Subgroups — C₂₃×#12060 #19159686

Order 5888.

204 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2183	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₈×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1367	C₂	1	1	yes	maximal, characteristic
5	C₂₃×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂₃×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1312	C₂	1	1	yes	maximal, characteristic
8	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₄×Q₁₆×C₂₃	C₄	1	1	yes	characteristic
11	C₂₃×C₄⋊C₁₆	—	2	1	no	—
12	C₂₃×C₄⋊C₁₆	—	2	1	no	—
13	C₂₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂₃×#346	—	2	1	no	—
16	C₄×C₁₆×C₂₃	—	2	1	no	—
17	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#335	C₄	1	1	yes	characteristic
20	C₂₃×Q₈⋊C₈	C₄	1	1	yes	characteristic
21	C₂₃×Q₈⋊C₈	C₄	1	1	yes	characteristic
22	C₈²×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×Q₁₆×C₂₃	C₈	1	1	yes	characteristic
24	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₂₃	—	2	1	no	—
27	C₂×C₁₆×C₂₃	—	2	1	no	—
28	C₂×C₁₆×C₂₃	—	2	1	no	—
29	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
31	C₂₃×#107	C₈	1	1	yes	characteristic
32	C₂₃×C₄⋊C₈	—	2	1	no	—
33	C₂₃×C₄⋊C₈	—	2	1	no	—
34	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₂₃×#103	C₈	1	1	yes	characteristic
37	C₂₃×#103	C₈	1	1	yes	characteristic
38	C₄×C₈×C₂₃	—	2	1	no	—
39	C₄×C₈×C₂₃	—	2	1	no	—
40	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₂₃	C₂×C₈	1	1	yes	characteristic
44	C₂×Q₈×C₂₃	C₂×C₈	1	1	yes	characteristic
45	Q₁₆×C₂₃	—	2	2	no	—
46	C₂×C₈×C₂₃	—	2	1	no	—
47	C₂×C₈×C₂₃	—	2	1	no	—
48	C₂×C₈×C₂₃	—	2	1	no	—
49	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
50	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
51	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
53	C₄⋊C₄×C₂₃	—	2	1	no	—
54	C₄⋊C₄×C₂₃	—	2	1	no	—
55	C₄⋊C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
56	C₄⋊C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
57	C₄²×C₂₃	—	2	1	no	—
58	C₄²×C₂₃	—	2	1	no	—
59	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
60	C₁₆×C₂₃	—	2	1	no	—
61	C₁₆×C₂₃	—	2	1	no	—
62	C₁₆×C₂₃	—	4	1	no	—
63	C₁₆×C₂₃	—	4	1	no	—
64	C₁₆×C₂₃	—	4	1	no	—
65	#12060	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	Q₈×C₂₃	—	2	1	no	—
67	Q₈×C₂₃	—	2	1	no	—
68	Q₈×C₂₃	—	2	1	no	—
69	Q₈×C₂₃	M₃₂	1	2	yes	—
70	C₂×C₄×C₂₃	—	2	1	no	—
71	C₂×C₄×C₂₃	—	2	1	no	—
72	C₂×C₄×C₂₃	—	2	1	no	—
73	C₂×C₄×C₂₃	—	2	1	no	—
74	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₂₃	#131	1	1	yes	characteristic
77	C₈×C₂₃	—	2	2	no	—
78	C₈×C₂₃	—	2	1	no	—
79	C₈×C₂₃	—	2	1	no	—
80	C₈×C₂₃	—	2	1	no	—
81	C₈×C₂₃	—	4	1	no	—
82	C₈×C₂₃	—	4	1	no	—
83	C₈×C₂₃	#137	1	1	yes	characteristic
84	C₈×C₂₃	#137	1	1	yes	characteristic
85	#2183	C₂×C₂₃	1	1	yes	characteristic
86	Q₈×C₁₆	C₂×C₂₃	1	1	yes	characteristic
87	C₈×Q₁₆	C₂×C₂₃	1	1	yes	characteristic
88	#1367	C₂×C₂₃	1	1	yes	characteristic
89	Q₈⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic
90	Q₈⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic
91	#1312	C₂×C₂₃	1	1	yes	characteristic
92	C₂²×C₂₃	C₈×D₄	1	1	yes	characteristic
93	C₄×C₂₃	—	2	2	no	—
94	C₄×C₂₃	—	2	1	no	—
95	C₄×C₂₃	—	4	1	no	—
96	C₄×C₂₃	—	4	1	no	—
97	C₄×C₂₃	—	4	1	no	—
98	C₄×C₂₃	#441	1	1	yes	characteristic
99	C₄×C₂₃	#444	1	1	yes	characteristic
100	C₄×C₂₃	C₂×M₃₂	1	1	yes	characteristic
101	C₄×C₂₃	#504	1	1	yes	characteristic
102	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
103	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
104	C₄×Q₁₆	C₄×C₂₃	1	1	yes	characteristic
105	C₄⋊C₁₆	—	2	1	no	—
106	C₄⋊C₁₆	—	2	1	no	—
107	C₄⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
108	C₄⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
109	#346	—	2	1	no	—
110	C₄×C₁₆	—	2	1	no	—
111	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
112	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
113	#335	C₄×C₂₃	1	1	yes	characteristic
114	Q₈⋊C₈	C₄×C₂₃	1	1	yes	characteristic
115	Q₈⋊C₈	C₄×C₂₃	1	1	yes	characteristic
116	C₈²	C₂²×C₂₃	1	1	yes	characteristic
117	C₂×C₂₃	#1579	1	1	yes	characteristic
118	C₂×C₂₃	#2168	1	1	yes	characteristic
119	C₂×C₂₃	#2170	1	1	yes	characteristic
120	C₂×Q₁₆	C₈×C₂₃	1	1	yes	characteristic
121	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
122	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
123	C₂×C₁₆	—	2	1	no	—
124	C₂×C₁₆	—	2	1	no	—
125	C₂×C₁₆	—	2	1	no	—
126	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
127	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
128	#107	C₈×C₂₃	1	1	yes	characteristic
129	C₄⋊C₈	—	2	1	no	—
130	C₄⋊C₈	—	2	1	no	—
131	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
132	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
133	#103	C₈×C₂₃	1	1	yes	characteristic
134	#103	C₈×C₂₃	1	1	yes	characteristic
135	C₄×C₈	—	2	1	no	—
136	C₄×C₈	—	2	1	no	—
137	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
138	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
139	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
140	C₂₃	#12060	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×Q₈	C₂×C₈×C₂₃	1	1	yes	characteristic
142	C₂×Q₈	C₂×C₈×C₂₃	1	1	yes	characteristic
143	Q₁₆	—	2	2	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
148	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
149	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
150	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
151	C₄⋊C₄	—	2	1	no	—
152	C₄⋊C₄	—	2	1	no	—
153	C₄⋊C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
154	C₄⋊C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
155	C₄²	—	2	1	no	—
156	C₄²	—	2	1	no	—
157	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
158	C₁₆	—	2	1	no	—
159	C₁₆	—	2	1	no	—
160	C₁₆	—	4	1	no	—
161	C₁₆	—	4	1	no	—
162	C₁₆	—	4	1	no	—
163	Q₈	—	2	1	no	—
164	Q₈	—	2	1	no	—
165	Q₈	—	2	1	no	—
166	Q₈	C₂₃×M₃₂	1	2	yes	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
172	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
173	C₂×C₄	C₂₃×#131	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	2	1	no	—
178	C₈	—	4	1	no	—
179	C₈	—	4	1	no	—
180	C₈	C₂₃×#137	1	1	yes	characteristic
181	C₈	C₂₃×#137	1	1	yes	characteristic
182	C₂²	C₈×D₄×C₂₃	1	1	yes	characteristic
183	C₄	—	2	2	no	—
184	C₄	—	2	1	no	—
185	C₄	—	4	1	no	—
186	C₄	—	4	1	no	—
187	C₄	—	4	1	no	—
188	C₄	C₂₃×#441	1	1	yes	characteristic
189	C₄	C₂₃×#444	1	1	yes	characteristic
190	C₄	C₂×C₂₃×M₃₂	1	1	yes	characteristic
191	C₄	C₂₃×#504	1	1	yes	characteristic
192	C₂	C₂₃×#1579	1	1	yes	characteristic
193	C₂	C₂₃×#2168	1	1	yes	characteristic
194	C₂	C₂₃×#2170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle, prime radical	C₂²×C₂₃	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ C₄×C₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
65. #12060	C₂₃	direct (N × H)
140. C₂₃	#12060	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#2183	2
Q₈×C₁₆×C₂₃	2
C₈×Q₁₆×C₂₃	2
C₂₃×#1367	2
C₂₃×Q₈⋊C₁₆	2
C₂₃×#1312	2
#12060	23

Subgroups — C23×#12060 #19159686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12060 #19159686