Subgroups — C₂₃×#20355 #19167981

Order 5888.

2018 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#519	C₂	1	2	yes	maximal
2	C₂×C₂₃×#519	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1740	C₂	1	12	yes	maximal
4	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
5	C₂₃×#519	—	2	8	no	—
6	C₂₃×#519	—	2	8	no	—
7	C₂₃×#519	C₄	1	16	yes	complemented
8	C₂×C₄×Q₈×C₂₃	—	2	8	no	—
9	C₂×C₄×Q₈×C₂₃	C₄	1	8	yes	complemented
10	C₂×C₂₃×#117	C₂²	1	12	yes	—
11	C₂×C₂₃×#117	C₂²	1	6	yes	—
12	C₂₃×#342	C₂²	1	16	yes	—
13	C₂²×Q₈×C₂₃	C₂×C₄	1	4	yes	—
14	C₂²×Q₈×C₂₃	C₂×C₄	1	8	yes	—
15	C₂²×Q₈×C₂₃	D₄	1	8	yes	—
16	C₂²×Q₈×C₂₃	D₄	1	4	yes	—
17	C₂²×Q₈×C₂₃	Q₈	1	4	yes	—
18	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
19	C₄×Q₈×C₂₃	—	2	64	no	—
20	C₄×Q₈×C₂₃	—	2	32	no	—
21	C₂₃×#117	—	2	24	no	—
22	C₂₃×#117	—	2	24	no	—
23	C₂₃×#117	C₂×C₄	1	48	yes	—
24	C₂×C₄⋊C₄×C₂₃	—	2	12	no	—
25	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	12	yes	—
26	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
27	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
28	C₂×C₄²×C₂₃	—	2	12	no	—
29	C₂×C₄²×C₂₃	C₂×C₄	1	12	yes	—
30	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
31	C₂×Q₈×C₂₃	C₄²	1	16	yes	—
32	C₂×Q₈×C₂₃	C₂²⋊C₄	1	32	yes	—
33	C₂×Q₈×C₂₃	C₂²⋊C₄	1	16	yes	—
34	C₂×Q₈×C₂₃	C₄⋊C₄	1	16	yes	—
35	C₂×Q₈×C₂₃	C₄⋊C₄	1	32	yes	—
36	C₂²×C₄×C₂₃	—	2	8	no	—
37	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
38	C₂²×C₄×C₂₃	C₂²×C₄	1	6	yes	—
39	C₂²×C₄×C₂₃	C₂²×C₄	1	12	yes	—
40	C₂²×C₄×C₂₃	C₂×D₄	1	12	yes	—
41	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
42	C₂²×C₄×C₂₃	C₂×Q₈	1	6	yes	—
43	C₄⋊C₄×C₂₃	—	2	48	no	—
44	C₄⋊C₄×C₂₃	—	2	24	no	—
45	C₂²⋊C₄×C₂₃	—	2	8	no	—
46	C₂²⋊C₄×C₂₃	—	2	8	no	—
47	C₂²⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
48	C₄²×C₂₃	—	2	48	no	—
49	C₄²×C₂₃	—	2	24	no	—
50	#20355	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂³×C₂₃	—	4	2	no	—
52	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
53	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
54	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
55	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
56	C₂³×C₂₃	C₂²×Q₈	1	1	yes	characteristic
57	Q₈×C₂₃	#95	1	64	yes	—
58	C₂×C₄×C₂₃	—	2	16	no	—
59	C₂×C₄×C₂₃	—	2	8	no	—
60	C₂×C₄×C₂₃	—	4	16	no	—
61	C₂×C₄×C₂₃	—	4	8	no	—
62	C₂×C₄×C₂₃	C₂×C₄²	1	12	yes	—
63	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	24	yes	—
64	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	12	yes	—
65	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	12	yes	—
66	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	24	yes	—
67	C₂×#519	C₂×C₂₃	1	2	yes	—
68	C₂×#519	C₂×C₂₃	1	1	yes	characteristic
69	#1740	C₂×C₂₃	1	12	yes	—
70	C₂²×C₂₃	—	2	2	no	—
71	C₂²×C₂₃	—	2	4	no	—
72	C₂²×C₂₃	—	4	2	no	—
73	C₂²×C₂₃	—	4	1	no	—
74	C₂²×C₂₃	—	4	1	no	—
75	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
76	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	2	yes	—
77	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₂₃	C₂²×C₄⋊C₄	1	2	yes	—
80	C₄×C₂₃	—	4	16	no	—
81	C₄×C₂₃	—	4	8	no	—
82	C₄×C₂₃	C₂×#95	1	24	yes	—
83	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
84	#519	—	2	8	no	—
85	#519	—	2	8	no	—
86	#519	C₄×C₂₃	1	16	yes	complemented
87	C₂×C₄×Q₈	—	2	8	no	—
88	C₂×C₄×Q₈	C₄×C₂₃	1	8	yes	complemented
89	C₂×#117	C₂²×C₂₃	1	12	yes	—
90	C₂×#117	C₂²×C₂₃	1	6	yes	—
91	#342	C₂²×C₂₃	1	16	yes	—
92	C₂×C₂₃	—	2	2	no	—
93	C₂×C₂₃	—	2	1	no	—
94	C₂×C₂₃	—	2	2	no	—
95	C₂×C₂₃	—	2	1	no	—
96	C₂×C₂₃	C₂²×#95	1	1	yes	characteristic
97	C₂×C₂₃	#2884	1	2	yes	—
98	C₂²×Q₈	C₂×C₄×C₂₃	1	4	yes	—
99	C₂²×Q₈	C₂×C₄×C₂₃	1	8	yes	—
100	C₂²×Q₈	D₄×C₂₃	1	8	yes	—
101	C₂²×Q₈	D₄×C₂₃	1	4	yes	—
102	C₂²×Q₈	Q₈×C₂₃	1	4	yes	—
103	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
104	C₄×Q₈	—	2	64	no	—
105	C₄×Q₈	—	2	32	no	—
106	#117	—	2	24	no	—
107	#117	—	2	24	no	—
108	#117	C₂×C₄×C₂₃	1	48	yes	—
109	C₂×C₄⋊C₄	—	2	12	no	—
110	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	12	yes	—
111	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
112	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
113	C₂×C₄²	—	2	12	no	—
114	C₂×C₄²	C₂×C₄×C₂₃	1	12	yes	—
115	C₂₃	#20355	1	1	yes	characteristic, direct product factor, Sylow
116	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
117	C₂×Q₈	C₄²×C₂₃	1	16	yes	—
118	C₂×Q₈	C₂²⋊C₄×C₂₃	1	32	yes	—
119	C₂×Q₈	C₂²⋊C₄×C₂₃	1	16	yes	—
120	C₂×Q₈	C₄⋊C₄×C₂₃	1	16	yes	—
121	C₂×Q₈	C₄⋊C₄×C₂₃	1	32	yes	—
122	C₂²×C₄	—	2	8	no	—
123	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
124	C₂²×C₄	C₂²×C₄×C₂₃	1	6	yes	—
125	C₂²×C₄	C₂²×C₄×C₂₃	1	12	yes	—
126	C₂²×C₄	C₂×D₄×C₂₃	1	12	yes	—
127	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
128	C₂²×C₄	C₂×Q₈×C₂₃	1	6	yes	—
129	C₄⋊C₄	—	2	48	no	—
130	C₄⋊C₄	—	2	24	no	—
131	C₂²⋊C₄	—	2	8	no	—
132	C₂²⋊C₄	—	2	8	no	—
133	C₂²⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
134	C₄²	—	2	48	no	—
135	C₄²	—	2	24	no	—
136	C₂³	—	4	2	no	—
137	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
138	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
139	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
140	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
141	C₂³	C₂²×Q₈×C₂₃	1	1	yes	characteristic
142	Q₈	C₂₃×#95	1	64	yes	—
143	C₂×C₄	—	2	8	no	—
144	C₂×C₄	—	2	16	no	—
145	C₂×C₄	—	4	16	no	—
146	C₂×C₄	—	4	8	no	—
147	C₂×C₄	C₂×C₄²×C₂₃	1	12	yes	—
148	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	24	yes	—
149	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	12	yes	—
150	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	12	yes	—
151	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	24	yes	—
152	C₂²	—	2	2	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	4	2	no	—
155	C₂²	—	4	1	no	—
156	C₂²	—	4	1	no	—
157	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
158	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	2	yes	—
159	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
160	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
161	C₂²	C₂²×C₄⋊C₄×C₂₃	1	2	yes	—
162	C₄	—	4	16	no	—
163	C₄	—	4	8	no	—
164	C₄	C₂×C₂₃×#95	1	24	yes	—
165	C₂	—	2	2	no	—
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	—	2	2	no	—
169	C₂	C₂²×C₂₃×#95	1	1	yes	characteristic
170	C₂	C₂₃×#2884	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×C₂²⋊C₄
commutator	C₂²	C₂²×C₄²×C₂₃
Frattini, cosocle	C₂⁴	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂⁴
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂³×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₂₃×#519	C₄	semidirect
9. C₂×C₄×Q₈×C₂₃	C₄	semidirect
50. #20355	C₂₃	direct (N × H)
86. #519	C₄×C₂₃	semidirect
88. C₂×C₄×Q₈	C₄×C₂₃	semidirect
115. C₂₃	#20355	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×#519	2
C₂₃×#1740	2
#20355	23

Subgroups — C23×#20355 #19167981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20355 #19167981