Subgroups — #72264761

Order 2048.

At least 6981 conjugacy classes of subgroups; only the 320 normal subgroups are shown.

The subgroup lattice has 230 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72264754	C₂	1	1	yes	maximal, characteristic, complemented
2	#72255867	C₂	1	1	yes	maximal, characteristic, complemented
3	#72246292	C₂	1	1	yes	maximal, characteristic, complemented
4	#72238589	C₂	1	1	yes	maximal, characteristic, complemented
5	#72225647	C₂	1	1	yes	maximal, characteristic, complemented
6	#72219240	C₂	1	1	yes	maximal, characteristic, complemented
7	#72219163	C₂	1	1	yes	maximal, characteristic, complemented
8	#72219162	C₂	1	1	yes	maximal, characteristic, complemented
9	#72218521	C₂	1	1	yes	maximal, characteristic, complemented
10	#72218521	C₂	1	1	yes	maximal, characteristic, complemented
11	#72218114	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#139750	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#139743	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×#23900	C₂²	1	1	yes	characteristic
20	C₂×#23900	C₂²	1	1	yes	characteristic
21	C₂×#23900	C₂²	1	1	yes	characteristic
22	C₂×#23897	C₂²	1	1	yes	characteristic
23	C₂×#21776	C₂²	1	1	yes	characteristic
24	#6371558	C₂²	1	1	yes	characteristic
25	C₂×#15947	C₂²	1	1	yes	characteristic
26	C₂×#12707	C₂²	1	1	yes	characteristic
27	C₂×#12707	C₂²	1	1	yes	characteristic, complemented
28	C₂×#12705	C₂²	1	1	yes	characteristic
29	C₂×#12687	C₂²	1	1	yes	characteristic
30	C₂×#12687	C₂²	1	1	yes	characteristic, complemented
31	#382544	C₂²	1	1	yes	characteristic, complemented
32	#359549	C₂²	1	1	yes	characteristic
33	#359549	C₂²	1	1	yes	characteristic
34	#359549	C₂²	1	1	yes	characteristic, complemented
35	#359501	C₂²	1	1	yes	characteristic
36	#359501	C₂²	1	1	yes	characteristic
37	#359501	C₂²	1	1	yes	characteristic, complemented
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×#8534	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	#107130	C₂²	1	1	yes	characteristic, complemented
51	#62817	D₄	1	4	yes	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	#62652	D₄	1	1	yes	characteristic, complemented
58	C₂×#3026	D₄	1	2	yes	complemented
59	C₂×#2883	D₄	1	1	yes	characteristic, complemented
60	C₂×#2846	D₄	1	2	yes	complemented
61	#34171	C₂³	1	1	yes	characteristic
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	#20354	D₄	1	1	yes	characteristic, complemented
75	#14679	D₄	1	2	yes	complemented
76	C₂×#1438	C₂³	1	1	yes	characteristic
77	#12705	D₄	1	4	yes	complemented
78	#10754	D₄	1	2	yes	complemented
79	#10735	D₄	1	2	yes	complemented
80	#8534	D₄	1	4	yes	complemented
81	C₂⁷	C₂×D₄	1	1	yes	characteristic
82	C₂⁷	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	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
86	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
87	#3484	C₂×D₄	1	2	yes	—
88	#3484	C₂×D₄	1	2	yes	—
89	#3484	C₂×D₄	1	2	yes	—
90	#3484	C₂×D₄	1	2	yes	—
91	#3484	#41	1	2	yes	—
92	#3484	#41	1	2	yes	—
93	C₂×#534	C₂×D₄	1	2	yes	—
94	C₂²×#120	C₂×D₄	1	2	yes	—
95	C₂²×#120	C₂×D₄	1	2	yes	—
96	C₂²×#120	#41	1	2	yes	—
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	C₂×#518	C₂×D₄	1	1	yes	characteristic
102	C₂×#518	C₂×D₄	1	1	yes	characteristic
103	C₂×#518	C₂×D₄	1	1	yes	characteristic
104	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
105	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
106	C₂²×#99	C₂×D₄	1	2	yes	—
107	C₂²×#99	C₂×D₄	1	2	yes	—
108	#2308	C₂×D₄	1	2	yes	—
109	C₂×#379	C₂×D₄	1	2	yes	—
110	C₂×#379	C₂×D₄	1	2	yes	—
111	C₂×#379	C₂×D₄	1	2	yes	—
112	#1783	C₂×D₄	1	2	yes	—
113	#1781	C₂×D₄	1	2	yes	—
114	#1781	C₂×D₄	1	2	yes	—
115	#1438	C₂×D₄	1	2	yes	—
116	C₂⁶	#121	1	2	yes	complemented
117	C₂⁶	C₂²×D₄	1	1	yes	characteristic
118	C₂⁶	C₂²×D₄	1	1	yes	characteristic
119	C₂×H(5,𝔽₂)	#121	1	2	yes	—
120	C₂×H(5,𝔽₂)	#121	1	2	yes	complemented
121	C₂×H(5,𝔽₂)	#127	1	2	yes	—
122	C₂³×D₄	#120	1	2	yes	—
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	C₂×#41	1	1	yes	characteristic
130	C₂³×D₄	C₂×#41	1	1	yes	characteristic
131	C₂³×D₄	C₂×#41	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	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂⁴×C₄	H(5,𝔽₂)	1	1	yes	characteristic
136	#534	#120	1	4	yes	—
137	#534	#121	1	4	yes	—
138	C₂×#120	#120	1	4	yes	—
139	C₂×#120	#121	1	4	yes	—
140	#518	#121	1	2	yes	—
141	#518	#121	1	2	yes	complemented
142	#518	#127	1	2	yes	—
143	C₂²×C₂²⋊C₄	#120	1	2	yes	—
144	C₂²×C₂²⋊C₄	#120	1	2	yes	—
145	C₂²×C₂²⋊C₄	#120	1	2	yes	—
146	C₂²×C₂²⋊C₄	#121	1	2	yes	complemented
147	C₂⁵	B(4,𝔽₂)	1	2	yes	—
148	C₂⁵	C₂×#120	1	1	yes	characteristic
149	C₂⁵	C₂×#120	1	1	yes	characteristic
150	C₂⁵	C₂×#120	1	1	yes	characteristic
151	C₂⁵	C₂×#121	1	1	yes	characteristic
152	C₂⁵	C₂×#121	1	1	yes	characteristic
153	C₂⁵	#525	1	1	yes	characteristic
154	C₂⁵	D₄²	1	1	yes	characteristic
155	C₂⁵	D₄²	1	1	yes	characteristic
156	C₂⁵	D₄²	1	1	yes	characteristic
157	C₂⁵	D₄²	1	1	yes	characteristic
158	C₂⁵	D₄²	1	1	yes	characteristic
159	C₂⁵	#546	1	1	yes	characteristic
160	C₂⁵	#546	1	1	yes	characteristic
161	C₂²×D₄	#392	1	4	yes	—
162	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
163	C₂²×D₄	C₂×#120	1	1	yes	characteristic
164	C₂²×D₄	C₂×#121	1	1	yes	characteristic
165	C₂²×D₄	C₂×#121	1	1	yes	characteristic
166	C₂²×D₄	C₂×#121	1	1	yes	characteristic
167	C₂²×D₄	#525	1	1	yes	characteristic
168	C₂²×D₄	#525	1	1	yes	characteristic
169	C₂²×D₄	C₂×#127	1	1	yes	characteristic
170	C₂²×D₄	#534	1	1	yes	characteristic
171	C₂²×D₄	#535	1	1	yes	characteristic
172	C₂²×D₄	#538	1	1	yes	characteristic
173	C₂²×D₄	D₄²	1	1	yes	characteristic
174	C₂²×D₄	D₄²	1	1	yes	characteristic
175	C₂²×D₄	#546	1	1	yes	characteristic
176	C₂²×D₄	#546	1	1	yes	characteristic
177	C₂²×D₄	#546	1	1	yes	characteristic
178	C₂²×D₄	#546	1	1	yes	characteristic
179	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
180	C₂³×C₄	B(4,𝔽₂)	1	4	yes	complemented
181	C₂³×C₄	#534	1	1	yes	characteristic
182	C₂³×C₄	#534	1	1	yes	characteristic
183	C₂³×C₄	#538	1	1	yes	characteristic
184	C₂³×C₄	D₄²	1	1	yes	characteristic
185	C₂³×C₄	#546	1	1	yes	characteristic
186	C₂³×C₄	#546	1	1	yes	characteristic
187	C₂×C₂²⋊C₄	#392	1	4	yes	—
188	C₂⁴	#2021	1	4	yes	complemented
189	C₂⁴	C₂×#392	1	1	yes	characteristic
190	C₂⁴	#2403	1	1	yes	characteristic
191	C₂⁴	#2404	1	1	yes	characteristic
192	C₂⁴	#2410	1	1	yes	characteristic
193	C₂⁴	#2410	1	1	yes	characteristic
194	C₂⁴	#2410	1	1	yes	characteristic
195	C₂⁴	#2424	1	1	yes	characteristic
196	C₂⁴	#2424	1	1	yes	characteristic
197	C₂⁴	#2424	1	1	yes	characteristic
198	C₂⁴	#2433	1	1	yes	characteristic
199	C₂⁴	#2847	1	1	yes	characteristic
200	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
201	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
202	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
203	C₂⁴	#3027	1	2	yes	—
204	C₂²×C₄	#2021	1	4	yes	complemented
205	C₂²×C₄	#2410	1	1	yes	characteristic
206	C₂²×C₄	#2424	1	1	yes	characteristic
207	C₂²×C₄	#2539	1	1	yes	characteristic
208	C₂²×C₄	#2617	1	1	yes	characteristic
209	C₂³	#14716	1	1	yes	characteristic
210	C₂³	C₂×#2021	1	2	yes	—
211	C₂³	#23895	1	1	yes	characteristic
212	C₂³	#23896	1	1	yes	characteristic
213	C₂³	#23900	1	1	yes	characteristic
214	C₂³	#23900	1	1	yes	characteristic
215	C₂³	#23900	1	1	yes	characteristic
216	C₂³	#23904	1	1	yes	characteristic
217	C₂³	#23984	1	2	yes	—
218	C₂³	#25771	1	2	yes	—
219	C₂³	#32926	1	2	yes	—
220	C₂²	#245225	1	1	yes	characteristic
221	C₂²	#245229	1	1	yes	characteristic
222	C₂²	#245247	1	1	yes	characteristic
223	C₂²	#245252	1	1	yes	characteristic
224	C₂²	#394167	1	1	yes	characteristic
225	C₂²	#394173	1	1	yes	characteristic
226	C₂²	#394174	1	1	yes	characteristic
227	C₂	B(5,𝔽₂)	1	2	yes	—
228	C₂	#72237500	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. #72264754	C₂	semidirect
2. #72255867	C₂	semidirect
3. #72246292	C₂	semidirect
4. #72238589	C₂	semidirect
5. #72225647	C₂	semidirect
6. #72219240	C₂	semidirect
7. #72219163	C₂	semidirect
8. #72219162	C₂	semidirect
9. #72218521	C₂	semidirect
10. #72218521	C₂	semidirect
11. #72218114	C₂	semidirect
12. C₂×#139750	C₂	semidirect
13. C₂×#139743	C₂	semidirect
14. C₂×#347042	C₂	semidirect
15. C₂×#511790	C₂	semidirect
17. #7642244	C₂²	semidirect
27. C₂×#12707	C₂²	semidirect
30. C₂×#12687	C₂²	semidirect
31. #382544	C₂²	semidirect
34. #359549	C₂²	semidirect
37. #359501	C₂²	semidirect
39. C₂×#8541	C₂²	semidirect
42. C₂×#8534	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. #107130	C₂²	semidirect
51. #62817	D₄	semidirect
57. #62652	D₄	semidirect
58. C₂×#3026	D₄	semidirect
59. C₂×#2883	D₄	semidirect
60. C₂×#2846	D₄	semidirect
63. C₂×#1913	D₄	semidirect
64. C₂×#1898	D₄	semidirect
65. #20609	D₄	semidirect
73. #20354	D₄	semidirect
74. #20354	D₄	semidirect
75. #14679	D₄	semidirect
77. #12705	D₄	semidirect
78. #10754	D₄	semidirect
79. #10735	D₄	semidirect
80. #8534	D₄	semidirect
116. C₂⁶	#121	semidirect
120. C₂×H(5,𝔽₂)	#121	semidirect
141. #518	#121	semidirect
146. C₂²×C₂²⋊C₄	#121	semidirect
180. C₂³×C₄	B(4,𝔽₂)	semidirect
188. C₂⁴	#2021	semidirect
204. C₂²×C₄	#2021	semidirect

Maximal subgroups

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

subgroup	index
#72264754	2
#72255867	2
#72246292	2
#72238589	2
#72225647	2
#72219240	2
#72219163	2
#72219162	2
#72218521	2
#72218114	2
C₂×#139750	2
C₂×#139743	2
C₂×#347042	2
C₂×#511790	2