Subgroups — C₂₃×#2792 #15670411

Order 2944.

384 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#398	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#398	C₂	1	2	yes	maximal
3	C₂₃×#395	C₂	1	2	yes	maximal
4	C₂₃×#395	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#389	C₂	1	2	yes	maximal
6	C₂₃×#389	C₂	1	2	yes	maximal
7	C₂₃×#387	C₂	1	2	yes	maximal
8	C₂₃×#387	C₂	1	2	yes	maximal
9	C₂₃×#384	C₂	1	1	yes	maximal, characteristic
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	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
23	C₂₃×#95	C₂²	1	1	yes	characteristic
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	2	yes	—
28	C₂₃×#95	C₂²	1	2	yes	—
29	C₂₃×#95	C₂²	1	2	yes	—
30	C₂₃×#95	C₂²	1	1	yes	characteristic
31	C₂×Q₈×C₂₃	—	4	1	no	—
32	C₂×Q₈×C₂₃	—	4	2	no	—
33	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
34	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₂₃	—	2	4	no	—
43	C₄⋊C₄×C₂₃	—	4	2	no	—
44	C₄⋊C₄×C₂₃	—	4	2	no	—
45	C₄⋊C₄×C₂₃	—	4	2	no	—
46	C₄⋊C₄×C₂₃	—	4	1	no	—
47	C₄⋊C₄×C₂₃	—	4	2	no	—
48	C₄⋊C₄×C₂₃	—	4	1	no	—
49	C₄⋊C₄×C₂₃	—	4	1	no	—
50	C₄⋊C₄×C₂₃	—	4	2	no	—
51	C₄⋊C₄×C₂₃	Q₈	1	8	yes	complemented
52	C₄²×C₂₃	—	2	2	no	—
53	C₄²×C₂₃	—	4	2	no	—
54	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
55	Q₈×C₂₃	—	4	4	no	—
56	C₂×C₄×C₂₃	—	2	2	no	—
57	C₂×C₄×C₂₃	—	2	2	no	—
58	C₂×C₄×C₂₃	—	2	2	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	1	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	1	no	—
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	2	2	no	—
71	C₂×C₄×C₂₃	—	2	1	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
78	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
79	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
80	C₂×C₄×C₂₃	#41	1	2	yes	—
81	#2792	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂²×C₂₃	C₂²×Q₈	1	2	yes	—
83	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
84	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
85	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
86	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
87	C₄×C₂₃	—	2	2	no	—
88	C₄×C₂₃	—	2	4	no	—
89	C₄×C₂₃	—	2	4	no	—
90	C₄×C₂₃	—	2	4	no	—
91	C₄×C₂₃	—	4	2	no	—
92	C₄×C₂₃	—	4	2	no	—
93	C₄×C₂₃	—	4	1	no	—
94	C₄×C₂₃	—	4	1	no	—
95	C₄×C₂₃	—	4	2	no	—
96	#398	C₂×C₂₃	1	2	yes	—
97	#398	C₂×C₂₃	1	1	yes	characteristic
98	#395	C₂×C₂₃	1	1	yes	characteristic
99	#395	C₂×C₂₃	1	2	yes	—
100	#389	C₂×C₂₃	1	2	yes	—
101	#389	C₂×C₂₃	1	2	yes	—
102	#387	C₂×C₂₃	1	2	yes	—
103	#387	C₂×C₂₃	1	2	yes	—
104	#384	C₂×C₂₃	1	1	yes	characteristic
105	C₂×C₂₃	#539	1	1	yes	characteristic
106	C₂×C₂₃	#544	1	2	yes	—
107	C₂×C₂₃	#554	1	2	yes	—
108	C₂×C₂₃	Q₈²	1	1	yes	characteristic
109	C₂×C₂₃	#560	1	1	yes	characteristic
110	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
111	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
112	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
113	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
115	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
116	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
117	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
118	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
119	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
120	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
121	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
122	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
123	#95	C₂²×C₂₃	1	1	yes	characteristic
124	#95	C₂²×C₂₃	1	2	yes	—
125	#95	C₂²×C₂₃	1	2	yes	—
126	#95	C₂²×C₂₃	1	2	yes	—
127	#95	C₂²×C₂₃	1	2	yes	—
128	#95	C₂²×C₂₃	1	2	yes	—
129	#95	C₂²×C₂₃	1	2	yes	—
130	#95	C₂²×C₂₃	1	1	yes	characteristic
131	C₂₃	#2792	1	1	yes	characteristic, direct product factor, Sylow
132	C₂×Q₈	—	4	2	no	—
133	C₂×Q₈	—	4	1	no	—
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	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
142	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	1	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	1	no	—
150	C₄⋊C₄	—	4	1	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	Q₈×C₂₃	1	8	yes	complemented
153	C₄²	—	2	2	no	—
154	C₄²	—	4	2	no	—
155	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
156	Q₈	—	4	4	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	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	2	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	C₂×Q₈×C₂₃	1	4	yes	—
179	C₂×C₄	C₂×Q₈×C₂₃	1	4	yes	—
180	C₂×C₄	C₂×Q₈×C₂₃	1	4	yes	—
181	C₂×C₄	#41×C₂₃	1	2	yes	—
182	C₂²	C₂²×Q₈×C₂₃	1	2	yes	—
183	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
184	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
185	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
186	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
187	C₄	—	2	4	no	—
188	C₄	—	2	4	no	—
189	C₄	—	2	2	no	—
190	C₄	—	2	4	no	—
191	C₄	—	4	2	no	—
192	C₄	—	4	1	no	—
193	C₄	—	4	2	no	—
194	C₄	—	4	1	no	—
195	C₄	—	4	2	no	—
196	C₂	C₂₃×#539	1	1	yes	characteristic
197	C₂	C₂₃×#544	1	2	yes	—
198	C₂	C₂₃×#554	1	2	yes	—
199	C₂	Q₈²×C₂₃	1	1	yes	characteristic
200	C₂	C₂₃×#560	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. C₄⋊C₄×C₂₃	Q₈	semidirect
81. #2792	C₂₃	direct (N × H)
131. C₂₃	#2792	direct (N × H)
152. C₄⋊C₄	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×#398	2
C₂₃×#395	2
C₂₃×#389	2
C₂₃×#387	2
C₂₃×#384	2
#2792	23

Subgroups — C23×#2792 #15670411

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2792 #15670411