Subgroups — C₂₃×#33329 #19180955

Order 5888.

716 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3407	C₂	1	2	yes	maximal
2	C₂₃×#2875	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2110	C₂	1	6	yes	maximal, complemented
4	C₂×C₂₃×#347	C₂	1	6	yes	maximal, complemented
5	C₂×C₂₃×#131	C₄	1	1	yes	characteristic
6	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
7	C₄×#41×C₂₃	C₄	1	1	yes	characteristic
8	C₂₃×#504	C₄	1	16	yes	complemented
9	C₂×C₂₃×M₃₂	C₂²	1	12	yes	—
10	C₂₃×#405	C₄	1	3	yes	—
11	C₂₃×#405	C₂²	1	3	yes	—
12	C₄×M₁₆×C₂₃	C₄	1	3	yes	—
13	C₂×C₂₃×#97	C₂²	1	3	yes	—
14	C₂₃×#347	C₂²	1	4	yes	—
15	C₂₃×#347	C₂²	1	12	yes	—
16	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂₃×#131	—	2	4	no	—
18	C₂₃×#131	C₂×C₄	1	8	yes	—
19	C₂×M₁₆×C₂₃	C₂×C₄	1	3	yes	—
20	C₂×M₁₆×C₂₃	C₂×C₄	1	3	yes	—
21	C₂²×C₈×C₂₃	C₂×C₄	1	3	yes	—
22	C₂²×C₈×C₂₃	C₂³	1	3	yes	—
23	C₄×Q₈×C₂₃	—	2	1	no	—
24	C₄×D₄×C₂₃	—	2	3	no	—
25	C₂₃×#117	C₂×C₄	1	3	yes	—
26	C₂×C₄²×C₂₃	C₂×C₄	1	3	yes	—
27	C₂₃×M₃₂	C₂×C₄	1	24	yes	complemented
28	C₂₃×M₃₂	C₂³	1	8	yes	—
29	C₂×C₁₆×C₂₃	C₂×C₄	1	24	yes	complemented
30	C₂₃×#97	C₂×C₄	1	3	yes	—
31	C₂₃×#97	C₂×C₄	1	3	yes	—
32	C₂₃×#97	C₂³	1	1	yes	characteristic
33	C₂₃×#97	C₂³	1	3	yes	—
34	C₄×C₈×C₂₃	C₂×C₄	1	6	yes	—
35	#41×C₂₃	C₄²	1	4	yes	—
36	#41×C₂₃	M₁₆	1	4	yes	—
37	C₂×Q₈×C₂₃	M₁₆	1	1	yes	characteristic
38	C₂×D₄×C₂₃	M₁₆	1	3	yes	—
39	C₂²×C₄×C₂₃	—	2	3	no	—
40	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
41	M₁₆×C₂₃	—	2	6	no	—
42	M₁₆×C₂₃	C₄²	1	12	yes	—
43	C₂×C₈×C₂₃	—	2	6	no	—
44	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₂₃	C₂²×C₄	1	3	yes	—
46	C₂×C₈×C₂₃	C₂²×C₄	1	3	yes	—
47	C₂×C₈×C₂₃	C₂²×C₄	1	12	yes	—
48	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
49	C₄⋊C₄×C₂₃	—	2	3	no	—
50	C₂²⋊C₄×C₂₃	—	2	3	no	—
51	C₄²×C₂₃	—	2	3	no	—
52	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₂₃	C₂²×C₄	1	3	yes	—
54	C₁₆×C₂₃	C₂²×C₄	1	16	yes	—
55	#33329	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂³×C₂₃	C₂×M₁₆	1	3	yes	—
57	Q₈×C₂₃	#97	1	4	yes	—
58	D₄×C₂₃	#97	1	12	yes	—
59	C₂×C₄×C₂₃	—	2	1	no	—
60	C₂×C₄×C₂₃	—	2	3	no	—
61	C₂×C₄×C₂₃	—	8	3	no	—
62	C₂×C₄×C₂₃	C₂×C₄²	1	6	yes	—
63	C₂×C₄×C₂₃	C₂×M₁₆	1	6	yes	—
64	C₂×C₄×C₂₃	C₂×M₁₆	1	3	yes	—
65	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
66	C₈×C₂₃	—	2	1	no	—
67	C₈×C₂₃	—	2	3	no	—
68	C₈×C₂₃	C₂×C₄²	1	6	yes	—
69	C₈×C₂₃	C₂³×C₄	1	2	yes	—
70	#3407	C₂×C₂₃	1	2	yes	—
71	#2875	C₂×C₂₃	1	1	yes	characteristic
72	#2110	C₂×C₂₃	1	6	yes	complemented
73	C₂×#347	C₂×C₂₃	1	6	yes	complemented
74	C₂²×C₂₃	—	4	3	no	—
75	C₂²×C₂₃	C₂×#97	1	6	yes	—
76	C₂²×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
77	C₄×C₂₃	—	4	3	no	—
78	C₄×C₂₃	—	4	1	no	—
79	C₄×C₂₃	C₂×#97	1	6	yes	—
80	C₄×C₂₃	C₂²×C₄²	1	1	yes	characteristic
81	C₄×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
82	C₂×#131	C₄×C₂₃	1	1	yes	characteristic
83	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
84	C₄×#41	C₄×C₂₃	1	1	yes	characteristic
85	#504	C₄×C₂₃	1	16	yes	complemented
86	C₂×M₃₂	C₂²×C₂₃	1	12	yes	—
87	#405	C₄×C₂₃	1	3	yes	—
88	#405	C₂²×C₂₃	1	3	yes	—
89	C₄×M₁₆	C₄×C₂₃	1	3	yes	—
90	C₂×#97	C₂²×C₂₃	1	3	yes	—
91	#347	C₂²×C₂₃	1	4	yes	—
92	#347	C₂²×C₂₃	1	12	yes	—
93	C₂×C₂₃	—	2	1	no	—
94	C₂×C₂₃	—	2	6	no	—
95	C₂×C₂₃	C₂²×#97	1	1	yes	characteristic
96	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
97	#131	—	2	4	no	—
98	#131	C₂×C₄×C₂₃	1	8	yes	—
99	C₂×M₁₆	C₂×C₄×C₂₃	1	3	yes	—
100	C₂×M₁₆	C₂×C₄×C₂₃	1	3	yes	—
101	C₂²×C₈	C₂×C₄×C₂₃	1	3	yes	—
102	C₂²×C₈	C₂³×C₂₃	1	3	yes	—
103	C₄×Q₈	—	2	1	no	—
104	C₄×D₄	—	2	3	no	—
105	#117	C₂×C₄×C₂₃	1	3	yes	—
106	C₂×C₄²	C₂×C₄×C₂₃	1	3	yes	—
107	M₃₂	C₂×C₄×C₂₃	1	24	yes	complemented
108	M₃₂	C₂³×C₂₃	1	8	yes	—
109	C₂×C₁₆	C₂×C₄×C₂₃	1	24	yes	complemented
110	#97	C₂×C₄×C₂₃	1	3	yes	—
111	#97	C₂×C₄×C₂₃	1	3	yes	—
112	#97	C₂³×C₂₃	1	1	yes	characteristic
113	#97	C₂³×C₂₃	1	3	yes	—
114	C₄×C₈	C₂×C₄×C₂₃	1	6	yes	—
115	C₂₃	#33329	1	1	yes	characteristic, direct product factor, Sylow
116	#41	C₄²×C₂₃	1	4	yes	—
117	#41	M₁₆×C₂₃	1	4	yes	—
118	C₂×Q₈	M₁₆×C₂₃	1	1	yes	characteristic
119	C₂×D₄	M₁₆×C₂₃	1	3	yes	—
120	C₂²×C₄	—	2	3	no	—
121	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
122	M₁₆	—	2	6	no	—
123	M₁₆	C₄²×C₂₃	1	12	yes	—
124	C₂×C₈	—	2	6	no	—
125	C₂×C₈	C₂²×C₄×C₂₃	1	3	yes	—
126	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
127	C₂×C₈	C₂²×C₄×C₂₃	1	3	yes	—
128	C₂×C₈	C₂²×C₄×C₂₃	1	12	yes	—
129	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
130	C₄⋊C₄	—	2	3	no	—
131	C₂²⋊C₄	—	2	3	no	—
132	C₄²	—	2	3	no	—
133	C₄²	C₂²×C₄×C₂₃	1	3	yes	—
134	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
135	C₁₆	C₂²×C₄×C₂₃	1	16	yes	—
136	C₂³	C₂×M₁₆×C₂₃	1	3	yes	—
137	Q₈	C₂₃×#97	1	4	yes	—
138	D₄	C₂₃×#97	1	12	yes	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	3	no	—
141	C₂×C₄	—	8	3	no	—
142	C₂×C₄	C₂×C₄²×C₂₃	1	6	yes	—
143	C₂×C₄	C₂×M₁₆×C₂₃	1	6	yes	—
144	C₂×C₄	C₂×M₁₆×C₂₃	1	3	yes	—
145	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	2	3	no	—
148	C₈	C₂×C₄²×C₂₃	1	6	yes	—
149	C₈	C₂³×C₄×C₂₃	1	2	yes	—
150	C₂²	—	4	3	no	—
151	C₂²	C₂×C₂₃×#97	1	6	yes	—
152	C₂²	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
153	C₄	—	4	1	no	—
154	C₄	—	4	3	no	—
155	C₄	C₂×C₂₃×#97	1	6	yes	—
156	C₄	C₂²×C₄²×C₂₃	1	1	yes	characteristic
157	C₄	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	2	6	no	—
160	C₂	C₂²×C₂₃×#97	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	C₂²×C₄²
commutator	C₄	C₂²×C₄²×C₂₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
socle	C₂×C₂₃	C₂²×#97
prime radical	C₂×#41×C₂₃	C₂×C₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2110	C₂	semidirect
4. C₂×C₂₃×#347	C₂	semidirect
8. C₂₃×#504	C₄	semidirect
27. C₂₃×M₃₂	C₂×C₄	semidirect
29. C₂×C₁₆×C₂₃	C₂×C₄	semidirect
55. #33329	C₂₃	direct (N × H)
72. #2110	C₂×C₂₃	semidirect
73. C₂×#347	C₂×C₂₃	semidirect
85. #504	C₄×C₂₃	semidirect
107. M₃₂	C₂×C₄×C₂₃	semidirect
109. C₂×C₁₆	C₂×C₄×C₂₃	semidirect
115. C₂₃	#33329	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×#3407	2
C₂₃×#2875	2
C₂₃×#2110	2
C₂×C₂₃×#347	2
#33329	23

Subgroups — C23×#33329 #19180955

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33329 #19180955