Subgroups — C₂₃×#9831 #19157457

Order 5888.

494 conjugacy classes of subgroups.

The subgroup lattice has 216 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	C₂₃×#1990	C₂	1	2	yes	maximal
3	C₂₃×#1956	C₂	1	2	yes	maximal
4	C₂₃×#1834	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₃×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂₃×#500	—	2	2	no	—
8	C₂₃×#500	C₄	1	4	yes	complemented
9	C₂×C₂₃×#107	—	2	2	no	—
10	C₂×C₂₃×#103	—	2	4	no	—
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#384	C₂²	1	2	yes	—
15	C₂₃×#340	—	2	4	no	—
16	C₂₃×#340	—	2	2	no	—
17	C₂₃×#322	—	2	2	no	—
18	C₂²×Q₈×C₂₃	—	2	2	no	—
19	C₂²×C₈×C₂₃	D₄	1	2	yes	—
20	C₂²×C₈×C₂₃	D₄	1	2	yes	—
21	C₂²×C₈×C₂₃	D₄	1	2	yes	—
22	C₂₃×C₄⋊Q₈	—	2	2	no	—
23	C₂₃×C₄⋊Q₈	—	2	2	no	—
24	C₂₃×C₄⋊Q₈	—	2	2	no	—
25	C₂₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
31	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
32	C₂₃×#107	—	2	8	no	—
33	C₂₃×#103	—	4	8	no	—
34	C₄×C₈×C₂₃	—	2	1	no	—
35	C₄×C₈×C₂₃	—	2	1	no	—
36	C₄×C₈×C₂₃	—	2	1	no	—
37	C₄×C₈×C₂₃	—	2	1	no	—
38	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₂₃	D₄	1	2	yes	—
40	C₂×Q₈×C₂₃	—	2	4	no	—
41	C₂×Q₈×C₂₃	—	4	2	no	—
42	C₂×Q₈×C₂₃	—	4	4	no	—
43	C₂×Q₈×C₂₃	—	4	4	no	—
44	C₂²×C₄×C₂₃	—	4	2	no	—
45	C₂²×C₄×C₂₃	—	4	2	no	—
46	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₂₃	—	2	2	no	—
50	C₂×C₈×C₂₃	—	2	2	no	—
51	C₂×C₈×C₂₃	—	2	4	no	—
52	C₂×C₈×C₂₃	—	2	2	no	—
53	C₂×C₈×C₂₃	—	2	4	no	—
54	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
55	C₂×C₈×C₂₃	D₈	1	4	yes	—
56	C₄⋊C₄×C₂₃	—	2	4	no	—
57	C₄⋊C₄×C₂₃	—	2	4	no	—
58	C₄⋊C₄×C₂₃	—	4	2	no	—
59	C₄⋊C₄×C₂₃	—	4	2	no	—
60	C₄⋊C₄×C₂₃	—	4	4	no	—
61	C₄⋊C₄×C₂₃	—	4	4	no	—
62	C₄⋊C₄×C₂₃	—	4	4	no	—
63	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₂₃	#41	1	1	yes	characteristic
66	C₄²×C₂₃	#41	1	1	yes	characteristic
67	#9831	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂³×C₂₃	#127	1	1	yes	characteristic
69	Q₈×C₂₃	—	4	8	no	—
70	Q₈×C₂₃	—	8	4	no	—
71	C₂×C₄×C₂₃	—	4	4	no	—
72	C₂×C₄×C₂₃	—	8	2	no	—
73	C₂×C₄×C₂₃	—	8	2	no	—
74	C₂×C₄×C₂₃	—	8	2	no	—
75	C₂×C₄×C₂₃	—	8	2	no	—
76	C₂×C₄×C₂₃	—	8	2	no	—
77	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
81	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
82	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
83	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₂₃	#135	1	2	yes	—
85	C₂×C₄×C₂₃	#137	1	4	yes	—
86	C₂×C₄×C₂₃	#137	1	4	yes	—
87	C₈×C₂₃	—	2	4	no	—
88	C₈×C₂₃	—	2	4	no	—
89	C₈×C₂₃	—	4	2	no	—
90	C₂×#500	C₂×C₂₃	1	1	yes	characteristic
91	#1990	C₂×C₂₃	1	2	yes	—
92	#1956	C₂×C₂₃	1	2	yes	—
93	#1834	C₂×C₂₃	1	1	yes	characteristic
94	C₂×#322	C₂×C₂₃	1	1	yes	characteristic
95	C₂²×C₂₃	#390	1	1	yes	characteristic
96	C₂²×C₂₃	#493	1	1	yes	characteristic
97	C₂²×C₂₃	#495	1	1	yes	characteristic
98	C₂²×C₂₃	#497	1	2	yes	—
99	C₂²×C₂₃	#497	1	2	yes	—
100	C₄×C₂₃	—	8	4	no	—
101	C₄×C₂₃	—	16	2	no	—
102	C₄×C₂₃	#419	1	2	yes	—
103	C₄×C₂₃	C₄×D₈	1	2	yes	—
104	C₄×C₂₃	#441	1	2	yes	—
105	C₄×C₂₃	#466	1	2	yes	—
106	C₄×C₂₃	#470	1	2	yes	—
107	C₄×C₂₃	#491	1	2	yes	—
108	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
109	#500	—	2	2	no	—
110	#500	C₄×C₂₃	1	4	yes	complemented
111	C₂×#107	—	2	2	no	—
112	C₂×#103	—	2	4	no	—
113	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
116	#384	C₂²×C₂₃	1	2	yes	—
117	#340	—	2	2	no	—
118	#340	—	2	4	no	—
119	#322	—	2	2	no	—
120	C₂×C₂₃	#1718	1	2	yes	—
121	C₂×C₂₃	#1722	1	2	yes	—
122	C₂×C₂₃	#1960	1	1	yes	characteristic
123	C₂×C₂₃	#1970	1	1	yes	characteristic
124	C₂×C₂₃	#1971	1	1	yes	characteristic
125	C₂²×Q₈	—	2	2	no	—
126	C₂²×C₈	D₄×C₂₃	1	2	yes	—
127	C₂²×C₈	D₄×C₂₃	1	2	yes	—
128	C₂²×C₈	D₄×C₂₃	1	2	yes	—
129	C₄⋊Q₈	—	2	2	no	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
139	#107	—	2	8	no	—
140	#103	—	4	8	no	—
141	C₄×C₈	—	2	1	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	—	2	1	no	—
145	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
146	C₄×C₈	D₄×C₂₃	1	2	yes	—
147	C₂₃	#9831	1	1	yes	characteristic, direct product factor, Sylow
148	C₂×Q₈	—	2	4	no	—
149	C₂×Q₈	—	4	2	no	—
150	C₂×Q₈	—	4	4	no	—
151	C₂×Q₈	—	4	4	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	—	4	2	no	—
154	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
155	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
156	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
163	C₂×C₈	D₈×C₂₃	1	4	yes	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	4	4	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₄⋊C₄	—	4	4	no	—
171	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
172	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
173	C₄²	#41×C₂₃	1	1	yes	characteristic
174	C₄²	#41×C₂₃	1	1	yes	characteristic
175	C₂³	C₂₃×#127	1	1	yes	characteristic
176	Q₈	—	4	8	no	—
177	Q₈	—	8	4	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
185	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
186	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
187	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
188	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
189	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
190	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
191	C₂×C₄	C₂₃×#135	1	2	yes	—
192	C₂×C₄	C₂₃×#137	1	4	yes	—
193	C₂×C₄	C₂₃×#137	1	4	yes	—
194	C₈	—	2	4	no	—
195	C₈	—	2	4	no	—
196	C₈	—	4	2	no	—
197	C₂²	C₂₃×#390	1	1	yes	characteristic
198	C₂²	C₂₃×#493	1	1	yes	characteristic
199	C₂²	C₂₃×#495	1	1	yes	characteristic
200	C₂²	C₂₃×#497	1	2	yes	—
201	C₂²	C₂₃×#497	1	2	yes	—
202	C₄	—	8	4	no	—
203	C₄	—	16	2	no	—
204	C₄	C₂₃×#419	1	2	yes	—
205	C₄	C₄×D₈×C₂₃	1	2	yes	—
206	C₄	C₂₃×#441	1	2	yes	—
207	C₄	C₂₃×#466	1	2	yes	—
208	C₄	C₂₃×#470	1	2	yes	—
209	C₄	C₂₃×#491	1	2	yes	—
210	C₂	C₂₃×#1718	1	2	yes	—
211	C₂	C₂₃×#1722	1	2	yes	—
212	C₂	C₂₃×#1960	1	1	yes	characteristic
213	C₂	C₂₃×#1970	1	1	yes	characteristic
214	C₂	C₂₃×#1971	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	#127
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₂₃×#500	C₄	semidirect
67. #9831	C₂₃	direct (N × H)
110. #500	C₄×C₂₃	semidirect
147. C₂₃	#9831	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
C₂₃×#1990	2
C₂₃×#1956	2
C₂₃×#1834	2
C₂×C₂₃×#322	2
#9831	23

Subgroups — C23×#9831 #19157457

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9831 #19157457