Subgroups — C₂₃×#9598 #19157224

Order 5888.

378 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1939	C₂	1	2	yes	maximal
2	C₄×C₂₃×#107	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#322	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1279	C₂	1	2	yes	maximal
6	C₂×C₂₃×#107	C₄	1	2	yes	—
7	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
8	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
13	C₂₃×#340	C₄	1	1	yes	characteristic
14	C₂₃×#340	C₄	1	1	yes	characteristic
15	C₂₃×#336	—	2	2	no	—
16	C₂₃×#335	—	2	2	no	—
17	C₂₃×#322	—	2	2	no	—
18	C₂²×C₈×C₂₃	—	2	1	no	—
19	C₂²×C₈×C₂₃	—	2	2	no	—
20	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₂₃	—	2	2	no	—
27	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
28	C₂₃×#107	—	2	4	no	—
29	C₂₃×C₄⋊C₈	—	2	2	no	—
30	C₂₃×C₄⋊C₈	—	2	2	no	—
31	C₂₃×C₄⋊C₈	—	4	4	no	—
32	C₄×C₈×C₂₃	—	2	1	no	—
33	C₄×C₈×C₂₃	—	2	1	no	—
34	C₄×C₈×C₂₃	—	2	1	no	—
35	C₄×C₈×C₂₃	D₄	1	2	yes	—
36	C₄×C₈×C₂₃	Q₈	1	2	yes	—
37	C₄×C₈×C₂₃	Q₈	1	2	yes	—
38	C₂₃×#95	—	2	2	no	—
39	C₂²×C₄×C₂₃	—	2	2	no	—
40	C₂²×C₄×C₂₃	—	2	2	no	—
41	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
44	C₂×C₈×C₂₃	—	2	1	no	—
45	C₂×C₈×C₂₃	—	2	1	no	—
46	C₂×C₈×C₂₃	—	2	2	no	—
47	C₂×C₈×C₂₃	—	2	2	no	—
48	C₂×C₈×C₂₃	—	2	1	no	—
49	C₂×C₈×C₂₃	—	2	1	no	—
50	C₂×C₈×C₂₃	—	2	2	no	—
51	C₂×C₈×C₂₃	—	4	2	no	—
52	C₂×C₈×C₂₃	—	4	2	no	—
53	C₂×C₈×C₂₃	—	4	2	no	—
54	C₂×C₈×C₂₃	—	4	2	no	—
55	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₂₃	D₈	1	2	yes	—
58	C₂×C₈×C₂₃	Q₁₆	1	2	yes	—
59	C₄⋊C₄×C₂₃	—	2	2	no	—
60	C₄⋊C₄×C₂₃	—	2	2	no	—
61	C₄⋊C₄×C₂₃	—	2	2	no	—
62	C₄⋊C₄×C₂₃	M₁₆	1	4	yes	—
63	C₄²×C₂₃	—	4	2	no	—
64	C₄²×C₂₃	—	4	2	no	—
65	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₂₃	#41	1	1	yes	characteristic
69	#9598	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	4	no	—
73	C₂×C₄×C₂₃	—	4	2	no	—
74	C₂×C₄×C₂₃	—	4	2	no	—
75	C₂×C₄×C₂₃	—	4	2	no	—
76	C₂×C₄×C₂₃	—	4	2	no	—
77	C₂×C₄×C₂₃	—	4	2	no	—
78	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
81	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₂₃	#125	1	1	yes	characteristic
83	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
85	C₂×C₄×C₂₃	#131	1	2	yes	—
86	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
89	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
90	C₈×C₂₃	—	2	2	no	—
91	C₈×C₂₃	—	2	2	no	—
92	C₈×C₂₃	—	2	2	no	—
93	C₈×C₂₃	—	2	2	no	—
94	C₈×C₂₃	—	4	2	no	—
95	C₈×C₂₃	—	8	2	no	—
96	#1939	C₂×C₂₃	1	2	yes	—
97	C₄×#107	C₂×C₂₃	1	1	yes	characteristic
98	C₂×#335	C₂×C₂₃	1	1	yes	characteristic
99	C₂×#322	C₂×C₂₃	1	1	yes	characteristic
100	#1279	C₂×C₂₃	1	2	yes	—
101	C₂²×C₂₃	#389	1	1	yes	characteristic
102	C₂²×C₂₃	#435	1	2	yes	—
103	C₂²×C₂₃	C₄×D₈	1	1	yes	characteristic
104	C₂²×C₂₃	C₄×Q₁₆	1	1	yes	characteristic
105	C₂²×C₂₃	#444	1	2	yes	—
106	C₄×C₂₃	—	2	1	no	—
107	C₄×C₂₃	—	2	1	no	—
108	C₄×C₂₃	—	4	4	no	—
109	C₄×C₂₃	—	8	2	no	—
110	C₄×C₂₃	#423	1	1	yes	characteristic
111	C₄×C₂₃	#424	1	1	yes	characteristic
112	C₄×C₂₃	#446	1	2	yes	—
113	C₄×C₂₃	#466	1	1	yes	characteristic
114	C₄×C₂₃	#467	1	1	yes	characteristic
115	C₄×C₂₃	#499	1	1	yes	characteristic
116	C₄×C₂₃	#500	1	1	yes	characteristic
117	C₂×#107	C₄×C₂₃	1	2	yes	—
118	C₂×C₄⋊C₈	—	2	2	no	—
119	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
120	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
123	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
124	#340	C₄×C₂₃	1	1	yes	characteristic
125	#340	C₄×C₂₃	1	1	yes	characteristic
126	#336	—	2	2	no	—
127	#335	—	2	2	no	—
128	#322	—	2	2	no	—
129	C₂×C₂₃	#1581	1	2	yes	—
130	C₂×C₂₃	#1584	1	2	yes	—
131	C₂×C₂₃	#1940	1	1	yes	characteristic
132	C₂×C₂₃	#1942	1	1	yes	characteristic
133	C₂×C₂₃	#1946	1	1	yes	characteristic
134	C₂²×C₈	—	2	1	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
137	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
138	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
139	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
140	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
141	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
142	C₂×C₄²	—	2	2	no	—
143	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
144	#107	—	2	4	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	4	4	no	—
148	C₄×C₈	—	2	1	no	—
149	C₄×C₈	—	2	1	no	—
150	C₄×C₈	—	2	1	no	—
151	C₄×C₈	D₄×C₂₃	1	2	yes	—
152	C₄×C₈	Q₈×C₂₃	1	2	yes	—
153	C₄×C₈	Q₈×C₂₃	1	2	yes	—
154	#95	—	2	2	no	—
155	C₂₃	#9598	1	1	yes	characteristic, direct product factor, Sylow
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
159	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
160	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
173	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
174	C₂×C₈	D₈×C₂₃	1	2	yes	—
175	C₂×C₈	Q₁₆×C₂₃	1	2	yes	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	M₁₆×C₂₃	1	4	yes	—
180	C₄²	—	4	2	no	—
181	C₄²	—	4	2	no	—
182	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
183	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
184	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
185	C₄²	#41×C₂₃	1	1	yes	characteristic
186	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
195	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
196	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
197	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
198	C₂×C₄	C₂₃×#125	1	1	yes	characteristic
199	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
200	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
201	C₂×C₄	C₂₃×#131	1	2	yes	—
202	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
203	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
204	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
205	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	2	2	no	—
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₈	—	4	2	no	—
211	C₈	—	8	2	no	—
212	C₂²	C₂₃×#389	1	1	yes	characteristic
213	C₂²	C₂₃×#435	1	2	yes	—
214	C₂²	C₄×D₈×C₂₃	1	1	yes	characteristic
215	C₂²	C₄×Q₁₆×C₂₃	1	1	yes	characteristic
216	C₂²	C₂₃×#444	1	2	yes	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	4	4	no	—
220	C₄	—	8	2	no	—
221	C₄	C₂₃×#423	1	1	yes	characteristic
222	C₄	C₂₃×#424	1	1	yes	characteristic
223	C₄	C₂₃×#446	1	2	yes	—
224	C₄	C₂₃×#466	1	1	yes	characteristic
225	C₄	C₂₃×#467	1	1	yes	characteristic
226	C₄	C₂₃×#499	1	1	yes	characteristic
227	C₄	C₂₃×#500	1	1	yes	characteristic
228	C₂	C₂₃×#1581	1	2	yes	—
229	C₂	C₂₃×#1584	1	2	yes	—
230	C₂	C₂₃×#1940	1	1	yes	characteristic
231	C₂	C₂₃×#1942	1	1	yes	characteristic
232	C₂	C₂₃×#1946	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle, prime radical	C₂³×C₂₃	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
69. #9598	C₂₃	direct (N × H)
155. C₂₃	#9598	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×#1939	2
C₄×C₂₃×#107	2
C₂×C₂₃×#335	2
C₂×C₂₃×#322	2
C₂₃×#1279	2
#9598	23

Subgroups — C23×#9598 #19157224

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9598 #19157224