Subgroups — C₂₃×#13796 #19161422

Order 5888.

1958 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	C₂×C₄×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal
2	C₂²×C₂₃×#95	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1434	C₂	1	8	yes	maximal, complemented
4	C₂₃×C₂²⋊#95	C₂	1	4	yes	maximal
5	C₂⁴×C₄×C₂₃	C₂²	1	2	yes	—
6	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
8	C₂²×C₄²×C₂₃	C₂²	1	4	yes	—
9	C₄×C₂²⋊C₄×C₂₃	C₄	1	16	yes	complemented
10	C₂×C₂₃×#95	C₄	1	8	yes	complemented
11	C₂×C₂₃×#95	C₂²	1	8	yes	—
12	C₂×C₂₃×#95	C₂²	1	8	yes	—
13	C₂×C₂₃×#95	C₂²	1	4	yes	—
14	C₂×C₂₃×#95	C₂²	1	4	yes	—
15	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
16	C₂³×C₄×C₂₃	—	2	8	no	—
17	C₂³×C₄×C₂₃	—	4	4	no	—
18	C₂³×C₄×C₂₃	—	4	4	no	—
19	C₂³×C₄×C₂₃	—	4	4	no	—
20	C₂³×C₄×C₂₃	C₂×C₄	1	8	yes	—
21	C₂³×C₄×C₂₃	C₂×C₄	1	4	yes	—
22	C₂³×C₄×C₂₃	C₂×C₄	1	8	yes	—
23	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
24	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
25	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
26	C₂×C₂²⋊C₄×C₂₃	—	2	16	no	—
27	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	16	yes	—
28	C₂×C₄²×C₂₃	—	4	8	no	—
29	C₂×C₄²×C₂₃	C₂×C₄	1	16	yes	complemented
30	C₂₃×#95	—	4	8	no	—
31	C₂₃×#95	—	4	8	no	—
32	C₂₃×#95	C₂×C₄	1	16	yes	—
33	C₂₃×#95	C₂×C₄	1	16	yes	complemented
34	C₂⁴×C₂₃	—	2	4	no	—
35	C₂⁴×C₂₃	—	2	2	no	—
36	C₂⁴×C₂₃	—	2	2	no	—
37	C₂⁴×C₂₃	C₂²×C₄	1	4	yes	—
38	C₂⁴×C₂₃	C₂²×C₄	1	2	yes	—
39	C₂⁴×C₂₃	#41	1	8	yes	—
40	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	—	2	8	no	—
42	C₂²×C₄×C₂₃	—	2	16	no	—
43	C₂²×C₄×C₂₃	—	2	16	no	—
44	C₂²×C₄×C₂₃	—	2	8	no	—
45	C₂²×C₄×C₂₃	—	2	16	no	—
46	C₂²×C₄×C₂₃	—	2	8	no	—
47	C₂²×C₄×C₂₃	—	2	8	no	—
48	C₂²×C₄×C₂₃	—	2	8	no	—
49	C₂²×C₄×C₂₃	—	2	8	no	—
50	C₂²×C₄×C₂₃	—	4	16	no	—
51	C₂²×C₄×C₂₃	—	4	8	no	—
52	C₂²×C₄×C₂₃	—	4	8	no	—
53	C₂²×C₄×C₂₃	—	4	8	no	—
54	C₂²×C₄×C₂₃	—	4	32	no	—
55	C₂²×C₄×C₂₃	C₄²	1	16	yes	—
56	C₂²×C₄×C₂₃	C₂²×C₄	1	16	yes	—
57	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
58	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
61	C₂²⋊C₄×C₂₃	—	2	32	no	—
62	C₄²×C₂₃	—	4	16	no	—
63	#13796	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂³×C₂₃	—	2	2	no	—
65	C₂³×C₂₃	—	2	4	no	—
66	C₂³×C₂₃	—	2	8	no	—
67	C₂³×C₂₃	—	2	2	no	—
68	C₂³×C₂₃	—	2	8	no	—
69	C₂³×C₂₃	—	2	16	no	—
70	C₂³×C₂₃	—	2	16	no	—
71	C₂³×C₂₃	C₂×C₄²	1	4	yes	—
72	C₂³×C₂₃	#117	1	16	yes	—
73	C₂³×C₂₃	#117	1	8	yes	—
74	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
75	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
76	C₂³×C₂₃	C₂×#41	1	4	yes	—
77	C₂³×C₂₃	H(5,𝔽₂)	1	4	yes	—
78	C₂³×C₂₃	H(5,𝔽₂)	1	2	yes	—
79	C₂³×C₂₃	2₋¹⁺⁴	1	2	yes	—
80	C₂×C₄×C₂₃	—	2	8	no	—
81	C₂×C₄×C₂₃	—	2	8	no	—
82	C₂×C₄×C₂₃	—	2	8	no	—
83	C₂×C₄×C₂₃	—	2	16	no	—
84	C₂×C₄×C₂₃	—	2	8	no	—
85	C₂×C₄×C₂₃	—	2	16	no	—
86	C₂×C₄×C₂₃	—	2	8	no	—
87	C₂×C₄×C₂₃	—	2	8	no	—
88	C₂×C₄×C₂₃	—	2	16	no	—
89	C₂×C₄×C₂₃	—	4	16	no	—
90	C₂×C₄×C₂₃	—	4	32	no	—
91	C₂×C₄×C₂₃	—	4	16	no	—
92	C₂×C₄×C₂₃	—	4	16	no	—
93	C₂×C₄×C₂₃	—	4	16	no	—
94	C₂×C₄×C₂₃	C₂×C₄²	1	8	yes	—
95	C₂×C₄×C₂²⋊C₄	C₂×C₂₃	1	2	yes	—
96	C₂²×#95	C₂×C₂₃	1	1	yes	characteristic
97	#1434	C₂×C₂₃	1	8	yes	complemented
98	C₂²⋊#95	C₂×C₂₃	1	4	yes	—
99	C₂²×C₂₃	—	2	8	no	—
100	C₂²×C₂₃	—	2	8	no	—
101	C₂²×C₂₃	—	2	32	no	—
102	C₂²×C₂₃	—	2	4	no	—
103	C₂²×C₂₃	—	2	4	no	—
104	C₂²×C₂₃	#376	1	8	yes	—
105	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
106	C₂²×C₂₃	C₂×#117	1	2	yes	—
107	C₂²×C₂₃	C₂×#117	1	4	yes	—
108	C₂²×C₂₃	#518	1	1	yes	characteristic
109	C₂²×C₂₃	#518	1	2	yes	—
110	C₂²×C₂₃	#518	1	4	yes	—
111	C₂²×C₂₃	#519	1	1	yes	characteristic
112	C₂²×C₂₃	#520	1	4	yes	—
113	C₂²×C₂₃	#538	1	8	yes	—
114	C₂²×C₂₃	#539	1	8	yes	—
115	C₄×C₂₃	—	2	8	no	—
116	C₄×C₂₃	—	4	8	no	—
117	C₄×C₂₃	—	4	16	no	—
118	C₂⁴×C₄	C₂²×C₂₃	1	2	yes	—
119	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
120	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
121	C₂²×C₄²	C₂²×C₂₃	1	4	yes	—
122	C₄×C₂²⋊C₄	C₄×C₂₃	1	16	yes	complemented
123	C₂×#95	C₄×C₂₃	1	8	yes	complemented
124	C₂×#95	C₂²×C₂₃	1	8	yes	—
125	C₂×#95	C₂²×C₂₃	1	8	yes	—
126	C₂×#95	C₂²×C₂₃	1	4	yes	—
127	C₂×#95	C₂²×C₂₃	1	4	yes	—
128	C₂×C₂₃	—	2	8	no	—
129	C₂×C₂₃	C₂×#376	1	1	yes	characteristic
130	C₂×C₂₃	#2273	1	1	yes	characteristic
131	C₂×C₂₃	#2274	1	1	yes	characteristic
132	C₂×C₂₃	#2312	1	2	yes	—
133	C₂×C₂₃	#2313	1	2	yes	—
134	C₂×C₂₃	#2318	1	8	yes	—
135	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
136	C₂³×C₄	—	2	8	no	—
137	C₂³×C₄	—	4	4	no	—
138	C₂³×C₄	—	4	4	no	—
139	C₂³×C₄	—	4	4	no	—
140	C₂³×C₄	C₂×C₄×C₂₃	1	8	yes	—
141	C₂³×C₄	C₂×C₄×C₂₃	1	4	yes	—
142	C₂³×C₄	C₂×C₄×C₂₃	1	8	yes	—
143	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
144	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
145	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
146	C₂×C₂²⋊C₄	—	2	16	no	—
147	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	16	yes	—
148	C₂×C₄²	—	4	8	no	—
149	C₂×C₄²	C₂×C₄×C₂₃	1	16	yes	complemented
150	#95	—	4	8	no	—
151	#95	—	4	8	no	—
152	#95	C₂×C₄×C₂₃	1	16	yes	—
153	#95	C₂×C₄×C₂₃	1	16	yes	complemented
154	C₂₃	#13796	1	1	yes	characteristic, direct product factor, Sylow
155	C₂⁴	—	2	4	no	—
156	C₂⁴	—	2	2	no	—
157	C₂⁴	—	2	2	no	—
158	C₂⁴	C₂²×C₄×C₂₃	1	4	yes	—
159	C₂⁴	C₂²×C₄×C₂₃	1	2	yes	—
160	C₂⁴	#41×C₂₃	1	8	yes	—
161	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
162	C₂²×C₄	—	2	8	no	—
163	C₂²×C₄	—	2	16	no	—
164	C₂²×C₄	—	2	16	no	—
165	C₂²×C₄	—	2	8	no	—
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	—	2	16	no	—
168	C₂²×C₄	—	2	8	no	—
169	C₂²×C₄	—	2	8	no	—
170	C₂²×C₄	—	2	8	no	—
171	C₂²×C₄	—	4	16	no	—
172	C₂²×C₄	—	4	8	no	—
173	C₂²×C₄	—	4	8	no	—
174	C₂²×C₄	—	4	8	no	—
175	C₂²×C₄	—	4	32	no	—
176	C₂²×C₄	C₄²×C₂₃	1	16	yes	—
177	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
178	C₂²×C₄	C₂²×C₄×C₂₃	1	16	yes	—
179	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
180	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
181	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
182	C₂²⋊C₄	—	2	32	no	—
183	C₄²	—	4	16	no	—
184	C₂³	—	2	4	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	8	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	8	no	—
189	C₂³	—	2	16	no	—
190	C₂³	—	2	16	no	—
191	C₂³	C₂×C₄²×C₂₃	1	4	yes	—
192	C₂³	C₂₃×#117	1	16	yes	—
193	C₂³	C₂₃×#117	1	8	yes	—
194	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
195	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
196	C₂³	C₂×#41×C₂₃	1	4	yes	—
197	C₂³	C₂₃×H(5,𝔽₂)	1	4	yes	—
198	C₂³	C₂₃×H(5,𝔽₂)	1	2	yes	—
199	C₂³	C₂₃×2₋¹⁺⁴	1	2	yes	—
200	C₂×C₄	—	2	8	no	—
201	C₂×C₄	—	2	8	no	—
202	C₂×C₄	—	2	8	no	—
203	C₂×C₄	—	2	16	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	2	16	no	—
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	—	2	8	no	—
208	C₂×C₄	—	2	16	no	—
209	C₂×C₄	—	4	16	no	—
210	C₂×C₄	—	4	32	no	—
211	C₂×C₄	—	4	16	no	—
212	C₂×C₄	—	4	16	no	—
213	C₂×C₄	—	4	16	no	—
214	C₂×C₄	C₂×C₄²×C₂₃	1	8	yes	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	2	8	no	—
218	C₂²	—	2	8	no	—
219	C₂²	—	2	32	no	—
220	C₂²	C₂₃×#376	1	8	yes	—
221	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
222	C₂²	C₂×C₂₃×#117	1	2	yes	—
223	C₂²	C₂×C₂₃×#117	1	4	yes	—
224	C₂²	C₂₃×#518	1	2	yes	—
225	C₂²	C₂₃×#518	1	4	yes	—
226	C₂²	C₂₃×#518	1	1	yes	characteristic
227	C₂²	C₂₃×#519	1	1	yes	characteristic
228	C₂²	C₂₃×#520	1	4	yes	—
229	C₂²	C₂₃×#538	1	8	yes	—
230	C₂²	C₂₃×#539	1	8	yes	—
231	C₄	—	2	8	no	—
232	C₄	—	4	8	no	—
233	C₄	—	4	16	no	—
234	C₂	—	2	8	no	—
235	C₂	C₂×C₂₃×#376	1	1	yes	characteristic
236	C₂	C₂₃×#2273	1	1	yes	characteristic
237	C₂	C₂₃×#2274	1	1	yes	characteristic
238	C₂	C₂₃×#2312	1	2	yes	—
239	C₂	C₂₃×#2313	1	2	yes	—
240	C₂	C₂₃×#2318	1	8	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₂₃	C₂⁴
commutator	C₂²	C₂²×C₄²×C₂₃
Frattini, cosocle	C₂⁴	C₂⁴×C₂₃
prime radical	C₂⁵×C₂₃	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1434	C₂	semidirect
9. C₄×C₂²⋊C₄×C₂₃	C₄	semidirect
10. C₂×C₂₃×#95	C₄	semidirect
29. C₂×C₄²×C₂₃	C₂×C₄	semidirect
33. C₂₃×#95	C₂×C₄	semidirect
63. #13796	C₂₃	direct (N × H)
97. #1434	C₂×C₂₃	semidirect
122. C₄×C₂²⋊C₄	C₄×C₂₃	semidirect
123. C₂×#95	C₄×C₂₃	semidirect
149. C₂×C₄²	C₂×C₄×C₂₃	semidirect
153. #95	C₂×C₄×C₂₃	semidirect
154. C₂₃	#13796	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×C₄×C₂²⋊C₄×C₂₃	2
C₂²×C₂₃×#95	2
C₂₃×#1434	2
C₂₃×C₂²⋊#95	2
#13796	23

Subgroups — C23×#13796 #19161422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13796 #19161422