Subgroups — #72231877

Order 1024.

257 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#7332	C₂	1	1	yes	maximal, characteristic
2	#149495	C₂	1	1	yes	maximal, characteristic
3	#122214	C₂	1	1	yes	maximal, characteristic
4	#93668	C₂	1	4	yes	maximal, complemented
5	C₄×M₆₄	C₄	1	1	yes	characteristic
6	C₄×M₆₄	C₄	1	1	yes	characteristic
7	C₂×#1397	C₄	1	2	yes	—
8	#11410	C₄	1	2	yes	—
9	C₂×#1311	C₂²	1	1	yes	characteristic
10	#7332	C₄	1	2	yes	—
11	#7332	C₂²	1	2	yes	—
12	#7330	C₄	1	2	yes	—
13	#7330	C₂²	1	2	yes	—
14	#7053	C₄	1	2	yes	—
15	#7053	C₂²	1	2	yes	—
16	C₂×M₆₄	C₈	1	2	yes	—
17	C₂×M₆₄	C₈	1	2	yes	—
18	C₂²×C₃₂	—	2	2	no	—
19	#2162	—	2	2	no	—
20	C₂×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈²	C₂×C₄	1	1	yes	characteristic
23	#1397	—	2	2	no	—
24	#1397	C₂×C₄	1	4	yes	—
25	#1397	C₂×C₄	1	2	yes	—
26	#1397	C₂×C₄	1	2	yes	—
27	C₄×C₃₂	C₈	1	2	yes	—
28	C₄×C₃₂	C₈	1	2	yes	—
29	#1312	C₂×C₄	1	4	yes	—
30	#1311	C₂×C₄	1	1	yes	characteristic
31	#1311	C₂×C₄	1	1	yes	characteristic
32	#1311	C₂×C₄	1	1	yes	characteristic
33	#1311	C₂³	1	1	yes	characteristic
34	C₈×C₁₆	—	2	2	no	—
35	C₂²×C₁₆	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₁₆	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₁₆	M₁₆	1	2	yes	—
38	C₂×C₄×C₈	C₄²	1	1	yes	characteristic
39	C₂×C₄×C₈	M₁₆	1	2	yes	—
40	M₆₄	—	2	4	no	—
41	M₆₄	C₁₆	1	8	yes	complemented
42	C₂×C₃₂	—	2	4	no	—
43	C₂×C₃₂	—	4	4	no	—
44	C₂×C₃₂	C₂×C₈	1	4	yes	—
45	C₂×C₃₂	C₂×C₈	1	4	yes	—
46	C₄⋊C₁₆	—	4	2	no	—
47	C₂²⋊C₁₆	—	4	2	no	—
48	C₄×C₁₆	—	2	4	no	—
49	C₄×C₁₆	C₄²	1	1	yes	characteristic
50	C₄×C₁₆	C₄²	1	1	yes	characteristic
51	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
52	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
53	C₄×C₁₆	M₁₆	1	2	yes	—
54	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
55	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
56	C₈²	C₄²	1	1	yes	characteristic
57	C₈²	M₁₆	1	2	yes	—
58	C₈²	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₈	—	2	2	no	—
60	C₂²×C₈	C₄×C₈	1	1	yes	characteristic
61	C₂²×C₈	#97	1	1	yes	characteristic
62	C₂×C₄²	#97	1	1	yes	characteristic
63	C₂×C₁₆	—	2	4	no	—
64	C₂×C₁₆	—	4	4	no	—
65	C₂×C₁₆	C₄×C₈	1	1	yes	characteristic
66	C₂×C₁₆	C₄×C₈	1	1	yes	characteristic
67	C₂×C₁₆	#97	1	2	yes	—
68	C₂×C₁₆	C₂×C₁₆	1	4	yes	—
69	C₂×C₁₆	M₃₂	1	4	yes	—
70	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₁₆	C₂×M₁₆	1	2	yes	—
73	C₄×C₈	—	2	2	no	—
74	C₄×C₈	C₄×C₈	1	1	yes	characteristic
75	C₄×C₈	#97	1	1	yes	characteristic
76	C₄×C₈	#97	1	2	yes	—
77	C₄×C₈	#97	1	1	yes	characteristic
78	C₄×C₈	C₂×C₄²	1	1	yes	characteristic
79	C₄×C₈	C₂×M₁₆	1	2	yes	—
80	C₃₂	—	2	4	no	—
81	C₃₂	—	4	4	no	—
82	C₃₂	C₂×C₁₆	1	8	yes	—
83	C₂²×C₄	#322	1	1	yes	characteristic
84	C₂²×C₄	#347	1	2	yes	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	8	2	no	—
88	C₂×C₈	#322	1	1	yes	characteristic
89	C₂×C₈	#322	1	1	yes	characteristic
90	C₂×C₈	C₄×C₁₆	1	2	yes	—
91	C₂×C₈	#346	1	2	yes	—
92	C₂×C₈	#347	1	4	yes	—
93	C₂×C₈	C₂×C₄×C₈	1	1	yes	characteristic
94	C₂×C₈	C₂×#97	1	1	yes	characteristic
95	C₄²	#322	1	1	yes	characteristic
96	C₄²	#347	1	2	yes	—
97	C₄²	C₂×#97	1	1	yes	characteristic
98	C₁₆	—	2	2	no	—
99	C₁₆	—	2	2	no	—
100	C₁₆	—	8	4	no	—
101	C₁₆	C₄×C₁₆	1	2	yes	—
102	C₁₆	#346	1	2	yes	—
103	C₁₆	C₂²×C₁₆	1	2	yes	—
104	C₁₆	C₂×M₃₂	1	2	yes	—
105	C₂³	#1313	1	1	yes	characteristic
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	#1312	1	2	yes	—
108	C₂×C₄	#1313	1	2	yes	—
109	C₂×C₄	#1313	1	2	yes	—
110	C₂×C₄	#1313	1	1	yes	characteristic
111	C₂×C₄	C₂×#322	1	1	yes	characteristic
112	C₂×C₄	C₂×#347	1	2	yes	—
113	C₈	—	4	2	no	—
114	C₈	—	4	2	no	—
115	C₈	#1312	1	2	yes	—
116	C₈	#1313	1	2	yes	—
117	C₈	C₂×C₄×C₁₆	1	1	yes	characteristic
118	C₈	C₂×#346	1	1	yes	characteristic
119	C₈	C₂×#347	1	2	yes	—
120	C₂²	—	2	2	no	—
121	C₂²	#7053	1	2	yes	—
122	C₂²	C₂×#1313	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	#7053	1	2	yes	—
126	C₄	C₂×#1312	1	1	yes	characteristic
127	C₄	C₂×#1313	1	1	yes	characteristic
128	C₂	—	2	2	no	—
129	C₂	C₂×#7053	1	1	yes	characteristic
130	C₂	#149504	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	#322
commutator	C₈	C₂×C₄×C₁₆
Frattini, cosocle	#1311	C₂³
socle	C₂²	C₂×#1313
prime radical	C₂³	#1313
LCS	G ▷ C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#1311 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #93668	C₂	semidirect
41. M₆₄	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#7332	2
#149495	2
#122214	2
#93668	2