Subgroups — C₂₃×#8824 #19156450

Order 5888.

322 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1837	C₂	1	2	yes	maximal
2	C₂₃×#1567	C₂	1	2	yes	maximal
3	C₂₃×#1566	C₂	1	2	yes	maximal
4	C₂×C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#423	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#341	—	2	4	no	—
10	C₂₃×#335	—	2	4	no	—
11	C₂₃×#334	—	2	4	no	—
12	C₈²×C₂₃	C₄	1	1	yes	characteristic
13	C₈²×C₂₃	C₄	1	1	yes	characteristic
14	C₈²×C₂₃	C₄	1	1	yes	characteristic
15	C₈²×C₂₃	C₄	1	1	yes	characteristic
16	C₂×M₁₆×C₂₃	—	2	4	no	—
17	C₂²×C₈×C₂₃	D₄	1	2	yes	—
18	C₂²×C₈×C₂₃	Q₈	1	2	yes	—
19	C₂²×C₈×C₂₃	Q₈	1	2	yes	—
20	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
21	C₂₃×C₄⋊C₈	—	2	4	no	—
22	C₂₃×C₄⋊C₈	—	2	4	no	—
23	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₂₃	D₄	1	2	yes	—
30	C₄×C₈×C₂₃	D₄	1	2	yes	—
31	C₄×C₈×C₂₃	Q₈	1	2	yes	—
32	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
35	M₁₆×C₂₃	—	4	8	no	—
36	C₂×C₈×C₂₃	—	2	4	no	—
37	C₂×C₈×C₂₃	—	2	4	no	—
38	C₂×C₈×C₂₃	—	2	4	no	—
39	C₂×C₈×C₂₃	—	4	4	no	—
40	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
46	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
50	#8824	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂³×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₄×C₂₃	—	2	2	no	—
53	C₂×C₄×C₂₃	—	2	2	no	—
54	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
58	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₂₃	#135	1	4	yes	—
61	C₈×C₂₃	—	2	2	no	—
62	C₈×C₂₃	—	2	2	no	—
63	C₈×C₂₃	—	2	2	no	—
64	C₈×C₂₃	—	2	2	no	—
65	C₈×C₂₃	—	8	4	no	—
66	C₈×C₂₃	#108	1	4	yes	—
67	C₈×C₂₃	#108	1	4	yes	—
68	#1837	C₂×C₂₃	1	2	yes	—
69	#1567	C₂×C₂₃	1	2	yes	—
70	#1566	C₂×C₂₃	1	2	yes	—
71	C₂×C₈²	C₂×C₂₃	1	1	yes	characteristic
72	C₂²×C₂₃	—	2	1	no	—
73	C₂²×C₂₃	—	2	1	no	—
74	C₂²×C₂₃	#384	1	1	yes	characteristic
75	C₂²×C₂₃	#499	1	2	yes	—
76	C₄×C₂₃	—	2	1	no	—
77	C₄×C₂₃	—	2	1	no	—
78	C₄×C₂₃	#427	1	2	yes	—
79	C₄×C₂₃	C₂×#108	1	2	yes	—
80	C₄×C₂₃	C₂×#108	1	2	yes	—
81	C₄×C₂₃	#495	1	2	yes	—
82	#423	C₂²×C₂₃	1	4	yes	—
83	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
84	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
85	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
86	#341	—	2	4	no	—
87	#335	—	2	4	no	—
88	#334	—	2	4	no	—
89	C₈²	C₄×C₂₃	1	1	yes	characteristic
90	C₈²	C₄×C₂₃	1	1	yes	characteristic
91	C₈²	C₄×C₂₃	1	1	yes	characteristic
92	C₈²	C₄×C₂₃	1	1	yes	characteristic
93	C₂×C₂₃	—	2	2	no	—
94	C₂×C₂₃	#1846	1	1	yes	characteristic
95	C₂×C₂₃	#1848	1	1	yes	characteristic
96	C₂×C₂₃	#1848	1	1	yes	characteristic
97	C₂×M₁₆	—	2	4	no	—
98	C₂²×C₈	D₄×C₂₃	1	2	yes	—
99	C₂²×C₈	Q₈×C₂₃	1	2	yes	—
100	C₂²×C₈	Q₈×C₂₃	1	2	yes	—
101	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
102	C₄⋊C₈	—	2	4	no	—
103	C₄⋊C₈	—	2	4	no	—
104	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
105	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
106	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
107	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
108	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
109	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
110	C₄×C₈	D₄×C₂₃	1	2	yes	—
111	C₄×C₈	D₄×C₂₃	1	2	yes	—
112	C₄×C₈	Q₈×C₂₃	1	2	yes	—
113	C₂₃	#8824	1	1	yes	characteristic, direct product factor, Sylow
114	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
115	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
116	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
117	M₁₆	—	4	8	no	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	—	2	4	no	—
120	C₂×C₈	—	2	4	no	—
121	C₂×C₈	—	4	4	no	—
122	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
123	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
124	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
125	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
126	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
127	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
128	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
129	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
130	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
131	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
132	C₂³	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
138	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
139	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
140	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
141	C₂×C₄	C₂₃×#135	1	4	yes	—
142	C₈	—	2	2	no	—
143	C₈	—	2	2	no	—
144	C₈	—	2	2	no	—
145	C₈	—	2	2	no	—
146	C₈	—	8	4	no	—
147	C₈	C₂₃×#108	1	4	yes	—
148	C₈	C₂₃×#108	1	4	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	C₂₃×#384	1	1	yes	characteristic
152	C₂²	C₂₃×#499	1	2	yes	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	C₂₃×#427	1	2	yes	—
156	C₄	C₂×C₂₃×#108	1	2	yes	—
157	C₄	C₂×C₂₃×#108	1	2	yes	—
158	C₄	C₂₃×#495	1	2	yes	—
159	C₂	—	2	2	no	—
160	C₂	C₂₃×#1846	1	1	yes	characteristic
161	C₂	C₂₃×#1848	1	1	yes	characteristic
162	C₂	C₂₃×#1848	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#127
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#384
prime radical	C₂³×C₂₃	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
50. #8824	C₂₃	direct (N × H)
113. C₂₃	#8824	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×#1837	2
C₂₃×#1567	2
C₂₃×#1566	2
C₂×C₈²×C₂₃	2
#8824	23

Subgroups — C23×#8824 #19156450

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8824 #19156450