Subgroups — C₂₃×#1862 #15669481

Order 2944.

362 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#102	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#102	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#340	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#336	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂³×C₄×C₂₃	—	2	1	no	—
10	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₄×D₄×C₂₃	—	2	2	no	—
13	C₄×D₄×C₂₃	—	2	2	no	—
14	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
19	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#107	—	2	2	no	—
21	C₂₃×#102	—	2	2	no	—
22	C₂₃×#102	C₄	1	4	yes	complemented
23	C₂₃×#95	—	2	1	no	—
24	C₂⁴×C₂₃	—	2	1	no	—
25	C₂×D₄×C₂₃	—	2	1	no	—
26	C₂×D₄×C₂₃	—	2	1	no	—
27	C₂×D₄×C₂₃	—	2	1	no	—
28	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
29	C₂×D₄×C₂₃	D₄	1	2	yes	—
30	C₂×D₄×C₂₃	Q₈	1	2	yes	—
31	C₂²×C₄×C₂₃	—	2	4	no	—
32	C₂²×C₄×C₂₃	—	2	1	no	—
33	C₂²×C₄×C₂₃	—	2	1	no	—
34	C₂²×C₄×C₂₃	—	2	1	no	—
35	C₂²×C₄×C₂₃	—	4	2	no	—
36	C₂²×C₄×C₂₃	—	4	2	no	—
37	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₂₃	—	2	1	no	—
41	C₂×C₈×C₂₃	—	2	1	no	—
42	C₂×C₈×C₂₃	—	2	1	no	—
43	C₂×C₈×C₂₃	—	2	1	no	—
44	C₂×C₈×C₂₃	—	2	1	no	—
45	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	complemented
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	—	2	1	no	—
50	C₄⋊C₄×C₂₃	—	4	2	no	—
51	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	complemented
52	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
53	C₂²⋊C₄×C₂₃	—	2	4	no	—
54	C₄²×C₂₃	—	2	1	no	—
55	C₄²×C₂₃	—	2	1	no	—
56	C₂³×C₂₃	—	2	2	no	—
57	C₂³×C₂₃	—	2	2	no	—
58	C₂³×C₂₃	—	2	2	no	—
59	C₂³×C₂₃	—	4	1	no	—
60	C₂³×C₂₃	—	4	1	no	—
61	C₂³×C₂₃	—	4	1	no	—
62	C₂³×C₂₃	—	4	1	no	—
63	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
64	D₄×C₂₃	—	2	2	no	—
65	D₄×C₂₃	—	2	2	no	—
66	D₄×C₂₃	—	2	2	no	—
67	D₄×C₂₃	C₄⋊C₄	1	4	yes	complemented
68	C₂×C₄×C₂₃	—	2	1	no	—
69	C₂×C₄×C₂₃	—	2	1	no	—
70	C₂×C₄×C₂₃	—	2	2	no	—
71	C₂×C₄×C₂₃	—	2	1	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	1	no	—
74	C₂×C₄×C₂₃	—	2	1	no	—
75	C₂×C₄×C₂₃	—	4	8	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	—	4	1	no	—
81	C₂×C₄×C₂₃	—	4	1	no	—
82	C₂×C₄×C₂₃	—	4	1	no	—
83	C₂×C₄×C₂₃	D₈	1	2	yes	—
84	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
89	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
90	C₈×C₂₃	—	2	2	no	—
91	C₈×C₂₃	—	4	1	no	—
92	#1862	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂²×C₂₃	—	2	4	no	—
94	C₂²×C₂₃	—	4	2	no	—
95	C₂²×C₂₃	—	4	2	no	—
96	C₂²×C₂₃	—	4	2	no	—
97	C₂²×C₂₃	—	4	2	no	—
98	C₂²×C₂₃	—	4	2	no	—
99	C₂²×C₂₃	—	4	2	no	—
100	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
101	C₂²×C₂₃	#120	1	1	yes	characteristic
102	C₂²×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂²×C₂₃	C₂×D₈	1	1	yes	characteristic
104	C₂²×C₂₃	#135	1	1	yes	characteristic
105	C₂²×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
106	C₂²×C₂₃	#137	1	1	yes	characteristic
107	C₄×C₂₃	—	2	2	no	—
108	C₄×C₂₃	—	4	2	no	—
109	C₄×C₂₃	—	4	2	no	—
110	C₄×C₂₃	—	4	1	no	—
111	C₄×C₂₃	—	8	1	no	—
112	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
114	C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
115	C₄×C₂₃	#123	1	1	yes	characteristic
116	C₂×C₄×D₄	C₂×C₂₃	1	1	yes	characteristic
117	C₂×#107	C₂×C₂₃	1	1	yes	characteristic, complemented
118	C₂×#102	C₂×C₂₃	1	1	yes	characteristic
119	C₂×#102	C₂×C₂₃	1	1	yes	characteristic
120	#389	C₂×C₂₃	1	1	yes	characteristic, complemented
121	#340	C₂×C₂₃	1	1	yes	characteristic, complemented
122	#336	C₂×C₂₃	1	1	yes	characteristic, complemented
123	C₂×C₂₃	—	4	4	no	—
124	C₂×C₂₃	#385	1	1	yes	characteristic
125	C₂×C₂₃	C₄×D₈	1	1	yes	characteristic
126	C₂×C₂₃	#440	1	1	yes	characteristic
127	C₂×C₂₃	#447	1	1	yes	characteristic
128	C₂×C₂₃	#452	1	1	yes	characteristic
129	C₂×C₂₃	#474	1	1	yes	characteristic
130	C₂×C₂₃	#478	1	1	yes	characteristic
131	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
132	C₂³×C₄	—	2	1	no	—
133	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
134	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
135	C₄×D₄	—	2	2	no	—
136	C₄×D₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	1	no	—
138	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
139	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
140	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
141	C₂×C₂²⋊C₄	—	2	1	no	—
142	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
143	#107	—	2	2	no	—
144	#102	—	2	2	no	—
145	#102	C₄×C₂₃	1	4	yes	complemented
146	#95	—	2	1	no	—
147	C₂₃	#1862	1	1	yes	characteristic, direct product factor, Sylow
148	C₂⁴	—	2	1	no	—
149	C₂×D₄	—	2	1	no	—
150	C₂×D₄	—	2	1	no	—
151	C₂×D₄	—	2	1	no	—
152	C₂×D₄	C₂×C₄×C₂₃	1	2	yes	—
153	C₂×D₄	D₄×C₂₃	1	2	yes	—
154	C₂×D₄	Q₈×C₂₃	1	2	yes	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	—	4	2	no	—
161	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
162	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	C₂×C₄×C₂₃	1	2	yes	complemented
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	complemented
176	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
177	C₂²⋊C₄	—	2	4	no	—
178	C₄²	—	2	1	no	—
179	C₄²	—	2	1	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	4	1	no	—
184	C₂³	—	4	1	no	—
185	C₂³	—	4	1	no	—
186	C₂³	—	4	1	no	—
187	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
188	D₄	—	2	2	no	—
189	D₄	—	2	2	no	—
190	D₄	—	2	2	no	—
191	D₄	C₄⋊C₄×C₂₃	1	4	yes	complemented
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	8	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	D₈×C₂₃	1	2	yes	—
208	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
209	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
210	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
211	C₂×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
212	C₂×C₄	#41×C₂₃	1	1	yes	characteristic
213	C₂×C₄	#41×C₂₃	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	4	1	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	C₄×D₄×C₂₃	1	1	yes	characteristic
224	C₂²	C₂₃×#120	1	1	yes	characteristic
225	C₂²	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
226	C₂²	C₂×D₈×C₂₃	1	1	yes	characteristic
227	C₂²	C₂₃×#135	1	1	yes	characteristic
228	C₂²	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
229	C₂²	C₂₃×#137	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	2	no	—
233	C₄	—	4	1	no	—
234	C₄	—	8	1	no	—
235	C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
236	C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
237	C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
238	C₄	C₂₃×#123	1	1	yes	characteristic
239	C₂	—	4	4	no	—
240	C₂	C₂₃×#385	1	1	yes	characteristic
241	C₂	C₄×D₈×C₂₃	1	1	yes	characteristic
242	C₂	C₂₃×#440	1	1	yes	characteristic
243	C₂	C₂₃×#447	1	1	yes	characteristic
244	C₂	C₂₃×#452	1	1	yes	characteristic
245	C₂	C₂₃×#474	1	1	yes	characteristic
246	C₂	C₂₃×#478	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	C₂²×D₄×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#107	C₂	semidirect
5. C₂₃×#389	C₂	semidirect
6. C₂₃×#340	C₂	semidirect
7. C₂₃×#336	C₂	semidirect
22. C₂₃×#102	C₄	semidirect
45. C₂×C₈×C₂₃	C₂×C₄	semidirect
51. C₄⋊C₄×C₂₃	C₂×C₄	semidirect
67. D₄×C₂₃	C₄⋊C₄	semidirect
92. #1862	C₂₃	direct (N × H)
117. C₂×#107	C₂×C₂₃	semidirect
120. #389	C₂×C₂₃	semidirect
121. #340	C₂×C₂₃	semidirect
122. #336	C₂×C₂₃	semidirect
145. #102	C₄×C₂₃	semidirect
147. C₂₃	#1862	direct (N × H)
169. C₂×C₈	C₂×C₄×C₂₃	semidirect
175. C₄⋊C₄	C₂×C₄×C₂₃	semidirect
191. D₄	C₄⋊C₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₂₃	2
C₂×C₂₃×#107	2
C₂×C₂₃×#102	2
C₂₃×#389	2
C₂₃×#340	2
C₂₃×#336	2
#1862	23

Subgroups — C23×#1862 #15669481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#1862 #15669481