Subgroups — C₄×C₂₃×#384 #19161398

Order 5888.

2094 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#384	C₂	1	2	yes	maximal
2	C₂×C₄×C₄⋊C₄×C₂₃	C₂	1	6	yes	maximal
3	C₂×C₄³×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×C₄⋊#95	C₂	1	6	yes	maximal
5	C₂²×C₄⋊C₄×C₂₃	C₂²	1	12	yes	—
6	C₂²×C₄²×C₂₃	C₂²	1	4	yes	—
7	C₂²×C₄²×C₂₃	C₂²	1	3	yes	—
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₂₃	C₂²	1	3	yes	—
10	C₂₃×#384	C₄	1	16	yes	direct product factor
11	C₄×C₄⋊C₄×C₂₃	—	2	24	no	—
12	C₂×C₂₃×#95	C₂²	1	12	yes	—
13	C₄³×C₂₃	C₄	1	8	yes	complemented
14	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
15	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
16	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
17	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₃	—	2	24	no	—
19	C₂×C₄⋊C₄×C₂₃	—	2	24	no	—
20	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	48	yes	—
21	C₂×C₄²×C₂₃	—	2	12	no	—
22	C₂×C₄²×C₂₃	—	2	24	no	—
23	C₂×C₄²×C₂₃	C₂×C₄	1	12	yes	—
24	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	—
26	C₂×C₄²×C₂₃	C₂×C₄	1	12	yes	—
27	C₂×C₄²×C₂₃	D₄	1	12	yes	—
28	C₂×C₄²×C₂₃	Q₈	1	12	yes	—
29	C₂₃×#95	—	2	24	no	—
30	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	—	2	12	no	—
32	C₂²×C₄×C₂₃	—	2	24	no	—
33	C₂²×C₄×C₂₃	—	2	24	no	—
34	C₂²×C₄×C₂₃	C₂²×C₄	1	16	yes	—
35	C₂²×C₄×C₂₃	C₂²×C₄	1	6	yes	—
36	C₂²×C₄×C₂₃	C₂²×C₄	1	6	yes	—
37	C₂²×C₄×C₂₃	C₂²×C₄	1	12	yes	—
38	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
39	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
40	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
41	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
42	C₂²×C₄×C₂₃	C₂×Q₈	1	6	yes	—
43	C₂²×C₄×C₂₃	C₂×Q₈	1	6	yes	—
44	C₂²×C₄×C₂₃	C₂×Q₈	1	6	yes	—
45	C₂²×C₄×C₂₃	#41	1	12	yes	—
46	C₄⋊C₄×C₂₃	—	2	96	no	—
47	C₄²×C₂₃	—	2	24	no	—
48	C₄²×C₂₃	—	4	16	no	—
49	C₄²×C₂₃	C₄²	1	16	yes	complemented
50	C₄²×C₂₃	C₄⋊C₄	1	48	yes	complemented
51	C₄×#384	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
53	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
54	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
55	C₂³×C₂₃	C₂²×Q₈	1	3	yes	—
56	C₂³×C₂₃	C₂×#41	1	6	yes	—
57	C₂×C₄×C₂₃	—	2	12	no	—
58	C₂×C₄×C₂₃	—	2	12	no	—
59	C₂×C₄×C₂₃	—	2	12	no	—
60	C₂×C₄×C₂₃	—	2	48	no	—
61	C₂×C₄×C₂₃	—	4	8	no	—
62	C₂×C₄×C₂₃	—	4	24	no	—
63	C₂×C₄×C₂₃	C₂×C₄²	1	12	yes	—
64	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	12	yes	—
65	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	12	yes	—
66	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	12	yes	—
67	C₂×C₄×C₂₃	#117	1	12	yes	—
68	C₂×C₄×C₂₃	C₄×D₄	1	24	yes	—
69	C₂×C₄×C₂₃	C₄×Q₈	1	24	yes	—
70	C₂×C₄×C₂₃	#127	1	4	yes	—
71	C₂×C₄×C₂₃	C₄⋊Q₈	1	12	yes	—
72	C₂×#384	C₂×C₂₃	1	2	yes	—
73	C₂×C₄×C₄⋊C₄	C₂×C₂₃	1	6	yes	—
74	C₂×C₄³	C₂×C₂₃	1	1	yes	characteristic
75	C₄⋊#95	C₂×C₂₃	1	6	yes	—
76	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
77	C₂²×C₂₃	C₂²×C₄⋊C₄	1	3	yes	—
78	C₂²×C₂₃	C₂×#117	1	3	yes	—
79	C₂²×C₂₃	C₂×C₄×D₄	1	6	yes	—
80	C₂²×C₂₃	C₂×C₄×Q₈	1	6	yes	—
81	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
82	C₂²×C₂₃	C₂×C₄⋊Q₈	1	3	yes	—
83	C₂²×C₂₃	#532	1	6	yes	—
84	C₂²×C₂₃	#533	1	6	yes	—
85	C₄×C₂₃	—	2	12	no	—
86	C₄×C₂₃	—	4	16	no	—
87	C₄×C₂₃	C₄×C₄⋊C₄	1	24	yes	complemented
88	C₄×C₂₃	#384	1	8	yes	direct product factor
89	C₂²×C₄⋊C₄	C₂²×C₂₃	1	12	yes	—
90	C₂²×C₄²	C₂²×C₂₃	1	4	yes	—
91	C₂²×C₄²	C₂²×C₂₃	1	3	yes	—
92	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
93	C₂²×C₄²	C₂²×C₂₃	1	3	yes	—
94	#384	C₄×C₂₃	1	16	yes	direct product factor
95	C₄×C₄⋊C₄	—	2	24	no	—
96	C₂×#95	C₂²×C₂₃	1	12	yes	—
97	C₄³	C₄×C₂₃	1	8	yes	complemented
98	C₂×C₂₃	C₂×C₄×C₄⋊C₄	1	3	yes	—
99	C₂×C₂₃	C₂×#384	1	1	yes	characteristic
100	C₂×C₂₃	#2285	1	3	yes	—
101	C₂×C₂₃	C₄×#127	1	2	yes	—
102	C₂×C₂₃	C₄×C₄⋊Q₈	1	6	yes	—
103	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
104	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
105	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
106	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	2	24	no	—
108	C₂×C₄⋊C₄	—	2	24	no	—
109	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	48	yes	—
110	C₂×C₄²	—	2	12	no	—
111	C₂×C₄²	—	2	24	no	—
112	C₂×C₄²	C₂×C₄×C₂₃	1	12	yes	—
113	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
114	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	—
115	C₂×C₄²	C₂×C₄×C₂₃	1	12	yes	—
116	C₂×C₄²	D₄×C₂₃	1	12	yes	—
117	C₂×C₄²	Q₈×C₂₃	1	12	yes	—
118	#95	—	2	24	no	—
119	C₂₃	C₄×#384	1	1	yes	characteristic, direct product factor, Sylow
120	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
121	C₂²×C₄	—	2	12	no	—
122	C₂²×C₄	—	2	24	no	—
123	C₂²×C₄	—	2	24	no	—
124	C₂²×C₄	C₂²×C₄×C₂₃	1	16	yes	—
125	C₂²×C₄	C₂²×C₄×C₂₃	1	6	yes	—
126	C₂²×C₄	C₂²×C₄×C₂₃	1	6	yes	—
127	C₂²×C₄	C₂²×C₄×C₂₃	1	12	yes	—
128	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
129	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
130	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
131	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
132	C₂²×C₄	C₂×Q₈×C₂₃	1	6	yes	—
133	C₂²×C₄	C₂×Q₈×C₂₃	1	6	yes	—
134	C₂²×C₄	C₂×Q₈×C₂₃	1	6	yes	—
135	C₂²×C₄	#41×C₂₃	1	12	yes	—
136	C₄⋊C₄	—	2	96	no	—
137	C₄²	—	2	24	no	—
138	C₄²	—	4	16	no	—
139	C₄²	C₄²×C₂₃	1	16	yes	complemented
140	C₄²	C₄⋊C₄×C₂₃	1	48	yes	complemented
141	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
142	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
143	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
144	C₂³	C₂²×Q₈×C₂₃	1	3	yes	—
145	C₂³	C₂×#41×C₂₃	1	6	yes	—
146	C₂×C₄	—	2	12	no	—
147	C₂×C₄	—	2	12	no	—
148	C₂×C₄	—	2	12	no	—
149	C₂×C₄	—	2	48	no	—
150	C₂×C₄	—	4	8	no	—
151	C₂×C₄	—	4	24	no	—
152	C₂×C₄	C₂×C₄²×C₂₃	1	12	yes	—
153	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	12	yes	—
154	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	12	yes	—
155	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	12	yes	—
156	C₂×C₄	C₂₃×#117	1	12	yes	—
157	C₂×C₄	C₄×D₄×C₂₃	1	24	yes	—
158	C₂×C₄	C₄×Q₈×C₂₃	1	24	yes	—
159	C₂×C₄	C₂₃×#127	1	4	yes	—
160	C₂×C₄	C₂₃×C₄⋊Q₈	1	12	yes	—
161	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
162	C₂²	C₂²×C₄⋊C₄×C₂₃	1	3	yes	—
163	C₂²	C₂×C₂₃×#117	1	3	yes	—
164	C₂²	C₂×C₄×D₄×C₂₃	1	6	yes	—
165	C₂²	C₂×C₄×Q₈×C₂₃	1	6	yes	—
166	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
167	C₂²	C₂×C₂₃×C₄⋊Q₈	1	3	yes	—
168	C₂²	C₂₃×#532	1	6	yes	—
169	C₂²	C₂₃×#533	1	6	yes	—
170	C₄	—	2	12	no	—
171	C₄	—	4	16	no	—
172	C₄	C₄×C₄⋊C₄×C₂₃	1	24	yes	complemented
173	C₄	C₂₃×#384	1	8	yes	direct product factor
174	C₂	C₂×C₄×C₄⋊C₄×C₂₃	1	3	yes	—
175	C₂	C₂×C₂₃×#384	1	1	yes	characteristic
176	C₂	C₂₃×#2285	1	3	yes	—
177	C₂	C₄×C₂₃×#127	1	2	yes	—
178	C₂	C₄×C₂₃×C₄⋊Q₈	1	6	yes	—

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

Complements

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

N	H ≅ G/N	type
10. C₂₃×#384	C₄	direct (N × H)
13. C₄³×C₂₃	C₄	semidirect
49. C₄²×C₂₃	C₄²	semidirect
50. C₄²×C₂₃	C₄⋊C₄	semidirect
51. C₄×#384	C₂₃	direct (N × H)
87. C₄×C₂₃	C₄×C₄⋊C₄	semidirect
88. C₄×C₂₃	#384	direct (N × H)
94. #384	C₄×C₂₃	direct (N × H)
97. C₄³	C₄×C₂₃	semidirect
119. C₂₃	C₄×#384	direct (N × H)
139. C₄²	C₄²×C₂₃	semidirect
140. C₄²	C₄⋊C₄×C₂₃	semidirect
172. C₄	C₄×C₄⋊C₄×C₂₃	semidirect
173. C₄	C₂₃×#384	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×#384	2
C₂×C₄×C₄⋊C₄×C₂₃	2
C₂×C₄³×C₂₃	2
C₂₃×C₄⋊#95	2
C₄×#384	23

Subgroups — C4×C23×#384 #19161398

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂₃×#384 #19161398