Subgroups — C₂₃×#12025 #19159651

Order 5888.

224 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1568	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1567	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1372	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1371	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×M₃₂	C₄	1	2	yes	—
7	C₂₃×#432	—	2	1	no	—
8	C₂₃×#423	—	2	1	no	—
9	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×#346	C₄	1	1	yes	characteristic
11	C₂₃×#346	C₄	1	1	yes	characteristic
12	C₄×C₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂₃×#335	—	2	1	no	—
14	C₂₃×#335	C₂²	1	2	yes	—
15	C₂₃×#335	C₂²	1	1	yes	characteristic
16	C₂₃×#335	C₂²	1	1	yes	characteristic
17	C₂₃×#334	—	2	1	no	—
18	C₂×M₁₆×C₂₃	—	4	1	no	—
19	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₂₃	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
22	C₂₃×M₃₂	—	2	4	no	—
23	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
25	C₂₃×C₄⋊C₈	—	2	1	no	—
26	C₂₃×C₄⋊C₈	—	2	1	no	—
27	C₂₃×C₄⋊C₈	—	2	2	no	—
28	C₂₃×C₄⋊C₈	—	4	1	no	—
29	C₂₃×C₂²⋊C₈	—	4	1	no	—
30	C₂₃×#97	—	4	1	no	—
31	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	Q₁₆	1	1	yes	characteristic
38	M₁₆×C₂₃	—	8	2	no	—
39	C₂×C₈×C₂₃	—	2	2	no	—
40	C₂×C₈×C₂₃	—	2	2	no	—
41	C₂×C₈×C₂₃	—	4	2	no	—
42	C₂×C₈×C₂₃	—	4	1	no	—
43	C₂×C₈×C₂₃	—	4	1	no	—
44	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₂₃	D₈	1	1	yes	characteristic
50	C₄²×C₂₃	Q₁₆	1	1	yes	characteristic
51	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
52	C₁₆×C₂₃	—	2	2	no	—
53	C₁₆×C₂₃	M₁₆	1	4	yes	complemented
54	#12025	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂³×C₂₃	#107	1	1	yes	characteristic
56	C₂×C₄×C₂₃	—	2	1	no	—
57	C₂×C₄×C₂₃	—	2	1	no	—
58	C₂×C₄×C₂₃	—	2	1	no	—
59	C₂×C₄×C₂₃	—	2	1	no	—
60	C₂×C₄×C₂₃	—	4	1	no	—
61	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
62	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
63	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
64	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
67	C₈×C₂₃	—	2	1	no	—
68	C₈×C₂₃	—	2	1	no	—
69	C₈×C₂₃	—	8	1	no	—
70	C₈×C₂₃	—	8	2	no	—
71	C₈×C₂₃	—	8	1	no	—
72	C₈×C₂₃	#108	1	2	yes	—
73	C₈×C₂₃	C₂×M₁₆	1	2	yes	—
74	C₄×M₃₂	C₂×C₂₃	1	1	yes	characteristic
75	#1568	C₂×C₂₃	1	1	yes	characteristic
76	#1567	C₂×C₂₃	1	1	yes	characteristic
77	#1372	C₂×C₂₃	1	2	yes	complemented
78	#1371	C₂×C₂₃	1	2	yes	complemented
79	C₂²×C₂₃	—	2	1	no	—
80	C₂²×C₂₃	—	2	1	no	—
81	C₂²×C₂₃	—	2	1	no	—
82	C₂²×C₂₃	C₂×#107	1	1	yes	characteristic
83	C₄×C₂₃	—	2	1	no	—
84	C₄×C₂₃	—	2	1	no	—
85	C₄×C₂₃	—	2	1	no	—
86	C₄×C₂₃	—	2	1	no	—
87	C₄×C₂₃	#423	1	1	yes	characteristic
88	C₄×C₂₃	C₂×#108	1	1	yes	characteristic
89	C₄×C₂₃	#509	1	1	yes	characteristic
90	C₄×C₂₃	#510	1	1	yes	characteristic
91	C₂×M₃₂	C₄×C₂₃	1	2	yes	—
92	#432	—	2	1	no	—
93	#423	—	2	1	no	—
94	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
95	#346	C₄×C₂₃	1	1	yes	characteristic
96	#346	C₄×C₂₃	1	1	yes	characteristic
97	C₄×C₁₆	C₂²×C₂₃	1	2	yes	—
98	#335	—	2	1	no	—
99	#335	C₂²×C₂₃	1	2	yes	—
100	#335	C₂²×C₂₃	1	1	yes	characteristic
101	#335	C₂²×C₂₃	1	1	yes	characteristic
102	#334	—	2	1	no	—
103	C₂×C₂₃	—	4	1	no	—
104	C₂×C₂₃	#1567	1	1	yes	characteristic
105	C₂×C₂₃	#2159	1	1	yes	characteristic
106	C₂×C₂₃	#2161	1	1	yes	characteristic
107	C₂×M₁₆	—	4	1	no	—
108	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
109	C₂²×C₈	Q₈×C₂₃	1	1	yes	characteristic
110	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
111	M₃₂	—	2	4	no	—
112	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
113	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
114	C₄⋊C₈	—	2	1	no	—
115	C₄⋊C₈	—	2	1	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	4	1	no	—
118	C₂²⋊C₈	—	4	1	no	—
119	#97	—	4	1	no	—
120	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
121	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
122	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
123	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
124	C₂₃	#12025	1	1	yes	characteristic, direct product factor, Sylow
125	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
126	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
127	C₂²×C₄	Q₁₆×C₂₃	1	1	yes	characteristic
128	M₁₆	—	8	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	4	1	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
135	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
136	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
137	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
138	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
139	C₄²	D₈×C₂₃	1	1	yes	characteristic
140	C₄²	Q₁₆×C₂₃	1	1	yes	characteristic
141	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
142	C₁₆	—	2	2	no	—
143	C₁₆	M₁₆×C₂₃	1	4	yes	complemented
144	C₂³	C₂₃×#107	1	1	yes	characteristic
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
151	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
152	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
154	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
155	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	8	2	no	—
159	C₈	—	8	1	no	—
160	C₈	—	8	1	no	—
161	C₈	C₂₃×#108	1	2	yes	—
162	C₈	C₂×M₁₆×C₂₃	1	2	yes	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₂×C₂₃×#107	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	C₂₃×#423	1	1	yes	characteristic
172	C₄	C₂×C₂₃×#108	1	1	yes	characteristic
173	C₄	C₂₃×#509	1	1	yes	characteristic
174	C₄	C₂₃×#510	1	1	yes	characteristic
175	C₂	—	4	1	no	—
176	C₂	C₂₃×#1567	1	1	yes	characteristic
177	C₂	C₂₃×#2159	1	1	yes	characteristic
178	C₂	C₂₃×#2161	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×#107
prime radical	C₂³×C₂₃	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1372	C₂	semidirect
5. C₂₃×#1371	C₂	semidirect
53. C₁₆×C₂₃	M₁₆	semidirect
54. #12025	C₂₃	direct (N × H)
77. #1372	C₂×C₂₃	semidirect
78. #1371	C₂×C₂₃	semidirect
124. C₂₃	#12025	direct (N × H)
143. C₁₆	M₁₆×C₂₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₂₃×M₃₂	2
C₂₃×#1568	2
C₂₃×#1567	2
C₂₃×#1372	2
C₂₃×#1371	2
#12025	23

Subgroups — C23×#12025 #19159651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12025 #19159651