Subgroups — C₂₃×#13172 #19160798

Order 5888.

274 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×#368	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#367	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2108	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1952	C₂	1	2	yes	maximal
5	C₂₃×#1387	C₂	1	2	yes	maximal
6	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×#108	C₂²	1	2	yes	—
9	C₂₃×#428	—	2	2	no	—
10	C₂×C₂₃×#107	C₂²	1	2	yes	—
11	C₂₃×#405	C₂²	1	1	yes	characteristic
12	C₂₃×#368	C₄	1	4	yes	complemented
13	C₂₃×#367	—	2	1	no	—
14	C₂₃×#367	—	2	1	no	—
15	C₂₃×#346	—	2	1	no	—
16	C₄×C₁₆×C₂₃	—	2	1	no	—
17	C₂₃×#343	—	2	2	no	—
18	C₂×M₁₆×C₂₃	—	2	2	no	—
19	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
21	C₂₃×#117	D₄	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
23	C₂₃×M₃₂	—	2	1	no	—
24	C₂₃×M₃₂	Q₈	1	2	yes	—
25	C₂×C₁₆×C₂₃	—	2	1	no	—
26	C₂×C₁₆×C₂₃	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
29	C₂×C₁₆×C₂₃	Q₈	1	2	yes	—
30	C₂₃×#108	—	2	2	no	—
31	C₂₃×#108	C₂×C₄	1	4	yes	—
32	C₂₃×#107	—	2	2	no	—
33	C₂₃×#107	—	2	2	no	—
34	C₂₃×#107	—	4	2	no	—
35	C₂₃×#106	—	4	2	no	—
36	C₂₃×#97	—	2	1	no	—
37	C₄×C₈×C₂₃	—	2	1	no	—
38	C₂²×C₄×C₂₃	—	4	2	no	—
39	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
40	M₁₆×C₂₃	—	2	1	no	—
41	M₁₆×C₂₃	—	2	4	no	—
42	M₁₆×C₂₃	—	4	2	no	—
43	M₁₆×C₂₃	Q₁₆	1	2	yes	—
44	C₂×C₈×C₂₃	—	2	1	no	—
45	C₂×C₈×C₂₃	—	4	2	no	—
46	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
51	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
52	C₄⋊C₄×C₂₃	—	4	2	no	—
53	C₄⋊C₄×C₂₃	—	4	2	no	—
54	C₄⋊C₄×C₂₃	D₈	1	2	yes	—
55	C₂²⋊C₄×C₂₃	—	2	1	no	—
56	C₄²×C₂₃	—	2	1	no	—
57	C₁₆×C₂₃	—	2	2	no	—
58	C₁₆×C₂₃	C₄⋊C₄	1	2	yes	—
59	C₁₆×C₂₃	C₄⋊C₄	1	2	yes	—
60	#13172	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₂₃	#135	1	1	yes	characteristic
62	C₂×C₄×C₂₃	—	2	2	no	—
63	C₂×C₄×C₂₃	—	8	2	no	—
64	C₂×C₄×C₂₃	—	16	2	no	—
65	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
66	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
67	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
71	C₈×C₂₃	—	2	2	no	—
72	C₈×C₂₃	—	4	4	no	—
73	C₈×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₈×C₂₃	C₄×Q₈	1	1	yes	characteristic
75	C₈×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₈×C₂₃	#125	1	1	yes	characteristic
77	C₂×#368	C₂×C₂₃	1	1	yes	characteristic
78	C₂×#367	C₂×C₂₃	1	1	yes	characteristic
79	#2108	C₂×C₂₃	1	1	yes	characteristic
80	#1952	C₂×C₂₃	1	2	yes	—
81	#1387	C₂×C₂₃	1	2	yes	—
82	C₂²×C₂₃	—	2	2	no	—
83	C₂²×C₂₃	C₄×Q₁₆	1	1	yes	characteristic
84	C₂²×C₂₃	#466	1	1	yes	characteristic
85	C₂²×C₂₃	#499	1	1	yes	characteristic
86	C₄×C₂₃	—	4	2	no	—
87	C₄×C₂₃	—	16	2	no	—
88	C₄×C₂₃	#389	1	1	yes	characteristic
89	C₄×C₂₃	C₄×D₈	1	1	yes	characteristic
90	C₄×C₂₃	#467	1	1	yes	characteristic
91	C₄×C₂₃	#500	1	1	yes	characteristic
92	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
93	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
94	C₂×#108	C₂²×C₂₃	1	2	yes	—
95	#428	—	2	2	no	—
96	C₂×#107	C₂²×C₂₃	1	2	yes	—
97	#405	C₂²×C₂₃	1	1	yes	characteristic
98	#368	C₄×C₂₃	1	4	yes	complemented
99	#367	—	2	1	no	—
100	#367	—	2	1	no	—
101	#346	—	2	1	no	—
102	C₄×C₁₆	—	2	1	no	—
103	#343	—	2	2	no	—
104	C₂×C₂₃	—	2	1	no	—
105	C₂×C₂₃	—	2	1	no	—
106	C₂×C₂₃	#1942	1	1	yes	characteristic
107	C₂×C₂₃	#2223	1	2	yes	—
108	C₂×M₁₆	—	2	2	no	—
109	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
110	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
111	#117	D₄×C₂₃	1	1	yes	characteristic
112	C₂×C₄⋊C₄	—	2	2	no	—
113	M₃₂	—	2	1	no	—
114	M₃₂	Q₈×C₂₃	1	2	yes	—
115	C₂×C₁₆	—	2	1	no	—
116	C₂×C₁₆	C₂×C₄×C₂₃	1	2	yes	—
117	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
118	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
119	C₂×C₁₆	Q₈×C₂₃	1	2	yes	—
120	#108	—	2	2	no	—
121	#108	C₂×C₄×C₂₃	1	4	yes	—
122	#107	—	2	2	no	—
123	#107	—	2	2	no	—
124	#107	—	4	2	no	—
125	#106	—	4	2	no	—
126	#97	—	2	1	no	—
127	C₄×C₈	—	2	1	no	—
128	C₂₃	#13172	1	1	yes	characteristic, direct product factor, Sylow
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
131	M₁₆	—	2	1	no	—
132	M₁₆	—	2	4	no	—
133	M₁₆	—	4	2	no	—
134	M₁₆	Q₁₆×C₂₃	1	2	yes	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
138	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
139	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
140	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
141	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
142	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	D₈×C₂₃	1	2	yes	—
146	C₂²⋊C₄	—	2	1	no	—
147	C₄²	—	2	1	no	—
148	C₁₆	—	2	2	no	—
149	C₁₆	C₄⋊C₄×C₂₃	1	2	yes	—
150	C₁₆	C₄⋊C₄×C₂₃	1	2	yes	—
151	C₂³	C₂₃×#135	1	1	yes	characteristic
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	8	2	no	—
154	C₂×C₄	—	16	2	no	—
155	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
156	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
157	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
158	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
159	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
160	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	—	4	4	no	—
163	C₈	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
164	C₈	C₄×Q₈×C₂₃	1	1	yes	characteristic
165	C₈	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
166	C₈	C₂₃×#125	1	1	yes	characteristic
167	C₂²	—	2	2	no	—
168	C₂²	C₄×Q₁₆×C₂₃	1	1	yes	characteristic
169	C₂²	C₂₃×#466	1	1	yes	characteristic
170	C₂²	C₂₃×#499	1	1	yes	characteristic
171	C₄	—	4	2	no	—
172	C₄	—	16	2	no	—
173	C₄	C₂₃×#389	1	1	yes	characteristic
174	C₄	C₄×D₈×C₂₃	1	1	yes	characteristic
175	C₄	C₂₃×#467	1	1	yes	characteristic
176	C₄	C₂₃×#500	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	C₂₃×#1942	1	1	yes	characteristic
180	C₂	C₂₃×#2223	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	#466
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂³×C₂₃	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
12. C₂₃×#368	C₄	semidirect
60. #13172	C₂₃	direct (N × H)
98. #368	C₄×C₂₃	semidirect
128. C₂₃	#13172	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×#368	2
C₂×C₂₃×#367	2
C₂₃×#2108	2
C₂₃×#1952	2
C₂₃×#1387	2
#13172	23

Subgroups — C23×#13172 #19160798

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13172 #19160798