Subgroups — C₂₃×#33499 #19181125

Order 5888.

452 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2960	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2183	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2183	C₂	1	4	yes	maximal, complemented
4	Q₈×C₁₆×C₂₃	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2164	C₂	1	4	yes	maximal
6	C₂₃×#2163	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#2162	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#533	C₄	1	1	yes	characteristic
9	C₂₃×#446	C₄	1	2	yes	—
10	C₂₃×#446	C₄	1	2	yes	—
11	C₈×Q₈×C₂₃	C₂²	1	2	yes	—
12	C₈×Q₈×C₂₃	C₂²	1	2	yes	—
13	C₂₃×#424	C₄	1	2	yes	—
14	C₂₃×#423	C₄	1	1	yes	characteristic
15	C₂₃×#405	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
18	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
19	C₂₃×C₄⋊C₁₆	C₂²	1	4	yes	—
20	C₂₃×C₄⋊C₁₆	C₂²	1	4	yes	—
21	C₂₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
22	C₂₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
23	C₂₃×#346	C₂²	1	2	yes	—
24	C₂₃×#346	C₂²	1	4	yes	—
25	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
26	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
27	C₄×C₁₆×C₂₃	C₂²	1	4	yes	—
28	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
31	C₂₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
32	C₂₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
33	C₂₃×#125	C₈	1	2	yes	—
34	C₂₃×C₂²⋊Q₈	C₈	1	2	yes	—
35	C₂₃×C₂²⋊Q₈	C₈	1	2	yes	—
36	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
37	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
38	C₂₃×#117	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
41	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
42	C₂×C₁₆×C₂₃	C₂³	1	4	yes	—
43	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
44	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
45	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
46	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
47	C₂₃×#97	C₂×C₄	1	4	yes	—
48	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₂₃	C₂³	1	4	yes	—
52	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₂×Q₈×C₂₃	C₂×C₈	1	2	yes	—
54	C₂×Q₈×C₂₃	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
58	M₁₆×C₂₃	—	4	2	no	—
59	C₂×C₈×C₂₃	—	2	2	no	—
60	C₂×C₈×C₂₃	—	4	1	no	—
61	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
66	C₄⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
67	C₄⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
68	C₄⋊C₄×C₂₃	C₂×C₈	1	4	yes	—
69	C₄⋊C₄×C₂₃	C₂×C₈	1	4	yes	—
70	C₄⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
71	C₄⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
72	C₂²⋊C₄×C₂₃	C₂×C₈	1	4	yes	—
73	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
74	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
75	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
78	C₁₆×C₂₃	—	2	2	no	—
79	C₁₆×C₂₃	—	2	2	no	—
80	C₁₆×C₂₃	—	2	4	no	—
81	#33499	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₂₃	C₂²×C₈	1	1	yes	characteristic
83	Q₈×C₂₃	—	2	4	no	—
84	Q₈×C₂₃	—	4	2	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	1	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	4	1	no	—
89	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂²×C₈	1	4	yes	—
94	C₂×C₄×C₂₃	C₂²×C₈	1	4	yes	—
95	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
97	C₈×C₂₃	—	2	1	no	—
98	C₈×C₂₃	—	2	1	no	—
99	C₈×C₂₃	—	2	1	no	—
100	C₈×C₂₃	—	2	4	no	—
101	C₈×C₂₃	2₋¹⁺⁴	1	2	yes	—
102	#2960	C₂×C₂₃	1	1	yes	characteristic
103	#2183	C₂×C₂₃	1	2	yes	complemented
104	#2183	C₂×C₂₃	1	4	yes	complemented
105	Q₈×C₁₆	C₂×C₂₃	1	2	yes	complemented
106	#2164	C₂×C₂₃	1	4	yes	—
107	#2163	C₂×C₂₃	1	1	yes	characteristic
108	#2162	C₂×C₂₃	1	1	yes	characteristic
109	C₂²×C₂₃	—	2	1	no	—
110	C₂²×C₂₃	—	2	1	no	—
111	C₂²×C₂₃	—	2	1	no	—
112	C₂²×C₂₃	C₂³×C₈	1	1	yes	characteristic
113	C₄×C₂₃	—	2	1	no	—
114	C₄×C₂₃	—	2	4	no	—
115	C₄×C₂₃	—	2	1	no	—
116	C₄×C₂₃	—	2	1	no	—
117	C₄×C₂₃	—	2	4	no	—
118	C₄×C₂₃	—	2	1	no	—
119	C₄×C₂₃	#504	1	2	yes	—
120	C₄×C₂₃	#519	1	1	yes	characteristic
121	C₄×C₂₃	#568	1	1	yes	characteristic
122	#533	C₄×C₂₃	1	1	yes	characteristic
123	#446	C₄×C₂₃	1	2	yes	—
124	#446	C₄×C₂₃	1	2	yes	—
125	C₈×Q₈	C₂²×C₂₃	1	2	yes	—
126	C₈×Q₈	C₂²×C₂₃	1	2	yes	—
127	#424	C₄×C₂₃	1	2	yes	—
128	#423	C₄×C₂₃	1	1	yes	characteristic
129	#405	C₂²×C₂₃	1	2	yes	—
130	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
131	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	—
132	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	—
133	C₄⋊C₁₆	C₂²×C₂₃	1	4	yes	—
134	C₄⋊C₁₆	C₂²×C₂₃	1	4	yes	—
135	C₂²⋊C₁₆	C₂²×C₂₃	1	2	yes	—
136	C₂²⋊C₁₆	C₂²×C₂₃	1	2	yes	—
137	#346	C₂²×C₂₃	1	2	yes	—
138	#346	C₂²×C₂₃	1	4	yes	—
139	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
140	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
141	C₄×C₁₆	C₂²×C₂₃	1	4	yes	—
142	C₂×C₂₃	—	4	1	no	—
143	C₂×C₂₃	#2982	1	1	yes	characteristic
144	C₂×C₂₃	C₂×#504	1	1	yes	characteristic
145	C₂×C₂₃	#3407	1	1	yes	characteristic
146	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
147	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
148	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
149	C₄⋊Q₈	C₈×C₂₃	1	1	yes	characteristic
150	C₄⋊Q₈	C₈×C₂₃	1	1	yes	characteristic
151	#125	C₈×C₂₃	1	2	yes	—
152	C₂²⋊Q₈	C₈×C₂₃	1	2	yes	—
153	C₂²⋊Q₈	C₈×C₂₃	1	2	yes	—
154	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
155	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
156	#117	C₂×C₄×C₂₃	1	2	yes	—
157	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
158	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
159	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
160	C₂×C₁₆	C₂³×C₂₃	1	4	yes	—
161	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
162	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
163	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
164	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
165	#97	C₂×C₄×C₂₃	1	4	yes	—
166	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
167	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
168	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
169	C₄×C₈	C₂³×C₂₃	1	4	yes	—
170	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
171	C₂₃	#33499	1	1	yes	characteristic, direct product factor, Sylow
172	C₂×Q₈	C₂×C₈×C₂₃	1	2	yes	—
173	C₂×Q₈	C₂×C₈×C₂₃	1	2	yes	—
174	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
176	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
177	M₁₆	—	4	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
181	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
182	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
183	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
184	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
185	C₄⋊C₄	C₂×C₈×C₂₃	1	2	yes	—
186	C₄⋊C₄	C₂×C₈×C₂₃	1	2	yes	—
187	C₄⋊C₄	C₂×C₈×C₂₃	1	4	yes	—
188	C₄⋊C₄	C₂×C₈×C₂₃	1	4	yes	—
189	C₄⋊C₄	C₂×C₈×C₂₃	1	2	yes	—
190	C₄⋊C₄	C₂×C₈×C₂₃	1	2	yes	—
191	C₂²⋊C₄	C₂×C₈×C₂₃	1	4	yes	—
192	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
193	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
194	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
195	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
196	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
197	C₁₆	—	2	2	no	—
198	C₁₆	—	2	2	no	—
199	C₁₆	—	2	4	no	—
200	C₂³	C₂²×C₈×C₂₃	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	Q₈	—	4	2	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
208	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
209	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
210	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
211	C₂×C₄	C₂²×C₈×C₂₃	1	4	yes	—
212	C₂×C₄	C₂²×C₈×C₂₃	1	4	yes	—
213	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
214	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	4	no	—
218	C₈	—	2	1	no	—
219	C₈	C₂₃×2₋¹⁺⁴	1	2	yes	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	C₂³×C₈×C₂₃	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	2	4	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	4	no	—
230	C₄	C₂₃×#504	1	2	yes	—
231	C₄	C₂₃×#519	1	1	yes	characteristic
232	C₄	C₂₃×#568	1	1	yes	characteristic
233	C₂	—	4	1	no	—
234	C₂	C₂₃×#2982	1	1	yes	characteristic
235	C₂	C₂×C₂₃×#504	1	1	yes	characteristic
236	C₂	C₂₃×#3407	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂⁴
commutator	C₂²	C₂³×C₈×C₂₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂³×C₈
prime radical	C₂³×C₂₃	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2183	C₂	semidirect
3. C₂₃×#2183	C₂	semidirect
4. Q₈×C₁₆×C₂₃	C₂	semidirect
81. #33499	C₂₃	direct (N × H)
103. #2183	C₂×C₂₃	semidirect
104. #2183	C₂×C₂₃	semidirect
105. Q₈×C₁₆	C₂×C₂₃	semidirect
171. C₂₃	#33499	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×#2960	2
C₂₃×#2183	2
Q₈×C₁₆×C₂₃	2
C₂₃×#2164	2
C₂₃×#2163	2
C₂₃×#2162	2
#33499	23

Subgroups — C23×#33499 #19181125

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33499 #19181125