Subgroups — C₂₃×#2548 #15670167

Order 2944.

676 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#124	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#394	C₂	1	4	yes	maximal, complemented
5	C₂₃×#390	C₂	1	2	yes	maximal, complemented
6	C₂₃×#390	C₂	1	4	yes	maximal, complemented
7	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
10	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
11	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
12	C₂₃×#127	—	2	4	no	—
13	C₂₃×#124	—	2	8	no	—
14	C₄×Q₈×C₂₃	—	2	4	no	—
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₂₃	C₂²	1	8	yes	—
19	C₂×C₂²⋊C₄×C₂₃	C₂²	1	8	yes	—
20	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#95	C₂²	1	2	yes	—
24	C₂⁴×C₂₃	C₂³	1	4	yes	—
25	C₂×Q₈×C₂₃	—	2	2	no	—
26	C₂×Q₈×C₂₃	—	2	2	no	—
27	C₂×Q₈×C₂₃	D₄	1	4	yes	complemented
28	C₂×D₄×C₂₃	—	2	4	no	—
29	C₂×D₄×C₂₃	—	2	4	no	—
30	C₂×D₄×C₂₃	—	2	4	no	—
31	C₂×D₄×C₂₃	—	2	4	no	—
32	C₂×D₄×C₂₃	—	2	4	no	—
33	C₂×D₄×C₂₃	—	2	8	no	—
34	C₂×D₄×C₂₃	—	4	4	no	—
35	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
36	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
37	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₂₃	—	2	4	no	—
42	C₄⋊C₄×C₂₃	—	2	2	no	—
43	C₄⋊C₄×C₂₃	—	2	4	no	—
44	C₂²⋊C₄×C₂₃	—	2	16	no	—
45	C₂²⋊C₄×C₂₃	—	4	8	no	—
46	C₄²×C₂₃	—	2	4	no	—
47	C₄²×C₂₃	—	2	4	no	—
48	C₄²×C₂₃	D₄	1	4	yes	—
49	C₂³×C₂₃	—	2	4	no	—
50	C₂³×C₂₃	—	2	4	no	—
51	C₂³×C₂₃	—	2	4	no	—
52	C₂³×C₂₃	—	2	8	no	—
53	C₂³×C₂₃	—	2	8	no	—
54	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
55	Q₈×C₂₃	—	2	8	no	—
56	D₄×C₂₃	—	4	8	no	—
57	D₄×C₂₃	—	4	8	no	—
58	D₄×C₂₃	—	4	8	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	4	no	—
63	C₂×C₄×C₂₃	—	2	2	no	—
64	C₂×C₄×C₂₃	—	2	4	no	—
65	C₂×C₄×C₂₃	—	2	4	no	—
66	C₂×C₄×C₂₃	—	2	4	no	—
67	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
69	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₂₃	#41	1	2	yes	—
73	C₂×C₄×C₂₃	#41	1	4	yes	—
74	#2548	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂²×C₂₃	—	4	4	no	—
76	C₂²×C₂₃	—	4	4	no	—
77	C₂²×C₂₃	—	4	4	no	—
78	C₂²×C₂₃	—	4	8	no	—
79	C₂²×C₂₃	—	4	8	no	—
80	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
81	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
82	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
83	C₂²×C₂₃	C₂×#41	1	2	yes	—
84	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
85	C₄×C₂₃	—	2	4	no	—
86	C₄×C₂₃	—	2	2	no	—
87	C₄×C₂₃	—	4	2	no	—
88	C₄×C₂₃	—	4	4	no	—
89	C₄×C₂₃	#121	1	4	yes	—
90	C₄×C₂₃	#124	1	4	yes	complemented
91	C₂×#127	C₂×C₂₃	1	1	yes	characteristic
92	C₂×#124	C₂×C₂₃	1	2	yes	complemented
93	C₂×C₄×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
94	#394	C₂×C₂₃	1	4	yes	complemented
95	#390	C₂×C₂₃	1	4	yes	complemented
96	#390	C₂×C₂₃	1	2	yes	complemented
97	C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
98	C₂×C₂₃	—	8	4	no	—
99	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
100	C₂×C₂₃	C₂×#124	1	1	yes	characteristic
101	C₂×C₂₃	#535	1	1	yes	characteristic
102	C₂×C₂₃	#550	1	2	yes	—
103	C₂×C₂₃	#559	1	2	yes	—
104	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
105	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
106	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
107	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
108	#127	—	2	4	no	—
109	#124	—	2	8	no	—
110	C₄×Q₈	—	2	4	no	—
111	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
112	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
113	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
114	C₂×C₂²⋊C₄	C₂²×C₂₃	1	8	yes	—
115	C₂×C₂²⋊C₄	C₂²×C₂₃	1	8	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	C₂₃	#2548	1	1	yes	characteristic, direct product factor, Sylow
121	C₂⁴	C₂³×C₂₃	1	4	yes	—
122	C₂×Q₈	—	2	2	no	—
123	C₂×Q₈	—	2	2	no	—
124	C₂×Q₈	D₄×C₂₃	1	4	yes	complemented
125	C₂×D₄	—	2	4	no	—
126	C₂×D₄	—	2	4	no	—
127	C₂×D₄	—	2	4	no	—
128	C₂×D₄	—	2	4	no	—
129	C₂×D₄	—	2	4	no	—
130	C₂×D₄	—	2	8	no	—
131	C₂×D₄	—	4	4	no	—
132	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
133	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
134	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
135	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
136	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
137	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	2	2	no	—
140	C₄⋊C₄	—	2	4	no	—
141	C₂²⋊C₄	—	2	16	no	—
142	C₂²⋊C₄	—	4	8	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	4	no	—
145	C₄²	D₄×C₂₃	1	4	yes	—
146	C₂³	—	2	4	no	—
147	C₂³	—	2	4	no	—
148	C₂³	—	2	8	no	—
149	C₂³	—	2	4	no	—
150	C₂³	—	2	8	no	—
151	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
152	Q₈	—	2	8	no	—
153	D₄	—	4	8	no	—
154	D₄	—	4	8	no	—
155	D₄	—	4	8	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
165	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
166	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
167	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
168	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
169	C₂×C₄	#41×C₂₃	1	2	yes	—
170	C₂×C₄	#41×C₂₃	1	4	yes	—
171	C₂²	—	4	8	no	—
172	C₂²	—	4	8	no	—
173	C₂²	—	4	4	no	—
174	C₂²	—	4	4	no	—
175	C₂²	—	4	4	no	—
176	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
177	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
178	C₂²	C₂×#41×C₂₃	1	2	yes	—
179	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
180	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
181	C₄	—	2	4	no	—
182	C₄	—	2	2	no	—
183	C₄	—	4	4	no	—
184	C₄	—	4	2	no	—
185	C₄	C₂₃×#121	1	4	yes	—
186	C₄	C₂₃×#124	1	4	yes	complemented
187	C₂	—	8	4	no	—
188	C₂	C₂×C₂₃×#121	1	1	yes	characteristic
189	C₂	C₂×C₂₃×#124	1	1	yes	characteristic
190	C₂	C₂₃×#535	1	1	yes	characteristic
191	C₂	C₂₃×#550	1	2	yes	—
192	C₂	C₂₃×#559	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#127	C₂
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
2. C₂×C₂₃×#124	C₂	semidirect
3. C₂×C₄×Q₈×C₂₃	C₂	semidirect
4. C₂₃×#394	C₂	semidirect
5. C₂₃×#390	C₂	semidirect
6. C₂₃×#390	C₂	semidirect
7. C₄×C₄⋊C₄×C₂₃	C₂	semidirect
27. C₂×Q₈×C₂₃	D₄	semidirect
74. #2548	C₂₃	direct (N × H)
90. C₄×C₂₃	#124	semidirect
92. C₂×#124	C₂×C₂₃	semidirect
93. C₂×C₄×Q₈	C₂×C₂₃	semidirect
94. #394	C₂×C₂₃	semidirect
95. #390	C₂×C₂₃	semidirect
96. #390	C₂×C₂₃	semidirect
97. C₄×C₄⋊C₄	C₂×C₂₃	semidirect
120. C₂₃	#2548	direct (N × H)
124. C₂×Q₈	D₄×C₂₃	semidirect
186. C₄	C₂₃×#124	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#127	2
C₂×C₂₃×#124	2
C₂×C₄×Q₈×C₂₃	2
C₂₃×#394	2
C₂₃×#390	2
C₄×C₄⋊C₄×C₂₃	2
#2548	23

Subgroups — C23×#2548 #15670167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2548 #15670167