Subgroups — C₄×C₂₃×#125 #15669924

Order 2944.

484 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#125	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#389	C₂	1	2	yes	maximal
3	C₂₃×#387	C₂	1	4	yes	maximal
4	C₂₃×#382	C₂	1	1	yes	maximal, characteristic
5	C₄×C₄⋊C₄×C₂₃	C₂	1	4	yes	maximal
6	C₄×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal
7	C₄³×C₂₃	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#125	C₄	1	8	yes	direct product factor
9	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
11	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
12	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₂₃	C₂²	1	4	yes	—
17	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#95	C₂²	1	4	yes	—
20	C₂₃×#95	C₂²	1	4	yes	—
21	C₂₃×#95	C₂²	1	4	yes	—
22	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
23	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
24	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
25	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
27	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₂₃	—	2	8	no	—
30	C₄⋊C₄×C₂₃	—	2	4	no	—
31	C₄⋊C₄×C₂₃	C₂×C₄	1	16	yes	—
32	C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
33	C₄²×C₂₃	—	2	4	no	—
34	C₄²×C₂₃	—	2	2	no	—
35	C₄²×C₂₃	—	2	4	no	—
36	C₄²×C₂₃	—	2	8	no	—
37	C₄²×C₂₃	C₂×C₄	1	4	yes	—
38	C₄²×C₂₃	Q₈	1	4	yes	—
39	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
40	C₂×C₄×C₂₃	—	2	2	no	—
41	C₂×C₄×C₂₃	—	2	4	no	—
42	C₂×C₄×C₂₃	—	2	2	no	—
43	C₂×C₄×C₂₃	—	2	2	no	—
44	C₂×C₄×C₂₃	—	2	2	no	—
45	C₂×C₄×C₂₃	—	2	2	no	—
46	C₂×C₄×C₂₃	—	2	4	no	—
47	C₂×C₄×C₂₃	—	2	4	no	—
48	C₂×C₄×C₂₃	—	2	4	no	—
49	C₂×C₄×C₂₃	—	2	4	no	—
50	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
51	C₂×C₄×C₂₃	C₂²×C₄	1	8	yes	—
52	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
56	C₂×C₄×C₂₃	#41	1	4	yes	—
57	C₂×C₄×C₂₃	#41	1	2	yes	—
58	C₂×C₄×C₂₃	#41	1	4	yes	—
59	C₄×#125	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
61	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
62	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
63	C₂²×C₂₃	C₂×#41	1	2	yes	—
64	C₂²×C₂₃	C₂×#41	1	2	yes	—
65	C₄×C₂₃	—	2	4	no	—
66	C₄×C₂₃	—	2	8	no	—
67	C₄×C₂₃	—	2	4	no	—
68	C₄×C₂₃	—	2	2	no	—
69	C₄×C₂₃	—	4	4	no	—
70	C₄×C₂₃	C₄×Q₈	1	4	yes	—
71	C₄×C₂₃	#125	1	4	yes	direct product factor
72	C₂×#125	C₂×C₂₃	1	1	yes	characteristic
73	#389	C₂×C₂₃	1	2	yes	—
74	#387	C₂×C₂₃	1	4	yes	—
75	#382	C₂×C₂₃	1	1	yes	characteristic
76	C₄×C₄⋊C₄	C₂×C₂₃	1	4	yes	—
77	C₄×C₄⋊C₄	C₂×C₂₃	1	2	yes	—
78	C₄³	C₂×C₂₃	1	1	yes	characteristic
79	C₂×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
80	C₂×C₂₃	C₄×#41	1	2	yes	—
81	C₂×C₂₃	C₂×#125	1	1	yes	characteristic
82	C₂×C₂₃	#529	1	2	yes	—
83	C₂×C₂₃	#533	1	1	yes	characteristic
84	#125	C₄×C₂₃	1	8	yes	direct product factor
85	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
86	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
87	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
88	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
89	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
90	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
91	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
92	C₂×C₄²	C₂²×C₂₃	1	4	yes	—
93	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
94	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
95	#95	C₂²×C₂₃	1	4	yes	—
96	#95	C₂²×C₂₃	1	4	yes	—
97	#95	C₂²×C₂₃	1	4	yes	—
98	C₂₃	C₄×#125	1	1	yes	characteristic, direct product factor, Sylow
99	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
100	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
101	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
102	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
103	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
104	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
105	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
106	C₄⋊C₄	—	2	8	no	—
107	C₄⋊C₄	—	2	4	no	—
108	C₄⋊C₄	C₂×C₄×C₂₃	1	16	yes	—
109	C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
110	C₄²	—	2	4	no	—
111	C₄²	—	2	2	no	—
112	C₄²	—	2	4	no	—
113	C₄²	—	2	8	no	—
114	C₄²	C₂×C₄×C₂₃	1	4	yes	—
115	C₄²	Q₈×C₂₃	1	4	yes	—
116	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
128	C₂×C₄	C₂²×C₄×C₂₃	1	8	yes	—
129	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
130	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
131	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
132	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
133	C₂×C₄	#41×C₂₃	1	4	yes	—
134	C₂×C₄	#41×C₂₃	1	2	yes	—
135	C₂×C₄	#41×C₂₃	1	4	yes	—
136	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
137	C₂²	C₂²×Q₈×C₂₃	1	1	yes	characteristic
138	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
139	C₂²	C₂×#41×C₂₃	1	2	yes	—
140	C₂²	C₂×#41×C₂₃	1	2	yes	—
141	C₄	—	2	4	no	—
142	C₄	—	2	8	no	—
143	C₄	—	2	4	no	—
144	C₄	—	2	2	no	—
145	C₄	—	4	4	no	—
146	C₄	C₄×Q₈×C₂₃	1	4	yes	—
147	C₄	C₂₃×#125	1	4	yes	direct product factor
148	C₂	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
149	C₂	C₄×#41×C₂₃	1	2	yes	—
150	C₂	C₂×C₂₃×#125	1	1	yes	characteristic
151	C₂	C₂₃×#529	1	2	yes	—
152	C₂	C₂₃×#533	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂³
commutator	C₂²	C₂³×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁴×C₂₃
socle, prime radical	C₂³×C₂₃	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₂₃×#125	C₄	direct (N × H)
59. C₄×#125	C₂₃	direct (N × H)
71. C₄×C₂₃	#125	direct (N × H)
84. #125	C₄×C₂₃	direct (N × H)
98. C₂₃	C₄×#125	direct (N × H)
147. C₄	C₂₃×#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×C₂₃×#125	2
C₂₃×#389	2
C₂₃×#387	2
C₂₃×#382	2
C₄×C₄⋊C₄×C₂₃	2
C₄³×C₂₃	2
C₄×#125	23

Subgroups — C4×C23×#125 #15669924

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂₃×#125 #15669924