Subgroups — #72266715

Order 2048.

At least 6123 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

The subgroup lattice has 203 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72266710	C₂	1	1	yes	maximal, characteristic, complemented
2	#72266699	C₂	1	1	yes	maximal, characteristic, complemented
3	#72266566	C₂	1	1	yes	maximal, characteristic, complemented
4	#72266436	C₂	1	1	yes	maximal, characteristic, complemented
5	#72266351	C₂	1	1	yes	maximal, characteristic, complemented
6	#72266040	C₂	1	1	yes	maximal, characteristic, complemented
7	#72263060	C₂	1	1	yes	maximal, characteristic, complemented
8	#72259157	C₂	1	1	yes	maximal, characteristic, complemented
9	#72258995	C₂	1	1	yes	maximal, characteristic, complemented
10	#72238699	C₂	1	1	yes	maximal, characteristic, complemented
11	#72238481	C₂	1	1	yes	maximal, characteristic, complemented
12	#72237598	C₂	1	1	yes	maximal, characteristic, complemented
13	#72235716	C₂	1	1	yes	maximal, characteristic, complemented
14	#72234757	C₂	1	1	yes	maximal, characteristic, complemented
15	#72226097	C₂	1	1	yes	maximal, characteristic, complemented
16	#6499870	C₂²	1	1	yes	characteristic
17	#6499868	C₂²	1	1	yes	characteristic
18	#5063442	C₂²	1	1	yes	characteristic
19	#5025789	C₂²	1	1	yes	characteristic
20	#5021140	C₂²	1	1	yes	characteristic
21	#4466055	C₂²	1	1	yes	characteristic
22	#2108865	C₂²	1	1	yes	characteristic
23	#2039146	C₂²	1	1	yes	characteristic
24	#621871	C₂²	1	1	yes	characteristic
25	#347042	C₂²	1	1	yes	characteristic
26	#346456	C₂²	1	1	yes	characteristic
27	#346446	C₂²	1	1	yes	characteristic
28	#346446	C₂²	1	1	yes	characteristic
29	#187934	C₂²	1	1	yes	characteristic
30	#139720	C₂²	1	1	yes	characteristic
31	#139718	C₂²	1	1	yes	characteristic
32	#139718	C₂²	1	1	yes	characteristic
33	#139716	C₂²	1	1	yes	characteristic
34	#139716	C₂²	1	1	yes	characteristic
35	#139716	C₂²	1	1	yes	characteristic
36	#139504	C₂²	1	1	yes	characteristic
37	#139504	C₂²	1	1	yes	characteristic
38	#139504	C₂²	1	1	yes	characteristic
39	#139504	C₂²	1	1	yes	characteristic
40	#139486	C₂²	1	1	yes	characteristic
41	#139486	C₂²	1	1	yes	characteristic
42	#135561	C₂²	1	1	yes	characteristic
43	#135507	C₂²	1	1	yes	characteristic
44	#134036	C₂²	1	1	yes	characteristic
45	#134034	C₂²	1	1	yes	characteristic
46	#133825	C₂²	1	1	yes	characteristic
47	#133823	C₂²	1	1	yes	characteristic
48	#119550	C₂²	1	1	yes	characteristic
49	#115692	C₂²	1	1	yes	characteristic
50	#115631	C₂²	1	1	yes	characteristic
51	#62817	C₂³	1	1	yes	characteristic
52	C₂²×#560	D₄	1	1	yes	characteristic
53	C₂²×#560	D₄	1	1	yes	characteristic, complemented
54	C₂×#2760	D₄	1	1	yes	characteristic, complemented
55	C₂×#2760	D₄	1	1	yes	characteristic, complemented
56	C₂×#2849	C₂³	1	1	yes	characteristic
57	C₂×#2613	C₂³	1	1	yes	characteristic
58	C₂×#2308	D₄	1	1	yes	characteristic, complemented
59	C₂²×#386	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	#17137	D₄	1	1	yes	characteristic, complemented
64	#17102	D₄	1	1	yes	characteristic, complemented
65	#15948	D₄	1	1	yes	characteristic, complemented
66	#15947	D₄	1	1	yes	characteristic
67	#15947	C₂³	1	1	yes	characteristic
68	#15930	D₄	1	1	yes	characteristic, complemented
69	#15636	D₄	1	1	yes	characteristic, complemented
70	#15561	D₄	1	1	yes	characteristic, complemented
71	#15467	D₄	1	1	yes	characteristic, complemented
72	#15267	D₄	1	1	yes	characteristic, complemented
73	#15261	D₄	1	1	yes	characteristic, complemented
74	#15259	D₄	1	1	yes	characteristic, complemented
75	#15259	C₂³	1	1	yes	characteristic
76	#15258	D₄	1	1	yes	characteristic
77	#15258	C₂³	1	1	yes	characteristic
78	#15258	C₂³	1	1	yes	characteristic
79	#14681	D₄	1	1	yes	characteristic, complemented
80	#14658	D₄	1	1	yes	characteristic, complemented
81	#14657	D₄	1	1	yes	characteristic, complemented
82	#14533	D₄	1	1	yes	characteristic, complemented
83	#14507	D₄	1	1	yes	characteristic, complemented
84	#14507	D₄	1	1	yes	characteristic, complemented
85	#14492	D₄	1	1	yes	characteristic, complemented
86	#14262	D₄	1	1	yes	characteristic, complemented
87	#14255	C₂³	1	1	yes	characteristic
88	#14207	D₄	1	1	yes	characteristic, complemented
89	#14167	D₄	1	1	yes	characteristic
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	C₂⁵×C₄	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×#124	C₂×D₄	1	1	yes	characteristic
99	C₂²×#123	C₂×D₄	1	1	yes	characteristic
100	C₂²×#123	C₂×D₄	1	1	yes	characteristic
101	C₂²×#123	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	C₂×D₄	1	1	yes	characteristic
105	C₂²×#120	C₂×D₄	1	1	yes	characteristic
106	C₂²×#120	C₂⁴	1	1	yes	characteristic
107	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
108	C₂×#393	C₂×D₄	1	1	yes	characteristic
109	C₂×#393	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×#392	C₂×D₄	1	1	yes	characteristic
113	C₂×#392	C₂×D₄	1	1	yes	characteristic
114	C₂×#381	C₂×D₄	1	1	yes	characteristic
115	C₂×#381	C₂×D₄	1	1	yes	characteristic
116	C₂×#381	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	C₂²×D₄	1	1	yes	characteristic
140	C₂³×Q₈	#120	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	C₂⁴×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	C₂×#120	1	1	yes	characteristic
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	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
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	C₂³	#33551	1	2	yes	—
192	C₂²	#347043	1	1	yes	characteristic
193	C₂²	#511779	1	1	yes	characteristic
194	C₂²	#511790	1	1	yes	characteristic
195	C₂²	#511791	1	1	yes	characteristic
196	C₂²	#511809	1	1	yes	characteristic
197	C₂²	#511857	1	1	yes	characteristic
198	C₂²	#511857	1	1	yes	characteristic
199	C₂	#72249398	1	1	yes	characteristic
200	C₂	#72265771	1	1	yes	characteristic
201	C₂	#72265982	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. #72266710	C₂	semidirect
2. #72266699	C₂	semidirect
3. #72266566	C₂	semidirect
4. #72266436	C₂	semidirect
5. #72266351	C₂	semidirect
6. #72266040	C₂	semidirect
7. #72263060	C₂	semidirect
8. #72259157	C₂	semidirect
9. #72258995	C₂	semidirect
10. #72238699	C₂	semidirect
11. #72238481	C₂	semidirect
12. #72237598	C₂	semidirect
13. #72235716	C₂	semidirect
14. #72234757	C₂	semidirect
15. #72226097	C₂	semidirect
53. C₂²×#560	D₄	semidirect
54. C₂×#2760	D₄	semidirect
55. C₂×#2760	D₄	semidirect
58. C₂×#2308	D₄	semidirect
59. C₂²×#386	D₄	semidirect
63. #17137	D₄	semidirect
64. #17102	D₄	semidirect
65. #15948	D₄	semidirect
68. #15930	D₄	semidirect
69. #15636	D₄	semidirect
70. #15561	D₄	semidirect
71. #15467	D₄	semidirect
72. #15267	D₄	semidirect
73. #15261	D₄	semidirect
74. #15259	D₄	semidirect
79. #14681	D₄	semidirect
80. #14658	D₄	semidirect
81. #14657	D₄	semidirect
82. #14533	D₄	semidirect
83. #14507	D₄	semidirect
84. #14507	D₄	semidirect
85. #14492	D₄	semidirect
86. #14262	D₄	semidirect
88. #14207	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72266710	2
#72266699	2
#72266566	2
#72266436	2
#72266351	2
#72266040	2
#72263060	2
#72259157	2
#72258995	2
#72238699	2
#72238481	2
#72237598	2
#72235716	2
#72234757	2
#72226097	2