Subgroups — C₂₃×#9771 #19157397

Order 5888.

414 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×C₂₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1848	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1549	C₂	1	2	yes	maximal
4	C₂₃×#1548	C₂	1	2	yes	maximal
5	C₂₃×#1450	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂₃×#500	—	2	1	no	—
8	C₂₃×#500	—	2	1	no	—
9	C₂₃×#500	C₄	1	4	yes	—
10	C₂₃×#431	C₂²	1	1	yes	characteristic
11	C₂₃×#431	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×#108	—	2	2	no	—
13	C₂×C₂₃×#107	—	2	2	no	—
14	C₂₃×#423	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#333	—	2	4	no	—
17	C₂₃×#332	—	2	4	no	—
18	C₂₃×#322	—	2	1	no	—
19	C₈²×C₂₃	—	2	1	no	—
20	C₂²×Q₈×C₂₃	—	2	2	no	—
21	C₂×M₁₆×C₂₃	—	2	2	no	—
22	C₂²×C₈×C₂₃	D₄	1	2	yes	—
23	C₂₃×C₄⋊Q₈	—	2	2	no	—
24	C₂₃×C₄⋊Q₈	—	2	2	no	—
25	C₂₃×C₄⋊Q₈	—	2	2	no	—
26	C₂₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
29	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
30	C₂₃×#108	—	4	4	no	—
31	C₂₃×#107	—	2	4	no	—
32	C₂₃×#107	—	2	4	no	—
33	C₂₃×C₄⋊C₈	—	2	2	no	—
34	C₂₃×C₄⋊C₈	—	2	2	no	—
35	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
36	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
37	C₂₃×C₂²⋊C₈	—	2	1	no	—
38	C₂₃×C₂²⋊C₈	—	2	1	no	—
39	C₄×C₈×C₂₃	—	2	1	no	—
40	C₄×C₈×C₂₃	—	2	1	no	—
41	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₂₃	—	2	4	no	—
45	C₂×Q₈×C₂₃	—	4	2	no	—
46	C₂×Q₈×C₂₃	—	8	4	no	—
47	C₂²×C₄×C₂₃	—	4	2	no	—
48	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
51	M₁₆×C₂₃	—	4	4	no	—
52	C₂×C₈×C₂₃	—	2	2	no	—
53	C₂×C₈×C₂₃	—	2	2	no	—
54	C₂×C₈×C₂₃	—	4	2	no	—
55	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₂₃	SD₁₆	1	4	yes	—
58	C₂×C₈×C₂₃	Q₁₆	1	4	yes	—
59	C₄⋊C₄×C₂₃	—	2	4	no	—
60	C₄⋊C₄×C₂₃	—	2	4	no	—
61	C₄⋊C₄×C₂₃	—	4	2	no	—
62	C₄⋊C₄×C₂₃	—	4	2	no	—
63	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
67	#9771	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂³×C₂₃	#124	1	1	yes	characteristic
69	Q₈×C₂₃	—	8	4	no	—
70	Q₈×C₂₃	—	8	4	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	4	4	no	—
74	C₂×C₄×C₂₃	—	8	4	no	—
75	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
78	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
79	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
80	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
81	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
82	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
83	C₈×C₂₃	—	4	2	no	—
84	C₈×C₂₃	—	4	2	no	—
85	C₈×C₂₃	—	8	2	no	—
86	C₈×C₂₃	#103	1	4	yes	—
87	C₈×C₂₃	#103	1	4	yes	—
88	C₂×#500	C₂×C₂₃	1	1	yes	characteristic
89	#1848	C₂×C₂₃	1	1	yes	characteristic
90	#1549	C₂×C₂₃	1	2	yes	—
91	#1548	C₂×C₂₃	1	2	yes	—
92	#1450	C₂×C₂₃	1	1	yes	characteristic
93	C₂²×C₂₃	—	2	1	no	—
94	C₂²×C₂₃	—	2	1	no	—
95	C₂²×C₂₃	#390	1	1	yes	characteristic
96	C₂²×C₂₃	#487	1	2	yes	—
97	C₄×C₂₃	—	2	1	no	—
98	C₄×C₂₃	—	2	1	no	—
99	C₄×C₂₃	—	8	4	no	—
100	C₄×C₂₃	C₂×#103	1	2	yes	—
101	C₄×C₂₃	#472	1	2	yes	—
102	C₄×C₂₃	#473	1	2	yes	—
103	C₄×C₂₃	#492	1	1	yes	characteristic
104	C₄×C₂₃	#494	1	1	yes	characteristic
105	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
106	#500	—	2	1	no	—
107	#500	—	2	1	no	—
108	#500	C₄×C₂₃	1	4	yes	—
109	#431	C₂²×C₂₃	1	1	yes	characteristic
110	#431	C₂²×C₂₃	1	1	yes	characteristic
111	C₂×#108	—	2	2	no	—
112	C₂×#107	—	2	2	no	—
113	#423	C₂²×C₂₃	1	2	yes	—
114	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
115	#333	—	2	4	no	—
116	#332	—	2	4	no	—
117	#322	—	2	1	no	—
118	C₈²	—	2	1	no	—
119	C₂×C₂₃	—	2	2	no	—
120	C₂×C₂₃	#1956	1	1	yes	characteristic
121	C₂×C₂₃	#1978	1	1	yes	characteristic
122	C₂×C₂₃	#1979	1	1	yes	characteristic
123	C₂²×Q₈	—	2	2	no	—
124	C₂×M₁₆	—	2	2	no	—
125	C₂²×C₈	D₄×C₂₃	1	2	yes	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	—	2	2	no	—
129	C₄⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
133	#108	—	4	4	no	—
134	#107	—	2	4	no	—
135	#107	—	2	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	C₂²⋊C₈	—	2	1	no	—
141	C₂²⋊C₈	—	2	1	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
145	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
146	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
147	C₂₃	#9771	1	1	yes	characteristic, direct product factor, Sylow
148	C₂×Q₈	—	2	4	no	—
149	C₂×Q₈	—	4	2	no	—
150	C₂×Q₈	—	8	4	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
153	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
154	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
155	M₁₆	—	4	4	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
160	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
161	C₂×C₈	SD₁₆×C₂₃	1	4	yes	—
162	C₂×C₈	Q₁₆×C₂₃	1	4	yes	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
168	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
169	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
170	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
171	C₂³	C₂₃×#124	1	1	yes	characteristic
172	Q₈	—	8	4	no	—
173	Q₈	—	8	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	—	8	4	no	—
178	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
179	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
180	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
181	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
182	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
183	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
184	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
185	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
186	C₈	—	4	2	no	—
187	C₈	—	4	2	no	—
188	C₈	—	8	2	no	—
189	C₈	C₂₃×#103	1	4	yes	—
190	C₈	C₂₃×#103	1	4	yes	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	C₂₃×#390	1	1	yes	characteristic
194	C₂²	C₂₃×#487	1	2	yes	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	8	4	no	—
198	C₄	C₂×C₂₃×#103	1	2	yes	—
199	C₄	C₂₃×#472	1	2	yes	—
200	C₄	C₂₃×#473	1	2	yes	—
201	C₄	C₂₃×#492	1	1	yes	characteristic
202	C₄	C₂₃×#494	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	C₂₃×#1956	1	1	yes	characteristic
205	C₂	C₂₃×#1978	1	1	yes	characteristic
206	C₂	C₂₃×#1979	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#390
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂³×C₂₃	#124
LCS	G ▷ C₄² ▷ 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
67. #9771	C₂₃	direct (N × H)
147. C₂₃	#9771	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#500	2
C₂₃×#1848	2
C₂₃×#1549	2
C₂₃×#1548	2
C₂₃×#1450	2
#9771	23

Subgroups — C23×#9771 #19157397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9771 #19157397