Subgroups — C₂₃×#23814 #19171440

Order 5888.

926 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#441	C₂	1	2	yes	maximal
2	C₂×C₄×Q₁₆×C₂₃	C₂	1	2	yes	maximal
3	C₂₃×#2327	C₂	1	2	yes	maximal
4	C₂₃×#2097	C₂	1	4	yes	maximal
5	C₂₃×#2016	C₂	1	4	yes	maximal
6	C₂₃×#1746	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
9	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₂₃	C₂²	1	4	yes	—
11	C₂₃×#441	—	2	8	no	—
12	C₄×Q₁₆×C₂₃	—	2	8	no	—
13	C₂×C₂₃×#107	C₂²	1	2	yes	—
14	C₂×C₂₃×#106	C₂²	1	2	yes	—
15	C₂×C₂₃×#103	C₂²	1	4	yes	—
16	C₂×C₂₃×#103	C₂²	1	4	yes	—
17	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#391	C₂²	1	4	yes	—
20	C₂₃×#387	—	2	8	no	—
21	C₂₃×#382	C₂²	1	2	yes	—
22	C₂₃×#376	C₂²	1	1	yes	characteristic
23	C₂₃×#336	C₂²	1	4	yes	—
24	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
25	C₂×Q₁₆×C₂₃	—	2	8	no	—
26	C₂×Q₁₆×C₂₃	—	2	8	no	—
27	C₂×Q₁₆×C₂₃	—	2	4	no	—
28	C₂×Q₁₆×C₂₃	—	2	4	no	—
29	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
30	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
31	C₂₃×C₄⋊Q₈	—	2	8	no	—
32	C₄×Q₈×C₂₃	—	2	8	no	—
33	C₄×Q₈×C₂₃	—	2	8	no	—
34	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
35	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
36	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
37	C₂×C₄²×C₂₃	—	2	4	no	—
38	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
39	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
40	C₂₃×#107	—	2	4	no	—
41	C₂₃×#106	—	2	4	no	—
42	C₂₃×#103	—	2	8	no	—
43	C₂₃×#103	—	2	8	no	—
44	C₂₃×#97	D₄	1	4	yes	—
45	C₄×C₈×C₂₃	D₄	1	4	yes	—
46	C₂₃×#95	—	2	4	no	—
47	C₂₃×#95	—	2	8	no	—
48	C₂₃×#95	D₄	1	4	yes	—
49	C₂×Q₈×C₂₃	—	2	4	no	—
50	C₂×Q₈×C₂₃	—	2	4	no	—
51	C₂×Q₈×C₂₃	—	2	4	no	—
52	C₂×Q₈×C₂₃	—	2	4	no	—
53	C₂×Q₈×C₂₃	—	2	4	no	—
54	C₂×Q₈×C₂₃	#41	1	8	yes	—
55	C₂²×C₄×C₂₃	—	2	4	no	—
56	C₂²×C₄×C₂₃	—	2	4	no	—
57	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
58	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
60	Q₁₆×C₂₃	—	2	16	no	—
61	Q₁₆×C₂₃	—	2	16	no	—
62	C₂×C₈×C₂₃	—	2	4	no	—
63	C₂×C₈×C₂₃	—	2	4	no	—
64	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
65	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
66	C₄⋊C₄×C₂₃	—	2	16	no	—
67	C₄⋊C₄×C₂₃	—	2	4	no	—
68	C₄⋊C₄×C₂₃	—	2	4	no	—
69	C₄⋊C₄×C₂₃	—	4	8	no	—
70	C₄²×C₂₃	—	2	4	no	—
71	C₄²×C₂₃	—	4	8	no	—
72	C₄²×C₂₃	C₂×D₄	1	2	yes	—
73	C₄²×C₂₃	C₂×D₄	1	2	yes	—
74	#23814	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
76	Q₈×C₂₃	—	2	8	no	—
77	Q₈×C₂₃	—	2	8	no	—
78	Q₈×C₂₃	—	4	4	no	—
79	Q₈×C₂₃	—	4	4	no	—
80	C₂×C₄×C₂₃	—	2	4	no	—
81	C₂×C₄×C₂₃	—	2	4	no	—
82	C₂×C₄×C₂₃	—	2	8	no	—
83	C₂×C₄×C₂₃	—	2	8	no	—
84	C₂×C₄×C₂₃	—	4	4	no	—
85	C₂×C₄×C₂₃	—	4	4	no	—
86	C₂×C₄×C₂₃	—	4	4	no	—
87	C₂×C₄×C₂₃	—	4	8	no	—
88	C₂×C₄×C₂₃	#127	1	4	yes	—
89	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
92	C₈×C₂₃	—	2	4	no	—
93	C₈×C₂₃	—	2	4	no	—
94	C₂×#441	C₂×C₂₃	1	2	yes	—
95	C₂×C₄×Q₁₆	C₂×C₂₃	1	2	yes	—
96	#2327	C₂×C₂₃	1	2	yes	—
97	#2097	C₂×C₂₃	1	4	yes	—
98	#2016	C₂×C₂₃	1	4	yes	—
99	#1746	C₂×C₂₃	1	1	yes	characteristic
100	C₂²×C₂₃	#525	1	2	yes	—
101	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
102	C₂²×C₂₃	#578	1	4	yes	—
103	C₄×C₂₃	—	2	4	no	—
104	C₄×C₂₃	—	4	4	no	—
105	C₄×C₂₃	—	4	8	no	—
106	C₄×C₂₃	—	8	4	no	—
107	C₄×C₂₃	#525	1	2	yes	—
108	C₄×C₂₃	#525	1	2	yes	—
109	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
110	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
111	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
112	C₂×C₄×Q₈	C₂²×C₂₃	1	4	yes	—
113	#441	—	2	8	no	—
114	C₄×Q₁₆	—	2	8	no	—
115	C₂×#107	C₂²×C₂₃	1	2	yes	—
116	C₂×#106	C₂²×C₂₃	1	2	yes	—
117	C₂×#103	C₂²×C₂₃	1	4	yes	—
118	C₂×#103	C₂²×C₂₃	1	4	yes	—
119	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
121	#391	C₂²×C₂₃	1	4	yes	—
122	#387	—	2	8	no	—
123	#382	C₂²×C₂₃	1	2	yes	—
124	#376	C₂²×C₂₃	1	1	yes	characteristic
125	#336	C₂²×C₂₃	1	4	yes	—
126	C₂×C₂₃	#2397	1	1	yes	characteristic
127	C₂×C₂₃	#3127	1	4	yes	—
128	C₂×C₂₃	#3156	1	2	yes	—
129	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
130	C₂×Q₁₆	—	2	8	no	—
131	C₂×Q₁₆	—	2	8	no	—
132	C₂×Q₁₆	—	2	4	no	—
133	C₂×Q₁₆	—	2	4	no	—
134	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
135	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
136	C₄⋊Q₈	—	2	8	no	—
137	C₄×Q₈	—	2	8	no	—
138	C₄×Q₈	—	2	8	no	—
139	C₂×C₄⋊C₄	—	2	4	no	—
140	C₂×C₄⋊C₄	—	2	4	no	—
141	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
142	C₂×C₄²	—	2	4	no	—
143	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
144	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
145	#107	—	2	4	no	—
146	#106	—	2	4	no	—
147	#103	—	2	8	no	—
148	#103	—	2	8	no	—
149	#97	D₄×C₂₃	1	4	yes	—
150	C₄×C₈	D₄×C₂₃	1	4	yes	—
151	#95	—	2	4	no	—
152	#95	—	2	8	no	—
153	#95	D₄×C₂₃	1	4	yes	—
154	C₂₃	#23814	1	1	yes	characteristic, direct product factor, Sylow
155	C₂×Q₈	—	2	4	no	—
156	C₂×Q₈	—	2	4	no	—
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	—	2	4	no	—
159	C₂×Q₈	—	2	4	no	—
160	C₂×Q₈	#41×C₂₃	1	8	yes	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
164	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
165	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
166	Q₁₆	—	2	16	no	—
167	Q₁₆	—	2	16	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
171	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
172	C₄⋊C₄	—	2	16	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	—	4	8	no	—
176	C₄²	—	2	4	no	—
177	C₄²	—	4	8	no	—
178	C₄²	C₂×D₄×C₂₃	1	2	yes	—
179	C₄²	C₂×D₄×C₂₃	1	2	yes	—
180	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
181	Q₈	—	2	8	no	—
182	Q₈	—	2	8	no	—
183	Q₈	—	4	4	no	—
184	Q₈	—	4	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	8	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	8	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	4	8	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	C₂₃×#127	1	4	yes	—
194	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
195	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
196	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
197	C₈	—	2	4	no	—
198	C₈	—	2	4	no	—
199	C₂²	C₂₃×#525	1	2	yes	—
200	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
201	C₂²	C₂₃×#578	1	4	yes	—
202	C₄	—	2	4	no	—
203	C₄	—	4	8	no	—
204	C₄	—	4	4	no	—
205	C₄	—	8	4	no	—
206	C₄	C₂₃×#525	1	2	yes	—
207	C₄	C₂₃×#525	1	2	yes	—
208	C₂	C₂₃×#2397	1	1	yes	characteristic
209	C₂	C₂₃×#3127	1	4	yes	—
210	C₂	C₂₃×#3156	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
74. #23814	C₂₃	direct (N × H)
154. C₂₃	#23814	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×#441	2
C₂×C₄×Q₁₆×C₂₃	2
C₂₃×#2327	2
C₂₃×#2097	2
C₂₃×#2016	2
C₂₃×#1746	2
#23814	23

Subgroups — C23×#23814 #19171440

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#23814 #19171440