Subgroups — C₂₃×#13276 #19160902

Order 5888.

398 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₃×#107	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1380	C₂	1	4	yes	maximal
5	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
6	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂₃×#500	—	2	2	no	—
8	C₂₃×#500	C₄	1	4	yes	complemented
9	C₂×C₂₃×#107	—	2	1	no	—
10	C₂×C₂₃×#107	C₂²	1	2	yes	—
11	C₂×C₂₃×#107	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
14	C₂₃×C₄⋊C₁₆	—	2	2	no	—
15	C₂₃×#340	—	2	1	no	—
16	C₂²×Q₈×C₂₃	—	4	1	no	—
17	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
19	C₂₃×C₄⋊Q₈	—	2	4	no	—
20	C₂₃×C₄⋊Q₈	—	4	2	no	—
21	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
22	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
23	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
24	C₂×C₄²×C₂₃	—	4	1	no	—
25	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₂₃	—	2	2	no	—
27	C₂×C₁₆×C₂₃	—	2	4	no	—
28	C₂₃×#107	—	2	4	no	—
29	C₂₃×#107	—	2	2	no	—
30	C₂₃×#107	—	2	2	no	—
31	C₂₃×#107	C₂×C₄	1	4	yes	—
32	C₂₃×#107	Q₈	1	4	yes	complemented
33	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₂₃	D₄	1	2	yes	—
35	C₂₃×#95	—	4	1	no	—
36	C₂×Q₈×C₂₃	—	4	4	no	—
37	C₂×Q₈×C₂₃	—	8	4	no	—
38	C₂²×C₄×C₂₃	—	4	2	no	—
39	C₂²×C₄×C₂₃	—	4	2	no	—
40	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₂₃	—	2	2	no	—
44	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₂₃	#41	1	2	yes	—
49	C₄⋊C₄×C₂₃	—	2	4	no	—
50	C₄⋊C₄×C₂₃	—	2	4	no	—
51	C₄⋊C₄×C₂₃	—	4	4	no	—
52	C₄⋊C₄×C₂₃	—	4	2	no	—
53	C₄⋊C₄×C₂₃	—	4	2	no	—
54	C₄²×C₂₃	—	8	2	no	—
55	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₂₃	SD₁₆	1	2	yes	—
57	C₁₆×C₂₃	—	4	2	no	—
58	#13276	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
60	Q₈×C₂₃	—	8	8	no	—
61	C₂×C₄×C₂₃	—	2	2	no	—
62	C₂×C₄×C₂₃	—	4	4	no	—
63	C₂×C₄×C₂₃	—	4	4	no	—
64	C₂×C₄×C₂₃	—	8	2	no	—
65	C₂×C₄×C₂₃	—	8	2	no	—
66	C₂×C₄×C₂₃	—	8	4	no	—
67	C₂×C₄×C₂₃	#102	1	2	yes	—
68	C₂×C₄×C₂₃	#102	1	2	yes	—
69	C₂×C₄×C₂₃	SD₃₂	1	2	yes	—
70	C₂×C₄×C₂₃	Q₃₂	1	2	yes	—
71	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
73	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₂₃	#137	1	2	yes	—
75	C₈×C₂₃	—	2	2	no	—
76	C₈×C₂₃	C₄×Q₈	1	2	yes	—
77	C₈×C₂₃	#124	1	1	yes	characteristic
78	C₈×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×#500	C₂×C₂₃	1	1	yes	characteristic
80	C₂×C₄⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic
81	C₄×#107	C₂×C₂₃	1	1	yes	characteristic
82	#1380	C₂×C₂₃	1	4	yes	—
83	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
84	C₂²×C₂₃	#474	1	2	yes	—
85	C₂²×C₂₃	C₂×SD₃₂	1	1	yes	characteristic
86	C₂²×C₂₃	C₂×Q₃₂	1	1	yes	characteristic
87	C₂²×C₂₃	#509	1	2	yes	—
88	C₄×C₂₃	—	2	2	no	—
89	C₄×C₂₃	—	8	4	no	—
90	C₄×C₂₃	—	8	4	no	—
91	C₄×C₂₃	#358	1	4	yes	—
92	C₄×C₂₃	#391	1	1	yes	characteristic
93	C₄×C₂₃	#419	1	1	yes	characteristic
94	C₄×C₂₃	#475	1	2	yes	—
95	C₂×C₄⋊Q₈	—	2	1	no	—
96	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
97	#500	—	2	2	no	—
98	#500	C₄×C₂₃	1	4	yes	complemented
99	C₂×#107	—	2	1	no	—
100	C₂×#107	C₂²×C₂₃	1	2	yes	—
101	C₂×#107	C₂²×C₂₃	1	2	yes	—
102	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
103	C₄×C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₁₆	—	2	2	no	—
105	#340	—	2	1	no	—
106	C₂×C₂₃	#1988	1	1	yes	characteristic
107	C₂×C₂₃	C₂×#358	1	1	yes	characteristic
108	C₂×C₂₃	#2140	1	1	yes	characteristic
109	C₂×C₂₃	#2226	1	2	yes	—
110	C₂×C₂₃	#2227	1	2	yes	—
111	C₂²×Q₈	—	4	1	no	—
112	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
113	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
114	C₄⋊Q₈	—	2	4	no	—
115	C₄⋊Q₈	—	4	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	4	1	no	—
119	C₂×C₄²	—	4	1	no	—
120	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	—	2	4	no	—
123	#107	—	2	4	no	—
124	#107	—	2	2	no	—
125	#107	—	2	2	no	—
126	#107	C₂×C₄×C₂₃	1	4	yes	—
127	#107	Q₈×C₂₃	1	4	yes	complemented
128	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
129	C₄×C₈	D₄×C₂₃	1	2	yes	—
130	#95	—	4	1	no	—
131	C₂₃	#13276	1	1	yes	characteristic, direct product factor, Sylow
132	C₂×Q₈	—	4	4	no	—
133	C₂×Q₈	—	8	4	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	4	2	no	—
136	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
137	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
141	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
142	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
143	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
144	C₂×C₈	#41×C₂₃	1	2	yes	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄²	—	8	2	no	—
151	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
152	C₄²	SD₁₆×C₂₃	1	2	yes	—
153	C₁₆	—	4	2	no	—
154	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
155	Q₈	—	8	8	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	4	4	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	C₂₃×#102	1	2	yes	—
163	C₂×C₄	C₂₃×#102	1	2	yes	—
164	C₂×C₄	C₂₃×SD₃₂	1	2	yes	—
165	C₂×C₄	C₂₃×Q₃₂	1	2	yes	—
166	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
167	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
168	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
169	C₂×C₄	C₂₃×#137	1	2	yes	—
170	C₈	—	2	2	no	—
171	C₈	C₄×Q₈×C₂₃	1	2	yes	—
172	C₈	C₂₃×#124	1	1	yes	characteristic
173	C₈	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
174	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
175	C₂²	C₂₃×#474	1	2	yes	—
176	C₂²	C₂×C₂₃×SD₃₂	1	1	yes	characteristic
177	C₂²	C₂×C₂₃×Q₃₂	1	1	yes	characteristic
178	C₂²	C₂₃×#509	1	2	yes	—
179	C₄	—	2	2	no	—
180	C₄	—	8	4	no	—
181	C₄	—	8	4	no	—
182	C₄	C₂₃×#358	1	4	yes	—
183	C₄	C₂₃×#391	1	1	yes	characteristic
184	C₄	C₂₃×#419	1	1	yes	characteristic
185	C₄	C₂₃×#475	1	2	yes	—
186	C₂	C₂₃×#1988	1	1	yes	characteristic
187	C₂	C₂×C₂₃×#358	1	1	yes	characteristic
188	C₂	C₂₃×#2140	1	1	yes	characteristic
189	C₂	C₂₃×#2226	1	2	yes	—
190	C₂	C₂₃×#2227	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
8. C₂₃×#500	C₄	semidirect
32. C₂₃×#107	Q₈	semidirect
58. #13276	C₂₃	direct (N × H)
98. #500	C₄×C₂₃	semidirect
127. #107	Q₈×C₂₃	semidirect
131. C₂₃	#13276	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×#500	2
C₂×C₂₃×C₄⋊C₁₆	2
C₄×C₂₃×#107	2
C₂₃×#1380	2
#13276	23

Subgroups — C23×#13276 #19160902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13276 #19160902