Subgroups — C₂₃×#13279 #19160905

Order 5888.

366 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2150	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₃×#108	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1387	C₂	1	4	yes	maximal
5	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
6	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
7	C₂₃×#500	—	2	1	no	—
8	C₂₃×#500	—	2	1	no	—
9	C₂₃×#500	C₄	1	2	yes	—
10	C₂₃×#500	C₄	1	2	yes	—
11	C₂×C₂₃×#108	C₂²	1	2	yes	—
12	C₂×C₂₃×#107	—	2	1	no	—
13	C₂×C₂₃×#107	C₂²	1	2	yes	—
14	C₄×M₁₆×C₂₃	—	2	1	no	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×C₄⋊C₁₆	—	2	1	no	—
17	C₂₃×C₄⋊C₁₆	—	2	1	no	—
18	C₂₃×#341	—	2	1	no	—
19	C₂²×Q₈×C₂₃	—	4	1	no	—
20	C₂×M₁₆×C₂₃	—	2	2	no	—
21	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
23	C₂₃×C₄⋊Q₈	—	2	2	no	—
24	C₂₃×C₄⋊Q₈	—	2	2	no	—
25	C₂₃×C₄⋊Q₈	—	4	1	no	—
26	C₂₃×C₄⋊Q₈	—	4	1	no	—
27	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
29	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
30	C₂₃×M₃₂	—	2	4	no	—
31	C₂×C₁₆×C₂₃	—	2	2	no	—
32	C₂₃×#108	—	2	2	no	—
33	C₂₃×#108	Q₈	1	4	yes	complemented
34	C₂₃×#107	—	2	2	no	—
35	C₂₃×#107	—	2	2	no	—
36	C₂₃×#107	—	2	2	no	—
37	C₂₃×#107	C₂×C₄	1	4	yes	—
38	C₂₃×#97	—	4	1	no	—
39	C₄×C₈×C₂₃	—	4	1	no	—
40	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₂₃	—	4	2	no	—
45	C₂×Q₈×C₂₃	—	4	2	no	—
46	C₂×Q₈×C₂₃	—	8	4	no	—
47	C₂²×C₄×C₂₃	—	4	2	no	—
48	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
51	M₁₆×C₂₃	—	2	4	no	—
52	M₁₆×C₂₃	—	4	2	no	—
53	C₂×C₈×C₂₃	—	2	2	no	—
54	C₂×C₈×C₂₃	—	4	2	no	—
55	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₂₃	#41	1	2	yes	—
61	C₄⋊C₄×C₂₃	—	2	4	no	—
62	C₄⋊C₄×C₂₃	—	4	2	no	—
63	C₄⋊C₄×C₂₃	—	4	2	no	—
64	C₄⋊C₄×C₂₃	—	4	2	no	—
65	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₂₃	D₈	1	1	yes	characteristic
68	C₄²×C₂₃	D₈	1	1	yes	characteristic
69	C₁₆×C₂₃	—	4	2	no	—
70	#13279	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
72	Q₈×C₂₃	—	8	4	no	—
73	Q₈×C₂₃	—	8	4	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	4	4	no	—
77	C₂×C₄×C₂₃	—	8	4	no	—
78	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
79	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
80	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
81	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
82	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
84	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₂₃	#137	1	2	yes	—
86	C₈×C₂₃	—	2	1	no	—
87	C₈×C₂₃	—	2	1	no	—
88	C₈×C₂₃	—	4	4	no	—
89	C₈×C₂₃	C₄×Q₈	1	2	yes	—
90	C₈×C₂₃	#124	1	1	yes	characteristic
91	C₈×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
92	C₂×#500	C₂×C₂₃	1	1	yes	characteristic
93	#2150	C₂×C₂₃	1	1	yes	characteristic
94	C₄×#108	C₂×C₂₃	1	1	yes	characteristic
95	#1387	C₂×C₂₃	1	4	yes	—
96	C₂²×C₂₃	—	2	1	no	—
97	C₂²×C₂₃	—	2	1	no	—
98	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
99	C₂²×C₂₃	#475	1	2	yes	—
100	C₄×C₂₃	—	2	1	no	—
101	C₄×C₂₃	—	2	1	no	—
102	C₄×C₂₃	—	8	4	no	—
103	C₄×C₂₃	#361	1	2	yes	—
104	C₄×C₂₃	#362	1	2	yes	—
105	C₄×C₂₃	#391	1	1	yes	characteristic
106	C₄×C₂₃	#419	1	1	yes	characteristic
107	C₄×C₂₃	#474	1	2	yes	—
108	C₂×C₄⋊Q₈	—	2	1	no	—
109	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
110	#500	—	2	1	no	—
111	#500	—	2	1	no	—
112	#500	C₄×C₂₃	1	2	yes	—
113	#500	C₄×C₂₃	1	2	yes	—
114	C₂×#108	C₂²×C₂₃	1	2	yes	—
115	C₂×#107	—	2	1	no	—
116	C₂×#107	C₂²×C₂₃	1	2	yes	—
117	C₄×M₁₆	—	2	1	no	—
118	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
119	C₄⋊C₁₆	—	2	1	no	—
120	C₄⋊C₁₆	—	2	1	no	—
121	#341	—	2	1	no	—
122	C₂×C₂₃	—	2	2	no	—
123	C₂×C₂₃	#1988	1	1	yes	characteristic
124	C₂×C₂₃	C₂×#361	1	1	yes	characteristic
125	C₂×C₂₃	C₂×#362	1	1	yes	characteristic
126	C₂²×Q₈	—	4	1	no	—
127	C₂×M₁₆	—	2	2	no	—
128	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
129	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	4	1	no	—
133	C₄⋊Q₈	—	4	1	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	4	1	no	—
136	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
137	M₃₂	—	2	4	no	—
138	C₂×C₁₆	—	2	2	no	—
139	#108	—	2	2	no	—
140	#108	Q₈×C₂₃	1	4	yes	complemented
141	#107	—	2	2	no	—
142	#107	—	2	2	no	—
143	#107	—	2	2	no	—
144	#107	C₂×C₄×C₂₃	1	4	yes	—
145	#97	—	4	1	no	—
146	C₄×C₈	—	4	1	no	—
147	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
148	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
149	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
150	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
151	C₂₃	#13279	1	1	yes	characteristic, direct product factor, Sylow
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	8	4	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
157	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
158	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
159	M₁₆	—	2	4	no	—
160	M₁₆	—	4	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
164	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
165	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
166	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
167	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
168	C₂×C₈	#41×C₂₃	1	2	yes	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
174	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
175	C₄²	D₈×C₂₃	1	1	yes	characteristic
176	C₄²	D₈×C₂₃	1	1	yes	characteristic
177	C₁₆	—	4	2	no	—
178	C₂³	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
179	Q₈	—	8	4	no	—
180	Q₈	—	8	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	8	4	no	—
185	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
186	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
187	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
188	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
190	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
191	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
192	C₂×C₄	C₂₃×#137	1	2	yes	—
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	4	4	no	—
196	C₈	C₄×Q₈×C₂₃	1	2	yes	—
197	C₈	C₂₃×#124	1	1	yes	characteristic
198	C₈	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
202	C₂²	C₂₃×#475	1	2	yes	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	8	4	no	—
206	C₄	C₂₃×#361	1	2	yes	—
207	C₄	C₂₃×#362	1	2	yes	—
208	C₄	C₂₃×#391	1	1	yes	characteristic
209	C₄	C₂₃×#419	1	1	yes	characteristic
210	C₄	C₂₃×#474	1	2	yes	—
211	C₂	—	2	2	no	—
212	C₂	C₂₃×#1988	1	1	yes	characteristic
213	C₂	C₂×C₂₃×#361	1	1	yes	characteristic
214	C₂	C₂×C₂₃×#362	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂³×C₂₃	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
33. C₂₃×#108	Q₈	semidirect
70. #13279	C₂₃	direct (N × H)
140. #108	Q₈×C₂₃	semidirect
151. C₂₃	#13279	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×#500	2
C₂₃×#2150	2
C₄×C₂₃×#108	2
C₂₃×#1387	2
#13279	23

Subgroups — C23×#13279 #19160905

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13279 #19160905