Subgroups — C₂₃×#60191 #19207817

Order 5888.

1810 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3458	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2961	C₂	1	12	yes	maximal, complemented
3	C₂₃×#2960	C₂	1	12	yes	maximal, complemented
4	C₂₃×#2905	C₂	1	3	yes	maximal
5	C₂×C₂₃×#405	C₂	1	3	yes	maximal
6	C₂²×M₁₆×C₂₃	C₂²	1	3	yes	—
7	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#544	C₄	1	12	yes	—
9	C₂₃×#543	C₄	1	4	yes	—
10	C₂₃×#533	C₂²	1	12	yes	—
11	C₂×C₂₃×#117	C₂²	1	3	yes	—
12	C₂₃×#446	C₂²	1	48	yes	complemented
13	C₈×Q₈×C₂₃	C₂²	1	16	yes	complemented
14	C₂₃×#424	C₂²	1	24	yes	—
15	C₂₃×#423	C₂²	1	12	yes	—
16	C₂₃×#405	C₂²	1	12	yes	—
17	C₂₃×#405	C₂²	1	12	yes	—
18	C₂×C₂₃×#97	C₂²	1	6	yes	—
19	C₂×C₄×C₈×C₂₃	C₂²	1	6	yes	—
20	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₂₃	—	2	12	no	—
22	C₂×M₁₆×C₂₃	C₂³	1	12	yes	—
23	C₂²×C₈×C₂₃	Q₈	1	8	yes	complemented
24	C₂²×C₈×C₂₃	C₂³	1	6	yes	—
25	C₂²×C₈×C₂₃	C₂³	1	3	yes	—
26	C₂²×C₈×C₂₃	C₂³	1	3	yes	—
27	C₂₃×C₄⋊Q₈	C₂×C₄	1	24	yes	—
28	C₂₃×#125	C₂×C₄	1	24	yes	—
29	C₂₃×C₂²⋊Q₈	C₂×C₄	1	48	yes	—
30	C₄×Q₈×C₂₃	C₂³	1	16	yes	—
31	C₂₃×#117	C₂×C₄	1	12	yes	—
32	C₂₃×#117	C₂³	1	12	yes	—
33	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	6	yes	—
34	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	6	yes	—
35	C₂×C₄²×C₂₃	C₂³	1	6	yes	—
36	C₂₃×C₄⋊C₈	C₂³	1	48	yes	—
37	C₂₃×#97	C₂³	1	24	yes	—
38	C₄×C₈×C₂₃	C₂³	1	12	yes	—
39	C₄×C₈×C₂₃	C₂³	1	12	yes	—
40	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×Q₈×C₂₃	C₂²×C₄	1	16	yes	—
42	C₂²×C₄×C₂₃	C₂²×C₄	1	12	yes	—
43	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
44	C₂²×C₄×C₂₃	C₂×Q₈	1	4	yes	—
45	C₂²×C₄×C₂₃	#41	1	4	yes	—
46	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
47	M₁₆×C₂₃	—	2	24	no	—
48	C₂×C₈×C₂₃	—	2	12	no	—
49	C₂×C₈×C₂₃	C₂×Q₈	1	12	yes	—
50	C₂×C₈×C₂₃	C₂×Q₈	1	12	yes	—
51	C₂×C₈×C₂₃	C₂⁴	1	12	yes	—
52	C₂×C₈×C₂₃	C₂⁴	1	3	yes	—
53	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
54	C₄⋊C₄×C₂₃	C₂²×C₄	1	24	yes	—
55	C₄⋊C₄×C₂₃	C₂²×C₄	1	48	yes	—
56	C₂²⋊C₄×C₂₃	C₂²×C₄	1	24	yes	—
57	C₄²×C₂₃	C₂²×C₄	1	12	yes	—
58	C₄²×C₂₃	C₂⁴	1	12	yes	—
59	#60191	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂³×C₂₃	—	4	2	no	—
61	C₂³×C₂₃	C₄×Q₈	1	4	yes	—
62	C₂³×C₂₃	C₂³×C₄	1	3	yes	—
63	Q₈×C₂₃	—	4	16	no	—
64	C₂×C₄×C₂₃	—	4	12	no	—
65	C₂×C₄×C₂₃	C₄×Q₈	1	12	yes	—
66	C₂×C₄×C₂₃	C₂³×C₄	1	24	yes	—
67	C₂×C₄×C₂₃	C₂³×C₄	1	3	yes	—
68	C₂×C₄×C₂₃	C₂²×Q₈	1	6	yes	—
69	C₂×C₄×C₂₃	C₂×#41	1	6	yes	—
70	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
71	C₈×C₂₃	—	2	12	no	—
72	C₈×C₂₃	C₂²×Q₈	1	6	yes	—
73	C₈×C₂₃	C₂²×Q₈	1	2	yes	—
74	#3458	C₂×C₂₃	1	1	yes	characteristic
75	#2961	C₂×C₂₃	1	12	yes	complemented
76	#2960	C₂×C₂₃	1	12	yes	complemented
77	#2905	C₂×C₂₃	1	3	yes	—
78	C₂×#405	C₂×C₂₃	1	3	yes	—
79	C₂²×C₂₃	—	2	6	no	—
80	C₂²×C₂₃	—	4	4	no	—
81	C₂²×C₂₃	C₂×C₄×Q₈	1	6	yes	—
82	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
83	C₄×C₂₃	—	2	24	no	—
84	C₄×C₂₃	C₂×C₄×Q₈	1	6	yes	—
85	C₄×C₂₃	C₂³×Q₈	1	1	yes	characteristic
86	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
87	C₂²×M₁₆	C₂²×C₂₃	1	3	yes	—
88	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
89	#544	C₄×C₂₃	1	12	yes	—
90	#543	C₄×C₂₃	1	4	yes	—
91	#533	C₂²×C₂₃	1	12	yes	—
92	C₂×#117	C₂²×C₂₃	1	3	yes	—
93	#446	C₂²×C₂₃	1	48	yes	complemented
94	C₈×Q₈	C₂²×C₂₃	1	16	yes	complemented
95	#424	C₂²×C₂₃	1	24	yes	—
96	#423	C₂²×C₂₃	1	12	yes	—
97	#405	C₂²×C₂₃	1	12	yes	—
98	#405	C₂²×C₂₃	1	12	yes	—
99	C₂×#97	C₂²×C₂₃	1	6	yes	—
100	C₂×C₄×C₈	C₂²×C₂₃	1	6	yes	—
101	C₂×C₂₃	—	2	6	no	—
102	C₂×C₂₃	C₂²×C₄×Q₈	1	1	yes	characteristic
103	C₂×C₂₃	#3573	1	2	yes	—
104	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
105	C₂×M₁₆	—	2	12	no	—
106	C₂×M₁₆	C₂³×C₂₃	1	12	yes	—
107	C₂²×C₈	Q₈×C₂₃	1	8	yes	complemented
108	C₂²×C₈	C₂³×C₂₃	1	6	yes	—
109	C₂²×C₈	C₂³×C₂₃	1	3	yes	—
110	C₂²×C₈	C₂³×C₂₃	1	3	yes	—
111	C₄⋊Q₈	C₂×C₄×C₂₃	1	24	yes	—
112	#125	C₂×C₄×C₂₃	1	24	yes	—
113	C₂²⋊Q₈	C₂×C₄×C₂₃	1	48	yes	—
114	C₄×Q₈	C₂³×C₂₃	1	16	yes	—
115	#117	C₂×C₄×C₂₃	1	12	yes	—
116	#117	C₂³×C₂₃	1	12	yes	—
117	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	6	yes	—
118	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	6	yes	—
119	C₂×C₄²	C₂³×C₂₃	1	6	yes	—
120	C₄⋊C₈	C₂³×C₂₃	1	48	yes	—
121	#97	C₂³×C₂₃	1	24	yes	—
122	C₄×C₈	C₂³×C₂₃	1	12	yes	—
123	C₄×C₈	C₂³×C₂₃	1	12	yes	—
124	C₂₃	#60191	1	1	yes	characteristic, direct product factor, Sylow
125	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
126	C₂×Q₈	C₂²×C₄×C₂₃	1	16	yes	—
127	C₂²×C₄	C₂²×C₄×C₂₃	1	12	yes	—
128	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
129	C₂²×C₄	C₂×Q₈×C₂₃	1	4	yes	—
130	C₂²×C₄	#41×C₂₃	1	4	yes	—
131	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
132	M₁₆	—	2	24	no	—
133	C₂×C₈	—	2	12	no	—
134	C₂×C₈	C₂×Q₈×C₂₃	1	12	yes	—
135	C₂×C₈	C₂×Q₈×C₂₃	1	12	yes	—
136	C₂×C₈	C₂⁴×C₂₃	1	12	yes	—
137	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
138	C₂×C₈	C₂⁴×C₂₃	1	3	yes	—
139	C₄⋊C₄	C₂²×C₄×C₂₃	1	24	yes	—
140	C₄⋊C₄	C₂²×C₄×C₂₃	1	48	yes	—
141	C₂²⋊C₄	C₂²×C₄×C₂₃	1	24	yes	—
142	C₄²	C₂²×C₄×C₂₃	1	12	yes	—
143	C₄²	C₂⁴×C₂₃	1	12	yes	—
144	C₂³	—	4	2	no	—
145	C₂³	C₄×Q₈×C₂₃	1	4	yes	—
146	C₂³	C₂³×C₄×C₂₃	1	3	yes	—
147	Q₈	—	4	16	no	—
148	C₂×C₄	—	4	12	no	—
149	C₂×C₄	C₄×Q₈×C₂₃	1	12	yes	—
150	C₂×C₄	C₂³×C₄×C₂₃	1	24	yes	—
151	C₂×C₄	C₂³×C₄×C₂₃	1	3	yes	—
152	C₂×C₄	C₂²×Q₈×C₂₃	1	6	yes	—
153	C₂×C₄	C₂×#41×C₂₃	1	6	yes	—
154	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
155	C₈	—	2	12	no	—
156	C₈	C₂²×Q₈×C₂₃	1	2	yes	—
157	C₈	C₂²×Q₈×C₂₃	1	6	yes	—
158	C₂²	—	2	6	no	—
159	C₂²	—	4	4	no	—
160	C₂²	C₂×C₄×Q₈×C₂₃	1	6	yes	—
161	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
162	C₄	—	2	24	no	—
163	C₄	C₂×C₄×Q₈×C₂₃	1	6	yes	—
164	C₄	C₂³×Q₈×C₂₃	1	1	yes	characteristic
165	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
166	C₂	—	2	6	no	—
167	C₂	C₂²×C₄×Q₈×C₂₃	1	1	yes	characteristic
168	C₂	C₂₃×#3573	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
socle	C₂²×C₂₃	C₂⁴×C₄
prime radical	C₂⁴×C₂₃	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2961	C₂	semidirect
3. C₂₃×#2960	C₂	semidirect
12. C₂₃×#446	C₂²	semidirect
13. C₈×Q₈×C₂₃	C₂²	semidirect
23. C₂²×C₈×C₂₃	Q₈	semidirect
59. #60191	C₂₃	direct (N × H)
75. #2961	C₂×C₂₃	semidirect
76. #2960	C₂×C₂₃	semidirect
93. #446	C₂²×C₂₃	semidirect
94. C₈×Q₈	C₂²×C₂₃	semidirect
107. C₂²×C₈	Q₈×C₂₃	semidirect
124. C₂₃	#60191	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3458	2
C₂₃×#2961	2
C₂₃×#2960	2
C₂₃×#2905	2
C₂×C₂₃×#405	2
#60191	23

Subgroups — C23×#60191 #19207817

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60191 #19207817