Subgroups — C₂₃×#2296 #15669915

Order 2944.

656 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#389	C₂	1	4	yes	maximal, complemented
5	C₂₃×#385	C₂	1	4	yes	maximal
6	C₄×C₄⋊C₄×C₂₃	C₂	1	4	yes	maximal, complemented
7	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
8	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#117	—	2	4	no	—
10	C₂₃×#117	C₄	1	8	yes	complemented
11	C₂×C₄⋊C₄×C₂₃	C₂²	1	8	yes	—
12	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
14	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₂₃	—	2	4	no	—
17	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
18	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
20	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
21	C₂₃×#95	C₂²	1	8	yes	—
22	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₂₃	—	2	4	no	—
24	C₂²×C₄×C₂₃	—	2	8	no	—
25	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₂₃	D₄	1	4	yes	—
27	C₂²×C₄×C₂₃	Q₈	1	4	yes	—
28	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
29	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
30	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
31	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₂₃	—	2	4	no	—
34	C₄⋊C₄×C₂₃	—	2	16	no	—
35	C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	complemented
36	C₂²⋊C₄×C₂₃	—	2	4	no	—
37	C₂²⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
38	C₄²×C₂₃	—	2	4	no	—
39	C₄²×C₂₃	—	2	4	no	—
40	C₄²×C₂₃	—	2	4	no	—
41	C₄²×C₂₃	C₂×C₄	1	8	yes	complemented
42	C₂³×C₂₃	—	2	4	no	—
43	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
44	C₂³×C₂₃	C₂×D₄	1	2	yes	—
45	C₂³×C₂₃	C₂×Q₈	1	2	yes	—
46	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
47	C₂×C₄×C₂₃	—	2	16	no	—
48	C₂×C₄×C₂₃	—	2	4	no	—
49	C₂×C₄×C₂₃	—	2	8	no	—
50	C₂×C₄×C₂₃	—	2	4	no	—
51	C₂×C₄×C₂₃	—	2	4	no	—
52	C₂×C₄×C₂₃	—	2	8	no	—
53	C₂×C₄×C₂₃	C₄⋊C₄	1	16	yes	complemented
54	C₂×C₄×C₂₃	C₂²×C₄	1	8	yes	—
55	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
60	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₂₃	#41	1	8	yes	—
62	#2296	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂²×C₂₃	—	2	4	no	—
64	C₂²×C₂₃	—	2	4	no	—
65	C₂²×C₂₃	—	2	4	no	—
66	C₂²×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
67	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
68	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
69	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
70	C₂²×C₂₃	C₂×#41	1	4	yes	—
71	C₄×C₂₃	—	2	8	no	—
72	C₄×C₂₃	—	2	8	no	—
73	C₄×C₂₃	—	4	4	no	—
74	C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
75	C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
76	C₂×#117	C₂×C₂₃	1	1	yes	characteristic
77	C₂×#117	C₂×C₂₃	1	1	yes	characteristic
78	C₂²×C₄²	C₂×C₂₃	1	1	yes	characteristic
79	#389	C₂×C₂₃	1	4	yes	complemented
80	#385	C₂×C₂₃	1	4	yes	—
81	C₄×C₄⋊C₄	C₂×C₂₃	1	4	yes	complemented
82	C₂×C₂₃	—	2	4	no	—
83	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₂₃	C₄×#41	1	2	yes	—
85	C₂×C₂₃	#525	1	2	yes	—
86	C₂×C₂₃	#533	1	2	yes	—
87	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
88	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
89	#117	—	2	4	no	—
90	#117	C₄×C₂₃	1	8	yes	complemented
91	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
92	C₂×C₄⋊C₄	C₂²×C₂₃	1	8	yes	—
93	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
94	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
95	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
96	C₂×C₄²	—	2	4	no	—
97	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
98	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
99	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
100	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
101	#95	C₂²×C₂₃	1	8	yes	—
102	C₂₃	#2296	1	1	yes	characteristic, direct product factor, Sylow
103	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
104	C₂²×C₄	—	2	4	no	—
105	C₂²×C₄	—	2	8	no	—
106	C₂²×C₄	C₂×C₄×C₂₃	1	4	yes	—
107	C₂²×C₄	D₄×C₂₃	1	4	yes	—
108	C₂²×C₄	Q₈×C₂₃	1	4	yes	—
109	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
110	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
111	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
112	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
113	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	16	no	—
116	C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	complemented
117	C₂²⋊C₄	—	2	4	no	—
118	C₂²⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
119	C₄²	—	2	4	no	—
120	C₄²	—	2	4	no	—
121	C₄²	—	2	4	no	—
122	C₄²	C₂×C₄×C₂₃	1	8	yes	complemented
123	C₂³	—	2	4	no	—
124	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
125	C₂³	C₂×D₄×C₂₃	1	2	yes	—
126	C₂³	C₂×Q₈×C₂₃	1	2	yes	—
127	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
128	C₂×C₄	—	2	16	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	8	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	8	no	—
134	C₂×C₄	C₄⋊C₄×C₂₃	1	16	yes	complemented
135	C₂×C₄	C₂²×C₄×C₂₃	1	8	yes	—
136	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
137	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
138	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
139	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
140	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
141	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
142	C₂×C₄	#41×C₂₃	1	8	yes	—
143	C₂²	—	2	4	no	—
144	C₂²	—	2	4	no	—
145	C₂²	—	2	4	no	—
146	C₂²	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
147	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
148	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
149	C₂²	C₂²×Q₈×C₂₃	1	1	yes	characteristic
150	C₂²	C₂×#41×C₂₃	1	4	yes	—
151	C₄	—	2	8	no	—
152	C₄	—	2	8	no	—
153	C₄	—	4	4	no	—
154	C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
155	C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
156	C₂	—	2	4	no	—
157	C₂	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
158	C₂	C₄×#41×C₂₃	1	2	yes	—
159	C₂	C₂₃×#525	1	2	yes	—
160	C₂	C₂₃×#533	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₂₃×#389	C₂	semidirect
6. C₄×C₄⋊C₄×C₂₃	C₂	semidirect
10. C₂₃×#117	C₄	semidirect
35. C₄⋊C₄×C₂₃	C₂×C₄	semidirect
41. C₄²×C₂₃	C₂×C₄	semidirect
53. C₂×C₄×C₂₃	C₄⋊C₄	semidirect
62. #2296	C₂₃	direct (N × H)
79. #389	C₂×C₂₃	semidirect
81. C₄×C₄⋊C₄	C₂×C₂₃	semidirect
90. #117	C₄×C₂₃	semidirect
102. C₂₃	#2296	direct (N × H)
116. C₄⋊C₄	C₂×C₄×C₂₃	semidirect
122. C₄²	C₂×C₄×C₂₃	semidirect
134. C₂×C₄	C₄⋊C₄×C₂₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₃×#117	2
C₂²×C₄²×C₂₃	2
C₂₃×#389	2
C₂₃×#385	2
C₄×C₄⋊C₄×C₂₃	2
#2296	23

Subgroups — C23×#2296 #15669915

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2296 #15669915