Subgroups — C₂₃×#23837 #19171463

Order 5888.

778 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×C₄⋊Q₈	C₂	1	2	yes	maximal
2	C₂₃×#2084	C₂	1	4	yes	maximal
3	C₂₃×#2058	C₂	1	4	yes	maximal
4	C₄×C₂₃×#106	C₂	1	4	yes	maximal
5	C₂₃×#1744	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×C₄×Q₈×C₂₃	—	2	2	no	—
8	C₂×C₂₃×#106	C₂²	1	8	yes	—
9	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
10	C₂₃×#391	—	2	2	no	—
11	C₂₃×#389	—	2	2	no	—
12	C₂₃×#389	C₂²	1	4	yes	—
13	C₂₃×#384	C₂²	1	2	yes	—
14	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
15	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
16	C₄³×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#340	C₂²	1	4	yes	—
18	C₂₃×#340	C₂²	1	4	yes	—
19	C₂₃×#336	C₂²	1	4	yes	—
20	C₂²×Q₈×C₂₃	—	2	2	no	—
21	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
22	C₂₃×C₄⋊Q₈	—	2	8	no	—
23	C₄×Q₈×C₂₃	—	4	8	no	—
24	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
26	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
28	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
29	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
30	C₂×C₄²×C₂₃	—	2	4	no	—
31	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₂₃	D₄	1	2	yes	—
34	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
35	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
36	C₂₃×#106	—	2	16	no	—
37	C₄×C₈×C₂₃	Q₈	1	8	yes	complemented
38	C₂₃×#95	—	2	4	no	—
39	C₂₃×#95	—	2	4	no	—
40	C₂×Q₈×C₂₃	—	2	8	no	—
41	C₂×Q₈×C₂₃	—	4	4	no	—
42	C₂×Q₈×C₂₃	—	4	4	no	—
43	C₂²×C₄×C₂₃	—	2	4	no	—
44	C₂²×C₄×C₂₃	—	2	4	no	—
45	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
50	C₂×C₈×C₂₃	—	2	8	no	—
51	C₂×C₈×C₂₃	C₂×Q₈	1	8	yes	—
52	C₄⋊C₄×C₂₃	—	2	8	no	—
53	C₄⋊C₄×C₂₃	—	2	4	no	—
54	C₄⋊C₄×C₂₃	—	2	8	no	—
55	C₄⋊C₄×C₂₃	—	4	4	no	—
56	C₄⋊C₄×C₂₃	—	4	8	no	—
57	C₄⋊C₄×C₂₃	—	4	4	no	—
58	C₄⋊C₄×C₂₃	#41	1	8	yes	—
59	C₄²×C₂₃	—	2	4	no	—
60	C₄²×C₂₃	—	4	1	no	—
61	C₄²×C₂₃	—	4	1	no	—
62	C₄²×C₂₃	—	4	8	no	—
63	C₄²×C₂₃	SD₁₆	1	8	yes	—
64	C₄²×C₂₃	C₂×Q₈	1	4	yes	—
65	#23837	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
67	Q₈×C₂₃	—	4	16	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	2	2	no	—
71	C₂×C₄×C₂₃	—	2	1	no	—
72	C₂×C₄×C₂₃	—	2	1	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	8	no	—
75	C₂×C₄×C₂₃	—	4	4	no	—
76	C₂×C₄×C₂₃	—	4	4	no	—
77	C₂×C₄×C₂₃	—	4	4	no	—
78	C₂×C₄×C₂₃	—	4	4	no	—
79	C₂×C₄×C₂₃	—	4	4	no	—
80	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
81	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
82	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
83	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
84	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
85	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
86	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
87	C₈×C₂₃	—	2	8	no	—
88	C₄×C₄⋊Q₈	C₂×C₂₃	1	2	yes	—
89	#2084	C₂×C₂₃	1	4	yes	—
90	#2058	C₂×C₂₃	1	4	yes	—
91	C₄×#106	C₂×C₂₃	1	4	yes	—
92	#1744	C₂×C₂₃	1	1	yes	characteristic
93	C₂²×C₂₃	#525	1	2	yes	—
94	C₂²×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
95	C₂²×C₂₃	C₂²×SD₁₆	1	2	yes	—
96	C₂²×C₂₃	#575	1	1	yes	characteristic
97	C₂²×C₂₃	#575	1	1	yes	characteristic
98	C₄×C₂₃	—	2	4	no	—
99	C₄×C₂₃	—	2	4	no	—
100	C₄×C₂₃	—	2	2	no	—
101	C₄×C₂₃	—	4	8	no	—
102	C₄×C₂₃	—	8	4	no	—
103	C₄×C₂₃	#498	1	4	yes	—
104	C₄×C₂₃	#533	1	4	yes	—
105	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
106	C₂×C₄×Q₈	—	2	2	no	—
107	C₂×#106	C₂²×C₂₃	1	8	yes	—
108	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
109	#391	—	2	2	no	—
110	#389	—	2	2	no	—
111	#389	C₂²×C₂₃	1	4	yes	—
112	#384	C₂²×C₂₃	1	2	yes	—
113	C₄×C₄⋊C₄	—	2	2	no	—
114	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
115	C₄³	C₂²×C₂₃	1	1	yes	characteristic
116	#340	C₂²×C₂₃	1	4	yes	—
117	#340	C₂²×C₂₃	1	4	yes	—
118	#336	C₂²×C₂₃	1	4	yes	—
119	C₂×C₂₃	#2399	1	1	yes	characteristic
120	C₂×C₂₃	#3101	1	2	yes	—
121	C₂×C₂₃	#3103	1	2	yes	—
122	C₂×C₂₃	C₂×#498	1	1	yes	characteristic
123	C₂×C₂₃	#3162	1	1	yes	characteristic
124	C₂²×Q₈	—	2	2	no	—
125	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
126	C₄⋊Q₈	—	2	8	no	—
127	C₄×Q₈	—	4	8	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	4	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
133	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
134	C₂×C₄²	—	2	4	no	—
135	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
136	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
137	C₂×C₄²	D₄×C₂₃	1	2	yes	—
138	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
139	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
140	#106	—	2	16	no	—
141	C₄×C₈	Q₈×C₂₃	1	8	yes	complemented
142	#95	—	2	4	no	—
143	#95	—	2	4	no	—
144	C₂₃	#23837	1	1	yes	characteristic, direct product factor, Sylow
145	C₂×Q₈	—	2	8	no	—
146	C₂×Q₈	—	4	4	no	—
147	C₂×Q₈	—	4	4	no	—
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
152	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
153	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
154	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
155	C₂×C₈	—	2	8	no	—
156	C₂×C₈	C₂×Q₈×C₂₃	1	8	yes	—
157	C₄⋊C₄	—	2	8	no	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	2	8	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄⋊C₄	—	4	8	no	—
162	C₄⋊C₄	—	4	4	no	—
163	C₄⋊C₄	#41×C₂₃	1	8	yes	—
164	C₄²	—	2	4	no	—
165	C₄²	—	4	1	no	—
166	C₄²	—	4	1	no	—
167	C₄²	—	4	8	no	—
168	C₄²	SD₁₆×C₂₃	1	8	yes	—
169	C₄²	C₂×Q₈×C₂₃	1	4	yes	—
170	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
171	Q₈	—	4	16	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
185	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
186	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
187	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
188	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
189	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
190	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
191	C₈	—	2	8	no	—
192	C₂²	C₂₃×#525	1	2	yes	—
193	C₂²	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
194	C₂²	C₂²×SD₁₆×C₂₃	1	2	yes	—
195	C₂²	C₂₃×#575	1	1	yes	characteristic
196	C₂²	C₂₃×#575	1	1	yes	characteristic
197	C₄	—	2	4	no	—
198	C₄	—	2	4	no	—
199	C₄	—	2	2	no	—
200	C₄	—	4	8	no	—
201	C₄	—	8	4	no	—
202	C₄	C₂₃×#498	1	4	yes	—
203	C₄	C₂₃×#533	1	4	yes	—
204	C₂	C₂₃×#2399	1	1	yes	characteristic
205	C₂	C₂₃×#3101	1	2	yes	—
206	C₂	C₂₃×#3103	1	2	yes	—
207	C₂	C₂×C₂₃×#498	1	1	yes	characteristic
208	C₂	C₂₃×#3162	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle, prime radical	C₂³×C₂₃	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
37. C₄×C₈×C₂₃	Q₈	semidirect
65. #23837	C₂₃	direct (N × H)
141. C₄×C₈	Q₈×C₂₃	semidirect
144. C₂₃	#23837	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₂₃×C₄⋊Q₈	2
C₂₃×#2084	2
C₂₃×#2058	2
C₄×C₂₃×#106	2
C₂₃×#1744	2
#23837	23

Subgroups — C23×#23837 #19171463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#23837 #19171463