Subgroups — C₂₃×#23836 #19171462

Order 5888.

674 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	C₄×C₂₃×#125	C₂	1	2	yes	maximal
2	C₂₃×#2085	C₂	1	4	yes	maximal
3	C₂₃×#2059	C₂	1	4	yes	maximal
4	C₄×C₂₃×#107	C₂	1	2	yes	maximal
5	C₄×C₂₃×#106	C₂	1	2	yes	maximal
6	C₂₃×#1744	C₂	1	1	yes	maximal, characteristic
7	C₂×C₂₃×#125	C₂²	1	2	yes	—
8	C₂×C₂₃×#107	C₂²	1	4	yes	—
9	C₂×C₂₃×#106	C₂²	1	4	yes	—
10	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#389	C₂²	1	4	yes	—
13	C₂₃×#387	—	2	4	no	—
14	C₂₃×#382	C₂²	1	2	yes	—
15	C₄×C₄⋊C₄×C₂₃	—	2	4	no	—
16	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
17	C₄³×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#340	C₂²	1	2	yes	—
19	C₂₃×#340	C₂²	1	2	yes	—
20	C₂₃×#340	C₂²	1	2	yes	—
21	C₂₃×#340	C₂²	1	2	yes	—
22	C₂₃×#336	C₂²	1	4	yes	—
23	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
24	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
25	C₂₃×#125	—	2	8	no	—
26	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
27	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
28	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
29	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
30	C₂×C₄²×C₂₃	—	2	4	no	—
31	C₂×C₄²×C₂₃	D₄	1	2	yes	—
32	C₂×C₄²×C₂₃	D₄	1	2	yes	—
33	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
34	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
35	C₂₃×#107	—	2	8	no	—
36	C₂₃×#106	—	2	8	no	—
37	C₄×C₈×C₂₃	Q₈	1	4	yes	—
38	C₄×C₈×C₂₃	Q₈	1	4	yes	—
39	C₂₃×#95	—	2	4	no	—
40	C₂₃×#95	—	2	4	no	—
41	C₂₃×#95	—	2	4	no	—
42	C₂²×C₄×C₂₃	—	2	4	no	—
43	C₂²×C₄×C₂₃	—	2	4	no	—
44	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
45	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
48	C₂×C₈×C₂₃	—	2	4	no	—
49	C₂×C₈×C₂₃	—	2	4	no	—
50	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
51	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
52	C₄⋊C₄×C₂₃	—	2	16	no	—
53	C₄⋊C₄×C₂₃	—	2	4	no	—
54	C₄⋊C₄×C₂₃	—	2	4	no	—
55	C₄⋊C₄×C₂₃	—	2	4	no	—
56	C₄⋊C₄×C₂₃	—	4	8	no	—
57	C₄⋊C₄×C₂₃	#41	1	8	yes	—
58	C₄²×C₂₃	—	2	4	no	—
59	C₄²×C₂₃	—	2	4	no	—
60	C₄²×C₂₃	—	4	2	no	—
61	C₄²×C₂₃	—	4	8	no	—
62	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
63	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
64	#23836	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
66	C₂×C₄×C₂₃	—	2	2	no	—
67	C₂×C₄×C₂₃	—	2	4	no	—
68	C₂×C₄×C₂₃	—	2	8	no	—
69	C₂×C₄×C₂₃	—	2	4	no	—
70	C₂×C₄×C₂₃	—	2	4	no	—
71	C₂×C₄×C₂₃	—	4	4	no	—
72	C₂×C₄×C₂₃	—	4	4	no	—
73	C₂×C₄×C₂₃	—	4	4	no	—
74	C₂×C₄×C₂₃	—	4	4	no	—
75	C₂×C₄×C₂₃	—	4	4	no	—
76	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
77	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
78	C₂×C₄×C₂₃	#135	1	4	yes	—
79	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
81	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
82	C₈×C₂₃	—	2	4	no	—
83	C₈×C₂₃	—	2	4	no	—
84	C₄×#125	C₂×C₂₃	1	2	yes	—
85	#2085	C₂×C₂₃	1	4	yes	—
86	#2059	C₂×C₂₃	1	4	yes	—
87	C₄×#107	C₂×C₂₃	1	2	yes	—
88	C₄×#106	C₂×C₂₃	1	2	yes	—
89	#1744	C₂×C₂₃	1	1	yes	characteristic
90	C₂²×C₂₃	#525	1	2	yes	—
91	C₂²×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
92	C₂²×C₂₃	C₂×#135	1	2	yes	—
93	C₂²×C₂₃	#575	1	2	yes	—
94	C₄×C₂₃	—	2	2	no	—
95	C₄×C₂₃	—	4	4	no	—
96	C₄×C₂₃	—	4	8	no	—
97	C₄×C₂₃	—	8	4	no	—
98	C₄×C₂₃	#499	1	4	yes	—
99	C₄×C₂₃	#533	1	2	yes	—
100	C₄×C₂₃	#533	1	2	yes	—
101	C₂×#125	C₂²×C₂₃	1	2	yes	—
102	C₂×#107	C₂²×C₂₃	1	4	yes	—
103	C₂×#106	C₂²×C₂₃	1	4	yes	—
104	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
106	#389	C₂²×C₂₃	1	4	yes	—
107	#387	—	2	4	no	—
108	#382	C₂²×C₂₃	1	2	yes	—
109	C₄×C₄⋊C₄	—	2	4	no	—
110	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
111	C₄³	C₂²×C₂₃	1	1	yes	characteristic
112	#340	C₂²×C₂₃	1	2	yes	—
113	#340	C₂²×C₂₃	1	2	yes	—
114	#340	C₂²×C₂₃	1	2	yes	—
115	#340	C₂²×C₂₃	1	2	yes	—
116	#336	C₂²×C₂₃	1	4	yes	—
117	C₂×C₂₃	#2399	1	1	yes	characteristic
118	C₂×C₂₃	#3102	1	4	yes	—
119	C₂×C₂₃	C₂×#499	1	1	yes	characteristic
120	C₂×C₂₃	#3162	1	1	yes	characteristic
121	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
122	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
123	#125	—	2	8	no	—
124	C₂×C₄⋊C₄	—	2	4	no	—
125	C₂×C₄⋊C₄	—	2	4	no	—
126	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
127	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
128	C₂×C₄²	—	2	4	no	—
129	C₂×C₄²	D₄×C₂₃	1	2	yes	—
130	C₂×C₄²	D₄×C₂₃	1	2	yes	—
131	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
132	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
133	#107	—	2	8	no	—
134	#106	—	2	8	no	—
135	C₄×C₈	Q₈×C₂₃	1	4	yes	—
136	C₄×C₈	Q₈×C₂₃	1	4	yes	—
137	#95	—	2	4	no	—
138	#95	—	2	4	no	—
139	#95	—	2	4	no	—
140	C₂₃	#23836	1	1	yes	characteristic, direct product factor, Sylow
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
144	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
145	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
146	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
150	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
151	C₄⋊C₄	—	2	16	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	8	no	—
156	C₄⋊C₄	#41×C₂₃	1	8	yes	—
157	C₄²	—	2	4	no	—
158	C₄²	—	2	4	no	—
159	C₄²	—	4	2	no	—
160	C₄²	—	4	8	no	—
161	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
162	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
163	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
175	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
176	C₂×C₄	C₂₃×#135	1	4	yes	—
177	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
178	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
179	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
180	C₈	—	2	4	no	—
181	C₈	—	2	4	no	—
182	C₂²	C₂₃×#525	1	2	yes	—
183	C₂²	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
184	C₂²	C₂×C₂₃×#135	1	2	yes	—
185	C₂²	C₂₃×#575	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	4	4	no	—
188	C₄	—	4	8	no	—
189	C₄	—	8	4	no	—
190	C₄	C₂₃×#499	1	4	yes	—
191	C₄	C₂₃×#533	1	2	yes	—
192	C₄	C₂₃×#533	1	2	yes	—
193	C₂	C₂₃×#2399	1	1	yes	characteristic
194	C₂	C₂₃×#3102	1	4	yes	—
195	C₂	C₂×C₂₃×#499	1	1	yes	characteristic
196	C₂	C₂₃×#3162	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle, prime radical	C₂³×C₂₃	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
64. #23836	C₂₃	direct (N × H)
140. C₂₃	#23836	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₃×#125	2
C₂₃×#2085	2
C₂₃×#2059	2
C₄×C₂₃×#107	2
C₄×C₂₃×#106	2
C₂₃×#1744	2
#23836	23

Subgroups — C23×#23836 #19171462

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#23836 #19171462