Subgroups — C₂₃×#11681 #19159307

Order 5888.

324 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3170	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1566	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1334	C₂	1	4	yes	maximal, complemented
5	C₂₃×#533	—	2	1	no	—
6	C₂×C₂₃×M₃₂	—	2	1	no	—
7	C₂₃×#500	C₄	1	1	yes	characteristic
8	C₂₃×#494	C₄	1	1	yes	characteristic
9	C₂₃×#487	—	2	1	no	—
10	C₂₃×#467	—	2	1	no	—
11	C₂₃×#467	C₄	1	2	yes	—
12	C₄×Q₁₆×C₂₃	—	2	1	no	—
13	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#423	—	2	1	no	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#346	C₂²	1	2	yes	—
17	C₄×C₁₆×C₂₃	—	2	1	no	—
18	C₂₃×#334	—	2	1	no	—
19	C₂₃×#334	C₂²	1	2	yes	—
20	C₂×Q₁₆×C₂₃	—	2	1	no	—
21	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
22	C₂×M₁₆×C₂₃	—	4	1	no	—
23	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
25	C₂₃×C₄⋊Q₈	—	2	1	no	—
26	C₂₃×C₄⋊Q₈	—	2	1	no	—
27	C₂₃×#125	—	4	1	no	—
28	C₂₃×C₂²⋊Q₈	—	2	2	no	—
29	C₂₃×C₂²⋊Q₈	—	4	1	no	—
30	C₄×Q₈×C₂₃	—	2	2	no	—
31	C₄×Q₈×C₂₃	—	4	1	no	—
32	C₂₃×#117	—	4	1	no	—
33	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
34	C₂₃×M₃₂	—	2	4	no	—
35	C₂×C₁₆×C₂₃	—	2	2	no	—
36	C₂₃×#107	—	2	1	no	—
37	C₂₃×#107	C₂×C₄	1	2	yes	—
38	C₂₃×C₄⋊C₈	—	2	2	no	—
39	C₂₃×C₄⋊C₈	—	4	1	no	—
40	C₂₃×#103	—	2	2	no	—
41	C₂₃×#103	—	2	2	no	—
42	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₂₃	—	2	2	no	—
47	C₂×Q₈×C₂₃	—	4	1	no	—
48	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
51	Q₁₆×C₂₃	—	2	4	no	—
52	Q₁₆×C₂₃	—	4	2	no	—
53	M₁₆×C₂₃	—	8	2	no	—
54	C₂×C₈×C₂₃	—	2	2	no	—
55	C₂×C₈×C₂₃	—	2	2	no	—
56	C₂×C₈×C₂₃	—	4	2	no	—
57	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₂₃	—	2	2	no	—
62	C₄⋊C₄×C₂₃	—	4	1	no	—
63	C₄⋊C₄×C₂₃	—	4	1	no	—
64	C₄⋊C₄×C₂₃	—	4	2	no	—
65	C₄⋊C₄×C₂₃	—	4	1	no	—
66	C₄⋊C₄×C₂₃	—	4	2	no	—
67	C₂²⋊C₄×C₂₃	—	4	2	no	—
68	C₄²×C₂₃	—	4	2	no	—
69	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₂₃	D₈	1	1	yes	characteristic
71	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
72	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
73	C₁₆×C₂₃	—	2	4	no	—
74	#11681	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₂₃	#102	1	1	yes	characteristic
76	Q₈×C₂₃	—	4	2	no	—
77	Q₈×C₂₃	—	4	2	no	—
78	Q₈×C₂₃	—	8	2	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	1	no	—
82	C₂×C₄×C₂₃	—	2	1	no	—
83	C₂×C₄×C₂₃	—	4	2	no	—
84	C₂×C₄×C₂₃	—	4	2	no	—
85	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
86	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
87	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
88	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
91	C₈×C₂₃	—	2	1	no	—
92	C₈×C₂₃	—	2	1	no	—
93	C₈×C₂₃	—	8	2	no	—
94	C₈×C₂₃	#104	1	2	yes	—
95	C₈×C₂₃	#131	1	2	yes	—
96	#3170	C₂×C₂₃	1	1	yes	characteristic
97	C₄×M₃₂	C₂×C₂₃	1	1	yes	characteristic
98	#1566	C₂×C₂₃	1	1	yes	characteristic
99	#1334	C₂×C₂₃	1	4	yes	complemented
100	C₂²×C₂₃	—	2	1	no	—
101	C₂²×C₂₃	—	2	1	no	—
102	C₂²×C₂₃	—	2	1	no	—
103	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
104	C₄×C₂₃	—	2	1	no	—
105	C₄×C₂₃	—	2	1	no	—
106	C₄×C₂₃	—	2	1	no	—
107	C₄×C₂₃	—	8	2	no	—
108	C₄×C₂₃	—	8	2	no	—
109	C₄×C₂₃	#360	1	2	yes	—
110	C₄×C₂₃	#362	1	2	yes	—
111	C₄×C₂₃	#408	1	1	yes	characteristic
112	C₄×C₂₃	C₂×#104	1	1	yes	characteristic
113	#533	—	2	1	no	—
114	C₂×M₃₂	—	2	1	no	—
115	#500	C₄×C₂₃	1	1	yes	characteristic
116	#494	C₄×C₂₃	1	1	yes	characteristic
117	#487	—	2	1	no	—
118	#467	—	2	1	no	—
119	#467	C₄×C₂₃	1	2	yes	—
120	C₄×Q₁₆	—	2	1	no	—
121	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
122	#423	—	2	1	no	—
123	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
124	#346	C₂²×C₂₃	1	2	yes	—
125	C₄×C₁₆	—	2	1	no	—
126	#334	—	2	1	no	—
127	#334	C₂²×C₂₃	1	2	yes	—
128	C₂×C₂₃	—	4	1	no	—
129	C₂×C₂₃	#1492	1	1	yes	characteristic
130	C₂×C₂₃	C₂×#360	1	1	yes	characteristic
131	C₂×C₂₃	C₂×#362	1	1	yes	characteristic
132	C₂×Q₁₆	—	2	1	no	—
133	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
134	C₂×M₁₆	—	4	1	no	—
135	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
136	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
137	C₄⋊Q₈	—	2	1	no	—
138	C₄⋊Q₈	—	2	1	no	—
139	#125	—	4	1	no	—
140	C₂²⋊Q₈	—	2	2	no	—
141	C₂²⋊Q₈	—	4	1	no	—
142	C₄×Q₈	—	2	2	no	—
143	C₄×Q₈	—	4	1	no	—
144	#117	—	4	1	no	—
145	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
146	M₃₂	—	2	4	no	—
147	C₂×C₁₆	—	2	2	no	—
148	#107	—	2	1	no	—
149	#107	C₂×C₄×C₂₃	1	2	yes	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	4	1	no	—
152	#103	—	2	2	no	—
153	#103	—	2	2	no	—
154	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
155	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
156	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
157	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
158	C₂₃	#11681	1	1	yes	characteristic, direct product factor, Sylow
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	4	1	no	—
161	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
162	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
163	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
164	Q₁₆	—	2	4	no	—
165	Q₁₆	—	4	2	no	—
166	M₁₆	—	8	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
171	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
172	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
173	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	4	1	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₂²⋊C₄	—	4	2	no	—
181	C₄²	—	4	2	no	—
182	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
183	C₄²	D₈×C₂₃	1	1	yes	characteristic
184	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
185	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
186	C₁₆	—	2	4	no	—
187	C₂³	C₂₃×#102	1	1	yes	characteristic
188	Q₈	—	4	2	no	—
189	Q₈	—	4	2	no	—
190	Q₈	—	8	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
198	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
199	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
200	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
201	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
202	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
203	C₈	—	2	1	no	—
204	C₈	—	2	1	no	—
205	C₈	—	8	2	no	—
206	C₈	C₂₃×#104	1	2	yes	—
207	C₈	C₂₃×#131	1	2	yes	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	8	2	no	—
216	C₄	—	8	2	no	—
217	C₄	C₂₃×#360	1	2	yes	—
218	C₄	C₂₃×#362	1	2	yes	—
219	C₄	C₂₃×#408	1	1	yes	characteristic
220	C₄	C₂×C₂₃×#104	1	1	yes	characteristic
221	C₂	—	4	1	no	—
222	C₂	C₂₃×#1492	1	1	yes	characteristic
223	C₂	C₂×C₂₃×#360	1	1	yes	characteristic
224	C₂	C₂×C₂₃×#362	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
prime radical	C₂³×C₂₃	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
4. C₂₃×#1334	C₂	semidirect
74. #11681	C₂₃	direct (N × H)
99. #1334	C₂×C₂₃	semidirect
158. C₂₃	#11681	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×#3170	2
C₄×C₂₃×M₃₂	2
C₂₃×#1566	2
C₂₃×#1334	2
#11681	23

Subgroups — C23×#11681 #19159307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11681 #19159307