Subgroups — C₂₃×#8038 #19155664

Order 5888.

650 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#431	C₂	1	2	yes	maximal
2	C₂₃×#2285	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1296	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₃×#117	C₄	1	2	yes	—
5	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
6	C₂₃×#431	—	2	4	no	—
7	C₂₃×#431	C₄	1	8	yes	complemented
8	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
9	C₂×C₂₃×C₂²⋊C₈	—	2	2	no	—
10	C₂×C₄×C₈×C₂₃	—	2	2	no	—
11	C₂₃×#384	C₂²	1	2	yes	—
12	C₂₃×#382	C₄	1	2	yes	—
13	C₂₃×#380	—	2	2	no	—
14	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
15	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₂₃	D₄	1	2	yes	—
17	C₂²×C₈×C₂₃	—	2	4	no	—
18	C₂₃×#117	—	4	4	no	—
19	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
20	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
21	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
22	C₂×C₄²×C₂₃	—	2	2	no	—
23	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₂₃	D₄	1	2	yes	—
26	C₂×C₄²×C₂₃	D₄	1	2	yes	—
27	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
28	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
29	C₂₃×C₄⋊C₈	—	2	4	no	—
30	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
31	C₂₃×C₂²⋊C₈	—	2	8	no	—
32	C₄×C₈×C₂₃	—	2	8	no	—
33	C₂₃×#95	—	2	4	no	—
34	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	—	2	2	no	—
36	C₂²×C₄×C₂₃	—	2	2	no	—
37	C₂²×C₄×C₂₃	—	2	8	no	—
38	C₂²×C₄×C₂₃	—	2	4	no	—
39	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
40	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
42	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₂₃	—	2	8	no	—
48	C₂×C₈×C₂₃	—	4	8	no	—
49	C₄⋊C₄×C₂₃	—	4	4	no	—
50	C₄⋊C₄×C₂₃	—	4	8	no	—
51	C₂²⋊C₄×C₂₃	—	4	4	no	—
52	C₄²×C₂₃	—	4	4	no	—
53	C₄²×C₂₃	C₄²	1	2	yes	—
54	C₄²×C₂₃	C₂²⋊C₄	1	4	yes	—
55	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
56	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
57	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
58	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
60	#8038	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₂₃	—	2	1	no	—
62	C₂³×C₂₃	—	2	1	no	—
63	C₂³×C₂₃	—	2	4	no	—
64	C₂³×C₂₃	#95	1	2	yes	—
65	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₂₃	—	2	1	no	—
67	C₂×C₄×C₂₃	—	2	1	no	—
68	C₂×C₄×C₂₃	—	2	4	no	—
69	C₂×C₄×C₂₃	—	2	4	no	—
70	C₂×C₄×C₂₃	—	2	4	no	—
71	C₂×C₄×C₂₃	—	2	8	no	—
72	C₂×C₄×C₂₃	—	2	8	no	—
73	C₂×C₄×C₂₃	—	4	4	no	—
74	C₂×C₄×C₂₃	—	4	8	no	—
75	C₂×C₄×C₂₃	#95	1	4	yes	—
76	C₂×C₄×C₂₃	#95	1	2	yes	—
77	C₂×C₄×C₂₃	#100	1	2	yes	—
78	C₂×C₄×C₂₃	#101	1	2	yes	—
79	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
81	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
83	C₂×C₄×C₂₃	#135	1	8	yes	—
84	C₈×C₂₃	—	4	8	no	—
85	C₂×#431	C₂×C₂₃	1	2	yes	—
86	#2285	C₂×C₂₃	1	1	yes	characteristic
87	#1296	C₂×C₂₃	1	4	yes	complemented
88	C₂²×C₂₃	—	2	4	no	—
89	C₂²×C₂₃	—	2	2	no	—
90	C₂²×C₂₃	—	4	2	no	—
91	C₂²×C₂₃	—	4	2	no	—
92	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
93	C₂²×C₂₃	C₂×#100	1	1	yes	characteristic
94	C₂²×C₂₃	C₂×#101	1	1	yes	characteristic
95	C₂²×C₂₃	#416	1	4	yes	—
96	C₄×C₂₃	—	2	4	no	—
97	C₄×C₂₃	—	2	1	no	—
98	C₄×C₂₃	—	2	1	no	—
99	C₄×C₂₃	—	8	4	no	—
100	C₄×C₂₃	#343	1	4	yes	—
101	C₄×C₂₃	#416	1	4	yes	—
102	C₄×C₂₃	#427	1	4	yes	—
103	C₂×#117	C₄×C₂₃	1	2	yes	—
104	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
105	#431	—	2	4	no	—
106	#431	C₄×C₂₃	1	8	yes	complemented
107	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
108	C₂×C₂²⋊C₈	—	2	2	no	—
109	C₂×C₄×C₈	—	2	2	no	—
110	#384	C₂²×C₂₃	1	2	yes	—
111	#382	C₄×C₂₃	1	2	yes	—
112	#380	—	2	2	no	—
113	C₄×C₄⋊C₄	—	2	2	no	—
114	C₂×C₂₃	—	4	2	no	—
115	C₂×C₂₃	#1543	1	2	yes	—
116	C₂×C₂₃	#1544	1	2	yes	—
117	C₂×C₂₃	#1735	1	2	yes	—
118	C₂×C₂₃	C₂×#343	1	1	yes	characteristic
119	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
120	C₂³×C₄	D₄×C₂₃	1	2	yes	—
121	C₂²×C₈	—	2	4	no	—
122	#117	—	4	4	no	—
123	C₂×C₄⋊C₄	—	2	4	no	—
124	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
125	C₂×C₂²⋊C₄	—	2	2	no	—
126	C₂×C₄²	—	2	2	no	—
127	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
128	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
129	C₂×C₄²	D₄×C₂₃	1	2	yes	—
130	C₂×C₄²	D₄×C₂₃	1	2	yes	—
131	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
132	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
133	C₄⋊C₈	—	2	4	no	—
134	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
135	C₂²⋊C₈	—	2	8	no	—
136	C₄×C₈	—	2	8	no	—
137	#95	—	2	4	no	—
138	C₂₃	#8038	1	1	yes	characteristic, direct product factor, Sylow
139	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	2	8	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
145	C₂²×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
146	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
147	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
148	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
149	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
150	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
152	C₂×C₈	—	2	8	no	—
153	C₂×C₈	—	4	8	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄⋊C₄	—	4	8	no	—
156	C₂²⋊C₄	—	4	4	no	—
157	C₄²	—	4	4	no	—
158	C₄²	C₄²×C₂₃	1	2	yes	—
159	C₄²	C₂²⋊C₄×C₂₃	1	4	yes	—
160	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
161	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
162	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
163	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
164	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
165	C₂³	—	2	1	no	—
166	C₂³	—	2	1	no	—
167	C₂³	—	2	4	no	—
168	C₂³	C₂₃×#95	1	2	yes	—
169	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	8	no	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	C₂₃×#95	1	4	yes	—
180	C₂×C₄	C₂₃×#95	1	2	yes	—
181	C₂×C₄	C₂₃×#100	1	2	yes	—
182	C₂×C₄	C₂₃×#101	1	2	yes	—
183	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
184	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
185	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
187	C₂×C₄	C₂₃×#135	1	8	yes	—
188	C₈	—	4	8	no	—
189	C₂²	—	2	4	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	4	2	no	—
193	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
194	C₂²	C₂×C₂₃×#100	1	1	yes	characteristic
195	C₂²	C₂×C₂₃×#101	1	1	yes	characteristic
196	C₂²	C₂₃×#416	1	4	yes	—
197	C₄	—	2	4	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	—	8	4	no	—
201	C₄	C₂₃×#343	1	4	yes	—
202	C₄	C₂₃×#416	1	4	yes	—
203	C₄	C₂₃×#427	1	4	yes	—
204	C₂	—	4	2	no	—
205	C₂	C₂₃×#1543	1	2	yes	—
206	C₂	C₂₃×#1544	1	2	yes	—
207	C₂	C₂₃×#1735	1	2	yes	—
208	C₂	C₂×C₂₃×#343	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₂₃	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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
3. C₂₃×#1296	C₂	semidirect
7. C₂₃×#431	C₄	semidirect
60. #8038	C₂₃	direct (N × H)
87. #1296	C₂×C₂₃	semidirect
106. #431	C₄×C₂₃	semidirect
138. C₂₃	#8038	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×#431	2
C₂₃×#2285	2
C₂₃×#1296	2
#8038	23

Subgroups — C23×#8038 #19155664

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8038 #19155664