Subgroups — C₄×C₂₃×#340 #19158328

Order 5888.

1182 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	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal
3	C₂×C₂₃×#340	C₂	1	4	yes	maximal
4	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
5	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
6	C₂²×C₄²×C₂₃	—	2	2	no	—
7	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₂₃	—	2	4	no	—
9	C₂×C₄×C₈×C₂₃	C₄	1	4	yes	complemented
10	C₄×C₄⋊C₄×C₂₃	—	2	4	no	—
11	C₄×C₄⋊C₄×C₂₃	C₄	1	8	yes	complemented
12	C₂×C₂₃×#95	—	2	2	no	—
13	C₂₃×#340	C₄	1	16	yes	direct product factor
14	C₂³×C₄×C₂₃	—	2	4	no	—
15	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
17	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	—	2	4	no	—
19	C₂²×C₈×C₂₃	—	2	4	no	—
20	C₂²×C₈×C₂₃	C₂×C₄	1	8	yes	—
21	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
22	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
23	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
24	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	16	yes	—
25	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
26	C₂×C₄²×C₂₃	—	2	8	no	—
27	C₂×C₄²×C₂₃	—	4	4	no	—
28	C₂×C₄²×C₂₃	—	4	4	no	—
29	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₂₃	D₄	1	4	yes	—
32	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
33	C₄×C₈×C₂₃	—	2	8	no	—
34	C₂₃×#95	—	2	8	no	—
35	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	—	2	4	no	—
37	C₂²×C₄×C₂₃	—	2	8	no	—
38	C₂²×C₄×C₂₃	—	2	16	no	—
39	C₂²×C₄×C₂₃	—	4	4	no	—
40	C₂²×C₄×C₂₃	—	4	4	no	—
41	C₂²×C₄×C₂₃	—	4	8	no	—
42	C₂²×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
43	C₂²×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
44	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₂₃	D₈	1	2	yes	—
47	C₂²×C₄×C₂₃	SD₁₆	1	4	yes	—
48	C₂²×C₄×C₂₃	Q₁₆	1	2	yes	—
49	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	#41	1	4	yes	—
55	C₂×C₈×C₂₃	—	2	4	no	—
56	C₂×C₈×C₂₃	—	2	4	no	—
57	C₂×C₈×C₂₃	—	2	16	no	—
58	C₂×C₈×C₂₃	C₄²	1	8	yes	complemented
59	C₄⋊C₄×C₂₃	—	2	8	no	—
60	C₄⋊C₄×C₂₃	—	2	16	no	—
61	C₄⋊C₄×C₂₃	C₄²	1	16	yes	complemented
62	C₄²×C₂₃	—	4	16	no	—
63	C₄²×C₂₃	C₄²	1	4	yes	—
64	C₄²×C₂₃	C₂²⋊C₄	1	4	yes	—
65	C₄²×C₂₃	C₄⋊C₄	1	8	yes	—
66	C₄×#340	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
68	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂³×C₂₃	C₄×D₄	1	4	yes	—
70	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
71	C₂³×C₂₃	C₂×SD₁₆	1	2	yes	—
72	C₂³×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
73	C₂³×C₂₃	#135	1	4	yes	—
74	C₂×C₄×C₂₃	—	2	8	no	—
75	C₂×C₄×C₂₃	—	2	4	no	—
76	C₂×C₄×C₂₃	—	2	16	no	—
77	C₂×C₄×C₂₃	—	4	8	no	—
78	C₂×C₄×C₂₃	—	4	8	no	—
79	C₂×C₄×C₂₃	—	4	16	no	—
80	C₂×C₄×C₂₃	—	4	16	no	—
81	C₂×C₄×C₂₃	#95	1	4	yes	—
82	C₂×C₄×C₂₃	#95	1	4	yes	—
83	C₂×C₄×C₂₃	#102	1	8	yes	—
84	C₂×C₄×C₂₃	#103	1	8	yes	—
85	C₂×C₄×C₂₃	#106	1	4	yes	—
86	C₂×C₄×C₂₃	#107	1	4	yes	—
87	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
89	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
90	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₂₃	#117	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	C₈×C₂₃	—	2	8	no	—
97	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
98	C₂×C₄×C₄⋊C₄	C₂×C₂₃	1	2	yes	—
99	C₂×#340	C₂×C₂₃	1	4	yes	—
100	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
101	C₂²×C₂₃	C₄×C₂²⋊C₄	1	4	yes	—
102	C₂²×C₂₃	C₄×C₄⋊C₄	1	2	yes	—
103	C₂²×C₂₃	C₂×#102	1	2	yes	—
104	C₂²×C₂₃	C₂×#103	1	2	yes	—
105	C₂²×C₂₃	#416	1	4	yes	—
106	C₂²×C₂₃	C₂×#106	1	1	yes	characteristic
107	C₂²×C₂₃	C₂×#107	1	1	yes	characteristic
108	C₂²×C₂₃	#427	1	2	yes	—
109	C₂²×C₂₃	C₄×D₈	1	4	yes	—
110	C₂²×C₂₃	C₄×SD₁₆	1	8	yes	—
111	C₂²×C₂₃	C₄×Q₁₆	1	4	yes	—
112	C₄×C₂₃	—	2	4	no	—
113	C₄×C₂₃	—	4	16	no	—
114	C₄×C₂₃	#340	1	8	yes	direct product factor
115	C₄×C₂₃	C₄³	1	1	yes	characteristic
116	C₄×C₂₃	#376	1	1	yes	characteristic
117	C₄×C₂₃	C₄×C₂²⋊C₄	1	2	yes	—
118	C₄×C₂₃	C₄×C₄⋊C₄	1	4	yes	—
119	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
120	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
121	C₂²×C₄²	—	2	2	no	—
122	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
123	C₂×C₄×C₈	—	2	4	no	—
124	C₂×C₄×C₈	C₄×C₂₃	1	4	yes	complemented
125	C₄×C₄⋊C₄	—	2	4	no	—
126	C₄×C₄⋊C₄	C₄×C₂₃	1	8	yes	complemented
127	C₂×#95	—	2	2	no	—
128	#340	C₄×C₂₃	1	16	yes	direct product factor
129	C₂×C₂₃	C₄×#95	1	1	yes	characteristic
130	C₂×C₂₃	C₂×#340	1	1	yes	characteristic
131	C₂×C₂₃	#1735	1	1	yes	characteristic
132	C₂×C₂₃	C₄×#102	1	4	yes	—
133	C₂×C₂₃	C₄×#103	1	4	yes	—
134	C₂×C₂₃	C₄×#106	1	2	yes	—
135	C₂×C₂₃	C₄×#107	1	2	yes	—
136	C₂³×C₄	—	2	4	no	—
137	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
138	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
139	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
140	C₂²×C₈	—	2	4	no	—
141	C₂²×C₈	—	2	4	no	—
142	C₂²×C₈	C₂×C₄×C₂₃	1	8	yes	—
143	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
144	C₂×C₄⋊C₄	—	2	4	no	—
145	C₂×C₄⋊C₄	—	2	8	no	—
146	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	16	yes	—
147	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
148	C₂×C₄²	—	2	8	no	—
149	C₂×C₄²	—	4	4	no	—
150	C₂×C₄²	—	4	4	no	—
151	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
152	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
153	C₂×C₄²	D₄×C₂₃	1	4	yes	—
154	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
155	C₄×C₈	—	2	8	no	—
156	#95	—	2	8	no	—
157	C₂₃	C₄×#340	1	1	yes	characteristic, direct product factor, Sylow
158	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	—	2	16	no	—
162	C₂²×C₄	—	4	4	no	—
163	C₂²×C₄	—	4	4	no	—
164	C₂²×C₄	—	4	8	no	—
165	C₂²×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
166	C₂²×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
167	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
168	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
169	C₂²×C₄	D₈×C₂₃	1	2	yes	—
170	C₂²×C₄	SD₁₆×C₂₃	1	4	yes	—
171	C₂²×C₄	Q₁₆×C₂₃	1	2	yes	—
172	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
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	1	yes	characteristic
177	C₂²×C₄	#41×C₂₃	1	4	yes	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	16	no	—
181	C₂×C₈	C₄²×C₂₃	1	8	yes	complemented
182	C₄⋊C₄	—	2	8	no	—
183	C₄⋊C₄	—	2	16	no	—
184	C₄⋊C₄	C₄²×C₂₃	1	16	yes	complemented
185	C₄²	—	4	16	no	—
186	C₄²	C₄²×C₂₃	1	4	yes	—
187	C₄²	C₂²⋊C₄×C₂₃	1	4	yes	—
188	C₄²	C₄⋊C₄×C₂₃	1	8	yes	—
189	C₂³	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
190	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
191	C₂³	C₄×D₄×C₂₃	1	4	yes	—
192	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
193	C₂³	C₂×SD₁₆×C₂₃	1	2	yes	—
194	C₂³	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
195	C₂³	C₂₃×#135	1	4	yes	—
196	C₂×C₄	—	2	8	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	16	no	—
199	C₂×C₄	—	4	8	no	—
200	C₂×C₄	—	4	8	no	—
201	C₂×C₄	—	4	16	no	—
202	C₂×C₄	—	4	16	no	—
203	C₂×C₄	C₂₃×#95	1	4	yes	—
204	C₂×C₄	C₂₃×#95	1	4	yes	—
205	C₂×C₄	C₂₃×#102	1	8	yes	—
206	C₂×C₄	C₂₃×#103	1	8	yes	—
207	C₂×C₄	C₂₃×#106	1	4	yes	—
208	C₂×C₄	C₂₃×#107	1	4	yes	—
209	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
211	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
212	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
214	C₂×C₄	C₂₃×#117	1	2	yes	—
215	C₂×C₄	C₂₃×#117	1	4	yes	—
216	C₂×C₄	C₄×D₄×C₂₃	1	8	yes	—
217	C₂×C₄	C₄×Q₈×C₂₃	1	4	yes	—
218	C₈	—	2	8	no	—
219	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
220	C₂²	C₄×C₂²⋊C₄×C₂₃	1	4	yes	—
221	C₂²	C₄×C₄⋊C₄×C₂₃	1	2	yes	—
222	C₂²	C₂×C₂₃×#102	1	2	yes	—
223	C₂²	C₂×C₂₃×#103	1	2	yes	—
224	C₂²	C₂₃×#416	1	4	yes	—
225	C₂²	C₂×C₂₃×#106	1	1	yes	characteristic
226	C₂²	C₂×C₂₃×#107	1	1	yes	characteristic
227	C₂²	C₂₃×#427	1	2	yes	—
228	C₂²	C₄×D₈×C₂₃	1	4	yes	—
229	C₂²	C₄×SD₁₆×C₂₃	1	8	yes	—
230	C₂²	C₄×Q₁₆×C₂₃	1	4	yes	—
231	C₄	—	2	4	no	—
232	C₄	—	4	16	no	—
233	C₄	C₂₃×#340	1	8	yes	direct product factor
234	C₄	C₄³×C₂₃	1	1	yes	characteristic
235	C₄	C₂₃×#376	1	1	yes	characteristic
236	C₄	C₄×C₂²⋊C₄×C₂₃	1	2	yes	—
237	C₄	C₄×C₄⋊C₄×C₂₃	1	4	yes	—
238	C₂	C₄×C₂₃×#95	1	1	yes	characteristic
239	C₂	C₂×C₂₃×#340	1	1	yes	characteristic
240	C₂	C₂₃×#1735	1	1	yes	characteristic
241	C₂	C₄×C₂₃×#102	1	4	yes	—
242	C₂	C₄×C₂₃×#103	1	4	yes	—
243	C₂	C₄×C₂₃×#106	1	2	yes	—
244	C₂	C₄×C₂₃×#107	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₂₃	D₄
commutator	C₄	C₄³×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle, prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₂×C₄×C₈×C₂₃	C₄	semidirect
11. C₄×C₄⋊C₄×C₂₃	C₄	semidirect
13. C₂₃×#340	C₄	direct (N × H)
58. C₂×C₈×C₂₃	C₄²	semidirect
61. C₄⋊C₄×C₂₃	C₄²	semidirect
66. C₄×#340	C₂₃	direct (N × H)
114. C₄×C₂₃	#340	direct (N × H)
124. C₂×C₄×C₈	C₄×C₂₃	semidirect
126. C₄×C₄⋊C₄	C₄×C₂₃	semidirect
128. #340	C₄×C₂₃	direct (N × H)
157. C₂₃	C₄×#340	direct (N × H)
181. C₂×C₈	C₄²×C₂₃	semidirect
184. C₄⋊C₄	C₄²×C₂₃	semidirect
233. C₄	C₂₃×#340	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×C₈×C₂₃	2
C₂×C₄×C₄⋊C₄×C₂₃	2
C₂×C₂₃×#340	2
C₄×#340	23

Subgroups — C4×C23×#340 #19158328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂₃×#340 #19158328