Subgroups — C₂₃×#9845 #19157471

Order 5888.

478 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×#3170	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1994	C₂	1	2	yes	maximal
3	C₄²×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1567	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1493	C₂	1	2	yes	maximal
6	C₂₃×#533	C₂²	1	2	yes	—
7	C₂₃×#500	C₄	1	1	yes	characteristic, complemented
8	C₂₃×#494	C₄	1	1	yes	characteristic, complemented
9	C₂₃×#487	C₄	1	2	yes	complemented
10	C₂₃×#467	—	2	2	no	—
11	C₄×Q₁₆×C₂₃	—	2	2	no	—
12	C₂₃×#423	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₂₃	—	2	2	no	—
14	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₄³×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#339	—	2	4	no	—
18	C₂₃×#335	—	2	2	no	—
19	C₂₃×Q₈⋊C₈	—	2	4	no	—
20	C₂×Q₁₆×C₂₃	—	2	2	no	—
21	C₂×M₁₆×C₂₃	—	2	2	no	—
22	C₂²×C₈×C₂₃	—	2	1	no	—
23	C₂²×C₈×C₂₃	D₄	1	2	yes	—
24	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
25	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
26	C₂₃×#125	—	2	2	no	—
27	C₂₃×C₂²⋊Q₈	—	2	2	no	—
28	C₂₃×C₂²⋊Q₈	—	2	2	no	—
29	C₄×Q₈×C₂₃	—	2	2	no	—
30	C₄×Q₈×C₂₃	—	2	2	no	—
31	C₂₃×#117	—	2	2	no	—
32	C₂×C₄²×C₂₃	—	2	1	no	—
33	C₂×C₄²×C₂₃	—	2	1	no	—
34	C₂×C₄²×C₂₃	D₄	1	2	yes	—
35	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
36	C₂₃×#107	—	2	2	no	—
37	C₂₃×C₄⋊C₈	—	2	2	no	—
38	C₂₃×C₄⋊C₈	—	2	2	no	—
39	C₂₃×#103	—	2	4	no	—
40	C₄×C₈×C₂₃	—	2	8	no	—
41	C₄×C₈×C₂₃	—	2	1	no	—
42	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₂₃	D₄	1	2	yes	—
45	C₄×C₈×C₂₃	D₄	1	2	yes	—
46	C₂×Q₈×C₂₃	—	2	2	no	—
47	C₂×Q₈×C₂₃	—	2	2	no	—
48	C₂²×C₄×C₂₃	—	2	2	no	—
49	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
52	Q₁₆×C₂₃	—	4	4	no	—
53	M₁₆×C₂₃	—	4	4	no	—
54	C₂×C₈×C₂₃	—	2	4	no	—
55	C₂×C₈×C₂₃	—	2	2	no	—
56	C₂×C₈×C₂₃	—	2	4	no	—
57	C₂×C₈×C₂₃	—	4	2	no	—
58	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
60	C₄⋊C₄×C₂₃	—	2	2	no	—
61	C₄⋊C₄×C₂₃	—	2	2	no	—
62	C₄⋊C₄×C₂₃	—	2	2	no	—
63	C₄⋊C₄×C₂₃	—	2	2	no	—
64	C₄⋊C₄×C₂₃	—	4	2	no	—
65	C₄⋊C₄×C₂₃	—	4	2	no	—
66	C₂²⋊C₄×C₂₃	—	4	2	no	—
67	C₄²×C₂₃	—	2	4	no	—
68	C₄²×C₂₃	—	2	4	no	—
69	C₄²×C₂₃	—	2	2	no	—
70	C₄²×C₂₃	—	4	2	no	—
71	C₄²×C₂₃	Q₁₆	1	4	yes	complemented
72	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₂₃	#41	1	1	yes	characteristic
76	#9845	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₂₃	#121	1	1	yes	characteristic
78	Q₈×C₂₃	—	4	4	no	—
79	Q₈×C₂₃	—	4	4	no	—
80	C₂×C₄×C₂₃	—	2	4	no	—
81	C₂×C₄×C₂₃	—	2	4	no	—
82	C₂×C₄×C₂₃	—	2	4	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	4	2	no	—
86	C₂×C₄×C₂₃	—	4	2	no	—
87	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
91	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
92	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
94	C₂×C₄×C₂₃	#135	1	2	yes	—
95	C₈×C₂₃	—	2	4	no	—
96	C₈×C₂₃	—	2	2	no	—
97	C₈×C₂₃	—	8	2	no	—
98	C₈×C₂₃	#104	1	4	yes	—
99	#3170	C₂×C₂₃	1	1	yes	characteristic
100	#1994	C₂×C₂₃	1	2	yes	—
101	C₄²×C₈	C₂×C₂₃	1	1	yes	characteristic
102	#1567	C₂×C₂₃	1	1	yes	characteristic
103	#1493	C₂×C₂₃	1	2	yes	—
104	C₂²×C₂₃	—	2	1	no	—
105	C₂²×C₂₃	—	2	1	no	—
106	C₂²×C₂₃	#390	1	1	yes	characteristic
107	C₂²×C₂₃	#467	1	2	yes	—
108	C₄×C₂₃	—	2	8	no	—
109	C₄×C₂₃	—	2	1	no	—
110	C₄×C₂₃	—	2	1	no	—
111	C₄×C₂₃	—	8	2	no	—
112	C₄×C₂₃	—	8	2	no	—
113	C₄×C₂₃	C₂×#104	1	2	yes	—
114	C₄×C₂₃	C₄×Q₁₆	1	2	yes	—
115	C₄×C₂₃	#443	1	2	yes	—
116	C₄×C₂₃	#494	1	1	yes	characteristic
117	C₄×C₂₃	#495	1	1	yes	characteristic
118	#533	C₂²×C₂₃	1	2	yes	—
119	#500	C₄×C₂₃	1	1	yes	characteristic, complemented
120	#494	C₄×C₂₃	1	1	yes	characteristic, complemented
121	#487	C₄×C₂₃	1	2	yes	complemented
122	#467	—	2	2	no	—
123	C₄×Q₁₆	—	2	2	no	—
124	#423	C₂²×C₂₃	1	2	yes	—
125	C₂×C₄×C₈	—	2	2	no	—
126	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
127	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
128	C₄³	C₂²×C₂₃	1	1	yes	characteristic
129	#339	—	2	4	no	—
130	#335	—	2	2	no	—
131	Q₈⋊C₈	—	2	4	no	—
132	C₂×C₂₃	—	2	2	no	—
133	C₂×C₂₃	#1962	1	1	yes	characteristic
134	C₂×C₂₃	#1969	1	1	yes	characteristic
135	C₂×C₂₃	#1974	1	1	yes	characteristic
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×M₁₆	—	2	2	no	—
138	C₂²×C₈	—	2	1	no	—
139	C₂²×C₈	D₄×C₂₃	1	2	yes	—
140	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
141	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
142	#125	—	2	2	no	—
143	C₂²⋊Q₈	—	2	2	no	—
144	C₂²⋊Q₈	—	2	2	no	—
145	C₄×Q₈	—	2	2	no	—
146	C₄×Q₈	—	2	2	no	—
147	#117	—	2	2	no	—
148	C₂×C₄²	—	2	1	no	—
149	C₂×C₄²	—	2	1	no	—
150	C₂×C₄²	D₄×C₂₃	1	2	yes	—
151	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
152	#107	—	2	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	#103	—	2	4	no	—
156	C₄×C₈	—	2	8	no	—
157	C₄×C₈	—	2	1	no	—
158	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
159	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
160	C₄×C₈	D₄×C₂₃	1	2	yes	—
161	C₄×C₈	D₄×C₂₃	1	2	yes	—
162	C₂₃	#9845	1	1	yes	characteristic, direct product factor, Sylow
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
168	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
169	Q₁₆	—	4	4	no	—
170	M₁₆	—	4	4	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
176	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₄²	—	2	4	no	—
185	C₄²	—	2	4	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	4	2	no	—
188	C₄²	Q₁₆×C₂₃	1	4	yes	complemented
189	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
190	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
191	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
192	C₄²	#41×C₂₃	1	1	yes	characteristic
193	C₂³	C₂₃×#121	1	1	yes	characteristic
194	Q₈	—	4	4	no	—
195	Q₈	—	4	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
204	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
205	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
206	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
207	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
208	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
209	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
210	C₂×C₄	C₂₃×#135	1	2	yes	—
211	C₈	—	2	4	no	—
212	C₈	—	2	2	no	—
213	C₈	—	8	2	no	—
214	C₈	C₂₃×#104	1	4	yes	—
215	C₂²	—	2	1	no	—
216	C₂²	—	2	1	no	—
217	C₂²	C₂₃×#390	1	1	yes	characteristic
218	C₂²	C₂₃×#467	1	2	yes	—
219	C₄	—	2	8	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	8	2	no	—
223	C₄	—	8	2	no	—
224	C₄	C₂×C₂₃×#104	1	2	yes	—
225	C₄	C₄×Q₁₆×C₂₃	1	2	yes	—
226	C₄	C₂₃×#443	1	2	yes	—
227	C₄	C₂₃×#494	1	1	yes	characteristic
228	C₄	C₂₃×#495	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	C₂₃×#1962	1	1	yes	characteristic
231	C₂	C₂₃×#1969	1	1	yes	characteristic
232	C₂	C₂₃×#1974	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#127
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#390
prime radical	C₂³×C₂₃	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₂₃×#500	C₄	semidirect
8. C₂₃×#494	C₄	semidirect
9. C₂₃×#487	C₄	semidirect
71. C₄²×C₂₃	Q₁₆	semidirect
76. #9845	C₂₃	direct (N × H)
119. #500	C₄×C₂₃	semidirect
120. #494	C₄×C₂₃	semidirect
121. #487	C₄×C₂₃	semidirect
162. C₂₃	#9845	direct (N × H)
188. C₄²	Q₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3170	2
C₂₃×#1994	2
C₄²×C₈×C₂₃	2
C₂₃×#1567	2
C₂₃×#1493	2
#9845	23

Subgroups — C23×#9845 #19157471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9845 #19157471