Subgroups — C₂₃×#22277 #19169903

Order 5888.

826 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#494	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2327	C₂	1	2	yes	maximal
3	C₂₃×#1960	C₂	1	2	yes	maximal
4	C₂₃×#1878	C₂	1	8	yes	maximal
5	C₂₃×#1835	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1746	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₂₃	C₂²	1	4	yes	—
8	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₄×Q₈×C₂₃	C₂²	1	4	yes	—
10	C₂₃×#494	C₄	1	8	yes	complemented
11	C₂×C₂₃×#103	C₂²	1	8	yes	—
12	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#391	C₂²	1	4	yes	—
15	C₂₃×#387	—	2	8	no	—
16	C₂₃×#382	C₂²	1	2	yes	—
17	C₂₃×#376	C₂²	1	1	yes	characteristic
18	C₂₃×#340	C₂²	1	4	yes	—
19	C₂₃×#336	C₂²	1	4	yes	—
20	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
21	C₂×Q₁₆×C₂₃	—	4	8	no	—
22	C₂×Q₁₆×C₂₃	C₂×C₄	1	16	yes	—
23	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
24	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
25	C₂₃×C₄⋊Q₈	—	2	4	no	—
26	C₂₃×C₄⋊Q₈	C₂×C₄	1	8	yes	—
27	C₄×Q₈×C₂₃	—	4	8	no	—
28	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
29	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
30	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
31	C₂×C₄²×C₂₃	—	2	4	no	—
32	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
33	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
34	C₂₃×#103	—	4	8	no	—
35	C₂₃×#97	—	2	2	no	—
36	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
37	C₂₃×#95	—	2	4	no	—
38	C₂₃×#95	—	2	8	no	—
39	C₂₃×#95	D₄	1	4	yes	—
40	C₂×Q₈×C₂₃	—	2	4	no	—
41	C₂×Q₈×C₂₃	—	2	8	no	—
42	C₂×Q₈×C₂₃	—	2	8	no	—
43	C₂×Q₈×C₂₃	C₂²×C₄	1	8	yes	—
44	C₂²×C₄×C₂₃	—	2	4	no	—
45	C₂²×C₄×C₂₃	—	2	4	no	—
46	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
47	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
49	Q₁₆×C₂₃	—	4	16	no	—
50	C₂×C₈×C₂₃	—	2	4	no	—
51	C₂×C₈×C₂₃	—	2	2	no	—
52	C₂×C₈×C₂₃	—	2	4	no	—
53	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
54	C₄⋊C₄×C₂₃	—	2	16	no	—
55	C₄⋊C₄×C₂₃	—	2	8	no	—
56	C₄⋊C₄×C₂₃	—	4	8	no	—
57	C₄²×C₂₃	—	2	4	no	—
58	C₄²×C₂₃	—	4	8	no	—
59	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
60	C₄²×C₂₃	#41	1	2	yes	—
61	#22277	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
63	Q₈×C₂₃	—	2	16	no	—
64	Q₈×C₂₃	—	4	8	no	—
65	C₂×C₄×C₂₃	—	2	4	no	—
66	C₂×C₄×C₂₃	—	2	8	no	—
67	C₂×C₄×C₂₃	—	2	4	no	—
68	C₂×C₄×C₂₃	—	2	8	no	—
69	C₂×C₄×C₂₃	—	4	4	no	—
70	C₂×C₄×C₂₃	—	4	4	no	—
71	C₂×C₄×C₂₃	—	4	4	no	—
72	C₂×C₄×C₂₃	—	4	8	no	—
73	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
74	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
75	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
76	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
77	C₈×C₂₃	—	2	4	no	—
78	C₈×C₂₃	—	4	2	no	—
79	C₂×#494	C₂×C₂₃	1	1	yes	characteristic
80	#2327	C₂×C₂₃	1	2	yes	—
81	#1960	C₂×C₂₃	1	2	yes	—
82	#1878	C₂×C₂₃	1	8	yes	—
83	#1835	C₂×C₂₃	1	1	yes	characteristic
84	#1746	C₂×C₂₃	1	1	yes	characteristic
85	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
86	C₂²×C₂₃	#535	1	2	yes	—
87	C₂²×C₂₃	#578	1	4	yes	—
88	C₄×C₂₃	—	2	4	no	—
89	C₄×C₂₃	—	4	8	no	—
90	C₄×C₂₃	—	4	4	no	—
91	C₄×C₂₃	—	8	4	no	—
92	C₄×C₂₃	#518	1	2	yes	—
93	C₄×C₂₃	#540	1	2	yes	—
94	C₂²×Q₁₆	C₂²×C₂₃	1	4	yes	—
95	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
96	C₂×C₄×Q₈	C₂²×C₂₃	1	4	yes	—
97	#494	C₄×C₂₃	1	8	yes	complemented
98	C₂×#103	C₂²×C₂₃	1	8	yes	—
99	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
100	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
101	#391	C₂²×C₂₃	1	4	yes	—
102	#387	—	2	8	no	—
103	#382	C₂²×C₂₃	1	2	yes	—
104	#376	C₂²×C₂₃	1	1	yes	characteristic
105	#340	C₂²×C₂₃	1	4	yes	—
106	#336	C₂²×C₂₃	1	4	yes	—
107	C₂×C₂₃	#2328	1	1	yes	characteristic
108	C₂×C₂₃	#2950	1	2	yes	—
109	C₂×C₂₃	#3223	1	4	yes	—
110	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
111	C₂×Q₁₆	—	4	8	no	—
112	C₂×Q₁₆	C₂×C₄×C₂₃	1	16	yes	—
113	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
114	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
115	C₄⋊Q₈	—	2	4	no	—
116	C₄⋊Q₈	C₂×C₄×C₂₃	1	8	yes	—
117	C₄×Q₈	—	4	8	no	—
118	C₂×C₄⋊C₄	—	2	4	no	—
119	C₂×C₄⋊C₄	—	2	4	no	—
120	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
121	C₂×C₄²	—	2	4	no	—
122	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
123	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
124	#103	—	4	8	no	—
125	#97	—	2	2	no	—
126	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
127	#95	—	2	4	no	—
128	#95	—	2	8	no	—
129	#95	D₄×C₂₃	1	4	yes	—
130	C₂₃	#22277	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×Q₈	—	2	4	no	—
132	C₂×Q₈	—	2	8	no	—
133	C₂×Q₈	—	2	8	no	—
134	C₂×Q₈	C₂²×C₄×C₂₃	1	8	yes	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
138	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
139	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
140	Q₁₆	—	4	16	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
145	C₄⋊C₄	—	2	16	no	—
146	C₄⋊C₄	—	2	8	no	—
147	C₄⋊C₄	—	4	8	no	—
148	C₄²	—	2	4	no	—
149	C₄²	—	4	8	no	—
150	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
151	C₄²	#41×C₂₃	1	2	yes	—
152	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
153	Q₈	—	2	16	no	—
154	Q₈	—	4	8	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	2	8	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	4	8	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
164	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
165	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
166	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
167	C₈	—	2	4	no	—
168	C₈	—	4	2	no	—
169	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
170	C₂²	C₂₃×#535	1	2	yes	—
171	C₂²	C₂₃×#578	1	4	yes	—
172	C₄	—	2	4	no	—
173	C₄	—	4	4	no	—
174	C₄	—	4	8	no	—
175	C₄	—	8	4	no	—
176	C₄	C₂₃×#518	1	2	yes	—
177	C₄	C₂₃×#540	1	2	yes	—
178	C₂	C₂₃×#2328	1	1	yes	characteristic
179	C₂	C₂₃×#2950	1	2	yes	—
180	C₂	C₂₃×#3223	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
10. C₂₃×#494	C₄	semidirect
61. #22277	C₂₃	direct (N × H)
97. #494	C₄×C₂₃	semidirect
130. C₂₃	#22277	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#494	2
C₂₃×#2327	2
C₂₃×#1960	2
C₂₃×#1878	2
C₂₃×#1835	2
C₂₃×#1746	2
#22277	23

Subgroups — C23×#22277 #19169903

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22277 #19169903