Subgroups — #72263331

Order 2048.

At least 6573 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

The subgroup lattice has 202 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72263330	C₂	1	1	yes	maximal, characteristic, complemented
2	#72263145	C₂	1	1	yes	maximal, characteristic, complemented
3	#72263077	C₂	1	1	yes	maximal, characteristic, complemented
4	#72262844	C₂	1	1	yes	maximal, characteristic, complemented
5	#72262842	C₂	1	1	yes	maximal, characteristic, complemented
6	#72256396	C₂	1	1	yes	maximal, characteristic, complemented
7	#72238493	C₂	1	1	yes	maximal, characteristic, complemented
8	#72238438	C₂	1	1	yes	maximal, characteristic, complemented
9	#72237443	C₂	1	1	yes	maximal, characteristic, complemented
10	#72234622	C₂	1	1	yes	maximal, characteristic, complemented
11	#72233792	C₂	1	1	yes	maximal, characteristic, complemented
12	#72232807	C₂	1	1	yes	maximal, characteristic, complemented
13	#72232567	C₂	1	1	yes	maximal, characteristic, complemented
14	#72219512	C₂	1	1	yes	maximal, characteristic, complemented
15	#72213719	C₂	1	1	yes	maximal, characteristic, complemented
16	#6500010	C₂²	1	1	yes	characteristic, complemented
17	#6499870	C₂²	1	1	yes	characteristic, complemented
18	#5236713	C₂²	1	1	yes	characteristic
19	#5228708	C₂²	1	1	yes	characteristic
20	#5024691	C₂²	1	1	yes	characteristic
21	#4466035	C₂²	1	1	yes	characteristic
22	#4379766	C₂²	1	1	yes	characteristic
23	#1462974	C₂²	1	1	yes	characteristic
24	#1418393	C₂²	1	1	yes	characteristic
25	#347042	C₂²	1	1	yes	characteristic, complemented
26	#346466	C₂²	1	1	yes	characteristic, complemented
27	#346446	C₂²	1	1	yes	characteristic
28	#346446	C₂²	1	1	yes	characteristic, complemented
29	#187949	C₂²	1	1	yes	characteristic, complemented
30	#139830	C₂²	1	1	yes	characteristic, complemented
31	#139718	C₂²	1	1	yes	characteristic
32	#139718	C₂²	1	1	yes	characteristic, complemented
33	#139716	C₂²	1	1	yes	characteristic, complemented
34	#139716	C₂²	1	1	yes	characteristic, complemented
35	#139716	C₂²	1	1	yes	characteristic, complemented
36	#139506	C₂²	1	1	yes	characteristic, complemented
37	#139504	C₂²	1	1	yes	characteristic
38	#139504	C₂²	1	1	yes	characteristic
39	#139504	C₂²	1	1	yes	characteristic, complemented
40	#139504	C₂²	1	1	yes	characteristic, complemented
41	#139484	C₂²	1	1	yes	characteristic, complemented
42	#135573	C₂²	1	1	yes	characteristic, complemented
43	#135524	C₂²	1	1	yes	characteristic, complemented
44	#134083	C₂²	1	1	yes	characteristic, complemented
45	#134036	C₂²	1	1	yes	characteristic, complemented
46	#133893	C₂²	1	1	yes	characteristic, complemented
47	#133825	C₂²	1	1	yes	characteristic, complemented
48	#119556	C₂²	1	1	yes	characteristic, complemented
49	#115631	C₂²	1	1	yes	characteristic, complemented
50	#115163	C₂²	1	1	yes	characteristic, complemented
51	#62817	C₂³	1	1	yes	characteristic
52	C₂²×D₄²	D₄	1	1	yes	characteristic, complemented
53	C₂×#2697	D₄	1	1	yes	characteristic, complemented
54	C₂×#2668	D₄	1	1	yes	characteristic, complemented
55	C₂×#2410	C₂³	1	1	yes	characteristic
56	C₂×#2403	D₄	1	1	yes	characteristic, complemented
57	C₂×#2403	D₄	1	1	yes	characteristic, complemented
58	C₂×#2613	C₂³	1	1	yes	characteristic
59	C₂×#2308	D₄	1	1	yes	characteristic, complemented
60	#21746	C₂³	1	1	yes	characteristic
61	#21746	C₂³	1	1	yes	characteristic
62	#21746	C₂³	1	1	yes	characteristic
63	#17272	D₄	1	1	yes	characteristic, complemented
64	#17147	D₄	1	1	yes	characteristic, complemented
65	#15947	D₄	1	1	yes	characteristic, complemented
66	#15947	C₂³	1	1	yes	characteristic
67	#15657	D₄	1	1	yes	characteristic, complemented
68	#15635	D₄	1	1	yes	characteristic, complemented
69	#15561	D₄	1	1	yes	characteristic, complemented
70	#15561	D₄	1	1	yes	characteristic, complemented
71	#15406	D₄	1	1	yes	characteristic, complemented
72	#15313	D₄	1	1	yes	characteristic, complemented
73	#15283	C₂³	1	1	yes	characteristic
74	#15281	D₄	1	1	yes	characteristic, complemented
75	#15266	D₄	1	1	yes	characteristic, complemented
76	#15258	D₄	1	1	yes	characteristic, complemented
77	#15258	C₂³	1	1	yes	characteristic
78	#15258	C₂³	1	1	yes	characteristic
79	#14657	C₂³	1	1	yes	characteristic
80	#14613	D₄	1	1	yes	characteristic, complemented
81	#14533	D₄	1	1	yes	characteristic, complemented
82	#14506	D₄	1	1	yes	characteristic, complemented
83	#14419	D₄	1	1	yes	characteristic, complemented
84	#14358	D₄	1	1	yes	characteristic, complemented
85	#14358	D₄	1	1	yes	characteristic, complemented
86	#14262	D₄	1	1	yes	characteristic, complemented
87	#13867	D₄	1	1	yes	characteristic, complemented
88	#13827	D₄	1	1	yes	characteristic, complemented
89	C₂²⋊C₄²	D₄	1	1	yes	characteristic, complemented
90	#8547	C₂³	1	1	yes	characteristic
91	#8547	C₂³	1	1	yes	characteristic
92	#8547	C₂³	1	1	yes	characteristic
93	#8541	C₂³	1	1	yes	characteristic
94	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
95	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
96	C₂²×#127	C₂×D₄	1	1	yes	characteristic
97	C₂²×#127	C₂×D₄	1	1	yes	characteristic
98	C₂²×#123	C₂×D₄	1	1	yes	characteristic
99	C₂²×#120	C₂×D₄	1	1	yes	characteristic
100	C₂²×#120	C₂×D₄	1	1	yes	characteristic
101	C₂²×#120	C₂×D₄	1	1	yes	characteristic
102	C₂²×#120	C₂×D₄	1	1	yes	characteristic
103	C₂²×#120	C₂×D₄	1	1	yes	characteristic
104	C₂²×#120	C₂⁴	1	1	yes	characteristic
105	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
106	C₂×#399	C₂×D₄	1	1	yes	characteristic
107	C₂×#397	C₂×D₄	1	1	yes	characteristic
108	C₂×#392	C₂×D₄	1	1	yes	characteristic
109	C₂×#392	C₂×D₄	1	1	yes	characteristic
110	C₂×#392	C₂×D₄	1	1	yes	characteristic
111	C₂×#392	C₂×D₄	1	1	yes	characteristic
112	C₂×#390	C₂×D₄	1	1	yes	characteristic
113	C₂×#386	C₂×D₄	1	1	yes	characteristic
114	C₂×#385	C₂×D₄	1	1	yes	characteristic
115	C₂×#381	C₂×D₄	1	1	yes	characteristic
116	C₂×#379	C₂×D₄	1	1	yes	characteristic
117	C₂×#379	C₂×D₄	1	1	yes	characteristic
118	C₂×#379	C₂×D₄	1	1	yes	characteristic
119	C₂×#379	C₂×D₄	1	1	yes	characteristic
120	C₂×#379	C₂×D₄	1	1	yes	characteristic
121	#1440	C₂×D₄	1	1	yes	characteristic
122	#1440	C₂×D₄	1	1	yes	characteristic
123	#1438	C₂×D₄	1	1	yes	characteristic
124	#1438	C₂×D₄	1	1	yes	characteristic
125	#1438	C₂×D₄	1	1	yes	characteristic
126	#1438	C₂×D₄	1	1	yes	characteristic
127	#1438	C₂×D₄	1	1	yes	characteristic
128	#1438	C₂×D₄	1	1	yes	characteristic
129	#1438	C₂×D₄	1	1	yes	characteristic
130	#1438	C₂×D₄	1	1	yes	characteristic
131	#1437	C₂×D₄	1	1	yes	characteristic
132	#1437	C₂×D₄	1	1	yes	characteristic
133	#1437	C₂×D₄	1	1	yes	characteristic
134	#1437	C₂×D₄	1	1	yes	characteristic
135	#1437	C₂×D₄	1	1	yes	characteristic
136	#1437	C₂×D₄	1	1	yes	characteristic
137	C₂⁶	#120	1	1	yes	characteristic
138	C₂⁶	#120	1	1	yes	characteristic
139	C₂⁶	#120	1	1	yes	characteristic
140	C₂⁶	C₂²×D₄	1	1	yes	characteristic
141	C₂³×D₄	#120	1	1	yes	characteristic
142	C₂³×D₄	#120	1	1	yes	characteristic
143	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
144	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
145	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
146	C₂⁴×C₄	#120	1	1	yes	characteristic
147	C₂⁴×C₄	#120	1	1	yes	characteristic
148	C₂²×C₄⋊C₄	#120	1	1	yes	characteristic
149	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
150	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
151	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
152	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
153	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
154	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
155	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
156	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
157	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
158	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
159	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
160	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
161	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
162	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
163	C₂²×C₄²	#120	1	1	yes	characteristic
164	C₂²×C₄²	#120	1	1	yes	characteristic
165	C₂²×C₄²	#120	1	1	yes	characteristic
166	C₂×#95	#120	1	1	yes	characteristic
167	C₂×#95	#120	1	1	yes	characteristic
168	C₂×#95	#120	1	1	yes	characteristic
169	C₂×#95	#120	1	1	yes	characteristic
170	C₂×#95	#120	1	1	yes	characteristic
171	C₂×#95	#120	1	1	yes	characteristic
172	C₂⁵	B(4,𝔽₂)	1	2	yes	—
173	C₂⁵	C₂×#120	1	1	yes	characteristic
174	C₂⁵	C₂×#120	1	1	yes	characteristic
175	C₂⁵	C₂×#120	1	1	yes	characteristic
176	C₂⁵	C₂×#120	1	1	yes	characteristic
177	C₂³×C₄	C₂×#120	1	1	yes	characteristic
178	C₂³×C₄	C₂×#120	1	1	yes	characteristic
179	C₂³×C₄	C₂×#120	1	1	yes	characteristic
180	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
181	C₂⁴	#2846	1	1	yes	characteristic
182	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
183	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
184	C₂⁴	#3025	1	1	yes	characteristic
185	C₂⁴	#3026	1	1	yes	characteristic
186	C₂⁴	#3026	1	1	yes	characteristic
187	C₂⁴	#3027	1	1	yes	characteristic
188	C₂³	#32898	1	1	yes	characteristic
189	C₂³	#32898	1	1	yes	characteristic
190	C₂³	#32899	1	1	yes	characteristic
191	C₂²	#347043	1	1	yes	characteristic
192	C₂²	#511790	1	1	yes	characteristic
193	C₂²	#511793	1	1	yes	characteristic
194	C₂²	#511810	1	1	yes	characteristic
195	C₂²	#511811	1	1	yes	characteristic
196	C₂²	#511853	1	1	yes	characteristic
197	C₂²	#511853	1	1	yes	characteristic
198	C₂	#72234413	1	1	yes	characteristic
199	C₂	#72249365	1	1	yes	characteristic
200	C₂	#72256072	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#347043
commutator, Frattini, cosocle	C₂²×#120	C₂⁴
LCS	G ▷ C₂²×#120 ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#120 ◁ G
LECS	G ▷ C₂²×#120 ▷ C₂⁴ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72263330	C₂	semidirect
2. #72263145	C₂	semidirect
3. #72263077	C₂	semidirect
4. #72262844	C₂	semidirect
5. #72262842	C₂	semidirect
6. #72256396	C₂	semidirect
7. #72238493	C₂	semidirect
8. #72238438	C₂	semidirect
9. #72237443	C₂	semidirect
10. #72234622	C₂	semidirect
11. #72233792	C₂	semidirect
12. #72232807	C₂	semidirect
13. #72232567	C₂	semidirect
14. #72219512	C₂	semidirect
15. #72213719	C₂	semidirect
16. #6500010	C₂²	semidirect
17. #6499870	C₂²	semidirect
25. #347042	C₂²	semidirect
26. #346466	C₂²	semidirect
28. #346446	C₂²	semidirect
29. #187949	C₂²	semidirect
30. #139830	C₂²	semidirect
32. #139718	C₂²	semidirect
33. #139716	C₂²	semidirect
34. #139716	C₂²	semidirect
35. #139716	C₂²	semidirect
36. #139506	C₂²	semidirect
39. #139504	C₂²	semidirect
40. #139504	C₂²	semidirect
41. #139484	C₂²	semidirect
42. #135573	C₂²	semidirect
43. #135524	C₂²	semidirect
44. #134083	C₂²	semidirect
45. #134036	C₂²	semidirect
46. #133893	C₂²	semidirect
47. #133825	C₂²	semidirect
48. #119556	C₂²	semidirect
49. #115631	C₂²	semidirect
50. #115163	C₂²	semidirect
52. C₂²×D₄²	D₄	semidirect
53. C₂×#2697	D₄	semidirect
54. C₂×#2668	D₄	semidirect
56. C₂×#2403	D₄	semidirect
57. C₂×#2403	D₄	semidirect
59. C₂×#2308	D₄	semidirect
63. #17272	D₄	semidirect
64. #17147	D₄	semidirect
65. #15947	D₄	semidirect
67. #15657	D₄	semidirect
68. #15635	D₄	semidirect
69. #15561	D₄	semidirect
70. #15561	D₄	semidirect
71. #15406	D₄	semidirect
72. #15313	D₄	semidirect
74. #15281	D₄	semidirect
75. #15266	D₄	semidirect
76. #15258	D₄	semidirect
80. #14613	D₄	semidirect
81. #14533	D₄	semidirect
82. #14506	D₄	semidirect
83. #14419	D₄	semidirect
84. #14358	D₄	semidirect
85. #14358	D₄	semidirect
86. #14262	D₄	semidirect
87. #13867	D₄	semidirect
88. #13827	D₄	semidirect
89. C₂²⋊C₄²	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72263330	2
#72263145	2
#72263077	2
#72262844	2
#72262842	2
#72256396	2
#72238493	2
#72238438	2
#72237443	2
#72234622	2
#72233792	2
#72232807	2
#72232567	2
#72219512	2
#72213719	2