Subgroups — C₂₃×#8928 #19156554

Order 5888.

476 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×#2369	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1956	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1948	C₂	1	2	yes	maximal
4	C₂×C₂₃×#333	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1279	C₂	1	2	yes	maximal
6	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×#106	—	2	2	no	—
8	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₂₃×#103	—	2	2	no	—
10	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#391	—	2	2	no	—
12	C₂₃×#389	—	2	2	no	—
13	C₂₃×#384	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
15	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
16	C₂₃×#340	—	2	2	no	—
17	C₂₃×#333	C₄	1	4	yes	complemented
18	C₂²×Q₈×C₂₃	D₄	1	2	yes	—
19	C₂²×C₈×C₂₃	—	2	2	no	—
20	C₂²×C₈×C₂₃	—	2	1	no	—
21	C₂₃×C₄⋊Q₈	—	2	2	no	—
22	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
23	C₂₃×C₄⋊Q₈	Q₈	1	2	yes	—
24	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
26	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
27	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
29	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
30	C₂×C₄²×C₂₃	—	2	2	no	—
31	C₂×C₄²×C₂₃	—	2	2	no	—
32	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
33	C₂₃×#106	—	4	4	no	—
34	C₂₃×C₄⋊C₈	—	2	2	no	—
35	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
36	C₂₃×#103	—	4	4	no	—
37	C₄×C₈×C₂₃	—	2	2	no	—
38	C₂₃×#95	—	2	4	no	—
39	C₂₃×#95	—	2	2	no	—
40	C₂×Q₈×C₂₃	—	2	2	no	—
41	C₂×Q₈×C₂₃	—	4	2	no	—
42	C₂×Q₈×C₂₃	—	4	2	no	—
43	C₂×Q₈×C₂₃	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₂₃	—	2	2	no	—
45	C₂²×C₄×C₂₃	—	2	2	no	—
46	C₂²×C₄×C₂₃	—	2	2	no	—
47	C₂²×C₄×C₂₃	—	4	1	no	—
48	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
50	C₂×C₈×C₂₃	—	2	2	no	—
51	C₂×C₈×C₂₃	—	2	4	no	—
52	C₂×C₈×C₂₃	—	4	1	no	—
53	C₂×C₈×C₂₃	—	4	1	no	—
54	C₂×C₈×C₂₃	—	4	2	no	—
55	C₂×C₈×C₂₃	—	4	2	no	—
56	C₄⋊C₄×C₂₃	—	2	4	no	—
57	C₄⋊C₄×C₂₃	—	2	4	no	—
58	C₄⋊C₄×C₂₃	—	2	2	no	—
59	C₄⋊C₄×C₂₃	—	2	2	no	—
60	C₄⋊C₄×C₂₃	—	2	2	no	—
61	C₄⋊C₄×C₂₃	—	4	2	no	—
62	C₄⋊C₄×C₂₃	—	4	2	no	—
63	C₄⋊C₄×C₂₃	—	4	2	no	—
64	C₄⋊C₄×C₂₃	—	8	2	no	—
65	C₄⋊C₄×C₂₃	SD₁₆	1	4	yes	—
66	C₄²×C₂₃	—	4	2	no	—
67	C₄²×C₂₃	—	4	2	no	—
68	C₄²×C₂₃	—	8	2	no	—
69	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₂₃	#41	1	2	yes	—
72	#8928	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₂₃	#120	1	1	yes	characteristic
74	Q₈×C₂₃	—	4	4	no	—
75	Q₈×C₂₃	—	8	2	no	—
76	C₂×C₄×C₂₃	—	2	1	no	—
77	C₂×C₄×C₂₃	—	2	1	no	—
78	C₂×C₄×C₂₃	—	2	4	no	—
79	C₂×C₄×C₂₃	—	2	4	no	—
80	C₂×C₄×C₂₃	—	4	2	no	—
81	C₂×C₄×C₂₃	—	4	2	no	—
82	C₂×C₄×C₂₃	—	4	2	no	—
83	C₂×C₄×C₂₃	—	4	2	no	—
84	C₂×C₄×C₂₃	—	4	2	no	—
85	C₂×C₄×C₂₃	—	4	2	no	—
86	C₂×C₄×C₂₃	—	4	2	no	—
87	C₂×C₄×C₂₃	—	8	2	no	—
88	C₂×C₄×C₂₃	—	8	1	no	—
89	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
92	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
94	C₂×C₄×C₂₃	#135	1	2	yes	—
95	C₂×C₄×C₂₃	#137	1	2	yes	—
96	C₂×C₄×C₂₃	#137	1	2	yes	—
97	C₈×C₂₃	—	4	2	no	—
98	C₈×C₂₃	—	4	4	no	—
99	#2369	C₂×C₂₃	1	1	yes	characteristic
100	#1956	C₂×C₂₃	1	1	yes	characteristic
101	#1948	C₂×C₂₃	1	2	yes	—
102	C₂×#333	C₂×C₂₃	1	1	yes	characteristic
103	#1279	C₂×C₂₃	1	2	yes	—
104	C₂²×C₂₃	#385	1	1	yes	characteristic
105	C₂²×C₂₃	#448	1	2	yes	—
106	C₂²×C₂₃	#452	1	2	yes	—
107	C₂²×C₂₃	#453	1	1	yes	characteristic
108	C₂²×C₂₃	#455	1	1	yes	characteristic
109	C₄×C₂₃	—	2	1	no	—
110	C₄×C₂₃	—	2	1	no	—
111	C₄×C₂₃	—	4	4	no	—
112	C₄×C₂₃	—	4	4	no	—
113	C₄×C₂₃	—	8	2	no	—
114	C₄×C₂₃	—	16	1	no	—
115	C₄×C₂₃	C₄×SD₁₆	1	2	yes	—
116	C₄×C₂₃	#441	1	2	yes	—
117	C₄×C₂₃	#475	1	2	yes	—
118	C₄×C₂₃	#479	1	2	yes	—
119	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
120	C₂×#106	—	2	2	no	—
121	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
122	C₂×#103	—	2	2	no	—
123	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
124	#391	—	2	2	no	—
125	#389	—	2	2	no	—
126	#384	C₂²×C₂₃	1	1	yes	characteristic
127	C₄×C₄⋊C₄	—	2	2	no	—
128	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
129	#340	—	2	2	no	—
130	#333	C₄×C₂₃	1	4	yes	complemented
131	C₂×C₂₃	#1623	1	1	yes	characteristic
132	C₂×C₂₃	#1634	1	1	yes	characteristic
133	C₂×C₂₃	#1643	1	2	yes	—
134	C₂×C₂₃	#1856	1	1	yes	characteristic
135	C₂×C₂₃	#1865	1	2	yes	—
136	C₂²×Q₈	D₄×C₂₃	1	2	yes	—
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	—	2	1	no	—
139	C₄⋊Q₈	—	2	2	no	—
140	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
141	C₄⋊Q₈	Q₈×C₂₃	1	2	yes	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	—	2	2	no	—
144	C₂×C₄⋊C₄	—	2	1	no	—
145	C₂×C₄⋊C₄	—	2	2	no	—
146	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
147	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
148	C₂×C₄²	—	2	2	no	—
149	C₂×C₄²	—	2	2	no	—
150	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
151	#106	—	4	4	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
154	#103	—	4	4	no	—
155	C₄×C₈	—	2	2	no	—
156	#95	—	2	4	no	—
157	#95	—	2	2	no	—
158	C₂₃	#8928	1	1	yes	characteristic, direct product factor, Sylow
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂×Q₈	C₄⋊C₄×C₂₃	1	4	yes	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	4	1	no	—
167	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
168	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	4	1	no	—
172	C₂×C₈	—	4	1	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	4	2	no	—
175	C₄⋊C₄	—	2	4	no	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	2	no	—
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	2	no	—
183	C₄⋊C₄	—	8	2	no	—
184	C₄⋊C₄	SD₁₆×C₂₃	1	4	yes	—
185	C₄²	—	4	2	no	—
186	C₄²	—	4	2	no	—
187	C₄²	—	8	2	no	—
188	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
189	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
190	C₄²	#41×C₂₃	1	2	yes	—
191	C₂³	C₂₃×#120	1	1	yes	characteristic
192	Q₈	—	4	4	no	—
193	Q₈	—	8	2	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	8	2	no	—
206	C₂×C₄	—	8	1	no	—
207	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
208	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
209	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
210	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
211	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
212	C₂×C₄	C₂₃×#135	1	2	yes	—
213	C₂×C₄	C₂₃×#137	1	2	yes	—
214	C₂×C₄	C₂₃×#137	1	2	yes	—
215	C₈	—	4	2	no	—
216	C₈	—	4	4	no	—
217	C₂²	C₂₃×#385	1	1	yes	characteristic
218	C₂²	C₂₃×#448	1	2	yes	—
219	C₂²	C₂₃×#452	1	2	yes	—
220	C₂²	C₂₃×#453	1	1	yes	characteristic
221	C₂²	C₂₃×#455	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	4	4	no	—
225	C₄	—	4	4	no	—
226	C₄	—	8	2	no	—
227	C₄	—	16	1	no	—
228	C₄	C₄×SD₁₆×C₂₃	1	2	yes	—
229	C₄	C₂₃×#441	1	2	yes	—
230	C₄	C₂₃×#475	1	2	yes	—
231	C₄	C₂₃×#479	1	2	yes	—
232	C₂	C₂₃×#1623	1	1	yes	characteristic
233	C₂	C₂₃×#1634	1	1	yes	characteristic
234	C₂	C₂₃×#1643	1	2	yes	—
235	C₂	C₂₃×#1856	1	1	yes	characteristic
236	C₂	C₂₃×#1865	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	#120
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
17. C₂₃×#333	C₄	semidirect
72. #8928	C₂₃	direct (N × H)
130. #333	C₄×C₂₃	semidirect
158. C₂₃	#8928	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×#2369	2
C₂₃×#1956	2
C₂₃×#1948	2
C₂×C₂₃×#333	2
C₂₃×#1279	2
#8928	23

Subgroups — C23×#8928 #19156554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8928 #19156554