Subgroups — C₂×C₂₃×#2351 #19181739

Order 5888.

3158 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₂₃	C₂	1	2	yes	maximal
2	C₂²×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2351	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#389	C₂	1	6	yes	maximal
5	C₂×C₂₃×#382	C₂	1	3	yes	maximal
6	C₂×C₄×C₄⋊C₄×C₂₃	C₂	1	3	yes	maximal
7	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₂₃	—	2	16	no	—
9	C₂×C₄×Q₈×C₂₃	C₄	1	16	yes	complemented
10	C₂×C₄×Q₈×C₂₃	C₂²	1	16	yes	—
11	C₂×C₄×Q₈×C₂₃	C₂²	1	8	yes	—
12	C₂²×C₄⋊C₄×C₂₃	C₂²	1	6	yes	—
13	C₂²×C₄⋊C₄×C₂₃	C₂²	1	6	yes	—
14	C₂²×C₄⋊C₄×C₂₃	C₂²	1	3	yes	—
15	C₂²×C₄²×C₂₃	C₂²	1	6	yes	—
16	C₂²×C₄²×C₂₃	C₂²	1	3	yes	—
17	C₂₃×#389	C₂²	1	48	yes	—
18	C₂₃×#382	C₂²	1	24	yes	—
19	C₄×C₄⋊C₄×C₂₃	C₂²	1	24	yes	—
20	C₂×C₂₃×#95	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#95	C₂²	1	3	yes	—
22	C₂×C₂₃×#95	C₂²	1	6	yes	—
23	C₂²×Q₈×C₂₃	C₂×C₄	1	8	yes	—
24	C₂²×Q₈×C₂₃	D₄	1	8	yes	—
25	C₂²×Q₈×C₂₃	Q₈	1	8	yes	—
26	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
27	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
28	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
29	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
30	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
31	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
32	C₄×Q₈×C₂₃	—	2	64	no	—
33	C₄×Q₈×C₂₃	C₂×C₄	1	64	yes	complemented
34	C₂×C₄⋊C₄×C₂₃	—	2	24	no	—
35	C₂×C₄⋊C₄×C₂₃	—	2	24	no	—
36	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	24	yes	—
37	C₂×C₄⋊C₄×C₂₃	C₂³	1	24	yes	—
38	C₂×C₄⋊C₄×C₂₃	C₂³	1	24	yes	—
39	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
40	C₂×C₄²×C₂₃	—	2	24	no	—
41	C₂×C₄²×C₂₃	C₂×C₄	1	24	yes	—
42	C₂×C₄²×C₂₃	C₂³	1	24	yes	—
43	C₂×C₄²×C₂₃	C₂³	1	12	yes	—
44	C₂₃×#95	C₂³	1	4	yes	—
45	C₂₃×#95	C₂³	1	12	yes	—
46	C₂₃×#95	C₂³	1	24	yes	—
47	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
48	C₂×Q₈×C₂₃	C₄⋊C₄	1	64	yes	complemented
49	C₂×Q₈×C₂₃	C₂²×C₄	1	16	yes	—
50	C₂×Q₈×C₂₃	C₂×D₄	1	16	yes	—
51	C₂×Q₈×C₂₃	C₂×Q₈	1	16	yes	—
52	C₂²×C₄×C₂₃	—	2	4	no	—
53	C₂²×C₄×C₂₃	—	2	12	no	—
54	C₂²×C₄×C₂₃	—	2	4	no	—
55	C₂²×C₄×C₂₃	—	2	4	no	—
56	C₂²×C₄×C₂₃	—	2	12	no	—
57	C₂²×C₄×C₂₃	—	2	12	no	—
58	C₂²×C₄×C₂₃	—	2	12	no	—
59	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₂₃	C₂²×C₄	1	12	yes	—
61	C₂²×C₄×C₂₃	C₂²×C₄	1	12	yes	—
62	C₂²×C₄×C₂₃	C₂×D₄	1	12	yes	—
63	C₂²×C₄×C₂₃	C₂×Q₈	1	12	yes	—
64	C₂²×C₄×C₂₃	#41	1	8	yes	—
65	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
66	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
67	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
68	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
69	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
70	C₄⋊C₄×C₂₃	—	2	48	no	—
71	C₄⋊C₄×C₂₃	—	2	48	no	—
72	C₄⋊C₄×C₂₃	C₂²×C₄	1	48	yes	—
73	C₄²×C₂₃	—	2	48	no	—
74	C₄²×C₂₃	C₂²×C₄	1	48	yes	—
75	C₂×#2351	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
77	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
78	C₂³×C₂₃	C₂²×Q₈	1	2	yes	—
79	C₂³×C₂₃	C₂×#41	1	4	yes	—
80	C₂³×C₂₃	2₋¹⁺⁴	1	4	yes	—
81	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
82	Q₈×C₂₃	C₂×C₄⋊C₄	1	64	yes	complemented
83	C₂×C₄×C₂₃	—	2	24	no	—
84	C₂×C₄×C₂₃	—	2	8	no	—
85	C₂×C₄×C₂₃	—	2	16	no	—
86	C₂×C₄×C₂₃	—	2	4	no	—
87	C₂×C₄×C₂₃	—	2	12	no	—
88	C₂×C₄×C₂₃	—	2	4	no	—
89	C₂×C₄×C₂₃	—	2	4	no	—
90	C₂×C₄×C₂₃	—	2	12	no	—
91	C₂×C₄×C₂₃	—	2	12	no	—
92	C₂×C₄×C₂₃	—	2	12	no	—
93	C₂×C₄×C₂₃	—	4	24	no	—
94	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	48	yes	—
95	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
96	C₂×C₄×C₂₃	C₂³×C₄	1	12	yes	—
97	C₂×C₄×C₂₃	C₂³×C₄	1	12	yes	—
98	C₂×C₄×C₂₃	C₂²×D₄	1	12	yes	—
99	C₂×C₄×C₂₃	C₂²×Q₈	1	12	yes	—
100	C₂×C₄×C₂₃	C₂×#41	1	8	yes	—
101	C₂²×C₄×Q₈	C₂×C₂₃	1	2	yes	—
102	C₂²×C₄×Q₈	C₂×C₂₃	1	1	yes	characteristic
103	#2351	C₂×C₂₃	1	16	yes	direct product factor
104	C₂×#389	C₂×C₂₃	1	6	yes	—
105	C₂×#382	C₂×C₂₃	1	3	yes	—
106	C₂×C₄×C₄⋊C₄	C₂×C₂₃	1	3	yes	—
107	C₂²×C₂₃	C₂²×C₄⋊C₄	1	4	yes	—
108	C₂²×C₂₃	C₄×#41	1	4	yes	—
109	C₂²×C₂₃	#519	1	4	yes	—
110	C₂²×C₂₃	#548	1	8	yes	—
111	C₂²×C₂₃	#557	1	8	yes	—
112	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
113	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
114	C₂²×C₂₃	C₂³×Q₈	1	1	yes	characteristic
115	C₂²×C₂₃	C₂²×#41	1	2	yes	—
116	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	2	yes	—
117	C₄×C₂₃	—	2	12	no	—
118	C₄×C₂₃	—	2	4	no	—
119	C₄×C₂₃	—	2	8	no	—
120	C₄×C₂₃	—	4	12	no	—
121	C₄×C₂₃	C₂²×C₄⋊C₄	1	24	yes	—
122	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
123	C₂×C₄×Q₈	—	2	16	no	—
124	C₂×C₄×Q₈	C₄×C₂₃	1	16	yes	complemented
125	C₂×C₄×Q₈	C₂²×C₂₃	1	16	yes	—
126	C₂×C₄×Q₈	C₂²×C₂₃	1	8	yes	—
127	C₂²×C₄⋊C₄	C₂²×C₂₃	1	6	yes	—
128	C₂²×C₄⋊C₄	C₂²×C₂₃	1	6	yes	—
129	C₂²×C₄⋊C₄	C₂²×C₂₃	1	3	yes	—
130	C₂²×C₄²	C₂²×C₂₃	1	6	yes	—
131	C₂²×C₄²	C₂²×C₂₃	1	3	yes	—
132	#389	C₂²×C₂₃	1	48	yes	—
133	#382	C₂²×C₂₃	1	24	yes	—
134	C₄×C₄⋊C₄	C₂²×C₂₃	1	24	yes	—
135	C₂×#95	C₂²×C₂₃	1	1	yes	characteristic
136	C₂×#95	C₂²×C₂₃	1	3	yes	—
137	C₂×#95	C₂²×C₂₃	1	6	yes	—
138	C₂×C₂₃	#2351	1	8	yes	direct product factor
139	C₂×C₂₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₂₃	C₂×C₄×#41	1	1	yes	characteristic
141	C₂×C₂₃	C₂×#519	1	1	yes	characteristic
142	C₂×C₂₃	C₂×#548	1	2	yes	—
143	C₂×C₂₃	C₂×#557	1	2	yes	—
144	C₂²×Q₈	C₂×C₄×C₂₃	1	8	yes	—
145	C₂²×Q₈	D₄×C₂₃	1	8	yes	—
146	C₂²×Q₈	Q₈×C₂₃	1	8	yes	—
147	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
148	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
149	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
150	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
151	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
152	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
153	C₄×Q₈	—	2	64	no	—
154	C₄×Q₈	C₂×C₄×C₂₃	1	64	yes	complemented
155	C₂×C₄⋊C₄	—	2	24	no	—
156	C₂×C₄⋊C₄	—	2	24	no	—
157	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	24	yes	—
158	C₂×C₄⋊C₄	C₂³×C₂₃	1	24	yes	—
159	C₂×C₄⋊C₄	C₂³×C₂₃	1	24	yes	—
160	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
161	C₂×C₄²	—	2	24	no	—
162	C₂×C₄²	C₂×C₄×C₂₃	1	24	yes	—
163	C₂×C₄²	C₂³×C₂₃	1	24	yes	—
164	C₂×C₄²	C₂³×C₂₃	1	12	yes	—
165	#95	C₂³×C₂₃	1	4	yes	—
166	#95	C₂³×C₂₃	1	12	yes	—
167	#95	C₂³×C₂₃	1	24	yes	—
168	C₂₃	C₂×#2351	1	1	yes	characteristic, direct product factor, Sylow
169	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
170	C₂×Q₈	C₄⋊C₄×C₂₃	1	64	yes	complemented
171	C₂×Q₈	C₂²×C₄×C₂₃	1	16	yes	—
172	C₂×Q₈	C₂×D₄×C₂₃	1	16	yes	—
173	C₂×Q₈	C₂×Q₈×C₂₃	1	16	yes	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	2	12	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	12	no	—
179	C₂²×C₄	—	2	12	no	—
180	C₂²×C₄	—	2	12	no	—
181	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
182	C₂²×C₄	C₂²×C₄×C₂₃	1	12	yes	—
183	C₂²×C₄	C₂²×C₄×C₂₃	1	12	yes	—
184	C₂²×C₄	C₂×D₄×C₂₃	1	12	yes	—
185	C₂²×C₄	C₂×Q₈×C₂₃	1	12	yes	—
186	C₂²×C₄	#41×C₂₃	1	8	yes	—
187	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
188	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
189	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
190	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
191	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
192	C₄⋊C₄	—	2	48	no	—
193	C₄⋊C₄	—	2	48	no	—
194	C₄⋊C₄	C₂²×C₄×C₂₃	1	48	yes	—
195	C₄²	—	2	48	no	—
196	C₄²	C₂²×C₄×C₂₃	1	48	yes	—
197	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
198	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
199	C₂³	C₂²×Q₈×C₂₃	1	2	yes	—
200	C₂³	C₂×#41×C₂₃	1	4	yes	—
201	C₂³	C₂₃×2₋¹⁺⁴	1	4	yes	—
202	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
203	Q₈	C₂×C₄⋊C₄×C₂₃	1	64	yes	complemented
204	C₂×C₄	—	2	16	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	12	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	24	no	—
210	C₂×C₄	—	2	8	no	—
211	C₂×C₄	—	2	12	no	—
212	C₂×C₄	—	2	12	no	—
213	C₂×C₄	—	2	12	no	—
214	C₂×C₄	—	4	24	no	—
215	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	48	yes	—
216	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
217	C₂×C₄	C₂³×C₄×C₂₃	1	12	yes	—
218	C₂×C₄	C₂³×C₄×C₂₃	1	12	yes	—
219	C₂×C₄	C₂²×D₄×C₂₃	1	12	yes	—
220	C₂×C₄	C₂²×Q₈×C₂₃	1	12	yes	—
221	C₂×C₄	C₂×#41×C₂₃	1	8	yes	—
222	C₂²	C₂²×C₄⋊C₄×C₂₃	1	4	yes	—
223	C₂²	C₄×#41×C₂₃	1	4	yes	—
224	C₂²	C₂₃×#519	1	4	yes	—
225	C₂²	C₂₃×#548	1	8	yes	—
226	C₂²	C₂₃×#557	1	8	yes	—
227	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
228	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
229	C₂²	C₂³×Q₈×C₂₃	1	1	yes	characteristic
230	C₂²	C₂²×#41×C₂₃	1	2	yes	—
231	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	2	yes	—
232	C₄	—	2	12	no	—
233	C₄	—	2	4	no	—
234	C₄	—	2	8	no	—
235	C₄	—	4	12	no	—
236	C₄	C₂²×C₄⋊C₄×C₂₃	1	24	yes	—
237	C₂	C₂₃×#2351	1	8	yes	direct product factor
238	C₂	C₂³×C₄⋊C₄×C₂₃	1	1	yes	characteristic
239	C₂	C₂×C₄×#41×C₂₃	1	1	yes	characteristic
240	C₂	C₂×C₂₃×#519	1	1	yes	characteristic
241	C₂	C₂×C₂₃×#548	1	2	yes	—
242	C₂	C₂×C₂₃×#557	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₂₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2351	C₂	direct (N × H)
9. C₂×C₄×Q₈×C₂₃	C₄	semidirect
33. C₄×Q₈×C₂₃	C₂×C₄	semidirect
48. C₂×Q₈×C₂₃	C₄⋊C₄	semidirect
75. C₂×#2351	C₂₃	direct (N × H)
82. Q₈×C₂₃	C₂×C₄⋊C₄	semidirect
103. #2351	C₂×C₂₃	direct (N × H)
124. C₂×C₄×Q₈	C₄×C₂₃	semidirect
138. C₂×C₂₃	#2351	direct (N × H)
154. C₄×Q₈	C₂×C₄×C₂₃	semidirect
168. C₂₃	C₂×#2351	direct (N × H)
170. C₂×Q₈	C₄⋊C₄×C₂₃	semidirect
203. Q₈	C₂×C₄⋊C₄×C₂₃	semidirect
237. C₂	C₂₃×#2351	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₂₃	2
C₂₃×#2351	2
C₂×C₂₃×#389	2
C₂×C₂₃×#382	2
C₂×C₄×C₄⋊C₄×C₂₃	2
C₂×#2351	23

Subgroups — C2×C23×#2351 #19181739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2351 #19181739