Subgroups — C₂₃×#20797 #19168423

Order 5888.

1126 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#427	C₂	1	4	yes	maximal
2	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2285	C₂	1	2	yes	maximal
4	C₂₃×#1846	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1845	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1844	C₂	1	2	yes	maximal, complemented
7	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
8	C₂×C₂₃×#117	C₂²	1	4	yes	—
9	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×#427	—	2	16	no	—
11	C₂×C₂₃×#107	C₂²	1	8	yes	—
12	C₂×C₂₃×#106	C₂²	1	8	yes	—
13	C₂×C₄×C₈×C₂₃	C₄	1	8	yes	complemented
14	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
17	C₂₃×#384	C₂²	1	4	yes	—
18	C₂₃×#382	C₂²	1	4	yes	—
19	C₂₃×#380	—	2	4	no	—
20	C₄×C₄⋊C₄×C₂₃	—	2	4	no	—
21	C₂³×C₄×C₂₃	D₄	1	2	yes	—
22	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₂₃	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₂₃	D₄	1	8	yes	—
25	C₂²×C₈×C₂₃	Q₈	1	8	yes	—
26	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
27	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
28	C₂₃×#117	—	2	16	no	—
29	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
30	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
31	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
32	C₂×C₄²×C₂₃	—	2	4	no	—
33	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₂₃	D₄	1	2	yes	—
35	C₂×C₄²×C₂₃	Q₈	1	4	yes	—
36	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
38	C₂₃×#107	—	2	16	no	—
39	C₂₃×#106	—	2	16	no	—
40	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
41	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	—
43	C₂₃×#95	—	2	8	no	—
44	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	—	2	8	no	—
46	C₂²×C₄×C₂₃	—	2	8	no	—
47	C₂²×C₄×C₂₃	C₄⋊C₄	1	8	yes	—
48	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
51	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₂×Q₈	1	4	yes	—
54	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
55	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
56	C₂×C₈×C₂₃	C₄⋊C₄	1	32	yes	complemented
57	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
58	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
59	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
60	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
61	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
62	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
63	C₄⋊C₄×C₂₃	—	2	16	no	—
64	C₄⋊C₄×C₂₃	—	4	16	no	—
65	C₂²⋊C₄×C₂₃	—	4	8	no	—
66	C₄²×C₂₃	—	4	8	no	—
67	C₄²×C₂₃	C₄⋊C₄	1	8	yes	—
68	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
69	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
70	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
71	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
72	#20797	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₂₃	—	2	4	no	—
74	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
75	C₂³×C₂₃	#127	1	2	yes	—
76	C₂³×C₂₃	C₄⋊Q₈	1	2	yes	—
77	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
78	C₂×C₄×C₂₃	—	2	4	no	—
79	C₂×C₄×C₂₃	—	2	16	no	—
80	C₂×C₄×C₂₃	—	4	8	no	—
81	C₂×C₄×C₂₃	—	4	16	no	—
82	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
83	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	8	yes	—
84	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
85	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
86	C₂×C₄×C₂₃	#127	1	2	yes	—
87	C₂×C₄×C₂₃	C₄⋊Q₈	1	8	yes	—
88	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
89	C₂×C₄×C₂₃	#135	1	8	yes	—
90	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
92	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
93	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
94	C₈×C₂₃	C₂×C₄⋊C₄	1	8	yes	—
95	C₈×C₂₃	C₂×C₄⋊C₄	1	8	yes	—
96	C₂×#427	C₂×C₂₃	1	4	yes	—
97	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
98	#2285	C₂×C₂₃	1	2	yes	—
99	#1846	C₂×C₂₃	1	4	yes	complemented
100	#1845	C₂×C₂₃	1	2	yes	complemented
101	#1844	C₂×C₂₃	1	2	yes	complemented
102	C₂²×C₂₃	—	2	4	no	—
103	C₂²×C₂₃	—	2	4	no	—
104	C₂²×C₂₃	—	2	4	no	—
105	C₂²×C₂₃	#384	1	4	yes	—
106	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
108	C₂²×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
109	C₂²×C₂₃	C₂×#135	1	4	yes	—
110	C₄×C₂₃	—	2	4	no	—
111	C₄×C₂₃	—	8	8	no	—
112	C₄×C₂₃	#384	1	4	yes	—
113	C₄×C₂₃	#427	1	8	yes	complemented
114	C₄×C₂₃	C₂²×C₄⋊C₄	1	2	yes	—
115	C₄×C₂₃	C₂×C₄⋊Q₈	1	2	yes	—
116	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
117	C₂×#117	C₂²×C₂₃	1	4	yes	—
118	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
119	#427	—	2	16	no	—
120	C₂×#107	C₂²×C₂₃	1	8	yes	—
121	C₂×#106	C₂²×C₂₃	1	8	yes	—
122	C₂×C₄×C₈	C₄×C₂₃	1	8	yes	complemented
123	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
124	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
126	#384	C₂²×C₂₃	1	4	yes	—
127	#382	C₂²×C₂₃	1	4	yes	—
128	#380	—	2	4	no	—
129	C₄×C₄⋊C₄	—	2	4	no	—
130	C₂×C₂₃	—	2	4	no	—
131	C₂×C₂₃	C₂×#384	1	1	yes	characteristic
132	C₂×C₂₃	C₂×#427	1	2	yes	—
133	C₂×C₂₃	#3147	1	2	yes	—
134	C₂×C₂₃	#3160	1	2	yes	—
135	C₂³×C₄	D₄×C₂₃	1	2	yes	—
136	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
137	C₂²×C₈	C₂×C₄×C₂₃	1	8	yes	—
138	C₂²×C₈	D₄×C₂₃	1	8	yes	—
139	C₂²×C₈	Q₈×C₂₃	1	8	yes	—
140	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
141	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
142	#117	—	2	16	no	—
143	C₂×C₄⋊C₄	—	2	8	no	—
144	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
145	C₂×C₂²⋊C₄	—	2	4	no	—
146	C₂×C₄²	—	2	4	no	—
147	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
148	C₂×C₄²	D₄×C₂₃	1	2	yes	—
149	C₂×C₄²	Q₈×C₂₃	1	4	yes	—
150	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
151	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
152	#107	—	2	16	no	—
153	#106	—	2	16	no	—
154	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
155	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
156	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	—
157	#95	—	2	8	no	—
158	C₂₃	#20797	1	1	yes	characteristic, direct product factor, Sylow
159	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	—	2	8	no	—
162	C₂²×C₄	C₄⋊C₄×C₂₃	1	8	yes	—
163	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
164	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
165	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
167	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
168	C₂²×C₄	C₂×Q₈×C₂₃	1	4	yes	—
169	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
170	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
171	C₂×C₈	C₄⋊C₄×C₂₃	1	32	yes	complemented
172	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
173	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
174	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
175	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
176	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
177	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
178	C₄⋊C₄	—	2	16	no	—
179	C₄⋊C₄	—	4	16	no	—
180	C₂²⋊C₄	—	4	8	no	—
181	C₄²	—	4	8	no	—
182	C₄²	C₄⋊C₄×C₂₃	1	8	yes	—
183	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
184	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
185	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
186	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
187	C₂³	—	2	4	no	—
188	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
189	C₂³	C₂₃×#127	1	2	yes	—
190	C₂³	C₂₃×C₄⋊Q₈	1	2	yes	—
191	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	16	no	—
194	C₂×C₄	—	4	16	no	—
195	C₂×C₄	—	4	8	no	—
196	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
197	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
198	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	8	yes	—
199	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
200	C₂×C₄	C₂₃×#127	1	2	yes	—
201	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
202	C₂×C₄	C₂₃×C₄⋊Q₈	1	8	yes	—
203	C₂×C₄	C₂₃×#135	1	8	yes	—
204	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
205	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
206	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
207	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
208	C₈	C₂×C₄⋊C₄×C₂₃	1	8	yes	—
209	C₈	C₂×C₄⋊C₄×C₂₃	1	8	yes	—
210	C₂²	—	2	4	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	4	no	—
213	C₂²	C₂₃×#384	1	4	yes	—
214	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
215	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
216	C₂²	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
217	C₂²	C₂×C₂₃×#135	1	4	yes	—
218	C₄	—	2	4	no	—
219	C₄	—	8	8	no	—
220	C₄	C₂₃×#384	1	4	yes	—
221	C₄	C₂₃×#427	1	8	yes	complemented
222	C₄	C₂²×C₄⋊C₄×C₂₃	1	2	yes	—
223	C₄	C₂×C₂₃×C₄⋊Q₈	1	2	yes	—
224	C₂	—	2	4	no	—
225	C₂	C₂×C₂₃×#384	1	1	yes	characteristic
226	C₂	C₂×C₂₃×#427	1	2	yes	—
227	C₂	C₂₃×#3147	1	2	yes	—
228	C₂	C₂₃×#3160	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₂²×D₄
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1846	C₂	semidirect
5. C₂₃×#1845	C₂	semidirect
6. C₂₃×#1844	C₂	semidirect
13. C₂×C₄×C₈×C₂₃	C₄	semidirect
56. C₂×C₈×C₂₃	C₄⋊C₄	semidirect
72. #20797	C₂₃	direct (N × H)
99. #1846	C₂×C₂₃	semidirect
100. #1845	C₂×C₂₃	semidirect
101. #1844	C₂×C₂₃	semidirect
113. C₄×C₂₃	#427	semidirect
122. C₂×C₄×C₈	C₄×C₂₃	semidirect
158. C₂₃	#20797	direct (N × H)
171. C₂×C₈	C₄⋊C₄×C₂₃	semidirect
221. C₄	C₂₃×#427	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#427	2
C₂²×C₄×C₈×C₂₃	2
C₂₃×#2285	2
C₂₃×#1846	2
C₂₃×#1845	2
C₂₃×#1844	2
#20797	23

Subgroups — C23×#20797 #19168423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20797 #19168423