Subgroups — C₂₃×#13275 #19160901

Order 5888.

398 conjugacy classes of subgroups.

The subgroup lattice has 196 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	2	no	—
38	C₂×Q₈×C₂₃	—	8	2	no	—
39	C₂²×C₄×C₂₃	—	4	2	no	—
40	C₂²×C₄×C₂₃	—	4	2	no	—
41	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₂₃	—	2	2	no	—
45	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
49	C₂×C₈×C₂₃	#41	1	2	yes	—
50	C₄⋊C₄×C₂₃	—	2	4	no	—
51	C₄⋊C₄×C₂₃	—	2	4	no	—
52	C₄⋊C₄×C₂₃	—	4	4	no	—
53	C₄⋊C₄×C₂₃	—	4	2	no	—
54	C₄⋊C₄×C₂₃	—	4	2	no	—
55	C₄²×C₂₃	—	8	2	no	—
56	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₂₃	SD₁₆	1	2	yes	—
58	C₁₆×C₂₃	—	4	2	no	—
59	#13275	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
61	Q₈×C₂₃	—	8	8	no	—
62	C₂×C₄×C₂₃	—	2	2	no	—
63	C₂×C₄×C₂₃	—	4	4	no	—
64	C₂×C₄×C₂₃	—	4	4	no	—
65	C₂×C₄×C₂₃	—	8	2	no	—
66	C₂×C₄×C₂₃	—	8	2	no	—
67	C₂×C₄×C₂₃	—	8	2	no	—
68	C₂×C₄×C₂₃	—	8	2	no	—
69	C₂×C₄×C₂₃	#102	1	2	yes	—
70	C₂×C₄×C₂₃	#102	1	2	yes	—
71	C₂×C₄×C₂₃	D₁₆	1	2	yes	—
72	C₂×C₄×C₂₃	SD₃₂	1	2	yes	—
73	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
75	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₂₃	#137	1	2	yes	—
77	C₈×C₂₃	—	2	2	no	—
78	C₈×C₂₃	C₄×Q₈	1	2	yes	—
79	C₈×C₂₃	#124	1	1	yes	characteristic
80	C₈×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×#500	C₂×C₂₃	1	1	yes	characteristic
82	C₂×C₄⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic
83	C₄×#107	C₂×C₂₃	1	1	yes	characteristic
84	#1380	C₂×C₂₃	1	4	yes	—
85	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
86	C₂²×C₂₃	#474	1	2	yes	—
87	C₂²×C₂₃	C₂×D₁₆	1	1	yes	characteristic
88	C₂²×C₂₃	C₂×SD₃₂	1	1	yes	characteristic
89	C₂²×C₂₃	#510	1	2	yes	—
90	C₄×C₂₃	—	2	2	no	—
91	C₄×C₂₃	—	8	4	no	—
92	C₄×C₂₃	—	8	4	no	—
93	C₄×C₂₃	#357	1	4	yes	—
94	C₄×C₂₃	#391	1	1	yes	characteristic
95	C₄×C₂₃	#419	1	1	yes	characteristic
96	C₄×C₂₃	#475	1	2	yes	—
97	C₂×C₄⋊Q₈	—	2	1	no	—
98	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
99	#500	—	2	2	no	—
100	#500	C₄×C₂₃	1	4	yes	complemented
101	C₂×#107	—	2	1	no	—
102	C₂×#107	C₂²×C₂₃	1	2	yes	—
103	C₂×#107	C₂²×C₂₃	1	2	yes	—
104	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
105	C₄×C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₁₆	—	2	2	no	—
107	#340	—	2	1	no	—
108	C₂×C₂₃	#1988	1	1	yes	characteristic
109	C₂×C₂₃	C₂×#357	1	1	yes	characteristic
110	C₂×C₂₃	#2141	1	1	yes	characteristic
111	C₂×C₂₃	#2224	1	2	yes	—
112	C₂×C₂₃	#2225	1	2	yes	—
113	C₂²×Q₈	—	4	1	no	—
114	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
115	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
116	C₄⋊Q₈	—	2	4	no	—
117	C₄⋊Q₈	—	4	2	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	4	1	no	—
121	C₂×C₄²	—	4	1	no	—
122	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	2	4	no	—
125	#107	—	2	4	no	—
126	#107	—	2	2	no	—
127	#107	—	2	2	no	—
128	#107	C₂×C₄×C₂₃	1	4	yes	—
129	#107	Q₈×C₂₃	1	4	yes	complemented
130	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
131	C₄×C₈	D₄×C₂₃	1	2	yes	—
132	#95	—	4	1	no	—
133	C₂₃	#13275	1	1	yes	characteristic, direct product factor, Sylow
134	C₂×Q₈	—	4	4	no	—
135	C₂×Q₈	—	8	2	no	—
136	C₂×Q₈	—	8	2	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
140	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
144	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
145	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
146	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
147	C₂×C₈	#41×C₂₃	1	2	yes	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄²	—	8	2	no	—
154	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
155	C₄²	SD₁₆×C₂₃	1	2	yes	—
156	C₁₆	—	4	2	no	—
157	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
158	Q₈	—	8	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	C₂₃×#102	1	2	yes	—
167	C₂×C₄	C₂₃×#102	1	2	yes	—
168	C₂×C₄	C₂₃×D₁₆	1	2	yes	—
169	C₂×C₄	C₂₃×SD₃₂	1	2	yes	—
170	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
171	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
172	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
173	C₂×C₄	C₂₃×#137	1	2	yes	—
174	C₈	—	2	2	no	—
175	C₈	C₄×Q₈×C₂₃	1	2	yes	—
176	C₈	C₂₃×#124	1	1	yes	characteristic
177	C₈	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
178	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
179	C₂²	C₂₃×#474	1	2	yes	—
180	C₂²	C₂×C₂₃×D₁₆	1	1	yes	characteristic
181	C₂²	C₂×C₂₃×SD₃₂	1	1	yes	characteristic
182	C₂²	C₂₃×#510	1	2	yes	—
183	C₄	—	2	2	no	—
184	C₄	—	8	4	no	—
185	C₄	—	8	4	no	—
186	C₄	C₂₃×#357	1	4	yes	—
187	C₄	C₂₃×#391	1	1	yes	characteristic
188	C₄	C₂₃×#419	1	1	yes	characteristic
189	C₄	C₂₃×#475	1	2	yes	—
190	C₂	C₂₃×#1988	1	1	yes	characteristic
191	C₂	C₂×C₂₃×#357	1	1	yes	characteristic
192	C₂	C₂₃×#2141	1	1	yes	characteristic
193	C₂	C₂₃×#2224	1	2	yes	—
194	C₂	C₂₃×#2225	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
59. #13275	C₂₃	direct (N × H)
100. #500	C₄×C₂₃	semidirect
129. #107	Q₈×C₂₃	semidirect
133. C₂₃	#13275	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
#13275	23

Subgroups — C23×#13275 #19160901

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13275 #19160901