Subgroups — C₂₃×#2794 #15670413

Order 2944.

344 conjugacy classes of subgroups.

The subgroup lattice has 190 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	2	yes	maximal
2	C₂₃×#400	C₂	1	2	yes	maximal
3	C₂₃×#400	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#398	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#387	C₂	1	2	yes	maximal
6	C₂₃×#387	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	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
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	1	yes	characteristic
18	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
20	C₂₃×#95	C₂²	1	2	yes	—
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	2	yes	—
28	C₂₃×#95	C₂²	1	2	yes	—
29	C₂₃×#95	C₂²	1	1	yes	characteristic
30	C₂₃×#95	C₂²	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
32	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
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	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
40	C₄⋊C₄×C₂₃	—	2	4	no	—
41	C₄⋊C₄×C₂₃	—	2	4	no	—
42	C₄⋊C₄×C₂₃	—	2	4	no	—
43	C₄⋊C₄×C₂₃	—	2	4	no	—
44	C₄⋊C₄×C₂₃	—	4	1	no	—
45	C₄⋊C₄×C₂₃	—	4	1	no	—
46	C₄⋊C₄×C₂₃	—	4	1	no	—
47	C₄⋊C₄×C₂₃	—	4	1	no	—
48	C₄²×C₂₃	—	2	2	no	—
49	C₄²×C₂₃	—	4	2	no	—
50	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
51	C₂×C₄×C₂₃	—	2	2	no	—
52	C₂×C₄×C₂₃	—	2	1	no	—
53	C₂×C₄×C₂₃	—	2	1	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	2	no	—
59	C₂×C₄×C₂₃	—	2	2	no	—
60	C₂×C₄×C₂₃	—	2	2	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	1	no	—
67	C₂×C₄×C₂₃	—	2	1	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	2	2	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	4	no	—
73	C₂×C₄×C₂₃	#41	1	2	yes	—
74	C₂×C₄×C₂₃	#41	1	4	yes	—
75	#2794	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
77	C₂²×C₂₃	C₂×#41	1	2	yes	—
78	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
79	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
80	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
81	C₄×C₂₃	—	2	2	no	—
82	C₄×C₂₃	—	2	4	no	—
83	C₄×C₂₃	—	4	2	no	—
84	C₄×C₂₃	—	4	2	no	—
85	C₄×C₂₃	—	4	2	no	—
86	C₄×C₂₃	—	4	2	no	—
87	C₄×C₂₃	—	4	1	no	—
88	C₄×C₂₃	—	4	1	no	—
89	C₄×C₂₃	—	4	2	no	—
90	#400	C₂×C₂₃	1	2	yes	—
91	#400	C₂×C₂₃	1	2	yes	—
92	#400	C₂×C₂₃	1	1	yes	characteristic
93	#398	C₂×C₂₃	1	1	yes	characteristic
94	#387	C₂×C₂₃	1	2	yes	—
95	#387	C₂×C₂₃	1	2	yes	—
96	#387	C₂×C₂₃	1	2	yes	—
97	#387	C₂×C₂₃	1	2	yes	—
98	#384	C₂×C₂₃	1	1	yes	characteristic
99	C₂×C₂₃	#539	1	1	yes	characteristic
100	C₂×C₂₃	#541	1	2	yes	—
101	C₂×C₂₃	#552	1	2	yes	—
102	C₂×C₂₃	#559	1	1	yes	characteristic
103	C₂×C₂₃	#563	1	1	yes	characteristic
104	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
105	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
106	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
107	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
108	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
109	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
110	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
111	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
112	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
113	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
114	#95	C₂²×C₂₃	1	2	yes	—
115	#95	C₂²×C₂₃	1	2	yes	—
116	#95	C₂²×C₂₃	1	2	yes	—
117	#95	C₂²×C₂₃	1	2	yes	—
118	#95	C₂²×C₂₃	1	2	yes	—
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	1	yes	characteristic
124	#95	C₂²×C₂₃	1	1	yes	characteristic
125	C₂₃	#2794	1	1	yes	characteristic, direct product factor, Sylow
126	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
127	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
128	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
129	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
130	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
131	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
132	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
133	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
134	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
135	C₄⋊C₄	—	2	4	no	—
136	C₄⋊C₄	—	2	4	no	—
137	C₄⋊C₄	—	2	4	no	—
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	4	1	no	—
140	C₄⋊C₄	—	4	1	no	—
141	C₄⋊C₄	—	4	1	no	—
142	C₄⋊C₄	—	4	1	no	—
143	C₄²	—	2	2	no	—
144	C₄²	—	4	2	no	—
145	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	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	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	#41×C₂₃	1	2	yes	—
169	C₂×C₄	#41×C₂₃	1	4	yes	—
170	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
171	C₂²	C₂×#41×C₂₃	1	2	yes	—
172	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
173	C₂²	C₂₃×2₋¹⁺⁴	1	2	yes	—
174	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
175	C₄	—	2	4	no	—
176	C₄	—	2	2	no	—
177	C₄	—	4	2	no	—
178	C₄	—	4	2	no	—
179	C₄	—	4	1	no	—
180	C₄	—	4	1	no	—
181	C₄	—	4	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₂	C₂₃×#539	1	1	yes	characteristic
185	C₂	C₂₃×#541	1	2	yes	—
186	C₂	C₂₃×#552	1	2	yes	—
187	C₂	C₂₃×#559	1	1	yes	characteristic
188	C₂	C₂₃×#563	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
75. #2794	C₂₃	direct (N × H)
125. C₂₃	#2794	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×#400	2
C₂₃×#398	2
C₂₃×#387	2
C₂₃×#384	2
#2794	23

Subgroups — C23×#2794 #15670413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2794 #15670413