Subgroups — C₂×C₂₃×#1544 #19165325

Order 5888.

994 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#431	C₂	1	2	yes	maximal
3	C₂₃×#1544	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₃×#332	C₂	1	4	yes	maximal, complemented
5	C₂₃×#533	—	2	4	no	—
6	C₂₃×#533	C₄	1	4	yes	—
7	C₂₃×#533	C₂²	1	4	yes	—
8	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₂₃×#125	C₄	1	2	yes	—
10	C₂×C₂₃×C₂²⋊Q₈	—	2	2	no	—
11	C₂×C₄×Q₈×C₂₃	—	2	2	no	—
12	C₂×C₂₃×#117	C₄	1	2	yes	—
13	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#431	—	2	4	no	—
15	C₂₃×#431	C₂²	1	8	yes	—
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
17	C₂×C₂₃×C₂²⋊C₈	—	2	2	no	—
18	C₂×C₄×C₈×C₂₃	—	2	2	no	—
19	C₂₃×#332	C₂²	1	16	yes	complemented
20	C₂²×Q₈×C₂₃	—	2	2	no	—
21	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₂₃	D₄	1	2	yes	—
23	C₂²×C₈×C₂₃	—	2	4	no	—
24	C₂₃×C₄⋊Q₈	—	2	2	no	—
25	C₂₃×C₄⋊Q₈	—	2	2	no	—
26	C₂₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
27	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
28	C₂₃×#125	—	2	4	no	—
29	C₂₃×#125	C₂×C₄	1	8	yes	—
30	C₂₃×C₂²⋊Q₈	—	2	8	no	—
31	C₂₃×C₂²⋊Q₈	—	2	8	no	—
32	C₄×Q₈×C₂₃	—	2	8	no	—
33	C₄×Q₈×C₂₃	—	2	8	no	—
34	C₂₃×#117	—	2	4	no	—
35	C₂₃×#117	C₂×C₄	1	8	yes	—
36	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
37	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
39	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
40	C₂×C₄²×C₂₃	—	2	2	no	—
41	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₂₃	D₄	1	4	yes	—
44	C₂×C₄²×C₂₃	D₄	1	2	yes	—
45	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
47	C₂₃×C₄⋊C₈	—	2	4	no	—
48	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
49	C₂₃×C₂²⋊C₈	—	2	8	no	—
50	C₄×C₈×C₂₃	—	2	8	no	—
51	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×Q₈×C₂₃	—	2	4	no	—
53	C₂×Q₈×C₂₃	—	2	4	no	—
54	C₂×Q₈×C₂₃	—	4	8	no	—
55	C₂²×C₄×C₂₃	—	2	4	no	—
56	C₂²×C₄×C₂₃	—	2	8	no	—
57	C₂²×C₄×C₂₃	—	2	4	no	—
58	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
60	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₂₃	—	2	8	no	—
67	C₂×C₈×C₂₃	—	4	8	no	—
68	C₄⋊C₄×C₂₃	—	2	16	no	—
69	C₄⋊C₄×C₂₃	—	2	4	no	—
70	C₄⋊C₄×C₂₃	—	2	4	no	—
71	C₄⋊C₄×C₂₃	—	2	4	no	—
72	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
73	C₂²⋊C₄×C₂₃	—	2	8	no	—
74	C₄²×C₂₃	—	2	8	no	—
75	C₄²×C₂₃	C₂²⋊C₄	1	4	yes	—
76	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
78	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₂₃	C₂×D₄	1	2	yes	—
80	C₄²×C₂₃	C₂×D₄	1	4	yes	—
81	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
82	C₂×#1544	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂³×C₂₃	—	2	2	no	—
84	C₂³×C₂₃	—	2	4	no	—
85	C₂³×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
86	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	Q₈×C₂₃	—	4	8	no	—
88	Q₈×C₂₃	—	4	8	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	4	no	—
91	C₂×C₄×C₂₃	—	2	8	no	—
92	C₂×C₄×C₂₃	—	2	4	no	—
93	C₂×C₄×C₂₃	—	2	4	no	—
94	C₂×C₄×C₂₃	—	2	8	no	—
95	C₂×C₄×C₂₃	—	2	8	no	—
96	C₂×C₄×C₂₃	—	4	8	no	—
97	C₂×C₄×C₂₃	#101	1	4	yes	—
98	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
99	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
101	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
102	C₂×C₄×C₂₃	#135	1	8	yes	—
103	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
105	C₈×C₂₃	—	4	8	no	—
106	C₂×#533	C₂×C₂₃	1	1	yes	characteristic
107	C₂×#431	C₂×C₂₃	1	2	yes	—
108	#1544	C₂×C₂₃	1	8	yes	direct product factor
109	C₂×#332	C₂×C₂₃	1	4	yes	complemented
110	C₂²×C₂₃	—	2	2	no	—
111	C₂²×C₂₃	—	2	4	no	—
112	C₂²×C₂₃	—	4	4	no	—
113	C₂²×C₂₃	C₂×#101	1	2	yes	—
114	C₂²×C₂₃	#416	1	4	yes	—
115	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
116	C₄×C₂₃	—	2	1	no	—
117	C₄×C₂₃	—	2	1	no	—
118	C₄×C₂₃	—	2	4	no	—
119	C₄×C₂₃	—	4	8	no	—
120	C₄×C₂₃	C₂×#101	1	4	yes	—
121	C₄×C₂₃	#416	1	4	yes	—
122	C₄×C₂₃	C₂×#135	1	4	yes	—
123	#533	—	2	4	no	—
124	#533	C₄×C₂₃	1	4	yes	—
125	#533	C₂²×C₂₃	1	4	yes	—
126	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
127	C₂×#125	C₄×C₂₃	1	2	yes	—
128	C₂×C₂²⋊Q₈	—	2	2	no	—
129	C₂×C₄×Q₈	—	2	2	no	—
130	C₂×#117	C₄×C₂₃	1	2	yes	—
131	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
132	#431	—	2	4	no	—
133	#431	C₂²×C₂₃	1	8	yes	—
134	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
135	C₂×C₂²⋊C₈	—	2	2	no	—
136	C₂×C₄×C₈	—	2	2	no	—
137	#332	C₂²×C₂₃	1	16	yes	complemented
138	C₂×C₂₃	—	4	2	no	—
139	C₂×C₂₃	#1544	1	4	yes	direct product factor
140	C₂×C₂₃	C₂²×#101	1	1	yes	characteristic
141	C₂×C₂₃	C₂×#416	1	2	yes	—
142	C₂²×Q₈	—	2	2	no	—
143	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
144	C₂³×C₄	D₄×C₂₃	1	2	yes	—
145	C₂²×C₈	—	2	4	no	—
146	C₄⋊Q₈	—	2	2	no	—
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
149	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
150	#125	—	2	4	no	—
151	#125	C₂×C₄×C₂₃	1	8	yes	—
152	C₂²⋊Q₈	—	2	8	no	—
153	C₂²⋊Q₈	—	2	8	no	—
154	C₄×Q₈	—	2	8	no	—
155	C₄×Q₈	—	2	8	no	—
156	#117	—	2	4	no	—
157	#117	C₂×C₄×C₂₃	1	8	yes	—
158	C₂×C₄⋊C₄	—	2	4	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
161	C₂×C₂²⋊C₄	—	2	2	no	—
162	C₂×C₄²	—	2	2	no	—
163	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
164	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
165	C₂×C₄²	D₄×C₂₃	1	2	yes	—
166	C₂×C₄²	D₄×C₂₃	1	4	yes	—
167	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
168	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
169	C₄⋊C₈	—	2	4	no	—
170	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
171	C₂²⋊C₈	—	2	8	no	—
172	C₄×C₈	—	2	8	no	—
173	C₂₃	C₂×#1544	1	1	yes	characteristic, direct product factor, Sylow
174	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
175	C₂×Q₈	—	2	4	no	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	4	8	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	8	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
182	C₂²×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
183	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
184	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
185	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
186	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
187	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
188	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
189	C₂×C₈	—	2	8	no	—
190	C₂×C₈	—	4	8	no	—
191	C₄⋊C₄	—	2	16	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
196	C₂²⋊C₄	—	2	8	no	—
197	C₄²	—	2	8	no	—
198	C₄²	C₂²⋊C₄×C₂₃	1	4	yes	—
199	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
200	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
201	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
202	C₄²	C₂×D₄×C₂₃	1	4	yes	—
203	C₄²	C₂×D₄×C₂₃	1	2	yes	—
204	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
205	C₂³	—	2	2	no	—
206	C₂³	—	2	4	no	—
207	C₂³	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
208	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
209	Q₈	—	4	8	no	—
210	Q₈	—	4	8	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	8	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	4	8	no	—
219	C₂×C₄	C₂₃×#101	1	4	yes	—
220	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
221	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
222	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
223	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
224	C₂×C₄	C₂₃×#135	1	8	yes	—
225	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
226	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
227	C₈	—	4	8	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	4	4	no	—
231	C₂²	C₂×C₂₃×#101	1	2	yes	—
232	C₂²	C₂₃×#416	1	4	yes	—
233	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	4	no	—
237	C₄	—	4	8	no	—
238	C₄	C₂×C₂₃×#101	1	4	yes	—
239	C₄	C₂₃×#416	1	4	yes	—
240	C₄	C₂×C₂₃×#135	1	4	yes	—
241	C₂	—	4	2	no	—
242	C₂	C₂₃×#1544	1	4	yes	direct product factor
243	C₂	C₂²×C₂₃×#101	1	1	yes	characteristic
244	C₂	C₂×C₂₃×#416	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₂₃	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1544	C₂	direct (N × H)
4. C₂×C₂₃×#332	C₂	semidirect
19. C₂₃×#332	C₂²	semidirect
82. C₂×#1544	C₂₃	direct (N × H)
108. #1544	C₂×C₂₃	direct (N × H)
109. C₂×#332	C₂×C₂₃	semidirect
137. #332	C₂²×C₂₃	semidirect
139. C₂×C₂₃	#1544	direct (N × H)
173. C₂₃	C₂×#1544	direct (N × H)
242. C₂	C₂₃×#1544	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#533	2
C₂×C₂₃×#431	2
C₂₃×#1544	2
C₂×C₂₃×#332	2
C₂×#1544	23

Subgroups — C2×C23×#1544 #19165325

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1544 #19165325