Subgroups — C₂₃×#12617 #19160243

Order 5888.

382 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#504	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2115	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2110	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1757	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1379	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
7	C₂₃×#504	—	2	4	no	—
8	C₂×C₂₃×M₃₂	C₄	1	2	yes	complemented
9	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
10	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₂₃	C₄	1	2	yes	complemented
12	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#413	C₄	1	1	yes	characteristic
14	C₂₃×#410	C₄	1	1	yes	characteristic
15	C₂₃×#405	C₂²	1	2	yes	—
16	C₂₃×#349	C₄	1	4	yes	complemented
17	C₂₃×#347	—	2	4	no	—
18	C₂₃×#344	C₄	1	2	yes	—
19	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂₃×#131	—	2	2	no	—
21	C₂₃×#131	—	2	2	no	—
22	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
23	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
27	C₂₃×#117	C₂×C₄	1	2	yes	—
28	C₂₃×M₃₂	—	2	2	no	—
29	C₂₃×M₃₂	—	2	4	no	—
30	C₂₃×M₃₂	C₂×C₄	1	4	yes	—
31	C₂₃×M₃₂	D₄	1	4	yes	—
32	C₂×C₁₆×C₂₃	—	2	2	no	—
33	C₂×C₁₆×C₂₃	—	2	2	no	—
34	C₂×C₁₆×C₂₃	—	2	4	no	—
35	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
38	C₂₃×#101	—	2	1	no	—
39	C₂₃×#100	—	2	1	no	—
40	C₂₃×#99	—	2	2	no	—
41	C₂₃×#97	—	2	2	no	—
42	C₄×C₈×C₂₃	—	2	2	no	—
43	#41×C₂₃	—	2	2	no	—
44	#41×C₂₃	—	2	2	no	—
45	C₂×Q₈×C₂₃	M₁₆	1	1	yes	characteristic
46	C₂×D₄×C₂₃	M₁₆	1	2	yes	—
47	C₂×D₄×C₂₃	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
49	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
50	M₁₆×C₂₃	—	2	2	no	—
51	M₁₆×C₂₃	—	2	4	no	—
52	M₁₆×C₂₃	—	2	2	no	—
53	M₁₆×C₂₃	—	2	2	no	—
54	C₂×C₈×C₂₃	—	2	2	no	—
55	C₂×C₈×C₂₃	—	2	2	no	—
56	C₂×C₈×C₂₃	—	2	2	no	—
57	C₂×C₈×C₂₃	C₄²	1	2	yes	—
58	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₂₃	#41	1	2	yes	—
62	C₄⋊C₄×C₂₃	—	2	2	no	—
63	C₂²⋊C₄×C₂₃	—	2	2	no	—
64	C₄²×C₂₃	—	2	2	no	—
65	C₁₆×C₂₃	—	2	1	no	—
66	C₁₆×C₂₃	—	2	1	no	—
67	C₁₆×C₂₃	—	2	4	no	—
68	C₁₆×C₂₃	C₂²⋊C₄	1	4	yes	—
69	#12617	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₂₃	#97	1	2	yes	—
71	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
72	Q₈×C₂₃	—	2	2	no	—
73	D₄×C₂₃	—	2	4	no	—
74	D₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	4	2	no	—
78	C₂×C₄×C₂₃	#97	1	2	yes	—
79	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₂₃	#131	1	2	yes	—
83	C₈×C₂₃	—	2	2	no	—
84	C₈×C₂₃	—	4	2	no	—
85	C₈×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₈×C₂₃	#117	1	1	yes	characteristic
87	C₈×C₂₃	C₄×D₄	1	2	yes	—
88	C₂×#504	C₂×C₂₃	1	1	yes	characteristic
89	#2115	C₂×C₂₃	1	1	yes	characteristic
90	#2110	C₂×C₂₃	1	2	yes	complemented
91	#1757	C₂×C₂₃	1	1	yes	characteristic
92	#1379	C₂×C₂₃	1	2	yes	complemented
93	C₂²×C₂₃	—	2	2	no	—
94	C₂²×C₂₃	—	4	1	no	—
95	C₂²×C₂₃	—	4	2	no	—
96	C₂²×C₂₃	C₂×#97	1	1	yes	characteristic
97	C₂²×C₂₃	#435	1	2	yes	—
98	C₄×C₂₃	—	2	2	no	—
99	C₄×C₂₃	—	8	2	no	—
100	C₄×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
101	C₄×C₂₃	#405	1	1	yes	characteristic
102	C₄×C₂₃	#435	1	2	yes	—
103	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
104	#504	—	2	4	no	—
105	C₂×M₃₂	C₄×C₂₃	1	2	yes	complemented
106	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
107	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
108	C₂²×C₁₆	C₄×C₂₃	1	2	yes	complemented
109	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
110	#413	C₄×C₂₃	1	1	yes	characteristic
111	#410	C₄×C₂₃	1	1	yes	characteristic
112	#405	C₂²×C₂₃	1	2	yes	—
113	#349	C₄×C₂₃	1	4	yes	complemented
114	#347	—	2	4	no	—
115	#344	C₄×C₂₃	1	2	yes	—
116	C₂×C₂₃	—	2	2	no	—
117	C₂×C₂₃	—	2	1	no	—
118	C₂×C₂₃	—	4	2	no	—
119	C₂×C₂₃	#1752	1	1	yes	characteristic
120	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
121	#131	—	2	2	no	—
122	#131	—	2	2	no	—
123	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
124	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
125	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
126	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
127	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
128	#117	C₂×C₄×C₂₃	1	2	yes	—
129	M₃₂	—	2	2	no	—
130	M₃₂	—	2	4	no	—
131	M₃₂	C₂×C₄×C₂₃	1	4	yes	—
132	M₃₂	D₄×C₂₃	1	4	yes	—
133	C₂×C₁₆	—	2	2	no	—
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	2	4	no	—
136	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
137	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
138	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
139	#101	—	2	1	no	—
140	#100	—	2	1	no	—
141	#99	—	2	2	no	—
142	#97	—	2	2	no	—
143	C₄×C₈	—	2	2	no	—
144	C₂₃	#12617	1	1	yes	characteristic, direct product factor, Sylow
145	#41	—	2	2	no	—
146	#41	—	2	2	no	—
147	C₂×Q₈	M₁₆×C₂₃	1	1	yes	characteristic
148	C₂×D₄	M₁₆×C₂₃	1	1	yes	characteristic
149	C₂×D₄	M₁₆×C₂₃	1	2	yes	—
150	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
151	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	2	4	no	—
154	M₁₆	—	2	2	no	—
155	M₁₆	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	C₄²×C₂₃	1	2	yes	—
160	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
161	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
162	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
163	C₂×C₈	#41×C₂₃	1	2	yes	—
164	C₄⋊C₄	—	2	2	no	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₄²	—	2	2	no	—
167	C₁₆	—	2	1	no	—
168	C₁₆	—	2	1	no	—
169	C₁₆	—	2	4	no	—
170	C₁₆	C₂²⋊C₄×C₂₃	1	4	yes	—
171	C₂³	C₂₃×#97	1	2	yes	—
172	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
173	Q₈	—	2	2	no	—
174	D₄	—	2	4	no	—
175	D₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	C₂₃×#97	1	2	yes	—
180	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
181	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
182	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
183	C₂×C₄	C₂₃×#131	1	2	yes	—
184	C₈	—	2	2	no	—
185	C₈	—	4	2	no	—
186	C₈	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
187	C₈	C₂₃×#117	1	1	yes	characteristic
188	C₈	C₄×D₄×C₂₃	1	2	yes	—
189	C₂²	—	2	2	no	—
190	C₂²	—	4	1	no	—
191	C₂²	—	4	2	no	—
192	C₂²	C₂×C₂₃×#97	1	1	yes	characteristic
193	C₂²	C₂₃×#435	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	8	2	no	—
196	C₄	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
197	C₄	C₂₃×#405	1	1	yes	characteristic
198	C₄	C₂₃×#435	1	2	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	2	1	no	—
201	C₂	—	4	2	no	—
202	C₂	C₂₃×#1752	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₂₃	#117
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂×C₂₃	#1752
prime radical	C₂×#41×C₂₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₃ ◁ C₂×C₂₃×M₃₂ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2110	C₂	semidirect
5. C₂₃×#1379	C₂	semidirect
8. C₂×C₂₃×M₃₂	C₄	semidirect
11. C₂²×C₁₆×C₂₃	C₄	semidirect
16. C₂₃×#349	C₄	semidirect
69. #12617	C₂₃	direct (N × H)
90. #2110	C₂×C₂₃	semidirect
92. #1379	C₂×C₂₃	semidirect
105. C₂×M₃₂	C₄×C₂₃	semidirect
108. C₂²×C₁₆	C₄×C₂₃	semidirect
113. #349	C₄×C₂₃	semidirect
144. C₂₃	#12617	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×#504	2
C₂₃×#2115	2
C₂₃×#2110	2
C₂₃×#1757	2
C₂₃×#1379	2
#12617	23

Subgroups — C23×#12617 #19160243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12617 #19160243