Subgroups — C₂₃×#9899 #19157525

Order 5888.

430 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	C₂₃×#2859	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1831	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1296	C₂	1	4	yes	maximal
5	C₂×C₂₃×#125	—	2	1	no	—
6	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₂₃	—	2	1	no	—
9	C₂₃×#389	—	2	2	no	—
10	C₂₃×#389	C₄	1	4	yes	—
11	C₂₃×#384	C₂²	1	2	yes	—
12	C₂₃×#382	—	2	1	no	—
13	C₄³×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#336	—	2	1	no	—
15	C₂₃×#336	—	2	2	no	—
16	C₂²×C₈×C₂₃	—	2	2	no	—
17	C₂²×C₈×C₂₃	—	2	2	no	—
18	C₂₃×#125	—	4	2	no	—
19	C₂₃×#125	—	4	2	no	—
20	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
21	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
22	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
23	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₂₃	D₄	1	2	yes	—
27	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
29	C₂₃×C₄⋊C₈	—	2	2	no	—
30	C₂₃×C₄⋊C₈	—	2	4	no	—
31	C₂₃×C₄⋊C₈	Q₈	1	4	yes	—
32	C₄×C₈×C₂₃	—	4	2	no	—
33	C₂₃×#95	—	2	4	no	—
34	C₂²×C₄×C₂₃	—	2	4	no	—
35	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₂₃	—	2	4	no	—
41	C₂×C₈×C₂₃	—	4	2	no	—
42	C₂×C₈×C₂₃	—	4	2	no	—
43	C₂×C₈×C₂₃	—	4	2	no	—
44	C₂×C₈×C₂₃	—	4	4	no	—
45	C₄⋊C₄×C₂₃	—	4	4	no	—
46	C₄⋊C₄×C₂₃	—	4	8	no	—
47	C₄⋊C₄×C₂₃	—	4	2	no	—
48	C₄⋊C₄×C₂₃	—	4	2	no	—
49	C₄⋊C₄×C₂₃	—	4	4	no	—
50	C₄²×C₂₃	—	2	4	no	—
51	C₄²×C₂₃	—	2	2	no	—
52	C₄²×C₂₃	—	4	2	no	—
53	C₄²×C₂₃	D₈	1	2	yes	—
54	C₄²×C₂₃	SD₁₆	1	2	yes	—
55	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
56	C₄²×C₂₃	#41	1	2	yes	—
57	#9899	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
63	C₂×C₄×C₂₃	—	2	2	no	—
64	C₂×C₄×C₂₃	—	2	2	no	—
65	C₂×C₄×C₂₃	—	4	4	no	—
66	C₂×C₄×C₂₃	—	4	4	no	—
67	C₂×C₄×C₂₃	—	4	4	no	—
68	C₂×C₄×C₂₃	#102	1	4	yes	—
69	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
70	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
72	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₂₃	#135	1	4	yes	—
76	C₂×C₄×C₂₃	#137	1	2	yes	—
77	C₈×C₂₃	—	4	4	no	—
78	C₈×C₂₃	—	8	2	no	—
79	#2859	C₂×C₂₃	1	1	yes	characteristic
80	#1831	C₂×C₂₃	1	1	yes	characteristic
81	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
82	#1296	C₂×C₂₃	1	4	yes	—
83	C₂²×C₂₃	#391	1	1	yes	characteristic
84	C₂²×C₂₃	#413	1	2	yes	—
85	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
86	C₂²×C₂₃	#416	1	2	yes	—
87	C₂²×C₂₃	#419	1	1	yes	characteristic
88	C₄×C₂₃	—	2	4	no	—
89	C₄×C₂₃	—	2	2	no	—
90	C₄×C₂₃	—	4	2	no	—
91	C₄×C₂₃	—	8	4	no	—
92	C₄×C₂₃	#474	1	2	yes	—
93	C₄×C₂₃	#475	1	2	yes	—
94	C₄×C₂₃	#488	1	2	yes	—
95	C₄×C₂₃	#499	1	2	yes	—
96	C₂×#125	—	2	1	no	—
97	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
98	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
99	C₂×C₄×C₈	—	2	1	no	—
100	#389	—	2	2	no	—
101	#389	C₄×C₂₃	1	4	yes	—
102	#384	C₂²×C₂₃	1	2	yes	—
103	#382	—	2	1	no	—
104	C₄³	C₂²×C₂₃	1	1	yes	characteristic
105	#336	—	2	2	no	—
106	#336	—	2	1	no	—
107	C₂×C₂₃	#1540	1	2	yes	—
108	C₂×C₂₃	#1554	1	2	yes	—
109	C₂×C₂₃	#1988	1	1	yes	characteristic
110	C₂×C₂₃	#1991	1	1	yes	characteristic
111	C₂×C₂₃	#1998	1	1	yes	characteristic
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	—	2	2	no	—
114	#125	—	4	2	no	—
115	#125	—	4	2	no	—
116	C₂×C₄⋊C₄	—	2	4	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
120	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
121	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
122	C₂×C₄²	D₄×C₂₃	1	2	yes	—
123	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
124	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	2	4	no	—
127	C₄⋊C₈	Q₈×C₂₃	1	4	yes	—
128	C₄×C₈	—	4	2	no	—
129	#95	—	2	4	no	—
130	C₂₃	#9899	1	1	yes	characteristic, direct product factor, Sylow
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
133	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
134	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
135	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
136	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	4	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄⋊C₄	—	4	8	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄²	—	2	4	no	—
148	C₄²	—	2	2	no	—
149	C₄²	—	4	2	no	—
150	C₄²	D₈×C₂₃	1	2	yes	—
151	C₄²	SD₁₆×C₂₃	1	2	yes	—
152	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
153	C₄²	#41×C₂₃	1	2	yes	—
154	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	C₂₃×#102	1	4	yes	—
165	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
166	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
167	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
168	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
169	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
170	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
171	C₂×C₄	C₂₃×#135	1	4	yes	—
172	C₂×C₄	C₂₃×#137	1	2	yes	—
173	C₈	—	4	4	no	—
174	C₈	—	8	2	no	—
175	C₂²	C₂₃×#391	1	1	yes	characteristic
176	C₂²	C₂₃×#413	1	2	yes	—
177	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
178	C₂²	C₂₃×#416	1	2	yes	—
179	C₂²	C₂₃×#419	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	4	no	—
182	C₄	—	4	2	no	—
183	C₄	—	8	4	no	—
184	C₄	C₂₃×#474	1	2	yes	—
185	C₄	C₂₃×#475	1	2	yes	—
186	C₄	C₂₃×#488	1	2	yes	—
187	C₄	C₂₃×#499	1	2	yes	—
188	C₂	C₂₃×#1540	1	2	yes	—
189	C₂	C₂₃×#1554	1	2	yes	—
190	C₂	C₂₃×#1988	1	1	yes	characteristic
191	C₂	C₂₃×#1991	1	1	yes	characteristic
192	C₂	C₂₃×#1998	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
57. #9899	C₂₃	direct (N × H)
130. C₂₃	#9899	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×#2859	2
C₂₃×#1831	2
C₄×C₂₃×C₄⋊C₈	2
C₂₃×#1296	2
#9899	23

Subgroups — C23×#9899 #19157525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9899 #19157525