Subgroups — C₂₃×#10705 #19158331

Order 5888.

870 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₂₃	C₂	1	2	yes	maximal
2	C₂²×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#341	C₂	1	4	yes	maximal
4	C₂²×M₁₆×C₂₃	C₂²	1	4	yes	—
5	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
6	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₄×M₁₆×C₂₃	—	2	4	no	—
8	C₄×M₁₆×C₂₃	C₄	1	8	yes	—
9	C₂×C₂₃×#97	—	2	4	no	—
10	C₂×C₂₃×#97	—	2	2	no	—
11	C₂×C₂₃×#97	C₄	1	4	yes	—
12	C₂×C₄×C₈×C₂₃	—	2	2	no	—
13	C₂₃×#341	C₄	1	16	yes	complemented
14	C₂³×C₄×C₂₃	Q₈	1	2	yes	—
15	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
16	C₂×M₁₆×C₂₃	—	2	4	no	—
17	C₂×M₁₆×C₂₃	—	2	8	no	—
18	C₂×M₁₆×C₂₃	C₂×C₄	1	16	yes	—
19	C₂×M₁₆×C₂₃	C₂×C₄	1	8	yes	—
20	C₂²×C₈×C₂₃	—	2	4	no	—
21	C₂²×C₈×C₂₃	—	2	4	no	—
22	C₂²×C₈×C₂₃	—	2	4	no	—
23	C₂²×C₈×C₂₃	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₂₃	D₄	1	4	yes	—
28	C₂×C₄²×C₂₃	D₄	1	2	yes	—
29	C₂₃×#97	—	2	8	no	—
30	C₂₃×#97	—	2	8	no	—
31	C₄×C₈×C₂₃	—	2	8	no	—
32	C₂⁴×C₂₃	C₂×Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	—	4	4	no	—
34	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
35	C₂²×C₄×C₂₃	C₄⋊C₄	1	8	yes	—
36	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
39	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
40	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	#41	1	4	yes	—
42	M₁₆×C₂₃	—	2	8	no	—
43	M₁₆×C₂₃	—	2	16	no	—
44	M₁₆×C₂₃	C₄²	1	16	yes	—
45	C₂×C₈×C₂₃	—	2	4	no	—
46	C₂×C₈×C₂₃	—	2	8	no	—
47	C₂×C₈×C₂₃	—	2	4	no	—
48	C₂×C₈×C₂₃	—	2	16	no	—
49	C₂×C₈×C₂₃	—	2	16	no	—
50	C₂×C₈×C₂₃	C₄²	1	8	yes	—
51	C₄²×C₂₃	C₄²	1	4	yes	—
52	C₄²×C₂₃	C₂²⋊C₄	1	4	yes	—
53	C₄²×C₂₃	C₂²⋊C₄	1	4	yes	—
54	C₄²×C₂₃	C₂²⋊C₄	1	4	yes	—
55	#10705	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂³×C₂₃	—	2	4	no	—
57	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
58	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂³×C₂₃	C₄×Q₈	1	4	yes	—
60	C₂×C₄×C₂₃	—	2	8	no	—
61	C₂×C₄×C₂₃	—	4	8	no	—
62	C₂×C₄×C₂₃	#95	1	8	yes	—
63	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
65	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
66	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
68	C₂×C₄×C₂₃	#117	1	2	yes	—
69	C₂×C₄×C₂₃	#117	1	4	yes	—
70	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
71	C₂×C₄×C₂₃	C₄×D₄	1	8	yes	—
72	C₈×C₂₃	—	2	8	no	—
73	C₈×C₂₃	—	2	16	no	—
74	C₂×C₄×M₁₆	C₂×C₂₃	1	2	yes	—
75	C₂²×#97	C₂×C₂₃	1	1	yes	characteristic
76	C₂×#341	C₂×C₂₃	1	4	yes	—
77	C₂²×C₂₃	—	2	4	no	—
78	C₂²×C₂₃	—	2	8	no	—
79	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
80	C₂²×C₂₃	C₄×C₄⋊C₄	1	4	yes	—
81	C₂²×C₂₃	C₄×C₄⋊C₄	1	2	yes	—
82	C₂²×C₂₃	#430	1	4	yes	—
83	C₄×C₂₃	—	2	8	no	—
84	C₄×C₂₃	C₄³	1	1	yes	characteristic
85	C₄×C₂₃	#376	1	1	yes	characteristic
86	C₄×C₂₃	C₄×C₂²⋊C₄	1	2	yes	—
87	C₄×C₂₃	C₄×C₂²⋊C₄	1	4	yes	—
88	C₂²×M₁₆	C₂²×C₂₃	1	4	yes	—
89	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
90	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
91	C₄×M₁₆	—	2	4	no	—
92	C₄×M₁₆	C₄×C₂₃	1	8	yes	—
93	C₂×#97	—	2	4	no	—
94	C₂×#97	—	2	2	no	—
95	C₂×#97	C₄×C₂₃	1	4	yes	—
96	C₂×C₄×C₈	—	2	2	no	—
97	#341	C₄×C₂₃	1	16	yes	complemented
98	C₂×C₂₃	—	2	4	no	—
99	C₂×C₂₃	C₄×#95	1	1	yes	characteristic
100	C₂×C₂₃	#1738	1	2	yes	—
101	C₂×C₂₃	#1778	1	4	yes	—
102	C₂³×C₄	Q₈×C₂₃	1	2	yes	—
103	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
104	C₂×M₁₆	—	2	4	no	—
105	C₂×M₁₆	—	2	8	no	—
106	C₂×M₁₆	C₂×C₄×C₂₃	1	16	yes	—
107	C₂×M₁₆	C₂×C₄×C₂₃	1	8	yes	—
108	C₂²×C₈	—	2	4	no	—
109	C₂²×C₈	—	2	4	no	—
110	C₂²×C₈	—	2	4	no	—
111	C₂²×C₈	C₂×C₄×C₂₃	1	8	yes	—
112	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
113	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
114	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
115	C₂×C₄²	D₄×C₂₃	1	2	yes	—
116	C₂×C₄²	D₄×C₂₃	1	4	yes	—
117	#97	—	2	8	no	—
118	#97	—	2	8	no	—
119	C₄×C₈	—	2	8	no	—
120	C₂₃	#10705	1	1	yes	characteristic, direct product factor, Sylow
121	C₂⁴	C₂×Q₈×C₂₃	1	1	yes	characteristic
122	C₂²×C₄	—	4	4	no	—
123	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
124	C₂²×C₄	C₄⋊C₄×C₂₃	1	8	yes	—
125	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
126	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
127	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
128	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
129	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
130	C₂²×C₄	#41×C₂₃	1	4	yes	—
131	M₁₆	—	2	8	no	—
132	M₁₆	—	2	16	no	—
133	M₁₆	C₄²×C₂₃	1	16	yes	—
134	C₂×C₈	—	2	8	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	2	16	no	—
138	C₂×C₈	—	2	16	no	—
139	C₂×C₈	C₄²×C₂₃	1	8	yes	—
140	C₄²	C₄²×C₂₃	1	4	yes	—
141	C₄²	C₂²⋊C₄×C₂₃	1	4	yes	—
142	C₄²	C₂²⋊C₄×C₂₃	1	4	yes	—
143	C₄²	C₂²⋊C₄×C₂₃	1	4	yes	—
144	C₂³	—	2	4	no	—
145	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
146	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
147	C₂³	C₄×Q₈×C₂₃	1	4	yes	—
148	C₂×C₄	—	2	8	no	—
149	C₂×C₄	—	4	8	no	—
150	C₂×C₄	C₂₃×#95	1	8	yes	—
151	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
153	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
154	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
155	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
156	C₂×C₄	C₂₃×#117	1	2	yes	—
157	C₂×C₄	C₂₃×#117	1	4	yes	—
158	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
159	C₂×C₄	C₄×D₄×C₂₃	1	8	yes	—
160	C₈	—	2	8	no	—
161	C₈	—	2	16	no	—
162	C₂²	—	2	4	no	—
163	C₂²	—	2	8	no	—
164	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
165	C₂²	C₄×C₄⋊C₄×C₂₃	1	4	yes	—
166	C₂²	C₄×C₄⋊C₄×C₂₃	1	2	yes	—
167	C₂²	C₂₃×#430	1	4	yes	—
168	C₄	—	2	8	no	—
169	C₄	C₄³×C₂₃	1	1	yes	characteristic
170	C₄	C₂₃×#376	1	1	yes	characteristic
171	C₄	C₄×C₂²⋊C₄×C₂₃	1	2	yes	—
172	C₄	C₄×C₂²⋊C₄×C₂₃	1	4	yes	—
173	C₂	—	2	4	no	—
174	C₂	C₄×C₂₃×#95	1	1	yes	characteristic
175	C₂	C₂₃×#1738	1	2	yes	—
176	C₂	C₂₃×#1778	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₄	C₄³×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₂₃	C₂×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂₃×#341	C₄	semidirect
55. #10705	C₂₃	direct (N × H)
97. #341	C₄×C₂₃	semidirect
120. C₂₃	#10705	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×C₄×M₁₆×C₂₃	2
C₂²×C₂₃×#97	2
C₂×C₂₃×#341	2
#10705	23

Subgroups — C23×#10705 #19158331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10705 #19158331