Subgroups — C₂₃×#13270 #19160896

Order 5888.

316 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	C₂₃×#2960	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2153	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2110	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1378	C₂	1	2	yes	maximal
5	C₂₃×#1376	C₂	1	2	yes	maximal
6	C₂₃×#533	C₄	1	1	yes	characteristic
7	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
8	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
9	C₂₃×#446	—	2	2	no	—
10	C₈×Q₈×C₂₃	—	2	2	no	—
11	C₂₃×#424	C₄	1	1	yes	characteristic
12	C₂₃×#424	C₄	1	1	yes	characteristic
13	C₂₃×#423	C₄	1	1	yes	characteristic
14	C₂₃×#405	C₂²	1	1	yes	characteristic
15	C₂₃×#405	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#364	—	2	2	no	—
18	C₂₃×#347	—	2	2	no	—
19	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
23	C₂₃×C₄⋊Q₈	—	2	1	no	—
24	C₂₃×#125	—	2	1	no	—
25	C₂₃×C₂²⋊Q₈	—	2	2	no	—
26	C₄×Q₈×C₂₃	—	2	2	no	—
27	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
28	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂₃×M₃₂	—	2	2	no	—
31	C₂₃×M₃₂	—	2	2	no	—
32	C₂₃×M₃₂	—	2	2	no	—
33	C₂₃×M₃₂	Q₈	1	4	yes	complemented
34	C₂×C₁₆×C₂₃	—	2	2	no	—
35	C₂×C₁₆×C₂₃	—	2	2	no	—
36	C₂₃×C₄⋊C₈	—	2	2	no	—
37	C₂₃×C₄⋊C₈	—	2	2	no	—
38	C₂₃×C₄⋊C₈	—	2	2	no	—
39	C₂₃×#97	—	2	1	no	—
40	C₂₃×#97	D₄	1	2	yes	—
41	C₄×C₈×C₂₃	—	2	1	no	—
42	C₄×C₈×C₂₃	—	2	1	no	—
43	C₄×C₈×C₂₃	—	2	1	no	—
44	C₄×C₈×C₂₃	D₄	1	2	yes	—
45	C₂×Q₈×C₂₃	—	2	2	no	—
46	C₂²×C₄×C₂₃	—	2	1	no	—
47	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
48	M₁₆×C₂₃	—	2	2	no	—
49	M₁₆×C₂₃	—	2	2	no	—
50	C₂×C₈×C₂₃	—	2	1	no	—
51	C₂×C₈×C₂₃	—	2	1	no	—
52	C₂×C₈×C₂₃	—	2	2	no	—
53	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₂₃	#41	1	2	yes	—
58	C₄⋊C₄×C₂₃	—	2	4	no	—
59	C₄⋊C₄×C₂₃	—	2	2	no	—
60	C₄⋊C₄×C₂₃	—	2	1	no	—
61	C₄⋊C₄×C₂₃	—	2	1	no	—
62	C₂²⋊C₄×C₂₃	—	2	1	no	—
63	C₂²⋊C₄×C₂₃	M₁₆	1	2	yes	—
64	C₄²×C₂₃	—	2	1	no	—
65	C₄²×C₂₃	—	2	1	no	—
66	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₂₃	M₁₆	1	1	yes	characteristic
68	C₄²×C₂₃	M₁₆	1	1	yes	characteristic
69	C₁₆×C₂₃	—	2	4	no	—
70	C₁₆×C₂₃	—	4	2	no	—
71	#13270	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
73	Q₈×C₂₃	—	4	4	no	—
74	C₂×C₄×C₂₃	—	2	1	no	—
75	C₂×C₄×C₂₃	—	2	1	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	4	2	no	—
79	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₄×Q₈	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	1	no	—
85	C₈×C₂₃	—	4	1	no	—
86	C₈×C₂₃	C₂²⋊Q₈	1	2	yes	—
87	C₈×C₂₃	#124	1	1	yes	characteristic
88	C₈×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
89	#2960	C₂×C₂₃	1	1	yes	characteristic
90	#2153	C₂×C₂₃	1	1	yes	characteristic
91	#2110	C₂×C₂₃	1	1	yes	characteristic
92	#1378	C₂×C₂₃	1	2	yes	—
93	#1376	C₂×C₂₃	1	2	yes	—
94	C₂²×C₂₃	—	4	1	no	—
95	C₂²×C₂₃	#407	1	1	yes	characteristic
96	C₂²×C₂₃	#446	1	2	yes	—
97	C₄×C₂₃	—	2	2	no	—
98	C₄×C₂₃	—	4	1	no	—
99	C₄×C₂₃	—	4	2	no	—
100	C₄×C₂₃	—	4	2	no	—
101	C₄×C₂₃	#391	1	1	yes	characteristic
102	C₄×C₂₃	#408	1	1	yes	characteristic
103	C₄×C₂₃	#446	1	2	yes	—
104	#533	C₄×C₂₃	1	1	yes	characteristic
105	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
106	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
107	#446	—	2	2	no	—
108	C₈×Q₈	—	2	2	no	—
109	#424	C₄×C₂₃	1	1	yes	characteristic
110	#424	C₄×C₂₃	1	1	yes	characteristic
111	#423	C₄×C₂₃	1	1	yes	characteristic
112	#405	C₂²×C₂₃	1	1	yes	characteristic
113	#405	C₂²×C₂₃	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
115	#364	—	2	2	no	—
116	#347	—	2	2	no	—
117	C₂×C₂₃	—	2	1	no	—
118	C₂×C₂₃	—	2	2	no	—
119	C₂×C₂₃	#1985	1	1	yes	characteristic
120	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
121	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
122	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
123	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
124	C₄⋊Q₈	—	2	1	no	—
125	#125	—	2	1	no	—
126	C₂²⋊Q₈	—	2	2	no	—
127	C₄×Q₈	—	2	2	no	—
128	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
129	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
130	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
131	M₃₂	—	2	2	no	—
132	M₃₂	—	2	2	no	—
133	M₃₂	—	2	2	no	—
134	M₃₂	Q₈×C₂₃	1	4	yes	complemented
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	2	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	—	2	2	no	—
140	#97	—	2	1	no	—
141	#97	D₄×C₂₃	1	2	yes	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	—	2	1	no	—
145	C₄×C₈	D₄×C₂₃	1	2	yes	—
146	C₂₃	#13270	1	1	yes	characteristic, direct product factor, Sylow
147	C₂×Q₈	—	2	2	no	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	—	2	2	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
156	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
157	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
158	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
159	C₂×C₈	#41×C₂₃	1	2	yes	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₂²⋊C₄	—	2	1	no	—
165	C₂²⋊C₄	M₁₆×C₂₃	1	2	yes	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
169	C₄²	M₁₆×C₂₃	1	1	yes	characteristic
170	C₄²	M₁₆×C₂₃	1	1	yes	characteristic
171	C₁₆	—	2	4	no	—
172	C₁₆	—	4	2	no	—
173	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
174	Q₈	—	4	4	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
181	C₂×C₄	C₄×Q₈×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	1	no	—
186	C₈	—	4	1	no	—
187	C₈	C₂₃×C₂²⋊Q₈	1	2	yes	—
188	C₈	C₂₃×#124	1	1	yes	characteristic
189	C₈	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
190	C₂²	—	4	1	no	—
191	C₂²	C₂₃×#407	1	1	yes	characteristic
192	C₂²	C₂₃×#446	1	2	yes	—
193	C₄	—	2	2	no	—
194	C₄	—	4	1	no	—
195	C₄	—	4	2	no	—
196	C₄	—	4	2	no	—
197	C₄	C₂₃×#391	1	1	yes	characteristic
198	C₄	C₂₃×#408	1	1	yes	characteristic
199	C₄	C₂₃×#446	1	2	yes	—
200	C₂	—	2	1	no	—
201	C₂	—	2	2	no	—
202	C₂	C₂₃×#1985	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₂₃	#124
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂×C₂₃	#1985
prime radical	C₂³×C₂₃	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
33. C₂₃×M₃₂	Q₈	semidirect
71. #13270	C₂₃	direct (N × H)
134. M₃₂	Q₈×C₂₃	semidirect
146. C₂₃	#13270	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×#2960	2
C₂₃×#2153	2
C₂₃×#2110	2
C₂₃×#1378	2
C₂₃×#1376	2
#13270	23

Subgroups — C23×#13270 #19160896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13270 #19160896