Subgroups — C₂₃×#11638 #19159264

Order 5888.

300 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2960	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2162	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2162	C₂	1	1	yes	maximal, characteristic
4	C₂₃×Q₈⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₂₃×#533	C₄	1	1	yes	characteristic
6	C₂₃×#446	—	2	1	no	—
7	C₂₃×#446	C₄	1	2	yes	—
8	C₈×Q₈×C₂₃	—	2	1	no	—
9	C₈×Q₈×C₂₃	C₂²	1	2	yes	—
10	C₂₃×#424	—	2	1	no	—
11	C₂₃×#423	C₄	1	1	yes	characteristic
12	C₂₃×#405	—	2	1	no	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×C₄⋊C₁₆	—	2	1	no	—
15	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₂₃×C₂²⋊C₁₆	—	2	1	no	—
17	C₂₃×C₂²⋊C₁₆	—	2	1	no	—
18	C₄×C₁₆×C₂₃	—	2	1	no	—
19	C₄×C₁₆×C₂₃	C₂²	1	2	yes	—
20	C₂×M₁₆×C₂₃	—	2	1	no	—
21	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
23	C₂₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
24	C₂₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
25	C₂₃×#125	—	2	1	no	—
26	C₂₃×C₂²⋊Q₈	—	2	1	no	—
27	C₂₃×C₂²⋊Q₈	C₈	1	2	yes	—
28	C₄×Q₈×C₂₃	—	2	1	no	—
29	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
30	C₂₃×#117	—	2	1	no	—
31	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₂₃	—	2	2	no	—
33	C₂×C₁₆×C₂₃	—	2	2	no	—
34	C₂₃×C₄⋊C₈	—	2	2	no	—
35	C₂₃×C₄⋊C₈	—	2	2	no	—
36	C₂₃×C₄⋊C₈	—	2	1	no	—
37	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
38	C₂₃×#97	—	2	2	no	—
39	C₄×C₈×C₂₃	—	2	2	no	—
40	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₂₃	—	2	1	no	—
45	C₂×Q₈×C₂₃	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
49	M₁₆×C₂₃	—	4	2	no	—
50	C₂×C₈×C₂₃	—	2	2	no	—
51	C₂×C₈×C₂₃	—	2	2	no	—
52	C₂×C₈×C₂₃	—	4	1	no	—
53	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₄⋊C₄×C₂₃	—	2	1	no	—
58	C₄⋊C₄×C₂₃	—	2	1	no	—
59	C₄⋊C₄×C₂₃	—	2	1	no	—
60	C₄⋊C₄×C₂₃	—	2	2	no	—
61	C₄⋊C₄×C₂₃	—	2	2	no	—
62	C₄⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
63	C₂²⋊C₄×C₂₃	—	2	2	no	—
64	C₄²×C₂₃	—	2	2	no	—
65	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
67	C₄²×C₂₃	M₁₆	1	1	yes	characteristic
68	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
69	C₁₆×C₂₃	—	4	2	no	—
70	C₁₆×C₂₃	—	4	2	no	—
71	#11638	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂³×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
73	Q₈×C₂₃	—	2	4	no	—
74	Q₈×C₂₃	—	4	2	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	1	no	—
77	C₂×C₄×C₂₃	—	2	1	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
82	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
87	C₈×C₂₃	—	2	1	no	—
88	C₈×C₂₃	—	2	1	no	—
89	C₈×C₂₃	—	2	1	no	—
90	C₈×C₂₃	—	4	2	no	—
91	C₈×C₂₃	#137	1	2	yes	—
92	#2960	C₂×C₂₃	1	1	yes	characteristic
93	#2162	C₂×C₂₃	1	1	yes	characteristic
94	#2162	C₂×C₂₃	1	1	yes	characteristic
95	Q₈⋊C₁₆	C₂×C₂₃	1	4	yes	complemented
96	C₂²×C₂₃	—	2	1	no	—
97	C₂²×C₂₃	—	2	1	no	—
98	C₂²×C₂₃	—	2	1	no	—
99	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
100	C₄×C₂₃	—	2	1	no	—
101	C₄×C₂₃	—	2	1	no	—
102	C₄×C₂₃	—	2	1	no	—
103	C₄×C₂₃	—	4	2	no	—
104	C₄×C₂₃	—	4	2	no	—
105	C₄×C₂₃	#350	1	2	yes	—
106	C₄×C₂₃	#419	1	1	yes	characteristic
107	C₄×C₂₃	#421	1	1	yes	characteristic
108	C₄×C₂₃	#504	1	2	yes	—
109	#533	C₄×C₂₃	1	1	yes	characteristic
110	#446	—	2	1	no	—
111	#446	C₄×C₂₃	1	2	yes	—
112	C₈×Q₈	—	2	1	no	—
113	C₈×Q₈	C₂²×C₂₃	1	2	yes	—
114	#424	—	2	1	no	—
115	#423	C₄×C₂₃	1	1	yes	characteristic
116	#405	—	2	1	no	—
117	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
118	C₄⋊C₁₆	—	2	1	no	—
119	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	—
120	C₂²⋊C₁₆	—	2	1	no	—
121	C₂²⋊C₁₆	—	2	1	no	—
122	C₄×C₁₆	—	2	1	no	—
123	C₄×C₁₆	C₂²×C₂₃	1	2	yes	—
124	C₂×C₂₃	—	4	1	no	—
125	C₂×C₂₃	#1479	1	1	yes	characteristic
126	C₂×C₂₃	#2113	1	1	yes	characteristic
127	C₂×C₂₃	C₂×#350	1	1	yes	characteristic
128	C₂×M₁₆	—	2	1	no	—
129	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
130	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
131	C₄⋊Q₈	C₈×C₂₃	1	1	yes	characteristic
132	C₄⋊Q₈	C₈×C₂₃	1	1	yes	characteristic
133	#125	—	2	1	no	—
134	C₂²⋊Q₈	—	2	1	no	—
135	C₂²⋊Q₈	C₈×C₂₃	1	2	yes	—
136	C₄×Q₈	—	2	1	no	—
137	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
138	#117	—	2	1	no	—
139	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
140	C₂×C₁₆	—	2	2	no	—
141	C₂×C₁₆	—	2	2	no	—
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	2	2	no	—
144	C₄⋊C₈	—	2	1	no	—
145	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
146	#97	—	2	2	no	—
147	C₄×C₈	—	2	2	no	—
148	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
149	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
150	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
151	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
152	C₂₃	#11638	1	1	yes	characteristic, direct product factor, Sylow
153	C₂×Q₈	—	2	1	no	—
154	C₂×Q₈	C₂×C₈×C₂₃	1	2	yes	—
155	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
157	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
158	M₁₆	—	4	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	4	1	no	—
162	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
163	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
164	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
165	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	C₂×C₈×C₂₃	1	2	yes	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
175	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
176	C₄²	M₁₆×C₂₃	1	1	yes	characteristic
177	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
178	C₁₆	—	4	2	no	—
179	C₁₆	—	4	2	no	—
180	C₂³	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
181	Q₈	—	2	4	no	—
182	Q₈	—	4	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
190	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
191	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
192	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
193	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
194	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
195	C₈	—	2	1	no	—
196	C₈	—	2	1	no	—
197	C₈	—	2	1	no	—
198	C₈	—	4	2	no	—
199	C₈	C₂₃×#137	1	2	yes	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	4	2	no	—
208	C₄	—	4	2	no	—
209	C₄	C₂₃×#350	1	2	yes	—
210	C₄	C₂₃×#419	1	1	yes	characteristic
211	C₄	C₂₃×#421	1	1	yes	characteristic
212	C₄	C₂₃×#504	1	2	yes	—
213	C₂	—	4	1	no	—
214	C₂	C₂₃×#1479	1	1	yes	characteristic
215	C₂	C₂₃×#2113	1	1	yes	characteristic
216	C₂	C₂×C₂₃×#350	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₂₃×Q₈⋊C₁₆	C₂	semidirect
71. #11638	C₂₃	direct (N × H)
95. Q₈⋊C₁₆	C₂×C₂₃	semidirect
152. C₂₃	#11638	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×#2960	2
C₂₃×#2162	2
C₂₃×Q₈⋊C₁₆	2
#11638	23

Subgroups — C23×#11638 #19159264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11638 #19159264