Subgroups — C₂₃×#23366 #19170992

Order 5888.

616 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1985	C₂	1	4	yes	maximal
3	C₂₃×#1917	C₂	1	8	yes	maximal
4	C₂₃×#1839	C₂	1	2	yes	maximal
5	C₂×C₂₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
7	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
8	C₂×C₂₃×#97	C₂²	1	4	yes	—
9	C₂₃×#391	C₄	1	2	yes	—
10	C₂₃×#389	C₄	1	4	yes	—
11	C₂₃×#384	C₄	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
13	C₄³×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#336	C₂²	1	8	yes	—
15	C₂₃×#336	C₂²	1	8	yes	—
16	C₂₃×#336	C₂²	1	4	yes	—
17	C₂²×Q₈×C₂₃	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₂₃	C₂³	1	8	yes	—
19	C₂₃×C₄⋊Q₈	—	2	4	no	—
20	C₄×Q₈×C₂₃	—	4	2	no	—
21	C₄×Q₈×C₂₃	—	4	2	no	—
22	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
24	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
30	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
32	C₂₃×C₄⋊C₈	—	4	4	no	—
33	C₂₃×#97	—	4	4	no	—
34	C₂₃×#95	C₂×C₄	1	8	yes	—
35	C₂×Q₈×C₂₃	—	2	4	no	—
36	C₂×Q₈×C₂₃	—	4	2	no	—
37	C₂×Q₈×C₂₃	—	4	2	no	—
38	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
40	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
41	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
42	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
45	C₂×C₈×C₂₃	—	2	8	no	—
46	C₂×C₈×C₂₃	—	2	16	no	—
47	C₄⋊C₄×C₂₃	—	2	8	no	—
48	C₄⋊C₄×C₂₃	—	4	4	no	—
49	C₄⋊C₄×C₂₃	—	4	2	no	—
50	C₄⋊C₄×C₂₃	—	4	4	no	—
51	C₄²×C₂₃	—	2	4	no	—
52	C₄²×C₂₃	—	2	2	no	—
53	C₄²×C₂₃	—	2	4	no	—
54	C₄²×C₂₃	—	2	4	no	—
55	C₄²×C₂₃	—	4	2	no	—
56	C₄²×C₂₃	—	4	1	no	—
57	C₄²×C₂₃	M₁₆	1	4	yes	—
58	#23366	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
60	Q₈×C₂₃	—	4	8	no	—
61	C₂×C₄×C₂₃	—	2	1	no	—
62	C₂×C₄×C₂₃	—	2	2	no	—
63	C₂×C₄×C₂₃	—	2	2	no	—
64	C₂×C₄×C₂₃	—	2	1	no	—
65	C₂×C₄×C₂₃	—	2	1	no	—
66	C₂×C₄×C₂₃	—	2	4	no	—
67	C₂×C₄×C₂₃	—	2	4	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	4	no	—
70	C₂×C₄×C₂₃	—	2	2	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	8	no	—
74	C₂×C₄×C₂₃	—	2	4	no	—
75	C₂×C₄×C₂₃	C₂×M₁₆	1	4	yes	—
76	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
78	C₂×C₄×C₂₃	2₋¹⁺⁴	1	4	yes	—
79	C₈×C₂₃	—	4	8	no	—
80	C₄×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
81	#1985	C₂×C₂₃	1	4	yes	—
82	#1917	C₂×C₂₃	1	8	yes	—
83	#1839	C₂×C₂₃	1	2	yes	—
84	C₂²×C₂₃	#518	1	1	yes	characteristic
85	C₂²×C₂₃	#519	1	2	yes	—
86	C₂²×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
87	C₂²×C₂₃	#568	1	1	yes	characteristic
88	C₂²×C₂₃	#568	1	2	yes	—
89	C₄×C₂₃	—	2	2	no	—
90	C₄×C₂₃	—	2	4	no	—
91	C₄×C₂₃	—	4	4	no	—
92	C₄×C₂₃	—	4	1	no	—
93	C₄×C₂₃	—	4	4	no	—
94	C₄×C₂₃	—	4	2	no	—
95	C₄×C₂₃	#563	1	4	yes	—
96	C₂×C₄⋊Q₈	C₄×C₂₃	1	1	yes	characteristic
97	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
98	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
99	C₂×#97	C₂²×C₂₃	1	4	yes	—
100	#391	C₄×C₂₃	1	2	yes	—
101	#389	C₄×C₂₃	1	4	yes	—
102	#384	C₄×C₂₃	1	1	yes	characteristic
103	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
104	C₄³	C₂²×C₂₃	1	1	yes	characteristic
105	#336	C₂²×C₂₃	1	8	yes	—
106	#336	C₂²×C₂₃	1	4	yes	—
107	#336	C₂²×C₂₃	1	8	yes	—
108	C₂×C₂₃	#2381	1	1	yes	characteristic
109	C₂×C₂₃	#2975	1	1	yes	characteristic
110	C₂×C₂₃	#2980	1	1	yes	characteristic
111	C₂×C₂₃	#2983	1	2	yes	—
112	C₂×C₂₃	#2987	1	2	yes	—
113	C₂²×Q₈	C₂×C₄×C₂₃	1	2	yes	—
114	C₂²×C₈	C₂³×C₂₃	1	8	yes	—
115	C₄⋊Q₈	—	2	4	no	—
116	C₄×Q₈	—	4	2	no	—
117	C₄×Q₈	—	4	2	no	—
118	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
119	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
120	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
121	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
122	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
123	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
124	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
125	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
126	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
127	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
128	C₄⋊C₈	—	4	4	no	—
129	#97	—	4	4	no	—
130	#95	C₂×C₄×C₂₃	1	8	yes	—
131	C₂₃	#23366	1	1	yes	characteristic, direct product factor, Sylow
132	C₂×Q₈	—	2	4	no	—
133	C₂×Q₈	—	4	2	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
136	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
137	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
138	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
139	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
140	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
141	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
142	C₂×C₈	—	2	8	no	—
143	C₂×C₈	—	2	16	no	—
144	C₄⋊C₄	—	2	8	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄²	—	2	4	no	—
149	C₄²	—	2	4	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	4	no	—
152	C₄²	—	4	2	no	—
153	C₄²	—	4	1	no	—
154	C₄²	M₁₆×C₂₃	1	4	yes	—
155	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
156	Q₈	—	4	8	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	C₂×M₁₆×C₂₃	1	4	yes	—
172	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
173	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
174	C₂×C₄	C₂₃×2₋¹⁺⁴	1	4	yes	—
175	C₈	—	4	8	no	—
176	C₂²	C₂₃×#518	1	1	yes	characteristic
177	C₂²	C₂₃×#519	1	2	yes	—
178	C₂²	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
179	C₂²	C₂₃×#568	1	1	yes	characteristic
180	C₂²	C₂₃×#568	1	2	yes	—
181	C₄	—	2	2	no	—
182	C₄	—	2	4	no	—
183	C₄	—	4	4	no	—
184	C₄	—	4	1	no	—
185	C₄	—	4	4	no	—
186	C₄	—	4	2	no	—
187	C₄	C₂₃×#563	1	4	yes	—
188	C₂	C₂₃×#2381	1	1	yes	characteristic
189	C₂	C₂₃×#2975	1	1	yes	characteristic
190	C₂	C₂₃×#2980	1	1	yes	characteristic
191	C₂	C₂₃×#2983	1	2	yes	—
192	C₂	C₂₃×#2987	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
58. #23366	C₂₃	direct (N × H)
131. C₂₃	#23366	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₃×C₄⋊Q₈	2
C₂₃×#1985	2
C₂₃×#1917	2
C₂₃×#1839	2
#23366	23

Subgroups — C23×#23366 #19170992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#23366 #19170992