Subgroups — C₂₃×#10865 #19158491

Order 5888.

766 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#108	C₂	1	2	yes	maximal
2	C₂²×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#341	C₂	1	2	yes	maximal
4	C₂₃×#1728	C₂	1	2	yes	maximal
5	C₂²×M₁₆×C₂₃	C₂²	1	4	yes	—
6	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#108	—	2	4	no	—
10	C₂×C₂₃×#108	—	2	4	no	—
11	C₂×C₂₃×#108	C₄	1	8	yes	complemented
12	C₂×C₂₃×#97	—	2	4	no	—
13	C₂×C₂₃×#97	C₄	1	4	yes	—
14	C₂₃×#341	—	2	4	no	—
15	C₂₃×#336	—	2	4	no	—
16	C₂³×C₄×C₂₃	Q₈	1	2	yes	—
17	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₂₃	—	2	4	no	—
19	C₂×M₁₆×C₂₃	—	2	4	no	—
20	C₂×M₁₆×C₂₃	—	2	4	no	—
21	C₂×M₁₆×C₂₃	—	2	4	no	—
22	C₂×M₁₆×C₂₃	—	2	4	no	—
23	C₂×M₁₆×C₂₃	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₂₃	—	2	2	no	—
25	C₂²×C₈×C₂₃	—	2	2	no	—
26	C₂²×C₈×C₂₃	—	2	4	no	—
27	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₂₃	D₄	1	2	yes	—
31	C₂²×C₈×C₂₃	D₄	1	4	yes	—
32	C₂²×C₈×C₂₃	D₄	1	2	yes	—
33	C₂²×C₈×C₂₃	Q₈	1	2	yes	—
34	C₂²×C₈×C₂₃	Q₈	1	2	yes	—
35	C₂×C₄²×C₂₃	—	2	2	no	—
36	C₂×C₄²×C₂₃	—	2	2	no	—
37	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₂₃	D₄	1	2	yes	—
39	C₂₃×#108	—	2	16	no	—
40	C₂₃×#97	—	2	8	no	—
41	C₂⁴×C₂₃	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	—	2	4	no	—
43	C₂²×C₄×C₂₃	—	4	4	no	—
44	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	#41	1	2	yes	—
54	C₂²×C₄×C₂₃	#41	1	2	yes	—
55	M₁₆×C₂₃	—	2	8	no	—
56	M₁₆×C₂₃	—	2	8	no	—
57	M₁₆×C₂₃	—	4	4	no	—
58	M₁₆×C₂₃	—	4	4	no	—
59	C₂×C₈×C₂₃	—	2	2	no	—
60	C₂×C₈×C₂₃	—	2	2	no	—
61	C₂×C₈×C₂₃	—	2	4	no	—
62	C₂×C₈×C₂₃	—	2	4	no	—
63	C₂×C₈×C₂₃	—	2	8	no	—
64	C₂×C₈×C₂₃	—	4	4	no	—
65	C₂×C₈×C₂₃	—	4	4	no	—
66	C₂×C₈×C₂₃	C₄²	1	4	yes	—
67	C₂×C₈×C₂₃	C₂²⋊C₄	1	8	yes	—
68	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
69	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
70	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
71	C₂×C₈×C₂₃	C₄⋊C₄	1	8	yes	—
72	C₄²×C₂₃	—	2	2	no	—
73	C₄²×C₂₃	—	2	2	no	—
74	C₄²×C₂₃	—	2	4	no	—
75	#10865	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₂₃	—	2	4	no	—
77	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂³×C₂₃	C₄×Q₈	1	2	yes	—
79	C₂³×C₂₃	C₂²⋊Q₈	1	2	yes	—
80	C₂³×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
81	C₂³×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₂₃	—	2	4	no	—
83	C₂×C₄×C₂₃	—	2	8	no	—
84	C₂×C₄×C₂₃	—	4	8	no	—
85	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
88	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
92	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
96	C₂×C₄×C₂₃	#121	1	2	yes	—
97	C₂×C₄×C₂₃	#121	1	2	yes	—
98	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
99	C₂×C₄×C₂₃	#124	1	2	yes	—
100	C₂×C₄×C₂₃	#125	1	2	yes	—
101	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
102	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
103	C₈×C₂₃	—	2	4	no	—
104	C₈×C₂₃	—	4	8	no	—
105	C₈×C₂₃	#95	1	8	yes	—
106	C₂²×#108	C₂×C₂₃	1	2	yes	—
107	C₂²×#97	C₂×C₂₃	1	1	yes	characteristic
108	C₂×#341	C₂×C₂₃	1	2	yes	—
109	#1728	C₂×C₂₃	1	2	yes	—
110	C₂²×C₂₃	—	2	4	no	—
111	C₂²×C₂₃	—	2	2	no	—
112	C₂²×C₂₃	—	2	2	no	—
113	C₂²×C₂₃	—	2	4	no	—
114	C₂²×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₂₃	#380	1	1	yes	characteristic
116	C₂²×C₂₃	#384	1	1	yes	characteristic
117	C₂²×C₂₃	#389	1	2	yes	—
118	C₂²×C₂₃	#391	1	2	yes	—
119	C₂²×C₂₃	#430	1	4	yes	—
120	C₄×C₂₃	—	4	4	no	—
121	C₄×C₂₃	C₂×#95	1	1	yes	characteristic
122	C₄×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
123	C₄×C₂₃	#380	1	1	yes	characteristic
124	C₄×C₂₃	#382	1	1	yes	characteristic
125	C₄×C₂₃	#389	1	2	yes	—
126	C₄×C₂₃	#390	1	2	yes	—
127	C₂²×M₁₆	C₂²×C₂₃	1	4	yes	—
128	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
129	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
130	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
131	C₂×#108	—	2	4	no	—
132	C₂×#108	—	2	4	no	—
133	C₂×#108	C₄×C₂₃	1	8	yes	complemented
134	C₂×#97	—	2	4	no	—
135	C₂×#97	C₄×C₂₃	1	4	yes	—
136	#341	—	2	4	no	—
137	#336	—	2	4	no	—
138	C₂×C₂₃	—	2	2	no	—
139	C₂×C₂₃	—	2	2	no	—
140	C₂×C₂₃	C₄⋊#95	1	1	yes	characteristic
141	C₂×C₂₃	#1778	1	2	yes	—
142	C₂×C₂₃	#1811	1	2	yes	—
143	C₂×C₂₃	#1849	1	2	yes	—
144	C₂³×C₄	Q₈×C₂₃	1	2	yes	—
145	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
146	C₂×M₁₆	—	2	4	no	—
147	C₂×M₁₆	—	2	4	no	—
148	C₂×M₁₆	—	2	4	no	—
149	C₂×M₁₆	—	2	4	no	—
150	C₂×M₁₆	—	2	4	no	—
151	C₂×M₁₆	C₂×C₄×C₂₃	1	8	yes	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	4	no	—
155	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
156	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
157	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
158	C₂²×C₈	D₄×C₂₃	1	2	yes	—
159	C₂²×C₈	D₄×C₂₃	1	4	yes	—
160	C₂²×C₈	D₄×C₂₃	1	2	yes	—
161	C₂²×C₈	Q₈×C₂₃	1	2	yes	—
162	C₂²×C₈	Q₈×C₂₃	1	2	yes	—
163	C₂×C₄²	—	2	2	no	—
164	C₂×C₄²	—	2	2	no	—
165	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
166	C₂×C₄²	D₄×C₂₃	1	2	yes	—
167	#108	—	2	16	no	—
168	#97	—	2	8	no	—
169	C₂₃	#10865	1	1	yes	characteristic, direct product factor, Sylow
170	C₂⁴	C₂×Q₈×C₂₃	1	1	yes	characteristic
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	4	4	no	—
173	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
174	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
175	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
176	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
177	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
178	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
180	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
181	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
182	C₂²×C₄	#41×C₂₃	1	2	yes	—
183	C₂²×C₄	#41×C₂₃	1	2	yes	—
184	M₁₆	—	2	8	no	—
185	M₁₆	—	2	8	no	—
186	M₁₆	—	4	4	no	—
187	M₁₆	—	4	4	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	8	no	—
193	C₂×C₈	—	4	4	no	—
194	C₂×C₈	—	4	4	no	—
195	C₂×C₈	C₄²×C₂₃	1	4	yes	—
196	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
197	C₂×C₈	C₂²⋊C₄×C₂₃	1	8	yes	—
198	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
199	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
200	C₂×C₈	C₄⋊C₄×C₂₃	1	8	yes	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	4	no	—
204	C₂³	—	2	4	no	—
205	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
206	C₂³	C₄×Q₈×C₂₃	1	2	yes	—
207	C₂³	C₂₃×C₂²⋊Q₈	1	2	yes	—
208	C₂³	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
209	C₂³	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	8	no	—
212	C₂×C₄	—	4	8	no	—
213	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
214	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
215	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
216	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
217	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
218	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
219	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
220	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
221	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
222	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
223	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
224	C₂×C₄	C₂₃×#121	1	2	yes	—
225	C₂×C₄	C₂₃×#121	1	2	yes	—
226	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
227	C₂×C₄	C₂₃×#124	1	2	yes	—
228	C₂×C₄	C₂₃×#125	1	2	yes	—
229	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
230	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
231	C₈	—	2	4	no	—
232	C₈	—	4	8	no	—
233	C₈	C₂₃×#95	1	8	yes	—
234	C₂²	—	2	4	no	—
235	C₂²	—	2	4	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
239	C₂²	C₂₃×#380	1	1	yes	characteristic
240	C₂²	C₂₃×#384	1	1	yes	characteristic
241	C₂²	C₂₃×#389	1	2	yes	—
242	C₂²	C₂₃×#391	1	2	yes	—
243	C₂²	C₂₃×#430	1	4	yes	—
244	C₄	—	4	4	no	—
245	C₄	C₂×C₂₃×#95	1	1	yes	characteristic
246	C₄	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
247	C₄	C₂₃×#380	1	1	yes	characteristic
248	C₄	C₂₃×#382	1	1	yes	characteristic
249	C₄	C₂₃×#389	1	2	yes	—
250	C₄	C₂₃×#390	1	2	yes	—
251	C₂	—	2	2	no	—
252	C₂	—	2	2	no	—
253	C₂	C₂₃×C₄⋊#95	1	1	yes	characteristic
254	C₂	C₂₃×#1778	1	2	yes	—
255	C₂	C₂₃×#1811	1	2	yes	—
256	C₂	C₂₃×#1849	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₂₃	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₂×C₂₃×#108	C₄	semidirect
75. #10865	C₂₃	direct (N × H)
133. C₂×#108	C₄×C₂₃	semidirect
169. C₂₃	#10865	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#108	2
C₂²×C₂₃×#97	2
C₂×C₂₃×#341	2
C₂₃×#1728	2
#10865	23

Subgroups — C23×#10865 #19158491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10865 #19158491