Subgroups — C₂₃×#12915 #19160541

Order 5888.

374 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3407	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2151	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2113	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1757	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1378	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
7	C₂₃×#504	—	2	4	no	—
8	C₂×C₂₃×M₃₂	—	2	2	no	—
9	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#413	C₄	1	1	yes	characteristic
13	C₂₃×#410	C₄	1	1	yes	characteristic
14	C₂₃×#405	C₂²	1	2	yes	—
15	C₂₃×C₄⋊C₁₆	—	2	4	no	—
16	C₂₃×C₂²⋊C₁₆	—	2	2	no	—
17	C₂₃×#344	C₄	1	2	yes	—
18	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂₃×#131	—	2	2	no	—
20	C₂₃×#131	—	2	2	no	—
21	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
22	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×M₁₆×C₂₃	Q₈	1	2	yes	—
24	C₂²×C₈×C₂₃	D₄	1	2	yes	—
25	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
26	C₂₃×#117	C₂×C₄	1	2	yes	—
27	C₂₃×M₃₂	—	2	4	no	—
28	C₂₃×M₃₂	—	2	2	no	—
29	C₂₃×M₃₂	—	2	2	no	—
30	C₂₃×M₃₂	—	2	2	no	—
31	C₂×C₁₆×C₂₃	—	2	4	no	—
32	C₂×C₁₆×C₂₃	—	2	1	no	—
33	C₂×C₁₆×C₂₃	—	2	1	no	—
34	C₂×C₁₆×C₂₃	—	2	2	no	—
35	C₂×C₁₆×C₂₃	D₄	1	4	yes	—
36	C₂₃×#101	—	2	1	no	—
37	C₂₃×#100	—	2	1	no	—
38	C₂₃×#99	C₈	1	4	yes	—
39	C₂₃×#97	—	2	2	no	—
40	C₄×C₈×C₂₃	—	2	2	no	—
41	#41×C₂₃	—	2	2	no	—
42	#41×C₂₃	—	2	2	no	—
43	C₂×Q₈×C₂₃	M₁₆	1	1	yes	characteristic
44	C₂×D₄×C₂₃	C₂×C₈	1	2	yes	—
45	C₂×D₄×C₂₃	M₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
48	M₁₆×C₂₃	—	2	2	no	—
49	M₁₆×C₂₃	—	2	4	no	—
50	M₁₆×C₂₃	—	2	2	no	—
51	M₁₆×C₂₃	—	2	2	no	—
52	C₂×C₈×C₂₃	—	2	2	no	—
53	C₂×C₈×C₂₃	—	2	2	no	—
54	C₂×C₈×C₂₃	—	2	2	no	—
55	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₂₃	#41	1	2	yes	—
60	C₄⋊C₄×C₂₃	—	2	2	no	—
61	C₂²⋊C₄×C₂₃	C₂×C₈	1	4	yes	—
62	C₄²×C₂₃	—	2	2	no	—
63	C₁₆×C₂₃	—	2	4	no	—
64	C₁₆×C₂₃	—	4	1	no	—
65	C₁₆×C₂₃	—	4	1	no	—
66	#12915	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₂₃	C₄⋊C₈	1	2	yes	—
68	C₂³×C₂₃	C₂²×C₈	1	1	yes	characteristic
69	Q₈×C₂₃	—	2	2	no	—
70	D₄×C₂₃	—	2	4	no	—
71	D₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	4	no	—
75	C₂×C₄×C₂₃	C₄⋊C₈	1	2	yes	—
76	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₂₃	#131	1	2	yes	—
80	C₈×C₂₃	—	2	2	no	—
81	C₈×C₂₃	—	4	2	no	—
82	C₈×C₂₃	#120	1	1	yes	characteristic
83	C₈×C₂₃	C₂²⋊Q₈	1	2	yes	—
84	C₈×C₂₃	#123	1	1	yes	characteristic
85	#3407	C₂×C₂₃	1	1	yes	characteristic
86	#2151	C₂×C₂₃	1	2	yes	complemented
87	#2113	C₂×C₂₃	1	1	yes	characteristic
88	#1757	C₂×C₂₃	1	1	yes	characteristic
89	#1378	C₂×C₂₃	1	2	yes	complemented
90	C₂²×C₂₃	—	2	2	no	—
91	C₂²×C₂₃	—	4	1	no	—
92	C₂²×C₂₃	—	4	2	no	—
93	C₂²×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
94	C₂²×C₂₃	C₈×D₄	1	2	yes	—
95	C₄×C₂₃	—	2	2	no	—
96	C₄×C₂₃	—	4	4	no	—
97	C₄×C₂₃	#385	1	1	yes	characteristic
98	C₄×C₂₃	#424	1	1	yes	characteristic
99	C₄×C₂₃	#435	1	2	yes	—
100	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
101	#504	—	2	4	no	—
102	C₂×M₃₂	—	2	2	no	—
103	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
104	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
105	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
106	#413	C₄×C₂₃	1	1	yes	characteristic
107	#410	C₄×C₂₃	1	1	yes	characteristic
108	#405	C₂²×C₂₃	1	2	yes	—
109	C₄⋊C₁₆	—	2	4	no	—
110	C₂²⋊C₁₆	—	2	2	no	—
111	#344	C₄×C₂₃	1	2	yes	—
112	C₂×C₂₃	—	2	2	no	—
113	C₂×C₂₃	—	2	1	no	—
114	C₂×C₂₃	—	4	2	no	—
115	C₂×C₂₃	#1850	1	1	yes	characteristic
116	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
117	#131	—	2	2	no	—
118	#131	—	2	2	no	—
119	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
120	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
121	C₂×M₁₆	Q₈×C₂₃	1	2	yes	—
122	C₂²×C₈	D₄×C₂₃	1	2	yes	—
123	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
124	#117	C₂×C₄×C₂₃	1	2	yes	—
125	M₃₂	—	2	4	no	—
126	M₃₂	—	2	2	no	—
127	M₃₂	—	2	2	no	—
128	M₃₂	—	2	2	no	—
129	C₂×C₁₆	—	2	4	no	—
130	C₂×C₁₆	—	2	1	no	—
131	C₂×C₁₆	—	2	1	no	—
132	C₂×C₁₆	—	2	2	no	—
133	C₂×C₁₆	D₄×C₂₃	1	4	yes	—
134	#101	—	2	1	no	—
135	#100	—	2	1	no	—
136	#99	C₈×C₂₃	1	4	yes	—
137	#97	—	2	2	no	—
138	C₄×C₈	—	2	2	no	—
139	C₂₃	#12915	1	1	yes	characteristic, direct product factor, Sylow
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	M₁₆×C₂₃	1	1	yes	characteristic
143	C₂×D₄	C₂×C₈×C₂₃	1	2	yes	—
144	C₂×D₄	M₁₆×C₂₃	1	1	yes	characteristic
145	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
146	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
147	M₁₆	—	2	2	no	—
148	M₁₆	—	2	4	no	—
149	M₁₆	—	2	2	no	—
150	M₁₆	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
155	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
156	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
157	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
158	C₂×C₈	#41×C₂₃	1	2	yes	—
159	C₄⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	C₂×C₈×C₂₃	1	4	yes	—
161	C₄²	—	2	2	no	—
162	C₁₆	—	2	4	no	—
163	C₁₆	—	4	1	no	—
164	C₁₆	—	4	1	no	—
165	C₂³	C₂₃×C₄⋊C₈	1	2	yes	—
166	C₂³	C₂²×C₈×C₂₃	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	C₂₃×C₄⋊C₈	1	2	yes	—
174	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
175	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
176	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
177	C₂×C₄	C₂₃×#131	1	2	yes	—
178	C₈	—	2	2	no	—
179	C₈	—	4	2	no	—
180	C₈	C₂₃×#120	1	1	yes	characteristic
181	C₈	C₂₃×C₂²⋊Q₈	1	2	yes	—
182	C₈	C₂₃×#123	1	1	yes	characteristic
183	C₂²	—	2	2	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	4	2	no	—
186	C₂²	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
187	C₂²	C₈×D₄×C₂₃	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	4	4	no	—
190	C₄	C₂₃×#385	1	1	yes	characteristic
191	C₄	C₂₃×#424	1	1	yes	characteristic
192	C₄	C₂₃×#435	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	—	2	1	no	—
195	C₂	—	4	2	no	—
196	C₂	C₂₃×#1850	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₂₃	#120
commutator	C₂³	C₂²×C₈×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂×C₂₃	#1850
prime radical	C₂×#41×C₂₃	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2151	C₂	semidirect
5. C₂₃×#1378	C₂	semidirect
66. #12915	C₂₃	direct (N × H)
86. #2151	C₂×C₂₃	semidirect
89. #1378	C₂×C₂₃	semidirect
139. C₂₃	#12915	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3407	2
C₂₃×#2151	2
C₂₃×#2113	2
C₂₃×#1757	2
C₂₃×#1378	2
#12915	23

Subgroups — C23×#12915 #19160541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12915 #19160541