Subgroups — C₂⁴×#382 #72325641

Order 1024.

184469 conjugacy classes of subgroups.

The subgroup lattice has 145 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄⋊C₄	C₂	1	2	yes	maximal
2	C₂⁵×C₄²	C₂	1	1	yes	maximal, characteristic
3	C₂³×#382	C₂	1	120	yes	maximal, direct product factor
4	C₂⁴×#95	C₂	1	4	yes	maximal
5	C₂⁶×C₄	C₂²	1	4	yes	—
6	C₂⁶×C₄	C₂²	1	2	yes	—
7	C₂⁶×C₄	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄⋊C₄	—	2	64	no	—
9	C₂⁴×C₄⋊C₄	C₂²	1	120	yes	—
10	C₂⁴×C₄²	C₄	1	64	yes	complemented
11	C₂⁴×C₄²	C₂²	1	60	yes	—
12	C₂²×#382	C₂²	1	2240	yes	direct product factor
13	C₂³×#95	C₂²	1	240	yes	—
14	C₂⁷	C₂³	1	1	yes	characteristic
15	C₂⁵×C₄	—	2	64	no	—
16	C₂⁵×C₄	—	2	64	no	—
17	C₂⁵×C₄	—	2	128	no	—
18	C₂⁵×C₄	C₂×C₄	1	64	yes	—
19	C₂⁵×C₄	C₂×C₄	1	32	yes	—
20	C₂⁵×C₄	D₄	1	32	yes	—
21	C₂⁵×C₄	Q₈	1	32	yes	—
22	C₂⁵×C₄	C₂³	1	120	yes	—
23	C₂⁵×C₄	C₂³	1	60	yes	—
24	C₂⁵×C₄	C₂³	1	30	yes	—
25	C₂³×C₄⋊C₄	—	2	1920	no	—
26	C₂³×C₄⋊C₄	C₂³	1	1120	yes	—
27	C₂³×C₄²	C₂×C₄	1	1920	yes	complemented
28	C₂³×C₄²	C₂³	1	560	yes	—
29	C₂×#382	C₂³	1	7680	yes	direct product factor
30	C₂²×#95	C₂³	1	2240	yes	—
31	C₂⁶	C₂²×C₄	1	16	yes	—
32	C₂⁶	C₂×D₄	1	16	yes	—
33	C₂⁶	C₂×Q₈	1	16	yes	—
34	C₂⁶	#41	1	64	yes	—
35	C₂⁶	C₂⁴	1	15	yes	—
36	C₂⁴×C₄	—	2	1024	no	—
37	C₂⁴×C₄	—	2	960	no	—
38	C₂⁴×C₄	—	2	960	no	—
39	C₂⁴×C₄	—	2	1920	no	—
40	C₂⁴×C₄	—	4	1024	no	—
41	C₂⁴×C₄	C₄⋊C₄	1	1024	yes	—
42	C₂⁴×C₄	C₂²×C₄	1	960	yes	—
43	C₂⁴×C₄	C₂²×C₄	1	480	yes	—
44	C₂⁴×C₄	C₂×D₄	1	480	yes	—
45	C₂⁴×C₄	C₂×Q₈	1	480	yes	—
46	C₂⁴×C₄	C₂⁴	1	560	yes	—
47	C₂⁴×C₄	C₂⁴	1	280	yes	—
48	C₂⁴×C₄	C₂⁴	1	140	yes	—
49	C₂²×C₄⋊C₄	—	2	8960	no	—
50	C₂²×C₄⋊C₄	C₂⁴	1	1920	yes	—
51	C₂²×C₄²	C₂²×C₄	1	8960	yes	complemented
52	C₂²×C₄²	C₂⁴	1	960	yes	—
53	#382	C₂⁴	1	4096	yes	direct product factor
54	C₂×#95	C₂⁴	1	3840	yes	—
55	C₂⁵	C₂×C₄⋊C₄	1	256	yes	—
56	C₂⁵	#117	1	512	yes	—
57	C₂⁵	#124	1	512	yes	—
58	C₂⁵	#125	1	512	yes	—
59	C₂⁵	C₂³×C₄	1	120	yes	—
60	C₂⁵	C₂²×D₄	1	120	yes	—
61	C₂⁵	C₂²×Q₈	1	120	yes	—
62	C₂⁵	C₂×#41	1	480	yes	—
63	C₂⁵	C₂⁵	1	35	yes	—
64	C₂³×C₄	—	2	7680	no	—
65	C₂³×C₄	—	2	2240	no	—
66	C₂³×C₄	—	2	2240	no	—
67	C₂³×C₄	—	2	4480	no	—
68	C₂³×C₄	—	4	7680	no	—
69	C₂³×C₄	C₂×C₄⋊C₄	1	7680	yes	—
70	C₂³×C₄	C₂³×C₄	1	2240	yes	—
71	C₂³×C₄	C₂³×C₄	1	1120	yes	—
72	C₂³×C₄	C₂²×D₄	1	1120	yes	—
73	C₂³×C₄	C₂²×Q₈	1	1120	yes	—
74	C₂³×C₄	C₂⁵	1	480	yes	—
75	C₂³×C₄	C₂⁵	1	240	yes	—
76	C₂³×C₄	C₂⁵	1	120	yes	—
77	C₂×C₄⋊C₄	—	2	7680	no	—
78	C₂×C₄⋊C₄	C₂⁵	1	512	yes	—
79	C₂×C₄²	C₂³×C₄	1	7680	yes	complemented
80	C₂×C₄²	C₂⁵	1	256	yes	—
81	#95	C₂⁵	1	1024	yes	—
82	C₂⁴	#382	1	4096	yes	direct product factor
83	C₂⁴	C₂²×C₄⋊C₄	1	960	yes	—
84	C₂⁴	C₂×#117	1	1920	yes	—
85	C₂⁴	C₂×#124	1	1920	yes	—
86	C₂⁴	C₂×#125	1	1920	yes	—
87	C₂⁴	C₂⁴×C₄	1	140	yes	—
88	C₂⁴	C₂³×D₄	1	140	yes	—
89	C₂⁴	C₂³×Q₈	1	140	yes	—
90	C₂⁴	C₂²×#41	1	560	yes	—
91	C₂⁴	C₂⁶	1	15	yes	—
92	C₂²×C₄	—	2	960	no	—
93	C₂²×C₄	—	2	8960	no	—
94	C₂²×C₄	—	2	960	no	—
95	C₂²×C₄	—	2	1920	no	—
96	C₂²×C₄	—	4	8960	no	—
97	C₂²×C₄	C₂²×C₄⋊C₄	1	8960	yes	—
98	C₂²×C₄	C₂⁴×C₄	1	960	yes	—
99	C₂²×C₄	C₂⁴×C₄	1	480	yes	—
100	C₂²×C₄	C₂³×D₄	1	480	yes	—
101	C₂²×C₄	C₂³×Q₈	1	480	yes	—
102	C₂²×C₄	C₂⁶	1	64	yes	—
103	C₂²×C₄	C₂⁶	1	32	yes	—
104	C₂²×C₄	C₂⁶	1	16	yes	—
105	C₄⋊C₄	—	2	1024	no	—
106	C₄²	C₂⁴×C₄	1	1024	yes	complemented
107	C₂³	C₂×#382	1	7680	yes	direct product factor
108	C₂³	C₂³×C₄⋊C₄	1	560	yes	—
109	C₂³	C₂²×#117	1	1120	yes	—
110	C₂³	C₂²×#124	1	1120	yes	—
111	C₂³	C₂²×#125	1	1120	yes	—
112	C₂³	C₂⁵×C₄	1	30	yes	—
113	C₂³	C₂⁴×D₄	1	30	yes	—
114	C₂³	C₂⁴×Q₈	1	30	yes	—
115	C₂³	C₂³×#41	1	120	yes	—
116	C₂³	C₂⁷	1	1	yes	characteristic
117	C₂×C₄	—	2	64	no	—
118	C₂×C₄	—	2	1920	no	—
119	C₂×C₄	—	2	64	no	—
120	C₂×C₄	—	2	128	no	—
121	C₂×C₄	—	4	1920	no	—
122	C₂×C₄	C₂³×C₄⋊C₄	1	1920	yes	—
123	C₂×C₄	C₂⁵×C₄	1	64	yes	—
124	C₂×C₄	C₂⁵×C₄	1	32	yes	—
125	C₂×C₄	C₂⁴×D₄	1	32	yes	—
126	C₂×C₄	C₂⁴×Q₈	1	32	yes	—
127	C₂²	C₂²×#382	1	2240	yes	direct product factor
128	C₂²	C₂⁴×C₄⋊C₄	1	60	yes	—
129	C₂²	C₂³×#117	1	120	yes	—
130	C₂²	C₂³×#124	1	120	yes	—
131	C₂²	C₂³×#125	1	120	yes	—
132	C₂²	C₂⁶×C₄	1	1	yes	characteristic
133	C₂²	C₂⁵×D₄	1	1	yes	characteristic
134	C₂²	C₂⁵×Q₈	1	1	yes	characteristic
135	C₂²	C₂⁴×#41	1	4	yes	—
136	C₄	—	2	64	no	—
137	C₄	—	4	64	no	—
138	C₄	C₂⁴×C₄⋊C₄	1	64	yes	—
139	C₂	C₂³×#382	1	120	yes	direct product factor
140	C₂	C₂⁵×C₄⋊C₄	1	1	yes	characteristic
141	C₂	C₂⁴×#117	1	2	yes	—
142	C₂	C₂⁴×#124	1	2	yes	—
143	C₂	C₂⁴×#125	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁷	C₂³
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. C₂³×#382	C₂	direct (N × H)
10. C₂⁴×C₄²	C₄	semidirect
12. C₂²×#382	C₂²	direct (N × H)
27. C₂³×C₄²	C₂×C₄	semidirect
29. C₂×#382	C₂³	direct (N × H)
51. C₂²×C₄²	C₂²×C₄	semidirect
53. #382	C₂⁴	direct (N × H)
79. C₂×C₄²	C₂³×C₄	semidirect
82. C₂⁴	#382	direct (N × H)
106. C₄²	C₂⁴×C₄	semidirect
107. C₂³	C₂×#382	direct (N × H)
127. C₂²	C₂²×#382	direct (N × H)
139. C₂	C₂³×#382	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×C₄⋊C₄	2
C₂⁵×C₄²	2
C₂³×#382	2
C₂⁴×#95	2

Subgroups — C24×#382 #72325641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#382 #72325641