Subgroups — C₂³×#2273 #72295990

Order 1024.

292245 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₂²⋊C₄	C₂	1	3	yes	maximal
2	C₂²×#2273	C₂	1	112	yes	maximal, direct product factor
3	C₂³×C₄×C₂²⋊C₄	C₂	1	12	yes	maximal, complemented
4	C₂⁸	C₂²	1	1	yes	characteristic
5	C₂⁶×C₄	C₂²	1	6	yes	—
6	C₂⁴×C₂²⋊C₄	C₄	1	192	yes	complemented
7	C₂⁴×C₂²⋊C₄	C₂²	1	12	yes	—
8	C₂⁴×C₂²⋊C₄	C₂²	1	168	yes	—
9	C₂⁴×C₄²	C₂²	1	8	yes	complemented
10	C₂×#2273	C₂²	1	1792	yes	direct product factor
11	C₂²×C₄×C₂²⋊C₄	C₂²	1	672	yes	complemented
12	C₂³×#95	C₂²	1	8	yes	complemented
13	C₂⁷	—	4	32	no	—
14	C₂⁷	C₂×C₄	1	96	yes	—
15	C₂⁷	C₂³	1	3	yes	—
16	C₂⁷	C₂³	1	28	yes	—
17	C₂⁵×C₄	—	4	96	no	—
18	C₂⁵×C₄	C₂×C₄	1	192	yes	complemented
19	C₂⁵×C₄	C₂³	1	12	yes	—
20	C₂⁵×C₄	C₂³	1	168	yes	—
21	C₂³×C₂²⋊C₄	C₂×C₄	1	384	yes	complemented
22	C₂³×C₂²⋊C₄	C₂×C₄	1	5376	yes	complemented
23	C₂³×C₂²⋊C₄	C₂³	1	336	yes	—
24	C₂³×C₂²⋊C₄	C₂³	1	1344	yes	—
25	C₂³×C₄²	—	4	64	no	—
26	C₂³×C₄²	C₂³	1	224	yes	complemented
27	#2273	C₂³	1	4096	yes	direct product factor
28	C₂×C₄×C₂²⋊C₄	C₂³	1	5376	yes	complemented
29	C₂²×#95	C₂³	1	224	yes	complemented
30	C₂⁶	—	2	96	no	—
31	C₂⁶	—	4	1536	no	—
32	C₂⁶	—	4	448	no	—
33	C₂⁶	C₄²	1	1024	yes	complemented
34	C₂⁶	C₂²×C₄	1	48	yes	—
35	C₂⁶	C₂²×C₄	1	96	yes	—
36	C₂⁶	C₂²×C₄	1	1344	yes	—
37	C₂⁶	C₂⁴	1	42	yes	—
38	C₂⁶	C₂⁴	1	112	yes	—
39	C₂⁶	C₂⁴	1	1	yes	characteristic
40	C₂⁴×C₄	—	2	192	no	—
41	C₂⁴×C₄	—	4	1536	no	—
42	C₂⁴×C₄	—	4	1344	no	—
43	C₂⁴×C₄	C₂²×C₄	1	192	yes	—
44	C₂⁴×C₄	C₂²×C₄	1	2688	yes	complemented
45	C₂⁴×C₄	C₂⁴	1	168	yes	—
46	C₂⁴×C₄	C₂⁴	1	672	yes	—
47	C₂²×C₂²⋊C₄	C₂²×C₄	1	5376	yes	complemented
48	C₂²×C₂²⋊C₄	C₂²×C₄	1	21504	yes	complemented
49	C₂²×C₂²⋊C₄	C₂⁴	1	1344	yes	—
50	C₂²×C₂²⋊C₄	C₂⁴	1	1536	yes	—
51	C₂²×C₄²	—	4	896	no	—
52	C₂²×C₄²	C₂⁴	1	896	yes	complemented
53	C₄×C₂²⋊C₄	C₂⁴	1	6144	yes	complemented
54	C₂×#95	C₂⁴	1	896	yes	complemented
55	C₂⁵	—	2	768	no	—
56	C₂⁵	—	2	1536	no	—
57	C₂⁵	—	2	672	no	—
58	C₂⁵	—	4	8192	no	—
59	C₂⁵	—	4	10752	no	—
60	C₂⁵	—	4	896	no	—
61	C₂⁵	C₂×C₄²	1	768	yes	—
62	C₂⁵	C₂×C₄²	1	7168	yes	complemented
63	C₂⁵	C₂³×C₄	1	672	yes	—
64	C₂⁵	C₂³×C₄	1	336	yes	—
65	C₂⁵	C₂³×C₄	1	2688	yes	—
66	C₂⁵	C₂³×C₄	1	24	yes	—
67	C₂⁵	H(5,𝔽₂)	1	32	yes	—
68	C₂⁵	C₂⁵	1	84	yes	—
69	C₂⁵	C₂⁵	1	64	yes	—
70	C₂⁵	C₂⁵	1	7	yes	—
71	C₂³×C₄	—	2	1536	no	—
72	C₂³×C₄	—	2	1344	no	—
73	C₂³×C₄	—	4	10752	no	—
74	C₂³×C₄	—	4	2688	no	—
75	C₂³×C₄	C₂³×C₄	1	1344	yes	—
76	C₂³×C₄	C₂³×C₄	1	5376	yes	complemented
77	C₂³×C₄	C₂⁵	1	336	yes	—
78	C₂³×C₄	C₂⁵	1	384	yes	—
79	C₂×C₂²⋊C₄	C₂³×C₄	1	10752	yes	complemented
80	C₂×C₂²⋊C₄	C₂³×C₄	1	12288	yes	complemented
81	C₂×C₂²⋊C₄	C₂⁵	1	768	yes	—
82	C₂×C₄²	—	4	1792	no	—
83	C₂×C₄²	C₂⁵	1	512	yes	complemented
84	#95	C₂⁵	1	512	yes	complemented
85	C₂⁴	—	2	672	no	—
86	C₂⁴	—	2	2688	no	—
87	C₂⁴	—	2	5376	no	—
88	C₂⁴	—	2	6144	no	—
89	C₂⁴	—	4	28672	no	—
90	C₂⁴	—	4	10752	no	—
91	C₂⁴	—	4	256	no	—
92	C₂⁴	C₂²×C₄²	1	2688	yes	—
93	C₂⁴	C₂²×C₄²	1	64	yes	—
94	C₂⁴	C₂²×C₄²	1	7168	yes	complemented
95	C₂⁴	#518	1	384	yes	—
96	C₂⁴	C₂⁴×C₄	1	672	yes	—
97	C₂⁴	C₂⁴×C₄	1	336	yes	—
98	C₂⁴	C₂⁴×C₄	1	768	yes	—
99	C₂⁴	C₂⁴×C₄	1	84	yes	—
100	C₂⁴	C₂×H(5,𝔽₂)	1	112	yes	—
101	C₂⁴	C₂⁶	1	24	yes	—
102	C₂⁴	C₂⁶	1	7	yes	—
103	C₂²×C₄	—	2	1344	no	—
104	C₂²×C₄	—	2	5376	no	—
105	C₂²×C₄	—	4	10752	no	—
106	C₂²×C₄	—	4	768	no	—
107	C₂²×C₄	C₂⁴×C₄	1	1344	yes	—
108	C₂²×C₄	C₂⁴×C₄	1	1536	yes	complemented
109	C₂²×C₄	C₂⁶	1	96	yes	—
110	C₂²⋊C₄	C₂⁴×C₄	1	3072	yes	complemented
111	C₄²	—	4	512	no	—
112	C₂³	—	2	96	no	—
113	C₂³	—	2	1344	no	—
114	C₂³	—	2	2688	no	—
115	C₂³	—	2	10752	no	—
116	C₂³	—	4	1536	no	—
117	C₂³	—	4	14336	no	—
118	C₂³	#2273	1	512	yes	direct product factor
119	C₂³	C₂³×C₄²	1	1344	yes	—
120	C₂³	C₂³×C₄²	1	112	yes	—
121	C₂³	C₂³×C₄²	1	1024	yes	complemented
122	C₂³	C₂×#518	1	672	yes	—
123	C₂³	C₂⁵×C₄	1	96	yes	—
124	C₂³	C₂⁵×C₄	1	48	yes	—
125	C₂³	C₂⁵×C₄	1	42	yes	—
126	C₂³	C₂²×H(5,𝔽₂)	1	56	yes	—
127	C₂³	C₂⁷	1	1	yes	characteristic
128	C₂×C₄	—	2	192	no	—
129	C₂×C₄	—	2	2688	no	—
130	C₂×C₄	—	4	1536	no	—
131	C₂×C₄	C₂⁵×C₄	1	192	yes	—
132	C₂²	—	2	96	no	—
133	C₂²	—	2	192	no	—
134	C₂²	—	2	2688	no	—
135	C₂²	—	4	1024	no	—
136	C₂²	C₂×#2273	1	448	yes	direct product factor
137	C₂²	C₂⁴×C₄²	1	96	yes	—
138	C₂²	C₂⁴×C₄²	1	28	yes	—
139	C₂²	C₂²×#518	1	168	yes	—
140	C₂²	C₂⁶×C₄	1	3	yes	—
141	C₂²	C₂³×H(5,𝔽₂)	1	4	yes	—
142	C₄	—	2	192	no	—
143	C₂	—	2	96	no	—
144	C₂	C₂²×#2273	1	56	yes	direct product factor
145	C₂	C₂⁵×C₄²	1	1	yes	characteristic
146	C₂	C₂³×#518	1	6	yes	—

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂	C₂⁵×C₄²
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁸	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂²×#2273	C₂	direct (N × H)
3. C₂³×C₄×C₂²⋊C₄	C₂	semidirect
6. C₂⁴×C₂²⋊C₄	C₄	semidirect
9. C₂⁴×C₄²	C₂²	semidirect
10. C₂×#2273	C₂²	direct (N × H)
11. C₂²×C₄×C₂²⋊C₄	C₂²	semidirect
12. C₂³×#95	C₂²	semidirect
18. C₂⁵×C₄	C₂×C₄	semidirect
21. C₂³×C₂²⋊C₄	C₂×C₄	semidirect
22. C₂³×C₂²⋊C₄	C₂×C₄	semidirect
26. C₂³×C₄²	C₂³	semidirect
27. #2273	C₂³	direct (N × H)
28. C₂×C₄×C₂²⋊C₄	C₂³	semidirect
29. C₂²×#95	C₂³	semidirect
33. C₂⁶	C₄²	semidirect
44. C₂⁴×C₄	C₂²×C₄	semidirect
47. C₂²×C₂²⋊C₄	C₂²×C₄	semidirect
48. C₂²×C₂²⋊C₄	C₂²×C₄	semidirect
52. C₂²×C₄²	C₂⁴	semidirect
53. C₄×C₂²⋊C₄	C₂⁴	semidirect
54. C₂×#95	C₂⁴	semidirect
62. C₂⁵	C₂×C₄²	semidirect
76. C₂³×C₄	C₂³×C₄	semidirect
79. C₂×C₂²⋊C₄	C₂³×C₄	semidirect
80. C₂×C₂²⋊C₄	C₂³×C₄	semidirect
83. C₂×C₄²	C₂⁵	semidirect
84. #95	C₂⁵	semidirect
94. C₂⁴	C₂²×C₄²	semidirect
108. C₂²×C₄	C₂⁴×C₄	semidirect
110. C₂²⋊C₄	C₂⁴×C₄	semidirect
118. C₂³	#2273	direct (N × H)
121. C₂³	C₂³×C₄²	semidirect
136. C₂²	C₂×#2273	direct (N × H)
144. C₂	C₂²×#2273	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₂²⋊C₄	2
C₂²×#2273	2
C₂³×C₄×C₂²⋊C₄	2

Subgroups — C23×#2273 #72295990

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#2273 #72295990