Subgroups — C₄×C₂₃×#364 #19160281

Order 5888.

398 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#364	C₂	1	2	yes	maximal
2	C₄×C₂₃×M₃₂	C₂	1	2	yes	maximal
3	C₄²×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1376	C₂	1	2	yes	maximal
5	C₂×C₂₃×M₃₂	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₄³×C₂₃	C₄	1	1	yes	characteristic
13	C₂₃×#364	C₄	1	8	yes	direct product factor
14	C₂₃×#346	—	2	2	no	—
15	C₄×C₁₆×C₂₃	—	2	2	no	—
16	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₂₃	—	2	2	no	—
21	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂₃×M₃₂	—	2	4	no	—
25	C₂₃×M₃₂	C₂×C₄	1	8	yes	—
26	C₂×C₁₆×C₂₃	—	2	4	no	—
27	C₄×C₈×C₂₃	—	2	2	no	—
28	C₄×C₈×C₂₃	—	2	2	no	—
29	C₄×C₈×C₂₃	C₈	1	4	yes	—
30	C₄×C₈×C₂₃	C₈	1	4	yes	—
31	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
34	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
35	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	—	2	1	no	—
37	C₂²×C₄×C₂₃	—	2	1	no	—
38	C₂²×C₄×C₂₃	M₁₆	1	2	yes	—
39	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₂₃	—	2	1	no	—
41	C₂×C₈×C₂₃	—	2	1	no	—
42	C₂×C₈×C₂₃	C₄²	1	2	yes	—
43	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
50	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
51	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₂₃	#41	1	2	yes	—
54	C₄²×C₂₃	—	2	8	no	—
55	C₄²×C₂₃	—	2	1	no	—
56	C₄²×C₂₃	—	2	1	no	—
57	C₄²×C₂₃	C₄²	1	2	yes	—
58	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
59	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
60	C₄²×C₂₃	C₂×C₈	1	2	yes	—
61	C₄²×C₂₃	C₂×C₈	1	2	yes	—
62	C₁₆×C₂₃	—	2	8	no	—
63	C₄×#364	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂³×C₂₃	C₂×M₁₆	1	1	yes	characteristic
65	C₂×C₄×C₂₃	—	2	4	no	—
66	C₂×C₄×C₂₃	—	2	4	no	—
67	C₂×C₄×C₂₃	—	2	2	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	4	no	—
70	C₂×C₄×C₂₃	C₄⋊C₈	1	2	yes	—
71	C₂×C₄×C₂₃	C₄⋊C₈	1	2	yes	—
72	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
75	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
78	C₈×C₂₃	—	2	2	no	—
79	C₈×C₂₃	C₄×C₈	1	4	yes	—
80	C₈×C₂₃	C₄⋊C₈	1	2	yes	—
81	C₈×C₂₃	C₄⋊C₈	1	2	yes	—
82	C₈×C₂₃	C₄×D₄	1	2	yes	—
83	C₈×C₂₃	C₄×Q₈	1	2	yes	—
84	C₂×#364	C₂×C₂₃	1	2	yes	—
85	C₄×M₃₂	C₂×C₂₃	1	2	yes	—
86	C₄²×C₈	C₂×C₂₃	1	1	yes	characteristic
87	#1376	C₂×C₂₃	1	2	yes	—
88	C₂²×C₂₃	—	2	2	no	—
89	C₂²×C₂₃	C₄×M₁₆	1	1	yes	characteristic
90	C₂²×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
91	C₂²×C₂₃	#431	1	1	yes	characteristic
92	C₄×C₂₃	—	2	4	no	—
93	C₄×C₂₃	—	2	4	no	—
94	C₄×C₂₃	—	2	2	no	—
95	C₄×C₂₃	#364	1	4	yes	direct product factor
96	C₄×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
97	C₄×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
98	C₄×C₂₃	C₂×C₄⋊C₈	1	1	yes	characteristic
99	C₄×C₂₃	#431	1	1	yes	characteristic
100	C₂×M₃₂	C₂²×C₂₃	1	4	yes	—
101	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic
102	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic
103	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
107	C₄³	C₄×C₂₃	1	1	yes	characteristic
108	#364	C₄×C₂₃	1	8	yes	direct product factor
109	#346	—	2	2	no	—
110	C₄×C₁₆	—	2	2	no	—
111	C₂×C₂₃	—	2	1	no	—
112	C₂×C₂₃	—	2	1	no	—
113	C₂×C₂₃	C₄×C₄⋊C₈	1	1	yes	characteristic
114	C₂×C₂₃	C₂×#364	1	2	yes	—
115	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
116	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
117	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
118	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
119	C₂×C₄²	—	2	2	no	—
120	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
121	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
122	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
123	M₃₂	—	2	4	no	—
124	M₃₂	C₂×C₄×C₂₃	1	8	yes	—
125	C₂×C₁₆	—	2	4	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	C₈×C₂₃	1	4	yes	—
129	C₄×C₈	C₈×C₂₃	1	4	yes	—
130	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
131	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
132	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
133	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
134	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
135	C₂₃	C₄×#364	1	1	yes	characteristic, direct product factor, Sylow
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	M₁₆×C₂₃	1	2	yes	—
139	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	C₄²×C₂₃	1	2	yes	—
143	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
144	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
145	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
146	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
147	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
148	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
149	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
150	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
151	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
152	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
153	C₂×C₈	#41×C₂₃	1	2	yes	—
154	C₄²	—	2	8	no	—
155	C₄²	—	2	1	no	—
156	C₄²	—	2	1	no	—
157	C₄²	C₄²×C₂₃	1	2	yes	—
158	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
159	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
160	C₄²	C₂×C₈×C₂₃	1	2	yes	—
161	C₄²	C₂×C₈×C₂₃	1	2	yes	—
162	C₁₆	—	2	8	no	—
163	C₂³	C₂×M₁₆×C₂₃	1	1	yes	characteristic
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	C₂₃×C₄⋊C₈	1	2	yes	—
170	C₂×C₄	C₂₃×C₄⋊C₈	1	2	yes	—
171	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
173	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
174	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
175	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
176	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
177	C₈	—	2	2	no	—
178	C₈	C₄×C₈×C₂₃	1	4	yes	—
179	C₈	C₂₃×C₄⋊C₈	1	2	yes	—
180	C₈	C₂₃×C₄⋊C₈	1	2	yes	—
181	C₈	C₄×D₄×C₂₃	1	2	yes	—
182	C₈	C₄×Q₈×C₂₃	1	2	yes	—
183	C₂²	—	2	2	no	—
184	C₂²	C₄×M₁₆×C₂₃	1	1	yes	characteristic
185	C₂²	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
186	C₂²	C₂₃×#431	1	1	yes	characteristic
187	C₄	—	2	4	no	—
188	C₄	—	2	4	no	—
189	C₄	—	2	2	no	—
190	C₄	C₂₃×#364	1	4	yes	direct product factor
191	C₄	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
192	C₄	C₂×C₄×C₈×C₂₃	1	1	yes	characteristic
193	C₄	C₂×C₂₃×C₄⋊C₈	1	1	yes	characteristic
194	C₄	C₂₃×#431	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	C₄×C₂₃×C₄⋊C₈	1	1	yes	characteristic
198	C₂	C₂×C₂₃×#364	1	2	yes	—

	subgroup	G/H
center	C₄×C₈×C₂₃	D₄
commutator	C₄	C₂×C₄×C₈×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×C₄⋊C₈
prime radical	C₂³×C₂₃	C₂×M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂₃×#364	C₄	direct (N × H)
63. C₄×#364	C₂₃	direct (N × H)
95. C₄×C₂₃	#364	direct (N × H)
108. #364	C₄×C₂₃	direct (N × H)
135. C₂₃	C₄×#364	direct (N × H)
190. C₄	C₂₃×#364	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×#364	2
C₄×C₂₃×M₃₂	2
C₄²×C₈×C₂₃	2
C₂₃×#1376	2
C₄×#364	23

Subgroups — C4×C23×#364 #19160281

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂₃×#364 #19160281