Subgroups — C₂₃×#3386 #15671005

Order 2944.

360 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×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#501	C₂	1	2	yes	maximal, complemented
4	C₂₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#478	C₂	1	2	yes	maximal
6	C₂₃×#477	C₂	1	2	yes	maximal, complemented
7	C₂₃×#475	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#474	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#440	C₂	1	2	yes	maximal
11	C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₂₃	—	2	1	no	—
13	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×C₄⋊Q₈	—	2	1	no	—
15	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂₃×#125	—	2	2	no	—
19	C₂₃×#125	C₂²	1	2	yes	—
20	C₂₃×C₂²⋊Q₈	—	2	1	no	—
21	C₂₃×C₂²⋊Q₈	—	2	2	no	—
22	C₄×Q₈×C₂₃	—	2	1	no	—
23	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
25	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₂₃	C₂²	1	2	yes	—
28	C₂₃×#107	C₂²	1	2	yes	—
29	C₂₃×#107	C₂²	1	2	yes	—
30	C₂₃×#107	C₂²	1	2	yes	—
31	C₂₃×#106	C₂²	1	1	yes	characteristic
32	C₂₃×#106	C₂²	1	2	yes	—
33	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
35	C₂₃×#103	C₂²	1	1	yes	characteristic
36	C₂₃×#103	C₂²	1	2	yes	—
37	C₂₃×#102	C₂²	1	1	yes	characteristic
38	C₂₃×#102	C₂²	1	2	yes	—
39	C₂₃×#97	C₂²	1	2	yes	—
40	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₂₃	—	2	1	no	—
42	C₂×Q₈×C₂₃	—	2	2	no	—
43	C₂×Q₈×C₂₃	—	2	1	no	—
44	C₂×Q₈×C₂₃	—	4	4	no	—
45	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
46	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	—	2	2	no	—
49	C₂²×C₄×C₂₃	—	2	1	no	—
50	SD₁₆×C₂₃	Q₈	1	4	yes	complemented
51	C₂×C₈×C₂₃	C₂³	1	2	yes	—
52	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₂₃	—	2	2	no	—
55	C₄⋊C₄×C₂₃	—	2	1	no	—
56	C₄⋊C₄×C₂₃	—	2	2	no	—
57	C₄⋊C₄×C₂₃	—	2	2	no	—
58	C₄⋊C₄×C₂₃	—	2	1	no	—
59	C₄⋊C₄×C₂₃	—	2	1	no	—
60	C₄⋊C₄×C₂₃	—	2	2	no	—
61	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
63	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
65	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
67	C₂²⋊C₄×C₂₃	—	2	2	no	—
68	C₂²⋊C₄×C₂₃	—	2	1	no	—
69	C₄²×C₂₃	—	2	2	no	—
70	C₄²×C₂₃	—	2	1	no	—
71	C₄²×C₂₃	C₂³	1	2	yes	—
72	C₄²×C₂₃	C₂³	1	1	yes	characteristic
73	C₂³×C₂₃	—	2	1	no	—
74	Q₈×C₂₃	—	2	2	no	—
75	Q₈×C₂₃	—	2	1	no	—
76	Q₈×C₂₃	—	4	4	no	—
77	Q₈×C₂₃	—	4	2	no	—
78	Q₈×C₂₃	—	4	2	no	—
79	Q₈×C₂₃	C₂×Q₈	1	2	yes	complemented
80	D₄×C₂₃	—	2	1	no	—
81	D₄×C₂₃	C₂×Q₈	1	2	yes	—
82	C₂×C₄×C₂₃	—	2	1	no	—
83	C₂×C₄×C₂₃	—	2	1	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	2	1	no	—
87	C₂×C₄×C₂₃	—	4	4	no	—
88	C₂×C₄×C₂₃	—	4	2	no	—
89	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
94	C₈×C₂₃	—	2	2	no	—
95	C₈×C₂₃	—	2	1	no	—
96	C₈×C₂₃	C₂×Q₈	1	2	yes	complemented
97	#3386	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂²×C₂₃	—	2	2	no	—
99	C₂²×C₂₃	—	4	1	no	—
100	C₂²×C₂₃	—	4	1	no	—
101	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
102	C₄×C₂₃	—	2	2	no	—
103	C₄×C₂₃	—	2	2	no	—
104	C₄×C₂₃	—	2	2	no	—
105	C₄×C₂₃	—	2	1	no	—
106	C₄×C₂₃	—	4	2	no	—
107	C₄×C₂₃	—	4	1	no	—
108	C₄×C₂₃	—	4	2	no	—
109	C₄×C₂₃	—	4	1	no	—
110	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
111	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
112	C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
113	#557	C₂×C₂₃	1	1	yes	characteristic, complemented
114	D₄×Q₈	C₂×C₂₃	1	1	yes	characteristic
115	#501	C₂×C₂₃	1	2	yes	complemented
116	#500	C₂×C₂₃	1	1	yes	characteristic, complemented
117	#478	C₂×C₂₃	1	2	yes	—
118	#477	C₂×C₂₃	1	2	yes	complemented
119	#475	C₂×C₂₃	1	1	yes	characteristic, complemented
120	#474	C₂×C₂₃	1	1	yes	characteristic
121	#446	C₂×C₂₃	1	1	yes	characteristic, complemented
122	#440	C₂×C₂₃	1	2	yes	—
123	C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic
124	C₂×C₂₃	—	4	2	no	—
125	C₂×C₂₃	D₄×Q₈	1	1	yes	characteristic
126	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₂₃	#578	1	1	yes	characteristic
128	C₂²×Q₈	—	2	1	no	—
129	C₂×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
130	C₄⋊Q₈	—	2	1	no	—
131	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
132	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
133	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
134	#125	—	2	2	no	—
135	#125	C₂²×C₂₃	1	2	yes	—
136	C₂²⋊Q₈	—	2	1	no	—
137	C₂²⋊Q₈	—	2	2	no	—
138	C₄×Q₈	—	2	1	no	—
139	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
140	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
141	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
142	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
143	C₄×D₄	C₂²×C₂₃	1	2	yes	—
144	#107	C₂²×C₂₃	1	2	yes	—
145	#107	C₂²×C₂₃	1	2	yes	—
146	#107	C₂²×C₂₃	1	2	yes	—
147	#106	C₂²×C₂₃	1	1	yes	characteristic
148	#106	C₂²×C₂₃	1	2	yes	—
149	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
150	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
151	#103	C₂²×C₂₃	1	1	yes	characteristic
152	#103	C₂²×C₂₃	1	2	yes	—
153	#102	C₂²×C₂₃	1	1	yes	characteristic
154	#102	C₂²×C₂₃	1	2	yes	—
155	#97	C₂²×C₂₃	1	2	yes	—
156	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
157	C₂₃	#3386	1	1	yes	characteristic, direct product factor, Sylow
158	C₂×Q₈	—	2	1	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	2	1	no	—
161	C₂×Q₈	—	4	4	no	—
162	C₂×Q₈	D₄×C₂₃	1	1	yes	characteristic
163	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
164	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	1	no	—
167	SD₁₆	Q₈×C₂₃	1	4	yes	complemented
168	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
169	C₂×C₈	C₂³×C₂₃	1	2	yes	—
170	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
179	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
180	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
181	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
182	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
183	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	1	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	2	1	no	—
188	C₄²	C₂³×C₂₃	1	1	yes	characteristic
189	C₄²	C₂³×C₂₃	1	2	yes	—
190	C₂³	—	2	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	2	1	no	—
193	Q₈	—	4	4	no	—
194	Q₈	—	4	2	no	—
195	Q₈	—	4	2	no	—
196	Q₈	C₂×Q₈×C₂₃	1	2	yes	complemented
197	D₄	—	2	1	no	—
198	D₄	C₂×Q₈×C₂₃	1	2	yes	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
207	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
208	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
209	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
210	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
211	C₈	—	2	1	no	—
212	C₈	—	2	2	no	—
213	C₈	C₂×Q₈×C₂₃	1	2	yes	complemented
214	C₂²	—	2	2	no	—
215	C₂²	—	4	1	no	—
216	C₂²	—	4	1	no	—
217	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	2	1	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	1	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	1	no	—
226	C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
227	C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
228	C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
229	C₂	—	4	2	no	—
230	C₂	D₄×Q₈×C₂₃	1	1	yes	characteristic
231	C₂	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
232	C₂	C₂₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂×D₄×C₂₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#557	C₂	semidirect
3. C₂₃×#501	C₂	semidirect
4. C₂₃×#500	C₂	semidirect
6. C₂₃×#477	C₂	semidirect
7. C₂₃×#475	C₂	semidirect
9. C₂₃×#446	C₂	semidirect
50. SD₁₆×C₂₃	Q₈	semidirect
79. Q₈×C₂₃	C₂×Q₈	semidirect
96. C₈×C₂₃	C₂×Q₈	semidirect
97. #3386	C₂₃	direct (N × H)
113. #557	C₂×C₂₃	semidirect
115. #501	C₂×C₂₃	semidirect
116. #500	C₂×C₂₃	semidirect
118. #477	C₂×C₂₃	semidirect
119. #475	C₂×C₂₃	semidirect
121. #446	C₂×C₂₃	semidirect
157. C₂₃	#3386	direct (N × H)
167. SD₁₆	Q₈×C₂₃	semidirect
196. Q₈	C₂×Q₈×C₂₃	semidirect
213. C₈	C₂×Q₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#557	2
D₄×Q₈×C₂₃	2
C₂₃×#501	2
C₂₃×#500	2
C₂₃×#478	2
C₂₃×#477	2
C₂₃×#475	2
C₂₃×#474	2
C₂₃×#446	2
C₂₃×#440	2
C₄×SD₁₆×C₂₃	2
#3386	23

Subgroups — C23×#3386 #15671005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3386 #15671005