Subgroups — C₂₃×#7171 #19154797

Order 5888.

282 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×#107	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆×C₂₃	—	2	1	no	—
5	C₂×C₂₃×#107	C₄	1	2	yes	—
6	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
7	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
9	C₄×C₁₆×C₂₃	—	2	1	no	—
10	C₄×C₁₆×C₂₃	C₄	1	2	yes	complemented
11	C₂₃×#340	—	2	1	no	—
12	C₂₃×#335	—	2	1	no	—
13	C₂₃×#335	C₄	1	2	yes	complemented
14	C₂²×C₈×C₂₃	—	4	1	no	—
15	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
18	C₂×C₄²×C₂₃	—	4	1	no	—
19	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
20	C₂×C₁₆×C₂₃	—	2	4	no	—
21	C₂×C₁₆×C₂₃	—	2	2	no	—
22	C₂₃×#107	—	2	2	no	—
23	C₂₃×#107	C₈	1	4	yes	complemented
24	C₂₃×C₄⋊C₈	—	2	2	no	—
25	C₂₃×C₄⋊C₈	—	4	1	no	—
26	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
30	C₂₃×#95	—	4	1	no	—
31	C₂²×C₄×C₂₃	—	4	2	no	—
32	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
35	C₂×C₈×C₂₃	—	2	2	no	—
36	C₂×C₈×C₂₃	—	4	2	no	—
37	C₂×C₈×C₂₃	—	8	1	no	—
38	C₂×C₈×C₂₃	—	8	1	no	—
39	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
45	C₄⋊C₄×C₂₃	—	2	4	no	—
46	C₄⋊C₄×C₂₃	—	4	2	no	—
47	C₄²×C₂₃	—	8	1	no	—
48	C₄²×C₂₃	—	8	1	no	—
49	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
52	C₄²×C₂₃	Q₁₆	1	1	yes	characteristic
53	C₁₆×C₂₃	—	2	4	no	—
54	#7171	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂³×C₂₃	#102	1	1	yes	characteristic
56	C₂×C₄×C₂₃	—	2	2	no	—
57	C₂×C₄×C₂₃	—	4	4	no	—
58	C₂×C₄×C₂₃	—	8	2	no	—
59	C₂×C₄×C₂₃	—	8	2	no	—
60	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
61	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
62	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
63	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
64	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
65	C₂×C₄×C₂₃	#104	1	2	yes	—
66	C₂×C₄×C₂₃	#106	1	1	yes	characteristic
67	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
68	C₂×C₄×C₂₃	D₁₆	1	1	yes	characteristic
69	C₂×C₄×C₂₃	SD₃₂	1	1	yes	characteristic
70	C₂×C₄×C₂₃	SD₃₂	1	1	yes	characteristic
71	C₂×C₄×C₂₃	Q₃₂	1	1	yes	characteristic
72	C₈×C₂₃	—	2	1	no	—
73	C₈×C₂₃	—	2	1	no	—
74	C₈×C₂₃	—	8	2	no	—
75	C₈×C₂₃	C₄×C₈	1	1	yes	characteristic
76	C₈×C₂₃	#97	1	1	yes	characteristic
77	C₈×C₂₃	C₄⋊C₈	1	2	yes	—
78	C₂×C₄×C₁₆	C₂×C₂₃	1	1	yes	characteristic
79	C₄×#107	C₂×C₂₃	1	1	yes	characteristic
80	C₂×#335	C₂×C₂₃	1	1	yes	characteristic
81	C₂²×C₂₃	D₄⋊C₈	1	2	yes	—
82	C₂²×C₂₃	#340	1	1	yes	characteristic
83	C₂²×C₂₃	#357	1	1	yes	characteristic
84	C₂²×C₂₃	#358	1	1	yes	characteristic
85	C₂²×C₂₃	#359	1	2	yes	—
86	C₄×C₂₃	—	2	1	no	—
87	C₄×C₂₃	—	2	1	no	—
88	C₄×C₂₃	—	8	4	no	—
89	C₄×C₂₃	Q₈⋊C₈	1	2	yes	—
90	C₄×C₂₃	#336	1	1	yes	characteristic
91	C₄×C₂₃	#339	1	1	yes	characteristic
92	C₄×C₂₃	#357	1	1	yes	characteristic
93	C₄×C₂₃	#358	1	1	yes	characteristic
94	C₄×C₂₃	#366	1	1	yes	characteristic
95	C₄×C₂₃	#367	1	1	yes	characteristic
96	C₂²×C₁₆	—	2	1	no	—
97	C₂×#107	C₄×C₂₃	1	2	yes	—
98	C₂×C₄⋊C₈	—	2	1	no	—
99	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
100	C₄×C₄⋊C₄	—	2	1	no	—
101	C₄×C₁₆	—	2	1	no	—
102	C₄×C₁₆	C₄×C₂₃	1	2	yes	complemented
103	#340	—	2	1	no	—
104	#335	—	2	1	no	—
105	#335	C₄×C₂₃	1	2	yes	complemented
106	C₂×C₂₃	#1279	1	1	yes	characteristic
107	C₂×C₂₃	#1331	1	2	yes	—
108	C₂×C₂₃	#1332	1	2	yes	—
109	C₂×C₂₃	#1380	1	1	yes	characteristic
110	C₂×C₂₃	#1382	1	1	yes	characteristic
111	C₂²×C₈	—	4	1	no	—
112	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
113	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄²	—	4	1	no	—
116	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
117	C₂×C₁₆	—	2	4	no	—
118	C₂×C₁₆	—	2	2	no	—
119	#107	—	2	2	no	—
120	#107	C₈×C₂₃	1	4	yes	complemented
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	—	4	1	no	—
123	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
124	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
125	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
126	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
127	#95	—	4	1	no	—
128	C₂₃	#7171	1	1	yes	characteristic, direct product factor, Sylow
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
131	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
132	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	8	1	no	—
136	C₂×C₈	—	8	1	no	—
137	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
138	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
139	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
140	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
141	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
142	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄²	—	8	1	no	—
146	C₄²	—	8	1	no	—
147	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
148	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
149	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
150	C₄²	Q₁₆×C₂₃	1	1	yes	characteristic
151	C₁₆	—	2	4	no	—
152	C₂³	C₂₃×#102	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
158	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
159	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
160	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
161	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
162	C₂×C₄	C₂₃×#104	1	2	yes	—
163	C₂×C₄	C₂₃×#106	1	1	yes	characteristic
164	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
165	C₂×C₄	C₂₃×D₁₆	1	1	yes	characteristic
166	C₂×C₄	C₂₃×SD₃₂	1	1	yes	characteristic
167	C₂×C₄	C₂₃×SD₃₂	1	1	yes	characteristic
168	C₂×C₄	C₂₃×Q₃₂	1	1	yes	characteristic
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	8	2	no	—
172	C₈	C₄×C₈×C₂₃	1	1	yes	characteristic
173	C₈	C₂₃×#97	1	1	yes	characteristic
174	C₈	C₂₃×C₄⋊C₈	1	2	yes	—
175	C₂²	C₂₃×D₄⋊C₈	1	2	yes	—
176	C₂²	C₂₃×#340	1	1	yes	characteristic
177	C₂²	C₂₃×#357	1	1	yes	characteristic
178	C₂²	C₂₃×#358	1	1	yes	characteristic
179	C₂²	C₂₃×#359	1	2	yes	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	8	4	no	—
183	C₄	C₂₃×Q₈⋊C₈	1	2	yes	—
184	C₄	C₂₃×#336	1	1	yes	characteristic
185	C₄	C₂₃×#339	1	1	yes	characteristic
186	C₄	C₂₃×#357	1	1	yes	characteristic
187	C₄	C₂₃×#358	1	1	yes	characteristic
188	C₄	C₂₃×#366	1	1	yes	characteristic
189	C₄	C₂₃×#367	1	1	yes	characteristic
190	C₂	C₂₃×#1279	1	1	yes	characteristic
191	C₂	C₂₃×#1331	1	2	yes	—
192	C₂	C₂₃×#1332	1	2	yes	—
193	C₂	C₂₃×#1380	1	1	yes	characteristic
194	C₂	C₂₃×#1382	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	D₈
commutator	C₈	C₄×C₈×C₂₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₂₃
socle, prime radical	C₂³×C₂₃	#102
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₄×C₁₆×C₂₃	C₄	semidirect
13. C₂₃×#335	C₄	semidirect
23. C₂₃×#107	C₈	semidirect
54. #7171	C₂₃	direct (N × H)
102. C₄×C₁₆	C₄×C₂₃	semidirect
105. #335	C₄×C₂₃	semidirect
120. #107	C₈×C₂₃	semidirect
128. C₂₃	#7171	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×C₁₆×C₂₃	2
C₄×C₂₃×#107	2
C₂×C₂₃×#335	2
#7171	23

Subgroups — C23×#7171 #19154797

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7171 #19154797