Subgroups — C₂₃×#17961 #19165587

Order 5888.

598 conjugacy classes of subgroups.

The subgroup lattice has 246 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×#1593	C₂	1	4	yes	maximal, complemented
4	C₂₃×#1579	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1566	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1493	C₂	1	2	yes	maximal
7	C₂×C₂₃×#322	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#533	C₂²	1	2	yes	—
9	C₂₃×#500	C₄	1	1	yes	characteristic
10	C₂₃×#494	C₄	1	1	yes	characteristic
11	C₂₃×#487	C₄	1	2	yes	—
12	C₂₃×#467	C₄	1	4	yes	—
13	C₂₃×#446	—	2	2	no	—
14	C₂₃×#446	C₂²	1	4	yes	—
15	C₈×Q₈×C₂₃	—	2	2	no	—
16	C₈×Q₈×C₂₃	C₂²	1	4	yes	—
17	C₄×Q₁₆×C₂₃	C₂²	1	4	yes	—
18	C₂₃×#424	—	2	2	no	—
19	C₂₃×#423	C₂²	1	2	yes	—
20	C₂₃×#405	—	2	2	no	—
21	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#334	C₂²	1	4	yes	—
25	C₂₃×Q₈⋊C₈	C₂²	1	8	yes	—
26	C₂₃×#322	C₂²	1	2	yes	—
27	C₂₃×#322	C₂²	1	2	yes	—
28	C₂×Q₁₆×C₂₃	C₂×C₄	1	4	yes	—
29	C₂×M₁₆×C₂₃	—	2	2	no	—
30	C₂²×C₈×C₂₃	—	2	1	no	—
31	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₂₃	D₄	1	2	yes	—
33	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
34	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
35	C₂₃×#125	—	2	2	no	—
36	C₂₃×C₂²⋊Q₈	—	2	2	no	—
37	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
38	C₄×Q₈×C₂₃	—	2	2	no	—
39	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
40	C₂₃×#117	—	2	2	no	—
41	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
42	C₂₃×#107	C₂×C₄	1	4	yes	—
43	C₂₃×C₄⋊C₈	—	2	4	no	—
44	C₂₃×C₄⋊C₈	—	2	4	no	—
45	C₂₃×C₄⋊C₈	—	2	2	no	—
46	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
47	C₂₃×#103	C₂×C₄	1	8	yes	—
48	C₂₃×#97	—	2	4	no	—
49	C₄×C₈×C₂₃	—	2	4	no	—
50	C₄×C₈×C₂₃	—	2	1	no	—
51	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₂₃	D₄	1	2	yes	—
54	C₄×C₈×C₂₃	C₂³	1	2	yes	—
55	C₄×C₈×C₂₃	C₂³	1	2	yes	—
56	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
58	C₂×Q₈×C₂₃	—	2	2	no	—
59	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
63	Q₁₆×C₂₃	—	2	8	no	—
64	M₁₆×C₂₃	—	4	4	no	—
65	C₂×C₈×C₂₃	—	2	4	no	—
66	C₂×C₈×C₂₃	—	2	4	no	—
67	C₂×C₈×C₂₃	—	2	4	no	—
68	C₂×C₈×C₂₃	—	2	2	no	—
69	C₂×C₈×C₂₃	—	4	2	no	—
70	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
74	C₄⋊C₄×C₂₃	—	2	2	no	—
75	C₄⋊C₄×C₂₃	—	2	2	no	—
76	C₄⋊C₄×C₂₃	—	2	2	no	—
77	C₄⋊C₄×C₂₃	—	2	4	no	—
78	C₄⋊C₄×C₂₃	—	2	4	no	—
79	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
80	C₂²⋊C₄×C₂₃	—	2	4	no	—
81	C₄²×C₂₃	—	2	4	no	—
82	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₂₃	#41	1	1	yes	characteristic
85	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
86	#17961	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
88	Q₈×C₂₃	—	2	8	no	—
89	Q₈×C₂₃	—	4	4	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	—	2	4	no	—
94	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₂₃	#137	1	4	yes	—
98	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
101	C₈×C₂₃	—	2	2	no	—
102	C₈×C₂₃	—	2	2	no	—
103	C₈×C₂₃	—	2	2	no	—
104	C₈×C₂₃	—	4	2	no	—
105	C₈×C₂₃	—	4	4	no	—
106	C₈×C₂₃	#131	1	4	yes	—
107	#3170	C₂×C₂₃	1	1	yes	characteristic
108	#2960	C₂×C₂₃	1	2	yes	—
109	#1593	C₂×C₂₃	1	4	yes	complemented
110	#1579	C₂×C₂₃	1	4	yes	complemented
111	#1566	C₂×C₂₃	1	1	yes	characteristic
112	#1493	C₂×C₂₃	1	2	yes	—
113	C₂×#322	C₂×C₂₃	1	1	yes	characteristic
114	C₂²×C₂₃	—	2	1	no	—
115	C₂²×C₂₃	—	2	1	no	—
116	C₂²×C₂₃	#441	1	2	yes	—
117	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
118	C₄×C₂₃	—	2	1	no	—
119	C₄×C₂₃	—	2	1	no	—
120	C₄×C₂₃	—	4	4	no	—
121	C₄×C₂₃	—	4	4	no	—
122	C₄×C₂₃	#441	1	2	yes	—
123	C₄×C₂₃	#444	1	2	yes	—
124	C₄×C₂₃	C₂×#131	1	2	yes	—
125	C₄×C₂₃	C₂×#137	1	2	yes	—
126	#533	C₂²×C₂₃	1	2	yes	—
127	#500	C₄×C₂₃	1	1	yes	characteristic
128	#494	C₄×C₂₃	1	1	yes	characteristic
129	#487	C₄×C₂₃	1	2	yes	—
130	#467	C₄×C₂₃	1	4	yes	—
131	#446	—	2	2	no	—
132	#446	C₂²×C₂₃	1	4	yes	—
133	C₈×Q₈	—	2	2	no	—
134	C₈×Q₈	C₂²×C₂₃	1	4	yes	—
135	C₄×Q₁₆	C₂²×C₂₃	1	4	yes	—
136	#424	—	2	2	no	—
137	#423	C₂²×C₂₃	1	2	yes	—
138	#405	—	2	2	no	—
139	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
141	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
142	#334	C₂²×C₂₃	1	4	yes	—
143	Q₈⋊C₈	C₂²×C₂₃	1	8	yes	—
144	#322	C₂²×C₂₃	1	2	yes	—
145	#322	C₂²×C₂₃	1	2	yes	—
146	C₂×C₂₃	—	2	2	no	—
147	C₂×C₂₃	#2931	1	1	yes	characteristic
148	C₂×C₂₃	C₂×#441	1	1	yes	characteristic
149	C₂×C₂₃	C₂×#444	1	1	yes	characteristic
150	C₂×Q₁₆	C₂×C₄×C₂₃	1	4	yes	—
151	C₂×M₁₆	—	2	2	no	—
152	C₂²×C₈	—	2	1	no	—
153	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
154	C₂²×C₈	D₄×C₂₃	1	2	yes	—
155	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
156	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
157	#125	—	2	2	no	—
158	C₂²⋊Q₈	—	2	2	no	—
159	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
160	C₄×Q₈	—	2	2	no	—
161	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
162	#117	—	2	2	no	—
163	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
164	#107	C₂×C₄×C₂₃	1	4	yes	—
165	C₄⋊C₈	—	2	4	no	—
166	C₄⋊C₈	—	2	4	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
169	#103	C₂×C₄×C₂₃	1	8	yes	—
170	#97	—	2	4	no	—
171	C₄×C₈	—	2	4	no	—
172	C₄×C₈	—	2	1	no	—
173	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
174	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
175	C₄×C₈	D₄×C₂₃	1	2	yes	—
176	C₄×C₈	C₂³×C₂₃	1	2	yes	—
177	C₄×C₈	C₂³×C₂₃	1	2	yes	—
178	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
179	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
180	C₂₃	#17961	1	1	yes	characteristic, direct product factor, Sylow
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
183	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
185	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
186	Q₁₆	—	2	8	no	—
187	M₁₆	—	4	4	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
194	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
195	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
196	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₄²	—	2	4	no	—
205	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
206	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
207	C₄²	#41×C₂₃	1	1	yes	characteristic
208	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
209	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
210	Q₈	—	2	8	no	—
211	Q₈	—	4	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
217	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
218	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
219	C₂×C₄	C₂₃×#137	1	4	yes	—
220	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
222	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	4	2	no	—
227	C₈	—	4	4	no	—
228	C₈	C₂₃×#131	1	4	yes	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	C₂₃×#441	1	2	yes	—
232	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	4	no	—
236	C₄	—	4	4	no	—
237	C₄	C₂₃×#441	1	2	yes	—
238	C₄	C₂₃×#444	1	2	yes	—
239	C₄	C₂×C₂₃×#131	1	2	yes	—
240	C₄	C₂×C₂₃×#137	1	2	yes	—
241	C₂	—	2	2	no	—
242	C₂	C₂₃×#2931	1	1	yes	characteristic
243	C₂	C₂×C₂₃×#441	1	1	yes	characteristic
244	C₂	C₂×C₂₃×#444	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×#1593	C₂	semidirect
4. C₂₃×#1579	C₂	semidirect
86. #17961	C₂₃	direct (N × H)
109. #1593	C₂×C₂₃	semidirect
110. #1579	C₂×C₂₃	semidirect
180. C₂₃	#17961	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×#1593	2
C₂₃×#1579	2
C₂₃×#1566	2
C₂₃×#1493	2
C₂×C₂₃×#322	2
#17961	23

Subgroups — C23×#17961 #19165587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17961 #19165587