Subgroups — C₂×#22575020 #31769728

Order 17152.

60570 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575020	C₂	1	64	yes	maximal, direct product factor
2	C₂×#16864233	C₂	1	12	yes	maximal, complemented
3	C₂×#16864232	C₂	1	15	yes	maximal, complemented
4	C₂²×#14524089	C₂	1	15	yes	maximal, complemented
5	C₂×H(5,𝔽₂)×D₆₇	C₂	1	20	yes	maximal, complemented
6	C₂×#585×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#16864233	C₂²	1	384	yes	complemented
8	#16864232	C₂²	1	480	yes	complemented
9	C₂×#14524091	C₂²	1	30	yes	complemented
10	C₂×#14524090	C₂²	1	20	yes	complemented
11	C₂×#14524089	C₂²	1	120	yes	complemented
12	C₂×#14524089	C₂²	1	60	yes	complemented
13	C₂×#14524089	C₂²	1	480	yes	complemented
14	C₂×#14524088	C₂²	1	90	yes	complemented
15	H(5,𝔽₂)×D₆₇	C₂²	1	640	yes	complemented
16	C₂²×#11761629	C₂²	1	30	yes	complemented
17	C₂²×#11761628	C₂²	1	45	yes	complemented
18	C₂×#41×D₆₇	C₂²	1	120	yes	complemented
19	C₂³×D₂₆₈	C₂²	1	15	yes	complemented
20	C₂²×D₄×D₆₇	C₂²	1	90	yes	complemented
21	#585×C₆₇	C₂²	1	32	yes	complemented
22	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	6	yes	complemented
23	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	10	yes	complemented
24	C₂²×#41×C₆₇	C₂²	1	15	yes	complemented
25	C₂⁴×D₆₇	C₂³	1	30	yes	complemented
26	#14524091	C₂³	1	480	yes	complemented
27	#14524090	C₂³	1	320	yes	complemented
28	#14524089	C₂³	1	1920	yes	complemented
29	#14524089	C₂³	1	960	yes	complemented
30	#14524088	C₂³	1	1440	yes	complemented
31	C₂×#11761629	C₂³	1	120	yes	complemented
32	C₂×#11761629	C₂³	1	480	yes	complemented
33	C₂×#11761628	C₂³	1	180	yes	complemented
34	C₂×#11761628	C₂³	1	60	yes	complemented
35	C₂×#11761628	C₂³	1	720	yes	complemented
36	C₂×#11761627	C₂³	1	120	yes	complemented
37	#41×D₆₇	C₂³	1	1920	yes	complemented
38	C₂²×Dic₁₃₄	C₂³	1	15	yes	complemented
39	C₂×Q₈×D₆₇	C₂³	1	40	yes	complemented
40	C₂²×D₂₆₈	C₂³	1	90	yes	complemented
41	C₂²×D₂₆₈	C₂³	1	45	yes	complemented
42	C₂²×D₂₆₈	C₂³	1	240	yes	complemented
43	C₂²×C₂²⋊D₆₇	C₂³	1	90	yes	complemented
44	C₂×D₄×D₆₇	C₂³	1	360	yes	complemented
45	C₂×D₄×D₆₇	C₂³	1	1440	yes	complemented
46	C₂²×C₄×D₆₇	C₂³	1	90	yes	complemented
47	2₋¹⁺⁴×C₆₇	C₂³	1	96	yes	complemented
48	H(5,𝔽₂)×C₆₇	C₂³	1	160	yes	complemented
49	C₂×#41×C₆₇	C₂³	1	20	yes	complemented
50	C₂×#41×C₆₇	C₂³	1	60	yes	complemented
51	C₂×#41×C₆₇	C₂³	1	240	yes	complemented
52	C₂²×Q₈×C₆₇	C₂³	1	15	yes	complemented
53	C₂²×D₄×C₆₇	C₂³	1	45	yes	complemented
54	C₂³×C₄×C₆₇	C₂³	1	15	yes	complemented
55	C₂³×D₆₇	—	8	60	no	—
56	C₂³×D₆₇	C₂⁴	1	90	yes	—
57	C₂³×D₆₇	C₂⁴	1	240	yes	complemented
58	#11761629	C₂⁴	1	960	yes	complemented
59	#11761628	C₂⁴	1	1440	yes	complemented
60	#11761628	C₂⁴	1	480	yes	complemented
61	#11761627	C₂⁴	1	960	yes	complemented
62	C₂×Dic₁₃₄	C₂⁴	1	30	yes	—
63	C₂×Dic₁₃₄	C₂⁴	1	6	yes	—
64	C₂×Dic₁₃₄	C₂⁴	1	120	yes	complemented
65	Q₈×D₆₇	C₂⁴	1	320	yes	complemented
66	C₂×D₂₆₈	C₂⁴	1	90	yes	—
67	C₂×D₂₆₈	C₂⁴	1	10	yes	—
68	C₂×D₂₆₈	C₂⁴	1	720	yes	complemented
69	C₂×D₂₆₈	C₂⁴	1	360	yes	complemented
70	C₂×C₂²⋊D₆₇	C₂⁴	1	120	yes	—
71	C₂×C₂²⋊D₆₇	C₂⁴	1	720	yes	complemented
72	D₄×D₆₇	C₂⁴	1	2880	yes	complemented
73	C₂²×Dic₆₇	C₂⁴	1	30	yes	—
74	C₂×C₄×D₆₇	C₂⁴	1	120	yes	—
75	C₂×C₄×D₆₇	C₂⁴	1	720	yes	complemented
76	C₂⁴×C₆₇	C₂⁴	1	15	yes	—
77	#41×C₆₇	C₂⁴	1	160	yes	complemented
78	#41×C₆₇	C₂⁴	1	480	yes	complemented
79	C₂×Q₈×C₆₇	C₂⁴	1	20	yes	—
80	C₂×Q₈×C₆₇	C₂⁴	1	120	yes	complemented
81	C₂×D₄×C₆₇	C₂⁴	1	60	yes	—
82	C₂×D₄×C₆₇	C₂⁴	1	360	yes	complemented
83	C₂²×C₄×C₆₇	C₂⁴	1	15	yes	—
84	C₂²×C₄×C₆₇	C₂⁴	1	45	yes	—
85	C₂²×C₄×C₆₇	C₂⁴	1	120	yes	complemented
86	C₂³×C₆₇	—	4	30	no	—
87	C₂³×C₆₇	C₂⁵	1	15	yes	—
88	C₂³×C₆₇	C₂⁵	1	60	yes	—
89	C₂²×D₆₇	—	4	180	no	—
90	C₂²×D₆₇	—	8	240	no	—
91	C₂²×D₆₇	C₂⁵	1	20	yes	—
92	C₂²×D₆₇	C₂⁵	1	360	yes	—
93	Q₈×C₆₇	C₂⁵	1	80	yes	—
94	D₄×C₆₇	C₂⁵	1	240	yes	—
95	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂⁵	1	15	yes	—
97	C₂×C₄×C₆₇	C₂⁵	1	60	yes	—
98	C₂×C₄×C₆₇	C₂⁵	1	180	yes	—
99	C₂²⋊D₆₇	C₂⁵	1	480	yes	—
100	C₂×Dic₆₇	C₂⁵	1	12	yes	—
101	C₂×Dic₆₇	C₂⁵	1	120	yes	—
102	D₂₆₈	C₂⁵	1	360	yes	—
103	D₂₆₈	C₂⁵	1	40	yes	—
104	C₄×D₆₇	C₂⁵	1	480	yes	—
105	Dic₁₃₄	C₂⁵	1	120	yes	—
106	Dic₁₃₄	C₂⁵	1	24	yes	—
107	C₂²×C₆₇	—	2	30	no	—
108	C₂²×C₆₇	—	4	60	no	—
109	C₂²×C₆₇	C₂⁶	1	30	yes	—
110	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
111	C₂×D₆₇	—	2	40	no	—
112	C₂×D₆₇	—	4	360	no	—
113	C₂×D₆₇	C₂⁶	1	40	yes	—
114	C₄×C₆₇	C₂⁶	1	2	yes	—
115	C₄×C₆₇	C₂⁶	1	30	yes	—
116	Dic₆₇	C₂⁶	1	24	yes	—
117	C₂×H(7,𝔽₂)	—	67	1	no	maximal, Sylow
118	C₂×C₆₇	—	2	30	no	—
119	C₂×C₆₇	H(7,𝔽₂)	1	2	yes	complemented
120	C₂×C₆₇	C₂⁷	1	1	yes	characteristic
121	D₆₇	—	2	40	no	—
122	H(7,𝔽₂)	—	67	64	no	—
123	C₂×#585	—	67	12	no	—
124	C₂×#585	—	67	15	no	—
125	C₂×#585	D₆₇	1	1	yes	characteristic, complemented
126	C₂²×H(5,𝔽₂)	—	67	20	no	—
127	C₂²×H(5,𝔽₂)	—	67	15	no	—
128	C₆₇	C₂×H(7,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
129	#585	—	67	384	no	—
130	#585	—	67	480	no	—
131	#585	C₂×D₆₇	1	32	yes	complemented
132	C₂×2₋¹⁺⁴	—	67	20	no	—
133	C₂×2₋¹⁺⁴	—	67	30	no	—
134	C₂×2₋¹⁺⁴	C₂×D₆₇	1	6	yes	complemented
135	C₂×H(5,𝔽₂)	—	67	640	no	—
136	C₂×H(5,𝔽₂)	—	67	120	no	—
137	C₂×H(5,𝔽₂)	—	67	60	no	—
138	C₂×H(5,𝔽₂)	—	67	480	no	—
139	C₂×H(5,𝔽₂)	—	67	90	no	—
140	C₂×H(5,𝔽₂)	C₂×D₆₇	1	10	yes	complemented
141	C₂²×#41	—	67	120	no	—
142	C₂²×#41	—	67	30	no	—
143	C₂²×#41	—	67	45	no	—
144	C₂²×#41	C₂×D₆₇	1	15	yes	complemented
145	C₂³×D₄	—	67	90	no	—
146	C₂³×D₄	—	67	15	no	—
147	C₂⁵	—	67	30	no	—
148	2₋¹⁺⁴	—	67	320	no	—
149	2₋¹⁺⁴	—	67	480	no	—
150	2₋¹⁺⁴	C₂²×D₆₇	1	96	yes	complemented
151	H(5,𝔽₂)	—	67	1920	no	—
152	H(5,𝔽₂)	—	67	960	no	—
153	H(5,𝔽₂)	—	67	1440	no	—
154	H(5,𝔽₂)	C₂²×D₆₇	1	160	yes	complemented
155	C₂×#41	—	67	1920	no	—
156	C₂×#41	—	67	120	no	—
157	C₂×#41	—	67	480	no	—
158	C₂×#41	—	67	120	no	—
159	C₂×#41	—	67	180	no	—
160	C₂×#41	—	67	60	no	—
161	C₂×#41	—	67	720	no	—
162	C₂×#41	C₂²×D₆₇	1	60	yes	complemented
163	C₂×#41	C₂²×D₆₇	1	20	yes	complemented
164	C₂×#41	C₂²×D₆₇	1	240	yes	complemented
165	C₂²×Q₈	—	67	40	no	—
166	C₂²×Q₈	—	67	15	no	—
167	C₂²×Q₈	C₂²×D₆₇	1	15	yes	complemented
168	C₂²×D₄	—	67	90	no	—
169	C₂²×D₄	—	67	360	no	—
170	C₂²×D₄	—	67	1440	no	—
171	C₂²×D₄	—	67	90	no	—
172	C₂²×D₄	—	67	45	no	—
173	C₂²×D₄	—	67	240	no	—
174	C₂²×D₄	C₂²×D₆₇	1	45	yes	complemented
175	C₂³×C₄	—	67	90	no	—
176	C₂³×C₄	C₂²×D₆₇	1	15	yes	complemented
177	C₂⁴	—	536	60	no	—
178	C₂⁴	—	67	90	no	—
179	C₂⁴	—	67	240	no	—
180	C₂⁴	C₂³×D₆₇	1	15	yes	—
181	#41	—	67	960	no	—
182	#41	—	67	960	no	—
183	#41	—	67	1440	no	—
184	#41	—	67	480	no	—
185	#41	C₂³×D₆₇	1	480	yes	complemented
186	#41	C₂³×D₆₇	1	160	yes	complemented
187	C₂×Q₈	—	67	320	no	—
188	C₂×Q₈	—	67	30	no	—
189	C₂×Q₈	—	67	6	no	—
190	C₂×Q₈	—	67	120	no	—
191	C₂×Q₈	C₂³×D₆₇	1	20	yes	—
192	C₂×Q₈	C₂³×D₆₇	1	120	yes	complemented
193	C₂×D₄	—	67	120	no	—
194	C₂×D₄	—	67	720	no	—
195	C₂×D₄	—	67	2880	no	—
196	C₂×D₄	—	67	90	no	—
197	C₂×D₄	—	67	10	no	—
198	C₂×D₄	—	67	720	no	—
199	C₂×D₄	—	67	360	no	—
200	C₂×D₄	C₂³×D₆₇	1	60	yes	—
201	C₂×D₄	C₂³×D₆₇	1	360	yes	complemented
202	C₂²×C₄	—	67	30	no	—
203	C₂²×C₄	—	67	120	no	—
204	C₂²×C₄	—	67	720	no	—
205	C₂²×C₄	C₂³×D₆₇	1	45	yes	—
206	C₂²×C₄	C₂³×D₆₇	1	15	yes	—
207	C₂²×C₄	C₂³×D₆₇	1	120	yes	complemented
208	C₂³	—	4	30	no	—
209	C₂³	—	268	180	no	—
210	C₂³	—	536	240	no	—
211	C₂³	—	67	360	no	—
212	C₂³	—	67	20	no	—
213	C₂³	C₂⁴×D₆₇	1	15	yes	—
214	C₂³	C₂⁴×D₆₇	1	60	yes	—
215	Q₈	—	67	120	no	—
216	Q₈	—	67	24	no	—
217	Q₈	C₂⁴×D₆₇	1	80	yes	—
218	D₄	—	67	480	no	—
219	D₄	—	67	360	no	—
220	D₄	—	67	40	no	—
221	D₄	C₂⁴×D₆₇	1	240	yes	—
222	C₂×C₄	—	67	120	no	—
223	C₂×C₄	—	67	12	no	—
224	C₂×C₄	—	67	480	no	—
225	C₂×C₄	C₂⁴×D₆₇	1	180	yes	—
226	C₂×C₄	C₂⁴×D₆₇	1	60	yes	—
227	C₂×C₄	C₂⁴×D₆₇	1	15	yes	—
228	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
229	C₂²	—	2	30	no	—
230	C₂²	—	4	60	no	—
231	C₂²	—	268	360	no	—
232	C₂²	—	67	40	no	—
233	C₂²	—	134	40	no	—
234	C₂²	C₂⁵×D₆₇	1	30	yes	—
235	C₂²	C₂⁵×D₆₇	1	1	yes	characteristic
236	C₄	—	67	24	no	—
237	C₄	C₂⁵×D₆₇	1	30	yes	—
238	C₄	C₂⁵×D₆₇	1	2	yes	—
239	C₂	—	2	30	no	—
240	C₂	—	134	40	no	—
241	C₂	#22575020	1	2	yes	direct product factor
242	C₂	C₂⁶×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂⁵×D₆₇
commutator, cosocle	C₂×C₆₇	C₂⁷
Frattini	C₂	C₂⁶×D₆₇
Fitting	C₂×#585×C₆₇	C₂
socle	C₂²×C₆₇	C₂⁶
hypercenter	C₂×#585	D₆₇
hypocenter	C₆₇	C₂×H(7,𝔽₂)
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×#585

Complements

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

N	H ≅ G/N	type
1. #22575020	C₂	direct (N × H)
2. C₂×#16864233	C₂	semidirect
3. C₂×#16864232	C₂	semidirect
4. C₂²×#14524089	C₂	semidirect
5. C₂×H(5,𝔽₂)×D₆₇	C₂	semidirect
6. C₂×#585×C₆₇	C₂	semidirect
7. #16864233	C₂²	semidirect
8. #16864232	C₂²	semidirect
9. C₂×#14524091	C₂²	semidirect
10. C₂×#14524090	C₂²	semidirect
11. C₂×#14524089	C₂²	semidirect
12. C₂×#14524089	C₂²	semidirect
13. C₂×#14524089	C₂²	semidirect
14. C₂×#14524088	C₂²	semidirect
15. H(5,𝔽₂)×D₆₇	C₂²	semidirect
16. C₂²×#11761629	C₂²	semidirect
17. C₂²×#11761628	C₂²	semidirect
18. C₂×#41×D₆₇	C₂²	semidirect
19. C₂³×D₂₆₈	C₂²	semidirect
20. C₂²×D₄×D₆₇	C₂²	semidirect
21. #585×C₆₇	C₂²	semidirect
22. C₂×2₋¹⁺⁴×C₆₇	C₂²	semidirect
23. C₂×H(5,𝔽₂)×C₆₇	C₂²	semidirect
24. C₂²×#41×C₆₇	C₂²	semidirect
25. C₂⁴×D₆₇	C₂³	semidirect
26. #14524091	C₂³	semidirect
27. #14524090	C₂³	semidirect
28. #14524089	C₂³	semidirect
29. #14524089	C₂³	semidirect
30. #14524088	C₂³	semidirect
31. C₂×#11761629	C₂³	semidirect
32. C₂×#11761629	C₂³	semidirect
33. C₂×#11761628	C₂³	semidirect
34. C₂×#11761628	C₂³	semidirect
35. C₂×#11761628	C₂³	semidirect
36. C₂×#11761627	C₂³	semidirect
37. #41×D₆₇	C₂³	semidirect
38. C₂²×Dic₁₃₄	C₂³	semidirect
39. C₂×Q₈×D₆₇	C₂³	semidirect
40. C₂²×D₂₆₈	C₂³	semidirect
41. C₂²×D₂₆₈	C₂³	semidirect
42. C₂²×D₂₆₈	C₂³	semidirect
43. C₂²×C₂²⋊D₆₇	C₂³	semidirect
44. C₂×D₄×D₆₇	C₂³	semidirect
45. C₂×D₄×D₆₇	C₂³	semidirect
46. C₂²×C₄×D₆₇	C₂³	semidirect
47. 2₋¹⁺⁴×C₆₇	C₂³	semidirect
48. H(5,𝔽₂)×C₆₇	C₂³	semidirect
49. C₂×#41×C₆₇	C₂³	semidirect
50. C₂×#41×C₆₇	C₂³	semidirect
51. C₂×#41×C₆₇	C₂³	semidirect
52. C₂²×Q₈×C₆₇	C₂³	semidirect
53. C₂²×D₄×C₆₇	C₂³	semidirect
54. C₂³×C₄×C₆₇	C₂³	semidirect
57. C₂³×D₆₇	C₂⁴	semidirect
58. #11761629	C₂⁴	semidirect
59. #11761628	C₂⁴	semidirect
60. #11761628	C₂⁴	semidirect
61. #11761627	C₂⁴	semidirect
64. C₂×Dic₁₃₄	C₂⁴	semidirect
65. Q₈×D₆₇	C₂⁴	semidirect
68. C₂×D₂₆₈	C₂⁴	semidirect
69. C₂×D₂₆₈	C₂⁴	semidirect
71. C₂×C₂²⋊D₆₇	C₂⁴	semidirect
72. D₄×D₆₇	C₂⁴	semidirect
75. C₂×C₄×D₆₇	C₂⁴	semidirect
77. #41×C₆₇	C₂⁴	semidirect
78. #41×C₆₇	C₂⁴	semidirect
80. C₂×Q₈×C₆₇	C₂⁴	semidirect
82. C₂×D₄×C₆₇	C₂⁴	semidirect
85. C₂²×C₄×C₆₇	C₂⁴	semidirect
119. C₂×C₆₇	H(7,𝔽₂)	semidirect
125. C₂×#585	D₆₇	semidirect
128. C₆₇	C₂×H(7,𝔽₂)	semidirect
131. #585	C₂×D₆₇	semidirect
134. C₂×2₋¹⁺⁴	C₂×D₆₇	semidirect
140. C₂×H(5,𝔽₂)	C₂×D₆₇	semidirect
144. C₂²×#41	C₂×D₆₇	semidirect
150. 2₋¹⁺⁴	C₂²×D₆₇	semidirect
154. H(5,𝔽₂)	C₂²×D₆₇	semidirect
162. C₂×#41	C₂²×D₆₇	semidirect
163. C₂×#41	C₂²×D₆₇	semidirect
164. C₂×#41	C₂²×D₆₇	semidirect
167. C₂²×Q₈	C₂²×D₆₇	semidirect
174. C₂²×D₄	C₂²×D₆₇	semidirect
176. C₂³×C₄	C₂²×D₆₇	semidirect
185. #41	C₂³×D₆₇	semidirect
186. #41	C₂³×D₆₇	semidirect
192. C₂×Q₈	C₂³×D₆₇	semidirect
201. C₂×D₄	C₂³×D₆₇	semidirect
207. C₂²×C₄	C₂³×D₆₇	semidirect
241. C₂	#22575020	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22575020	2
C₂×#16864233	2
C₂×#16864232	2
C₂²×#14524089	2
C₂×H(5,𝔽₂)×D₆₇	2
C₂×#585×C₆₇	2
C₂×H(7,𝔽₂)	67

Subgroups — C2×#22575020 #31769728

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22575020 #31769728