Subgroups — C₂₃×#20799 #19168425

Order 5888.

1086 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#427	C₂	1	4	yes	maximal
2	C₂²×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2285	C₂	1	2	yes	maximal
4	C₂₃×#1847	C₂	1	8	yes	maximal, complemented
5	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
6	C₂×C₂₃×#117	C₂²	1	4	yes	—
7	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#427	—	2	16	no	—
9	C₂×C₂₃×#107	C₂²	1	8	yes	—
10	C₂×C₂₃×#106	C₂²	1	8	yes	—
11	C₂×C₂₃×#97	C₄	1	8	yes	complemented
12	C₂×C₂₃×#97	C₂²	1	4	yes	—
13	C₂₃×#384	C₂²	1	4	yes	—
14	C₂₃×#382	C₂²	1	4	yes	—
15	C₂₃×#380	—	2	4	no	—
16	C₄×C₄⋊C₄×C₂₃	—	2	4	no	—
17	C₂³×C₄×C₂₃	D₄	1	2	yes	—
18	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂²×C₈×C₂₃	C₂×C₄	1	8	yes	—
20	C₂²×C₈×C₂₃	D₄	1	8	yes	—
21	C₂²×C₈×C₂₃	Q₈	1	8	yes	—
22	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
23	C₂₃×#117	—	2	16	no	—
24	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
25	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
26	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
27	C₂×C₄²×C₂₃	—	2	4	no	—
28	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₂₃	D₄	1	2	yes	—
30	C₂×C₄²×C₂₃	Q₈	1	4	yes	—
31	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
32	C₂₃×#107	—	2	16	no	—
33	C₂₃×#106	—	2	16	no	—
34	C₂₃×#97	C₂×C₄	1	16	yes	—
35	C₂₃×#95	—	2	8	no	—
36	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	—	2	8	no	—
38	C₂²×C₄×C₂₃	—	4	4	no	—
39	C₂²×C₄×C₂₃	C₄⋊C₄	1	8	yes	—
40	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
42	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
43	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂×Q₈	1	4	yes	—
45	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
46	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
47	C₂×C₈×C₂₃	C₄⋊C₄	1	32	yes	complemented
48	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
49	C₂×C₈×C₂₃	C₂×D₄	1	8	yes	—
50	C₂×C₈×C₂₃	C₂×Q₈	1	8	yes	—
51	C₄⋊C₄×C₂₃	—	2	16	no	—
52	C₄⋊C₄×C₂₃	—	4	16	no	—
53	C₂²⋊C₄×C₂₃	—	4	8	no	—
54	C₄²×C₂₃	—	4	8	no	—
55	C₄²×C₂₃	C₄⋊C₄	1	8	yes	—
56	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
57	C₄²×C₂₃	C₂×Q₈	1	4	yes	—
58	#20799	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₂₃	—	2	4	no	—
60	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
61	C₂³×C₂₃	#127	1	2	yes	—
62	C₂³×C₂₃	C₄⋊Q₈	1	2	yes	—
63	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
64	C₂×C₄×C₂₃	—	2	8	no	—
65	C₂×C₄×C₂₃	—	4	8	no	—
66	C₂×C₄×C₂₃	—	4	8	no	—
67	C₂×C₄×C₂₃	—	4	16	no	—
68	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	8	yes	—
69	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	8	yes	—
70	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₂₃	#127	1	2	yes	—
72	C₂×C₄×C₂₃	C₄⋊Q₈	1	8	yes	—
73	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
74	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
75	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
76	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
77	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
78	C₈×C₂₃	C₂×C₄⋊C₄	1	16	yes	—
79	C₂×#427	C₂×C₂₃	1	4	yes	—
80	C₂²×#97	C₂×C₂₃	1	1	yes	characteristic
81	#2285	C₂×C₂₃	1	2	yes	—
82	#1847	C₂×C₂₃	1	8	yes	complemented
83	C₂²×C₂₃	—	2	4	no	—
84	C₂²×C₂₃	—	2	4	no	—
85	C₂²×C₂₃	—	2	4	no	—
86	C₂²×C₂₃	#384	1	4	yes	—
87	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
89	C₂²×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
90	C₂²×C₂₃	#575	1	4	yes	—
91	C₄×C₂₃	—	2	8	no	—
92	C₄×C₂₃	—	8	8	no	—
93	C₄×C₂₃	#384	1	4	yes	—
94	C₄×C₂₃	C₂²×C₄⋊C₄	1	2	yes	—
95	C₄×C₂₃	C₂×C₄⋊Q₈	1	2	yes	—
96	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
97	C₂×#117	C₂²×C₂₃	1	4	yes	—
98	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
99	#427	—	2	16	no	—
100	C₂×#107	C₂²×C₂₃	1	8	yes	—
101	C₂×#106	C₂²×C₂₃	1	8	yes	—
102	C₂×#97	C₄×C₂₃	1	8	yes	complemented
103	C₂×#97	C₂²×C₂₃	1	4	yes	—
104	#384	C₂²×C₂₃	1	4	yes	—
105	#382	C₂²×C₂₃	1	4	yes	—
106	#380	—	2	4	no	—
107	C₄×C₄⋊C₄	—	2	4	no	—
108	C₂×C₂₃	—	2	4	no	—
109	C₂×C₂₃	C₂×#384	1	1	yes	characteristic
110	C₂×C₂₃	#2911	1	2	yes	—
111	C₂×C₂₃	#3150	1	2	yes	—
112	C₂×C₂₃	#3162	1	2	yes	—
113	C₂³×C₄	D₄×C₂₃	1	2	yes	—
114	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
115	C₂²×C₈	C₂×C₄×C₂₃	1	8	yes	—
116	C₂²×C₈	D₄×C₂₃	1	8	yes	—
117	C₂²×C₈	Q₈×C₂₃	1	8	yes	—
118	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
119	#117	—	2	16	no	—
120	C₂×C₄⋊C₄	—	2	8	no	—
121	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
122	C₂×C₂²⋊C₄	—	2	4	no	—
123	C₂×C₄²	—	2	4	no	—
124	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
125	C₂×C₄²	D₄×C₂₃	1	2	yes	—
126	C₂×C₄²	Q₈×C₂₃	1	4	yes	—
127	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
128	#107	—	2	16	no	—
129	#106	—	2	16	no	—
130	#97	C₂×C₄×C₂₃	1	16	yes	—
131	#95	—	2	8	no	—
132	C₂₃	#20799	1	1	yes	characteristic, direct product factor, Sylow
133	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
134	C₂²×C₄	—	2	8	no	—
135	C₂²×C₄	—	4	4	no	—
136	C₂²×C₄	C₄⋊C₄×C₂₃	1	8	yes	—
137	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
138	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
139	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
141	C₂²×C₄	C₂×Q₈×C₂₃	1	4	yes	—
142	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
143	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
144	C₂×C₈	C₄⋊C₄×C₂₃	1	32	yes	complemented
145	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
146	C₂×C₈	C₂×D₄×C₂₃	1	8	yes	—
147	C₂×C₈	C₂×Q₈×C₂₃	1	8	yes	—
148	C₄⋊C₄	—	2	16	no	—
149	C₄⋊C₄	—	4	16	no	—
150	C₂²⋊C₄	—	4	8	no	—
151	C₄²	—	4	8	no	—
152	C₄²	C₄⋊C₄×C₂₃	1	8	yes	—
153	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
154	C₄²	C₂×Q₈×C₂₃	1	4	yes	—
155	C₂³	—	2	4	no	—
156	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
157	C₂³	C₂₃×#127	1	2	yes	—
158	C₂³	C₂₃×C₄⋊Q₈	1	2	yes	—
159	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	4	8	no	—
162	C₂×C₄	—	4	16	no	—
163	C₂×C₄	—	4	8	no	—
164	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	8	yes	—
165	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
166	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	8	yes	—
167	C₂×C₄	C₂₃×#127	1	2	yes	—
168	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
169	C₂×C₄	C₂₃×C₄⋊Q₈	1	8	yes	—
170	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
171	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
172	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
173	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
174	C₈	C₂×C₄⋊C₄×C₂₃	1	16	yes	—
175	C₂²	—	2	4	no	—
176	C₂²	—	2	4	no	—
177	C₂²	—	2	4	no	—
178	C₂²	C₂₃×#384	1	4	yes	—
179	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
180	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
181	C₂²	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
182	C₂²	C₂₃×#575	1	4	yes	—
183	C₄	—	2	8	no	—
184	C₄	—	8	8	no	—
185	C₄	C₂₃×#384	1	4	yes	—
186	C₄	C₂²×C₄⋊C₄×C₂₃	1	2	yes	—
187	C₄	C₂×C₂₃×C₄⋊Q₈	1	2	yes	—
188	C₂	—	2	4	no	—
189	C₂	C₂×C₂₃×#384	1	1	yes	characteristic
190	C₂	C₂₃×#2911	1	2	yes	—
191	C₂	C₂₃×#3150	1	2	yes	—
192	C₂	C₂₃×#3162	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₂²×D₄
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1847	C₂	semidirect
11. C₂×C₂₃×#97	C₄	semidirect
47. C₂×C₈×C₂₃	C₄⋊C₄	semidirect
58. #20799	C₂₃	direct (N × H)
82. #1847	C₂×C₂₃	semidirect
102. C₂×#97	C₄×C₂₃	semidirect
132. C₂₃	#20799	direct (N × H)
144. C₂×C₈	C₄⋊C₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#427	2
C₂²×C₂₃×#97	2
C₂₃×#2285	2
C₂₃×#1847	2
#20799	23

Subgroups — C23×#20799 #19168425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20799 #19168425