Subgroups — C₂₃×#1854 #15669473

Order 2944.

316 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#380	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#340	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#340	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#117	—	2	2	no	—
12	C₂₃×#117	—	2	2	no	—
13	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
18	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
19	C₂×C₄²×C₂₃	—	2	1	no	—
20	C₂₃×#107	—	2	2	no	—
21	C₂₃×#106	—	2	2	no	—
22	C₂₃×C₂²⋊C₈	C₄	1	4	yes	complemented
23	C₂₃×#95	—	2	1	no	—
24	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₃	—	2	1	no	—
26	C₂²×C₄×C₂₃	—	2	1	no	—
27	C₂²×C₄×C₂₃	—	2	1	no	—
28	C₂²×C₄×C₂₃	—	2	1	no	—
29	C₂²×C₄×C₂₃	—	2	1	no	—
30	C₂²×C₄×C₂₃	—	2	1	no	—
31	C₂²×C₄×C₂₃	—	2	1	no	—
32	C₂²×C₄×C₂₃	—	2	1	no	—
33	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	Q₈	1	2	yes	—
36	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₂₃	—	2	1	no	—
38	C₂×C₈×C₂₃	—	2	1	no	—
39	C₂×C₈×C₂₃	—	2	1	no	—
40	C₂×C₈×C₂₃	—	2	1	no	—
41	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
42	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
43	C₄⋊C₄×C₂₃	—	2	1	no	—
44	C₄⋊C₄×C₂₃	—	2	1	no	—
45	C₄⋊C₄×C₂₃	—	2	1	no	—
46	C₄⋊C₄×C₂₃	—	2	1	no	—
47	C₄⋊C₄×C₂₃	—	2	1	no	—
48	C₄⋊C₄×C₂₃	—	2	1	no	—
49	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
50	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
51	C₂²⋊C₄×C₂₃	—	2	4	no	—
52	C₂²⋊C₄×C₂₃	—	4	2	no	—
53	C₄²×C₂₃	—	2	4	no	—
54	C₂³×C₂₃	—	2	1	no	—
55	C₂³×C₂₃	—	2	1	no	—
56	C₂³×C₂₃	—	2	1	no	—
57	C₂³×C₂₃	—	2	1	no	—
58	C₂³×C₂₃	—	2	1	no	—
59	C₂³×C₂₃	—	2	1	no	—
60	C₂³×C₂₃	C₄⋊C₄	1	2	yes	—
61	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₂₃	—	2	1	no	—
63	C₂×C₄×C₂₃	—	2	1	no	—
64	C₂×C₄×C₂₃	—	2	2	no	—
65	C₂×C₄×C₂₃	—	2	2	no	—
66	C₂×C₄×C₂₃	—	2	2	no	—
67	C₂×C₄×C₂₃	—	2	2	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	4	1	no	—
71	C₂×C₄×C₂₃	—	4	1	no	—
72	C₂×C₄×C₂₃	—	4	1	no	—
73	C₂×C₄×C₂₃	—	4	1	no	—
74	C₂×C₄×C₂₃	—	4	1	no	—
75	C₂×C₄×C₂₃	—	4	1	no	—
76	C₂×C₄×C₂₃	—	4	1	no	—
77	C₂×C₄×C₂₃	—	4	1	no	—
78	C₂×C₄×C₂₃	—	4	1	no	—
79	C₂×C₄×C₂₃	—	4	1	no	—
80	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
81	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
86	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
87	C₈×C₂₃	—	2	2	no	—
88	C₈×C₂₃	—	2	2	no	—
89	#1854	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂²×C₂₃	—	2	2	no	—
91	C₂²×C₂₃	—	2	2	no	—
92	C₂²×C₂₃	—	2	2	no	—
93	C₂²×C₂₃	—	4	1	no	—
94	C₂²×C₂₃	—	4	1	no	—
95	C₂²×C₂₃	—	4	1	no	—
96	C₂²×C₂₃	—	4	1	no	—
97	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₂₃	#120	1	1	yes	characteristic
99	C₂²×C₂₃	#123	1	1	yes	characteristic
100	C₂²×C₂₃	#135	1	1	yes	characteristic
101	C₂²×C₂₃	#135	1	1	yes	characteristic
102	C₂²×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂²×C₂₃	#137	1	1	yes	characteristic
104	C₄×C₂₃	—	2	2	no	—
105	C₄×C₂₃	—	4	2	no	—
106	C₄×C₂₃	—	4	2	no	—
107	C₄×C₂₃	—	8	1	no	—
108	C₄×C₂₃	—	8	1	no	—
109	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
110	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
111	C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
112	C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×#117	C₂×C₂₃	1	1	yes	characteristic
114	C₂×#107	C₂×C₂₃	1	1	yes	characteristic, complemented
115	C₂×#106	C₂×C₂₃	1	1	yes	characteristic, complemented
116	C₂×C₂²⋊C₈	C₂×C₂₃	1	1	yes	characteristic
117	#380	C₂×C₂₃	1	1	yes	characteristic
118	#340	C₂×C₂₃	1	1	yes	characteristic, complemented
119	#340	C₂×C₂₃	1	1	yes	characteristic, complemented
120	C₂×C₂₃	—	4	2	no	—
121	C₂×C₂₃	#385	1	1	yes	characteristic
122	C₂×C₂₃	#427	1	1	yes	characteristic
123	C₂×C₂₃	#428	1	1	yes	characteristic
124	C₂×C₂₃	#449	1	1	yes	characteristic
125	C₂×C₂₃	#452	1	1	yes	characteristic
126	C₂×C₂₃	#482	1	1	yes	characteristic
127	C₂×C₂₃	#485	1	1	yes	characteristic
128	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
129	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
130	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
131	#117	—	2	2	no	—
132	#117	—	2	2	no	—
133	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
134	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
135	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
136	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
137	C₂×C₂²⋊C₄	—	2	1	no	—
138	C₂×C₂²⋊C₄	—	2	1	no	—
139	C₂×C₄²	—	2	1	no	—
140	#107	—	2	2	no	—
141	#106	—	2	2	no	—
142	C₂²⋊C₈	C₄×C₂₃	1	4	yes	complemented
143	#95	—	2	1	no	—
144	C₂₃	#1854	1	1	yes	characteristic, direct product factor, Sylow
145	C₂⁴	D₄×C₂₃	1	1	yes	characteristic
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	—	2	1	no	—
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	—	2	1	no	—
154	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
155	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
156	C₂²×C₄	Q₈×C₂₃	1	2	yes	—
157	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	C₂×C₄×C₂₃	1	2	yes	—
163	C₂×C₈	C₂×C₄×C₂₃	1	2	yes	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
171	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
172	C₂²⋊C₄	—	2	4	no	—
173	C₂²⋊C₄	—	4	2	no	—
174	C₄²	—	2	4	no	—
175	C₂³	—	2	1	no	—
176	C₂³	—	2	1	no	—
177	C₂³	—	2	1	no	—
178	C₂³	—	2	1	no	—
179	C₂³	—	2	1	no	—
180	C₂³	—	2	1	no	—
181	C₂³	C₄⋊C₄×C₂₃	1	2	yes	—
182	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
202	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
203	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
204	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
205	C₂×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
206	C₂×C₄	#41×C₂₃	1	1	yes	characteristic
207	C₂×C₄	#41×C₂₃	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	1	no	—
214	C₂²	—	4	1	no	—
215	C₂²	—	4	1	no	—
216	C₂²	—	4	1	no	—
217	C₂²	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
218	C₂²	C₂₃×#120	1	1	yes	characteristic
219	C₂²	C₂₃×#123	1	1	yes	characteristic
220	C₂²	C₂₃×#135	1	1	yes	characteristic
221	C₂²	C₂₃×#135	1	1	yes	characteristic
222	C₂²	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
223	C₂²	C₂₃×#137	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	2	no	—
227	C₄	—	8	1	no	—
228	C₄	—	8	1	no	—
229	C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
230	C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
231	C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
232	C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
233	C₂	—	4	2	no	—
234	C₂	C₂₃×#385	1	1	yes	characteristic
235	C₂	C₂₃×#427	1	1	yes	characteristic
236	C₂	C₂₃×#428	1	1	yes	characteristic
237	C₂	C₂₃×#449	1	1	yes	characteristic
238	C₂	C₂₃×#452	1	1	yes	characteristic
239	C₂	C₂₃×#482	1	1	yes	characteristic
240	C₂	C₂₃×#485	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#107	C₂	semidirect
3. C₂×C₂₃×#106	C₂	semidirect
6. C₂₃×#340	C₂	semidirect
7. C₂₃×#340	C₂	semidirect
22. C₂₃×C₂²⋊C₈	C₄	semidirect
89. #1854	C₂₃	direct (N × H)
114. C₂×#107	C₂×C₂₃	semidirect
115. C₂×#106	C₂×C₂₃	semidirect
118. #340	C₂×C₂₃	semidirect
119. #340	C₂×C₂₃	semidirect
142. C₂²⋊C₈	C₄×C₂₃	semidirect
144. C₂₃	#1854	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×#117	2
C₂×C₂₃×#107	2
C₂×C₂₃×#106	2
C₂×C₂₃×C₂²⋊C₈	2
C₂₃×#380	2
C₂₃×#340	2
#1854	23

Subgroups — C23×#1854 #15669473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#1854 #15669473