Subgroups — #72264773

Order 2048.

At least 7138 conjugacy classes of subgroups; only the 272 normal subgroups are shown.

The subgroup lattice has 215 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	#72264765	C₂	1	1	yes	maximal, characteristic, complemented
3	#72264754	C₂	1	1	yes	maximal, characteristic, complemented
4	#72255867	C₂	1	1	yes	maximal, characteristic, complemented
5	#72246450	C₂	1	1	yes	maximal, characteristic, complemented
6	#72225635	C₂	1	1	yes	maximal, characteristic, complemented
7	#72219241	C₂	1	1	yes	maximal, characteristic, complemented
8	#72219158	C₂	1	1	yes	maximal, characteristic, complemented
9	#72219157	C₂	1	1	yes	maximal, characteristic, complemented
10	#72218521	C₂	1	1	yes	maximal, characteristic, complemented
11	#72218475	C₂	1	1	yes	maximal, characteristic, complemented
12	#72218104	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#144366	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#144378	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#347042	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#62817	C₂²	1	1	yes	characteristic
17	#7626202	C₂²	1	1	yes	characteristic, complemented
18	C₂×#23895	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
22	C₂×#23900	C₂²	1	1	yes	characteristic, complemented
23	#6431795	C₂²	1	1	yes	characteristic
24	C₂×#15947	C₂²	1	1	yes	characteristic
25	C₂×#12707	C₂²	1	1	yes	characteristic
26	C₂×#12707	C₂²	1	1	yes	characteristic, complemented
27	C₂×#12705	C₂²	1	1	yes	characteristic
28	C₂×#12687	C₂²	1	1	yes	characteristic
29	C₂×#12687	C₂²	1	1	yes	characteristic, complemented
30	#382544	C₂²	1	1	yes	characteristic, complemented
31	#382434	C₂²	1	1	yes	characteristic
32	#359558	C₂²	1	1	yes	characteristic
33	#359549	C₂²	1	1	yes	characteristic
34	#359549	C₂²	1	1	yes	characteristic
35	#359549	C₂²	1	1	yes	characteristic, complemented
36	#359501	C₂²	1	1	yes	characteristic
37	#359501	C₂²	1	1	yes	characteristic
38	C₂×#8541	C₂²	1	1	yes	characteristic
39	C₂×#8541	C₂²	1	1	yes	characteristic, complemented
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	#146479	C₂²	1	1	yes	characteristic, complemented
44	#146475	C₂²	1	1	yes	characteristic, complemented
45	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
46	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7307	C₂²	1	1	yes	characteristic, complemented
49	#108222	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	C₂³×#120	C₂³	1	1	yes	characteristic
56	C₂³×#120	C₂³	1	1	yes	characteristic
57	C₂×#2883	D₄	1	1	yes	characteristic, complemented
58	C₂×#2883	D₄	1	1	yes	characteristic, complemented
59	C₂×#2846	D₄	1	2	yes	complemented
60	#34171	C₂³	1	1	yes	characteristic
61	#34171	C₂³	1	1	yes	characteristic
62	#23900	D₄	1	4	yes	complemented
63	C₂×#1913	D₄	1	2	yes	complemented
64	C₂×#1898	D₄	1	2	yes	complemented
65	#20960	D₄	1	2	yes	complemented
66	#20609	D₄	1	2	yes	complemented
67	C₂×#1783	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×#1781	C₂³	1	1	yes	characteristic
74	#20354	D₄	1	1	yes	characteristic, complemented
75	#15947	D₄	1	4	yes	complemented
76	#14679	D₄	1	2	yes	complemented
77	C₂×#1438	C₂³	1	1	yes	characteristic
78	#10754	D₄	1	2	yes	complemented
79	#10735	D₄	1	2	yes	complemented
80	C₂⁷	C₂×D₄	1	1	yes	characteristic
81	C₂⁷	C₂×D₄	1	1	yes	characteristic
82	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
83	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
84	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
85	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
86	#3484	C₂×D₄	1	2	yes	—
87	#3484	#41	1	2	yes	—
88	C₂×#534	C₂×D₄	1	2	yes	—
89	C₂²×#120	C₂×D₄	1	2	yes	—
90	C₂²×#120	C₂×D₄	1	2	yes	—
91	C₂²×#120	C₂×D₄	1	2	yes	—
92	C₂²×#120	C₂×D₄	1	2	yes	—
93	C₂²×#120	#41	1	2	yes	—
94	C₂×#518	C₂×D₄	1	1	yes	characteristic
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	#1781	C₂×D₄	1	2	yes	—
110	#1438	C₂×D₄	1	2	yes	—
111	C₂⁶	#120	1	2	yes	complemented
112	C₂⁶	#121	1	2	yes	complemented
113	C₂⁶	C₂²×D₄	1	1	yes	characteristic
114	C₂⁶	C₂²×D₄	1	1	yes	characteristic
115	C₂×H(5,𝔽₂)	#121	1	2	yes	complemented
116	C₂³×D₄	#120	1	2	yes	—
117	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
118	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
119	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
120	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
121	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
122	C₂³×D₄	C₂×#41	1	1	yes	characteristic
123	C₂³×D₄	C₂×#41	1	1	yes	characteristic
124	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂⁴×C₄	H(5,𝔽₂)	1	1	yes	characteristic
130	#518	#121	1	2	yes	complemented
131	C₂²×C₂²⋊C₄	#120	1	2	yes	—
132	C₂²×C₂²⋊C₄	#120	1	2	yes	—
133	C₂²×C₂²⋊C₄	#120	1	2	yes	—
134	C₂²×C₂²⋊C₄	#120	1	2	yes	complemented
135	C₂²×C₂²⋊C₄	#121	1	2	yes	complemented
136	C₂⁵	B(4,𝔽₂)	1	2	yes	—
137	C₂⁵	C₂×#120	1	1	yes	characteristic
138	C₂⁵	C₂×#120	1	1	yes	characteristic
139	C₂⁵	C₂×#120	1	1	yes	characteristic
140	C₂⁵	C₂×#121	1	1	yes	characteristic
141	C₂⁵	#534	1	1	yes	characteristic
142	C₂⁵	#535	1	1	yes	characteristic
143	C₂⁵	D₄²	1	1	yes	characteristic
144	C₂⁵	D₄²	1	1	yes	characteristic
145	C₂⁵	D₄²	1	1	yes	characteristic
146	C₂⁵	D₄²	1	1	yes	characteristic
147	C₂⁵	#546	1	1	yes	characteristic
148	C₂⁵	#546	1	1	yes	characteristic
149	C₂⁵	#546	1	1	yes	characteristic
150	C₂²×D₄	C₂×#121	1	1	yes	characteristic
151	C₂²×D₄	#525	1	1	yes	characteristic
152	C₂²×D₄	#534	1	1	yes	characteristic
153	C₂²×D₄	#535	1	1	yes	characteristic
154	C₂²×D₄	#535	1	1	yes	characteristic
155	C₂²×D₄	#538	1	1	yes	characteristic
156	C₂²×D₄	D₄²	1	1	yes	characteristic
157	C₂²×D₄	D₄²	1	1	yes	characteristic
158	C₂²×D₄	D₄²	1	1	yes	characteristic
159	C₂²×D₄	D₄²	1	1	yes	characteristic
160	C₂²×D₄	D₄²	1	1	yes	characteristic
161	C₂²×D₄	#546	1	1	yes	characteristic
162	C₂²×D₄	#546	1	1	yes	characteristic
163	C₂²×D₄	#546	1	1	yes	characteristic
164	C₂²×D₄	#546	1	1	yes	characteristic
165	C₂²×D₄	#560	1	1	yes	characteristic
166	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
167	C₂³×C₄	B(4,𝔽₂)	1	4	yes	complemented
168	C₂³×C₄	#534	1	1	yes	characteristic
169	C₂³×C₄	#534	1	1	yes	characteristic
170	C₂³×C₄	#538	1	1	yes	characteristic
171	C₂³×C₄	D₄²	1	1	yes	characteristic
172	C₂³×C₄	#546	1	1	yes	characteristic
173	C₂³×C₄	#560	1	1	yes	characteristic
174	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	4	yes	—
175	C₂⁴	#2403	1	1	yes	characteristic
176	C₂⁴	#2403	1	1	yes	characteristic
177	C₂⁴	#2404	1	1	yes	characteristic
178	C₂⁴	#2408	1	1	yes	characteristic
179	C₂⁴	#2408	1	1	yes	characteristic
180	C₂⁴	#2410	1	1	yes	characteristic
181	C₂⁴	#2410	1	1	yes	characteristic
182	C₂⁴	#2424	1	1	yes	characteristic
183	C₂⁴	#2669	1	1	yes	characteristic
184	C₂⁴	#2678	1	1	yes	characteristic
185	C₂⁴	#2678	1	1	yes	characteristic
186	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
187	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
188	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
189	C₂⁴	#3025	1	2	yes	—
190	C₂²×C₄	#2424	1	1	yes	characteristic
191	C₂²×C₄	#2668	1	1	yes	characteristic
192	C₂²×C₄	#2668	1	1	yes	characteristic
193	C₂²×C₄	#2669	1	1	yes	characteristic
194	C₂³	#15257	1	1	yes	characteristic
195	C₂³	#23895	1	1	yes	characteristic
196	C₂³	#23895	1	1	yes	characteristic
197	C₂³	#23900	1	1	yes	characteristic
198	C₂³	#23900	1	1	yes	characteristic
199	C₂³	#23900	1	2	yes	—
200	C₂³	#23900	1	1	yes	characteristic
201	C₂³	#23902	1	1	yes	characteristic
202	C₂³	#23984	1	2	yes	—
203	C₂³	#24260	1	2	yes	—
204	C₂³	#24260	1	2	yes	—
205	C₂²	#245225	1	1	yes	characteristic
206	C₂²	#245228	1	1	yes	characteristic
207	C₂²	#245231	1	1	yes	characteristic
208	C₂²	#245233	1	1	yes	characteristic
209	C₂²	#417493	1	1	yes	characteristic
210	C₂²	#417526	1	1	yes	characteristic
211	C₂²	#417532	1	1	yes	characteristic
212	C₂	B(5,𝔽₂)	1	2	yes	—
213	C₂	#72237498	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. #72264765	C₂	semidirect
3. #72264754	C₂	semidirect
4. #72255867	C₂	semidirect
5. #72246450	C₂	semidirect
6. #72225635	C₂	semidirect
7. #72219241	C₂	semidirect
8. #72219158	C₂	semidirect
9. #72219157	C₂	semidirect
10. #72218521	C₂	semidirect
11. #72218475	C₂	semidirect
12. #72218104	C₂	semidirect
13. C₂×#144366	C₂	semidirect
14. C₂×#144378	C₂	semidirect
15. C₂×#347042	C₂	semidirect
17. #7626202	C₂²	semidirect
22. C₂×#23900	C₂²	semidirect
26. C₂×#12707	C₂²	semidirect
29. C₂×#12687	C₂²	semidirect
30. #382544	C₂²	semidirect
35. #359549	C₂²	semidirect
39. C₂×#8541	C₂²	semidirect
42. C₂×#8453	C₂²	semidirect
43. #146479	C₂²	semidirect
44. #146475	C₂²	semidirect
45. C₂×#7316	C₂²	semidirect
46. C₂×#7310	C₂²	semidirect
47. C₂×#7310	C₂²	semidirect
48. C₂×#7307	C₂²	semidirect
49. #108222	C₂²	semidirect
50. #108222	C₂²	semidirect
51. #62842	D₄	semidirect
57. C₂×#2883	D₄	semidirect
58. C₂×#2883	D₄	semidirect
59. C₂×#2846	D₄	semidirect
62. #23900	D₄	semidirect
63. C₂×#1913	D₄	semidirect
64. C₂×#1898	D₄	semidirect
65. #20960	D₄	semidirect
66. #20609	D₄	semidirect
74. #20354	D₄	semidirect
75. #15947	D₄	semidirect
76. #14679	D₄	semidirect
78. #10754	D₄	semidirect
79. #10735	D₄	semidirect
111. C₂⁶	#120	semidirect
112. C₂⁶	#121	semidirect
115. C₂×H(5,𝔽₂)	#121	semidirect
130. #518	#121	semidirect
134. C₂²×C₂²⋊C₄	#120	semidirect
135. C₂²×C₂²⋊C₄	#121	semidirect
167. 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
#72264765	2
#72264754	2
#72255867	2
#72246450	2
#72225635	2
#72219241	2
#72219158	2
#72219157	2
#72218521	2
#72218475	2
#72218104	2
C₂×#144366	2
C₂×#144378	2
C₂×#347042	2