Subgroups — C₂₃×#8617 #19156243

Order 5888.

398 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1801	C₂	1	2	yes	maximal
3	C₂₃×#1567	C₂	1	2	yes	maximal
4	C₂₃×#1292	C₂	1	2	yes	maximal
5	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
6	C₈×Q₈×C₂₃	—	2	2	no	—
7	C₈×Q₈×C₂₃	—	2	2	no	—
8	C₂₃×#423	C₂²	1	4	yes	—
9	C₂×C₂₃×C₄⋊C₈	C₄	1	2	yes	—
10	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#101	—	2	2	no	—
12	C₂×C₄×C₈×C₂₃	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#343	—	2	2	no	—
15	C₂₃×#335	—	2	2	no	—
16	C₂₃×#335	—	2	2	no	—
17	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
18	C₂×M₁₆×C₂₃	—	2	4	no	—
19	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₄×Q₈×C₂₃	—	2	2	no	—
23	C₄×Q₈×C₂₃	—	2	2	no	—
24	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₂₃	Q₈	1	2	yes	—
26	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
28	C₂₃×C₄⋊C₈	—	2	4	no	—
29	C₂₃×C₄⋊C₈	—	2	2	no	—
30	C₂₃×C₄⋊C₈	—	2	4	no	—
31	C₂₃×C₄⋊C₈	—	2	4	no	—
32	C₂₃×#101	—	4	4	no	—
33	C₄×C₈×C₂₃	—	2	2	no	—
34	C₄×C₈×C₂₃	—	2	2	no	—
35	C₄×C₈×C₂₃	D₄	1	2	yes	—
36	C₄×C₈×C₂₃	D₄	1	2	yes	—
37	C₂×Q₈×C₂₃	—	2	2	no	—
38	C₂×Q₈×C₂₃	—	2	2	no	—
39	C₂×Q₈×C₂₃	D₈	1	2	yes	—
40	C₂×Q₈×C₂₃	Q₁₆	1	2	yes	—
41	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
46	M₁₆×C₂₃	—	4	4	no	—
47	M₁₆×C₂₃	—	4	4	no	—
48	C₂×C₈×C₂₃	—	2	2	no	—
49	C₂×C₈×C₂₃	—	2	2	no	—
50	C₂×C₈×C₂₃	—	4	1	no	—
51	C₂×C₈×C₂₃	—	4	2	no	—
52	C₂×C₈×C₂₃	—	4	4	no	—
53	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₄⋊C₄×C₂₃	—	2	2	no	—
57	C₄⋊C₄×C₂₃	—	2	4	no	—
58	C₄²×C₂₃	—	2	2	no	—
59	C₄²×C₂₃	—	2	2	no	—
60	C₄²×C₂₃	C₂×D₄	1	2	yes	—
61	C₄²×C₂₃	#41	1	2	yes	—
62	#8617	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	Q₈×C₂₃	—	2	4	no	—
65	Q₈×C₂₃	—	2	4	no	—
66	C₂×C₄×C₂₃	—	2	2	no	—
67	C₂×C₄×C₂₃	—	2	2	no	—
68	C₂×C₄×C₂₃	—	2	1	no	—
69	C₂×C₄×C₂₃	—	2	1	no	—
70	C₂×C₄×C₂₃	—	2	2	no	—
71	C₂×C₄×C₂₃	—	4	1	no	—
72	C₂×C₄×C₂₃	—	4	2	no	—
73	C₂×C₄×C₂₃	#107	1	4	yes	—
74	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
76	C₂×C₄×C₂₃	#121	1	2	yes	—
77	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
78	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×C₄×C₂₃	#135	1	2	yes	—
81	C₈×C₂₃	—	2	1	no	—
82	C₈×C₂₃	—	2	2	no	—
83	C₈×C₂₃	—	4	2	no	—
84	C₈×C₂₃	—	8	4	no	—
85	C₈×C₂₃	#101	1	2	yes	complemented
86	C₂×C₈×Q₈	C₂×C₂₃	1	1	yes	characteristic
87	#1801	C₂×C₂₃	1	2	yes	—
88	#1567	C₂×C₂₃	1	2	yes	—
89	#1292	C₂×C₂₃	1	2	yes	—
90	C₂²×C₂₃	—	2	1	no	—
91	C₂²×C₂₃	—	2	1	no	—
92	C₂²×C₂₃	#380	1	1	yes	characteristic
93	C₂²×C₂₃	C₂×#107	1	1	yes	characteristic
94	C₂²×C₂₃	#427	1	1	yes	characteristic
95	C₄×C₂₃	—	2	1	no	—
96	C₄×C₂₃	—	2	4	no	—
97	C₄×C₂₃	—	2	2	no	—
98	C₄×C₂₃	—	4	2	no	—
99	C₄×C₂₃	C₂×#101	1	1	yes	characteristic
100	C₄×C₂₃	#413	1	1	yes	characteristic
101	C₄×C₂₃	#466	1	2	yes	—
102	C₄×C₂₃	#467	1	2	yes	—
103	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
104	C₈×Q₈	—	2	2	no	—
105	C₈×Q₈	—	2	2	no	—
106	#423	C₂²×C₂₃	1	4	yes	—
107	C₂×C₄⋊C₈	C₄×C₂₃	1	2	yes	—
108	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
109	C₂×#101	—	2	2	no	—
110	C₂×C₄×C₈	C₄×C₂₃	1	2	yes	—
111	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
112	#343	—	2	2	no	—
113	#335	—	2	2	no	—
114	#335	—	2	2	no	—
115	C₂×C₂₃	—	2	1	no	—
116	C₂×C₂₃	—	2	1	no	—
117	C₂×C₂₃	#1801	1	1	yes	characteristic
118	C₂×C₂₃	#1808	1	1	yes	characteristic
119	C₂×C₂₃	#1813	1	1	yes	characteristic
120	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
121	C₂×M₁₆	—	2	4	no	—
122	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
123	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
124	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
125	C₄×Q₈	—	2	2	no	—
126	C₄×Q₈	—	2	2	no	—
127	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
128	C₂×C₄⋊C₄	Q₈×C₂₃	1	2	yes	—
129	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
130	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
131	C₄⋊C₈	—	2	4	no	—
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	2	4	no	—
134	C₄⋊C₈	—	2	4	no	—
135	#101	—	4	4	no	—
136	C₄×C₈	—	2	2	no	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	D₄×C₂₃	1	2	yes	—
139	C₄×C₈	D₄×C₂₃	1	2	yes	—
140	C₂₃	#8617	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂×Q₈	D₈×C₂₃	1	2	yes	—
144	C₂×Q₈	Q₁₆×C₂₃	1	2	yes	—
145	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
146	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
147	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
148	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
149	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
150	M₁₆	—	4	4	no	—
151	M₁₆	—	4	4	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	—	4	4	no	—
157	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
158	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
159	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	C₂×D₄×C₂₃	1	2	yes	—
165	C₄²	#41×C₂₃	1	2	yes	—
166	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
167	Q₈	—	2	4	no	—
168	Q₈	—	2	4	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	C₂₃×#107	1	4	yes	—
177	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
178	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
179	C₂×C₄	C₂₃×#121	1	2	yes	—
180	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
181	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
182	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
183	C₂×C₄	C₂₃×#135	1	2	yes	—
184	C₈	—	2	1	no	—
185	C₈	—	2	2	no	—
186	C₈	—	4	2	no	—
187	C₈	—	8	4	no	—
188	C₈	C₂₃×#101	1	2	yes	complemented
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	C₂₃×#380	1	1	yes	characteristic
192	C₂²	C₂×C₂₃×#107	1	1	yes	characteristic
193	C₂²	C₂₃×#427	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	4	no	—
196	C₄	—	2	2	no	—
197	C₄	—	4	2	no	—
198	C₄	C₂×C₂₃×#101	1	1	yes	characteristic
199	C₄	C₂₃×#413	1	1	yes	characteristic
200	C₄	C₂₃×#466	1	2	yes	—
201	C₄	C₂₃×#467	1	2	yes	—
202	C₂	—	2	1	no	—
203	C₂	—	2	1	no	—
204	C₂	C₂₃×#1801	1	1	yes	characteristic
205	C₂	C₂₃×#1808	1	1	yes	characteristic
206	C₂	C₂₃×#1813	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	C₂×C₄⋊C₄
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
62. #8617	C₂₃	direct (N × H)
85. C₈×C₂₃	#101	semidirect
140. C₂₃	#8617	direct (N × H)
188. C₈	C₂₃×#101	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₈×Q₈×C₂₃	2
C₂₃×#1801	2
C₂₃×#1567	2
C₂₃×#1292	2
#8617	23

Subgroups — C23×#8617 #19156243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8617 #19156243