Subgroups — #72315351

Order 2048.

At least 6465 conjugacy classes of subgroups; only the 260 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	#72314739	C₂	1	1	yes	maximal, characteristic, complemented
2	#72310322	C₂	1	1	yes	maximal, characteristic, complemented
3	#72258368	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#139742	C₂	1	1	yes	maximal, characteristic, complemented
5	#72238974	C₂	1	1	yes	maximal, characteristic, complemented
6	#72225742	C₂	1	1	yes	maximal, characteristic, complemented
7	#72219438	C₂	1	1	yes	maximal, characteristic, complemented
8	#72219393	C₂	1	1	yes	maximal, characteristic, complemented
9	#72219293	C₂	1	1	yes	maximal, characteristic, complemented
10	#72218391	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#144366	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#139750	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#144382	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	C₂×#23895	C₂²	1	1	yes	characteristic
18	C₂×#23900	C₂²	1	1	yes	characteristic
19	C₂×#23900	C₂²	1	1	yes	characteristic
20	C₂×#23900	C₂²	1	1	yes	characteristic
21	C₂×#23900	C₂²	1	1	yes	characteristic, complemented
22	C₂×#23897	C₂²	1	1	yes	characteristic
23	C₂×#21776	C₂²	1	1	yes	characteristic, complemented
24	C₂×#15947	C₂²	1	1	yes	characteristic
25	C₂×#12707	C₂²	1	1	yes	characteristic
26	C₂×#12705	C₂²	1	1	yes	characteristic
27	C₂×#12705	C₂²	1	1	yes	characteristic
28	C₂×#12705	C₂²	1	1	yes	characteristic
29	C₂×#12705	C₂²	1	1	yes	characteristic, complemented
30	C₂×#12687	C₂²	1	1	yes	characteristic
31	C₂×#12687	C₂²	1	1	yes	characteristic
32	C₂×#12687	C₂²	1	1	yes	characteristic
33	C₂×#12687	C₂²	1	1	yes	characteristic, complemented
34	#392030	C₂²	1	1	yes	characteristic, complemented
35	#382436	C₂²	1	1	yes	characteristic
36	#382434	C₂²	1	1	yes	characteristic
37	#382422	C₂²	1	1	yes	characteristic, complemented
38	C₂×#8547	C₂²	1	1	yes	characteristic, complemented
39	C₂×#8541	C₂²	1	1	yes	characteristic
40	C₂×#8534	C₂²	1	1	yes	characteristic
41	C₂×#8534	C₂²	1	1	yes	characteristic
42	C₂×#8453	C₂²	1	1	yes	characteristic, complemented
43	C₂×#7304	C₂²	1	1	yes	characteristic, complemented
44	C₂×#7304	C₂²	1	1	yes	characteristic, complemented
45	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
46	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7311	C₂²	1	1	yes	characteristic, complemented
49	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
50	C₂×#7307	C₂²	1	1	yes	characteristic, complemented
51	C₂×#3484	C₂³	1	1	yes	characteristic
52	C₂×#3484	C₂³	1	1	yes	characteristic
53	C₂×#3484	C₂³	1	1	yes	characteristic
54	C₂³×#120	C₂³	1	1	yes	characteristic
55	C₂³×#120	C₂³	1	1	yes	characteristic
56	C₂×#3026	D₄	1	2	yes	complemented
57	C₂×#2846	D₄	1	2	yes	complemented
58	#23900	D₄	1	4	yes	complemented
59	#23897	D₄	1	4	yes	complemented
60	C₂×#1913	D₄	1	2	yes	complemented
61	C₂×#1898	D₄	1	2	yes	complemented
62	C₂×#1828	D₄	1	2	yes	complemented
63	C₂×#1783	C₂³	1	1	yes	characteristic
64	C₂×#1783	C₂³	1	1	yes	characteristic
65	C₂×#1783	C₂³	1	1	yes	characteristic
66	#20585	D₄	1	2	yes	complemented
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	C₂×#1438	C₂³	1	1	yes	characteristic
74	#8541	D₄	1	4	yes	complemented
75	#8534	D₄	1	4	yes	complemented
76	C₂⁷	C₂×D₄	1	1	yes	characteristic
77	C₂⁷	#41	1	1	yes	characteristic
78	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
79	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
80	C₂²×H(5,𝔽₂)	#41	1	1	yes	characteristic
81	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
82	#3484	C₂×D₄	1	2	yes	—
83	#3484	C₂×D₄	1	2	yes	—
84	#3484	C₂×D₄	1	2	yes	—
85	#3484	C₂×D₄	1	2	yes	—
86	C₂×#534	C₂×D₄	1	2	yes	—
87	C₂²×#120	C₂×D₄	1	2	yes	—
88	C₂²×#120	C₂×D₄	1	2	yes	—
89	C₂²×#120	C₂×D₄	1	2	yes	—
90	C₂×#518	C₂×D₄	1	1	yes	characteristic
91	C₂×#518	C₂×D₄	1	1	yes	characteristic
92	C₂×#518	#41	1	1	yes	characteristic
93	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
94	C₂³×C₂²⋊C₄	#41	1	1	yes	characteristic
95	C₂²×#99	C₂×D₄	1	2	yes	—
96	C₂²×#99	C₂×D₄	1	2	yes	—
97	C₂²×#99	C₂×D₄	1	2	yes	—
98	C₂×#379	C₂×D₄	1	2	yes	—
99	#1783	C₂×D₄	1	2	yes	—
100	#1781	C₂×D₄	1	2	yes	—
101	#1781	C₂×D₄	1	2	yes	—
102	#1781	C₂×D₄	1	2	yes	—
103	#1781	C₂×D₄	1	2	yes	—
104	#1438	C₂×D₄	1	2	yes	—
105	C₂⁶	#121	1	2	yes	complemented
106	C₂⁶	C₂²×D₄	1	1	yes	characteristic
107	C₂⁶	C₂×#41	1	1	yes	characteristic
108	C₂×H(5,𝔽₂)	#120	1	2	yes	—
109	C₂×H(5,𝔽₂)	#121	1	2	yes	complemented
110	C₂³×D₄	#120	1	2	yes	—
111	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
112	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
113	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
114	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
115	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
116	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
117	C₂³×D₄	C₂×#41	1	1	yes	characteristic
118	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
119	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂⁴×C₄	H(5,𝔽₂)	1	1	yes	characteristic
124	#518	#120	1	2	yes	—
125	#518	#121	1	2	yes	complemented
126	C₂²×C₂²⋊C₄	#120	1	2	yes	—
127	C₂²×C₂²⋊C₄	#121	1	2	yes	complemented
128	#379	#120	1	4	yes	—
129	C₂⁵	B(4,𝔽₂)	1	2	yes	—
130	C₂⁵	C₂×#120	1	1	yes	characteristic
131	C₂⁵	C₂×#120	1	1	yes	characteristic
132	C₂⁵	C₂×#121	1	1	yes	characteristic
133	C₂⁵	#525	1	1	yes	characteristic
134	C₂⁵	#534	1	1	yes	characteristic
135	C₂⁵	#538	1	1	yes	characteristic
136	C₂⁵	D₄²	1	1	yes	characteristic
137	C₂⁵	D₄²	1	1	yes	characteristic
138	C₂⁵	#546	1	1	yes	characteristic
139	C₂⁵	#546	1	1	yes	characteristic
140	C₂⁵	#546	1	1	yes	characteristic
141	C₂⁵	#546	1	1	yes	characteristic
142	C₂⁵	#546	1	1	yes	characteristic
143	C₂²×D₄	C₂×#120	1	1	yes	characteristic
144	C₂²×D₄	C₂×#121	1	1	yes	characteristic
145	C₂²×D₄	#534	1	1	yes	characteristic
146	C₂²×D₄	#534	1	1	yes	characteristic
147	C₂²×D₄	#535	1	1	yes	characteristic
148	C₂²×D₄	#538	1	1	yes	characteristic
149	C₂²×D₄	D₄²	1	1	yes	characteristic
150	C₂²×D₄	D₄²	1	1	yes	characteristic
151	C₂²×D₄	D₄²	1	1	yes	characteristic
152	C₂²×D₄	D₄²	1	1	yes	characteristic
153	C₂²×D₄	D₄²	1	1	yes	characteristic
154	C₂²×D₄	D₄²	1	1	yes	characteristic
155	C₂²×D₄	D₄²	1	1	yes	characteristic
156	C₂²×D₄	#546	1	1	yes	characteristic
157	C₂²×D₄	#546	1	1	yes	characteristic
158	C₂²×D₄	#560	1	1	yes	characteristic
159	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
160	C₂³×C₄	#534	1	1	yes	characteristic
161	C₂³×C₄	#535	1	1	yes	characteristic
162	C₂³×C₄	#535	1	1	yes	characteristic
163	C₂³×C₄	D₄²	1	1	yes	characteristic
164	C₂³×C₄	#546	1	1	yes	characteristic
165	C₂³×C₄	#560	1	1	yes	characteristic
166	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	4	yes	—
167	C₂⁴	#2403	1	1	yes	characteristic
168	C₂⁴	#2403	1	1	yes	characteristic
169	C₂⁴	#2404	1	1	yes	characteristic
170	C₂⁴	#2408	1	1	yes	characteristic
171	C₂⁴	#2408	1	1	yes	characteristic
172	C₂⁴	#2410	1	1	yes	characteristic
173	C₂⁴	#2424	1	1	yes	characteristic
174	C₂⁴	#2424	1	1	yes	characteristic
175	C₂⁴	#2668	1	1	yes	characteristic
176	C₂⁴	#2668	1	1	yes	characteristic
177	C₂⁴	#2678	1	1	yes	characteristic
178	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
179	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
180	C₂²×C₄	#2410	1	1	yes	characteristic
181	C₂²×C₄	#2669	1	1	yes	characteristic
182	C₂²×C₄	#2669	1	1	yes	characteristic
183	C₂²×C₄	#2678	1	1	yes	characteristic
184	C₂³	#15257	1	1	yes	characteristic
185	C₂³	#23900	1	2	yes	—
186	C₂³	#23900	1	2	yes	—
187	C₂³	#23984	1	1	yes	characteristic
188	C₂³	#23984	1	1	yes	characteristic
189	C₂³	#23996	1	1	yes	characteristic
190	C₂³	#23996	1	1	yes	characteristic
191	C₂³	#23996	1	1	yes	characteristic
192	C₂³	#23996	1	1	yes	characteristic
193	C₂²	#251827	1	1	yes	characteristic
194	C₂²	#251828	1	1	yes	characteristic
195	C₂²	#251859	1	1	yes	characteristic
196	C₂²	#251860	1	1	yes	characteristic
197	C₂²	#417493	1	1	yes	characteristic
198	C₂²	#417538	1	1	yes	characteristic
199	C₂²	#417542	1	1	yes	characteristic
200	C₂	#72216344	1	2	yes	—
201	C₂	#72241194	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#251827
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. #72314739	C₂	semidirect
2. #72310322	C₂	semidirect
3. #72258368	C₂	semidirect
4. C₂×#139742	C₂	semidirect
5. #72238974	C₂	semidirect
6. #72225742	C₂	semidirect
7. #72219438	C₂	semidirect
8. #72219393	C₂	semidirect
9. #72219293	C₂	semidirect
10. #72218391	C₂	semidirect
11. C₂×#144366	C₂	semidirect
12. C₂×#139750	C₂	semidirect
13. C₂×#144382	C₂	semidirect
14. C₂×#347042	C₂	semidirect
15. C₂×#511790	C₂	semidirect
21. C₂×#23900	C₂²	semidirect
23. C₂×#21776	C₂²	semidirect
29. C₂×#12705	C₂²	semidirect
33. C₂×#12687	C₂²	semidirect
34. #392030	C₂²	semidirect
37. #382422	C₂²	semidirect
38. C₂×#8547	C₂²	semidirect
42. C₂×#8453	C₂²	semidirect
43. C₂×#7304	C₂²	semidirect
44. C₂×#7304	C₂²	semidirect
45. C₂×#7316	C₂²	semidirect
46. C₂×#7314	C₂²	semidirect
47. C₂×#7314	C₂²	semidirect
48. C₂×#7311	C₂²	semidirect
49. C₂×#7310	C₂²	semidirect
50. C₂×#7307	C₂²	semidirect
56. C₂×#3026	D₄	semidirect
57. C₂×#2846	D₄	semidirect
58. #23900	D₄	semidirect
59. #23897	D₄	semidirect
60. C₂×#1913	D₄	semidirect
61. C₂×#1898	D₄	semidirect
62. C₂×#1828	D₄	semidirect
66. #20585	D₄	semidirect
74. #8541	D₄	semidirect
75. #8534	D₄	semidirect
105. C₂⁶	#121	semidirect
109. C₂×H(5,𝔽₂)	#121	semidirect
125. #518	#121	semidirect
127. C₂²×C₂²⋊C₄	#121	semidirect

Maximal subgroups

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

subgroup	index
#72314739	2
#72310322	2
#72258368	2
C₂×#139742	2
#72238974	2
#72225742	2
#72219438	2
#72219393	2
#72219293	2
#72218391	2
C₂×#144366	2
C₂×#139750	2
C₂×#144382	2
C₂×#347042	2
C₂×#511790	2