Subgroups — C₂₃×#7872 #19155498

Order 5888.

718 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	C₂×C₂₃×#431	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1277	C₂	1	4	yes	maximal, complemented
4	C₂³×C₈×C₂₃	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	—
8	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
9	C₂×C₂₃×C₂²⋊C₈	—	2	2	no	—
10	C₂×C₄×C₈×C₂₃	—	2	4	no	—
11	C₂×C₄×C₈×C₂₃	—	2	2	no	—
12	C₂×C₄×C₈×C₂₃	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
14	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₂₃	Q₈	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	—	2	4	no	—
18	C₂²×C₈×C₂₃	—	2	4	no	—
19	C₂²×C₈×C₂₃	—	2	4	no	—
20	C₂²×C₈×C₂₃	C₈	1	8	yes	complemented
21	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₂₃	D₄	1	4	yes	—
25	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₂₃	Q₈	1	1	yes	characteristic
27	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
28	C₂₃×C₄⋊C₈	—	2	4	no	—
29	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
30	C₂₃×C₂²⋊C₈	—	2	8	no	—
31	C₄×C₈×C₂₃	—	2	4	no	—
32	C₄×C₈×C₂₃	—	2	8	no	—
33	C₄×C₈×C₂₃	—	2	4	no	—
34	C₂⁴×C₂₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	—	2	4	no	—
36	C₂²×C₄×C₂₃	—	2	4	no	—
37	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
39	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₂₃	M₁₆	1	4	yes	—
46	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₂₃	—	2	16	no	—
50	C₂×C₈×C₂₃	—	2	4	no	—
51	C₂×C₈×C₂₃	—	2	4	no	—
52	C₂×C₈×C₂₃	—	2	8	no	—
53	C₂×C₈×C₂₃	—	4	8	no	—
54	C₂×C₈×C₂₃	C₂×C₈	1	8	yes	—
55	C₄²×C₂₃	C₄²	1	2	yes	—
56	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₂₃	C₂²⋊C₄	1	4	yes	—
58	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
59	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
60	C₄²×C₂₃	C₂×D₄	1	2	yes	—
61	#7872	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂³×C₂₃	—	2	2	no	—
63	C₂³×C₂₃	—	2	2	no	—
64	C₂³×C₂₃	#95	1	2	yes	—
65	C₂³×C₂₃	C₄⋊C₈	1	4	yes	—
66	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₃	—	2	2	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	8	no	—
70	C₂×C₄×C₂₃	—	2	8	no	—
71	C₂×C₄×C₂₃	#95	1	2	yes	—
72	C₂×C₄×C₂₃	#95	1	2	yes	—
73	C₂×C₄×C₂₃	#95	1	2	yes	—
74	C₂×C₄×C₂₃	C₄×C₈	1	4	yes	—
75	C₂×C₄×C₂₃	#97	1	4	yes	—
76	C₂×C₄×C₂₃	C₂²⋊C₈	1	8	yes	—
77	C₂×C₄×C₂₃	C₄⋊C₈	1	4	yes	—
78	C₂×C₄×C₂₃	#108	1	4	yes	—
79	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
82	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
83	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
84	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
85	C₂×C₄×C₂₃	#135	1	4	yes	—
86	C₈×C₂₃	—	2	8	no	—
87	C₈×C₂₃	—	4	8	no	—
88	C₂×#431	C₂×C₂₃	1	2	yes	—
89	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
90	#1277	C₂×C₂₃	1	4	yes	complemented
91	C₂²×C₂₃	—	2	8	no	—
92	C₂²×C₂₃	—	2	2	no	—
93	C₂²×C₂₃	—	2	2	no	—
94	C₂²×C₂₃	#336	1	4	yes	—
95	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
96	C₂²×C₂₃	C₂×C₄⋊C₈	1	2	yes	—
97	C₂²×C₂₃	#427	1	2	yes	—
98	C₂²×C₂₃	C₂×#108	1	2	yes	—
99	C₄×C₂₃	—	2	2	no	—
100	C₄×C₂₃	—	2	2	no	—
101	C₄×C₂₃	#336	1	4	yes	—
102	C₄×C₂₃	#341	1	4	yes	—
103	C₄×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
104	C₄×C₂₃	C₂×#97	1	1	yes	characteristic
105	C₄×C₂₃	C₂×C₂²⋊C₈	1	2	yes	—
106	C₄×C₂₃	#416	1	4	yes	—
107	C₂³×C₈	C₄×C₂₃	1	2	yes	—
108	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
109	#431	—	2	4	no	—
110	#431	C₄×C₂₃	1	8	yes	—
111	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
112	C₂×C₂²⋊C₈	—	2	2	no	—
113	C₂×C₄×C₈	—	2	4	no	—
114	C₂×C₄×C₈	—	2	2	no	—
115	C₂×C₄×C₈	C₄×C₂₃	1	2	yes	—
116	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
117	C₂×C₂₃	—	2	4	no	—
118	C₂×C₂₃	#1564	1	4	yes	—
119	C₂×C₂₃	C₂×#336	1	1	yes	characteristic
120	C₂×C₂₃	#1735	1	1	yes	characteristic
121	C₂×C₂₃	C₂×#341	1	1	yes	characteristic
122	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
123	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
124	C₂³×C₄	Q₈×C₂₃	1	1	yes	characteristic
125	C₂²×C₈	—	2	4	no	—
126	C₂²×C₈	—	2	4	no	—
127	C₂²×C₈	—	2	4	no	—
128	C₂²×C₈	C₈×C₂₃	1	8	yes	complemented
129	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
130	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
131	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
132	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
133	C₂×C₄²	D₄×C₂₃	1	4	yes	—
134	C₂×C₄²	Q₈×C₂₃	1	1	yes	characteristic
135	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
136	C₄⋊C₈	—	2	4	no	—
137	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
138	C₂²⋊C₈	—	2	8	no	—
139	C₄×C₈	—	2	4	no	—
140	C₄×C₈	—	2	8	no	—
141	C₄×C₈	—	2	4	no	—
142	C₂₃	#7872	1	1	yes	characteristic, direct product factor, Sylow
143	C₂⁴	C₄⋊C₄×C₂₃	1	1	yes	characteristic
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
147	C₂²×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
148	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
149	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
150	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
151	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
152	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₈×C₂₃	1	4	yes	—
154	C₂²×C₄	M₁₆×C₂₃	1	4	yes	—
155	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
156	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
157	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	16	no	—
161	C₂×C₈	—	2	8	no	—
162	C₂×C₈	—	4	8	no	—
163	C₂×C₈	C₂×C₈×C₂₃	1	8	yes	—
164	C₄²	C₄²×C₂₃	1	2	yes	—
165	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
166	C₄²	C₂²⋊C₄×C₂₃	1	4	yes	—
167	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
168	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
169	C₄²	C₂×D₄×C₂₃	1	2	yes	—
170	C₂³	—	2	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	C₂₃×#95	1	2	yes	—
173	C₂³	C₂₃×C₄⋊C₈	1	4	yes	—
174	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	C₂₃×#95	1	2	yes	—
180	C₂×C₄	C₂₃×#95	1	2	yes	—
181	C₂×C₄	C₂₃×#95	1	2	yes	—
182	C₂×C₄	C₄×C₈×C₂₃	1	4	yes	—
183	C₂×C₄	C₂₃×#97	1	4	yes	—
184	C₂×C₄	C₂₃×C₂²⋊C₈	1	8	yes	—
185	C₂×C₄	C₂₃×C₄⋊C₈	1	4	yes	—
186	C₂×C₄	C₂₃×#108	1	4	yes	—
187	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
188	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
190	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
191	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
192	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
193	C₂×C₄	C₂₃×#135	1	4	yes	—
194	C₈	—	2	8	no	—
195	C₈	—	4	8	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	8	no	—
199	C₂²	C₂₃×#336	1	4	yes	—
200	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
201	C₂²	C₂×C₂₃×C₄⋊C₈	1	2	yes	—
202	C₂²	C₂₃×#427	1	2	yes	—
203	C₂²	C₂×C₂₃×#108	1	2	yes	—
204	C₄	—	2	2	no	—
205	C₄	—	2	2	no	—
206	C₄	C₂₃×#336	1	4	yes	—
207	C₄	C₂₃×#341	1	4	yes	—
208	C₄	C₂×C₄×C₈×C₂₃	1	1	yes	characteristic
209	C₄	C₂×C₂₃×#97	1	1	yes	characteristic
210	C₄	C₂×C₂₃×C₂²⋊C₈	1	2	yes	—
211	C₄	C₂₃×#416	1	4	yes	—
212	C₂	—	2	4	no	—
213	C₂	C₂₃×#1564	1	4	yes	—
214	C₂	C₂×C₂₃×#336	1	1	yes	characteristic
215	C₂	C₂₃×#1735	1	1	yes	characteristic
216	C₂	C₂×C₂₃×#341	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1277	C₂	semidirect
20. C₂²×C₈×C₂₃	C₈	semidirect
61. #7872	C₂₃	direct (N × H)
90. #1277	C₂×C₂₃	semidirect
128. C₂²×C₈	C₈×C₂₃	semidirect
142. C₂₃	#7872	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
C₂²×C₄×C₈×C₂₃	2
C₂₃×#1277	2
#7872	23

Subgroups — C23×#7872 #19155498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7872 #19155498