Subgroups — C₂₃×#13179 #19160805

Order 5888.

286 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#368	C₂	1	2	yes	maximal
2	C₂×C₂₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1848	C₂	1	1	yes	maximal, characteristic
4	C₄×C₂₃×#108	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1381	C₂	1	2	yes	maximal
6	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂×C₂₃×#108	—	2	1	no	—
8	C₂×C₂₃×#108	C₂²	1	2	yes	—
9	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
11	C₂₃×#423	—	2	1	no	—
12	C₄×M₁₆×C₂₃	—	2	1	no	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#368	—	2	4	no	—
15	C₂₃×C₄⋊C₁₆	C₄	1	2	yes	—
16	C₂₃×C₄⋊C₁₆	C₄	1	2	yes	—
17	C₂₃×#341	—	2	1	no	—
18	C₂×M₁₆×C₂₃	—	2	1	no	—
19	C₂×M₁₆×C₂₃	—	2	1	no	—
20	C₂×M₁₆×C₂₃	—	2	2	no	—
21	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₂₃	—	2	2	no	—
25	C₂×C₁₆×C₂₃	—	2	2	no	—
26	C₂×C₁₆×C₂₃	C₂×C₄	1	4	yes	—
27	C₂₃×#108	—	2	2	no	—
28	C₂₃×#108	—	2	1	no	—
29	C₂₃×#108	—	2	1	no	—
30	C₂₃×#108	—	2	2	no	—
31	C₂₃×#108	—	4	2	no	—
32	C₂₃×#108	D₄	1	2	yes	—
33	C₂₃×#108	Q₈	1	2	yes	—
34	C₂₃×C₄⋊C₈	—	4	1	no	—
35	C₂₃×C₄⋊C₈	—	4	1	no	—
36	C₂₃×#97	—	4	1	no	—
37	C₄×C₈×C₂₃	—	4	1	no	—
38	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	Q₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	Q₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
45	M₁₆×C₂₃	—	2	2	no	—
46	M₁₆×C₂₃	—	2	2	no	—
47	M₁₆×C₂₃	—	4	1	no	—
48	M₁₆×C₂₃	—	4	1	no	—
49	M₁₆×C₂₃	—	4	2	no	—
50	M₁₆×C₂₃	—	4	2	no	—
51	C₂×C₈×C₂₃	—	2	1	no	—
52	C₂×C₈×C₂₃	—	2	1	no	—
53	C₂×C₈×C₂₃	—	4	1	no	—
54	C₂×C₈×C₂₃	—	4	1	no	—
55	C₂×C₈×C₂₃	—	4	2	no	—
56	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
63	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
64	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
65	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
66	C₄²×C₂₃	D₈	1	1	yes	characteristic
67	C₄²×C₂₃	D₈	1	1	yes	characteristic
68	C₁₆×C₂₃	—	2	4	no	—
69	#13179	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	4	1	no	—
73	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
74	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
75	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
76	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
83	C₈×C₂₃	—	2	1	no	—
84	C₈×C₂₃	—	2	1	no	—
85	C₈×C₂₃	—	4	2	no	—
86	C₈×C₂₃	—	4	2	no	—
87	C₈×C₂₃	—	8	2	no	—
88	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
89	C₈×C₂₃	C₄×Q₈	1	1	yes	characteristic
90	C₈×C₂₃	#125	1	1	yes	characteristic
91	C₈×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
92	C₂×#368	C₂×C₂₃	1	2	yes	—
93	C₂×C₄⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic
94	#1848	C₂×C₂₃	1	1	yes	characteristic
95	C₄×#108	C₂×C₂₃	1	1	yes	characteristic
96	#1381	C₂×C₂₃	1	2	yes	—
97	C₂²×C₂₃	—	2	1	no	—
98	C₂²×C₂₃	—	2	1	no	—
99	C₂²×C₂₃	C₂×#107	1	1	yes	characteristic
100	C₂²×C₂₃	#462	1	1	yes	characteristic
101	C₂²×C₂₃	#477	1	1	yes	characteristic
102	C₄×C₂₃	—	2	1	no	—
103	C₄×C₂₃	—	2	1	no	—
104	C₄×C₂₃	—	2	1	no	—
105	C₄×C₂₃	#368	1	2	yes	complemented
106	C₄×C₂₃	#389	1	1	yes	characteristic
107	C₄×C₂₃	#428	1	1	yes	characteristic
108	C₄×C₂₃	#459	1	1	yes	characteristic
109	C₄×C₂₃	#474	1	1	yes	characteristic
110	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
111	C₂×#108	—	2	1	no	—
112	C₂×#108	C₂²×C₂₃	1	2	yes	—
113	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
114	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
115	#423	—	2	1	no	—
116	C₄×M₁₆	—	2	1	no	—
117	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
118	#368	—	2	4	no	—
119	C₄⋊C₁₆	C₄×C₂₃	1	2	yes	—
120	C₄⋊C₁₆	C₄×C₂₃	1	2	yes	—
121	#341	—	2	1	no	—
122	C₂×C₂₃	—	2	1	no	—
123	C₂×C₂₃	—	2	1	no	—
124	C₂×C₂₃	#1947	1	1	yes	characteristic
125	C₂×C₂₃	C₂×#368	1	1	yes	characteristic
126	C₂×C₂₃	#2161	1	1	yes	characteristic
127	C₂×M₁₆	—	2	1	no	—
128	C₂×M₁₆	—	2	1	no	—
129	C₂×M₁₆	—	2	2	no	—
130	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
131	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
132	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
133	C₂×C₁₆	—	2	2	no	—
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	C₂×C₄×C₂₃	1	4	yes	—
136	#108	—	2	2	no	—
137	#108	—	2	1	no	—
138	#108	—	2	1	no	—
139	#108	—	2	2	no	—
140	#108	—	4	2	no	—
141	#108	D₄×C₂₃	1	2	yes	—
142	#108	Q₈×C₂₃	1	2	yes	—
143	C₄⋊C₈	—	4	1	no	—
144	C₄⋊C₈	—	4	1	no	—
145	#97	—	4	1	no	—
146	C₄×C₈	—	4	1	no	—
147	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
148	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
149	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
150	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
151	C₂₃	#13179	1	1	yes	characteristic, direct product factor, Sylow
152	C₂²×C₄	Q₁₆×C₂₃	1	1	yes	characteristic
153	C₂²×C₄	Q₁₆×C₂₃	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
155	M₁₆	—	2	2	no	—
156	M₁₆	—	2	2	no	—
157	M₁₆	—	4	1	no	—
158	M₁₆	—	4	1	no	—
159	M₁₆	—	4	2	no	—
160	M₁₆	—	4	2	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	—	4	1	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
167	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
168	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
169	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
170	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
171	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
172	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
173	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
174	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
175	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
176	C₄²	D₈×C₂₃	1	1	yes	characteristic
177	C₄²	D₈×C₂₃	1	1	yes	characteristic
178	C₁₆	—	2	4	no	—
179	C₂³	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
183	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
184	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
185	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
187	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
188	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
189	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
190	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
191	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
192	C₈	—	2	1	no	—
193	C₈	—	2	1	no	—
194	C₈	—	4	2	no	—
195	C₈	—	4	2	no	—
196	C₈	—	8	2	no	—
197	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
198	C₈	C₄×Q₈×C₂₃	1	1	yes	characteristic
199	C₈	C₂₃×#125	1	1	yes	characteristic
200	C₈	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	C₂×C₂₃×#107	1	1	yes	characteristic
204	C₂²	C₂₃×#462	1	1	yes	characteristic
205	C₂²	C₂₃×#477	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	C₂₃×#368	1	2	yes	complemented
210	C₄	C₂₃×#389	1	1	yes	characteristic
211	C₄	C₂₃×#428	1	1	yes	characteristic
212	C₄	C₂₃×#459	1	1	yes	characteristic
213	C₄	C₂₃×#474	1	1	yes	characteristic
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	C₂₃×#1947	1	1	yes	characteristic
217	C₂	C₂×C₂₃×#368	1	1	yes	characteristic
218	C₂	C₂₃×#2161	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×#107
prime radical	C₂³×C₂₃	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
69. #13179	C₂₃	direct (N × H)
105. C₄×C₂₃	#368	semidirect
151. C₂₃	#13179	direct (N × H)
209. C₄	C₂₃×#368	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#368	2
C₂×C₂₃×C₄⋊C₁₆	2
C₂₃×#1848	2
C₄×C₂₃×#108	2
C₂₃×#1381	2
#13179	23

Subgroups — C23×#13179 #19160805

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13179 #19160805