Subgroups — C₂₃×#17980 #19165606

Order 5888.

590 conjugacy classes of subgroups.

The subgroup lattice has 242 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	C₂₃×#2960	C₂	1	2	yes	maximal
3	C₂₃×#1587	C₂	1	8	yes	maximal, complemented
4	C₂₃×#1566	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1493	C₂	1	2	yes	maximal
6	C₂×C₂₃×#322	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#533	C₂²	1	2	yes	—
8	C₂₃×#500	C₄	1	1	yes	characteristic
9	C₂₃×#494	C₄	1	1	yes	characteristic
10	C₂₃×#487	C₄	1	2	yes	—
11	C₂₃×#467	C₄	1	4	yes	—
12	C₂₃×#446	—	2	2	no	—
13	C₂₃×#446	C₂²	1	4	yes	—
14	C₈×Q₈×C₂₃	—	2	2	no	—
15	C₈×Q₈×C₂₃	C₂²	1	4	yes	—
16	C₄×Q₁₆×C₂₃	C₂²	1	4	yes	—
17	C₂₃×#424	—	2	2	no	—
18	C₂₃×#423	C₂²	1	2	yes	—
19	C₂₃×#405	—	2	2	no	—
20	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#334	C₂²	1	4	yes	—
24	C₂₃×Q₈⋊C₈	C₂²	1	8	yes	—
25	C₂₃×#322	C₂²	1	4	yes	—
26	C₂×Q₁₆×C₂₃	C₂×C₄	1	4	yes	—
27	C₂×M₁₆×C₂₃	—	2	2	no	—
28	C₂²×C₈×C₂₃	—	2	1	no	—
29	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₂₃	D₄	1	2	yes	—
31	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
32	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
33	C₂₃×#125	—	2	2	no	—
34	C₂₃×C₂²⋊Q₈	—	2	2	no	—
35	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
36	C₄×Q₈×C₂₃	—	2	2	no	—
37	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
38	C₂₃×#117	—	2	2	no	—
39	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
40	C₂₃×#107	C₂×C₄	1	4	yes	—
41	C₂₃×C₄⋊C₈	—	2	4	no	—
42	C₂₃×C₄⋊C₈	—	2	4	no	—
43	C₂₃×C₄⋊C₈	—	2	2	no	—
44	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
45	C₂₃×#103	C₂×C₄	1	8	yes	—
46	C₂₃×#97	—	2	4	no	—
47	C₄×C₈×C₂₃	—	2	4	no	—
48	C₄×C₈×C₂₃	—	2	1	no	—
49	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₂₃	C₂³	1	2	yes	—
54	C₄×C₈×C₂₃	C₂³	1	2	yes	—
55	C₄×C₈×C₂₃	C₂³	1	2	yes	—
56	C₂×Q₈×C₂₃	—	2	2	no	—
57	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
61	Q₁₆×C₂₃	—	4	4	no	—
62	M₁₆×C₂₃	—	4	4	no	—
63	C₂×C₈×C₂₃	—	2	4	no	—
64	C₂×C₈×C₂₃	—	2	4	no	—
65	C₂×C₈×C₂₃	—	2	2	no	—
66	C₂×C₈×C₂₃	—	2	4	no	—
67	C₂×C₈×C₂₃	—	4	2	no	—
68	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
72	C₄⋊C₄×C₂₃	—	2	2	no	—
73	C₄⋊C₄×C₂₃	—	2	2	no	—
74	C₄⋊C₄×C₂₃	—	2	2	no	—
75	C₄⋊C₄×C₂₃	—	2	4	no	—
76	C₄⋊C₄×C₂₃	—	2	4	no	—
77	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
78	C₂²⋊C₄×C₂₃	—	2	4	no	—
79	C₄²×C₂₃	—	2	4	no	—
80	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₂₃	#41	1	1	yes	characteristic
83	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
84	#17980	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
86	Q₈×C₂₃	—	2	8	no	—
87	Q₈×C₂₃	—	4	4	no	—
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	4	no	—
91	C₂×C₄×C₂₃	—	2	4	no	—
92	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	#137	1	4	yes	—
96	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
99	C₈×C₂₃	—	2	4	no	—
100	C₈×C₂₃	—	2	2	no	—
101	C₈×C₂₃	—	2	2	no	—
102	C₈×C₂₃	—	2	2	no	—
103	C₈×C₂₃	—	2	2	no	—
104	C₈×C₂₃	—	4	4	no	—
105	#3170	C₂×C₂₃	1	1	yes	characteristic
106	#2960	C₂×C₂₃	1	2	yes	—
107	#1587	C₂×C₂₃	1	8	yes	complemented
108	#1566	C₂×C₂₃	1	1	yes	characteristic
109	#1493	C₂×C₂₃	1	2	yes	—
110	C₂×#322	C₂×C₂₃	1	1	yes	characteristic
111	C₂²×C₂₃	—	2	1	no	—
112	C₂²×C₂₃	—	2	1	no	—
113	C₂²×C₂₃	#441	1	2	yes	—
114	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
115	C₄×C₂₃	—	2	1	no	—
116	C₄×C₂₃	—	2	1	no	—
117	C₄×C₂₃	—	4	4	no	—
118	C₄×C₂₃	—	4	4	no	—
119	C₄×C₂₃	#441	1	2	yes	—
120	C₄×C₂₃	#443	1	2	yes	—
121	C₄×C₂₃	#568	1	2	yes	—
122	C₄×C₂₃	C₂×#137	1	2	yes	—
123	#533	C₂²×C₂₃	1	2	yes	—
124	#500	C₄×C₂₃	1	1	yes	characteristic
125	#494	C₄×C₂₃	1	1	yes	characteristic
126	#487	C₄×C₂₃	1	2	yes	—
127	#467	C₄×C₂₃	1	4	yes	—
128	#446	—	2	2	no	—
129	#446	C₂²×C₂₃	1	4	yes	—
130	C₈×Q₈	—	2	2	no	—
131	C₈×Q₈	C₂²×C₂₃	1	4	yes	—
132	C₄×Q₁₆	C₂²×C₂₃	1	4	yes	—
133	#424	—	2	2	no	—
134	#423	C₂²×C₂₃	1	2	yes	—
135	#405	—	2	2	no	—
136	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
137	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
139	#334	C₂²×C₂₃	1	4	yes	—
140	Q₈⋊C₈	C₂²×C₂₃	1	8	yes	—
141	#322	C₂²×C₂₃	1	4	yes	—
142	C₂×C₂₃	—	2	2	no	—
143	C₂×C₂₃	#2932	1	1	yes	characteristic
144	C₂×C₂₃	C₂×#441	1	1	yes	characteristic
145	C₂×C₂₃	C₂×#443	1	1	yes	characteristic
146	C₂×Q₁₆	C₂×C₄×C₂₃	1	4	yes	—
147	C₂×M₁₆	—	2	2	no	—
148	C₂²×C₈	—	2	1	no	—
149	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
150	C₂²×C₈	D₄×C₂₃	1	2	yes	—
151	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
152	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
153	#125	—	2	2	no	—
154	C₂²⋊Q₈	—	2	2	no	—
155	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
156	C₄×Q₈	—	2	2	no	—
157	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
158	#117	—	2	2	no	—
159	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
160	#107	C₂×C₄×C₂₃	1	4	yes	—
161	C₄⋊C₈	—	2	4	no	—
162	C₄⋊C₈	—	2	4	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
165	#103	C₂×C₄×C₂₃	1	8	yes	—
166	#97	—	2	4	no	—
167	C₄×C₈	—	2	4	no	—
168	C₄×C₈	—	2	1	no	—
169	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
170	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
171	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
172	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
173	C₄×C₈	C₂³×C₂₃	1	2	yes	—
174	C₄×C₈	C₂³×C₂₃	1	2	yes	—
175	C₄×C₈	C₂³×C₂₃	1	2	yes	—
176	C₂₃	#17980	1	1	yes	characteristic, direct product factor, Sylow
177	C₂×Q₈	—	2	2	no	—
178	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
179	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
181	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
182	Q₁₆	—	4	4	no	—
183	M₁₆	—	4	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
190	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
191	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
192	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
199	C₂²⋊C₄	—	2	4	no	—
200	C₄²	—	2	4	no	—
201	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
202	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
203	C₄²	#41×C₂₃	1	1	yes	characteristic
204	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
205	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
206	Q₈	—	2	8	no	—
207	Q₈	—	4	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
213	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
214	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
215	C₂×C₄	C₂₃×#137	1	4	yes	—
216	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
217	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
218	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
219	C₈	—	2	4	no	—
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	—	2	2	no	—
224	C₈	—	4	4	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	C₂₃×#441	1	2	yes	—
228	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	4	no	—
232	C₄	—	4	4	no	—
233	C₄	C₂₃×#441	1	2	yes	—
234	C₄	C₂₃×#443	1	2	yes	—
235	C₄	C₂₃×#568	1	2	yes	—
236	C₄	C₂×C₂₃×#137	1	2	yes	—
237	C₂	—	2	2	no	—
238	C₂	C₂₃×#2932	1	1	yes	characteristic
239	C₂	C₂×C₂₃×#441	1	1	yes	characteristic
240	C₂	C₂×C₂₃×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂³×C₂₃	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1587	C₂	semidirect
84. #17980	C₂₃	direct (N × H)
107. #1587	C₂×C₂₃	semidirect
176. C₂₃	#17980	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
C₂₃×#2960	2
C₂₃×#1587	2
C₂₃×#1566	2
C₂₃×#1493	2
C₂×C₂₃×#322	2
#17980	23

Subgroups — C23×#17980 #19165606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17980 #19165606