Subgroups — #72310388

Order 2048.

At least 6547 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

The subgroup lattice has 184 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72266073	C₂	1	1	yes	maximal, characteristic, complemented
2	#72225619	C₂	1	1	yes	maximal, characteristic
3	#72219414	C₂	1	1	yes	maximal, characteristic, complemented
4	#72218069	C₂	1	1	yes	maximal, characteristic, complemented
5	#72218069	C₂	1	1	yes	maximal, characteristic, complemented
6	#72218057	C₂	1	1	yes	maximal, characteristic, complemented
7	#72218057	C₂	1	1	yes	maximal, characteristic, complemented
8	#7630939	C₂²	1	1	yes	characteristic
9	#7630939	C₂²	1	1	yes	characteristic
10	C₂×#23895	C₄	1	1	yes	characteristic, complemented
11	C₂²×#1781	C₂²	1	1	yes	characteristic
12	C₂²×#1781	C₂²	1	1	yes	characteristic
13	C₂²×#1781	C₂²	1	1	yes	characteristic
14	#392113	C₂²	1	1	yes	characteristic
15	#392109	C₂²	1	1	yes	characteristic
16	#373637	C₄	1	1	yes	characteristic, complemented
17	#359549	C₄	1	1	yes	characteristic, complemented
18	#359549	C₄	1	1	yes	characteristic, complemented
19	C₂⁵×D₄	C₂³	1	1	yes	characteristic
20	C₂×#3484	C₂×C₄	1	1	yes	characteristic
21	C₂×#3484	C₂×C₄	1	1	yes	characteristic
22	C₂×#3484	D₄	1	1	yes	characteristic
23	C₂×#3484	D₄	1	1	yes	characteristic
24	C₂×#3484	D₄	1	1	yes	characteristic
25	C₂×#3484	D₄	1	1	yes	characteristic
26	C₂³×#99	D₄	1	2	yes	complemented
27	#34171	C₂×C₄	1	1	yes	characteristic
28	#34171	C₂×C₄	1	1	yes	characteristic
29	#34171	D₄	1	1	yes	characteristic
30	#34171	D₄	1	1	yes	characteristic
31	#34171	D₄	1	1	yes	characteristic
32	#34171	D₄	1	1	yes	characteristic
33	C₂²×#379	D₄	1	1	yes	characteristic
34	C₂²×#379	D₄	1	1	yes	characteristic
35	C₂×#1783	D₄	1	2	yes	complemented
36	C₂×#1781	C₂×C₄	1	1	yes	characteristic
37	C₂×#1781	C₂×C₄	1	1	yes	characteristic
38	C₂×#1781	D₄	1	1	yes	characteristic
39	C₂×#1781	D₄	1	1	yes	characteristic
40	C₂×#1781	D₄	1	2	yes	complemented
41	C₂×#1781	D₄	1	2	yes	complemented
42	C₂⁷	C₂×D₄	1	1	yes	characteristic
43	C₂⁷	#41	1	1	yes	characteristic
44	C₂²×H(5,𝔽₂)	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×H(5,𝔽₂)	C₂²⋊C₄	1	1	yes	characteristic
46	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
47	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
48	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
49	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
50	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
51	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
52	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
53	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
54	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
55	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
56	C₂⁴×D₄	#41	1	1	yes	characteristic
57	C₂⁴×D₄	#41	1	1	yes	characteristic
58	C₂⁵×C₄	#41	1	1	yes	characteristic
59	C₂×#518	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×#518	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×#518	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×#518	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×#518	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×#518	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×#379	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×#379	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×#379	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×#379	C₂²⋊C₄	1	1	yes	characteristic
69	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂⁶	C₄×D₄	1	1	yes	characteristic
71	C₂⁶	#120	1	1	yes	characteristic
72	C₂⁶	#120	1	1	yes	characteristic
73	C₂⁶	#121	1	1	yes	characteristic
74	C₂⁶	#121	1	1	yes	characteristic
75	C₂⁶	#121	1	1	yes	characteristic
76	C₂⁶	#121	1	1	yes	characteristic
77	C₂⁶	#121	1	1	yes	characteristic
78	C₂⁶	#123	1	1	yes	characteristic
79	C₂⁶	#123	1	1	yes	characteristic
80	C₂⁶	#124	1	1	yes	characteristic
81	C₂⁶	#127	1	1	yes	characteristic
82	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂³×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
85	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
86	C₂³×D₄	#120	1	1	yes	characteristic
87	C₂³×D₄	#120	1	1	yes	characteristic
88	C₂³×D₄	#120	1	1	yes	characteristic
89	C₂³×D₄	#120	1	1	yes	characteristic
90	C₂³×D₄	#120	1	1	yes	characteristic
91	C₂³×D₄	#120	1	1	yes	characteristic
92	C₂³×D₄	#121	1	1	yes	characteristic
93	C₂³×D₄	#121	1	1	yes	characteristic
94	C₂³×D₄	#121	1	1	yes	characteristic
95	C₂³×D₄	#121	1	1	yes	characteristic
96	C₂³×D₄	#124	1	1	yes	characteristic
97	C₂³×D₄	#127	1	1	yes	characteristic
98	C₂⁴×C₄	C₄×D₄	1	1	yes	characteristic
99	C₂⁴×C₄	#121	1	1	yes	characteristic
100	C₂⁴×C₄	#121	1	1	yes	characteristic
101	C₂⁴×C₄	#121	1	1	yes	characteristic
102	C₂⁴×C₄	#123	1	1	yes	characteristic
103	C₂⁴×C₄	#123	1	1	yes	characteristic
104	#379	#99	1	2	yes	—
105	#379	#99	1	2	yes	—
106	C₂⁵	#379	1	1	yes	characteristic
107	C₂⁵	#386	1	1	yes	characteristic
108	C₂⁵	#386	1	1	yes	characteristic
109	C₂⁵	#390	1	1	yes	characteristic
110	C₂⁵	#390	1	1	yes	characteristic
111	C₂⁵	#392	1	1	yes	characteristic
112	C₂⁵	#392	1	1	yes	characteristic
113	C₂⁵	#392	1	1	yes	characteristic
114	C₂⁵	#394	1	1	yes	characteristic
115	C₂⁵	#394	1	1	yes	characteristic
116	C₂⁵	#394	1	1	yes	characteristic
117	C₂⁵	C₂×#99	1	1	yes	characteristic
118	C₂⁵	C₂×#99	1	1	yes	characteristic
119	C₂⁵	#410	1	1	yes	characteristic
120	C₂⁵	#410	1	1	yes	characteristic
121	C₂⁵	B(4,𝔽₂)	1	1	yes	characteristic
122	C₂⁵	B(4,𝔽₂)	1	1	yes	characteristic
123	C₂⁵	B(4,𝔽₂)	1	1	yes	characteristic
124	C₂⁵	B(4,𝔽₂)	1	1	yes	characteristic
125	C₂⁵	#458	1	1	yes	characteristic
126	C₂⁵	#458	1	1	yes	characteristic
127	C₂⁵	#458	1	1	yes	characteristic
128	C₂⁵	#458	1	1	yes	characteristic
129	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
130	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
131	C₂³×C₄	#386	1	1	yes	characteristic
132	C₂³×C₄	#386	1	1	yes	characteristic
133	C₂³×C₄	#392	1	1	yes	characteristic
134	C₂³×C₄	#394	1	1	yes	characteristic
135	C₂⁴	#1438	1	1	yes	characteristic
136	C₂⁴	#1781	1	1	yes	characteristic
137	C₂⁴	#1782	1	1	yes	characteristic
138	C₂⁴	#1790	1	1	yes	characteristic
139	C₂⁴	#1790	1	1	yes	characteristic
140	C₂⁴	#1790	1	1	yes	characteristic
141	C₂⁴	#1790	1	1	yes	characteristic
142	C₂⁴	#1889	1	2	yes	—
143	C₂⁴	#1913	1	2	yes	—
144	C₂⁴	#1964	1	1	yes	characteristic
145	C₂⁴	#1964	1	1	yes	characteristic
146	C₂⁴	#1964	1	1	yes	characteristic
147	C₂⁴	#1964	1	1	yes	characteristic
148	C₂⁴	#2021	1	2	yes	—
149	C₂⁴	#2021	1	1	yes	characteristic
150	C₂⁴	#2021	1	1	yes	characteristic
151	C₂⁴	#2021	1	2	yes	—
152	C₂⁴	#2021	1	2	yes	—
153	C₂⁴	#2022	1	1	yes	characteristic
154	C₂⁴	#2022	1	1	yes	characteristic
155	C₂⁴	#2044	1	1	yes	characteristic
156	C₂⁴	#2044	1	1	yes	characteristic
157	C₂⁴	#2044	1	1	yes	characteristic
158	C₂⁴	#2044	1	1	yes	characteristic
159	C₂²×C₄	#1889	1	2	yes	—
160	C₂²×C₄	#1913	1	2	yes	—
161	C₂²×C₄	#2021	1	2	yes	—
162	C₂³	#8534	1	1	yes	characteristic
163	C₂³	#8534	1	1	yes	characteristic
164	C₂³	#8535	1	1	yes	characteristic
165	C₂³	#8535	1	1	yes	characteristic
166	C₂³	#10336	1	2	yes	—
167	C₂³	#10337	1	2	yes	—
168	C₂³	#10972	1	1	yes	characteristic
169	C₂³	#10972	1	1	yes	characteristic
170	C₂³	#10980	1	1	yes	characteristic
171	C₂³	#10991	1	1	yes	characteristic
172	C₂³	#10991	1	1	yes	characteristic
173	C₂³	#10995	1	1	yes	characteristic
174	C₂³	#10995	1	1	yes	characteristic
175	C₂²	#111458	1	1	yes	characteristic
176	C₂²	#111462	1	1	yes	characteristic
177	C₂²	#111467	1	1	yes	characteristic
178	C₂²	#115163	1	1	yes	characteristic
179	C₂²	#115163	1	1	yes	characteristic
180	C₂²	#115204	1	1	yes	characteristic
181	C₂²	#115204	1	1	yes	characteristic
182	C₂	#72258689	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72258689
commutator	C₂⁴×D₄	C₂²×C₄
Frattini, cosocle	C₂⁵×D₄	C₂³
prime radical	#72225619	C₂
LCS	G ▷ C₂⁴×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂⁵×D₄ ◁ G
LECS	G ▷ C₂⁵×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #72266073	C₂	semidirect
3. #72219414	C₂	semidirect
4. #72218069	C₂	semidirect
5. #72218069	C₂	semidirect
6. #72218057	C₂	semidirect
7. #72218057	C₂	semidirect
10. C₂×#23895	C₄	semidirect
16. #373637	C₄	semidirect
17. #359549	C₄	semidirect
18. #359549	C₄	semidirect
26. C₂³×#99	D₄	semidirect
35. C₂×#1783	D₄	semidirect
40. C₂×#1781	D₄	semidirect
41. C₂×#1781	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72266073	2
#72225619	2
#72219414	2
#72218069	2
#72218057	2