Subgroups — C₂₃×#12658 #19160284

Order 5888.

350 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2150	C₂	1	2	yes	maximal
2	C₂×C₂₃×#347	C₂	1	2	yes	maximal
3	C₄×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1375	C₂	1	2	yes	maximal
5	C₂×C₂₃×M₃₂	C₂²	1	4	yes	—
6	C₂×C₂₃×#97	C₄	1	1	yes	characteristic
7	C₂×C₂₃×#97	C₄	1	1	yes	characteristic
8	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₄³×C₂₃	C₄	1	1	yes	characteristic
13	C₂₃×C₄⋊C₁₆	C₄	1	8	yes	complemented
14	C₂₃×#347	—	2	4	no	—
15	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₂₃	—	2	2	no	—
20	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂₃×M₃₂	—	2	4	no	—
24	C₂₃×M₃₂	—	2	4	no	—
25	C₂×C₁₆×C₂₃	C₂×C₄	1	8	yes	—
26	C₂₃×#97	—	2	2	no	—
27	C₂₃×#97	—	2	2	no	—
28	C₂₃×#97	—	2	2	no	—
29	C₂₃×#97	D₄	1	2	yes	—
30	C₂₃×#97	Q₈	1	2	yes	—
31	C₄×C₈×C₂₃	—	2	1	no	—
32	C₄×C₈×C₂₃	—	2	1	no	—
33	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₂₃	—	2	1	no	—
35	C₂²×C₄×C₂₃	—	2	1	no	—
36	C₂²×C₄×C₂₃	M₁₆	1	2	yes	—
37	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₂₃	—	2	1	no	—
39	C₂×C₈×C₂₃	—	2	1	no	—
40	C₂×C₈×C₂₃	—	2	1	no	—
41	C₂×C₈×C₂₃	—	2	1	no	—
42	C₂×C₈×C₂₃	—	2	1	no	—
43	C₂×C₈×C₂₃	—	2	1	no	—
44	C₂×C₈×C₂₃	C₄²	1	2	yes	—
45	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
47	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₂₃	#41	1	2	yes	—
51	C₄²×C₂₃	—	2	1	no	—
52	C₄²×C₂₃	—	2	1	no	—
53	C₄²×C₂₃	—	4	4	no	—
54	C₄²×C₂₃	C₄²	1	2	yes	—
55	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
56	C₄²×C₂₃	M₁₆	1	2	yes	—
57	C₄²×C₂₃	M₁₆	1	1	yes	characteristic
58	C₄²×C₂₃	M₁₆	1	1	yes	characteristic
59	C₁₆×C₂₃	—	2	8	no	—
60	#12658	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
62	C₂×C₄×C₂₃	—	2	4	no	—
63	C₂×C₄×C₂₃	—	2	2	no	—
64	C₂×C₄×C₂₃	—	2	2	no	—
65	C₂×C₄×C₂₃	—	4	2	no	—
66	C₂×C₄×C₂₃	—	4	2	no	—
67	C₂×C₄×C₂₃	#97	1	2	yes	—
68	C₂×C₄×C₂₃	#97	1	2	yes	—
69	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
72	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
75	C₈×C₂₃	—	2	2	no	—
76	C₈×C₂₃	—	2	2	no	—
77	C₈×C₂₃	—	2	2	no	—
78	C₈×C₂₃	C₄×D₄	1	2	yes	—
79	C₈×C₂₃	C₄×Q₈	1	2	yes	—
80	#2150	C₂×C₂₃	1	2	yes	—
81	C₂×#347	C₂×C₂₃	1	2	yes	—
82	C₄×#97	C₂×C₂₃	1	1	yes	characteristic
83	#1375	C₂×C₂₃	1	2	yes	—
84	C₂²×C₂₃	—	2	2	no	—
85	C₂²×C₂₃	C₂×#97	1	1	yes	characteristic
86	C₂²×C₂₃	#423	1	1	yes	characteristic
87	C₂²×C₂₃	#432	1	1	yes	characteristic
88	C₄×C₂₃	—	2	2	no	—
89	C₄×C₂₃	—	4	2	no	—
90	C₄×C₂₃	—	4	2	no	—
91	C₄×C₂₃	#347	1	4	yes	complemented
92	C₄×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
93	C₄×C₂₃	C₄×M₁₆	1	1	yes	characteristic
94	C₄×C₂₃	#423	1	1	yes	characteristic
95	C₄×C₂₃	#432	1	1	yes	characteristic
96	C₂×M₃₂	C₂²×C₂₃	1	4	yes	—
97	C₂×#97	C₄×C₂₃	1	1	yes	characteristic
98	C₂×#97	C₄×C₂₃	1	1	yes	characteristic
99	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
100	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
101	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
103	C₄³	C₄×C₂₃	1	1	yes	characteristic
104	C₄⋊C₁₆	C₄×C₂₃	1	8	yes	complemented
105	#347	—	2	4	no	—
106	C₂×C₂₃	—	2	1	no	—
107	C₂×C₂₃	—	2	1	no	—
108	C₂×C₂₃	#1767	1	1	yes	characteristic
109	C₂×C₂₃	C₂×#347	1	2	yes	—
110	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
111	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
112	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
113	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
114	C₂×C₄²	—	2	2	no	—
115	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
116	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
117	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
118	M₃₂	—	2	4	no	—
119	M₃₂	—	2	4	no	—
120	C₂×C₁₆	C₂×C₄×C₂₃	1	8	yes	—
121	#97	—	2	2	no	—
122	#97	—	2	2	no	—
123	#97	—	2	2	no	—
124	#97	D₄×C₂₃	1	2	yes	—
125	#97	Q₈×C₂₃	1	2	yes	—
126	C₄×C₈	—	2	1	no	—
127	C₄×C₈	—	2	1	no	—
128	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
129	C₂₃	#12658	1	1	yes	characteristic, direct product factor, Sylow
130	C₂²×C₄	—	2	1	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	M₁₆×C₂₃	1	2	yes	—
133	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	C₄²×C₂₃	1	2	yes	—
141	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
142	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
143	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
144	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
145	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
146	C₂×C₈	#41×C₂₃	1	2	yes	—
147	C₄²	—	2	1	no	—
148	C₄²	—	2	1	no	—
149	C₄²	—	4	4	no	—
150	C₄²	C₄²×C₂₃	1	2	yes	—
151	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
152	C₄²	M₁₆×C₂₃	1	2	yes	—
153	C₄²	M₁₆×C₂₃	1	1	yes	characteristic
154	C₄²	M₁₆×C₂₃	1	1	yes	characteristic
155	C₁₆	—	2	8	no	—
156	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	C₂₃×#97	1	2	yes	—
163	C₂×C₄	C₂₃×#97	1	2	yes	—
164	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
165	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
166	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
167	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
168	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
169	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	2	2	no	—
172	C₈	—	2	2	no	—
173	C₈	C₄×D₄×C₂₃	1	2	yes	—
174	C₈	C₄×Q₈×C₂₃	1	2	yes	—
175	C₂²	—	2	2	no	—
176	C₂²	C₂×C₂₃×#97	1	1	yes	characteristic
177	C₂²	C₂₃×#423	1	1	yes	characteristic
178	C₂²	C₂₃×#432	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	4	2	no	—
181	C₄	—	4	2	no	—
182	C₄	C₂₃×#347	1	4	yes	complemented
183	C₄	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
184	C₄	C₄×M₁₆×C₂₃	1	1	yes	characteristic
185	C₄	C₂₃×#423	1	1	yes	characteristic
186	C₄	C₂₃×#432	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	C₂₃×#1767	1	1	yes	characteristic
190	C₂	C₂×C₂₃×#347	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×#97
prime radical	C₂³×C₂₃	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂₃×C₄⋊C₁₆	C₄	semidirect
60. #12658	C₂₃	direct (N × H)
91. C₄×C₂₃	#347	semidirect
104. C₄⋊C₁₆	C₄×C₂₃	semidirect
129. C₂₃	#12658	direct (N × H)
182. C₄	C₂₃×#347	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2150	2
C₂×C₂₃×#347	2
C₄×C₂₃×#97	2
C₂₃×#1375	2
#12658	23

Subgroups — C23×#12658 #19160284

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12658 #19160284