Subgroups — C₂₃×#7288 #19154914

Order 5888.

202 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	C₂₃×#1567	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#423	—	2	1	no	—
5	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
6	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×C₄⋊C₁₆	—	2	1	no	—
8	C₂₃×C₂²⋊C₁₆	C₄	1	2	yes	—
9	C₄×C₁₆×C₂₃	—	2	1	no	—
10	C₂₃×#335	—	2	1	no	—
11	C₂₃×#335	—	2	1	no	—
12	C₂₃×#335	C₄	1	2	yes	—
13	C₂₃×#335	C₄	1	2	yes	—
14	C₂×M₁₆×C₂₃	—	4	1	no	—
15	C₂²×C₈×C₂₃	—	4	1	no	—
16	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
19	C₂×C₁₆×C₂₃	—	2	2	no	—
20	C₂₃×C₄⋊C₈	—	2	2	no	—
21	C₂₃×C₄⋊C₈	—	2	2	no	—
22	C₂₃×C₄⋊C₈	—	4	1	no	—
23	C₂₃×C₄⋊C₈	—	4	1	no	—
24	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
30	C₂²×C₄×C₂₃	Q₁₆	1	1	yes	characteristic
31	M₁₆×C₂₃	—	8	1	no	—
32	M₁₆×C₂₃	—	8	1	no	—
33	C₂×C₈×C₂₃	—	2	2	no	—
34	C₂×C₈×C₂₃	—	4	1	no	—
35	C₂×C₈×C₂₃	—	4	2	no	—
36	C₂×C₈×C₂₃	—	4	2	no	—
37	C₂×C₈×C₂₃	—	8	1	no	—
38	C₂×C₈×C₂₃	—	8	1	no	—
39	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
46	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
47	C₁₆×C₂₃	—	4	2	no	—
48	#7288	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂³×C₂₃	#107	1	1	yes	characteristic
50	C₂×C₄×C₂₃	—	2	1	no	—
51	C₂×C₄×C₂₃	—	2	1	no	—
52	C₂×C₄×C₂₃	—	4	1	no	—
53	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
54	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
55	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
56	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
57	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
58	C₂×C₄×C₂₃	#106	1	1	yes	characteristic
59	C₂×C₄×C₂₃	D₁₆	1	1	yes	characteristic
60	C₂×C₄×C₂₃	SD₃₂	1	1	yes	characteristic
61	C₂×C₄×C₂₃	SD₃₂	1	1	yes	characteristic
62	C₂×C₄×C₂₃	Q₃₂	1	1	yes	characteristic
63	C₈×C₂₃	—	2	1	no	—
64	C₈×C₂₃	—	2	1	no	—
65	C₈×C₂₃	—	2	1	no	—
66	C₈×C₂₃	—	8	2	no	—
67	C₈×C₂₃	—	8	2	no	—
68	C₈×C₂₃	#108	1	2	yes	—
69	#2162	C₂×C₂₃	1	1	yes	characteristic
70	#1567	C₂×C₂₃	1	1	yes	characteristic
71	C₂×#335	C₂×C₂₃	1	1	yes	characteristic
72	C₂²×C₂₃	—	2	1	no	—
73	C₂²×C₂₃	—	2	1	no	—
74	C₂²×C₂₃	#340	1	1	yes	characteristic
75	C₂²×C₂₃	#366	1	1	yes	characteristic
76	C₂²×C₂₃	#367	1	1	yes	characteristic
77	C₄×C₂₃	—	2	1	no	—
78	C₄×C₂₃	—	2	1	no	—
79	C₄×C₂₃	—	2	1	no	—
80	C₄×C₂₃	#341	1	1	yes	characteristic
81	C₄×C₂₃	#344	1	1	yes	characteristic
82	C₄×C₂₃	#357	1	1	yes	characteristic
83	C₄×C₂₃	#357	1	1	yes	characteristic
84	C₄×C₂₃	#358	1	1	yes	characteristic
85	C₄×C₂₃	#358	1	1	yes	characteristic
86	#423	—	2	1	no	—
87	C₂×C₄⋊C₈	—	2	1	no	—
88	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
89	C₄⋊C₁₆	—	2	1	no	—
90	C₂²⋊C₁₆	C₄×C₂₃	1	2	yes	—
91	C₄×C₁₆	—	2	1	no	—
92	#335	—	2	1	no	—
93	#335	—	2	1	no	—
94	#335	C₄×C₂₃	1	2	yes	—
95	#335	C₄×C₂₃	1	2	yes	—
96	C₂×C₂₃	—	2	1	no	—
97	C₂×C₂₃	—	2	1	no	—
98	C₂×C₂₃	#1301	1	1	yes	characteristic
99	C₂×C₂₃	#1380	1	1	yes	characteristic
100	C₂×C₂₃	#1388	1	1	yes	characteristic
101	C₂×M₁₆	—	4	1	no	—
102	C₂²×C₈	—	4	1	no	—
103	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
104	C₂²×C₈	Q₈×C₂₃	1	1	yes	characteristic
105	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
106	C₂×C₁₆	—	2	2	no	—
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	4	1	no	—
110	C₄⋊C₈	—	4	1	no	—
111	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
112	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
113	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
114	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
115	C₂₃	#7288	1	1	yes	characteristic, direct product factor, Sylow
116	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
117	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
118	C₂²×C₄	Q₁₆×C₂₃	1	1	yes	characteristic
119	M₁₆	—	8	1	no	—
120	M₁₆	—	8	1	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	—	4	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	8	1	no	—
126	C₂×C₈	—	8	1	no	—
127	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
128	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
129	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
130	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
131	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
132	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
133	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
134	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
135	C₁₆	—	4	2	no	—
136	C₂³	C₂₃×#107	1	1	yes	characteristic
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
141	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
142	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
143	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
144	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
145	C₂×C₄	C₂₃×#106	1	1	yes	characteristic
146	C₂×C₄	C₂₃×D₁₆	1	1	yes	characteristic
147	C₂×C₄	C₂₃×SD₃₂	1	1	yes	characteristic
148	C₂×C₄	C₂₃×SD₃₂	1	1	yes	characteristic
149	C₂×C₄	C₂₃×Q₃₂	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	8	2	no	—
154	C₈	—	8	2	no	—
155	C₈	C₂₃×#108	1	2	yes	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	C₂₃×#340	1	1	yes	characteristic
159	C₂²	C₂₃×#366	1	1	yes	characteristic
160	C₂²	C₂₃×#367	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	C₂₃×#341	1	1	yes	characteristic
165	C₄	C₂₃×#344	1	1	yes	characteristic
166	C₄	C₂₃×#357	1	1	yes	characteristic
167	C₄	C₂₃×#357	1	1	yes	characteristic
168	C₄	C₂₃×#358	1	1	yes	characteristic
169	C₄	C₂₃×#358	1	1	yes	characteristic
170	C₂	—	2	1	no	—
171	C₂	—	2	1	no	—
172	C₂	C₂₃×#1301	1	1	yes	characteristic
173	C₂	C₂₃×#1380	1	1	yes	characteristic
174	C₂	C₂₃×#1388	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#102
commutator	C₂×C₈	C₄²×C₂₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₂₃
socle	C₂²×C₂₃	#340
prime radical	C₂³×C₂₃	#107
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
48. #7288	C₂₃	direct (N × H)
115. C₂₃	#7288	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×#2162	2
C₂₃×#1567	2
C₂×C₂₃×#335	2
#7288	23

Subgroups — C23×#7288 #19154914

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7288 #19154914