Subgroups — C₂₃×#7209 #19154835

Order 5888.

222 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2162	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆×C₂₃	C₄	1	2	yes	—
5	C₂₃×#431	C₄	1	1	yes	characteristic
6	C₄×M₁₆×C₂₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×C₄⋊C₁₆	—	2	1	no	—
9	C₂₃×C₄⋊C₁₆	—	2	1	no	—
10	C₂₃×C₄⋊C₁₆	—	2	1	no	—
11	C₂₃×C₂²⋊C₁₆	C₄	1	2	yes	—
12	C₄×C₁₆×C₂₃	—	2	1	no	—
13	C₂₃×#322	—	2	1	no	—
14	C₈²×C₂₃	—	2	1	no	—
15	C₂×M₁₆×C₂₃	C₈	1	2	yes	—
16	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆×C₂₃	—	2	2	no	—
20	C₂×C₁₆×C₂₃	—	2	2	no	—
21	C₂×C₁₆×C₂₃	—	4	2	no	—
22	C₂₃×C₄⋊C₈	—	2	1	no	—
23	C₂₃×C₂²⋊C₈	C₈	1	2	yes	—
24	C₂₃×#97	—	2	1	no	—
25	C₄×C₈×C₂₃	—	2	1	no	—
26	C₄×C₈×C₂₃	—	2	1	no	—
27	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	C₄²	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
34	M₁₆×C₂₃	—	2	2	no	—
35	M₁₆×C₂₃	C₁₆	1	4	yes	complemented
36	C₂×C₈×C₂₃	—	2	2	no	—
37	C₂×C₈×C₂₃	—	2	1	no	—
38	C₂×C₈×C₂₃	—	2	1	no	—
39	C₂×C₈×C₂₃	—	2	2	no	—
40	C₂×C₈×C₂₃	—	4	1	no	—
41	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₂₃	M₁₆	1	1	yes	characteristic
48	C₄²×C₂₃	M₁₆	1	1	yes	characteristic
49	C₁₆×C₂₃	—	4	2	no	—
50	C₁₆×C₂₃	—	4	2	no	—
51	#7209	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂³×C₂₃	C₄×C₈	1	1	yes	characteristic
53	C₂×C₄×C₂₃	—	2	1	no	—
54	C₂×C₄×C₂₃	—	2	1	no	—
55	C₂×C₄×C₂₃	—	4	1	no	—
56	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
57	C₂×C₄×C₂₃	#97	1	1	yes	characteristic
58	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₂₃	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₂₃	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₂₃	C₂×C₁₆	1	2	yes	—
63	C₂×C₄×C₂₃	M₃₂	1	2	yes	—
64	C₈×C₂₃	—	2	1	no	—
65	C₈×C₂₃	—	2	1	no	—
66	C₈×C₂₃	—	2	1	no	—
67	C₈×C₂₃	—	2	4	no	—
68	C₈×C₂₃	—	4	2	no	—
69	C₈×C₂₃	#100	1	1	yes	characteristic
70	C₈×C₂₃	#101	1	1	yes	characteristic
71	#2162	C₂×C₂₃	1	1	yes	characteristic
72	C₂×C₄⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic
73	C₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic
74	C₂²×C₂₃	—	2	1	no	—
75	C₂²×C₂₃	—	2	1	no	—
76	C₂²×C₂₃	#336	1	1	yes	characteristic
77	C₂²×C₂₃	C₄×C₁₆	1	1	yes	characteristic
78	C₂²×C₂₃	#346	1	1	yes	characteristic
79	C₄×C₂₃	—	2	1	no	—
80	C₄×C₂₃	—	2	1	no	—
81	C₄×C₂₃	—	2	1	no	—
82	C₄×C₂₃	#343	1	1	yes	characteristic
83	C₄×C₂₃	#344	1	1	yes	characteristic
84	C₄×C₂₃	C₂²⋊C₁₆	1	2	yes	—
85	C₄×C₂₃	C₄⋊C₁₆	1	2	yes	—
86	C₂²×C₁₆	C₄×C₂₃	1	2	yes	—
87	#431	C₄×C₂₃	1	1	yes	characteristic
88	C₄×M₁₆	C₄×C₂₃	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
90	C₄⋊C₁₆	—	2	1	no	—
91	C₄⋊C₁₆	—	2	1	no	—
92	C₄⋊C₁₆	—	2	1	no	—
93	C₂²⋊C₁₆	C₄×C₂₃	1	2	yes	—
94	C₄×C₁₆	—	2	1	no	—
95	#322	—	2	1	no	—
96	C₈²	—	2	1	no	—
97	C₂×C₂₃	—	2	1	no	—
98	C₂×C₂₃	—	2	1	no	—
99	C₂×C₂₃	#1283	1	1	yes	characteristic
100	C₂×C₂₃	#1374	1	1	yes	characteristic
101	C₂×C₂₃	#1378	1	1	yes	characteristic
102	C₂×M₁₆	C₈×C₂₃	1	2	yes	—
103	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
104	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
105	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
106	C₂×C₁₆	—	2	2	no	—
107	C₂×C₁₆	—	2	2	no	—
108	C₂×C₁₆	—	4	2	no	—
109	C₄⋊C₈	—	2	1	no	—
110	C₂²⋊C₈	C₈×C₂₃	1	2	yes	—
111	#97	—	2	1	no	—
112	C₄×C₈	—	2	1	no	—
113	C₄×C₈	—	2	1	no	—
114	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
115	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
116	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
117	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
118	C₂₃	#7209	1	1	yes	characteristic, direct product factor, Sylow
119	C₂²×C₄	C₄²×C₂₃	1	1	yes	characteristic
120	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
121	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
122	M₁₆	—	2	2	no	—
123	M₁₆	C₁₆×C₂₃	1	4	yes	complemented
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	4	1	no	—
129	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
130	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
131	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
132	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
133	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
134	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
135	C₄²	M₁₆×C₂₃	1	1	yes	characteristic
136	C₄²	M₁₆×C₂₃	1	1	yes	characteristic
137	C₁₆	—	4	2	no	—
138	C₁₆	—	4	2	no	—
139	C₂³	C₄×C₈×C₂₃	1	1	yes	characteristic
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
144	C₂×C₄	C₂₃×#97	1	1	yes	characteristic
145	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
146	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
147	C₂×C₄	C₂₃×C₄⋊C₈	1	1	yes	characteristic
148	C₂×C₄	C₂₃×C₄⋊C₈	1	1	yes	characteristic
149	C₂×C₄	C₂×C₁₆×C₂₃	1	2	yes	—
150	C₂×C₄	C₂₃×M₃₂	1	2	yes	—
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	2	4	no	—
155	C₈	—	4	2	no	—
156	C₈	C₂₃×#100	1	1	yes	characteristic
157	C₈	C₂₃×#101	1	1	yes	characteristic
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	C₂₃×#336	1	1	yes	characteristic
161	C₂²	C₄×C₁₆×C₂₃	1	1	yes	characteristic
162	C₂²	C₂₃×#346	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	C₂₃×#343	1	1	yes	characteristic
167	C₄	C₂₃×#344	1	1	yes	characteristic
168	C₄	C₂₃×C₂²⋊C₁₆	1	2	yes	—
169	C₄	C₂₃×C₄⋊C₁₆	1	2	yes	—
170	C₂	—	2	1	no	—
171	C₂	—	2	1	no	—
172	C₂	C₂₃×#1283	1	1	yes	characteristic
173	C₂	C₂₃×#1374	1	1	yes	characteristic
174	C₂	C₂₃×#1378	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂²⋊C₄
commutator	C₂²	C₄×C₁₆×C₂₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₂₃
socle	C₂²×C₂₃	#336
prime radical	C₂³×C₂₃	C₄×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
35. M₁₆×C₂₃	C₁₆	semidirect
51. #7209	C₂₃	direct (N × H)
118. C₂₃	#7209	direct (N × H)
123. M₁₆	C₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2162	2
C₂×C₂₃×C₄⋊C₁₆	2
C₈×M₁₆×C₂₃	2
#7209	23

Subgroups — C23×#7209 #19154835

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7209 #19154835