Subgroups — C₂₃×#13854 #19161480

Order 5888.

1422 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×C₄×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal
2	C₂×C₂₃×#376	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1442	C₂	1	8	yes	maximal
4	C₄×C₂₃×#95	C₂	1	4	yes	maximal
5	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
6	C₂²×C₄²×C₂₃	C₂²	1	4	yes	—
7	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₄×C₄⋊C₄×C₂₃	—	2	8	no	—
10	C₂₃×#376	C₄	1	8	yes	complemented
11	C₂×C₂₃×#95	C₂²	1	16	yes	—
12	C₂×C₂₃×#95	C₂²	1	4	yes	—
13	C₂×C₂₃×#95	C₂²	1	4	yes	—
14	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
15	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
16	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
17	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
19	C₂×C₄⋊C₄×C₂₃	—	2	16	no	—
20	C₂×C₄²×C₂₃	—	2	8	no	—
21	C₂×C₄²×C₂₃	—	2	8	no	—
22	C₂×C₄²×C₂₃	—	2	16	no	—
23	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	—
24	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₂₃	D₄	1	4	yes	—
26	C₂×C₄²×C₂₃	Q₈	1	4	yes	—
27	C₂₃×#95	—	2	8	no	—
28	C₂₃×#95	—	2	32	no	—
29	C₂₃×#95	C₂×C₄	1	16	yes	—
30	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	—	2	8	no	—
32	C₂²×C₄×C₂₃	—	2	8	no	—
33	C₂²×C₄×C₂₃	—	2	8	no	—
34	C₂²×C₄×C₂₃	—	2	16	no	—
35	C₂²×C₄×C₂₃	—	2	8	no	—
36	C₂²×C₄×C₂₃	—	2	16	no	—
37	C₂²×C₄×C₂₃	—	2	16	no	—
38	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
39	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
40	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
42	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂×Q₈	1	4	yes	—
44	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
45	C₂²×C₄×C₂₃	#41	1	16	yes	—
46	C₂²×C₄×C₂₃	#41	1	8	yes	—
47	C₄⋊C₄×C₂₃	—	2	32	no	—
48	C₄²×C₂₃	—	2	16	no	—
49	C₄²×C₂₃	—	4	16	no	—
50	C₄²×C₂₃	C₄⋊C₄	1	16	yes	—
51	#13854	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
53	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
54	C₂³×C₂₃	C₂²×Q₈	1	1	yes	characteristic
55	C₂³×C₂₃	C₂×#41	1	4	yes	—
56	C₂³×C₂₃	C₂×#41	1	8	yes	—
57	C₂×C₄×C₂₃	—	2	8	no	—
58	C₂×C₄×C₂₃	—	2	8	no	—
59	C₂×C₄×C₂₃	—	2	8	no	—
60	C₂×C₄×C₂₃	—	2	8	no	—
61	C₂×C₄×C₂₃	—	2	16	no	—
62	C₂×C₄×C₂₃	—	2	16	no	—
63	C₂×C₄×C₂₃	—	2	32	no	—
64	C₂×C₄×C₂₃	—	4	8	no	—
65	C₂×C₄×C₂₃	—	4	8	no	—
66	C₂×C₄×C₂₃	—	4	16	no	—
67	C₂×C₄×C₂₃	—	4	16	no	—
68	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	8	yes	—
69	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
70	C₂×C₄×C₂₃	#117	1	8	yes	—
71	C₂×C₄×C₂₃	#124	1	8	yes	—
72	C₂×C₄×C₂₃	#125	1	8	yes	—
73	C₂×C₄×C₄⋊C₄	C₂×C₂₃	1	2	yes	—
74	C₂×#376	C₂×C₂₃	1	1	yes	characteristic
75	#1442	C₂×C₂₃	1	8	yes	—
76	C₄×#95	C₂×C₂₃	1	4	yes	—
77	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₂₃	C₂×#117	1	2	yes	—
79	C₂²×C₂₃	C₄×#41	1	4	yes	—
80	C₂²×C₂₃	#525	1	4	yes	—
81	C₂²×C₂₃	C₂×#124	1	2	yes	—
82	C₂²×C₂₃	C₂×#125	1	2	yes	—
83	C₂²×C₂₃	#529	1	16	yes	—
84	C₂²×C₂₃	#533	1	4	yes	—
85	C₄×C₂₃	—	2	8	no	—
86	C₄×C₂₃	—	4	16	no	—
87	C₄×C₂₃	—	4	8	no	—
88	C₄×C₂₃	#382	1	8	yes	—
89	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
90	C₂²×C₄²	C₂²×C₂₃	1	4	yes	—
91	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
92	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
93	C₄×C₄⋊C₄	—	2	8	no	—
94	#376	C₄×C₂₃	1	8	yes	complemented
95	C₂×#95	C₂²×C₂₃	1	16	yes	—
96	C₂×#95	C₂²×C₂₃	1	4	yes	—
97	C₂×#95	C₂²×C₂₃	1	4	yes	—
98	C₂×C₂₃	C₂×#382	1	1	yes	characteristic
99	C₂×C₂₃	#2296	1	2	yes	—
100	C₂×C₂₃	#2298	1	4	yes	—
101	C₂×C₂₃	#2398	1	4	yes	—
102	C₂×C₂₃	#2402	1	4	yes	—
103	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
104	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
105	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
106	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	2	8	no	—
108	C₂×C₄⋊C₄	—	2	16	no	—
109	C₂×C₄²	—	2	8	no	—
110	C₂×C₄²	—	2	8	no	—
111	C₂×C₄²	—	2	16	no	—
112	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	—
113	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
114	C₂×C₄²	D₄×C₂₃	1	4	yes	—
115	C₂×C₄²	Q₈×C₂₃	1	4	yes	—
116	#95	—	2	8	no	—
117	#95	—	2	32	no	—
118	#95	C₂×C₄×C₂₃	1	16	yes	—
119	C₂₃	#13854	1	1	yes	characteristic, direct product factor, Sylow
120	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
121	C₂²×C₄	—	2	8	no	—
122	C₂²×C₄	—	2	8	no	—
123	C₂²×C₄	—	2	16	no	—
124	C₂²×C₄	—	2	16	no	—
125	C₂²×C₄	—	2	8	no	—
126	C₂²×C₄	—	2	16	no	—
127	C₂²×C₄	—	2	8	no	—
128	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
129	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
130	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
131	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
132	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
133	C₂²×C₄	C₂×Q₈×C₂₃	1	4	yes	—
134	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
135	C₂²×C₄	#41×C₂₃	1	16	yes	—
136	C₂²×C₄	#41×C₂₃	1	8	yes	—
137	C₄⋊C₄	—	2	32	no	—
138	C₄²	—	2	16	no	—
139	C₄²	—	4	16	no	—
140	C₄²	C₄⋊C₄×C₂₃	1	16	yes	—
141	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
142	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
143	C₂³	C₂²×Q₈×C₂₃	1	1	yes	characteristic
144	C₂³	C₂×#41×C₂₃	1	4	yes	—
145	C₂³	C₂×#41×C₂₃	1	8	yes	—
146	C₂×C₄	—	2	8	no	—
147	C₂×C₄	—	2	8	no	—
148	C₂×C₄	—	2	8	no	—
149	C₂×C₄	—	2	8	no	—
150	C₂×C₄	—	2	16	no	—
151	C₂×C₄	—	2	16	no	—
152	C₂×C₄	—	2	32	no	—
153	C₂×C₄	—	4	8	no	—
154	C₂×C₄	—	4	8	no	—
155	C₂×C₄	—	4	16	no	—
156	C₂×C₄	—	4	16	no	—
157	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	8	yes	—
158	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
159	C₂×C₄	C₂₃×#117	1	8	yes	—
160	C₂×C₄	C₂₃×#124	1	8	yes	—
161	C₂×C₄	C₂₃×#125	1	8	yes	—
162	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
163	C₂²	C₂×C₂₃×#117	1	2	yes	—
164	C₂²	C₄×#41×C₂₃	1	4	yes	—
165	C₂²	C₂₃×#525	1	4	yes	—
166	C₂²	C₂×C₂₃×#124	1	2	yes	—
167	C₂²	C₂×C₂₃×#125	1	2	yes	—
168	C₂²	C₂₃×#529	1	16	yes	—
169	C₂²	C₂₃×#533	1	4	yes	—
170	C₄	—	2	8	no	—
171	C₄	—	4	16	no	—
172	C₄	—	4	8	no	—
173	C₄	C₂₃×#382	1	8	yes	—
174	C₂	C₂×C₂₃×#382	1	1	yes	characteristic
175	C₂	C₂₃×#2296	1	2	yes	—
176	C₂	C₂₃×#2298	1	4	yes	—
177	C₂	C₂₃×#2398	1	4	yes	—
178	C₂	C₂₃×#2402	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₂₃	C₂³
commutator	C₂³	C₂³×C₄×C₂₃
Frattini, cosocle	C₂⁴	C₂⁴×C₂₃
socle, prime radical	C₂⁴×C₂₃	C₂⁴
LCS	G ▷ C₂³ ▷ 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×#376	C₄	semidirect
51. #13854	C₂₃	direct (N × H)
94. #376	C₄×C₂₃	semidirect
119. C₂₃	#13854	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×C₄⋊C₄×C₂₃	2
C₂×C₂₃×#376	2
C₂₃×#1442	2
C₄×C₂₃×#95	2
#13854	23

Subgroups — C23×#13854 #19161480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13854 #19161480