Subgroups — C₄×C₂₃×#343 #19158335

Order 5888.

994 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₂₃	C₂	1	2	yes	maximal
2	C₂×C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#343	C₂	1	4	yes	maximal
4	C₂²×M₁₆×C₂₃	C₂²	1	4	yes	—
5	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
6	C₂²×C₄²×C₂₃	C₄	1	2	yes	—
7	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₄×M₁₆×C₂₃	—	2	4	no	—
9	C₄×M₁₆×C₂₃	C₄	1	8	yes	complemented
10	C₂×C₂₃×#97	—	2	2	no	—
11	C₂×C₄×C₈×C₂₃	—	2	2	no	—
12	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
13	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
14	C₂×C₂₃×#95	C₄	1	2	yes	—
15	C₂₃×#343	C₄	1	16	yes	direct product factor
16	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₂₃	C₂×C₄	1	4	yes	—
18	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₂₃	—	2	4	no	—
21	C₂×M₁₆×C₂₃	—	2	8	no	—
22	C₂×M₁₆×C₂₃	C₂×C₄	1	16	yes	—
23	C₂×M₁₆×C₂₃	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₂₃	—	2	4	no	—
25	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
26	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
27	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
28	C₂×C₄²×C₂₃	—	2	2	no	—
29	C₂×C₄²×C₂₃	—	2	2	no	—
30	C₂×C₄²×C₂₃	—	4	4	no	—
31	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₂₃	D₄	1	2	yes	—
33	C₂×C₄²×C₂₃	D₄	1	2	yes	—
34	C₂×C₄²×C₂₃	D₄	1	2	yes	—
35	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
36	C₂₃×#97	—	2	8	no	—
37	C₄×C₈×C₂₃	—	2	8	no	—
38	C₂₃×#95	—	2	4	no	—
39	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	—	2	2	no	—
41	C₂²×C₄×C₂₃	—	2	2	no	—
42	C₂²×C₄×C₂₃	—	2	4	no	—
43	C₂²×C₄×C₂₃	—	2	4	no	—
44	C₂²×C₄×C₂₃	—	4	2	no	—
45	C₂²×C₄×C₂₃	—	4	4	no	—
46	C₂²×C₄×C₂₃	—	4	4	no	—
47	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
48	C₂²×C₄×C₂₃	C₄²	1	8	yes	—
49	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
50	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	#41	1	4	yes	—
61	M₁₆×C₂₃	—	2	8	no	—
62	M₁₆×C₂₃	—	2	16	no	—
63	M₁₆×C₂₃	C₄²	1	16	yes	complemented
64	C₂×C₈×C₂₃	—	2	8	no	—
65	C₂×C₈×C₂₃	—	2	16	no	—
66	C₄⋊C₄×C₂₃	—	2	16	no	—
67	C₄²×C₂₃	—	4	4	no	—
68	C₄²×C₂₃	—	4	4	no	—
69	C₄²×C₂₃	C₂²⋊C₄	1	4	yes	—
70	C₄²×C₂₃	C₂²⋊C₄	1	4	yes	—
71	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
72	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
73	C₄×#343	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₂₃	—	2	1	no	—
75	C₂³×C₂₃	—	2	1	no	—
76	C₂³×C₂₃	—	2	2	no	—
77	C₂³×C₂₃	C₂×C₄²	1	1	yes	characteristic
78	C₂³×C₂₃	C₂×C₄²	1	2	yes	—
79	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂³×C₂₃	#117	1	2	yes	—
82	C₂×C₄×C₂₃	—	2	1	no	—
83	C₂×C₄×C₂₃	—	2	1	no	—
84	C₂×C₄×C₂₃	—	2	4	no	—
85	C₂×C₄×C₂₃	—	2	4	no	—
86	C₂×C₄×C₂₃	—	2	4	no	—
87	C₂×C₄×C₂₃	—	2	8	no	—
88	C₂×C₄×C₂₃	—	4	2	no	—
89	C₂×C₄×C₂₃	—	4	2	no	—
90	C₂×C₄×C₂₃	—	4	8	no	—
91	C₂×C₄×C₂₃	—	4	4	no	—
92	C₂×C₄×C₂₃	—	4	4	no	—
93	C₂×C₄×C₂₃	—	4	8	no	—
94	C₂×C₄×C₂₃	#95	1	2	yes	—
95	C₂×C₄×C₂₃	#95	1	4	yes	—
96	C₂×C₄×C₂₃	#95	1	2	yes	—
97	C₂×C₄×C₂₃	#100	1	2	yes	—
98	C₂×C₄×C₂₃	#101	1	2	yes	—
99	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
100	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
101	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₂₃	#117	1	4	yes	—
103	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
104	C₂×C₄×C₂₃	C₄×D₄	1	8	yes	—
105	C₂×C₄×C₂₃	C₄×Q₈	1	4	yes	—
106	C₈×C₂₃	—	2	16	no	—
107	C₂×C₄×M₁₆	C₂×C₂₃	1	2	yes	—
108	C₂×C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
109	C₂×#343	C₂×C₂₃	1	4	yes	—
110	C₂²×C₂₃	—	2	1	no	—
111	C₂²×C₂₃	—	2	1	no	—
112	C₂²×C₂₃	—	2	1	no	—
113	C₂²×C₂₃	—	2	1	no	—
114	C₂²×C₂₃	—	2	2	no	—
115	C₂²×C₂₃	—	2	2	no	—
116	C₂²×C₂₃	—	2	2	no	—
117	C₂²×C₂₃	—	2	2	no	—
118	C₂²×C₂₃	C₄³	1	1	yes	characteristic
119	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
120	C₂²×C₂₃	#376	1	1	yes	characteristic
121	C₂²×C₂₃	C₄×C₂²⋊C₄	1	2	yes	—
122	C₂²×C₂₃	C₄×C₄⋊C₄	1	2	yes	—
123	C₂²×C₂₃	C₂×#100	1	1	yes	characteristic
124	C₂²×C₂₃	C₂×#101	1	1	yes	characteristic
125	C₂²×C₂₃	#413	1	2	yes	—
126	C₄×C₂₃	—	2	2	no	—
127	C₄×C₂₃	—	2	4	no	—
128	C₄×C₂₃	—	4	8	no	—
129	C₄×C₂₃	#343	1	4	yes	direct product factor
130	C₄×C₂₃	C₄×C₂²⋊C₄	1	4	yes	—
131	C₄×C₂₃	C₄×C₄⋊C₄	1	4	yes	—
132	C₂²×M₁₆	C₂²×C₂₃	1	4	yes	—
133	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
134	C₂²×C₄²	C₄×C₂₃	1	2	yes	—
135	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
136	C₄×M₁₆	—	2	4	no	—
137	C₄×M₁₆	C₄×C₂₃	1	8	yes	complemented
138	C₂×#97	—	2	2	no	—
139	C₂×C₄×C₈	—	2	2	no	—
140	C₄×C₄⋊C₄	—	2	2	no	—
141	C₄×C₄⋊C₄	—	2	2	no	—
142	C₂×#95	C₄×C₂₃	1	2	yes	—
143	#343	C₄×C₂₃	1	16	yes	direct product factor
144	C₂×C₂₃	—	2	1	no	—
145	C₂×C₂₃	—	2	1	no	—
146	C₂×C₂₃	—	2	2	no	—
147	C₂×C₂₃	C₄×#95	1	1	yes	characteristic
148	C₂×C₂₃	C₂×#343	1	1	yes	characteristic
149	C₂×C₂₃	#1742	1	1	yes	characteristic
150	C₂×C₂₃	C₄×#100	1	2	yes	—
151	C₂×C₂₃	C₄×#101	1	2	yes	—
152	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
153	C₂³×C₄	C₂×C₄×C₂₃	1	4	yes	—
154	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
155	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
156	C₂×M₁₆	—	2	4	no	—
157	C₂×M₁₆	—	2	8	no	—
158	C₂×M₁₆	C₂×C₄×C₂₃	1	16	yes	—
159	C₂×M₁₆	C₂×C₄×C₂₃	1	8	yes	—
160	C₂²×C₈	—	2	4	no	—
161	C₂×C₄⋊C₄	—	2	4	no	—
162	C₂×C₄⋊C₄	—	2	4	no	—
163	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
164	C₂×C₄²	—	2	2	no	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	—	4	4	no	—
167	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
168	C₂×C₄²	D₄×C₂₃	1	2	yes	—
169	C₂×C₄²	D₄×C₂₃	1	2	yes	—
170	C₂×C₄²	D₄×C₂₃	1	2	yes	—
171	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
172	#97	—	2	8	no	—
173	C₄×C₈	—	2	8	no	—
174	#95	—	2	4	no	—
175	C₂₃	C₄×#343	1	1	yes	characteristic, direct product factor, Sylow
176	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	—	4	4	no	—
183	C₂²×C₄	—	4	4	no	—
184	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
185	C₂²×C₄	C₄²×C₂₃	1	8	yes	—
186	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
187	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
188	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
190	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
191	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
192	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
193	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
194	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
196	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
197	C₂²×C₄	#41×C₂₃	1	4	yes	—
198	M₁₆	—	2	8	no	—
199	M₁₆	—	2	16	no	—
200	M₁₆	C₄²×C₂₃	1	16	yes	complemented
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	—	2	16	no	—
203	C₄⋊C₄	—	2	16	no	—
204	C₄²	—	4	4	no	—
205	C₄²	—	4	4	no	—
206	C₄²	C₂²⋊C₄×C₂₃	1	4	yes	—
207	C₄²	C₂²⋊C₄×C₂₃	1	4	yes	—
208	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
209	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
210	C₂³	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	2	no	—
213	C₂³	C₂×C₄²×C₂₃	1	1	yes	characteristic
214	C₂³	C₂×C₄²×C₂₃	1	2	yes	—
215	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
216	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
217	C₂³	C₂₃×#117	1	2	yes	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	8	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	4	8	no	—
230	C₂×C₄	C₂₃×#95	1	2	yes	—
231	C₂×C₄	C₂₃×#95	1	4	yes	—
232	C₂×C₄	C₂₃×#95	1	2	yes	—
233	C₂×C₄	C₂₃×#100	1	2	yes	—
234	C₂×C₄	C₂₃×#101	1	2	yes	—
235	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
236	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
237	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
238	C₂×C₄	C₂₃×#117	1	4	yes	—
239	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
240	C₂×C₄	C₄×D₄×C₂₃	1	8	yes	—
241	C₂×C₄	C₄×Q₈×C₂₃	1	4	yes	—
242	C₈	—	2	16	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	C₄³×C₂₃	1	1	yes	characteristic
252	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
253	C₂²	C₂₃×#376	1	1	yes	characteristic
254	C₂²	C₄×C₂²⋊C₄×C₂₃	1	2	yes	—
255	C₂²	C₄×C₄⋊C₄×C₂₃	1	2	yes	—
256	C₂²	C₂×C₂₃×#100	1	1	yes	characteristic
257	C₂²	C₂×C₂₃×#101	1	1	yes	characteristic
258	C₂²	C₂₃×#413	1	2	yes	—
259	C₄	—	2	2	no	—
260	C₄	—	2	4	no	—
261	C₄	—	4	8	no	—
262	C₄	C₂₃×#343	1	4	yes	direct product factor
263	C₄	C₄×C₂²⋊C₄×C₂₃	1	4	yes	—
264	C₄	C₄×C₄⋊C₄×C₂₃	1	4	yes	—
265	C₂	—	2	1	no	—
266	C₂	—	2	1	no	—
267	C₂	—	2	2	no	—
268	C₂	C₄×C₂₃×#95	1	1	yes	characteristic
269	C₂	C₂×C₂₃×#343	1	1	yes	characteristic
270	C₂	C₂₃×#1742	1	1	yes	characteristic
271	C₂	C₄×C₂₃×#100	1	2	yes	—
272	C₂	C₄×C₂₃×#101	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂²⋊C₄
commutator	C₂²	C₄³×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₂₃	C₂²×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ 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
9. C₄×M₁₆×C₂₃	C₄	semidirect
15. C₂₃×#343	C₄	direct (N × H)
63. M₁₆×C₂₃	C₄²	semidirect
73. C₄×#343	C₂₃	direct (N × H)
129. C₄×C₂₃	#343	direct (N × H)
137. C₄×M₁₆	C₄×C₂₃	semidirect
143. #343	C₄×C₂₃	direct (N × H)
175. C₂₃	C₄×#343	direct (N × H)
200. M₁₆	C₄²×C₂₃	semidirect
262. C₄	C₂₃×#343	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×C₄×M₁₆×C₂₃	2
C₂×C₄×C₄⋊C₄×C₂₃	2
C₂×C₂₃×#343	2
C₄×#343	23

Subgroups — C4×C23×#343 #19158335

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂₃×#343 #19158335