Subgroups — C₂²×C₉×#125 #11768158

Order 1152.

1788 conjugacy classes of subgroups.

The subgroup lattice has 126 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#125	C₂	1	24	yes	maximal, direct product factor
2	C₂²×C₉×C₄⋊C₄	C₂	1	4	yes	maximal
3	C₂²×C₉×C₄⋊C₄	C₂	1	2	yes	maximal
4	C₂²×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₃×#125	C₃	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₉	C₂²	1	2	yes	—
7	C₂³×C₄×C₉	C₂²	1	4	yes	—
8	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₉×#125	C₂²	1	64	yes	direct product factor
10	C₂×C₉×C₄⋊C₄	C₂²	1	48	yes	—
11	C₂×C₉×C₄⋊C₄	C₂²	1	24	yes	—
12	C₂×C₄²×C₉	C₂²	1	12	yes	—
13	C₂×C₃×#125	C₂×C₃	1	24	yes	—
14	C₂²×C₃×C₄⋊C₄	C₂×C₃	1	4	yes	—
15	C₂²×C₃×C₄⋊C₄	C₂×C₃	1	2	yes	—
16	C₂²×C₃×C₄²	C₂×C₃	1	1	yes	characteristic
17	C₂⁴×C₉	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₉	—	2	8	no	—
19	C₂²×C₄×C₉	—	2	16	no	—
20	C₂²×C₄×C₉	Q₈	1	8	yes	—
21	C₂²×C₄×C₉	C₂³	1	12	yes	—
22	C₂²×C₄×C₉	C₂³	1	24	yes	—
23	C₂²×C₄×C₉	C₂³	1	6	yes	—
24	C₉×C₄⋊C₄	C₂³	1	64	yes	—
25	C₉×C₄⋊C₄	C₂³	1	32	yes	—
26	C₄²×C₉	C₂³	1	16	yes	—
27	C₂²×#125	C₉	1	1	yes	characteristic, direct product factor, Sylow
28	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
29	C₂³×C₃×C₄	C₂²×C₃	1	4	yes	—
30	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
31	C₃×#125	C₂²×C₃	1	64	yes	—
32	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	48	yes	—
33	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	24	yes	—
34	C₂×C₃×C₄²	C₂²×C₃	1	12	yes	—
35	C₂³×C₉	C₂×Q₈	1	4	yes	—
36	C₂³×C₉	#41	1	8	yes	—
37	C₂³×C₉	C₂⁴	1	3	yes	—
38	C₂×C₄×C₉	—	2	24	no	—
39	C₂×C₄×C₉	—	2	48	no	—
40	C₂×C₄×C₉	C₂×Q₈	1	24	yes	—
41	C₂×C₄×C₉	C₂⁴	1	8	yes	—
42	C₂×C₄×C₉	C₂⁴	1	16	yes	—
43	C₂×C₄×C₉	C₂⁴	1	4	yes	—
44	C₂×#125	C₂×C₉	1	24	yes	direct product factor
45	C₂²×C₄⋊C₄	C₂×C₉	1	4	yes	—
46	C₂²×C₄⋊C₄	C₂×C₉	1	2	yes	—
47	C₂²×C₄²	C₂×C₉	1	1	yes	characteristic
48	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
49	C₂²×C₃×C₄	—	2	8	no	—
50	C₂²×C₃×C₄	—	2	16	no	—
51	C₂²×C₃×C₄	C₃×Q₈	1	8	yes	—
52	C₂²×C₃×C₄	C₂³×C₃	1	12	yes	—
53	C₂²×C₃×C₄	C₂³×C₃	1	24	yes	—
54	C₂²×C₃×C₄	C₂³×C₃	1	6	yes	—
55	C₃×C₄⋊C₄	C₂³×C₃	1	64	yes	—
56	C₃×C₄⋊C₄	C₂³×C₃	1	32	yes	—
57	C₃×C₄²	C₂³×C₃	1	16	yes	—
58	C₂²×C₉	#125	1	16	yes	direct product factor
59	C₂²×C₉	C₂²×Q₈	1	6	yes	—
60	C₂²×C₉	C₂×#41	1	12	yes	—
61	C₂²×C₉	C₂⁵	1	1	yes	characteristic
62	C₄×C₉	—	2	8	no	—
63	C₄×C₉	—	2	16	no	—
64	C₄×C₉	C₂²×Q₈	1	8	yes	—
65	C₂³×C₄	C₂²×C₉	1	4	yes	—
66	C₂³×C₄	C₂²×C₉	1	2	yes	—
67	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
68	#125	C₂²×C₉	1	64	yes	direct product factor
69	C₂×C₄⋊C₄	C₂²×C₉	1	48	yes	—
70	C₂×C₄⋊C₄	C₂²×C₉	1	24	yes	—
71	C₂×C₄²	C₂²×C₉	1	12	yes	—
72	C₂³×C₃	C₂×C₃×Q₈	1	4	yes	—
73	C₂³×C₃	C₃×#41	1	8	yes	—
74	C₂³×C₃	C₂⁴×C₃	1	3	yes	—
75	C₂×C₃×C₄	—	2	24	no	—
76	C₂×C₃×C₄	—	2	48	no	—
77	C₂×C₃×C₄	C₂×C₃×Q₈	1	24	yes	—
78	C₂×C₃×C₄	C₂⁴×C₃	1	8	yes	—
79	C₂×C₃×C₄	C₂⁴×C₃	1	16	yes	—
80	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
81	C₂×C₉	C₂×#125	1	12	yes	direct product factor
82	C₂×C₉	C₂³×Q₈	1	1	yes	characteristic
83	C₂×C₉	C₂²×#41	1	2	yes	—
84	C₂⁴	C₂³×C₉	1	1	yes	characteristic
85	C₂²×C₄	—	2	16	no	—
86	C₂²×C₄	—	2	8	no	—
87	C₂²×C₄	Q₈×C₉	1	8	yes	—
88	C₂²×C₄	C₂³×C₉	1	12	yes	—
89	C₂²×C₄	C₂³×C₉	1	24	yes	—
90	C₂²×C₄	C₂³×C₉	1	6	yes	—
91	C₄⋊C₄	C₂³×C₉	1	64	yes	—
92	C₄⋊C₄	C₂³×C₉	1	32	yes	—
93	C₄²	C₂³×C₉	1	16	yes	—
94	C₂²×C₃	C₃×#125	1	16	yes	—
95	C₂²×C₃	C₂²×C₃×Q₈	1	6	yes	—
96	C₂²×C₃	C₂×C₃×#41	1	12	yes	—
97	C₂²×C₃	C₂⁵×C₃	1	1	yes	characteristic
98	C₃×C₄	—	2	8	no	—
99	C₃×C₄	—	2	16	no	—
100	C₃×C₄	C₂²×C₃×Q₈	1	8	yes	—
101	C₉	C₂²×#125	1	1	yes	characteristic, direct product factor, Sylow
102	C₂³	C₂×Q₈×C₉	1	4	yes	—
103	C₂³	C₉×#41	1	8	yes	—
104	C₂³	C₂⁴×C₉	1	3	yes	—
105	C₂×C₄	—	2	24	no	—
106	C₂×C₄	—	2	48	no	—
107	C₂×C₄	C₂×Q₈×C₉	1	24	yes	—
108	C₂×C₄	C₂⁴×C₉	1	8	yes	—
109	C₂×C₄	C₂⁴×C₉	1	16	yes	—
110	C₂×C₄	C₂⁴×C₉	1	4	yes	—
111	C₂×C₃	C₂×C₃×#125	1	12	yes	—
112	C₂×C₃	C₂³×C₃×Q₈	1	1	yes	characteristic
113	C₂×C₃	C₂²×C₃×#41	1	2	yes	—
114	C₂²	C₉×#125	1	16	yes	direct product factor
115	C₂²	C₂²×Q₈×C₉	1	6	yes	—
116	C₂²	C₂×C₉×#41	1	12	yes	—
117	C₂²	C₂⁵×C₉	1	1	yes	characteristic
118	C₄	—	2	8	no	—
119	C₄	—	2	16	no	—
120	C₄	C₂²×Q₈×C₉	1	8	yes	—
121	C₃	C₂²×C₃×#125	1	1	yes	characteristic
122	C₂	C₂×C₉×#125	1	12	yes	direct product factor
123	C₂	C₂³×Q₈×C₉	1	1	yes	characteristic
124	C₂	C₂²×C₉×#41	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₉	C₂³
commutator	C₂²	C₂⁵×C₉
Frattini, cosocle	C₂²×C₃	C₂⁵×C₃
socle, prime radical	C₂⁴×C₃	C₂³×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₉×#125	C₂	direct (N × H)
9. C₉×#125	C₂²	direct (N × H)
27. C₂²×#125	C₉	direct (N × H)
44. C₂×#125	C₂×C₉	direct (N × H)
58. C₂²×C₉	#125	direct (N × H)
68. #125	C₂²×C₉	direct (N × H)
81. C₂×C₉	C₂×#125	direct (N × H)
101. C₉	C₂²×#125	direct (N × H)
114. C₂²	C₉×#125	direct (N × H)
122. C₂	C₂×C₉×#125	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₉×#125	2
C₂²×C₉×C₄⋊C₄	2
C₂²×C₄²×C₉	2
C₂²×C₃×#125	3

Subgroups — C22×C9×#125 #11768158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₉×#125 #11768158