Subgroups — C₂₃×#13274 #19160900

Order 5888.

338 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3160	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₃×#108	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1382	C₂	1	4	yes	maximal
5	C₂₃×#533	—	2	1	no	—
6	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂₃×#500	C₄	1	1	yes	characteristic
8	C₂₃×#500	C₄	1	1	yes	characteristic
9	C₂₃×#499	—	2	1	no	—
10	C₂₃×#499	C₄	1	2	yes	—
11	C₂₃×#498	—	2	1	no	—
12	C₂×C₂₃×#108	C₂²	1	2	yes	—
13	C₂₃×#427	—	2	1	no	—
14	C₂₃×#427	C₂²	1	2	yes	—
15	C₄×M₁₆×C₂₃	—	2	1	no	—
16	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×C₄⋊C₁₆	—	2	1	no	—
18	C₂₃×C₄⋊C₁₆	—	2	1	no	—
19	C₂₃×#341	—	2	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	1	no	—
24	C₂₃×C₄⋊Q₈	—	2	1	no	—
25	C₂₃×#125	—	2	2	no	—
26	C₂₃×C₂²⋊Q₈	—	4	1	no	—
27	C₂₃×C₂²⋊Q₈	—	4	1	no	—
28	C₄×Q₈×C₂₃	—	4	1	no	—
29	C₄×Q₈×C₂₃	—	4	1	no	—
30	C₂₃×#117	—	2	2	no	—
31	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₂₃	—	2	2	no	—
33	C₂×C₁₆×C₂₃	—	2	4	no	—
34	C₂₃×#108	—	2	2	no	—
35	C₂₃×#108	Q₈	1	4	yes	complemented
36	C₂₃×#107	—	2	1	no	—
37	C₂₃×#107	—	2	1	no	—
38	C₂₃×#107	C₂×C₄	1	2	yes	—
39	C₂₃×#107	C₂×C₄	1	2	yes	—
40	C₂₃×#106	—	2	2	no	—
41	C₂₃×#106	—	2	2	no	—
42	C₂₃×#97	—	4	1	no	—
43	C₄×C₈×C₂₃	—	4	1	no	—
44	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
48	C₂×Q₈×C₂₃	—	4	1	no	—
49	C₂×Q₈×C₂₃	—	4	1	no	—
50	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
53	M₁₆×C₂₃	—	2	4	no	—
54	M₁₆×C₂₃	—	4	2	no	—
55	C₂×C₈×C₂₃	—	2	2	no	—
56	C₂×C₈×C₂₃	—	4	2	no	—
57	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
62	C₂×C₈×C₂₃	#41	1	2	yes	—
63	C₄⋊C₄×C₂₃	—	2	2	no	—
64	C₄⋊C₄×C₂₃	—	2	2	no	—
65	C₄⋊C₄×C₂₃	—	4	2	no	—
66	C₄⋊C₄×C₂₃	—	4	2	no	—
67	C₄⋊C₄×C₂₃	—	4	1	no	—
68	C₄⋊C₄×C₂₃	—	4	1	no	—
69	C₂²⋊C₄×C₂₃	—	4	2	no	—
70	C₄²×C₂₃	—	4	2	no	—
71	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₂₃	D₈	1	1	yes	characteristic
74	C₄²×C₂₃	D₈	1	1	yes	characteristic
75	C₁₆×C₂₃	—	4	2	no	—
76	#13274	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
78	Q₈×C₂₃	—	8	2	no	—
79	Q₈×C₂₃	—	8	2	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	4	1	no	—
82	C₂×C₄×C₂₃	—	4	2	no	—
83	C₂×C₄×C₂₃	—	4	2	no	—
84	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
85	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
86	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
87	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
88	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
90	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₂₃	#137	1	2	yes	—
92	C₈×C₂₃	—	2	1	no	—
93	C₈×C₂₃	—	2	1	no	—
94	C₈×C₂₃	—	4	4	no	—
95	C₈×C₂₃	C₄×Q₈	1	2	yes	—
96	C₈×C₂₃	#124	1	1	yes	characteristic
97	C₈×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
98	#3160	C₂×C₂₃	1	1	yes	characteristic
99	C₂×C₄⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic
100	C₄×#108	C₂×C₂₃	1	1	yes	characteristic
101	#1382	C₂×C₂₃	1	4	yes	—
102	C₂²×C₂₃	—	2	1	no	—
103	C₂²×C₂₃	—	2	1	no	—
104	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
105	C₂²×C₂₃	#475	1	2	yes	—
106	C₄×C₂₃	—	2	1	no	—
107	C₄×C₂₃	—	2	1	no	—
108	C₄×C₂₃	—	2	1	no	—
109	C₄×C₂₃	—	8	2	no	—
110	C₄×C₂₃	—	8	2	no	—
111	C₄×C₂₃	#359	1	2	yes	—
112	C₄×C₂₃	#391	1	1	yes	characteristic
113	C₄×C₂₃	#419	1	1	yes	characteristic
114	C₄×C₂₃	#474	1	2	yes	—
115	#533	—	2	1	no	—
116	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
117	#500	C₄×C₂₃	1	1	yes	characteristic
118	#500	C₄×C₂₃	1	1	yes	characteristic
119	#499	—	2	1	no	—
120	#499	C₄×C₂₃	1	2	yes	—
121	#498	—	2	1	no	—
122	C₂×#108	C₂²×C₂₃	1	2	yes	—
123	#427	—	2	1	no	—
124	#427	C₂²×C₂₃	1	2	yes	—
125	C₄×M₁₆	—	2	1	no	—
126	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
127	C₄⋊C₁₆	—	2	1	no	—
128	C₄⋊C₁₆	—	2	1	no	—
129	#341	—	2	1	no	—
130	C₂×C₂₃	—	2	2	no	—
131	C₂×C₂₃	#1988	1	1	yes	characteristic
132	C₂×C₂₃	C₂×#359	1	1	yes	characteristic
133	C₂×C₂₃	#2143	1	1	yes	characteristic
134	C₂×M₁₆	—	2	2	no	—
135	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
136	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
137	C₄⋊Q₈	—	2	1	no	—
138	C₄⋊Q₈	—	2	1	no	—
139	#125	—	2	2	no	—
140	C₂²⋊Q₈	—	4	1	no	—
141	C₂²⋊Q₈	—	4	1	no	—
142	C₄×Q₈	—	4	1	no	—
143	C₄×Q₈	—	4	1	no	—
144	#117	—	2	2	no	—
145	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
146	C₂×C₁₆	—	2	2	no	—
147	C₂×C₁₆	—	2	4	no	—
148	#108	—	2	2	no	—
149	#108	Q₈×C₂₃	1	4	yes	complemented
150	#107	—	2	1	no	—
151	#107	—	2	1	no	—
152	#107	C₂×C₄×C₂₃	1	2	yes	—
153	#107	C₂×C₄×C₂₃	1	2	yes	—
154	#106	—	2	2	no	—
155	#106	—	2	2	no	—
156	#97	—	4	1	no	—
157	C₄×C₈	—	4	1	no	—
158	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
159	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
160	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
161	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
162	C₂₃	#13274	1	1	yes	characteristic, direct product factor, Sylow
163	C₂×Q₈	—	4	1	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
166	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
168	M₁₆	—	2	4	no	—
169	M₁₆	—	4	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
174	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
175	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
176	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
177	C₂×C₈	#41×C₂₃	1	2	yes	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₄⋊C₄	—	4	1	no	—
184	C₂²⋊C₄	—	4	2	no	—
185	C₄²	—	4	2	no	—
186	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
187	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
188	C₄²	D₈×C₂₃	1	1	yes	characteristic
189	C₄²	D₈×C₂₃	1	1	yes	characteristic
190	C₁₆	—	4	2	no	—
191	C₂³	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
192	Q₈	—	8	2	no	—
193	Q₈	—	8	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
199	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
200	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
201	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
202	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
203	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
204	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
205	C₂×C₄	C₂₃×#137	1	2	yes	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	4	4	no	—
209	C₈	C₄×Q₈×C₂₃	1	2	yes	—
210	C₈	C₂₃×#124	1	1	yes	characteristic
211	C₈	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
215	C₂²	C₂₃×#475	1	2	yes	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	8	2	no	—
220	C₄	—	8	2	no	—
221	C₄	C₂₃×#359	1	2	yes	—
222	C₄	C₂₃×#391	1	1	yes	characteristic
223	C₄	C₂₃×#419	1	1	yes	characteristic
224	C₄	C₂₃×#474	1	2	yes	—
225	C₂	—	2	2	no	—
226	C₂	C₂₃×#1988	1	1	yes	characteristic
227	C₂	C₂×C₂₃×#359	1	1	yes	characteristic
228	C₂	C₂₃×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×#102
prime radical	C₂³×C₂₃	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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
35. C₂₃×#108	Q₈	semidirect
76. #13274	C₂₃	direct (N × H)
149. #108	Q₈×C₂₃	semidirect
162. C₂₃	#13274	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3160	2
C₂×C₂₃×C₄⋊C₁₆	2
C₄×C₂₃×#108	2
C₂₃×#1382	2
#13274	23

Subgroups — C23×#13274 #19160900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13274 #19160900