Subgroups — C₂₃×#9701 #19157327

Order 5888.

486 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×#2862	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1842	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1767	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#333	C₂	1	2	yes	maximal
5	C₂×C₂₃×#332	C₂	1	2	yes	maximal
6	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₂₃×#97	—	2	1	no	—
12	C₂₃×#391	—	2	2	no	—
13	C₂₃×#391	C₄	1	4	yes	—
14	C₂₃×#382	—	2	1	no	—
15	C₂₃×#382	—	2	1	no	—
16	C₄³×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#336	—	2	1	no	—
18	C₂₃×#333	—	2	4	no	—
19	C₂₃×#332	—	2	4	no	—
20	C₂²×Q₈×C₂₃	—	2	1	no	—
21	C₂²×Q₈×C₂₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₂₃	—	2	2	no	—
23	C₂²×C₈×C₂₃	—	2	2	no	—
24	C₂₃×C₄⋊Q₈	—	2	2	no	—
25	C₂₃×C₄⋊Q₈	—	2	2	no	—
26	C₂₃×C₄⋊Q₈	—	2	2	no	—
27	C₂₃×C₄⋊Q₈	—	2	2	no	—
28	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
29	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
30	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
31	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₂₃	D₄	1	2	yes	—
35	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
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	2	no	—
40	C₂₃×C₄⋊C₈	—	4	4	no	—
41	C₂₃×C₄⋊C₈	D₄	1	4	yes	—
42	C₂₃×#97	—	4	2	no	—
43	C₂₃×#95	—	2	4	no	—
44	C₂₃×#95	—	2	4	no	—
45	C₂×Q₈×C₂₃	—	2	4	no	—
46	C₂×Q₈×C₂₃	—	4	4	no	—
47	C₂×Q₈×C₂₃	—	4	1	no	—
48	C₂×Q₈×C₂₃	—	4	1	no	—
49	C₂×Q₈×C₂₃	—	4	2	no	—
50	C₂×Q₈×C₂₃	—	4	2	no	—
51	C₂²×C₄×C₂₃	—	2	2	no	—
52	C₂²×C₄×C₂₃	—	2	2	no	—
53	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₂₃	—	2	4	no	—
59	C₂×C₈×C₂₃	—	4	2	no	—
60	C₂×C₈×C₂₃	—	4	2	no	—
61	C₂×C₈×C₂₃	—	4	2	no	—
62	C₂×C₈×C₂₃	—	4	4	no	—
63	C₄⋊C₄×C₂₃	—	2	4	no	—
64	C₄⋊C₄×C₂₃	—	4	2	no	—
65	C₄⋊C₄×C₂₃	—	4	2	no	—
66	C₄⋊C₄×C₂₃	—	4	1	no	—
67	C₄⋊C₄×C₂₃	—	4	1	no	—
68	C₄²×C₂₃	—	2	2	no	—
69	C₄²×C₂₃	—	2	2	no	—
70	C₄²×C₂₃	—	2	2	no	—
71	C₄²×C₂₃	—	4	2	no	—
72	C₄²×C₂₃	SD₁₆	1	2	yes	—
73	C₄²×C₂₃	Q₁₆	1	2	yes	—
74	C₄²×C₂₃	C₂×D₄	1	2	yes	—
75	C₄²×C₂₃	#41	1	2	yes	—
76	#9701	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	Q₈×C₂₃	—	8	4	no	—
79	Q₈×C₂₃	—	8	2	no	—
80	Q₈×C₂₃	—	8	1	no	—
81	Q₈×C₂₃	—	8	1	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	4	2	no	—
89	C₂×C₄×C₂₃	—	4	2	no	—
90	C₂×C₄×C₂₃	—	4	4	no	—
91	C₂×C₄×C₂₃	—	4	4	no	—
92	C₂×C₄×C₂₃	#103	1	4	yes	—
93	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₂₃	#121	1	2	yes	—
95	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
96	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
99	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
100	C₂×C₄×C₂₃	#137	1	2	yes	—
101	C₂×C₄×C₂₃	#137	1	2	yes	—
102	C₈×C₂₃	—	4	4	no	—
103	C₈×C₂₃	—	8	2	no	—
104	#2862	C₂×C₂₃	1	1	yes	characteristic
105	#1842	C₂×C₂₃	1	1	yes	characteristic
106	#1767	C₂×C₂₃	1	1	yes	characteristic
107	C₂×#333	C₂×C₂₃	1	2	yes	—
108	C₂×#332	C₂×C₂₃	1	2	yes	—
109	C₂²×C₂₃	#390	1	1	yes	characteristic
110	C₂²×C₂₃	#413	1	2	yes	—
111	C₂²×C₂₃	C₂×#103	1	1	yes	characteristic
112	C₂²×C₂₃	#417	1	2	yes	—
113	C₂²×C₂₃	#419	1	1	yes	characteristic
114	C₄×C₂₃	—	2	4	no	—
115	C₄×C₂₃	—	2	2	no	—
116	C₄×C₂₃	—	4	2	no	—
117	C₄×C₂₃	—	8	2	no	—
118	C₄×C₂₃	—	8	2	no	—
119	C₄×C₂₃	#461	1	2	yes	—
120	C₄×C₂₃	#462	1	2	yes	—
121	C₄×C₂₃	#489	1	2	yes	—
122	C₄×C₂₃	#496	1	1	yes	characteristic
123	C₄×C₂₃	#497	1	1	yes	characteristic
124	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
125	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
126	C₂×C₄⋊C₈	—	2	2	no	—
127	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
128	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
129	C₂×#97	—	2	1	no	—
130	#391	—	2	2	no	—
131	#391	C₄×C₂₃	1	4	yes	—
132	#382	—	2	1	no	—
133	#382	—	2	1	no	—
134	C₄³	C₂²×C₂₃	1	1	yes	characteristic
135	#336	—	2	1	no	—
136	#333	—	2	4	no	—
137	#332	—	2	4	no	—
138	C₂×C₂₃	#1550	1	2	yes	—
139	C₂×C₂₃	#1559	1	2	yes	—
140	C₂×C₂₃	#1959	1	1	yes	characteristic
141	C₂×C₂₃	#1967	1	1	yes	characteristic
142	C₂×C₂₃	#1981	1	1	yes	characteristic
143	C₂²×Q₈	—	2	1	no	—
144	C₂²×Q₈	C₂×C₄×C₂₃	1	2	yes	—
145	C₂²×C₈	—	2	2	no	—
146	C₂²×C₈	—	2	2	no	—
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	—	2	2	no	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
155	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
156	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
157	C₂×C₄²	D₄×C₂₃	1	2	yes	—
158	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
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	2	no	—
163	C₄⋊C₈	—	4	4	no	—
164	C₄⋊C₈	D₄×C₂₃	1	4	yes	—
165	#97	—	4	2	no	—
166	#95	—	2	4	no	—
167	#95	—	2	4	no	—
168	C₂₃	#9701	1	1	yes	characteristic, direct product factor, Sylow
169	C₂×Q₈	—	2	4	no	—
170	C₂×Q₈	—	4	4	no	—
171	C₂×Q₈	—	4	1	no	—
172	C₂×Q₈	—	4	1	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
178	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
179	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	4	4	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	1	no	—
191	C₄⋊C₄	—	4	1	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	4	2	no	—
196	C₄²	SD₁₆×C₂₃	1	2	yes	—
197	C₄²	Q₁₆×C₂₃	1	2	yes	—
198	C₄²	C₂×D₄×C₂₃	1	2	yes	—
199	C₄²	#41×C₂₃	1	2	yes	—
200	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
201	Q₈	—	8	4	no	—
202	Q₈	—	8	2	no	—
203	Q₈	—	8	1	no	—
204	Q₈	—	8	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	C₂₃×#103	1	4	yes	—
216	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
217	C₂×C₄	C₂₃×#121	1	2	yes	—
218	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
219	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
220	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
221	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
222	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
223	C₂×C₄	C₂₃×#137	1	2	yes	—
224	C₂×C₄	C₂₃×#137	1	2	yes	—
225	C₈	—	4	4	no	—
226	C₈	—	8	2	no	—
227	C₂²	C₂₃×#390	1	1	yes	characteristic
228	C₂²	C₂₃×#413	1	2	yes	—
229	C₂²	C₂×C₂₃×#103	1	1	yes	characteristic
230	C₂²	C₂₃×#417	1	2	yes	—
231	C₂²	C₂₃×#419	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	4	no	—
234	C₄	—	4	2	no	—
235	C₄	—	8	2	no	—
236	C₄	—	8	2	no	—
237	C₄	C₂₃×#461	1	2	yes	—
238	C₄	C₂₃×#462	1	2	yes	—
239	C₄	C₂₃×#489	1	2	yes	—
240	C₄	C₂₃×#496	1	1	yes	characteristic
241	C₄	C₂₃×#497	1	1	yes	characteristic
242	C₂	C₂₃×#1550	1	2	yes	—
243	C₂	C₂₃×#1559	1	2	yes	—
244	C₂	C₂₃×#1959	1	1	yes	characteristic
245	C₂	C₂₃×#1967	1	1	yes	characteristic
246	C₂	C₂₃×#1981	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
76. #9701	C₂₃	direct (N × H)
168. C₂₃	#9701	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×#2862	2
C₂₃×#1842	2
C₂₃×#1767	2
C₂×C₂₃×#333	2
C₂×C₂₃×#332	2
#9701	23

Subgroups — C23×#9701 #19157327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9701 #19157327