Subgroups — C₂₃×#2326 #15669945

Order 2944.

612 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×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#390	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#388	C₂	1	4	yes	maximal, complemented
7	C₂₃×#385	C₂	1	4	yes	maximal, complemented
8	C₂₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
13	C₂₃×#124	C₄	1	8	yes	complemented
14	C₄×Q₈×C₂₃	—	2	4	no	—
15	C₄×D₄×C₂₃	—	2	4	no	—
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
22	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
23	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
25	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#95	C₂²	1	2	yes	—
28	C₂₃×#95	C₂²	1	2	yes	—
29	C₂₃×#95	C₂²	1	4	yes	—
30	C₂⁴×C₂₃	C₂³	1	2	yes	—
31	C₂×Q₈×C₂₃	—	2	2	no	—
32	C₂×Q₈×C₂₃	—	2	2	no	—
33	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	complemented
34	C₂×D₄×C₂₃	—	2	2	no	—
35	C₂×D₄×C₂₃	—	2	2	no	—
36	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	—
37	C₂²×C₄×C₂₃	—	2	4	no	—
38	C₂²×C₄×C₂₃	—	4	4	no	—
39	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₂₃	—	2	2	no	—
48	C₄⋊C₄×C₂₃	—	2	4	no	—
49	C₄⋊C₄×C₂₃	—	2	2	no	—
50	C₄⋊C₄×C₂₃	—	4	4	no	—
51	C₂²⋊C₄×C₂₃	—	2	4	no	—
52	C₂²⋊C₄×C₂₃	—	4	4	no	—
53	C₂²⋊C₄×C₂₃	C₂×C₄	1	16	yes	complemented
54	C₄²×C₂₃	—	2	4	no	—
55	C₄²×C₂₃	—	2	2	no	—
56	C₄²×C₂₃	C₂×C₄	1	4	yes	complemented
57	C₄²×C₂₃	D₄	1	4	yes	—
58	C₂³×C₂₃	—	2	2	no	—
59	C₂³×C₂₃	—	2	2	no	—
60	C₂³×C₂₃	—	2	4	no	—
61	C₂³×C₂₃	—	2	4	no	—
62	C₂³×C₂₃	C₂²×C₄	1	4	yes	—
63	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
64	Q₈×C₂₃	—	2	8	no	—
65	D₄×C₂₃	—	2	8	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	—	2	2	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	4	no	—
77	C₂×C₄×C₂₃	—	2	4	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	4	8	no	—
80	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
81	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₂₃	#41	1	2	yes	—
87	#2326	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂²×C₂₃	—	2	4	no	—
89	C₂²×C₂₃	—	2	8	no	—
90	C₂²×C₂₃	—	4	2	no	—
91	C₂²×C₂₃	—	4	2	no	—
92	C₂²×C₂₃	—	4	4	no	—
93	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
94	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
95	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
96	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
97	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
98	C₄×C₂₃	—	2	4	no	—
99	C₄×C₂₃	—	2	4	no	—
100	C₄×C₂₃	—	2	2	no	—
101	C₄×C₂₃	—	2	2	no	—
102	C₄×C₂₃	—	4	2	no	—
103	C₄×C₂₃	—	4	2	no	—
104	C₄×C₂₃	—	4	2	no	—
105	C₄×C₂₃	C₄×D₄	1	4	yes	—
106	C₂×#124	C₂×C₂₃	1	1	yes	characteristic
107	C₂×C₄×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
108	C₂×C₄×D₄	C₂×C₂₃	1	1	yes	characteristic
109	#391	C₂×C₂₃	1	1	yes	characteristic, complemented
110	#390	C₂×C₂₃	1	1	yes	characteristic
111	#388	C₂×C₂₃	1	4	yes	complemented
112	#385	C₂×C₂₃	1	4	yes	complemented
113	#384	C₂×C₂₃	1	1	yes	characteristic, complemented
114	#376	C₂×C₂₃	1	1	yes	characteristic, complemented
115	C₂×C₂₃	—	4	4	no	—
116	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
117	C₂×C₂₃	#520	1	2	yes	—
118	C₂×C₂₃	#535	1	1	yes	characteristic
119	C₂×C₂₃	#536	1	1	yes	characteristic
120	C₂×C₂₃	#542	1	2	yes	—
121	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
122	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
123	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
124	#124	C₄×C₂₃	1	8	yes	complemented
125	C₄×Q₈	—	2	4	no	—
126	C₄×D₄	—	2	4	no	—
127	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
128	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
129	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
130	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
131	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
132	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
133	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
134	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
135	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
136	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
137	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
138	#95	C₂²×C₂₃	1	2	yes	—
139	#95	C₂²×C₂₃	1	2	yes	—
140	#95	C₂²×C₂₃	1	4	yes	—
141	C₂₃	#2326	1	1	yes	characteristic, direct product factor, Sylow
142	C₂⁴	C₂³×C₂₃	1	2	yes	—
143	C₂×Q₈	—	2	2	no	—
144	C₂×Q₈	—	2	2	no	—
145	C₂×Q₈	C₂×C₄×C₂₃	1	4	yes	complemented
146	C₂×D₄	—	2	2	no	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	C₂×C₄×C₂₃	1	4	yes	—
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	—	4	4	no	—
151	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
152	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
153	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
154	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
155	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
156	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
157	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	4	4	no	—
163	C₂²⋊C₄	—	2	4	no	—
164	C₂²⋊C₄	—	4	4	no	—
165	C₂²⋊C₄	C₂×C₄×C₂₃	1	16	yes	complemented
166	C₄²	—	2	4	no	—
167	C₄²	—	2	2	no	—
168	C₄²	C₂×C₄×C₂₃	1	4	yes	complemented
169	C₄²	D₄×C₂₃	1	4	yes	—
170	C₂³	—	2	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	2	4	no	—
173	C₂³	—	2	4	no	—
174	C₂³	C₂²×C₄×C₂₃	1	4	yes	—
175	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
176	Q₈	—	2	8	no	—
177	D₄	—	2	8	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	8	no	—
192	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
193	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
194	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
195	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
196	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
197	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
198	C₂×C₄	#41×C₂₃	1	2	yes	—
199	C₂²	—	2	4	no	—
200	C₂²	—	2	8	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	4	4	no	—
204	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
205	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
206	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
207	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
208	C₂²	C₂₃×2₋¹⁺⁴	1	2	yes	—
209	C₄	—	2	4	no	—
210	C₄	—	2	4	no	—
211	C₄	—	2	2	no	—
212	C₄	—	2	2	no	—
213	C₄	—	4	2	no	—
214	C₄	—	4	2	no	—
215	C₄	—	4	2	no	—
216	C₄	C₄×D₄×C₂₃	1	4	yes	—
217	C₂	—	4	4	no	—
218	C₂	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
219	C₂	C₂₃×#520	1	2	yes	—
220	C₂	C₂₃×#535	1	1	yes	characteristic
221	C₂	C₂₃×#536	1	1	yes	characteristic
222	C₂	C₂₃×#542	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₈×C₂₃	C₂	semidirect
4. C₂₃×#391	C₂	semidirect
6. C₂₃×#388	C₂	semidirect
7. C₂₃×#385	C₂	semidirect
8. C₂₃×#384	C₂	semidirect
9. C₂₃×#376	C₂	semidirect
13. C₂₃×#124	C₄	semidirect
33. C₂×Q₈×C₂₃	C₂×C₄	semidirect
53. C₂²⋊C₄×C₂₃	C₂×C₄	semidirect
56. C₄²×C₂₃	C₂×C₄	semidirect
87. #2326	C₂₃	direct (N × H)
107. C₂×C₄×Q₈	C₂×C₂₃	semidirect
109. #391	C₂×C₂₃	semidirect
111. #388	C₂×C₂₃	semidirect
112. #385	C₂×C₂₃	semidirect
113. #384	C₂×C₂₃	semidirect
114. #376	C₂×C₂₃	semidirect
124. #124	C₄×C₂₃	semidirect
141. C₂₃	#2326	direct (N × H)
145. C₂×Q₈	C₂×C₄×C₂₃	semidirect
165. C₂²⋊C₄	C₂×C₄×C₂₃	semidirect
168. C₄²	C₂×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×C₂₃×#124	2
C₂×C₄×Q₈×C₂₃	2
C₂×C₄×D₄×C₂₃	2
C₂₃×#391	2
C₂₃×#390	2
C₂₃×#388	2
C₂₃×#385	2
C₂₃×#384	2
C₂₃×#376	2
#2326	23

Subgroups — C23×#2326 #15669945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2326 #15669945