Subgroups — C₂₃×#13187 #19160813

Order 5888.

294 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#365	C₂	1	2	yes	maximal
2	C₂×C₂₃×#364	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1846	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1772	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1386	C₂	1	2	yes	maximal
6	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
7	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
8	C₂₃×#427	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#107	—	2	1	no	—
10	C₂×C₂₃×#106	C₂²	1	2	yes	—
11	C₂₃×#405	—	2	1	no	—
12	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#382	—	2	1	no	—
14	C₂₃×#365	—	2	4	no	—
15	C₂₃×#364	C₄	1	4	yes	complemented
16	C₂₃×#339	—	2	1	no	—
17	C₂×M₁₆×C₂₃	—	2	1	no	—
18	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
20	C₂₃×#117	—	2	1	no	—
21	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
22	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
23	C₂₃×M₃₂	—	2	2	no	—
24	C₂₃×M₃₂	C₂×C₄	1	4	yes	—
25	C₂×C₁₆×C₂₃	—	2	2	no	—
26	C₂₃×#108	—	2	1	no	—
27	C₂₃×#108	D₄	1	2	yes	—
28	C₂₃×#107	—	4	2	no	—
29	C₂₃×#107	Q₈	1	2	yes	—
30	C₂₃×#106	—	2	1	no	—
31	C₂₃×#106	—	2	2	no	—
32	C₂₃×#106	—	2	2	no	—
33	C₂₃×#97	—	4	1	no	—
34	C₄×C₈×C₂₃	—	4	1	no	—
35	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₂₃	Q₈	1	2	yes	—
37	C₂₃×#95	—	4	2	no	—
38	C₂²×C₄×C₂₃	—	2	1	no	—
39	C₂²×C₄×C₂₃	—	4	2	no	—
40	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
41	M₁₆×C₂₃	—	2	2	no	—
42	M₁₆×C₂₃	—	4	1	no	—
43	C₂×C₈×C₂₃	—	4	1	no	—
44	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
46	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
53	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
54	C₄⋊C₄×C₂₃	—	2	2	no	—
55	C₄⋊C₄×C₂₃	—	4	2	no	—
56	C₄⋊C₄×C₂₃	—	4	2	no	—
57	C₂²⋊C₄×C₂₃	—	4	1	no	—
58	C₄²×C₂₃	—	2	1	no	—
59	C₄²×C₂₃	—	4	1	no	—
60	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
61	C₁₆×C₂₃	—	2	4	no	—
62	#13187	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
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	—	8	4	no	—
68	C₂×C₄×C₂₃	—	8	2	no	—
69	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
71	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
75	C₈×C₂₃	—	4	2	no	—
76	C₈×C₂₃	#106	1	2	yes	—
77	C₈×C₂₃	#106	1	2	yes	—
78	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
79	C₈×C₂₃	C₄×Q₈	1	1	yes	characteristic
80	C₈×C₂₃	#125	1	1	yes	characteristic
81	C₈×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×#365	C₂×C₂₃	1	2	yes	—
83	C₂×#364	C₂×C₂₃	1	1	yes	characteristic
84	#1846	C₂×C₂₃	1	1	yes	characteristic
85	#1772	C₂×C₂₃	1	1	yes	characteristic
86	#1386	C₂×C₂₃	1	2	yes	—
87	C₂²×C₂₃	—	2	2	no	—
88	C₂²×C₂₃	#428	1	1	yes	characteristic
89	C₂²×C₂₃	#460	1	1	yes	characteristic
90	C₂²×C₂₃	#476	1	1	yes	characteristic
91	C₄×C₂₃	—	4	2	no	—
92	C₄×C₂₃	—	8	2	no	—
93	C₄×C₂₃	—	16	2	no	—
94	C₄×C₂₃	#389	1	1	yes	characteristic
95	C₄×C₂₃	C₂×#106	1	1	yes	characteristic
96	C₄×C₂₃	#461	1	1	yes	characteristic
97	C₄×C₂₃	#475	1	1	yes	characteristic
98	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
99	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
100	#427	C₂²×C₂₃	1	1	yes	characteristic
101	C₂×#107	—	2	1	no	—
102	C₂×#106	C₂²×C₂₃	1	2	yes	—
103	#405	—	2	1	no	—
104	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
105	#382	—	2	1	no	—
106	#365	—	2	4	no	—
107	#364	C₄×C₂₃	1	4	yes	complemented
108	#339	—	2	1	no	—
109	C₂×C₂₃	—	2	1	no	—
110	C₂×C₂₃	—	2	1	no	—
111	C₂×C₂₃	#1948	1	1	yes	characteristic
112	C₂×C₂₃	#2231	1	2	yes	—
113	C₂×M₁₆	—	2	1	no	—
114	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
115	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
116	#117	—	2	1	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
119	M₃₂	—	2	2	no	—
120	M₃₂	C₂×C₄×C₂₃	1	4	yes	—
121	C₂×C₁₆	—	2	2	no	—
122	#108	—	2	1	no	—
123	#108	D₄×C₂₃	1	2	yes	—
124	#107	—	4	2	no	—
125	#107	Q₈×C₂₃	1	2	yes	—
126	#106	—	2	1	no	—
127	#106	—	2	2	no	—
128	#106	—	2	2	no	—
129	#97	—	4	1	no	—
130	C₄×C₈	—	4	1	no	—
131	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
132	C₄×C₈	Q₈×C₂₃	1	2	yes	—
133	#95	—	4	2	no	—
134	C₂₃	#13187	1	1	yes	characteristic, direct product factor, Sylow
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
138	M₁₆	—	2	2	no	—
139	M₁₆	—	4	1	no	—
140	C₂×C₈	—	4	1	no	—
141	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
142	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
143	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
144	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
145	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
146	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
147	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
148	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
149	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
150	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₂²⋊C₄	—	4	1	no	—
155	C₄²	—	2	1	no	—
156	C₄²	—	4	1	no	—
157	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
158	C₁₆	—	2	4	no	—
159	C₂³	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	8	4	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
166	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
167	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
168	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
169	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
170	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
171	C₈	—	4	2	no	—
172	C₈	C₂₃×#106	1	2	yes	—
173	C₈	C₂₃×#106	1	2	yes	—
174	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
175	C₈	C₄×Q₈×C₂₃	1	1	yes	characteristic
176	C₈	C₂₃×#125	1	1	yes	characteristic
177	C₈	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
178	C₂²	—	2	2	no	—
179	C₂²	C₂₃×#428	1	1	yes	characteristic
180	C₂²	C₂₃×#460	1	1	yes	characteristic
181	C₂²	C₂₃×#476	1	1	yes	characteristic
182	C₄	—	4	2	no	—
183	C₄	—	8	2	no	—
184	C₄	—	16	2	no	—
185	C₄	C₂₃×#389	1	1	yes	characteristic
186	C₄	C₂×C₂₃×#106	1	1	yes	characteristic
187	C₄	C₂₃×#461	1	1	yes	characteristic
188	C₄	C₂₃×#475	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	C₂₃×#1948	1	1	yes	characteristic
192	C₂	C₂₃×#2231	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	#476
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂³×C₂₃	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₂₃×#364	C₄	semidirect
62. #13187	C₂₃	direct (N × H)
107. #364	C₄×C₂₃	semidirect
134. C₂₃	#13187	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×#365	2
C₂×C₂₃×#364	2
C₂₃×#1846	2
C₂₃×#1772	2
C₂₃×#1386	2
#13187	23

Subgroups — C23×#13187 #19160813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13187 #19160813