Subgroups — C₂₃×#12656 #19160282

Order 5888.

358 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#364	C₂	1	2	yes	maximal
2	C₄×C₂₃×M₃₂	C₂	1	2	yes	maximal
3	C₂₃×#1726	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1376	C₂	1	2	yes	maximal
5	C₂×C₂₃×M₃₂	C₂²	1	4	yes	—
6	C₂×C₂₃×#97	C₄	1	1	yes	characteristic
7	C₂×C₂₃×#97	C₄	1	1	yes	characteristic
8	C₂×C₂₃×#97	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#376	C₄	1	1	yes	characteristic
13	C₂₃×#364	C₄	1	8	yes	complemented
14	C₂₃×#346	—	2	2	no	—
15	C₄×C₁₆×C₂₃	—	2	2	no	—
16	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂₃×M₃₂	—	2	4	no	—
24	C₂₃×M₃₂	C₂×C₄	1	8	yes	—
25	C₂×C₁₆×C₂₃	—	2	4	no	—
26	C₂₃×#97	—	2	2	no	—
27	C₂₃×#97	C₈	1	4	yes	—
28	C₂₃×#97	C₈	1	4	yes	—
29	C₄×C₈×C₂₃	—	2	2	no	—
30	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₂₃	D₄	1	2	yes	—
32	C₄×C₈×C₂₃	Q₈	1	2	yes	—
33	C₂₃×#95	—	2	2	no	—
34	C₂²×C₄×C₂₃	—	2	1	no	—
35	C₂²×C₄×C₂₃	—	2	1	no	—
36	C₂²×C₄×C₂₃	M₁₆	1	2	yes	—
37	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₂₃	—	2	1	no	—
39	C₂×C₈×C₂₃	—	2	1	no	—
40	C₂×C₈×C₂₃	C₄²	1	2	yes	—
41	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
47	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₂₃	#41	1	2	yes	—
51	C₄²×C₂₃	—	2	1	no	—
52	C₄²×C₂₃	—	2	1	no	—
53	C₄²×C₂₃	C₄²	1	2	yes	—
54	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
55	C₄²×C₂₃	C₂×C₈	1	2	yes	—
56	C₄²×C₂₃	C₂×C₈	1	2	yes	—
57	C₁₆×C₂₃	—	2	8	no	—
58	#12656	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
60	C₂×C₄×C₂₃	—	2	2	no	—
61	C₂×C₄×C₂₃	—	2	2	no	—
62	C₂×C₄×C₂₃	—	4	2	no	—
63	C₂×C₄×C₂₃	—	4	2	no	—
64	C₂×C₄×C₂₃	—	4	2	no	—
65	C₂×C₄×C₂₃	C₄⋊C₈	1	4	yes	—
66	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
69	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
72	C₈×C₂₃	—	2	2	no	—
73	C₈×C₂₃	C₄×C₈	1	4	yes	—
74	C₈×C₂₃	C₄⋊C₈	1	4	yes	—
75	C₈×C₂₃	C₄×D₄	1	2	yes	—
76	C₈×C₂₃	C₄×Q₈	1	2	yes	—
77	C₂×#364	C₂×C₂₃	1	2	yes	—
78	C₄×M₃₂	C₂×C₂₃	1	2	yes	—
79	#1726	C₂×C₂₃	1	1	yes	characteristic
80	#1376	C₂×C₂₃	1	2	yes	—
81	C₂²×C₂₃	—	2	2	no	—
82	C₂²×C₂₃	C₄×M₁₆	1	1	yes	characteristic
83	C₂²×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
84	C₂²×C₂₃	#431	1	1	yes	characteristic
85	C₄×C₂₃	—	2	4	no	—
86	C₄×C₂₃	—	4	2	no	—
87	C₄×C₂₃	—	4	2	no	—
88	C₄×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
89	C₄×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
90	C₄×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
91	C₄×C₂₃	#431	1	1	yes	characteristic
92	C₂×M₃₂	C₂²×C₂₃	1	4	yes	—
93	C₂×#97	C₄×C₂₃	1	1	yes	characteristic
94	C₂×#97	C₄×C₂₃	1	1	yes	characteristic
95	C₂×#97	C₄×C₂₃	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
98	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
99	#376	C₄×C₂₃	1	1	yes	characteristic
100	#364	C₄×C₂₃	1	8	yes	complemented
101	#346	—	2	2	no	—
102	C₄×C₁₆	—	2	2	no	—
103	C₂×C₂₃	—	2	1	no	—
104	C₂×C₂₃	—	2	1	no	—
105	C₂×C₂₃	C₄×C₄⋊C₈	1	1	yes	characteristic
106	C₂×C₂₃	#2153	1	2	yes	—
107	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
108	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
109	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
110	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
111	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
112	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
113	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
114	M₃₂	—	2	4	no	—
115	M₃₂	C₂×C₄×C₂₃	1	8	yes	—
116	C₂×C₁₆	—	2	4	no	—
117	#97	—	2	2	no	—
118	#97	C₈×C₂₃	1	4	yes	—
119	#97	C₈×C₂₃	1	4	yes	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
122	C₄×C₈	D₄×C₂₃	1	2	yes	—
123	C₄×C₈	Q₈×C₂₃	1	2	yes	—
124	#95	—	2	2	no	—
125	C₂₃	#12656	1	1	yes	characteristic, direct product factor, Sylow
126	C₂²×C₄	—	2	1	no	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	M₁₆×C₂₃	1	2	yes	—
129	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	C₄²×C₂₃	1	2	yes	—
133	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
134	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
135	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
136	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
137	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
138	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
139	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
140	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
141	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
142	C₂×C₈	#41×C₂₃	1	2	yes	—
143	C₄²	—	2	1	no	—
144	C₄²	—	2	1	no	—
145	C₄²	C₄²×C₂₃	1	2	yes	—
146	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
147	C₄²	C₂×C₈×C₂₃	1	2	yes	—
148	C₄²	C₂×C₈×C₂₃	1	2	yes	—
149	C₁₆	—	2	8	no	—
150	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	C₂₃×C₄⋊C₈	1	4	yes	—
157	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
159	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
160	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
161	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
162	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
163	C₈	—	2	2	no	—
164	C₈	C₄×C₈×C₂₃	1	4	yes	—
165	C₈	C₂₃×C₄⋊C₈	1	4	yes	—
166	C₈	C₄×D₄×C₂₃	1	2	yes	—
167	C₈	C₄×Q₈×C₂₃	1	2	yes	—
168	C₂²	—	2	2	no	—
169	C₂²	C₄×M₁₆×C₂₃	1	1	yes	characteristic
170	C₂²	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
171	C₂²	C₂₃×#431	1	1	yes	characteristic
172	C₄	—	2	4	no	—
173	C₄	—	4	2	no	—
174	C₄	—	4	2	no	—
175	C₄	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
176	C₄	C₂×C₄×C₈×C₂₃	1	1	yes	characteristic
177	C₄	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
178	C₄	C₂₃×#431	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	C₄×C₂₃×C₄⋊C₈	1	1	yes	characteristic
182	C₂	C₂₃×#2153	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂×D₄
commutator	C₄	C₂×C₄×C₈×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×C₄⋊C₈
prime radical	C₂³×C₂₃	C₂×M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂₃×#364	C₄	semidirect
58. #12656	C₂₃	direct (N × H)
100. #364	C₄×C₂₃	semidirect
125. C₂₃	#12656	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×#364	2
C₄×C₂₃×M₃₂	2
C₂₃×#1726	2
C₂₃×#1376	2
#12656	23

Subgroups — C23×#12656 #19160282

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12656 #19160282