	—
80	D₄×C₄₇	—	2	1	no	—
81	D₄×C₄₇	C₂×D₄	1	12	yes	—
82	D₄×C₄₇	C₂⁴	1	2	yes	—
83	C₂×C₄×C₄₇	—	2	3	no	—
84	C₂×C₄×C₄₇	—	2	12	no	—
85	C₂×C₄×C₄₇	—	2	12	no	—
86	C₂×C₄×C₄₇	C₂×D₄	1	3	yes	—
87	C₂×C₄×C₄₇	C₂×D₄	1	6	yes	—
88	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₂⁴	1	6	yes	—
90	C₂²⋊D₄₇	—	2	12	no	—
91	C₂×Dic₄₇	—	2	3	no	—
92	C₂×Dic₄₇	—	2	12	no	—
93	D₁₈₈	—	2	1	no	—
94	D₁₈₈	C₂⁴	1	2	yes	—
95	C₄×D₄₇	—	2	12	no	—
96	Dic₉₄	—	2	3	no	—
97	Dic₉₄	—	2	12	no	—
98	Dic₉₄	C₂⁴	1	6	yes	—
99	C₄₇⋊C₈	C₂⁴	1	2	yes	—
100	C₄₇⋊C₈	C₂⁴	1	6	yes	—
101	C₂²×C₄₇	—	2	6	no	—
102	C₂²×C₄₇	—	2	2	no	—
103	C₂²×C₄₇	—	4	2	no	—
104	C₂²×C₄₇	C₂²×D₄	1	6	yes	—
105	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×D₄₇	—	2	2	no	—
107	C₂×D₄₇	—	4	2	no	—
108	C₄×C₄₇	—	2	6	no	—
109	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
110	C₄×C₄₇	C₂²×D₄	1	6	yes	—
111	C₄×C₄₇	C₂⁵	1	1	yes	characteristic
112	Dic₄₇	—	2	6	no	—
113	C₂×#578	—	47	1	no	maximal, Sylow
114	C₂×C₄₇	—	2	6	no	—
115	C₂×C₄₇	—	4	2	no	—
116	C₂×C₄₇	#578	1	2	yes	complemented
117	C₂×C₄₇	C₂³×D₄	1	1	yes	characteristic
118	D₄₇	—	4	2	no	—
119	C₂×2₋¹⁺⁴	—	47	1	no	—
120	C₂×2₋¹⁺⁴	D₄₇	1	1	yes	characteristic, complemented
121	#578	—	47	16	no	—
122	C₂×#137	—	47	3	no	—
123	C₂×#137	—	47	3	no	—
124	C₂×#135	—	47	3	no	—
125	C₂²×Q₁₆	—	47	3	no	—
126	C₂×#131	—	47	1	no	—
127	C₄₇	C₂×#578	1	1	yes	characteristic, complemented, Sylow
128	2₋¹⁺⁴	—	2	4	no	—
129	2₋¹⁺⁴	—	47	8	no	—
130	2₋¹⁺⁴	—	94	4	no	—
131	2₋¹⁺⁴	C₂×D₄₇	1	8	yes	complemented
132	C₂×#41	—	2	3	no	—
133	C₂×#41	—	47	3	no	—
134	C₂×#41	—	94	3	no	—
135	C₂×#41	C₂×D₄₇	1	3	yes	—
136	C₂×#41	C₂×D₄₇	1	1	yes	characteristic
137	C₂²×Q₈	—	2	1	no	—
138	C₂²×Q₈	—	47	3	no	—
139	C₂²×Q₈	—	94	1	no	—
140	C₂²×Q₈	C₂×D₄₇	1	3	yes	—
141	#137	—	47	24	no	—
142	#137	—	47	24	no	—
143	#135	—	47	24	no	—
144	C₂×Q₁₆	—	47	3	no	—
145	C₂×Q₁₆	—	47	6	no	—
146	C₂×Q₁₆	—	47	24	no	—
147	C₂×SD₁₆	—	47	3	no	—
148	C₂×SD₁₆	—	47	3	no	—
149	C₂×D₈	—	47	1	no	—
150	#131	—	47	8	no	—
151	C₂×M₁₆	—	47	3	no	—
152	C₂²×C₈	—	47	3	no	—
153	#41	—	2	24	no	—
154	#41	—	2	6	no	—
155	#41	—	47	12	no	—
156	#41	—	94	24	no	—
157	#41	—	94	6	no	—
158	#41	C₂²⋊D₄₇	1	4	yes	—
159	#41	C₂²×D₄₇	1	12	yes	—
160	#41	C₂²×D₄₇	1	4	yes	—
161	C₂×Q₈	—	2	8	no	—
162	C₂×Q₈	—	2	3	no	—
163	C₂×Q₈	—	2	6	no	—
164	C₂×Q₈	—	47	3	no	—
165	C₂×Q₈	—	47	12	no	—
166	C₂×Q₈	—	94	8	no	—
167	C₂×Q₈	—	94	3	no	—
168	C₂×Q₈	—	94	6	no	—
169	C₂×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
170	C₂×Q₈	C₂²×D₄₇	1	3	yes	—
171	C₂×Q₈	C₂²×D₄₇	1	12	yes	—
172	C₂×D₄	—	2	3	no	—
173	C₂×D₄	—	47	1	no	—
174	C₂×D₄	—	94	3	no	—
175	C₂×D₄	C₂²⋊D₄₇	1	3	yes	—
176	C₂×D₄	C₂²×D₄₇	1	1	yes	characteristic
177	C₂²×C₄	—	2	3	no	—
178	C₂²×C₄	—	2	3	no	—
179	C₂²×C₄	—	94	3	no	—
180	C₂²×C₄	—	94	3	no	—
181	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
182	Q₁₆	—	47	12	no	—
183	Q₁₆	—	47	24	no	—
184	SD₁₆	—	47	12	no	—
185	SD₁₆	—	47	12	no	—
186	D₈	—	47	4	no	—
187	M₁₆	—	47	12	no	—
188	C₂×C₈	—	47	1	no	—
189	C₂×C₈	—	47	3	no	—
190	C₂×C₈	—	47	12	no	—
191	C₂³	—	2	1	no	—
192	C₂³	—	94	1	no	—
193	C₂³	C₂×C₂²⋊D₄₇	1	3	yes	—
194	Q₈	—	2	12	no	—
195	Q₈	—	2	3	no	—
196	Q₈	—	47	6	no	—
197	Q₈	—	94	12	no	—
198	Q₈	—	94	3	no	—
199	Q₈	C₂×C₂²⋊D₄₇	1	4	yes	—
200	Q₈	C₂³×D₄₇	1	6	yes	—
201	D₄	—	2	12	no	—
202	D₄	—	2	1	no	—
203	D₄	—	47	2	no	—
204	D₄	—	94	12	no	—
205	D₄	—	94	1	no	—
206	D₄	C₂×C₂²⋊D₄₇	1	12	yes	—
207	D₄	C₂³×D₄₇	1	2	yes	—
208	C₂×C₄	—	2	12	no	—
209	C₂×C₄	—	2	3	no	—
210	C₂×C₄	—	2	12	no	—
211	C₂×C₄	—	94	12	no	—
212	C₂×C₄	—	94	3	no	—
213	C₂×C₄	—	94	12	no	—
214	C₂×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
215	C₂×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
216	C₂×C₄	C₂³×D₄₇	1	6	yes	—
217	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
218	C₈	—	47	2	no	—
219	C₈	—	47	6	no	—
220	C₂²	—	2	6	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	94	2	no	—
224	C₂²	—	188	2	no	—
225	C₂²	C₂²×C₂²⋊D₄₇	1	6	yes	—
226	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
227	C₄	—	2	6	no	—
228	C₄	—	94	6	no	—
229	C₄	C₂²×C₂²⋊D₄₇	1	6	yes	—
230	C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
231	C₄	C₂⁴×D₄₇	1	1	yes	characteristic
232	C₂	—	2	6	no	—
233	C₂	—	4	2	no	—
234	C₂	—	188	2	no	—
235	C₂	#15689938	1	2	yes	direct product factor
236	C₂	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂⁵
Frattini	C₄	C₂⁴×D₄₇
Fitting	C₂×2₋¹⁺⁴×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	C₂×2₋¹⁺⁴	D₄₇
hypocenter	C₄₇	C₂×#578
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₂×#13094071	C₂	semidirect
2. #15689938	C₂	direct (N × H)
3. C₂×#13094055	C₂	semidirect
4. C₂×#13094054	C₂	semidirect
5. C₂×#13094044	C₂	semidirect
6. C₂²×#10625337	C₂	semidirect
7. C₂×#13094031	C₂	semidirect
8. C₂×2₋¹⁺⁴×C₄₇	C₂	semidirect
10. #13094071	C₂²	semidirect
15. #13094055	C₂²	semidirect
16. #13094054	C₂²	semidirect
17. #13094044	C₂²	semidirect
20. C₂×#10625337	C₂²	semidirect
24. #13094031	C₂²	semidirect
28. 2₋¹⁺⁴×C₄₇	C₂²	semidirect
116. C₂×C₄₇	#578	semidirect
120. C₂×2₋¹⁺⁴	D₄₇	semidirect
127. C₄₇	C₂×#578	semidirect
131. 2₋¹⁺⁴	C₂×D₄₇	semidirect
235. C₂	#15689938	direct (N × H)

Subgroups — C2×#15689938 #20258153

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689938 #20258153