Subgroups — C₂₃×#2612 #15670231

Order 2944.

388 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×#400	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#398	C₂	1	2	yes	maximal
3	C₂₃×#395	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#391	C₂	1	2	yes	maximal
5	C₂₃×#389	C₂	1	2	yes	maximal
6	C₂₃×#387	C₂	1	2	yes	maximal
7	C₂₃×#387	C₂	1	2	yes	maximal
8	C₂₃×#382	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#376	C₂	1	2	yes	maximal
10	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
20	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×#95	C₂²	1	2	yes	—
22	C₂₃×#95	C₂²	1	2	yes	—
23	C₂₃×#95	C₂²	1	2	yes	—
24	C₂₃×#95	C₂²	1	2	yes	—
25	C₂₃×#95	C₂²	1	2	yes	—
26	C₂₃×#95	C₂²	1	2	yes	—
27	C₂₃×#95	C₂²	1	1	yes	characteristic
28	C₂₃×#95	C₂²	1	1	yes	characteristic
29	C₂₃×#95	C₂²	1	2	yes	—
30	C₂₃×#95	C₂²	1	2	yes	—
31	C₂×Q₈×C₂₃	—	2	4	no	—
32	C₂×Q₈×C₂₃	—	4	1	no	—
33	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
34	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
35	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
36	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
42	C₄⋊C₄×C₂₃	—	2	4	no	—
43	C₄⋊C₄×C₂₃	—	2	4	no	—
44	C₄⋊C₄×C₂₃	—	2	4	no	—
45	C₄⋊C₄×C₂₃	—	4	1	no	—
46	C₄⋊C₄×C₂₃	—	4	1	no	—
47	C₄⋊C₄×C₂₃	—	4	2	no	—
48	C₄⋊C₄×C₂₃	—	4	1	no	—
49	C₄²×C₂₃	—	2	4	no	—
50	C₄²×C₂₃	—	2	4	no	—
51	C₄²×C₂₃	Q₈	1	4	yes	complemented
52	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
53	Q₈×C₂₃	—	4	4	no	—
54	C₂×C₄×C₂₃	—	2	2	no	—
55	C₂×C₄×C₂₃	—	2	2	no	—
56	C₂×C₄×C₂₃	—	2	2	no	—
57	C₂×C₄×C₂₃	—	2	2	no	—
58	C₂×C₄×C₂₃	—	2	1	no	—
59	C₂×C₄×C₂₃	—	2	2	no	—
60	C₂×C₄×C₂₃	—	2	1	no	—
61	C₂×C₄×C₂₃	—	2	2	no	—
62	C₂×C₄×C₂₃	—	2	2	no	—
63	C₂×C₄×C₂₃	—	2	2	no	—
64	C₂×C₄×C₂₃	—	2	2	no	—
65	C₂×C₄×C₂₃	—	2	2	no	—
66	C₂×C₄×C₂₃	—	2	2	no	—
67	C₂×C₄×C₂₃	—	2	2	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	2	1	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	1	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
76	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
77	C₂×C₄×C₂₃	#41	1	4	yes	—
78	C₂×C₄×C₂₃	#41	1	4	yes	—
79	#2612	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
81	C₂²×C₂₃	C₂×#41	1	2	yes	—
82	C₂²×C₂₃	C₂×#41	1	2	yes	—
83	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
84	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
85	C₄×C₂₃	—	2	4	no	—
86	C₄×C₂₃	—	2	2	no	—
87	C₄×C₂₃	—	2	4	no	—
88	C₄×C₂₃	—	2	4	no	—
89	C₄×C₂₃	—	4	1	no	—
90	C₄×C₂₃	—	4	1	no	—
91	C₄×C₂₃	—	4	2	no	—
92	C₄×C₂₃	—	4	2	no	—
93	C₄×C₂₃	—	4	2	no	—
94	#400	C₂×C₂₃	1	1	yes	characteristic
95	#398	C₂×C₂₃	1	2	yes	—
96	#395	C₂×C₂₃	1	1	yes	characteristic
97	#391	C₂×C₂₃	1	2	yes	—
98	#389	C₂×C₂₃	1	2	yes	—
99	#387	C₂×C₂₃	1	2	yes	—
100	#387	C₂×C₂₃	1	2	yes	—
101	#382	C₂×C₂₃	1	1	yes	characteristic
102	#376	C₂×C₂₃	1	2	yes	—
103	C₂×C₂₃	#533	1	2	yes	—
104	C₂×C₂₃	#544	1	1	yes	characteristic
105	C₂×C₂₃	#551	1	1	yes	characteristic
106	C₂×C₂₃	#552	1	2	yes	—
107	C₂×C₂₃	#555	1	1	yes	characteristic
108	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
109	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
110	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
111	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
112	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
113	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
115	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
116	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
117	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
118	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
119	#95	C₂²×C₂₃	1	2	yes	—
120	#95	C₂²×C₂₃	1	2	yes	—
121	#95	C₂²×C₂₃	1	2	yes	—
122	#95	C₂²×C₂₃	1	2	yes	—
123	#95	C₂²×C₂₃	1	2	yes	—
124	#95	C₂²×C₂₃	1	2	yes	—
125	#95	C₂²×C₂₃	1	1	yes	characteristic
126	#95	C₂²×C₂₃	1	1	yes	characteristic
127	#95	C₂²×C₂₃	1	2	yes	—
128	#95	C₂²×C₂₃	1	2	yes	—
129	C₂₃	#2612	1	1	yes	characteristic, direct product factor, Sylow
130	C₂×Q₈	—	2	4	no	—
131	C₂×Q₈	—	4	1	no	—
132	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
133	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
134	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
135	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
136	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
137	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
138	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
139	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
140	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	4	1	no	—
145	C₄⋊C₄	—	4	1	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	1	no	—
148	C₄²	—	2	4	no	—
149	C₄²	—	2	4	no	—
150	C₄²	Q₈×C₂₃	1	4	yes	complemented
151	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
152	Q₈	—	4	4	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	C₂×Q₈×C₂₃	1	4	yes	—
175	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
176	C₂×C₄	#41×C₂₃	1	4	yes	—
177	C₂×C₄	#41×C₂₃	1	4	yes	—
178	C₂²	C₂²×Q₈×C₂₃	1	1	yes	characteristic
179	C₂²	C₂×#41×C₂₃	1	2	yes	—
180	C₂²	C₂×#41×C₂₃	1	2	yes	—
181	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
182	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
183	C₄	—	2	4	no	—
184	C₄	—	2	4	no	—
185	C₄	—	2	4	no	—
186	C₄	—	2	2	no	—
187	C₄	—	4	1	no	—
188	C₄	—	4	1	no	—
189	C₄	—	4	2	no	—
190	C₄	—	4	2	no	—
191	C₄	—	4	2	no	—
192	C₂	C₂₃×#533	1	2	yes	—
193	C₂	C₂₃×#544	1	1	yes	characteristic
194	C₂	C₂₃×#551	1	1	yes	characteristic
195	C₂	C₂₃×#552	1	2	yes	—
196	C₂	C₂₃×#555	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
51. C₄²×C₂₃	Q₈	semidirect
79. #2612	C₂₃	direct (N × H)
129. C₂₃	#2612	direct (N × H)
150. C₄²	Q₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#400	2
C₂₃×#398	2
C₂₃×#395	2
C₂₃×#391	2
C₂₃×#389	2
C₂₃×#387	2
C₂₃×#382	2
C₂₃×#376	2
#2612	23

Subgroups — C23×#2612 #15670231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2612 #15670231