Subgroups — C₂×#72210587 #72238591

Order 2048.

At least 6551 conjugacy classes of subgroups; only the 832 normal subgroups are shown.

The subgroup lattice has 261 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#139718	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#139700	C₂	1	2	yes	maximal, complemented
3	C₂×#144386	C₂	1	2	yes	maximal, complemented
4	C₂×#511771	C₂	1	2	yes	maximal, complemented
5	C₂×#346446	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#139716	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#647656	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#144366	C₂	1	2	yes	maximal, complemented
9	C₂×#133822	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#6499867	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#347042	C₂	1	1	yes	maximal, characteristic, complemented
12	#72210587	C₂	1	16	yes	maximal, direct product factor
13	C₂×#62817	C₂²	1	1	yes	characteristic, complemented
14	C₂³×D₄²	C₂²	1	1	yes	characteristic, complemented
15	C₂²×#2846	C₂²	1	1	yes	characteristic
16	#6499867	C₂²	1	8	yes	complemented
17	C₂×#23900	C₂²	1	2	yes	—
18	C₂×#23900	C₂²	1	2	yes	complemented
19	C₂×#23900	C₂²	1	2	yes	complemented
20	C₂×#23906	C₂²	1	2	yes	complemented
21	C₂×#21746	C₂²	1	1	yes	characteristic, complemented
22	#647656	C₂²	1	8	yes	complemented
23	C₂×#15947	C₂²	1	1	yes	characteristic
24	C₂×#15947	C₂²	1	1	yes	characteristic, complemented
25	C₂×#15258	C₂²	1	1	yes	characteristic
26	C₂×#15258	C₂²	1	1	yes	characteristic, complemented
27	C₂×#14134	C₂²	1	1	yes	characteristic, complemented
28	#511771	C₂²	1	16	yes	complemented
29	C₂×#12693	C₂²	1	2	yes	complemented
30	C₂×#12687	C₂²	1	2	yes	complemented
31	C₂×#12687	C₂²	1	2	yes	complemented
32	C₂×#12687	C₂²	1	2	yes	complemented
33	#347042	C₂²	1	8	yes	complemented
34	#346446	C₂²	1	8	yes	complemented
35	C₂×#8547	C₂²	1	1	yes	characteristic, complemented
36	C₂×#8541	C₂²	1	1	yes	characteristic, complemented
37	C₂×#8534	C₂²	1	2	yes	complemented
38	C₂×#8453	C₂²	1	2	yes	complemented
39	#144386	C₂²	1	16	yes	complemented
40	#144366	C₂²	1	16	yes	complemented
41	#139718	C₂²	1	8	yes	complemented
42	#139716	C₂²	1	8	yes	complemented
43	#139700	C₂²	1	16	yes	complemented
44	#133822	C₂²	1	8	yes	complemented
45	C₂×#7304	C₂²	1	2	yes	complemented
46	C₂×#7307	C₂²	1	2	yes	complemented
47	C₂⁸	D₄	1	1	yes	characteristic, complemented
48	#62817	C₂³	1	4	yes	complemented
49	C₂×#3484	C₂³	1	2	yes	—
50	C₂²×D₄²	C₂³	1	4	yes	complemented
51	C₂³×#127	C₂³	1	1	yes	characteristic
52	C₂³×#120	D₄	1	2	yes	complemented
53	C₂³×#120	C₂³	1	1	yes	characteristic
54	C₂³×#120	C₂³	1	1	yes	characteristic
55	C₂³×#120	C₂³	1	1	yes	characteristic
56	C₂³×#120	C₂³	1	1	yes	characteristic
57	C₂²×B(4,𝔽₂)	D₄	1	4	yes	complemented
58	C₂×#2846	D₄	1	4	yes	complemented
59	C₂×#2846	C₂³	1	4	yes	—
60	C₂²×#392	D₄	1	2	yes	complemented
61	C₂²×#379	D₄	1	1	yes	characteristic, complemented
62	C₂²×#379	D₄	1	1	yes	characteristic, complemented
63	C₂²×#379	D₄	1	1	yes	characteristic, complemented
64	#23906	C₂³	1	8	yes	complemented
65	#23900	D₄	1	8	yes	complemented
66	#23900	C₂³	1	8	yes	—
67	#23900	C₂³	1	8	yes	complemented
68	#23900	C₂³	1	8	yes	complemented
69	#21746	C₂³	1	4	yes	complemented
70	C₂×#1889	D₄	1	4	yes	complemented
71	C₂×#1783	C₂³	1	2	yes	—
72	C₂×#1781	C₂³	1	2	yes	—
73	C₂×#1781	C₂³	1	2	yes	—
74	#15947	D₄	1	4	yes	complemented
75	#15947	C₂³	1	4	yes	—
76	#15947	C₂³	1	4	yes	complemented
77	#15258	D₄	1	4	yes	complemented
78	#15258	C₂³	1	4	yes	—
79	#15258	C₂³	1	4	yes	complemented
80	#14134	C₂³	1	4	yes	complemented
81	C₂×#1438	D₄	1	2	yes	complemented
82	C₂×#1438	C₂³	1	1	yes	characteristic
83	C₂×#1438	C₂³	1	1	yes	characteristic
84	C₂×#1437	D₄	1	2	yes	complemented
85	#12693	C₂³	1	8	yes	complemented
86	#12687	C₂³	1	8	yes	complemented
87	#12687	C₂³	1	8	yes	complemented
88	#12687	C₂³	1	8	yes	complemented
89	#8547	C₂³	1	4	yes	complemented
90	#8541	C₂³	1	4	yes	complemented
91	#8534	C₂³	1	8	yes	complemented
92	#8453	C₂³	1	8	yes	complemented
93	#7307	C₂³	1	8	yes	complemented
94	#7304	C₂³	1	8	yes	complemented
95	C₂⁷	C₂×D₄	1	1	yes	characteristic
96	C₂⁷	C₂×D₄	1	1	yes	characteristic
97	C₂⁷	C₂×D₄	1	1	yes	characteristic
98	C₂⁷	C₂×D₄	1	1	yes	characteristic, complemented
99	C₂⁷	C₂×D₄	1	4	yes	complemented
100	C₂²×H(5,𝔽₂)	C₂×D₄	1	2	yes	complemented
101	C₂⁴×D₄	C₂×D₄	1	2	yes	—
102	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic, complemented
103	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
104	#3484	C₂×D₄	1	4	yes	complemented
105	#3484	C₂⁴	1	4	yes	—
106	C₂²×#127	C₂×D₄	1	2	yes	complemented
107	C₂²×#127	C₂⁴	1	2	yes	—
108	C₂²×#120	C₂×D₄	1	4	yes	—
109	C₂²×#120	C₂×D₄	1	4	yes	—
110	C₂²×#120	C₂×D₄	1	2	yes	—
111	C₂²×#120	C₂×D₄	1	2	yes	—
112	C₂²×#120	C₂×D₄	1	2	yes	—
113	C₂²×#120	C₂×D₄	1	2	yes	—
114	C₂²×#120	C₂×D₄	1	2	yes	—
115	C₂²×#120	C₂×D₄	1	2	yes	complemented
116	C₂²×#120	C₂×D₄	1	8	yes	complemented
117	C₂²×#120	C₂⁴	1	2	yes	—
118	C₂²×#120	C₂⁴	1	2	yes	—
119	C₂²×#120	C₂⁴	1	2	yes	—
120	C₂²×#120	C₂⁴	1	2	yes	—
121	C₂×#518	C₂×D₄	1	2	yes	complemented
122	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
123	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
124	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
125	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
126	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic, complemented
127	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	complemented
128	C₂³×C₄²	C₂×D₄	1	1	yes	characteristic, complemented
129	C₂×B(4,𝔽₂)	C₂×D₄	1	16	yes	complemented
130	C₂²×#99	C₂×D₄	1	4	yes	complemented
131	C₂×#392	C₂×D₄	1	8	yes	complemented
132	C₂×#379	C₂×D₄	1	4	yes	complemented
133	C₂×#379	C₂×D₄	1	4	yes	complemented
134	C₂×#379	C₂×D₄	1	4	yes	complemented
135	C₂²×#95	C₂×D₄	1	2	yes	complemented
136	#1889	C₂×D₄	1	16	yes	complemented
137	#1783	C₂⁴	1	4	yes	—
138	#1781	C₂×D₄	1	4	yes	complemented
139	#1781	C₂⁴	1	4	yes	—
140	#1781	C₂⁴	1	4	yes	—
141	#1438	C₂×D₄	1	2	yes	complemented
142	#1438	C₂×D₄	1	8	yes	complemented
143	#1438	C₂×D₄	1	2	yes	complemented
144	#1438	C₂⁴	1	2	yes	—
145	#1438	C₂⁴	1	2	yes	—
146	#1437	C₂×D₄	1	8	yes	complemented
147	C₂⁶	#120	1	2	yes	—
148	C₂⁶	#120	1	2	yes	—
149	C₂⁶	#120	1	2	yes	—
150	C₂⁶	#120	1	2	yes	—
151	C₂⁶	C₂²×D₄	1	1	yes	characteristic
152	C₂⁶	C₂²×D₄	1	2	yes	—
153	C₂⁶	C₂²×D₄	1	2	yes	—
154	C₂⁶	C₂²×D₄	1	1	yes	characteristic
155	C₂⁶	C₂²×D₄	1	2	yes	—
156	C₂⁶	C₂²×D₄	1	2	yes	complemented
157	C₂×H(5,𝔽₂)	C₂²×D₄	1	4	yes	complemented
158	C₂³×D₄	#120	1	4	yes	—
159	C₂³×D₄	C₂²×D₄	1	4	yes	—
160	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
161	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
162	C₂³×D₄	C₂²×D₄	1	2	yes	—
163	C₂³×D₄	C₂²×D₄	1	2	yes	—
164	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
165	C₂³×D₄	C₂²×D₄	1	2	yes	complemented
166	C₂³×D₄	H(5,𝔽₂)	1	2	yes	—
167	C₂³×D₄	H(5,𝔽₂)	1	2	yes	—
168	C₂³×D₄	C₂⁵	1	1	yes	characteristic
169	C₂⁴×C₄	#120	1	2	yes	—
170	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
171	C₂×#120	#120	1	8	yes	—
172	C₂×#120	C₂²×D₄	1	8	yes	—
173	#518	C₂²×D₄	1	4	yes	complemented
174	C₂²×C₂²⋊C₄	#120	1	2	yes	—
175	C₂²×C₂²⋊C₄	#120	1	4	yes	—
176	C₂²×C₂²⋊C₄	#120	1	2	yes	—
177	C₂²×C₂²⋊C₄	#120	1	2	yes	—
178	C₂²×C₂²⋊C₄	C₂²×D₄	1	4	yes	—
179	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
180	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
181	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
182	C₂²×C₂²⋊C₄	C₂²×D₄	1	2	yes	complemented
183	C₂²×C₂²⋊C₄	C₂²×D₄	1	4	yes	complemented
184	C₂²×C₄²	C₂²×D₄	1	2	yes	complemented
185	C₂×#99	C₂²×D₄	1	8	yes	complemented
186	C₂×#95	C₂²×D₄	1	4	yes	complemented
187	C₂⁵	B(4,𝔽₂)	1	2	yes	—
188	C₂⁵	B(4,𝔽₂)	1	4	yes	—
189	C₂⁵	B(4,𝔽₂)	1	2	yes	—
190	C₂⁵	B(4,𝔽₂)	1	8	yes	complemented
191	C₂⁵	C₂×#120	1	1	yes	characteristic
192	C₂⁵	C₂×#120	1	1	yes	characteristic
193	C₂⁵	C₂×#120	1	4	yes	—
194	C₂⁵	C₂×#120	1	1	yes	characteristic
195	C₂⁵	C₂×#120	1	1	yes	characteristic
196	C₂⁵	C₂×#120	1	4	yes	—
197	C₂⁵	C₂×#120	1	1	yes	characteristic
198	C₂⁵	#534	1	1	yes	characteristic
199	C₂⁵	#534	1	1	yes	characteristic
200	C₂⁵	D₄²	1	2	yes	—
201	C₂⁵	D₄²	1	2	yes	—
202	C₂⁵	C₂³×D₄	1	1	yes	characteristic
203	C₂⁵	C₂³×D₄	1	1	yes	characteristic
204	C₂²×D₄	C₂×#120	1	2	yes	—
205	C₂²×D₄	#534	1	2	yes	—
206	C₂²×D₄	D₄²	1	2	yes	—
207	C₂²×D₄	D₄²	1	2	yes	—
208	C₂²×D₄	D₄²	1	2	yes	—
209	C₂²×D₄	D₄²	1	2	yes	—
210	C₂²×D₄	C₂³×D₄	1	2	yes	—
211	C₂²×D₄	C₂×H(5,𝔽₂)	1	2	yes	—
212	C₂³×C₄	C₂×#120	1	4	yes	—
213	C₂³×C₄	C₂×#120	1	1	yes	characteristic
214	C₂³×C₄	D₄²	1	2	yes	—
215	C₂³×C₄	D₄²	1	2	yes	—
216	C₂³×C₄	C₂³×D₄	1	1	yes	characteristic
217	C₂×C₂²⋊C₄	C₂×#120	1	4	yes	—
218	C₂⁴	#2196	1	8	yes	complemented
219	C₂⁴	#2403	1	1	yes	characteristic
220	C₂⁴	#2403	1	1	yes	characteristic
221	C₂⁴	#2403	1	1	yes	characteristic
222	C₂⁴	#2403	1	2	yes	—
223	C₂⁴	#2403	1	1	yes	characteristic
224	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
225	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
226	C₂⁴	C₂×B(4,𝔽₂)	1	4	yes	—
227	C₂⁴	C₂×B(4,𝔽₂)	1	4	yes	—
228	C₂⁴	C₂×B(4,𝔽₂)	1	4	yes	—
229	C₂⁴	C₂×B(4,𝔽₂)	1	8	yes	complemented
230	C₂⁴	C₂²×#120	1	1	yes	characteristic
231	C₂⁴	C₂²×#120	1	1	yes	characteristic
232	C₂⁴	C₂×#534	1	1	yes	characteristic
233	C₂⁴	C₂×D₄²	1	2	yes	—
234	C₂²×C₄	#2403	1	2	yes	—
235	C₂²×C₄	C₂×D₄²	1	2	yes	—
236	C₂³	#23895	1	2	yes	—
237	C₂³	#23900	1	2	yes	—
238	C₂³	#23900	1	1	yes	characteristic
239	C₂³	#23900	1	2	yes	—
240	C₂³	#23900	1	2	yes	—
241	C₂³	#23900	1	1	yes	characteristic
242	C₂³	C₂×#2196	1	4	yes	—
243	C₂³	C₂×#2196	1	8	yes	complemented
244	C₂³	#33551	1	4	yes	—
245	C₂³	C₂×#2403	1	1	yes	characteristic
246	C₂³	C₂²×B(4,𝔽₂)	1	2	yes	—
247	C₂³	C₂²×B(4,𝔽₂)	1	2	yes	—
248	C₂²	#245225	1	2	yes	—
249	C₂²	#245225	1	2	yes	—
250	C₂²	#499787	1	4	yes	—
251	C₂²	#499787	1	4	yes	—
252	C₂²	C₂×#23900	1	1	yes	characteristic
253	C₂²	C₂×#23900	1	1	yes	characteristic
254	C₂²	C₂×#23895	1	1	yes	characteristic
255	C₂²	C₂²×#2196	1	2	yes	—
256	C₂²	C₂×#33551	1	2	yes	—
257	C₂	#72210587	1	4	yes	direct product factor
258	C₂	C₂×#245225	1	1	yes	characteristic
259	C₂	C₂×#499787	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	#23900
commutator, Frattini, cosocle	C₂³×D₄	C₂⁵
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ C₂³×#120 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#139718	C₂	semidirect
2. C₂×#139700	C₂	semidirect
3. C₂×#144386	C₂	semidirect
4. C₂×#511771	C₂	semidirect
5. C₂×#346446	C₂	semidirect
6. C₂×#139716	C₂	semidirect
7. C₂×#647656	C₂	semidirect
8. C₂×#144366	C₂	semidirect
9. C₂×#133822	C₂	semidirect
10. C₂×#6499867	C₂	semidirect
11. C₂×#347042	C₂	semidirect
12. #72210587	C₂	direct (N × H)
13. C₂×#62817	C₂²	semidirect
14. C₂³×D₄²	C₂²	semidirect
16. #6499867	C₂²	semidirect
18. C₂×#23900	C₂²	semidirect
19. C₂×#23900	C₂²	semidirect
20. C₂×#23906	C₂²	semidirect
21. C₂×#21746	C₂²	semidirect
22. #647656	C₂²	semidirect
24. C₂×#15947	C₂²	semidirect
26. C₂×#15258	C₂²	semidirect
27. C₂×#14134	C₂²	semidirect
28. #511771	C₂²	semidirect
29. C₂×#12693	C₂²	semidirect
30. C₂×#12687	C₂²	semidirect
31. C₂×#12687	C₂²	semidirect
32. C₂×#12687	C₂²	semidirect
33. #347042	C₂²	semidirect
34. #346446	C₂²	semidirect
35. C₂×#8547	C₂²	semidirect
36. C₂×#8541	C₂²	semidirect
37. C₂×#8534	C₂²	semidirect
38. C₂×#8453	C₂²	semidirect
39. #144386	C₂²	semidirect
40. #144366	C₂²	semidirect
41. #139718	C₂²	semidirect
42. #139716	C₂²	semidirect
43. #139700	C₂²	semidirect
44. #133822	C₂²	semidirect
45. C₂×#7304	C₂²	semidirect
46. C₂×#7307	C₂²	semidirect
47. C₂⁸	D₄	semidirect
48. #62817	C₂³	semidirect
50. C₂²×D₄²	C₂³	semidirect
52. C₂³×#120	D₄	semidirect
57. C₂²×B(4,𝔽₂)	D₄	semidirect
58. C₂×#2846	D₄	semidirect
60. C₂²×#392	D₄	semidirect
61. C₂²×#379	D₄	semidirect
62. C₂²×#379	D₄	semidirect
63. C₂²×#379	D₄	semidirect
64. #23906	C₂³	semidirect
65. #23900	D₄	semidirect
67. #23900	C₂³	semidirect
68. #23900	C₂³	semidirect
69. #21746	C₂³	semidirect
70. C₂×#1889	D₄	semidirect
74. #15947	D₄	semidirect
76. #15947	C₂³	semidirect
77. #15258	D₄	semidirect
79. #15258	C₂³	semidirect
80. #14134	C₂³	semidirect
81. C₂×#1438	D₄	semidirect
84. C₂×#1437	D₄	semidirect
85. #12693	C₂³	semidirect
86. #12687	C₂³	semidirect
87. #12687	C₂³	semidirect
88. #12687	C₂³	semidirect
89. #8547	C₂³	semidirect
90. #8541	C₂³	semidirect
91. #8534	C₂³	semidirect
92. #8453	C₂³	semidirect
93. #7307	C₂³	semidirect
94. #7304	C₂³	semidirect
98. C₂⁷	C₂×D₄	semidirect
99. C₂⁷	C₂×D₄	semidirect
100. C₂²×H(5,𝔽₂)	C₂×D₄	semidirect
102. C₂⁴×D₄	C₂×D₄	semidirect
104. #3484	C₂×D₄	semidirect
106. C₂²×#127	C₂×D₄	semidirect
115. C₂²×#120	C₂×D₄	semidirect
116. C₂²×#120	C₂×D₄	semidirect
121. C₂×#518	C₂×D₄	semidirect
126. C₂³×C₂²⋊C₄	C₂×D₄	semidirect
127. C₂³×C₂²⋊C₄	C₂×D₄	semidirect
128. C₂³×C₄²	C₂×D₄	semidirect
129. C₂×B(4,𝔽₂)	C₂×D₄	semidirect
130. C₂²×#99	C₂×D₄	semidirect
131. C₂×#392	C₂×D₄	semidirect
132. C₂×#379	C₂×D₄	semidirect
133. C₂×#379	C₂×D₄	semidirect
134. C₂×#379	C₂×D₄	semidirect
135. C₂²×#95	C₂×D₄	semidirect
136. #1889	C₂×D₄	semidirect
138. #1781	C₂×D₄	semidirect
141. #1438	C₂×D₄	semidirect
142. #1438	C₂×D₄	semidirect
143. #1438	C₂×D₄	semidirect
146. #1437	C₂×D₄	semidirect
156. C₂⁶	C₂²×D₄	semidirect
157. C₂×H(5,𝔽₂)	C₂²×D₄	semidirect
165. C₂³×D₄	C₂²×D₄	semidirect
173. #518	C₂²×D₄	semidirect
182. C₂²×C₂²⋊C₄	C₂²×D₄	semidirect
183. C₂²×C₂²⋊C₄	C₂²×D₄	semidirect
184. C₂²×C₄²	C₂²×D₄	semidirect
185. C₂×#99	C₂²×D₄	semidirect
186. C₂×#95	C₂²×D₄	semidirect
190. C₂⁵	B(4,𝔽₂)	semidirect
218. C₂⁴	#2196	semidirect
229. C₂⁴	C₂×B(4,𝔽₂)	semidirect
243. C₂³	C₂×#2196	semidirect
257. C₂	#72210587	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#139718	2
C₂×#139700	2
C₂×#144386	2
C₂×#511771	2
C₂×#346446	2
C₂×#139716	2
C₂×#647656	2
C₂×#144366	2
C₂×#133822	2
C₂×#6499867	2
C₂×#347042	2
#72210587	2

Subgroups — C2×#72210587 #72238591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#72210587 #72238591