Subgroups — C₂₃×#11644 #19159270

Order 5888.

306 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#446	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2163	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2163	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1338	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×Q₈×C₂₃	C₄	1	1	yes	characteristic
6	C₂₃×#446	—	2	1	no	—
7	C₂₃×#446	—	2	1	no	—
8	C₂₃×#446	C₄	1	2	yes	—
9	C₂₃×#446	C₂²	1	2	yes	—
10	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
11	C₂×C₂₃×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₂₃×#97	—	2	1	no	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×C₄⋊C₁₆	—	2	1	no	—
15	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₂₃×C₂²⋊C₁₆	—	2	1	no	—
17	C₂₃×C₂²⋊C₁₆	—	2	1	no	—
18	C₂₃×#346	—	2	1	no	—
19	C₂₃×#346	C₂²	1	2	yes	—
20	C₂²×Q₈×C₂₃	C₈	1	1	yes	characteristic
21	C₂²×C₈×C₂₃	—	2	1	no	—
22	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
24	C₄×Q₈×C₂₃	—	2	1	no	—
25	C₄×Q₈×C₂₃	—	2	1	no	—
26	C₄×Q₈×C₂₃	C₈	1	2	yes	—
27	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
29	C₂×C₄⋊C₄×C₂₃	C₈	1	1	yes	characteristic
30	C₂×C₄²×C₂₃	—	2	1	no	—
31	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₂₃	—	2	2	no	—
33	C₂×C₁₆×C₂₃	—	2	2	no	—
34	C₂₃×C₄⋊C₈	—	2	2	no	—
35	C₂₃×C₄⋊C₈	—	2	2	no	—
36	C₂₃×C₄⋊C₈	—	2	1	no	—
37	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
38	C₂₃×#97	—	2	2	no	—
39	C₂₃×#97	—	2	2	no	—
40	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₂₃	—	2	1	no	—
45	C₂×Q₈×C₂₃	—	4	2	no	—
46	C₂×Q₈×C₂₃	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₂₃	—	2	1	no	—
48	C₂²×C₄×C₂₃	—	2	1	no	—
49	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
52	C₂×C₈×C₂₃	—	2	2	no	—
53	C₂×C₈×C₂₃	—	2	2	no	—
54	C₂×C₈×C₂₃	—	4	1	no	—
55	C₂×C₈×C₂₃	—	4	2	no	—
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×D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₂₃	—	2	2	no	—
61	C₄⋊C₄×C₂₃	—	2	1	no	—
62	C₄⋊C₄×C₂₃	—	2	2	no	—
63	C₄⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
64	C₄²×C₂₃	—	2	2	no	—
65	C₄²×C₂₃	—	2	2	no	—
66	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
68	C₄²×C₂₃	M₁₆	1	1	yes	characteristic
69	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
70	C₁₆×C₂₃	—	4	2	no	—
71	C₁₆×C₂₃	—	4	2	no	—
72	#11644	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
74	Q₈×C₂₃	—	4	2	no	—
75	Q₈×C₂₃	—	4	2	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	2	1	no	—
78	C₂×C₄×C₂₃	—	2	1	no	—
79	C₂×C₄×C₂₃	—	2	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	C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
90	C₈×C₂₃	—	2	1	no	—
91	C₈×C₂₃	—	2	1	no	—
92	C₈×C₂₃	—	2	1	no	—
93	C₈×C₂₃	—	4	2	no	—
94	C₈×C₂₃	#137	1	2	yes	—
95	C₂×#446	C₂×C₂₃	1	1	yes	characteristic
96	#2163	C₂×C₂₃	1	1	yes	characteristic
97	#2163	C₂×C₂₃	1	1	yes	characteristic
98	#1338	C₂×C₂₃	1	4	yes	complemented
99	C₂²×C₂₃	—	2	1	no	—
100	C₂²×C₂₃	—	2	1	no	—
101	C₂²×C₂₃	#349	1	2	yes	—
102	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
103	C₄×C₂₃	—	2	1	no	—
104	C₄×C₂₃	—	2	1	no	—
105	C₄×C₂₃	—	2	1	no	—
106	C₄×C₂₃	—	2	1	no	—
107	C₄×C₂₃	—	2	1	no	—
108	C₄×C₂₃	—	4	2	no	—
109	C₄×C₂₃	—	4	2	no	—
110	C₄×C₂₃	#419	1	1	yes	characteristic
111	C₄×C₂₃	#421	1	1	yes	characteristic
112	C₂×C₄×Q₈	C₄×C₂₃	1	1	yes	characteristic
113	#446	—	2	1	no	—
114	#446	—	2	1	no	—
115	#446	C₄×C₂₃	1	2	yes	—
116	#446	C₂²×C₂₃	1	2	yes	—
117	C₂×C₄⋊C₈	—	2	1	no	—
118	C₂×C₄⋊C₈	C₄×C₂₃	1	1	yes	characteristic
119	C₂×#97	—	2	1	no	—
120	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
121	C₄⋊C₁₆	—	2	1	no	—
122	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	—
123	C₂²⋊C₁₆	—	2	1	no	—
124	C₂²⋊C₁₆	—	2	1	no	—
125	#346	—	2	1	no	—
126	#346	C₂²×C₂₃	1	2	yes	—
127	C₂×C₂₃	—	2	2	no	—
128	C₂×C₂₃	#1479	1	1	yes	characteristic
129	C₂×C₂₃	C₂×#349	1	1	yes	characteristic
130	C₂×C₂₃	#2117	1	1	yes	characteristic
131	C₂²×Q₈	C₈×C₂₃	1	1	yes	characteristic
132	C₂²×C₈	—	2	1	no	—
133	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
134	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
135	C₄×Q₈	—	2	1	no	—
136	C₄×Q₈	—	2	1	no	—
137	C₄×Q₈	C₈×C₂₃	1	2	yes	—
138	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
139	C₂×C₄⋊C₄	—	2	1	no	—
140	C₂×C₄⋊C₄	C₈×C₂₃	1	1	yes	characteristic
141	C₂×C₄²	—	2	1	no	—
142	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	—	2	2	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	2	1	no	—
148	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
149	#97	—	2	2	no	—
150	#97	—	2	2	no	—
151	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
152	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
153	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
154	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
155	C₂₃	#11644	1	1	yes	characteristic, direct product factor, Sylow
156	C₂×Q₈	—	2	1	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂×Q₈	C₂×C₈×C₂₃	1	2	yes	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
162	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
163	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
169	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
170	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
171	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	C₂×C₈×C₂₃	1	2	yes	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	2	no	—
178	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
179	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
180	C₄²	M₁₆×C₂₃	1	1	yes	characteristic
181	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
182	C₁₆	—	4	2	no	—
183	C₁₆	—	4	2	no	—
184	C₂³	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
185	Q₈	—	4	2	no	—
186	Q₈	—	4	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
196	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
197	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
198	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
199	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
200	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	4	2	no	—
205	C₈	C₂₃×#137	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	C₂₃×#349	1	2	yes	—
209	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	4	2	no	—
216	C₄	—	4	2	no	—
217	C₄	C₂₃×#419	1	1	yes	characteristic
218	C₄	C₂₃×#421	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	C₂₃×#1479	1	1	yes	characteristic
221	C₂	C₂×C₂₃×#349	1	1	yes	characteristic
222	C₂	C₂₃×#2117	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1338	C₂	semidirect
72. #11644	C₂₃	direct (N × H)
98. #1338	C₂×C₂₃	semidirect
155. C₂₃	#11644	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×#446	2
C₂₃×#2163	2
C₂₃×#1338	2
#11644	23

Subgroups — C23×#11644 #19159270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11644 #19159270