Subgroups — C₂₃×#8689 #19156315

Order 5888.

346 conjugacy classes of subgroups.

The subgroup lattice has 248 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×#1837	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1550	C₂	1	2	yes	maximal
5	C₂₃×#1292	C₂	1	2	yes	maximal
6	C₂₃×#533	C₂²	1	1	yes	characteristic
7	C₂₃×#446	—	2	1	no	—
8	C₂₃×#446	C₄	1	2	yes	—
9	C₈×Q₈×C₂₃	—	2	1	no	—
10	C₈×Q₈×C₂₃	C₄	1	2	yes	—
11	C₂₃×#424	—	2	1	no	—
12	C₂₃×#423	C₂²	1	2	yes	—
13	C₂₃×#423	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂₃×#405	—	2	1	no	—
16	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#341	—	2	1	no	—
18	C₂₃×#341	—	2	1	no	—
19	C₂₃×#335	—	2	1	no	—
20	C₂₃×#335	—	2	1	no	—
21	C₂₃×#333	—	2	2	no	—
22	C₂₃×#332	—	2	2	no	—
23	C₂×M₁₆×C₂₃	—	2	1	no	—
24	C₂×M₁₆×C₂₃	—	2	2	no	—
25	C₂²×C₈×C₂₃	—	2	2	no	—
26	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
28	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
29	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
30	C₂₃×#125	—	2	1	no	—
31	C₂₃×C₂²⋊Q₈	—	2	1	no	—
32	C₂₃×C₂²⋊Q₈	Q₈	1	2	yes	—
33	C₄×Q₈×C₂₃	—	2	1	no	—
34	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
35	C₂₃×#117	—	2	1	no	—
36	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
37	C₂₃×C₄⋊C₈	—	2	2	no	—
38	C₂₃×C₄⋊C₈	—	2	2	no	—
39	C₂₃×C₄⋊C₈	—	2	1	no	—
40	C₂₃×C₄⋊C₈	—	2	2	no	—
41	C₂₃×C₄⋊C₈	—	2	2	no	—
42	C₂₃×C₄⋊C₈	—	2	2	no	—
43	C₂₃×C₄⋊C₈	—	2	2	no	—
44	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
45	C₂₃×#97	—	2	2	no	—
46	C₄×C₈×C₂₃	—	2	2	no	—
47	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
51	C₂×Q₈×C₂₃	—	2	1	no	—
52	C₂×Q₈×C₂₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
56	M₁₆×C₂₃	—	4	2	no	—
57	M₁₆×C₂₃	—	4	2	no	—
58	M₁₆×C₂₃	—	4	2	no	—
59	C₂×C₈×C₂₃	—	2	2	no	—
60	C₂×C₈×C₂₃	—	2	2	no	—
61	C₂×C₈×C₂₃	—	2	2	no	—
62	C₂×C₈×C₂₃	—	4	2	no	—
63	C₂×C₈×C₂₃	—	4	2	no	—
64	C₂×C₈×C₂₃	—	8	2	no	—
65	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₄⋊C₄×C₂₃	—	2	2	no	—
70	C₄⋊C₄×C₂₃	—	2	1	no	—
71	C₄⋊C₄×C₂₃	—	2	1	no	—
72	C₄⋊C₄×C₂₃	—	2	1	no	—
73	C₄⋊C₄×C₂₃	—	2	2	no	—
74	C₄⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
75	C₂²⋊C₄×C₂₃	—	2	2	no	—
76	C₄²×C₂₃	—	2	2	no	—
77	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₂₃	#41	1	1	yes	characteristic
81	#8689	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
83	Q₈×C₂₃	—	4	2	no	—
84	Q₈×C₂₃	—	4	1	no	—
85	Q₈×C₂₃	—	4	1	no	—
86	C₂×C₄×C₂₃	—	2	1	no	—
87	C₂×C₄×C₂₃	—	2	1	no	—
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	4	1	no	—
91	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
94	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
95	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₂₃	#135	1	2	yes	—
98	C₂×C₄×C₂₃	#137	1	2	yes	—
99	C₈×C₂₃	—	2	1	no	—
100	C₈×C₂₃	—	2	1	no	—
101	C₈×C₂₃	—	2	1	no	—
102	C₈×C₂₃	—	2	1	no	—
103	C₈×C₂₃	—	4	2	no	—
104	C₈×C₂₃	—	8	2	no	—
105	C₈×C₂₃	—	8	2	no	—
106	#2960	C₂×C₂₃	1	1	yes	characteristic
107	#1837	C₂×C₂₃	1	1	yes	characteristic
108	C₂×#335	C₂×C₂₃	1	1	yes	characteristic
109	#1550	C₂×C₂₃	1	2	yes	—
110	#1292	C₂×C₂₃	1	2	yes	—
111	C₂²×C₂₃	—	2	1	no	—
112	C₂²×C₂₃	—	2	1	no	—
113	C₂²×C₂₃	#380	1	1	yes	characteristic
114	C₂²×C₂₃	#479	1	2	yes	—
115	C₄×C₂₃	—	2	1	no	—
116	C₄×C₂₃	—	2	1	no	—
117	C₄×C₂₃	—	2	1	no	—
118	C₄×C₂₃	—	4	2	no	—
119	C₄×C₂₃	—	4	2	no	—
120	C₄×C₂₃	#416	1	1	yes	characteristic
121	C₄×C₂₃	#419	1	1	yes	characteristic
122	C₄×C₂₃	#464	1	2	yes	—
123	C₄×C₂₃	#472	1	1	yes	characteristic
124	C₄×C₂₃	#473	1	1	yes	characteristic
125	#533	C₂²×C₂₃	1	1	yes	characteristic
126	#446	—	2	1	no	—
127	#446	C₄×C₂₃	1	2	yes	—
128	C₈×Q₈	—	2	1	no	—
129	C₈×Q₈	C₄×C₂₃	1	2	yes	—
130	#424	—	2	1	no	—
131	#423	C₂²×C₂₃	1	2	yes	—
132	#423	C₂²×C₂₃	1	1	yes	characteristic
133	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
134	#405	—	2	1	no	—
135	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
136	#341	—	2	1	no	—
137	#341	—	2	1	no	—
138	#335	—	2	1	no	—
139	#335	—	2	1	no	—
140	#333	—	2	2	no	—
141	#332	—	2	2	no	—
142	C₂×C₂₃	—	2	2	no	—
143	C₂×C₂₃	#1805	1	1	yes	characteristic
144	C₂×C₂₃	#1813	1	1	yes	characteristic
145	C₂×C₂₃	#1816	1	1	yes	characteristic
146	C₂×M₁₆	—	2	1	no	—
147	C₂×M₁₆	—	2	2	no	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
150	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
151	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
152	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
153	#125	—	2	1	no	—
154	C₂²⋊Q₈	—	2	1	no	—
155	C₂²⋊Q₈	Q₈×C₂₃	1	2	yes	—
156	C₄×Q₈	—	2	1	no	—
157	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
158	#117	—	2	1	no	—
159	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
160	C₄⋊C₈	—	2	2	no	—
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	—	2	1	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
168	#97	—	2	2	no	—
169	C₄×C₈	—	2	2	no	—
170	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
171	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
172	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
173	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
174	C₂₃	#8689	1	1	yes	characteristic, direct product factor, Sylow
175	C₂×Q₈	—	2	1	no	—
176	C₂×Q₈	C₄⋊C₄×C₂₃	1	2	yes	—
177	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
178	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
179	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
180	M₁₆	—	4	2	no	—
181	M₁₆	—	4	2	no	—
182	M₁₆	—	4	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	8	2	no	—
189	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
190	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
191	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
192	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
199	C₂²⋊C₄	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
202	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
203	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
204	C₄²	#41×C₂₃	1	1	yes	characteristic
205	C₂³	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
206	Q₈	—	4	2	no	—
207	Q₈	—	4	1	no	—
208	Q₈	—	4	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
216	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
217	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
218	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
219	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
220	C₂×C₄	C₂₃×#135	1	2	yes	—
221	C₂×C₄	C₂₃×#137	1	2	yes	—
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	2	1	no	—
226	C₈	—	4	2	no	—
227	C₈	—	8	2	no	—
228	C₈	—	8	2	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	C₂₃×#380	1	1	yes	characteristic
232	C₂²	C₂₃×#479	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	2	no	—
238	C₄	C₂₃×#416	1	1	yes	characteristic
239	C₄	C₂₃×#419	1	1	yes	characteristic
240	C₄	C₂₃×#464	1	2	yes	—
241	C₄	C₂₃×#472	1	1	yes	characteristic
242	C₄	C₂₃×#473	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	C₂₃×#1805	1	1	yes	characteristic
245	C₂	C₂₃×#1813	1	1	yes	characteristic
246	C₂	C₂₃×#1816	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#380
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂³×C₂₃	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
81. #8689	C₂₃	direct (N × H)
174. C₂₃	#8689	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×#1837	2
C₂×C₂₃×#335	2
C₂₃×#1550	2
C₂₃×#1292	2
#8689	23

Subgroups — C23×#8689 #19156315

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8689 #19156315