Subgroups — #72264779

Order 2048.

At least 7551 conjugacy classes of subgroups; only the 292 normal subgroups are shown.

The subgroup lattice has 221 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72264770	C₂	1	1	yes	maximal, characteristic, complemented
2	#72246299	C₂	1	1	yes	maximal, characteristic, complemented
3	#72246286	C₂	1	1	yes	maximal, characteristic, complemented
4	#72225644	C₂	1	1	yes	maximal, characteristic, complemented
5	#72219240	C₂	1	1	yes	maximal, characteristic, complemented
6	#72219160	C₂	1	1	yes	maximal, characteristic, complemented
7	#72219158	C₂	1	1	yes	maximal, characteristic, complemented
8	#72219157	C₂	1	1	yes	maximal, characteristic, complemented
9	#72218114	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#144366	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#144364	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#144378	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#144378	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#347042	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#511790	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#62817	C₂²	1	1	yes	characteristic
17	#7642244	C₂²	1	1	yes	characteristic, complemented
18	C₂×#23895	C₂²	1	1	yes	characteristic
19	C₂×#23895	C₂²	1	1	yes	characteristic
20	C₂×#23895	C₂²	1	1	yes	characteristic, complemented
21	C₂×#23900	C₂²	1	1	yes	characteristic
22	C₂×#23900	C₂²	1	1	yes	characteristic
23	C₂×#23900	C₂²	1	1	yes	characteristic
24	C₂×#23900	C₂²	1	1	yes	characteristic
25	C₂×#23900	C₂²	1	1	yes	characteristic, complemented
26	C₂×#23897	C₂²	1	1	yes	characteristic
27	#6431795	C₂²	1	1	yes	characteristic
28	C₂×#15947	C₂²	1	1	yes	characteristic
29	C₂×#12707	C₂²	1	1	yes	characteristic
30	C₂×#12707	C₂²	1	1	yes	characteristic, complemented
31	C₂×#12707	C₂²	1	1	yes	characteristic, complemented
32	C₂×#12705	C₂²	1	1	yes	characteristic
33	C₂×#12687	C₂²	1	1	yes	characteristic
34	C₂×#12687	C₂²	1	1	yes	characteristic, complemented
35	#382434	C₂²	1	1	yes	characteristic
36	#359549	C₂²	1	1	yes	characteristic
37	#359501	C₂²	1	1	yes	characteristic
38	#355096	C₂²	1	1	yes	characteristic, complemented
39	C₂×#8541	C₂²	1	1	yes	characteristic
40	C₂×#8541	C₂²	1	1	yes	characteristic, complemented
41	C₂×#8534	C₂²	1	1	yes	characteristic
42	C₂×#8534	C₂²	1	1	yes	characteristic
43	C₂×#8534	C₂²	1	1	yes	characteristic, complemented
44	C₂×#8453	C₂²	1	1	yes	characteristic, complemented
45	#146475	C₂²	1	1	yes	characteristic, complemented
46	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
49	C₂×#7307	C₂²	1	1	yes	characteristic, complemented
50	#108222	C₂²	1	1	yes	characteristic, complemented
51	#62842	D₄	1	1	yes	characteristic, complemented
52	C₂×#3484	C₂³	1	1	yes	characteristic
53	C₂×#3484	C₂³	1	1	yes	characteristic
54	C₂×#3484	C₂³	1	1	yes	characteristic
55	C₂×#3484	C₂³	1	1	yes	characteristic
56	C₂³×#120	C₂³	1	1	yes	characteristic
57	C₂³×#120	C₂³	1	1	yes	characteristic
58	C₂×#3026	D₄	1	2	yes	complemented
59	C₂×#2883	D₄	1	1	yes	characteristic, complemented
60	C₂×#2883	D₄	1	1	yes	characteristic, complemented
61	C₂×#2846	D₄	1	2	yes	complemented
62	#34171	C₂³	1	1	yes	characteristic
63	C₂×#1913	D₄	1	2	yes	complemented
64	C₂×#1898	D₄	1	2	yes	complemented
65	#20609	D₄	1	2	yes	complemented
66	C₂×#1783	C₂³	1	1	yes	characteristic
67	C₂×#1781	C₂³	1	1	yes	characteristic
68	C₂×#1781	C₂³	1	1	yes	characteristic
69	C₂×#1781	C₂³	1	1	yes	characteristic
70	C₂×#1781	C₂³	1	1	yes	characteristic
71	C₂×#1781	C₂³	1	1	yes	characteristic
72	C₂×#1781	C₂³	1	1	yes	characteristic
73	#20354	D₄	1	1	yes	characteristic, complemented
74	#14679	D₄	1	2	yes	complemented
75	C₂×#1438	C₂³	1	1	yes	characteristic
76	#12707	D₄	1	4	yes	complemented
77	#12687	D₄	1	4	yes	complemented
78	#10754	D₄	1	2	yes	complemented
79	#10735	D₄	1	2	yes	complemented
80	#8541	D₄	1	4	yes	complemented
81	#8534	D₄	1	4	yes	complemented
82	C₂⁷	C₂×D₄	1	1	yes	characteristic
83	C₂⁷	C₂×D₄	1	1	yes	characteristic
84	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
85	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
86	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
87	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
88	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
89	#3484	C₂×D₄	1	2	yes	—
90	#3484	C₂×D₄	1	2	yes	—
91	C₂×#534	C₂×D₄	1	2	yes	—
92	C₂²×#120	C₂×D₄	1	2	yes	—
93	C₂²×#120	C₂×D₄	1	2	yes	—
94	C₂²×#120	C₂×D₄	1	2	yes	—
95	C₂×#518	C₂×D₄	1	1	yes	characteristic
96	C₂×#518	C₂×D₄	1	1	yes	characteristic
97	C₂×#518	C₂×D₄	1	1	yes	characteristic
98	C₂×#518	C₂×D₄	1	1	yes	characteristic
99	C₂×#518	C₂×D₄	1	1	yes	characteristic
100	C₂×#518	C₂×D₄	1	1	yes	characteristic
101	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
102	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
103	C₂²×#99	C₂×D₄	1	2	yes	—
104	C₂²×#99	C₂×D₄	1	2	yes	—
105	#2308	C₂×D₄	1	2	yes	—
106	C₂×#379	C₂×D₄	1	2	yes	—
107	C₂×#379	C₂×D₄	1	2	yes	—
108	C₂×#379	C₂×D₄	1	2	yes	—
109	#1783	C₂×D₄	1	2	yes	—
110	#1781	C₂×D₄	1	2	yes	—
111	#1781	C₂×D₄	1	2	yes	—
112	#1781	C₂×D₄	1	2	yes	—
113	#1781	C₂×D₄	1	2	yes	—
114	#1781	C₂×D₄	1	2	yes	—
115	#1781	C₂×D₄	1	2	yes	—
116	#1438	C₂×D₄	1	2	yes	—
117	C₂⁶	#127	1	2	yes	complemented
118	C₂⁶	C₂²×D₄	1	1	yes	characteristic
119	C₂⁶	C₂²×D₄	1	1	yes	characteristic
120	C₂×H(5,𝔽₂)	#127	1	2	yes	complemented
121	C₂³×D₄	#120	1	2	yes	—
122	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
123	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
124	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
125	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
126	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
127	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
128	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
129	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
134	C₂⁴×C₄	H(5,𝔽₂)	1	1	yes	characteristic
135	#518	#127	1	2	yes	complemented
136	C₂²×C₂²⋊C₄	#120	1	2	yes	—
137	C₂²×C₂²⋊C₄	#120	1	2	yes	—
138	C₂²×C₂²⋊C₄	#120	1	2	yes	—
139	C₂²×C₂²⋊C₄	#127	1	2	yes	complemented
140	C₂×#99	#120	1	4	yes	complemented
141	#379	#120	1	4	yes	complemented
142	C₂⁵	B(4,𝔽₂)	1	2	yes	—
143	C₂⁵	C₂×#120	1	1	yes	characteristic
144	C₂⁵	C₂×#120	1	1	yes	characteristic
145	C₂⁵	C₂×#120	1	1	yes	characteristic
146	C₂⁵	C₂×#127	1	1	yes	characteristic
147	C₂⁵	#534	1	1	yes	characteristic
148	C₂⁵	#535	1	1	yes	characteristic
149	C₂⁵	D₄²	1	1	yes	characteristic
150	C₂⁵	D₄²	1	1	yes	characteristic
151	C₂⁵	D₄²	1	1	yes	characteristic
152	C₂⁵	D₄²	1	1	yes	characteristic
153	C₂⁵	D₄²	1	1	yes	characteristic
154	C₂⁵	D₄²	1	1	yes	characteristic
155	C₂⁵	D₄²	1	1	yes	characteristic
156	C₂²×D₄	C₂×#120	1	1	yes	characteristic
157	C₂²×D₄	C₂×#127	1	1	yes	characteristic
158	C₂²×D₄	#534	1	1	yes	characteristic
159	C₂²×D₄	#535	1	1	yes	characteristic
160	C₂²×D₄	#535	1	1	yes	characteristic
161	C₂²×D₄	#535	1	1	yes	characteristic
162	C₂²×D₄	D₄²	1	1	yes	characteristic
163	C₂²×D₄	D₄²	1	1	yes	characteristic
164	C₂²×D₄	D₄²	1	1	yes	characteristic
165	C₂²×D₄	D₄²	1	1	yes	characteristic
166	C₂²×D₄	D₄²	1	1	yes	characteristic
167	C₂²×D₄	D₄²	1	1	yes	characteristic
168	C₂²×D₄	D₄²	1	1	yes	characteristic
169	C₂²×D₄	D₄²	1	1	yes	characteristic
170	C₂²×D₄	D₄²	1	1	yes	characteristic
171	C₂²×D₄	#560	1	1	yes	characteristic
172	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
173	C₂³×C₄	B(4,𝔽₂)	1	4	yes	complemented
174	C₂³×C₄	#534	1	1	yes	characteristic
175	C₂³×C₄	#535	1	1	yes	characteristic
176	C₂³×C₄	#535	1	1	yes	characteristic
177	C₂³×C₄	D₄²	1	1	yes	characteristic
178	C₂³×C₄	D₄²	1	1	yes	characteristic
179	C₂³×C₄	#560	1	1	yes	characteristic
180	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	4	yes	—
181	C₂⁴	#2403	1	1	yes	characteristic
182	C₂⁴	#2403	1	1	yes	characteristic
183	C₂⁴	#2403	1	1	yes	characteristic
184	C₂⁴	#2433	1	1	yes	characteristic
185	C₂⁴	#2433	1	1	yes	characteristic
186	C₂⁴	#2433	1	1	yes	characteristic
187	C₂⁴	#2433	1	1	yes	characteristic
188	C₂⁴	#2433	1	1	yes	characteristic
189	C₂⁴	#2669	1	1	yes	characteristic
190	C₂⁴	#2669	1	1	yes	characteristic
191	C₂⁴	#2669	1	1	yes	characteristic
192	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
193	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
194	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
195	C₂⁴	#3027	1	2	yes	—
196	C₂²×C₄	#2433	1	1	yes	characteristic
197	C₂²×C₄	#2669	1	1	yes	characteristic
198	C₂²×C₄	#2669	1	1	yes	characteristic
199	C₂²×C₄	#2669	1	1	yes	characteristic
200	C₂³	#15258	1	1	yes	characteristic
201	C₂³	#23895	1	1	yes	characteristic
202	C₂³	#23895	1	1	yes	characteristic
203	C₂³	#23900	1	1	yes	characteristic
204	C₂³	#23900	1	1	yes	characteristic
205	C₂³	#23900	1	2	yes	—
206	C₂³	#23900	1	2	yes	—
207	C₂³	#23900	1	1	yes	characteristic
208	C₂³	#23904	1	1	yes	characteristic
209	C₂³	#25771	1	2	yes	—
210	C₂³	#25771	1	2	yes	—
211	C₂²	#245225	1	1	yes	characteristic
212	C₂²	#245233	1	1	yes	characteristic
213	C₂²	#245249	1	1	yes	characteristic
214	C₂²	#245252	1	1	yes	characteristic
215	C₂²	#417744	1	1	yes	characteristic
216	C₂²	#417821	1	1	yes	characteristic
217	C₂²	#417824	1	1	yes	characteristic
218	C₂	B(5,𝔽₂)	1	2	yes	—
219	C₂	#72237501	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#245225
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
LCS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72264770	C₂	semidirect
2. #72246299	C₂	semidirect
3. #72246286	C₂	semidirect
4. #72225644	C₂	semidirect
5. #72219240	C₂	semidirect
6. #72219160	C₂	semidirect
7. #72219158	C₂	semidirect
8. #72219157	C₂	semidirect
9. #72218114	C₂	semidirect
10. C₂×#144366	C₂	semidirect
11. C₂×#144364	C₂	semidirect
12. C₂×#144378	C₂	semidirect
13. C₂×#144378	C₂	semidirect
14. C₂×#347042	C₂	semidirect
15. C₂×#511790	C₂	semidirect
17. #7642244	C₂²	semidirect
20. C₂×#23895	C₂²	semidirect
25. C₂×#23900	C₂²	semidirect
30. C₂×#12707	C₂²	semidirect
31. C₂×#12707	C₂²	semidirect
34. C₂×#12687	C₂²	semidirect
38. #355096	C₂²	semidirect
40. C₂×#8541	C₂²	semidirect
43. C₂×#8534	C₂²	semidirect
44. C₂×#8453	C₂²	semidirect
45. #146475	C₂²	semidirect
46. C₂×#7316	C₂²	semidirect
47. C₂×#7310	C₂²	semidirect
48. C₂×#7310	C₂²	semidirect
49. C₂×#7307	C₂²	semidirect
50. #108222	C₂²	semidirect
51. #62842	D₄	semidirect
58. C₂×#3026	D₄	semidirect
59. C₂×#2883	D₄	semidirect
60. C₂×#2883	D₄	semidirect
61. C₂×#2846	D₄	semidirect
63. C₂×#1913	D₄	semidirect
64. C₂×#1898	D₄	semidirect
65. #20609	D₄	semidirect
73. #20354	D₄	semidirect
74. #14679	D₄	semidirect
76. #12707	D₄	semidirect
77. #12687	D₄	semidirect
78. #10754	D₄	semidirect
79. #10735	D₄	semidirect
80. #8541	D₄	semidirect
81. #8534	D₄	semidirect
117. C₂⁶	#127	semidirect
120. C₂×H(5,𝔽₂)	#127	semidirect
135. #518	#127	semidirect
139. C₂²×C₂²⋊C₄	#127	semidirect
140. C₂×#99	#120	semidirect
141. #379	#120	semidirect
173. C₂³×C₄	B(4,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
#72264770	2
#72246299	2
#72246286	2
#72225644	2
#72219240	2
#72219160	2
#72219158	2
#72219157	2
#72218114	2
C₂×#144366	2
C₂×#144364	2
C₂×#144378	2
C₂×#347042	2
C₂×#511790	2