Subgroups — C₂₃×#8635 #19156261

Order 5888.

320 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×#2960	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1568	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1568	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1537	C₂	1	2	yes	maximal
5	C₂₃×#1307	C₂	1	2	yes	maximal
6	C₂₃×#533	C₂²	1	1	yes	characteristic
7	C₂₃×#446	—	2	1	no	—
8	C₂₃×#446	—	2	1	no	—
9	C₈×Q₈×C₂₃	—	2	1	no	—
10	C₈×Q₈×C₂₃	—	2	1	no	—
11	C₂₃×#432	C₂²	1	2	yes	—
12	C₂₃×#432	C₂²	1	2	yes	—
13	C₂₃×#424	C₄	1	2	yes	—
14	C₂₃×#423	C₂²	1	1	yes	characteristic
15	C₂₃×#405	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#335	—	2	1	no	—
18	C₂₃×#335	—	2	1	no	—
19	C₂₃×#334	—	2	1	no	—
20	C₂₃×#334	—	2	1	no	—
21	C₂₃×#330	—	2	4	no	—
22	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂₃×C₄⋊Q₈	Q₈	1	1	yes	characteristic
26	C₂₃×C₄⋊Q₈	Q₈	1	1	yes	characteristic
27	C₂₃×#125	D₄	1	2	yes	—
28	C₂₃×C₂²⋊Q₈	—	2	1	no	—
29	C₂₃×C₂²⋊Q₈	—	2	1	no	—
30	C₄×Q₈×C₂₃	—	2	1	no	—
31	C₄×Q₈×C₂₃	—	2	1	no	—
32	C₂₃×#117	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
34	C₂₃×C₄⋊C₈	—	2	2	no	—
35	C₂₃×C₄⋊C₈	—	2	2	no	—
36	C₂₃×C₄⋊C₈	—	2	1	no	—
37	C₂₃×C₄⋊C₈	—	2	1	no	—
38	C₂₃×C₄⋊C₈	—	2	2	no	—
39	C₂₃×C₄⋊C₈	—	2	2	no	—
40	C₂₃×C₂²⋊C₈	—	2	2	no	—
41	C₂₃×C₂²⋊C₈	—	2	2	no	—
42	C₂₃×#97	—	2	2	no	—
43	C₂₃×#97	—	2	2	no	—
44	C₂₃×#97	—	2	2	no	—
45	C₄×C₈×C₂₃	—	2	2	no	—
46	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
48	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
50	C₂×Q₈×C₂₃	—	2	1	no	—
51	C₂×Q₈×C₂₃	—	2	1	no	—
52	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
55	M₁₆×C₂₃	—	4	2	no	—
56	C₂×C₈×C₂₃	—	2	2	no	—
57	C₂×C₈×C₂₃	—	2	2	no	—
58	C₂×C₈×C₂₃	—	4	1	no	—
59	C₂×C₈×C₂₃	—	4	2	no	—
60	C₂×C₈×C₂₃	—	4	2	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	1	yes	characteristic
64	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₄⋊C₄×C₂₃	—	2	2	no	—
66	C₄⋊C₄×C₂₃	—	2	2	no	—
67	C₄⋊C₄×C₂₃	—	2	1	no	—
68	C₄⋊C₄×C₂₃	—	2	1	no	—
69	C₄⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
70	C₄⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
71	C₂²⋊C₄×C₂₃	—	2	2	no	—
72	C₄²×C₂₃	—	2	2	no	—
73	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
77	#8635	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₂₃	#117	1	1	yes	characteristic
79	Q₈×C₂₃	—	4	2	no	—
80	Q₈×C₂₃	—	4	2	no	—
81	C₂×C₄×C₂₃	—	2	1	no	—
82	C₂×C₄×C₂₃	—	2	1	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	4	1	no	—
86	C₂×C₄×C₂₃	—	4	1	no	—
87	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
90	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
93	C₈×C₂₃	—	2	1	no	—
94	C₈×C₂₃	—	2	1	no	—
95	C₈×C₂₃	—	2	1	no	—
96	C₈×C₂₃	—	4	2	no	—
97	C₈×C₂₃	—	8	2	no	—
98	C₈×C₂₃	—	8	2	no	—
99	C₈×C₂₃	#101	1	2	yes	—
100	#2960	C₂×C₂₃	1	1	yes	characteristic
101	#1568	C₂×C₂₃	1	1	yes	characteristic
102	#1568	C₂×C₂₃	1	1	yes	characteristic
103	#1537	C₂×C₂₃	1	2	yes	—
104	#1307	C₂×C₂₃	1	2	yes	—
105	C₂²×C₂₃	—	2	1	no	—
106	C₂²×C₂₃	—	2	1	no	—
107	C₂²×C₂₃	—	2	1	no	—
108	C₂²×C₂₃	#380	1	1	yes	characteristic
109	C₄×C₂₃	—	2	1	no	—
110	C₄×C₂₃	—	2	1	no	—
111	C₄×C₂₃	—	2	1	no	—
112	C₄×C₂₃	—	2	1	no	—
113	C₄×C₂₃	—	2	1	no	—
114	C₄×C₂₃	—	4	2	no	—
115	C₄×C₂₃	—	4	2	no	—
116	C₄×C₂₃	C₂×#101	1	1	yes	characteristic
117	C₄×C₂₃	#413	1	1	yes	characteristic
118	#533	C₂²×C₂₃	1	1	yes	characteristic
119	#446	—	2	1	no	—
120	#446	—	2	1	no	—
121	C₈×Q₈	—	2	1	no	—
122	C₈×Q₈	—	2	1	no	—
123	#432	C₂²×C₂₃	1	2	yes	—
124	#432	C₂²×C₂₃	1	2	yes	—
125	#424	C₄×C₂₃	1	2	yes	—
126	#423	C₂²×C₂₃	1	1	yes	characteristic
127	#405	C₄×C₂₃	1	2	yes	—
128	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
129	#335	—	2	1	no	—
130	#335	—	2	1	no	—
131	#334	—	2	1	no	—
132	#334	—	2	1	no	—
133	#330	—	2	4	no	—
134	C₂×C₂₃	—	4	1	no	—
135	C₂×C₂₃	#1801	1	1	yes	characteristic
136	C₂×C₂₃	#1814	1	1	yes	characteristic
137	C₂×C₂₃	#1814	1	1	yes	characteristic
138	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
139	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
140	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
141	C₄⋊Q₈	Q₈×C₂₃	1	1	yes	characteristic
142	C₄⋊Q₈	Q₈×C₂₃	1	1	yes	characteristic
143	#125	D₄×C₂₃	1	2	yes	—
144	C₂²⋊Q₈	—	2	1	no	—
145	C₂²⋊Q₈	—	2	1	no	—
146	C₄×Q₈	—	2	1	no	—
147	C₄×Q₈	—	2	1	no	—
148	#117	C₂×C₄×C₂₃	1	2	yes	—
149	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	2	1	no	—
153	C₄⋊C₈	—	2	1	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₂²⋊C₈	—	2	2	no	—
157	C₂²⋊C₈	—	2	2	no	—
158	#97	—	2	2	no	—
159	#97	—	2	2	no	—
160	#97	—	2	2	no	—
161	C₄×C₈	—	2	2	no	—
162	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
163	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
164	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
165	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
166	C₂₃	#8635	1	1	yes	characteristic, direct product factor, Sylow
167	C₂×Q₈	—	2	1	no	—
168	C₂×Q₈	—	2	1	no	—
169	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
170	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
171	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
172	M₁₆	—	4	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
179	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
180	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
181	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
187	C₄⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
191	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
192	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
193	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
194	C₂³	C₂₃×#117	1	1	yes	characteristic
195	Q₈	—	4	2	no	—
196	Q₈	—	4	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
205	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
206	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
207	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
208	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	4	2	no	—
213	C₈	—	8	2	no	—
214	C₈	—	8	2	no	—
215	C₈	C₂₃×#101	1	2	yes	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	C₂₃×#380	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	2	no	—
227	C₄	C₂×C₂₃×#101	1	1	yes	characteristic
228	C₄	C₂₃×#413	1	1	yes	characteristic
229	C₂	—	4	1	no	—
230	C₂	C₂₃×#1801	1	1	yes	characteristic
231	C₂	C₂₃×#1814	1	1	yes	characteristic
232	C₂	C₂₃×#1814	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#380
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂³×C₂₃	#117
LCS	G ▷ C₂²×C₄ ▷ 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
77. #8635	C₂₃	direct (N × H)
166. C₂₃	#8635	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×#1568	2
C₂₃×#1537	2
C₂₃×#1307	2
#8635	23

Subgroups — C23×#8635 #19156261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8635 #19156261