Subgroups — C₂₃×#12839 #19160465

Order 5888.

362 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#504	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2158	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1812	C₂	1	2	yes	maximal
4	C₂₃×#1384	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
6	C₂₃×#504	—	2	2	no	—
7	C₂₃×#504	—	2	2	no	—
8	C₂×C₂₃×M₃₂	C₄	1	2	yes	complemented
9	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₂₃	C₄	1	2	yes	complemented
11	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#427	C₂²	1	4	yes	—
13	C₂₃×#410	—	2	2	no	—
14	C₂₃×#367	—	2	2	no	—
15	C₂₃×#366	—	2	2	no	—
16	C₂₃×H(3,ℤ/4)	—	2	2	no	—
17	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
18	C₂₃×#131	—	2	2	no	—
19	C₂₃×#131	—	2	2	no	—
20	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
21	C₂×M₁₆×C₂₃	Q₈	1	2	yes	—
22	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
24	C₂₃×#117	—	2	4	no	—
25	C₂₃×M₃₂	—	2	2	no	—
26	C₂₃×M₃₂	—	2	4	no	—
27	C₂×C₁₆×C₂₃	—	2	2	no	—
28	C₂×C₁₆×C₂₃	—	2	2	no	—
29	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₂₃	D₄	1	2	yes	—
33	C₂×C₁₆×C₂₃	D₄	1	2	yes	—
34	C₂₃×#107	—	2	4	no	—
35	C₂₃×#106	—	2	4	no	—
36	C₂₃×#99	—	4	4	no	—
37	#41×C₂₃	—	2	2	no	—
38	#41×C₂₃	—	2	2	no	—
39	C₂×Q₈×C₂₃	Q₁₆	1	1	yes	characteristic
40	C₂×D₄×C₂₃	D₈	1	2	yes	—
41	C₂×D₄×C₂₃	Q₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
44	M₁₆×C₂₃	—	2	2	no	—
45	M₁₆×C₂₃	—	2	4	no	—
46	C₂×C₈×C₂₃	—	2	2	no	—
47	C₂×C₈×C₂₃	—	2	2	no	—
48	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₂₃	#41	1	2	yes	—
53	C₄⋊C₄×C₂₃	—	4	4	no	—
54	C₂²⋊C₄×C₂₃	—	4	4	no	—
55	C₄²×C₂₃	—	4	4	no	—
56	C₁₆×C₂₃	—	2	2	no	—
57	C₁₆×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₁₆×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₁₆×C₂₃	C₂²⋊C₄	1	2	yes	—
60	#12839	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₂₃	#107	1	2	yes	—
62	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
63	Q₈×C₂₃	—	2	2	no	—
64	D₄×C₂₃	—	2	4	no	—
65	D₄×C₂₃	—	2	2	no	—
66	C₂×C₄×C₂₃	—	2	2	no	—
67	C₂×C₄×C₂₃	—	2	2	no	—
68	C₂×C₄×C₂₃	—	8	4	no	—
69	C₂×C₄×C₂₃	#107	1	2	yes	—
70	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₂₃	#121	1	2	yes	—
72	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
73	C₂×C₄×C₂₃	#135	1	2	yes	—
74	C₈×C₂₃	—	2	2	no	—
75	C₈×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₈×C₂₃	#117	1	1	yes	characteristic
77	C₈×C₂₃	C₂²⋊Q₈	1	2	yes	—
78	C₂×#504	C₂×C₂₃	1	1	yes	characteristic
79	#2158	C₂×C₂₃	1	2	yes	complemented
80	#1812	C₂×C₂₃	1	2	yes	—
81	#1384	C₂×C₂₃	1	2	yes	complemented
82	C₂²×C₂₃	—	2	2	no	—
83	C₂²×C₂₃	—	4	1	no	—
84	C₂²×C₂₃	—	4	2	no	—
85	C₂²×C₂₃	C₂×#107	1	1	yes	characteristic
86	C₂²×C₂₃	#466	1	2	yes	—
87	C₄×C₂₃	—	2	2	no	—
88	C₄×C₂₃	—	16	4	no	—
89	C₄×C₂₃	#380	1	1	yes	characteristic
90	C₄×C₂₃	#427	1	1	yes	characteristic
91	C₄×C₂₃	#467	1	2	yes	—
92	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
93	#504	—	2	2	no	—
94	#504	—	2	2	no	—
95	C₂×M₃₂	C₄×C₂₃	1	2	yes	complemented
96	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
97	C₂²×C₁₆	C₄×C₂₃	1	2	yes	complemented
98	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
99	#427	C₂²×C₂₃	1	4	yes	—
100	#410	—	2	2	no	—
101	#367	—	2	2	no	—
102	#366	—	2	2	no	—
103	H(3,ℤ/4)	—	2	2	no	—
104	C₂×C₂₃	—	2	2	no	—
105	C₂×C₂₃	—	2	1	no	—
106	C₂×C₂₃	—	4	2	no	—
107	C₂×C₂₃	#1808	1	1	yes	characteristic
108	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
109	#131	—	2	2	no	—
110	#131	—	2	2	no	—
111	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
112	C₂×M₁₆	Q₈×C₂₃	1	2	yes	—
113	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
114	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
115	#117	—	2	4	no	—
116	M₃₂	—	2	2	no	—
117	M₃₂	—	2	4	no	—
118	C₂×C₁₆	—	2	2	no	—
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
121	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
122	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
123	C₂×C₁₆	D₄×C₂₃	1	2	yes	—
124	C₂×C₁₆	D₄×C₂₃	1	2	yes	—
125	#107	—	2	4	no	—
126	#106	—	2	4	no	—
127	#99	—	4	4	no	—
128	C₂₃	#12839	1	1	yes	characteristic, direct product factor, Sylow
129	#41	—	2	2	no	—
130	#41	—	2	2	no	—
131	C₂×Q₈	Q₁₆×C₂₃	1	1	yes	characteristic
132	C₂×D₄	D₈×C₂₃	1	2	yes	—
133	C₂×D₄	Q₁₆×C₂₃	1	1	yes	characteristic
134	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
135	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
136	M₁₆	—	2	2	no	—
137	M₁₆	—	2	4	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
141	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
142	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
143	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
144	C₂×C₈	#41×C₂₃	1	2	yes	—
145	C₄⋊C₄	—	4	4	no	—
146	C₂²⋊C₄	—	4	4	no	—
147	C₄²	—	4	4	no	—
148	C₁₆	—	2	2	no	—
149	C₁₆	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
150	C₁₆	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
151	C₁₆	C₂²⋊C₄×C₂₃	1	2	yes	—
152	C₂³	C₂₃×#107	1	2	yes	—
153	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
154	Q₈	—	2	2	no	—
155	D₄	—	2	4	no	—
156	D₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	8	4	no	—
160	C₂×C₄	C₂₃×#107	1	2	yes	—
161	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
162	C₂×C₄	C₂₃×#121	1	2	yes	—
163	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
164	C₂×C₄	C₂₃×#135	1	2	yes	—
165	C₈	—	2	2	no	—
166	C₈	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
167	C₈	C₂₃×#117	1	1	yes	characteristic
168	C₈	C₂₃×C₂²⋊Q₈	1	2	yes	—
169	C₂²	—	2	2	no	—
170	C₂²	—	4	1	no	—
171	C₂²	—	4	2	no	—
172	C₂²	C₂×C₂₃×#107	1	1	yes	characteristic
173	C₂²	C₂₃×#466	1	2	yes	—
174	C₄	—	2	2	no	—
175	C₄	—	16	4	no	—
176	C₄	C₂₃×#380	1	1	yes	characteristic
177	C₄	C₂₃×#427	1	1	yes	characteristic
178	C₄	C₂₃×#467	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	—	2	1	no	—
181	C₂	—	4	2	no	—
182	C₂	C₂₃×#1808	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	#427
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂×C₂₃	#1808
prime radical	C₂×#41×C₂₃	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂×M₁₆×C₂₃ ◁ C₂×C₂₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2158	C₂	semidirect
4. C₂₃×#1384	C₂	semidirect
8. C₂×C₂₃×M₃₂	C₄	semidirect
10. C₂²×C₁₆×C₂₃	C₄	semidirect
60. #12839	C₂₃	direct (N × H)
79. #2158	C₂×C₂₃	semidirect
81. #1384	C₂×C₂₃	semidirect
95. C₂×M₃₂	C₄×C₂₃	semidirect
97. C₂²×C₁₆	C₄×C₂₃	semidirect
128. C₂₃	#12839	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×#504	2
C₂₃×#2158	2
C₂₃×#1812	2
C₂₃×#1384	2
#12839	23

Subgroups — C23×#12839 #19160465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12839 #19160465