Subgroups — C₂₃×#7152 #19154778

Order 5888.

230 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2150	C₂	1	2	yes	maximal
2	C₂×C₂₃×#322	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×M₃₂	—	2	2	no	—
4	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
5	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×C₄⋊C₁₆	—	2	2	no	—
8	C₂₃×C₄⋊C₁₆	C₄	1	4	yes	—
9	C₂₃×#322	—	2	2	no	—
10	C₂²×C₈×C₂₃	—	2	1	no	—
11	C₂²×C₈×C₂₃	C₈	1	2	yes	—
12	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
13	C₂²×C₈×C₂₃	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂₃×M₃₂	—	4	2	no	—
16	C₂₃×M₃₂	—	4	2	no	—
17	C₂×C₁₆×C₂₃	—	2	4	no	—
18	C₄×C₈×C₂₃	—	2	1	no	—
19	C₄×C₈×C₂₃	—	2	1	no	—
20	C₄×C₈×C₂₃	—	2	1	no	—
21	C₄×C₈×C₂₃	C₈	1	2	yes	—
22	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₂₃	D₄	1	2	yes	—
24	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
27	C₂×C₈×C₂₃	—	2	1	no	—
28	C₂×C₈×C₂₃	—	2	1	no	—
29	C₂×C₈×C₂₃	—	2	2	no	—
30	C₂×C₈×C₂₃	—	2	2	no	—
31	C₂×C₈×C₂₃	—	4	2	no	—
32	C₂×C₈×C₂₃	—	4	2	no	—
33	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
34	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
36	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
37	C₂×C₈×C₂₃	Q₁₆	1	1	yes	characteristic
38	C₄²×C₂₃	C₄²	1	1	yes	characteristic
39	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
41	C₄²×C₂₃	M₁₆	1	1	yes	characteristic
42	C₁₆×C₂₃	—	4	4	no	—
43	#7152	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂³×C₂₃	C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₂₃	—	2	2	no	—
46	C₂×C₄×C₂₃	—	2	2	no	—
47	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
48	C₂×C₄×C₂₃	C₄×C₈	1	1	yes	characteristic
49	C₂×C₄×C₂₃	#97	1	1	yes	characteristic
50	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₂₃	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₂₃	#106	1	1	yes	characteristic
54	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
55	C₂×C₄×C₂₃	#108	1	1	yes	characteristic
56	C₂×C₄×C₂₃	#108	1	1	yes	characteristic
57	C₈×C₂₃	—	2	2	no	—
58	C₈×C₂₃	—	4	2	no	—
59	C₈×C₂₃	—	4	2	no	—
60	C₈×C₂₃	#102	1	2	yes	—
61	C₈×C₂₃	#103	1	2	yes	—
62	#2150	C₂×C₂₃	1	2	yes	—
63	C₂×#322	C₂×C₂₃	1	1	yes	characteristic
64	C₂²×C₂₃	—	2	1	no	—
65	C₂²×C₂₃	—	2	1	no	—
66	C₂²×C₂₃	#334	1	1	yes	characteristic
67	C₂²×C₂₃	#335	1	1	yes	characteristic
68	C₂²×C₂₃	#336	1	1	yes	characteristic
69	C₄×C₂₃	—	2	1	no	—
70	C₄×C₂₃	—	2	1	no	—
71	C₄×C₂₃	#334	1	1	yes	characteristic
72	C₄×C₂₃	#335	1	1	yes	characteristic
73	C₄×C₂₃	#340	1	1	yes	characteristic
74	C₄×C₂₃	#341	1	1	yes	characteristic
75	C₄×C₂₃	#347	1	2	yes	—
76	C₄×C₂₃	#349	1	2	yes	—
77	C₂×M₃₂	—	2	2	no	—
78	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic
79	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic
80	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
81	C₄⋊C₁₆	—	2	2	no	—
82	C₄⋊C₁₆	C₄×C₂₃	1	4	yes	—
83	#322	—	2	2	no	—
84	C₂×C₂₃	—	2	1	no	—
85	C₂×C₂₃	—	2	1	no	—
86	C₂×C₂₃	#1277	1	1	yes	characteristic
87	C₂×C₂₃	#1375	1	1	yes	characteristic
88	C₂×C₂₃	#1377	1	1	yes	characteristic
89	C₂²×C₈	—	2	1	no	—
90	C₂²×C₈	C₈×C₂₃	1	2	yes	—
91	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
92	C₂²×C₈	Q₈×C₂₃	1	1	yes	characteristic
93	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
94	M₃₂	—	4	2	no	—
95	M₃₂	—	4	2	no	—
96	C₂×C₁₆	—	2	4	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	C₈×C₂₃	1	2	yes	—
101	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
102	C₄×C₈	D₄×C₂₃	1	2	yes	—
103	C₂₃	#7152	1	1	yes	characteristic, direct product factor, Sylow
104	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
105	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
106	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
114	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
115	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
116	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
117	C₂×C₈	Q₁₆×C₂₃	1	1	yes	characteristic
118	C₄²	C₄²×C₂₃	1	1	yes	characteristic
119	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
120	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
121	C₄²	M₁₆×C₂₃	1	1	yes	characteristic
122	C₁₆	—	4	4	no	—
123	C₂³	C₂₃×C₄⋊C₈	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
127	C₂×C₄	C₄×C₈×C₂₃	1	1	yes	characteristic
128	C₂×C₄	C₂₃×#97	1	1	yes	characteristic
129	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
130	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
131	C₂×C₄	C₂₃×C₄⋊C₈	1	1	yes	characteristic
132	C₂×C₄	C₂₃×#106	1	1	yes	characteristic
133	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
134	C₂×C₄	C₂₃×#108	1	1	yes	characteristic
135	C₂×C₄	C₂₃×#108	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	—	4	2	no	—
138	C₈	—	4	2	no	—
139	C₈	C₂₃×#102	1	2	yes	—
140	C₈	C₂₃×#103	1	2	yes	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	C₂₃×#334	1	1	yes	characteristic
144	C₂²	C₂₃×#335	1	1	yes	characteristic
145	C₂²	C₂₃×#336	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	C₂₃×#334	1	1	yes	characteristic
149	C₄	C₂₃×#335	1	1	yes	characteristic
150	C₄	C₂₃×#340	1	1	yes	characteristic
151	C₄	C₂₃×#341	1	1	yes	characteristic
152	C₄	C₂₃×#347	1	2	yes	—
153	C₄	C₂₃×#349	1	2	yes	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	C₂₃×#1277	1	1	yes	characteristic
157	C₂	C₂₃×#1375	1	1	yes	characteristic
158	C₂	C₂₃×#1377	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#95
commutator	C₂×C₄	C₄×C₈×C₂₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₂₃
socle	C₂²×C₂₃	#336
prime radical	C₂³×C₂₃	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
43. #7152	C₂₃	direct (N × H)
103. C₂₃	#7152	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×#2150	2
C₂×C₂₃×#322	2
#7152	23

Subgroups — C23×#7152 #19154778

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7152 #19154778