Subgroups — C₂₃×#7162 #19154788

Order 5888.

200 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	C₂×C₂₃×#364	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#364	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×M₃₂	—	2	1	no	—
5	C₂×C₂₃×M₃₂	—	2	1	no	—
6	C₂×C₂₃×M₃₂	C₄	1	1	yes	characteristic
7	C₂×C₂₃×M₃₂	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#364	—	2	1	no	—
10	C₂₃×#364	—	2	1	no	—
11	C₂₃×#364	C₄	1	2	yes	—
12	C₂₃×#364	C₄	1	2	yes	—
13	C₂₃×#346	—	2	1	no	—
14	C₄×C₁₆×C₂₃	—	2	1	no	—
15	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₂₃	Q₈	1	1	yes	characteristic
18	C₂₃×M₃₂	—	2	1	no	—
19	C₂₃×M₃₂	—	2	1	no	—
20	C₂₃×M₃₂	—	2	2	no	—
21	C₂₃×M₃₂	—	2	2	no	—
22	C₂₃×M₃₂	—	4	1	no	—
23	C₂₃×M₃₂	—	4	1	no	—
24	C₂₃×M₃₂	C₈	1	2	yes	—
25	C₂₃×M₃₂	C₈	1	2	yes	—
26	C₂×C₁₆×C₂₃	—	2	1	no	—
27	C₂×C₁₆×C₂₃	—	2	1	no	—
28	C₂×C₁₆×C₂₃	—	4	1	no	—
29	C₂×C₁₆×C₂₃	—	4	1	no	—
30	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	—	2	1	no	—
35	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₂₃	M₁₆	1	1	yes	characteristic
43	C₂×C₈×C₂₃	M₁₆	1	1	yes	characteristic
44	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
45	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₂₃	Q₁₆	1	1	yes	characteristic
48	C₄²×C₂₃	—	2	1	no	—
49	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
51	C₁₆×C₂₃	—	2	2	no	—
52	C₁₆×C₂₃	—	2	2	no	—
53	C₁₆×C₂₃	—	4	2	no	—
54	C₁₆×C₂₃	—	4	2	no	—
55	#7162	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂³×C₂₃	#108	1	1	yes	characteristic
57	C₂×C₄×C₂₃	—	2	1	no	—
58	C₂×C₄×C₂₃	—	2	1	no	—
59	C₂×C₄×C₂₃	—	4	2	no	—
60	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
61	C₂×C₄×C₂₃	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₂₃	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₂₃	#106	1	1	yes	characteristic
64	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
65	C₂×C₄×C₂₃	#108	1	1	yes	characteristic
66	C₈×C₂₃	C₄×C₈	1	1	yes	characteristic
67	C₈×C₂₃	#97	1	1	yes	characteristic
68	C₈×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
69	C₈×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
70	C₈×C₂₃	#102	1	1	yes	characteristic
71	C₈×C₂₃	#102	1	1	yes	characteristic
72	C₈×C₂₃	#103	1	1	yes	characteristic
73	C₈×C₂₃	#103	1	1	yes	characteristic
74	C₂×#364	C₂×C₂₃	1	1	yes	characteristic
75	C₂×#364	C₂×C₂₃	1	1	yes	characteristic
76	C₄×M₃₂	C₂×C₂₃	1	1	yes	characteristic
77	C₂²×C₂₃	—	4	1	no	—
78	C₂²×C₂₃	#334	1	1	yes	characteristic
79	C₂²×C₂₃	#335	1	1	yes	characteristic
80	C₂²×C₂₃	#341	1	1	yes	characteristic
81	C₄×C₂₃	—	2	2	no	—
82	C₄×C₂₃	—	4	1	no	—
83	C₄×C₂₃	#334	1	1	yes	characteristic
84	C₄×C₂₃	#335	1	1	yes	characteristic
85	C₄×C₂₃	#336	1	1	yes	characteristic
86	C₄×C₂₃	#340	1	1	yes	characteristic
87	C₂×M₃₂	—	2	1	no	—
88	C₂×M₃₂	—	2	1	no	—
89	C₂×M₃₂	C₄×C₂₃	1	1	yes	characteristic
90	C₂×M₃₂	C₄×C₂₃	1	1	yes	characteristic
91	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
92	#364	—	2	1	no	—
93	#364	—	2	1	no	—
94	#364	C₄×C₂₃	1	2	yes	—
95	#364	C₄×C₂₃	1	2	yes	—
96	#346	—	2	1	no	—
97	C₄×C₁₆	—	2	1	no	—
98	C₂×C₂₃	—	2	1	no	—
99	C₂×C₂₃	—	2	1	no	—
100	C₂×C₂₃	—	2	1	no	—
101	C₂×C₂₃	#1277	1	1	yes	characteristic
102	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
103	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
104	C₂×C₄²	Q₈×C₂₃	1	1	yes	characteristic
105	M₃₂	—	2	1	no	—
106	M₃₂	—	2	1	no	—
107	M₃₂	—	2	2	no	—
108	M₃₂	—	2	2	no	—
109	M₃₂	—	4	1	no	—
110	M₃₂	—	4	1	no	—
111	M₃₂	C₈×C₂₃	1	2	yes	—
112	M₃₂	C₈×C₂₃	1	2	yes	—
113	C₂×C₁₆	—	2	1	no	—
114	C₂×C₁₆	—	2	1	no	—
115	C₂×C₁₆	—	4	1	no	—
116	C₂×C₁₆	—	4	1	no	—
117	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
118	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
119	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
120	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
121	C₂₃	#7162	1	1	yes	characteristic, direct product factor, Sylow
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
124	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
125	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
126	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
127	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
128	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
129	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
130	C₂×C₈	M₁₆×C₂₃	1	1	yes	characteristic
131	C₂×C₈	M₁₆×C₂₃	1	1	yes	characteristic
132	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
133	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
134	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
135	C₂×C₈	Q₁₆×C₂₃	1	1	yes	characteristic
136	C₄²	—	2	1	no	—
137	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
138	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
139	C₁₆	—	2	2	no	—
140	C₁₆	—	2	2	no	—
141	C₁₆	—	4	2	no	—
142	C₁₆	—	4	2	no	—
143	C₂³	C₂₃×#108	1	1	yes	characteristic
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
148	C₂×C₄	C₂₃×C₄⋊C₈	1	1	yes	characteristic
149	C₂×C₄	C₂₃×C₄⋊C₈	1	1	yes	characteristic
150	C₂×C₄	C₂₃×#106	1	1	yes	characteristic
151	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
152	C₂×C₄	C₂₃×#108	1	1	yes	characteristic
153	C₈	C₄×C₈×C₂₃	1	1	yes	characteristic
154	C₈	C₂₃×#97	1	1	yes	characteristic
155	C₈	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
156	C₈	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
157	C₈	C₂₃×#102	1	1	yes	characteristic
158	C₈	C₂₃×#102	1	1	yes	characteristic
159	C₈	C₂₃×#103	1	1	yes	characteristic
160	C₈	C₂₃×#103	1	1	yes	characteristic
161	C₂²	—	4	1	no	—
162	C₂²	C₂₃×#334	1	1	yes	characteristic
163	C₂²	C₂₃×#335	1	1	yes	characteristic
164	C₂²	C₂₃×#341	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	4	1	no	—
167	C₄	C₂₃×#334	1	1	yes	characteristic
168	C₄	C₂₃×#335	1	1	yes	characteristic
169	C₄	C₂₃×#336	1	1	yes	characteristic
170	C₄	C₂₃×#340	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	C₂₃×#1277	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₂₃	#102
commutator	C₈	C₄×C₈×C₂₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₂₃
socle	C₂×C₂₃	#1277
prime radical	C₂³×C₂₃	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
55. #7162	C₂₃	direct (N × H)
121. C₂₃	#7162	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#364	2
C₄×C₂₃×M₃₂	2
#7162	23

Subgroups — C23×#7162 #19154788

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7162 #19154788