Subgroups — C₂₃×#13553 #19161179

Order 5888.

330 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×M₆₄	C₂	1	2	yes	maximal, complemented
3	C₂₃×C₂²⋊C₃₂	C₂	1	4	yes	maximal, complemented
4	C₂³×C₈×C₂₃	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₂₃	—	2	4	no	—
6	C₂²×C₁₆×C₂₃	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₂₃	C₄	1	2	yes	—
8	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
9	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×M₆₄	—	2	4	no	—
11	C₂×C₂₃×C₃₂	C₂²	1	4	yes	complemented
12	C₂³×C₄×C₂₃	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₂₃	—	2	2	no	—
14	C₂²×C₈×C₂₃	—	2	2	no	—
15	C₂²×C₈×C₂₃	C₈	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	C₈	1	2	yes	—
17	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₂₃	—	2	2	no	—
20	C₂×C₁₆×C₂₃	—	2	4	no	—
21	C₂×C₁₆×C₂₃	—	2	4	no	—
22	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₂₃	D₄	1	4	yes	—
25	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
26	C₂₃×C₃₂	—	2	4	no	—
27	C₂⁴×C₂₃	C₁₆	1	1	yes	characteristic
28	C₂²×C₄×C₂₃	—	2	2	no	—
29	C₂²×C₄×C₂₃	—	2	2	no	—
30	C₂²×C₄×C₂₃	C₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	C₁₆	1	2	yes	—
32	C₂²×C₄×C₂₃	C₂×C₈	1	2	yes	—
33	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₂₃	—	2	2	no	—
35	C₂×C₈×C₂₃	—	2	4	no	—
36	C₂×C₈×C₂₃	—	2	2	no	—
37	C₂×C₈×C₂₃	—	2	2	no	—
38	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
39	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
42	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
43	C₁₆×C₂₃	—	2	2	no	—
44	C₁₆×C₂₃	C₂²⋊C₄	1	2	yes	—
45	C₁₆×C₂₃	C₂×D₄	1	2	yes	—
46	#13553	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂³×C₂₃	—	2	2	no	—
48	C₂³×C₂₃	—	2	4	no	—
49	C₂³×C₂₃	C₂×C₁₆	1	1	yes	characteristic
50	C₂³×C₂₃	C₂×C₁₆	1	2	yes	—
51	C₂×C₄×C₂₃	—	2	2	no	—
52	C₂×C₄×C₂₃	—	2	4	no	—
53	C₂×C₄×C₂₃	—	2	2	no	—
54	C₂×C₄×C₂₃	—	2	2	no	—
55	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
56	C₂×C₄×C₂₃	C₂×C₁₆	1	2	yes	—
57	C₂×C₄×C₂₃	C₂×C₁₆	1	1	yes	characteristic
58	C₂×C₄×C₂₃	M₃₂	1	2	yes	—
59	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
60	C₈×C₂₃	—	2	2	no	—
61	C₈×C₂₃	C₂²⋊C₈	1	2	yes	—
62	C₈×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₈×C₂₃	C₂×M₁₆	1	1	yes	characteristic
64	C₂³×C₁₆	C₂×C₂₃	1	1	yes	characteristic
65	C₂×M₆₄	C₂×C₂₃	1	2	yes	complemented
66	C₂²⋊C₃₂	C₂×C₂₃	1	4	yes	complemented
67	C₂²×C₂₃	—	2	8	no	—
68	C₂²×C₂₃	—	2	2	no	—
69	C₂²×C₂₃	—	2	2	no	—
70	C₂²×C₂₃	—	2	2	no	—
71	C₂²×C₂₃	C₂²⋊C₁₆	1	2	yes	—
72	C₂²×C₂₃	M₆₄	1	4	yes	complemented
73	C₂²×C₂₃	C₂²×C₁₆	1	1	yes	characteristic
74	C₄×C₂₃	—	2	2	no	—
75	C₄×C₂₃	C₂²⋊C₁₆	1	2	yes	—
76	C₄×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
77	C₄×C₂₃	C₂×M₃₂	1	1	yes	characteristic
78	C₂³×C₈	C₄×C₂₃	1	1	yes	characteristic
79	C₂²×C₁₆	—	2	4	no	—
80	C₂²×C₁₆	C₄×C₂₃	1	1	yes	characteristic
81	C₂²×C₁₆	C₄×C₂₃	1	2	yes	—
82	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
83	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
84	M₆₄	—	2	4	no	—
85	C₂×C₃₂	C₂²×C₂₃	1	4	yes	complemented
86	C₂×C₂₃	—	2	2	no	—
87	C₂×C₂₃	—	2	4	no	—
88	C₂×C₂₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
89	C₂×C₂₃	C₂×M₆₄	1	2	yes	—
90	C₂³×C₄	C₈×C₂₃	1	1	yes	characteristic
91	C₂²×C₈	—	2	2	no	—
92	C₂²×C₈	—	2	2	no	—
93	C₂²×C₈	C₈×C₂₃	1	1	yes	characteristic
94	C₂²×C₈	C₈×C₂₃	1	2	yes	—
95	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
96	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
97	C₂×C₁₆	—	2	2	no	—
98	C₂×C₁₆	—	2	4	no	—
99	C₂×C₁₆	—	2	4	no	—
100	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
101	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
102	C₂×C₁₆	D₄×C₂₃	1	4	yes	—
103	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
104	C₃₂	—	2	4	no	—
105	C₂₃	#13553	1	1	yes	characteristic, direct product factor, Sylow
106	C₂⁴	C₁₆×C₂₃	1	1	yes	characteristic
107	C₂²×C₄	—	2	2	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	C₁₆×C₂₃	1	1	yes	characteristic
110	C₂²×C₄	C₁₆×C₂₃	1	2	yes	—
111	C₂²×C₄	C₂×C₈×C₂₃	1	2	yes	—
112	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
118	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
119	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
120	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
121	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
122	C₁₆	—	2	2	no	—
123	C₁₆	C₂²⋊C₄×C₂₃	1	2	yes	—
124	C₁₆	C₂×D₄×C₂₃	1	2	yes	—
125	C₂³	—	2	2	no	—
126	C₂³	—	2	4	no	—
127	C₂³	C₂×C₁₆×C₂₃	1	1	yes	characteristic
128	C₂³	C₂×C₁₆×C₂₃	1	2	yes	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
134	C₂×C₄	C₂×C₁₆×C₂₃	1	2	yes	—
135	C₂×C₄	C₂×C₁₆×C₂₃	1	1	yes	characteristic
136	C₂×C₄	C₂₃×M₃₂	1	2	yes	—
137	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
138	C₈	—	2	2	no	—
139	C₈	C₂₃×C₂²⋊C₈	1	2	yes	—
140	C₈	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
141	C₈	C₂×M₁₆×C₂₃	1	1	yes	characteristic
142	C₂²	—	2	8	no	—
143	C₂²	—	2	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	2	2	no	—
146	C₂²	C₂₃×C₂²⋊C₁₆	1	2	yes	—
147	C₂²	C₂₃×M₆₄	1	4	yes	complemented
148	C₂²	C₂²×C₁₆×C₂₃	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	C₂₃×C₂²⋊C₁₆	1	2	yes	—
151	C₄	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
152	C₄	C₂×C₂₃×M₃₂	1	1	yes	characteristic
153	C₂	—	2	2	no	—
154	C₂	—	2	4	no	—
155	C₂	C₂×C₂₃×C₂²⋊C₁₆	1	1	yes	characteristic
156	C₂	C₂×C₂₃×M₆₄	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆×C₂₃	C₂³
commutator	C₂²	C₂²×C₁₆×C₂₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₂₃
socle	C₂²×C₂₃	C₂²×C₁₆
prime radical	C₂⁴×C₂₃	C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×M₆₄	C₂	semidirect
3. C₂₃×C₂²⋊C₃₂	C₂	semidirect
11. C₂×C₂₃×C₃₂	C₂²	semidirect
46. #13553	C₂₃	direct (N × H)
65. C₂×M₆₄	C₂×C₂₃	semidirect
66. C₂²⋊C₃₂	C₂×C₂₃	semidirect
72. C₂²×C₂₃	M₆₄	semidirect
85. C₂×C₃₂	C₂²×C₂₃	semidirect
105. C₂₃	#13553	direct (N × H)
147. C₂²	C₂₃×M₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₂₃	2
C₂×C₂₃×M₆₄	2
C₂₃×C₂²⋊C₃₂	2
#13553	23

Subgroups — C23×#13553 #19161179

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13553 #19161179