Subgroups — C₂₃×#12850 #19160476

Order 5888.

430 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#349	C₂	1	2	yes	maximal
3	C₂₃×#1376	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1375	C₂	1	2	yes	maximal, complemented
5	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₂₃	C₄	1	1	yes	characteristic
7	C₂×C₂₃×#117	C₄	1	1	yes	characteristic
8	C₂×C₂₃×M₃₂	C₂²	1	4	yes	—
9	C₂₃×#405	—	2	4	no	—
10	C₂×C₂₃×#97	C₄	1	1	yes	characteristic
11	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#349	—	2	4	no	—
15	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂×M₁₆×C₂₃	—	2	2	no	—
17	C₂×M₁₆×C₂₃	—	2	2	no	—
18	C₂×M₁₆×C₂₃	D₄	1	4	yes	—
19	C₂²×C₈×C₂₃	—	2	2	no	—
20	C₂²×C₈×C₂₃	—	2	2	no	—
21	C₂²×C₈×C₂₃	—	2	2	no	—
22	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
26	C₂₃×#117	—	2	2	no	—
27	C₂₃×#117	C₈	1	4	yes	—
28	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
29	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
30	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂₃×M₃₂	—	2	4	no	—
33	C₂₃×M₃₂	—	2	4	no	—
34	C₂×C₁₆×C₂₃	—	2	4	no	—
35	C₂₃×#97	—	2	2	no	—
36	C₂₃×#97	—	2	2	no	—
37	C₄×C₈×C₂₃	—	2	2	no	—
38	C₄×C₈×C₂₃	—	2	2	no	—
39	C₂⁴×C₂₃	M₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	—	2	1	no	—
41	C₂²×C₄×C₂₃	—	2	1	no	—
42	C₂²×C₄×C₂₃	—	2	1	no	—
43	C₂²×C₄×C₂₃	—	2	1	no	—
44	C₂²×C₄×C₂₃	—	2	1	no	—
45	C₂²×C₄×C₂₃	—	2	1	no	—
46	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₂₃	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
50	M₁₆×C₂₃	—	2	8	no	—
51	C₂×C₈×C₂₃	—	2	8	no	—
52	C₂×C₈×C₂₃	—	2	1	no	—
53	C₂×C₈×C₂₃	—	2	1	no	—
54	C₂×C₈×C₂₃	—	2	1	no	—
55	C₂×C₈×C₂₃	—	2	1	no	—
56	C₂×C₈×C₂₃	—	2	1	no	—
57	C₂×C₈×C₂₃	—	2	1	no	—
58	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
60	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
61	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₂₃	#41	1	2	yes	—
63	C₂×C₈×C₂₃	#41	1	2	yes	—
64	C₄⋊C₄×C₂₃	—	2	4	no	—
65	C₂²⋊C₄×C₂₃	—	2	4	no	—
66	C₄²×C₂₃	—	2	1	no	—
67	C₄²×C₂₃	—	2	1	no	—
68	C₄²×C₂₃	C₂×C₈	1	2	yes	—
69	C₄²×C₂₃	C₂×C₈	1	2	yes	—
70	C₁₆×C₂₃	—	4	4	no	—
71	#12850	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂³×C₂₃	—	2	1	no	—
73	C₂³×C₂₃	—	2	1	no	—
74	C₂³×C₂₃	—	4	2	no	—
75	C₂³×C₂₃	C₂²⋊C₈	1	2	yes	—
76	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₂₃	—	2	1	no	—
78	C₂×C₄×C₂₃	—	2	1	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	4	2	no	—
86	C₂×C₄×C₂₃	—	4	2	no	—
87	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
88	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
90	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
94	C₈×C₂₃	—	2	2	no	—
95	C₈×C₂₃	—	2	2	no	—
96	C₈×C₂₃	—	2	2	no	—
97	C₈×C₂₃	#123	1	4	yes	—
98	C₂×#405	C₂×C₂₃	1	1	yes	characteristic
99	C₂×#349	C₂×C₂₃	1	2	yes	—
100	#1376	C₂×C₂₃	1	2	yes	complemented
101	#1375	C₂×C₂₃	1	2	yes	complemented
102	C₂²×C₂₃	—	2	2	no	—
103	C₂²×C₂₃	—	2	2	no	—
104	C₂²×C₂₃	—	4	2	no	—
105	C₂²×C₂₃	—	4	2	no	—
106	C₂²×C₂₃	—	4	2	no	—
107	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
108	C₂²×C₂₃	#432	1	1	yes	characteristic
109	C₂²×C₂₃	#432	1	1	yes	characteristic
110	C₄×C₂₃	—	2	2	no	—
111	C₄×C₂₃	—	4	2	no	—
112	C₄×C₂₃	—	4	2	no	—
113	C₄×C₂₃	#381	1	1	yes	characteristic
114	C₄×C₂₃	#407	1	1	yes	characteristic
115	C₄×C₂₃	#431	1	1	yes	characteristic
116	C₄×C₂₃	#431	1	1	yes	characteristic
117	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
118	C₂³×C₈	C₄×C₂₃	1	1	yes	characteristic
119	C₂×#117	C₄×C₂₃	1	1	yes	characteristic
120	C₂×M₃₂	C₂²×C₂₃	1	4	yes	—
121	#405	—	2	4	no	—
122	C₂×#97	C₄×C₂₃	1	1	yes	characteristic
123	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
124	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
126	#349	—	2	4	no	—
127	C₂×C₂₃	—	2	1	no	—
128	C₂×C₂₃	—	2	1	no	—
129	C₂×C₂₃	—	4	2	no	—
130	C₂×C₂₃	#1817	1	1	yes	characteristic
131	C₂×C₂₃	#2166	1	2	yes	—
132	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
133	C₂×M₁₆	—	2	2	no	—
134	C₂×M₁₆	—	2	2	no	—
135	C₂×M₁₆	D₄×C₂₃	1	4	yes	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
140	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
141	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
142	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
143	#117	—	2	2	no	—
144	#117	C₈×C₂₃	1	4	yes	—
145	C₂×C₄⋊C₄	—	2	1	no	—
146	C₂×C₂²⋊C₄	—	2	1	no	—
147	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
148	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
149	M₃₂	—	2	4	no	—
150	M₃₂	—	2	4	no	—
151	C₂×C₁₆	—	2	4	no	—
152	#97	—	2	2	no	—
153	#97	—	2	2	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₂₃	#12850	1	1	yes	characteristic, direct product factor, Sylow
157	C₂⁴	M₁₆×C₂₃	1	1	yes	characteristic
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
165	C₂²×C₄	C₂×C₈×C₂₃	1	2	yes	—
166	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
167	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
168	M₁₆	—	2	8	no	—
169	C₂×C₈	—	2	8	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
177	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
178	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
179	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
180	C₂×C₈	#41×C₂₃	1	2	yes	—
181	C₂×C₈	#41×C₂₃	1	2	yes	—
182	C₄⋊C₄	—	2	4	no	—
183	C₂²⋊C₄	—	2	4	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	2	1	no	—
186	C₄²	C₂×C₈×C₂₃	1	2	yes	—
187	C₄²	C₂×C₈×C₂₃	1	2	yes	—
188	C₁₆	—	4	4	no	—
189	C₂³	—	2	1	no	—
190	C₂³	—	2	1	no	—
191	C₂³	—	4	2	no	—
192	C₂³	C₂₃×C₂²⋊C₈	1	2	yes	—
193	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
205	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
206	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
207	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
208	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
209	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
210	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
211	C₈	—	2	2	no	—
212	C₈	—	2	2	no	—
213	C₈	—	2	2	no	—
214	C₈	C₂₃×#123	1	4	yes	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
221	C₂²	C₂₃×#432	1	1	yes	characteristic
222	C₂²	C₂₃×#432	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	2	no	—
226	C₄	C₂₃×#381	1	1	yes	characteristic
227	C₄	C₂₃×#407	1	1	yes	characteristic
228	C₄	C₂₃×#431	1	1	yes	characteristic
229	C₄	C₂₃×#431	1	1	yes	characteristic
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	—	4	2	no	—
233	C₂	C₂₃×#1817	1	1	yes	characteristic
234	C₂	C₂₃×#2166	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	#117
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	#432
prime radical	C₂⁴×C₂₃	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₂₃×#97 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1376	C₂	semidirect
4. C₂₃×#1375	C₂	semidirect
71. #12850	C₂₃	direct (N × H)
100. #1376	C₂×C₂₃	semidirect
101. #1375	C₂×C₂₃	semidirect
156. C₂₃	#12850	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×#405	2
C₂×C₂₃×#349	2
C₂₃×#1376	2
C₂₃×#1375	2
#12850	23

Subgroups — C23×#12850 #19160476

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12850 #19160476