Subgroups — #72262619

Order 2048.

At least 6616 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

The subgroup lattice has 225 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#135513	C₂	1	1	yes	maximal, characteristic, complemented
2	#72262540	C₂	1	1	yes	maximal, characteristic, complemented
3	#72256195	C₂	1	1	yes	maximal, characteristic, complemented
4	#72255778	C₂	1	1	yes	maximal, characteristic, complemented
5	#72255774	C₂	1	1	yes	maximal, characteristic, complemented
6	#72251501	C₂	1	1	yes	maximal, characteristic, complemented
7	#72251476	C₂	1	1	yes	maximal, characteristic, complemented
8	#72251475	C₂	1	1	yes	maximal, characteristic, complemented
9	#72241271	C₂	1	1	yes	maximal, characteristic, complemented
10	#72237037	C₂	1	1	yes	maximal, characteristic, complemented
11	#72236928	C₂	1	1	yes	maximal, characteristic, complemented
12	#72236636	C₂	1	1	yes	maximal, characteristic, complemented
13	#72225740	C₂	1	1	yes	maximal, characteristic, complemented
14	#72218475	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#511790	C₂	1	1	yes	maximal, characteristic
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×#23897	C₂²	1	1	yes	characteristic
20	C₂×#21746	C₂²	1	1	yes	characteristic
21	#556752	C₂²	1	1	yes	characteristic
22	C₂×#12707	C₂²	1	1	yes	characteristic
23	C₂×#12707	C₂²	1	1	yes	characteristic
24	C₂×#12705	C₂²	1	1	yes	characteristic
25	C₂×#12705	C₂²	1	1	yes	characteristic
26	C₂×#12687	C₂²	1	1	yes	characteristic
27	C₂×#12687	C₂²	1	1	yes	characteristic
28	#392030	C₂²	1	1	yes	characteristic, complemented
29	#382544	C₂²	1	1	yes	characteristic, complemented
30	#382434	C₂²	1	1	yes	characteristic, complemented
31	#359558	C₂²	1	1	yes	characteristic
32	#359549	C₂²	1	1	yes	characteristic
33	#359549	C₂²	1	1	yes	characteristic, complemented
34	#359501	C₂²	1	1	yes	characteristic
35	#187944	C₂²	1	1	yes	characteristic
36	#187944	C₂²	1	1	yes	characteristic
37	#187944	C₂²	1	1	yes	characteristic
38	#187944	C₂²	1	1	yes	characteristic
39	#187944	C₂²	1	1	yes	characteristic, complemented
40	#186835	C₂²	1	1	yes	characteristic, complemented
41	#128033	C₂²	1	1	yes	characteristic, complemented
42	#128033	C₂²	1	1	yes	characteristic, complemented
43	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
44	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
45	C₂×#7311	C₂²	1	1	yes	characteristic, complemented
46	C₂×#7310	C₂²	1	1	yes	characteristic, complemented
47	#128009	C₂²	1	1	yes	characteristic, complemented
48	#128009	C₂²	1	1	yes	characteristic, complemented
49	#108238	C₂²	1	1	yes	characteristic, complemented
50	#107114	C₂²	1	1	yes	characteristic, complemented
51	#62817	D₄	1	2	yes	—
52	#62817	D₄	1	2	yes	—
53	C₂×#3484	C₂³	1	1	yes	characteristic
54	C₂×#3484	C₂³	1	1	yes	characteristic
55	C₂×#3484	C₂³	1	1	yes	characteristic
56	C₂³×#120	C₂³	1	1	yes	characteristic
57	C₂²×B(4,𝔽₂)	D₄	1	2	yes	—
58	#34171	C₂³	1	1	yes	characteristic
59	C₂²×#379	D₄	1	2	yes	complemented
60	C₂×#1889	D₄	1	2	yes	—
61	C₂×#1889	D₄	1	2	yes	complemented
62	C₂×#1889	D₄	1	2	yes	complemented
63	C₂×#1819	D₄	1	2	yes	complemented
64	C₂×#1783	C₂³	1	1	yes	characteristic
65	C₂×#1783	C₂³	1	1	yes	characteristic
66	C₂×#1781	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	#14657	D₄	1	2	yes	complemented
73	#14335	C₂³	1	1	yes	characteristic
74	#14335	C₂³	1	1	yes	characteristic
75	#14134	D₄	1	2	yes	complemented
76	#12687	D₄	1	4	yes	—
77	C₂⁷	C₂×D₄	1	1	yes	characteristic
78	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
79	C₂²×H(5,𝔽₂)	#41	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	#41	1	2	yes	—
85	#3484	#41	1	2	yes	—
86	C₂²×#120	C₂×D₄	1	2	yes	—
87	C₂²×#120	C₂×D₄	1	2	yes	—
88	C₂²×#120	C₂×D₄	1	1	yes	characteristic
89	C₂²×#120	C₂×D₄	1	1	yes	characteristic
90	C₂²×#120	C₂×D₄	1	1	yes	characteristic
91	C₂²×#120	C₂×D₄	1	1	yes	characteristic
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	#41	1	1	yes	characteristic
98	C₂×#518	#41	1	1	yes	characteristic
99	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
100	C₂²×#99	C₂×D₄	1	2	yes	—
101	C₂²×#99	C₂×D₄	1	2	yes	—
102	C₂×#379	C₂×D₄	1	2	yes	—
103	C₂×#379	C₂×D₄	1	2	yes	—
104	#1783	C₂×D₄	1	2	yes	—
105	#1781	C₂×D₄	1	2	yes	—
106	#1438	C₂×D₄	1	2	yes	—
107	#1438	C₂×D₄	1	2	yes	—
108	C₂⁶	#120	1	1	yes	characteristic
109	C₂⁶	#120	1	1	yes	characteristic
110	C₂⁶	#120	1	2	yes	—
111	C₂⁶	#121	1	2	yes	—
112	C₂⁶	C₂²×D₄	1	1	yes	characteristic
113	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂×H(5,𝔽₂)	#123	1	2	yes	complemented
115	C₂×H(5,𝔽₂)	#123	1	2	yes	complemented
116	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
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	C₂×#41	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	C₂×#41	1	1	yes	characteristic
125	C₂³×D₄	C₂×#41	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	C₂⁴×C₄	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂×#120	#120	1	2	yes	—
130	C₂×#120	#120	1	2	yes	—
131	C₂×#120	#121	1	2	yes	—
132	C₂×#120	#121	1	2	yes	—
133	C₂×#120	#121	1	2	yes	—
134	C₂×#120	#121	1	2	yes	—
135	C₂×#120	#127	1	2	yes	—
136	C₂×#120	#127	1	2	yes	—
137	#518	#123	1	2	yes	complemented
138	#518	#123	1	2	yes	complemented
139	C₂²×C₂²⋊C₄	#120	1	2	yes	—
140	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
141	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
142	C₂²×C₂²⋊C₄	#121	1	2	yes	—
143	C₂⁵	#392	1	2	yes	—
144	C₂⁵	#392	1	2	yes	—
145	C₂⁵	B(4,𝔽₂)	1	2	yes	—
146	C₂⁵	B(4,𝔽₂)	1	2	yes	—
147	C₂⁵	B(4,𝔽₂)	1	2	yes	—
148	C₂⁵	B(4,𝔽₂)	1	2	yes	—
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	#525	1	1	yes	characteristic
153	C₂⁵	#525	1	1	yes	characteristic
154	C₂⁵	#534	1	1	yes	characteristic
155	C₂⁵	#534	1	1	yes	characteristic
156	C₂⁵	#535	1	1	yes	characteristic
157	C₂⁵	#535	1	1	yes	characteristic
158	C₂⁵	#546	1	1	yes	characteristic
159	C₂⁵	#546	1	1	yes	characteristic
160	C₂⁵	#546	1	1	yes	characteristic
161	C₂⁵	#551	1	1	yes	characteristic
162	C₂²×D₄	C₂×#120	1	1	yes	characteristic
163	C₂²×D₄	C₂×#121	1	1	yes	characteristic
164	C₂²×D₄	C₂×#121	1	1	yes	characteristic
165	C₂²×D₄	C₂×#123	1	1	yes	characteristic
166	C₂²×D₄	C₂×#123	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	#546	1	1	yes	characteristic
171	C₂²×D₄	#546	1	1	yes	characteristic
172	C₂²×D₄	#546	1	1	yes	characteristic
173	C₂²×D₄	#546	1	1	yes	characteristic
174	C₂²×D₄	#546	1	1	yes	characteristic
175	C₂²×D₄	#546	1	1	yes	characteristic
176	C₂²×D₄	#551	1	1	yes	characteristic
177	C₂²×D₄	#551	1	1	yes	characteristic
178	C₂³×C₄	#534	1	1	yes	characteristic
179	C₂³×C₄	#534	1	1	yes	characteristic
180	C₂³×C₄	#546	1	1	yes	characteristic
181	C₂³×C₄	#546	1	1	yes	characteristic
182	C₂³×C₄	#546	1	1	yes	characteristic
183	C₂³×C₄	#551	1	1	yes	characteristic
184	C₂×C₂²⋊C₄	#392	1	2	yes	—
185	C₂×C₂²⋊C₄	#392	1	2	yes	—
186	C₂⁴	#2021	1	2	yes	—
187	C₂⁴	#2021	1	2	yes	—
188	C₂⁴	#2021	1	2	yes	—
189	C₂⁴	#2021	1	2	yes	—
190	C₂⁴	C₂×#392	1	1	yes	characteristic
191	C₂⁴	#2404	1	1	yes	characteristic
192	C₂⁴	#2404	1	1	yes	characteristic
193	C₂⁴	#2405	1	1	yes	characteristic
194	C₂⁴	#2411	1	1	yes	characteristic
195	C₂⁴	#2411	1	1	yes	characteristic
196	C₂⁴	#2412	1	1	yes	characteristic
197	C₂⁴	#2534	1	1	yes	characteristic
198	C₂⁴	#2539	1	1	yes	characteristic
199	C₂⁴	#2613	1	1	yes	characteristic
200	C₂⁴	#2849	1	1	yes	characteristic
201	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
202	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
203	C₂²×C₄	#2405	1	1	yes	characteristic
204	C₂²×C₄	#2412	1	1	yes	characteristic
205	C₂²×C₄	#2534	1	1	yes	characteristic
206	C₂²×C₄	#2539	1	1	yes	characteristic
207	C₂³	#14911	1	1	yes	characteristic
208	C₂³	C₂×#2021	1	2	yes	—
209	C₂³	#23984	1	2	yes	—
210	C₂³	#23984	1	1	yes	characteristic
211	C₂³	#23984	1	1	yes	characteristic
212	C₂³	#23996	1	1	yes	characteristic
213	C₂³	#23996	1	1	yes	characteristic
214	C₂³	#23996	1	1	yes	characteristic
215	C₂³	#23997	1	1	yes	characteristic
216	C₂²	#251827	1	1	yes	characteristic
217	C₂²	#251837	1	1	yes	characteristic
218	C₂²	#251843	1	1	yes	characteristic
219	C₂²	#251854	1	1	yes	characteristic
220	C₂²	#405156	1	1	yes	characteristic
221	C₂²	#405268	1	1	yes	characteristic
222	C₂²	#405274	1	1	yes	characteristic
223	C₂	#72251614	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72251614
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	C₂×#511790	C₂
LCS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ 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. C₂×#135513	C₂	semidirect
2. #72262540	C₂	semidirect
3. #72256195	C₂	semidirect
4. #72255778	C₂	semidirect
5. #72255774	C₂	semidirect
6. #72251501	C₂	semidirect
7. #72251476	C₂	semidirect
8. #72251475	C₂	semidirect
9. #72241271	C₂	semidirect
10. #72237037	C₂	semidirect
11. #72236928	C₂	semidirect
12. #72236636	C₂	semidirect
13. #72225740	C₂	semidirect
14. #72218475	C₂	semidirect
28. #392030	C₂²	semidirect
29. #382544	C₂²	semidirect
30. #382434	C₂²	semidirect
33. #359549	C₂²	semidirect
39. #187944	C₂²	semidirect
40. #186835	C₂²	semidirect
41. #128033	C₂²	semidirect
42. #128033	C₂²	semidirect
43. C₂×#7316	C₂²	semidirect
44. C₂×#7316	C₂²	semidirect
45. C₂×#7311	C₂²	semidirect
46. C₂×#7310	C₂²	semidirect
47. #128009	C₂²	semidirect
48. #128009	C₂²	semidirect
49. #108238	C₂²	semidirect
50. #107114	C₂²	semidirect
59. C₂²×#379	D₄	semidirect
61. C₂×#1889	D₄	semidirect
62. C₂×#1889	D₄	semidirect
63. C₂×#1819	D₄	semidirect
72. #14657	D₄	semidirect
75. #14134	D₄	semidirect
114. C₂×H(5,𝔽₂)	#123	semidirect
115. C₂×H(5,𝔽₂)	#123	semidirect
137. #518	#123	semidirect
138. #518	#123	semidirect

Maximal subgroups

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

subgroup	index
C₂×#135513	2
#72262540	2
#72256195	2
#72255778	2
#72255774	2
#72251501	2
#72251476	2
#72251475	2
#72241271	2
#72237037	2
#72236928	2
#72236636	2
#72225740	2
#72218475	2
C₂×#511790	2