Subgroups — C₂₃×#21691 #19169317

Order 5888.

1062 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#441	C₂	1	4	yes	maximal
3	C₄×C₂₃×C₄⋊Q₈	C₂	1	2	yes	maximal
4	C₂₃×#1971	C₂	1	4	yes	maximal
5	C₂₃×#1956	C₂	1	2	yes	maximal
6	C₂₃×#1844	C₂	1	1	yes	maximal, characteristic
7	C₄×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₂₃	C₂²	1	4	yes	—
9	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₂₃	C₂²	1	4	yes	—
11	C₂₃×#494	C₄	1	8	yes	complemented
12	C₂₃×#441	—	2	16	no	—
13	C₂×C₂₃×#106	C₂²	1	4	yes	—
14	C₂×C₂₃×#103	C₂²	1	8	yes	—
15	C₂×C₂₃×#97	C₂²	1	2	yes	—
16	C₂×C₂₃×#97	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#391	C₂²	1	4	yes	—
20	C₂₃×#389	—	2	4	no	—
21	C₂₃×#384	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄×C₂₃	—	2	4	no	—
23	C₄³×C₂₃	C₂²	1	1	yes	characteristic
24	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
25	C₂×Q₁₆×C₂₃	—	2	8	no	—
26	C₂×Q₁₆×C₂₃	—	2	8	no	—
27	C₂×Q₁₆×C₂₃	C₂×C₄	1	16	yes	—
28	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
29	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
30	C₂₃×C₄⋊Q₈	—	2	4	no	—
31	C₂₃×C₄⋊Q₈	C₂×C₄	1	8	yes	—
32	C₄×Q₈×C₂₃	—	2	16	no	—
33	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
34	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
35	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
36	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
37	C₂×C₄²×C₂₃	—	2	4	no	—
38	C₂×C₄²×C₂₃	D₄	1	4	yes	—
39	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
42	C₂₃×#106	—	2	8	no	—
43	C₂₃×#103	—	2	16	no	—
44	C₂₃×#97	—	2	4	no	—
45	C₂₃×#97	D₄	1	8	yes	—
46	C₄×C₈×C₂₃	—	2	2	no	—
47	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
48	C₂₃×#95	—	2	8	no	—
49	C₂×Q₈×C₂₃	—	2	4	no	—
50	C₂×Q₈×C₂₃	—	2	8	no	—
51	C₂×Q₈×C₂₃	—	2	8	no	—
52	C₂×Q₈×C₂₃	C₂²×C₄	1	8	yes	—
53	C₂²×C₄×C₂₃	—	2	4	no	—
54	C₂²×C₄×C₂₃	—	2	4	no	—
55	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
59	Q₁₆×C₂₃	—	2	32	no	—
60	C₂×C₈×C₂₃	—	2	2	no	—
61	C₂×C₈×C₂₃	—	2	2	no	—
62	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
64	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
65	C₂×C₈×C₂₃	#41	1	4	yes	—
66	C₄⋊C₄×C₂₃	—	2	16	no	—
67	C₄⋊C₄×C₂₃	—	2	8	no	—
68	C₄⋊C₄×C₂₃	—	4	8	no	—
69	C₄⋊C₄×C₂₃	—	4	8	no	—
70	C₄²×C₂₃	—	2	8	no	—
71	C₄²×C₂₃	—	2	2	no	—
72	C₄²×C₂₃	—	4	8	no	—
73	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
74	C₄²×C₂₃	C₂×D₄	1	4	yes	—
75	C₄²×C₂₃	#41	1	2	yes	—
76	#21691	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
78	Q₈×C₂₃	—	2	16	no	—
79	Q₈×C₂₃	—	4	8	no	—
80	C₂×C₄×C₂₃	—	2	4	no	—
81	C₂×C₄×C₂₃	—	2	8	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	4	no	—
84	C₂×C₄×C₂₃	—	4	4	no	—
85	C₂×C₄×C₂₃	—	4	4	no	—
86	C₂×C₄×C₂₃	—	4	4	no	—
87	C₂×C₄×C₂₃	—	4	8	no	—
88	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
89	C₂×C₄×C₂₃	#127	1	4	yes	—
90	C₂×C₄×C₂₃	#137	1	8	yes	—
91	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
93	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
94	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
95	C₈×C₂₃	—	2	4	no	—
96	C₈×C₂₃	C₄×D₄	1	8	yes	—
97	C₂×#494	C₂×C₂₃	1	1	yes	characteristic
98	C₂×#441	C₂×C₂₃	1	4	yes	—
99	C₄×C₄⋊Q₈	C₂×C₂₃	1	2	yes	—
100	#1971	C₂×C₂₃	1	4	yes	—
101	#1956	C₂×C₂₃	1	2	yes	—
102	#1844	C₂×C₂₃	1	1	yes	characteristic
103	C₄×#97	C₂×C₂₃	1	1	yes	characteristic
104	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
105	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
106	C₂²×C₂₃	#532	1	1	yes	characteristic
107	C₂²×C₂₃	C₂×#137	1	4	yes	—
108	C₄×C₂₃	—	2	4	no	—
109	C₄×C₂₃	—	4	8	no	—
110	C₄×C₂₃	—	4	2	no	—
111	C₄×C₂₃	—	8	4	no	—
112	C₄×C₂₃	#441	1	8	yes	complemented
113	C₄×C₂₃	C₂×C₄×D₄	1	2	yes	—
114	C₄×C₂₃	#532	1	2	yes	—
115	C₂²×Q₁₆	C₂²×C₂₃	1	4	yes	—
116	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
117	C₂×C₄×Q₈	C₂²×C₂₃	1	4	yes	—
118	#494	C₄×C₂₃	1	8	yes	complemented
119	#441	—	2	16	no	—
120	C₂×#106	C₂²×C₂₃	1	4	yes	—
121	C₂×#103	C₂²×C₂₃	1	8	yes	—
122	C₂×#97	C₂²×C₂₃	1	2	yes	—
123	C₂×#97	C₂²×C₂₃	1	2	yes	—
124	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
126	#391	C₂²×C₂₃	1	4	yes	—
127	#389	—	2	4	no	—
128	#384	C₂²×C₂₃	1	2	yes	—
129	C₄×C₄⋊C₄	—	2	4	no	—
130	C₄³	C₂²×C₂₃	1	1	yes	characteristic
131	C₂×C₂₃	C₄×#127	1	1	yes	characteristic
132	C₂×C₂₃	C₂×#441	1	2	yes	—
133	C₂×C₂₃	#3152	1	2	yes	—
134	C₂×C₂₃	#3185	1	2	yes	—
135	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
136	C₂×Q₁₆	—	2	8	no	—
137	C₂×Q₁₆	—	2	8	no	—
138	C₂×Q₁₆	C₂×C₄×C₂₃	1	16	yes	—
139	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
140	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
141	C₄⋊Q₈	—	2	4	no	—
142	C₄⋊Q₈	C₂×C₄×C₂₃	1	8	yes	—
143	C₄×Q₈	—	2	16	no	—
144	C₂×C₄⋊C₄	—	2	4	no	—
145	C₂×C₄⋊C₄	—	2	4	no	—
146	C₂×C₄⋊C₄	—	2	4	no	—
147	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
148	C₂×C₄²	—	2	4	no	—
149	C₂×C₄²	D₄×C₂₃	1	4	yes	—
150	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
151	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
152	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
153	#106	—	2	8	no	—
154	#103	—	2	16	no	—
155	#97	—	2	4	no	—
156	#97	D₄×C₂₃	1	8	yes	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
159	#95	—	2	8	no	—
160	C₂₃	#21691	1	1	yes	characteristic, direct product factor, Sylow
161	C₂×Q₈	—	2	4	no	—
162	C₂×Q₈	—	2	8	no	—
163	C₂×Q₈	—	2	8	no	—
164	C₂×Q₈	C₂²×C₄×C₂₃	1	8	yes	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
168	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
169	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
170	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
171	Q₁₆	—	2	32	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
175	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
176	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
177	C₂×C₈	#41×C₂₃	1	4	yes	—
178	C₄⋊C₄	—	2	16	no	—
179	C₄⋊C₄	—	2	8	no	—
180	C₄⋊C₄	—	4	8	no	—
181	C₄⋊C₄	—	4	8	no	—
182	C₄²	—	2	8	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	4	8	no	—
185	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
186	C₄²	C₂×D₄×C₂₃	1	4	yes	—
187	C₄²	#41×C₂₃	1	2	yes	—
188	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
189	Q₈	—	2	16	no	—
190	Q₈	—	4	8	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	8	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	8	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
200	C₂×C₄	C₂₃×#127	1	4	yes	—
201	C₂×C₄	C₂₃×#137	1	8	yes	—
202	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
203	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
204	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
205	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
206	C₈	—	2	4	no	—
207	C₈	C₄×D₄×C₂₃	1	8	yes	—
208	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
209	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
210	C₂²	C₂₃×#532	1	1	yes	characteristic
211	C₂²	C₂×C₂₃×#137	1	4	yes	—
212	C₄	—	2	4	no	—
213	C₄	—	4	2	no	—
214	C₄	—	4	8	no	—
215	C₄	—	8	4	no	—
216	C₄	C₂₃×#441	1	8	yes	complemented
217	C₄	C₂×C₄×D₄×C₂₃	1	2	yes	—
218	C₄	C₂₃×#532	1	2	yes	—
219	C₂	C₄×C₂₃×#127	1	1	yes	characteristic
220	C₂	C₂×C₂₃×#441	1	2	yes	—
221	C₂	C₂₃×#3152	1	2	yes	—
222	C₂	C₂₃×#3185	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₂₃×#494	C₄	semidirect
76. #21691	C₂₃	direct (N × H)
112. C₄×C₂₃	#441	semidirect
118. #494	C₄×C₂₃	semidirect
160. C₂₃	#21691	direct (N × H)
216. C₄	C₂₃×#441	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#494	2
C₂×C₂₃×#441	2
C₄×C₂₃×C₄⋊Q₈	2
C₂₃×#1971	2
C₂₃×#1956	2
C₂₃×#1844	2
C₄×C₂₃×#97	2
#21691	23

Subgroups — C23×#21691 #19169317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21691 #19169317