Subgroups — C₂₃×#12914 #19160540

Order 5888.

420 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#364	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2112	C₂	1	1	yes	maximal, characteristic
4	C₈×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1376	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×M₃₂	—	2	1	no	—
8	C₂×C₂₃×M₃₂	—	2	4	no	—
9	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₂₃	—	2	1	no	—
13	C₂×C₂₃×C₂²⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
15	C₄×C₂²⋊C₄×C₂₃	C₄	1	1	yes	characteristic
16	C₂₃×#364	—	2	4	no	—
17	C₂₃×C₂²⋊C₁₆	—	2	1	no	—
18	C₂₃×C₂²⋊C₁₆	—	2	1	no	—
19	C₂₃×#336	C₄	1	2	yes	—
20	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₂₃	—	2	2	no	—
22	C₂²×C₈×C₂₃	—	2	2	no	—
23	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₂₃	Q₈	1	1	yes	characteristic
28	C₂²×C₈×C₂₃	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
31	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
32	C₂₃×M₃₂	—	2	4	no	—
33	C₂₃×M₃₂	—	2	2	no	—
34	C₂₃×M₃₂	—	2	2	no	—
35	C₂₃×M₃₂	—	2	2	no	—
36	C₂₃×M₃₂	D₄	1	4	yes	—
37	C₂×C₁₆×C₂₃	—	2	4	no	—
38	C₂×C₁₆×C₂₃	—	2	2	no	—
39	C₂×C₁₆×C₂₃	—	2	1	no	—
40	C₂×C₁₆×C₂₃	—	2	1	no	—
41	C₂₃×C₂²⋊C₈	C₈	1	2	yes	—
42	C₂₃×C₂²⋊C₈	C₈	1	2	yes	—
43	C₄×C₈×C₂₃	—	2	2	no	—
44	C₄×C₈×C₂₃	—	2	2	no	—
45	C₂₃×#95	—	2	1	no	—
46	C₂⁴×C₂₃	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	—	2	2	no	—
48	C₂²×C₄×C₂₃	—	2	2	no	—
49	C₂²×C₄×C₂₃	—	2	2	no	—
50	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₂₃	—	2	2	no	—
57	C₂×C₈×C₂₃	—	2	2	no	—
58	C₂×C₈×C₂₃	—	2	2	no	—
59	C₂×C₈×C₂₃	—	2	1	no	—
60	C₂×C₈×C₂₃	—	2	1	no	—
61	C₂×C₈×C₂₃	—	2	2	no	—
62	C₂×C₈×C₂₃	—	2	2	no	—
63	C₂×C₈×C₂₃	—	2	1	no	—
64	C₂×C₈×C₂₃	—	2	1	no	—
65	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₂₃	C₂×C₈	1	4	yes	—
68	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₈×C₂₃	#41	1	2	yes	—
72	C₂²⋊C₄×C₂₃	—	2	4	no	—
73	C₄²×C₂₃	—	2	2	no	—
74	C₄²×C₂₃	—	2	2	no	—
75	C₁₆×C₂₃	—	2	4	no	—
76	C₁₆×C₂₃	—	4	1	no	—
77	C₁₆×C₂₃	—	4	1	no	—
78	#12914	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂³×C₂₃	—	2	2	no	—
80	C₂³×C₂₃	—	2	2	no	—
81	C₂³×C₂₃	—	4	1	no	—
82	C₂³×C₂₃	C₄⋊C₈	1	1	yes	characteristic
83	C₂³×C₂₃	C₄⋊C₈	1	1	yes	characteristic
84	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	2	1	no	—
87	C₂×C₄×C₂₃	—	2	1	no	—
88	C₂×C₄×C₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	2	1	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	4	no	—
93	C₂×C₄×C₂₃	—	4	2	no	—
94	C₂×C₄×C₂₃	—	4	2	no	—
95	C₂×C₄×C₂₃	C₄⋊C₈	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₄⋊C₈	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₂₃	#131	1	2	yes	—
101	C₈×C₂₃	—	2	1	no	—
102	C₈×C₂₃	—	2	1	no	—
103	C₈×C₂₃	—	2	4	no	—
104	C₈×C₂₃	#120	1	1	yes	characteristic
105	C₈×C₂₃	C₂²⋊Q₈	1	2	yes	—
106	C₈×C₂₃	#123	1	1	yes	characteristic
107	C₂²×M₃₂	C₂×C₂₃	1	1	yes	characteristic
108	C₂×#364	C₂×C₂₃	1	2	yes	complemented
109	#2112	C₂×C₂₃	1	1	yes	characteristic
110	C₈×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic
111	#1376	C₂×C₂₃	1	2	yes	complemented
112	C₂²×C₂₃	—	2	1	no	—
113	C₂²×C₂₃	—	2	1	no	—
114	C₂²×C₂₃	—	2	1	no	—
115	C₂²×C₂₃	—	2	1	no	—
116	C₂²×C₂₃	—	2	1	no	—
117	C₂²×C₂₃	—	2	1	no	—
118	C₂²×C₂₃	—	2	1	no	—
119	C₂²×C₂₃	—	4	4	no	—
120	C₂²×C₂₃	#364	1	2	yes	complemented
121	C₂²×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
122	C₂²×C₂₃	#435	1	2	yes	—
123	C₄×C₂₃	—	2	1	no	—
124	C₄×C₂₃	—	2	1	no	—
125	C₄×C₂₃	—	4	4	no	—
126	C₄×C₂₃	#385	1	1	yes	characteristic
127	C₄×C₂₃	#424	1	1	yes	characteristic
128	C₄×C₂₃	C₈×D₄	1	2	yes	—
129	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
130	C₂×M₃₂	—	2	1	no	—
131	C₂×M₃₂	—	2	4	no	—
132	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
133	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
134	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
135	C₂²×C₁₆	—	2	1	no	—
136	C₂×C₂²⋊C₈	C₄×C₂₃	1	1	yes	characteristic
137	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
138	C₄×C₂²⋊C₄	C₄×C₂₃	1	1	yes	characteristic
139	#364	—	2	4	no	—
140	C₂²⋊C₁₆	—	2	1	no	—
141	C₂²⋊C₁₆	—	2	1	no	—
142	#336	C₄×C₂₃	1	2	yes	—
143	C₂×C₂₃	—	2	2	no	—
144	C₂×C₂₃	—	2	2	no	—
145	C₂×C₂₃	—	4	1	no	—
146	C₂×C₂₃	#1850	1	1	yes	characteristic
147	C₂×C₂₃	C₂×#364	1	1	yes	characteristic
148	C₂×C₂₃	#2153	1	1	yes	characteristic
149	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
153	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
154	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
155	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
156	C₂²×C₈	Q₈×C₂₃	1	1	yes	characteristic
157	C₂²×C₈	Q₈×C₂₃	1	1	yes	characteristic
158	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
159	C₂×C₂²⋊C₄	—	2	1	no	—
160	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
161	M₃₂	—	2	4	no	—
162	M₃₂	—	2	2	no	—
163	M₃₂	—	2	2	no	—
164	M₃₂	—	2	2	no	—
165	M₃₂	D₄×C₂₃	1	4	yes	—
166	C₂×C₁₆	—	2	4	no	—
167	C₂×C₁₆	—	2	2	no	—
168	C₂×C₁₆	—	2	1	no	—
169	C₂×C₁₆	—	2	1	no	—
170	C₂²⋊C₈	C₈×C₂₃	1	2	yes	—
171	C₂²⋊C₈	C₈×C₂₃	1	2	yes	—
172	C₄×C₈	—	2	2	no	—
173	C₄×C₈	—	2	2	no	—
174	#95	—	2	1	no	—
175	C₂₃	#12914	1	1	yes	characteristic, direct product factor, Sylow
176	C₂⁴	M₁₆×C₂₃	1	1	yes	characteristic
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
181	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
184	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
185	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
196	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
197	C₂×C₈	C₂×C₈×C₂₃	1	4	yes	—
198	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
199	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
200	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
201	C₂×C₈	#41×C₂₃	1	2	yes	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₁₆	—	2	4	no	—
206	C₁₆	—	4	1	no	—
207	C₁₆	—	4	1	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	4	1	no	—
211	C₂³	C₂₃×C₄⋊C₈	1	1	yes	characteristic
212	C₂³	C₂₃×C₄⋊C₈	1	1	yes	characteristic
213	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	C₂₃×C₄⋊C₈	1	1	yes	characteristic
225	C₂×C₄	C₂₃×C₄⋊C₈	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
227	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
228	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
229	C₂×C₄	C₂₃×#131	1	2	yes	—
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	2	4	no	—
233	C₈	C₂₃×#120	1	1	yes	characteristic
234	C₈	C₂₃×C₂²⋊Q₈	1	2	yes	—
235	C₈	C₂₃×#123	1	1	yes	characteristic
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	4	4	no	—
244	C₂²	C₂₃×#364	1	2	yes	complemented
245	C₂²	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
246	C₂²	C₂₃×#435	1	2	yes	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	4	no	—
250	C₄	C₂₃×#385	1	1	yes	characteristic
251	C₄	C₂₃×#424	1	1	yes	characteristic
252	C₄	C₈×D₄×C₂₃	1	2	yes	—
253	C₂	—	2	2	no	—
254	C₂	—	2	2	no	—
255	C₂	—	4	1	no	—
256	C₂	C₂₃×#1850	1	1	yes	characteristic
257	C₂	C₂×C₂₃×#364	1	1	yes	characteristic
258	C₂	C₂₃×#2153	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×C₄⋊C₈
prime radical	C₂⁴×C₂₃	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#364	C₂	semidirect
5. C₂₃×#1376	C₂	semidirect
78. #12914	C₂₃	direct (N × H)
108. C₂×#364	C₂×C₂₃	semidirect
111. #1376	C₂×C₂₃	semidirect
120. C₂²×C₂₃	#364	semidirect
175. C₂₃	#12914	direct (N × H)
244. C₂²	C₂₃×#364	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×M₃₂	2
C₂×C₂₃×#364	2
C₂₃×#2112	2
C₈×C₂²⋊C₄×C₂₃	2
C₂₃×#1376	2
#12914	23

Subgroups — C23×#12914 #19160540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12914 #19160540