Subgroups — C₄×C₂₃×#430 #19168084

Order 5888.

1034 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#430	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₂₃	C₂	1	2	yes	maximal
3	C₂×C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1778	C₂	1	4	yes	maximal, complemented
5	C₄×C₂₃×#108	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1738	C₂	1	2	yes	maximal
7	C₂²×M₁₆×C₂₃	C₂²	1	4	yes	—
8	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#430	C₄	1	16	yes	direct product factor
12	C₂×C₂₃×#108	C₂²	1	8	yes	—
13	C₄×M₁₆×C₂₃	—	2	2	no	—
14	C₄×M₁₆×C₂₃	—	2	2	no	—
15	C₄×M₁₆×C₂₃	C₄	1	8	yes	—
16	C₄×M₁₆×C₂₃	C₂²	1	4	yes	—
17	C₄×M₁₆×C₂₃	C₂²	1	4	yes	—
18	C₂×C₂₃×#97	—	2	2	no	—
19	C₂×C₂₃×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₂₃	—	2	2	no	—
21	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
22	C₂₃×#341	C₂²	1	8	yes	—
23	C₂³×C₄×C₂₃	Q₈	1	2	yes	—
24	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
25	C₂×M₁₆×C₂₃	—	2	4	no	—
26	C₂×M₁₆×C₂₃	—	2	8	no	—
27	C₂×M₁₆×C₂₃	—	2	4	no	—
28	C₂×M₁₆×C₂₃	C₂×C₄	1	16	yes	—
29	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
30	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
31	C₂×M₁₆×C₂₃	C₂×C₄	1	8	yes	—
32	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
33	C₂²×C₈×C₂₃	—	2	4	no	—
34	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
36	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₂₃	D₄	1	2	yes	—
38	C₂×C₄²×C₂₃	D₄	1	2	yes	—
39	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
40	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
42	C₂₃×#108	C₂×C₄	1	32	yes	complemented
43	C₂₃×#97	—	2	4	no	—
44	C₂₃×#97	—	2	4	no	—
45	C₂₃×#97	C₂×C₄	1	8	yes	—
46	C₄×C₈×C₂₃	—	2	4	no	—
47	C₄×C₈×C₂₃	—	2	4	no	—
48	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	—
49	C₂⁴×C₂₃	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	—	4	4	no	—
51	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
58	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	#41	1	4	yes	—
60	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
61	M₁₆×C₂₃	—	2	8	no	—
62	M₁₆×C₂₃	—	2	16	no	—
63	M₁₆×C₂₃	—	2	8	no	—
64	M₁₆×C₂₃	C₄²	1	16	yes	—
65	M₁₆×C₂₃	C₂²×C₄	1	16	yes	—
66	C₂×C₈×C₂₃	—	2	16	no	—
67	C₂×C₈×C₂₃	—	2	8	no	—
68	C₂×C₈×C₂₃	—	2	4	no	—
69	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
71	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
72	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
73	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
74	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
75	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
76	C₄²×C₂₃	C₂×D₄	1	2	yes	—
77	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₄×#430	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂³×C₂₃	—	4	2	no	—
81	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
82	C₂³×C₂₃	C₄×Q₈	1	4	yes	—
83	C₂³×C₂₃	C₂²×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₂₃	—	2	8	no	—
85	C₂×C₄×C₂₃	—	2	4	no	—
86	C₂×C₄×C₂₃	—	4	4	no	—
87	C₂×C₄×C₂₃	—	4	2	no	—
88	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
89	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
91	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₂₃	#117	1	4	yes	—
94	C₂×C₄×C₂₃	C₄×D₄	1	8	yes	—
95	C₂×C₄×C₂₃	C₄×Q₈	1	4	yes	—
96	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
98	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
100	C₈×C₂₃	—	2	4	no	—
101	C₈×C₂₃	—	2	16	no	—
102	C₈×C₂₃	C₂×C₄²	1	8	yes	—
103	C₂×#430	C₂×C₂₃	1	2	yes	—
104	C₂×C₄×M₁₆	C₂×C₂₃	1	2	yes	—
105	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic
106	#1778	C₂×C₂₃	1	4	yes	complemented
107	C₄×#108	C₂×C₂₃	1	4	yes	complemented
108	#1738	C₂×C₂₃	1	2	yes	—
109	C₂²×C₂₃	—	2	4	no	—
110	C₂²×C₂₃	—	2	2	no	—
111	C₂²×C₂₃	—	4	4	no	—
112	C₂²×C₂₃	C₄×C₄⋊C₄	1	4	yes	—
113	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₂₃	C₂×C₄×Q₈	1	2	yes	—
115	C₄×C₂₃	—	2	4	no	—
116	C₄×C₂₃	—	2	2	no	—
117	C₄×C₂₃	C₄×C₄⋊C₄	1	4	yes	—
118	C₄×C₂₃	#430	1	4	yes	direct product factor
119	C₄×C₂₃	C₂²×C₄²	1	1	yes	characteristic
120	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
121	C₄×C₂₃	C₂×C₄×D₄	1	2	yes	—
122	C₂²×M₁₆	C₂²×C₂₃	1	4	yes	—
123	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
124	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
125	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
126	#430	C₄×C₂₃	1	16	yes	direct product factor
127	C₂×#108	C₂²×C₂₃	1	8	yes	—
128	C₄×M₁₆	—	2	2	no	—
129	C₄×M₁₆	—	2	2	no	—
130	C₄×M₁₆	C₄×C₂₃	1	8	yes	—
131	C₄×M₁₆	C₂²×C₂₃	1	4	yes	—
132	C₄×M₁₆	C₂²×C₂₃	1	4	yes	—
133	C₂×#97	—	2	2	no	—
134	C₂×#97	C₂²×C₂₃	1	2	yes	—
135	C₂×C₄×C₈	—	2	2	no	—
136	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
137	#341	C₂²×C₂₃	1	8	yes	—
138	C₂×C₂₃	—	2	2	no	—
139	C₂×C₂₃	—	2	4	no	—
140	C₂×C₂₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₂₃	C₂×#430	1	2	yes	—
142	C₂³×C₄	Q₈×C₂₃	1	2	yes	—
143	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
144	C₂×M₁₆	—	2	4	no	—
145	C₂×M₁₆	—	2	8	no	—
146	C₂×M₁₆	—	2	4	no	—
147	C₂×M₁₆	C₂×C₄×C₂₃	1	16	yes	—
148	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
149	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
150	C₂×M₁₆	C₂×C₄×C₂₃	1	8	yes	—
151	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
152	C₂²×C₈	—	2	4	no	—
153	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
154	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
155	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
156	C₂×C₄²	D₄×C₂₃	1	2	yes	—
157	C₂×C₄²	D₄×C₂₃	1	2	yes	—
158	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
159	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
160	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
161	#108	C₂×C₄×C₂₃	1	32	yes	complemented
162	#97	—	2	4	no	—
163	#97	—	2	4	no	—
164	#97	C₂×C₄×C₂₃	1	8	yes	—
165	C₄×C₈	—	2	4	no	—
166	C₄×C₈	—	2	4	no	—
167	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	—
168	C₂₃	C₄×#430	1	1	yes	characteristic, direct product factor, Sylow
169	C₂⁴	C₂×Q₈×C₂₃	1	1	yes	characteristic
170	C₂²×C₄	—	4	4	no	—
171	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
172	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
173	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
174	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
175	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
176	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
177	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
178	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
179	C₂²×C₄	#41×C₂₃	1	4	yes	—
180	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
181	M₁₆	—	2	8	no	—
182	M₁₆	—	2	16	no	—
183	M₁₆	—	2	8	no	—
184	M₁₆	C₄²×C₂₃	1	16	yes	—
185	M₁₆	C₂²×C₄×C₂₃	1	16	yes	—
186	C₂×C₈	—	2	16	no	—
187	C₂×C₈	—	2	8	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
190	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
191	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
192	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
193	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
194	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
195	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
196	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
197	C₄²	C₂×D₄×C₂₃	1	2	yes	—
198	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
199	C₂³	—	4	2	no	—
200	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
201	C₂³	C₄×Q₈×C₂₃	1	4	yes	—
202	C₂³	C₂²×Q₈×C₂₃	1	1	yes	characteristic
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
208	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
209	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
210	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
211	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
212	C₂×C₄	C₂₃×#117	1	4	yes	—
213	C₂×C₄	C₄×D₄×C₂₃	1	8	yes	—
214	C₂×C₄	C₄×Q₈×C₂₃	1	4	yes	—
215	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
216	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
217	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
218	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
219	C₈	—	2	4	no	—
220	C₈	—	2	16	no	—
221	C₈	C₂×C₄²×C₂₃	1	8	yes	—
222	C₂²	—	2	4	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	4	no	—
225	C₂²	C₄×C₄⋊C₄×C₂₃	1	4	yes	—
226	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
227	C₂²	C₂×C₄×Q₈×C₂₃	1	2	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	C₄×C₄⋊C₄×C₂₃	1	4	yes	—
231	C₄	C₂₃×#430	1	4	yes	direct product factor
232	C₄	C₂²×C₄²×C₂₃	1	1	yes	characteristic
233	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
234	C₄	C₂×C₄×D₄×C₂₃	1	2	yes	—
235	C₂	—	2	4	no	—
236	C₂	—	2	2	no	—
237	C₂	C₂×C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
238	C₂	C₂×C₂₃×#430	1	2	yes	—

	subgroup	G/H
center	C₄²×C₂₃	C₂×D₄
commutator	C₄	C₂²×C₄²×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₂₃	C₂×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1778	C₂	semidirect
5. C₄×C₂₃×#108	C₂	semidirect
11. C₂₃×#430	C₄	direct (N × H)
42. C₂₃×#108	C₂×C₄	semidirect
79. C₄×#430	C₂₃	direct (N × H)
106. #1778	C₂×C₂₃	semidirect
107. C₄×#108	C₂×C₂₃	semidirect
118. C₄×C₂₃	#430	direct (N × H)
126. #430	C₄×C₂₃	direct (N × H)
161. #108	C₂×C₄×C₂₃	semidirect
168. C₂₃	C₄×#430	direct (N × H)
231. C₄	C₂₃×#430	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×#430	2
C₂×C₄×M₁₆×C₂₃	2
C₂₃×#1778	2
C₄×C₂₃×#108	2
C₂₃×#1738	2
C₄×#430	23

Subgroups — C4×C23×#430 #19168084

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂₃×#430 #19168084