Subgroups — C₂₃×#3387 #15671006

Order 2944.

326 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈²×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#557	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#501	C₂	1	2	yes	maximal
4	C₂₃×#498	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#479	C₂	1	2	yes	maximal
6	C₂₃×#477	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#475	C₂	1	2	yes	maximal
8	C₂₃×#475	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#446	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#441	C₂	1	2	yes	maximal
11	C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×C₄⋊Q₈	—	2	1	no	—
14	C₂₃×C₄⋊Q₈	—	2	1	no	—
15	C₂₃×C₄⋊Q₈	—	2	2	no	—
16	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
18	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂₃×#125	—	2	2	no	—
20	C₂₃×#125	C₂²	1	2	yes	—
21	C₄×Q₈×C₂₃	—	2	1	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×Q₈×C₂₃	C₂²	1	2	yes	—
27	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#107	C₂²	1	1	yes	characteristic
29	C₂₃×#107	C₂²	1	2	yes	—
30	C₂₃×#106	C₂²	1	2	yes	—
31	C₂₃×#106	C₂²	1	2	yes	—
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	1	yes	characteristic
37	C₂₃×#103	C₂²	1	2	yes	—
38	C₂₃×#103	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	D₄	1	1	yes	characteristic
45	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
47	Q₁₆×C₂₃	Q₈	1	4	yes	complemented
48	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₂₃	C₂³	1	2	yes	—
50	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₂₃	—	2	2	no	—
52	C₄⋊C₄×C₂₃	—	2	2	no	—
53	C₄⋊C₄×C₂₃	—	2	1	no	—
54	C₄⋊C₄×C₂₃	—	2	1	no	—
55	C₄⋊C₄×C₂₃	—	2	2	no	—
56	C₄⋊C₄×C₂₃	—	2	2	no	—
57	C₄⋊C₄×C₂₃	—	2	1	no	—
58	C₄⋊C₄×C₂₃	—	2	1	no	—
59	C₄⋊C₄×C₂₃	—	2	2	no	—
60	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
62	C₄⋊C₄×C₂₃	C₂³	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	C₄²×C₂₃	—	2	2	no	—
67	C₄²×C₂₃	—	2	1	no	—
68	C₄²×C₂₃	—	2	2	no	—
69	C₄²×C₂₃	—	2	1	no	—
70	C₄²×C₂₃	C₂³	1	2	yes	—
71	C₄²×C₂₃	C₂³	1	1	yes	characteristic
72	Q₈×C₂₃	—	2	2	no	—
73	Q₈×C₂₃	—	2	1	no	—
74	Q₈×C₂₃	—	2	1	no	—
75	Q₈×C₂₃	—	4	2	no	—
76	Q₈×C₂₃	—	8	2	no	—
77	Q₈×C₂₃	—	8	1	no	—
78	Q₈×C₂₃	C₂×Q₈	1	2	yes	—
79	Q₈×C₂₃	C₂×Q₈	1	2	yes	—
80	C₂×C₄×C₂₃	—	2	1	no	—
81	C₂×C₄×C₂₃	—	2	1	no	—
82	C₂×C₄×C₂₃	—	2	1	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
91	C₈×C₂₃	—	2	1	no	—
92	C₈×C₂₃	—	2	2	no	—
93	C₈×C₂₃	C₂×Q₈	1	2	yes	—
94	#3387	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
96	C₄×C₂₃	—	2	2	no	—
97	C₄×C₂₃	—	2	2	no	—
98	C₄×C₂₃	—	2	2	no	—
99	C₄×C₂₃	—	2	2	no	—
100	C₄×C₂₃	—	2	1	no	—
101	C₄×C₂₃	—	4	2	no	—
102	C₄×C₂₃	—	4	1	no	—
103	C₄×C₂₃	—	4	2	no	—
104	C₄×C₂₃	—	4	1	no	—
105	C₄×C₂₃	#137	1	2	yes	—
106	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
107	C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
108	Q₈²	C₂×C₂₃	1	1	yes	characteristic
109	#557	C₂×C₂₃	1	1	yes	characteristic
110	#501	C₂×C₂₃	1	2	yes	—
111	#498	C₂×C₂₃	1	1	yes	characteristic
112	#479	C₂×C₂₃	1	2	yes	—
113	#477	C₂×C₂₃	1	1	yes	characteristic
114	#475	C₂×C₂₃	1	2	yes	—
115	#475	C₂×C₂₃	1	1	yes	characteristic
116	#446	C₂×C₂₃	1	1	yes	characteristic
117	#441	C₂×C₂₃	1	2	yes	—
118	C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic
119	C₂×C₂₃	D₄×Q₈	1	1	yes	characteristic
120	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
121	C₂×C₂₃	#577	1	1	yes	characteristic
122	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
123	C₄⋊Q₈	—	2	1	no	—
124	C₄⋊Q₈	—	2	1	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
127	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
128	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
129	#125	—	2	2	no	—
130	#125	C₂²×C₂₃	1	2	yes	—
131	C₄×Q₈	—	2	1	no	—
132	C₄×Q₈	—	2	1	no	—
133	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
134	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
135	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
136	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
137	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
138	#107	C₂²×C₂₃	1	1	yes	characteristic
139	#107	C₂²×C₂₃	1	2	yes	—
140	#106	C₂²×C₂₃	1	2	yes	—
141	#106	C₂²×C₂₃	1	2	yes	—
142	#106	C₂²×C₂₃	1	2	yes	—
143	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
144	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
145	#103	C₂²×C₂₃	1	1	yes	characteristic
146	#103	C₂²×C₂₃	1	1	yes	characteristic
147	#103	C₂²×C₂₃	1	2	yes	—
148	#103	C₂²×C₂₃	1	2	yes	—
149	#97	C₂²×C₂₃	1	2	yes	—
150	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
151	C₂₃	#3387	1	1	yes	characteristic, direct product factor, Sylow
152	C₂×Q₈	—	2	1	no	—
153	C₂×Q₈	—	2	2	no	—
154	C₂×Q₈	—	2	1	no	—
155	C₂×Q₈	D₄×C₂₃	1	1	yes	characteristic
156	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
157	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
158	Q₁₆	Q₈×C₂₃	1	4	yes	complemented
159	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
160	C₂×C₈	C₂³×C₂₃	1	2	yes	—
161	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
172	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
173	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
174	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
175	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
176	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
177	C₄²	—	2	2	no	—
178	C₄²	—	2	1	no	—
179	C₄²	—	2	2	no	—
180	C₄²	—	2	1	no	—
181	C₄²	C₂³×C₂₃	1	1	yes	characteristic
182	C₄²	C₂³×C₂₃	1	2	yes	—
183	Q₈	—	2	2	no	—
184	Q₈	—	2	1	no	—
185	Q₈	—	2	1	no	—
186	Q₈	—	4	2	no	—
187	Q₈	—	8	2	no	—
188	Q₈	—	8	1	no	—
189	Q₈	C₂×Q₈×C₂₃	1	2	yes	—
190	Q₈	C₂×Q₈×C₂₃	1	2	yes	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
198	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
199	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
200	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
201	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
202	C₈	—	2	1	no	—
203	C₈	—	2	2	no	—
204	C₈	C₂×Q₈×C₂₃	1	2	yes	—
205	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	—	2	1	no	—
211	C₄	—	4	2	no	—
212	C₄	—	4	2	no	—
213	C₄	—	4	1	no	—
214	C₄	—	4	1	no	—
215	C₄	C₂₃×#137	1	2	yes	—
216	C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
217	C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
218	C₂	D₄×Q₈×C₂₃	1	1	yes	characteristic
219	C₂	C₂×C₂₃×#137	1	1	yes	characteristic
220	C₂	C₂₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×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
47. Q₁₆×C₂₃	Q₈	semidirect
94. #3387	C₂₃	direct (N × H)
151. C₂₃	#3387	direct (N × H)
158. Q₁₆	Q₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
Q₈²×C₂₃	2
C₂₃×#557	2
C₂₃×#501	2
C₂₃×#498	2
C₂₃×#479	2
C₂₃×#477	2
C₂₃×#475	2
C₂₃×#446	2
C₂₃×#441	2
C₄×Q₁₆×C₂₃	2
#3387	23

Subgroups — C23×#3387 #15671006

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3387 #15671006