Subgroups — C₂₃×#8030 #19155656

Order 5888.

638 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#432	C₂	1	2	yes	maximal
2	C₂×C₂₃×#382	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1295	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
5	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
6	C₂₃×#432	—	2	4	no	—
7	C₂₃×#432	C₄	1	8	yes	complemented
8	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
9	C₂×C₂₃×C₂²⋊C₈	—	2	2	no	—
10	C₂×C₂₃×#97	C₂²	1	4	yes	—
11	C₂₃×#382	—	2	2	no	—
12	C₂₃×#382	C₄	1	2	yes	—
13	C₂₃×#382	C₂²	1	2	yes	—
14	C₂×C₂₃×#95	—	2	2	no	—
15	C₂³×C₄×C₂₃	—	2	2	no	—
16	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₂₃	D₄	1	2	yes	—
18	C₂²×C₈×C₂₃	—	2	4	no	—
19	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
20	C₂×C₄⋊C₄×C₂₃	—	4	4	no	—
21	C₂×C₄⋊C₄×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	2	yes	—
25	C₂×C₄²×C₂₃	D₄	1	2	yes	—
26	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
27	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
28	C₂₃×C₄⋊C₈	—	2	8	no	—
29	C₂₃×C₂²⋊C₈	—	2	8	no	—
30	C₂₃×#97	—	2	4	no	—
31	C₂₃×#97	C₂×C₄	1	8	yes	complemented
32	C₂₃×#95	—	2	4	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	4	no	—
38	C₂²×C₄×C₂₃	—	4	4	no	—
39	C₂²×C₄×C₂₃	—	4	8	no	—
40	C₂²×C₄×C₂₃	—	4	4	no	—
41	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
45	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
47	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
49	C₂×C₈×C₂₃	—	2	8	no	—
50	C₂×C₈×C₂₃	—	4	8	no	—
51	C₄⋊C₄×C₂₃	—	4	4	no	—
52	C₄⋊C₄×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	#8030	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₂₃	—	2	4	no	—
62	C₂³×C₂₃	#95	1	2	yes	—
63	C₂³×C₂₃	#100	1	2	yes	—
64	C₂³×C₂₃	#101	1	2	yes	—
65	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₂₃	—	2	4	no	—
67	C₂×C₄×C₂₃	—	2	4	no	—
68	C₂×C₄×C₂₃	—	2	1	no	—
69	C₂×C₄×C₂₃	—	2	1	no	—
70	C₂×C₄×C₂₃	—	2	1	no	—
71	C₂×C₄×C₂₃	—	2	1	no	—
72	C₂×C₄×C₂₃	—	2	4	no	—
73	C₂×C₄×C₂₃	—	2	4	no	—
74	C₂×C₄×C₂₃	—	4	4	no	—
75	C₂×C₄×C₂₃	—	4	4	no	—
76	C₂×C₄×C₂₃	—	4	4	no	—
77	C₂×C₄×C₂₃	—	4	8	no	—
78	C₂×C₄×C₂₃	—	8	8	no	—
79	C₂×C₄×C₂₃	#95	1	4	yes	—
80	C₂×C₄×C₂₃	#95	1	2	yes	—
81	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
83	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
84	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₈×C₂₃	—	4	8	no	—
86	C₂×#432	C₂×C₂₃	1	2	yes	—
87	C₂×#382	C₂×C₂₃	1	1	yes	characteristic
88	#1295	C₂×C₂₃	1	4	yes	complemented
89	C₂²×C₂₃	—	2	4	no	—
90	C₂²×C₂₃	—	2	4	no	—
91	C₂²×C₂₃	—	2	4	no	—
92	C₂²×C₂₃	#343	1	4	yes	—
93	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
94	C₂²×C₂₃	C₂×#100	1	1	yes	characteristic
95	C₂²×C₂₃	C₂×#101	1	1	yes	characteristic
96	C₂²×C₂₃	#421	1	4	yes	—
97	C₄×C₂₃	—	2	2	no	—
98	C₄×C₂₃	—	2	4	no	—
99	C₄×C₂₃	—	2	2	no	—
100	C₄×C₂₃	—	2	2	no	—
101	C₄×C₂₃	—	2	1	no	—
102	C₄×C₂₃	—	2	1	no	—
103	C₄×C₂₃	—	8	4	no	—
104	C₂²×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
105	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
106	#432	—	2	4	no	—
107	#432	C₄×C₂₃	1	8	yes	complemented
108	C₂×C₄⋊C₈	—	2	2	no	—
109	C₂×C₂²⋊C₈	—	2	2	no	—
110	C₂×#97	C₂²×C₂₃	1	4	yes	—
111	#382	—	2	2	no	—
112	#382	C₄×C₂₃	1	2	yes	—
113	#382	C₂²×C₂₃	1	2	yes	—
114	C₂×#95	—	2	2	no	—
115	C₂×C₂₃	—	2	4	no	—
116	C₂×C₂₃	#1533	1	2	yes	—
117	C₂×C₂₃	#1534	1	2	yes	—
118	C₂×C₂₃	#1733	1	2	yes	—
119	C₂×C₂₃	C₂×#343	1	1	yes	characteristic
120	C₂³×C₄	—	2	2	no	—
121	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
122	C₂³×C₄	D₄×C₂₃	1	2	yes	—
123	C₂²×C₈	—	2	4	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	—	4	4	no	—
126	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
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	8	no	—
134	C₂²⋊C₈	—	2	8	no	—
135	#97	—	2	4	no	—
136	#97	C₂×C₄×C₂₃	1	8	yes	complemented
137	#95	—	2	4	no	—
138	#95	—	2	4	no	—
139	C₂₃	#8030	1	1	yes	characteristic, direct product factor, Sylow
140	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	—	4	4	no	—
145	C₂²×C₄	—	4	8	no	—
146	C₂²×C₄	—	4	4	no	—
147	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
148	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
149	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
150	C₂²×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
151	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
152	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
153	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
155	C₂×C₈	—	2	8	no	—
156	C₂×C₈	—	4	8	no	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	—	4	4	no	—
159	C₄²	C₄²×C₂₃	1	2	yes	—
160	C₄²	C₂²⋊C₄×C₂₃	1	4	yes	—
161	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
162	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
163	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
164	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
165	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
166	C₂³	—	2	4	no	—
167	C₂³	C₂₃×#95	1	2	yes	—
168	C₂³	C₂₃×#100	1	2	yes	—
169	C₂³	C₂₃×#101	1	2	yes	—
170	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	8	no	—
183	C₂×C₄	—	8	8	no	—
184	C₂×C₄	C₂₃×#95	1	4	yes	—
185	C₂×C₄	C₂₃×#95	1	2	yes	—
186	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
187	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
188	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
189	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
190	C₈	—	4	8	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	4	no	—
193	C₂²	—	2	4	no	—
194	C₂²	C₂₃×#343	1	4	yes	—
195	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
196	C₂²	C₂×C₂₃×#100	1	1	yes	characteristic
197	C₂²	C₂×C₂₃×#101	1	1	yes	characteristic
198	C₂²	C₂₃×#421	1	4	yes	—
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	4	no	—
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	8	4	no	—
206	C₂	—	2	4	no	—
207	C₂	C₂₃×#1533	1	2	yes	—
208	C₂	C₂₃×#1534	1	2	yes	—
209	C₂	C₂₃×#1733	1	2	yes	—
210	C₂	C₂×C₂₃×#343	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1295	C₂	semidirect
7. C₂₃×#432	C₄	semidirect
31. C₂₃×#97	C₂×C₄	semidirect
60. #8030	C₂₃	direct (N × H)
88. #1295	C₂×C₂₃	semidirect
107. #432	C₄×C₂₃	semidirect
136. #97	C₂×C₄×C₂₃	semidirect
139. C₂₃	#8030	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×#432	2
C₂×C₂₃×#382	2
C₂₃×#1295	2
#8030	23

Subgroups — C23×#8030 #19155656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8030 #19155656