Subgroups — C₂₃×#9825 #19157451

Order 5888.

558 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	C₂×C₂₃×#494	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1990	C₂	1	4	yes	maximal
3	C₂₃×#1845	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₂₃	—	2	2	no	—
6	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂₃×#494	—	2	2	no	—
8	C₂₃×#494	C₄	1	4	yes	complemented
9	C₂×C₂₃×#107	—	2	2	no	—
10	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#384	C₂²	1	2	yes	—
13	C₂₃×#340	—	2	8	no	—
14	C₈²×C₂₃	—	2	2	no	—
15	C₂²×Q₈×C₂₃	—	2	2	no	—
16	C₂×Q₁₆×C₂₃	—	2	8	no	—
17	C₂×Q₁₆×C₂₃	—	4	4	no	—
18	C₂×Q₁₆×C₂₃	—	4	4	no	—
19	C₂²×C₈×C₂₃	D₄	1	4	yes	—
20	C₂²×C₈×C₂₃	D₄	1	2	yes	—
21	C₂₃×C₄⋊Q₈	—	2	2	no	—
22	C₂₃×C₄⋊Q₈	—	2	4	no	—
23	C₂₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
24	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
25	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
26	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
27	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
28	C₂₃×#107	—	4	4	no	—
29	C₄×C₈×C₂₃	—	2	2	no	—
30	C₄×C₈×C₂₃	—	2	2	no	—
31	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₂₃	D₄	1	2	yes	—
33	C₂×Q₈×C₂₃	—	2	4	no	—
34	C₂×Q₈×C₂₃	—	4	2	no	—
35	C₂×Q₈×C₂₃	—	4	4	no	—
36	C₂×Q₈×C₂₃	—	4	4	no	—
37	C₂²×C₄×C₂₃	—	4	2	no	—
38	C₂²×C₄×C₂₃	—	4	2	no	—
39	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
41	Q₁₆×C₂₃	—	4	16	no	—
42	C₂×C₈×C₂₃	—	2	4	no	—
43	C₂×C₈×C₂₃	—	2	4	no	—
44	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
45	C₂×C₈×C₂₃	D₈	1	4	yes	—
46	C₂×C₈×C₂₃	SD₁₆	1	4	yes	—
47	C₂×C₈×C₂₃	Q₁₆	1	8	yes	complemented
48	C₄⋊C₄×C₂₃	—	2	8	no	—
49	C₄⋊C₄×C₂₃	—	4	4	no	—
50	C₄⋊C₄×C₂₃	—	4	4	no	—
51	C₄⋊C₄×C₂₃	—	4	8	no	—
52	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₂₃	#41	1	2	yes	—
55	#9825	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂³×C₂₃	#127	1	1	yes	characteristic
57	Q₈×C₂₃	—	4	8	no	—
58	Q₈×C₂₃	—	8	4	no	—
59	C₂×C₄×C₂₃	—	4	4	no	—
60	C₂×C₄×C₂₃	—	8	2	no	—
61	C₂×C₄×C₂₃	—	8	2	no	—
62	C₂×C₄×C₂₃	—	8	2	no	—
63	C₂×C₄×C₂₃	—	8	4	no	—
64	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
66	C₂×C₄×C₂₃	#121	1	2	yes	—
67	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
68	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
69	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
70	C₂×C₄×C₂₃	C₂×Q₁₆	1	4	yes	—
71	C₂×C₄×C₂₃	#135	1	4	yes	—
72	C₈×C₂₃	—	2	8	no	—
73	C₈×C₂₃	#102	1	8	yes	—
74	C₂×#494	C₂×C₂₃	1	1	yes	characteristic
75	#1990	C₂×C₂₃	1	4	yes	—
76	#1845	C₂×C₂₃	1	1	yes	characteristic
77	C₂×C₈²	C₂×C₂₃	1	1	yes	characteristic
78	C₂²×C₂₃	#390	1	1	yes	characteristic
79	C₂²×C₂₃	#492	1	1	yes	characteristic
80	C₂²×C₂₃	#493	1	1	yes	characteristic
81	C₂²×C₂₃	#494	1	2	yes	—
82	C₂²×C₂₃	#495	1	2	yes	—
83	C₄×C₂₃	—	8	4	no	—
84	C₄×C₂₃	—	16	2	no	—
85	C₄×C₂₃	C₂×#102	1	2	yes	—
86	C₄×C₂₃	C₄×Q₁₆	1	4	yes	—
87	C₄×C₂₃	#467	1	4	yes	—
88	C₄×C₂₃	#486	1	2	yes	—
89	C₂²×Q₁₆	—	2	2	no	—
90	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
91	#494	—	2	2	no	—
92	#494	C₄×C₂₃	1	4	yes	complemented
93	C₂×#107	—	2	2	no	—
94	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
95	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
96	#384	C₂²×C₂₃	1	2	yes	—
97	#340	—	2	8	no	—
98	C₈²	—	2	2	no	—
99	C₂×C₂₃	#1707	1	2	yes	—
100	C₂×C₂₃	#1709	1	1	yes	characteristic
101	C₂×C₂₃	#1713	1	1	yes	characteristic
102	C₂×C₂₃	#1957	1	1	yes	characteristic
103	C₂×C₂₃	#1969	1	2	yes	—
104	C₂²×Q₈	—	2	2	no	—
105	C₂×Q₁₆	—	2	8	no	—
106	C₂×Q₁₆	—	4	4	no	—
107	C₂×Q₁₆	—	4	4	no	—
108	C₂²×C₈	D₄×C₂₃	1	4	yes	—
109	C₂²×C₈	D₄×C₂₃	1	2	yes	—
110	C₄⋊Q₈	—	2	2	no	—
111	C₄⋊Q₈	—	2	4	no	—
112	C₄⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
113	C₂×C₄⋊C₄	—	2	4	no	—
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
117	#107	—	4	4	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
121	C₄×C₈	D₄×C₂₃	1	2	yes	—
122	C₂₃	#9825	1	1	yes	characteristic, direct product factor, Sylow
123	C₂×Q₈	—	2	4	no	—
124	C₂×Q₈	—	4	2	no	—
125	C₂×Q₈	—	4	4	no	—
126	C₂×Q₈	—	4	4	no	—
127	C₂²×C₄	—	4	2	no	—
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
130	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
131	Q₁₆	—	4	16	no	—
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
135	C₂×C₈	D₈×C₂₃	1	4	yes	—
136	C₂×C₈	SD₁₆×C₂₃	1	4	yes	—
137	C₂×C₈	Q₁₆×C₂₃	1	8	yes	complemented
138	C₄⋊C₄	—	2	8	no	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	4	4	no	—
141	C₄⋊C₄	—	4	8	no	—
142	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
143	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
144	C₄²	#41×C₂₃	1	2	yes	—
145	C₂³	C₂₃×#127	1	1	yes	characteristic
146	Q₈	—	4	8	no	—
147	Q₈	—	8	4	no	—
148	C₂×C₄	—	4	4	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	4	no	—
153	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
154	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
155	C₂×C₄	C₂₃×#121	1	2	yes	—
156	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
157	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
158	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
159	C₂×C₄	C₂×Q₁₆×C₂₃	1	4	yes	—
160	C₂×C₄	C₂₃×#135	1	4	yes	—
161	C₈	—	2	8	no	—
162	C₈	C₂₃×#102	1	8	yes	—
163	C₂²	C₂₃×#390	1	1	yes	characteristic
164	C₂²	C₂₃×#492	1	1	yes	characteristic
165	C₂²	C₂₃×#493	1	1	yes	characteristic
166	C₂²	C₂₃×#494	1	2	yes	—
167	C₂²	C₂₃×#495	1	2	yes	—
168	C₄	—	8	4	no	—
169	C₄	—	16	2	no	—
170	C₄	C₂×C₂₃×#102	1	2	yes	—
171	C₄	C₄×Q₁₆×C₂₃	1	4	yes	—
172	C₄	C₂₃×#467	1	4	yes	—
173	C₄	C₂₃×#486	1	2	yes	—
174	C₂	C₂₃×#1707	1	2	yes	—
175	C₂	C₂₃×#1709	1	1	yes	characteristic
176	C₂	C₂₃×#1713	1	1	yes	characteristic
177	C₂	C₂₃×#1957	1	1	yes	characteristic
178	C₂	C₂₃×#1969	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	#127
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₂₃×#494	C₄	semidirect
47. C₂×C₈×C₂₃	Q₁₆	semidirect
55. #9825	C₂₃	direct (N × H)
92. #494	C₄×C₂₃	semidirect
122. C₂₃	#9825	direct (N × H)
137. C₂×C₈	Q₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#494	2
C₂₃×#1990	2
C₂₃×#1845	2
C₂×C₈²×C₂₃	2
#9825	23

Subgroups — C23×#9825 #19157451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9825 #19157451