Subgroups — C₂₃×#12005 #19159631

Order 5888.

242 conjugacy classes of subgroups.

The subgroup lattice has 170 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	1	yes	maximal, characteristic
2	C₂×C₂₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1450	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1370	C₂	1	4	yes	maximal, complemented
5	C₂×C₂₃×M₃₂	—	2	1	no	—
6	C₂²×C₁₆×C₂₃	—	2	1	no	—
7	C₂₃×#431	C₄	1	1	yes	characteristic
8	C₂₃×#431	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×C₄⋊C₁₆	—	2	1	no	—
11	C₂₃×C₄⋊C₁₆	—	2	1	no	—
12	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
13	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
14	C₂₃×#322	C₄	1	2	yes	—
15	C₈²×C₂₃	C₂²	1	2	yes	—
16	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂₃×M₃₂	—	4	2	no	—
20	C₂×C₁₆×C₂₃	—	2	2	no	—
21	C₂×C₁₆×C₂₃	—	2	2	no	—
22	C₂×C₁₆×C₂₃	—	4	2	no	—
23	C₂₃×C₄⋊C₈	—	2	1	no	—
24	C₂₃×C₄⋊C₈	C₈	1	2	yes	—
25	C₂₃×C₂²⋊C₈	—	2	1	no	—
26	C₂₃×C₂²⋊C₈	C₈	1	2	yes	—
27	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
36	C₂×C₈×C₂₃	—	2	2	no	—
37	C₂×C₈×C₂₃	—	2	2	no	—
38	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₂₃	C₂×C₈	1	4	yes	—
41	C₂×C₈×C₂₃	D₈	1	2	yes	—
42	C₂×C₈×C₂₃	Q₁₆	1	2	yes	—
43	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
45	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₂₃	M₁₆	1	1	yes	characteristic
48	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
49	C₁₆×C₂₃	—	4	2	no	—
50	C₁₆×C₂₃	—	4	2	no	—
51	#12005	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂³×C₂₃	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₂₃	—	2	2	no	—
54	C₂×C₄×C₂₃	—	4	1	no	—
55	C₂×C₄×C₂₃	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₂₃	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₂₃	C₄⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₂₃	#107	1	2	yes	—
59	C₂×C₄×C₂₃	#108	1	2	yes	—
60	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
62	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
63	C₈×C₂₃	—	2	1	no	—
64	C₈×C₂₃	—	2	1	no	—
65	C₈×C₂₃	—	2	4	no	—
66	C₈×C₂₃	—	4	2	no	—
67	C₈×C₂₃	#107	1	2	yes	—
68	C₈×C₂₃	C₂×D₈	1	1	yes	characteristic
69	C₈×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
70	#2150	C₂×C₂₃	1	1	yes	characteristic
71	C₂×C₄⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic
72	#1450	C₂×C₂₃	1	1	yes	characteristic
73	#1370	C₂×C₂₃	1	4	yes	complemented
74	C₂²×C₂₃	—	2	1	no	—
75	C₂²×C₂₃	—	2	1	no	—
76	C₂²×C₂₃	#335	1	2	yes	—
77	C₂²×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
78	C₄×C₂₃	—	2	1	no	—
79	C₄×C₂₃	—	2	1	no	—
80	C₄×C₂₃	—	2	1	no	—
81	C₄×C₂₃	#335	1	2	yes	—
82	C₄×C₂₃	C₂×#107	1	1	yes	characteristic
83	C₄×C₂₃	C₂×#108	1	1	yes	characteristic
84	C₄×C₂₃	#504	1	2	yes	—
85	C₂×M₃₂	—	2	1	no	—
86	C₂²×C₁₆	—	2	1	no	—
87	#431	C₄×C₂₃	1	1	yes	characteristic
88	#431	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	C₂²×C₂₃	1	2	yes	—
93	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	—
94	#322	C₄×C₂₃	1	2	yes	—
95	C₈²	C₂²×C₂₃	1	2	yes	—
96	C₂×C₂₃	—	2	2	no	—
97	C₂×C₂₃	C₂×#335	1	1	yes	characteristic
98	C₂×C₂₃	#2151	1	1	yes	characteristic
99	C₂×C₂₃	#2153	1	1	yes	characteristic
100	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
101	C₂²×C₈	Q₈×C₂₃	1	1	yes	characteristic
102	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
103	M₃₂	—	4	2	no	—
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	—	4	2	no	—
107	C₄⋊C₈	—	2	1	no	—
108	C₄⋊C₈	C₈×C₂₃	1	2	yes	—
109	C₂²⋊C₈	—	2	1	no	—
110	C₂²⋊C₈	C₈×C₂₃	1	2	yes	—
111	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
112	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
113	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
114	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
115	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
116	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
117	C₂₃	#12005	1	1	yes	characteristic, direct product factor, Sylow
118	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
119	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
120	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
124	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
125	C₂×C₈	C₂×C₈×C₂₃	1	4	yes	—
126	C₂×C₈	D₈×C₂₃	1	2	yes	—
127	C₂×C₈	Q₁₆×C₂₃	1	2	yes	—
128	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
129	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
130	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
131	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
132	C₄²	M₁₆×C₂₃	1	1	yes	characteristic
133	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
134	C₁₆	—	4	2	no	—
135	C₁₆	—	4	2	no	—
136	C₂³	C₂₃×C₄⋊C₈	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	C₂₃×C₄⋊C₈	1	1	yes	characteristic
140	C₂×C₄	C₂₃×C₄⋊C₈	1	1	yes	characteristic
141	C₂×C₄	C₂₃×C₄⋊C₈	1	1	yes	characteristic
142	C₂×C₄	C₂₃×#107	1	2	yes	—
143	C₂×C₄	C₂₃×#108	1	2	yes	—
144	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
145	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
146	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	2	4	no	—
150	C₈	—	4	2	no	—
151	C₈	C₂₃×#107	1	2	yes	—
152	C₈	C₂×D₈×C₂₃	1	1	yes	characteristic
153	C₈	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	C₂₃×#335	1	2	yes	—
157	C₂²	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	C₂₃×#335	1	2	yes	—
162	C₄	C₂×C₂₃×#107	1	1	yes	characteristic
163	C₄	C₂×C₂₃×#108	1	1	yes	characteristic
164	C₄	C₂₃×#504	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	C₂×C₂₃×#335	1	1	yes	characteristic
167	C₂	C₂₃×#2151	1	1	yes	characteristic
168	C₂	C₂₃×#2153	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×C₄⋊C₈
prime radical	C₂³×C₂₃	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ 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
4. C₂₃×#1370	C₂	semidirect
51. #12005	C₂₃	direct (N × H)
73. #1370	C₂×C₂₃	semidirect
117. C₂₃	#12005	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×C₄⋊C₁₆	2
C₂₃×#1450	2
C₂₃×#1370	2
#12005	23

Subgroups — C23×#12005 #19159631

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12005 #19159631