Subgroups — C₂₃×#8812 #19156438

Order 5888.

382 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1842	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1831	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#335	C₂	1	2	yes	maximal
5	C₂×C₂₃×#334	C₂	1	2	yes	maximal
6	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
7	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
11	C₄³×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#336	—	2	2	no	—
13	C₂₃×#336	C₄	1	2	yes	—
14	C₂₃×#336	C₄	1	2	yes	—
15	C₂₃×#335	—	2	4	no	—
16	C₂₃×#334	—	2	4	no	—
17	C₂²×C₈×C₂₃	—	2	1	no	—
18	C₂²×C₈×C₂₃	—	2	1	no	—
19	C₂²×C₈×C₂₃	—	2	2	no	—
20	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₂₃	D₄	1	2	yes	—
24	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
25	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
26	C₂₃×C₄⋊C₈	—	2	2	no	—
27	C₂₃×C₄⋊C₈	—	2	2	no	—
28	C₂₃×C₄⋊C₈	—	2	2	no	—
29	C₂₃×C₄⋊C₈	—	2	2	no	—
30	C₂₃×C₄⋊C₈	—	4	4	no	—
31	C₄×C₈×C₂₃	D₄	1	4	yes	—
32	C₄×C₈×C₂₃	Q₈	1	4	yes	—
33	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₂₃	—	2	4	no	—
39	C₂×C₈×C₂₃	—	2	4	no	—
40	C₂×C₈×C₂₃	—	4	2	no	—
41	C₂×C₈×C₂₃	—	4	1	no	—
42	C₂×C₈×C₂₃	—	4	1	no	—
43	C₂×C₈×C₂₃	—	4	4	no	—
44	C₂×C₈×C₂₃	—	4	2	no	—
45	C₂×C₈×C₂₃	—	4	2	no	—
46	C₂×C₈×C₂₃	C₄⋊C₄	1	8	yes	—
47	C₄²×C₂₃	—	2	4	no	—
48	C₄²×C₂₃	—	2	2	no	—
49	C₄²×C₂₃	—	2	2	no	—
50	C₄²×C₂₃	—	2	2	no	—
51	C₄²×C₂₃	—	2	2	no	—
52	C₄²×C₂₃	C₂×D₄	1	2	yes	—
53	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
54	#8812	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₂₃	—	2	2	no	—
57	C₂×C₄×C₂₃	—	2	2	no	—
58	C₂×C₄×C₂₃	—	2	2	no	—
59	C₂×C₄×C₂₃	—	2	2	no	—
60	C₂×C₄×C₂₃	—	2	2	no	—
61	C₂×C₄×C₂₃	—	2	2	no	—
62	C₂×C₄×C₂₃	#108	1	4	yes	—
63	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
64	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
65	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
66	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₂₃	#131	1	2	yes	—
68	C₂×C₄×C₂₃	#131	1	2	yes	—
69	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
70	C₂×C₄×C₂₃	#137	1	2	yes	—
71	C₈×C₂₃	—	2	8	no	—
72	C₈×C₂₃	—	8	2	no	—
73	C₈×C₂₃	—	8	1	no	—
74	C₈×C₂₃	—	8	1	no	—
75	#1842	C₂×C₂₃	1	1	yes	characteristic
76	#1831	C₂×C₂₃	1	1	yes	characteristic
77	#1744	C₂×C₂₃	1	1	yes	characteristic
78	C₂×#335	C₂×C₂₃	1	2	yes	—
79	C₂×#334	C₂×C₂₃	1	2	yes	—
80	C₂²×C₂₃	#384	1	1	yes	characteristic
81	C₂²×C₂₃	#424	1	1	yes	characteristic
82	C₂²×C₂₃	#424	1	1	yes	characteristic
83	C₂²×C₂₃	#428	1	2	yes	—
84	C₂²×C₂₃	C₂×#108	1	2	yes	—
85	C₄×C₂₃	—	2	4	no	—
86	C₄×C₂₃	—	2	2	no	—
87	C₄×C₂₃	—	4	2	no	—
88	C₄×C₂₃	#424	1	4	yes	—
89	C₄×C₂₃	#496	1	1	yes	characteristic
90	C₄×C₂₃	#497	1	1	yes	characteristic
91	C₄×C₂₃	#501	1	2	yes	—
92	C₂×C₄⋊C₈	—	2	2	no	—
93	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
94	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
95	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
97	C₄³	C₂²×C₂₃	1	1	yes	characteristic
98	#336	—	2	2	no	—
99	#336	C₄×C₂₃	1	2	yes	—
100	#336	C₄×C₂₃	1	2	yes	—
101	#335	—	2	4	no	—
102	#334	—	2	4	no	—
103	C₂×C₂₃	#1573	1	2	yes	—
104	C₂×C₂₃	#1574	1	2	yes	—
105	C₂×C₂₃	#1843	1	1	yes	characteristic
106	C₂×C₂₃	#1847	1	1	yes	characteristic
107	C₂×C₂₃	#1848	1	1	yes	characteristic
108	C₂²×C₈	—	2	1	no	—
109	C₂²×C₈	—	2	1	no	—
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
112	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
113	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
114	C₂×C₄²	D₄×C₂₃	1	2	yes	—
115	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
116	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	4	4	no	—
122	C₄×C₈	D₄×C₂₃	1	4	yes	—
123	C₄×C₈	Q₈×C₂₃	1	4	yes	—
124	C₂₃	#8812	1	1	yes	characteristic, direct product factor, Sylow
125	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
126	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
127	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
128	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
129	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
130	C₂×C₈	—	2	4	no	—
131	C₂×C₈	—	2	4	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	4	1	no	—
134	C₂×C₈	—	4	1	no	—
135	C₂×C₈	—	4	4	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	C₄⋊C₄×C₂₃	1	8	yes	—
139	C₄²	—	2	4	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	C₂×D₄×C₂₃	1	2	yes	—
145	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
146	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	C₂₃×#108	1	4	yes	—
154	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
155	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
156	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
157	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
158	C₂×C₄	C₂₃×#131	1	2	yes	—
159	C₂×C₄	C₂₃×#131	1	2	yes	—
160	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
161	C₂×C₄	C₂₃×#137	1	2	yes	—
162	C₈	—	2	8	no	—
163	C₈	—	8	2	no	—
164	C₈	—	8	1	no	—
165	C₈	—	8	1	no	—
166	C₂²	C₂₃×#384	1	1	yes	characteristic
167	C₂²	C₂₃×#424	1	1	yes	characteristic
168	C₂²	C₂₃×#424	1	1	yes	characteristic
169	C₂²	C₂₃×#428	1	2	yes	—
170	C₂²	C₂×C₂₃×#108	1	2	yes	—
171	C₄	—	2	4	no	—
172	C₄	—	2	2	no	—
173	C₄	—	4	2	no	—
174	C₄	C₂₃×#424	1	4	yes	—
175	C₄	C₂₃×#496	1	1	yes	characteristic
176	C₄	C₂₃×#497	1	1	yes	characteristic
177	C₄	C₂₃×#501	1	2	yes	—
178	C₂	C₂₃×#1573	1	2	yes	—
179	C₂	C₂₃×#1574	1	2	yes	—
180	C₂	C₂₃×#1843	1	1	yes	characteristic
181	C₂	C₂₃×#1847	1	1	yes	characteristic
182	C₂	C₂₃×#1848	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle, prime radical	C₂³×C₂₃	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
54. #8812	C₂₃	direct (N × H)
124. C₂₃	#8812	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×#1842	2
C₂₃×#1831	2
C₂₃×#1744	2
C₂×C₂₃×#335	2
C₂×C₂₃×#334	2
#8812	23

Subgroups — C23×#8812 #19156438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8812 #19156438