Subgroups — C₂₃×#33311 #19180937

Order 5888.

830 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#499	C₂	1	4	yes	maximal
2	C₂₃×#2859	C₂	1	2	yes	maximal
3	C₂₃×#1846	C₂	1	4	yes	maximal
4	C₂₃×#1845	C₂	1	2	yes	maximal
5	C₂₃×#1844	C₂	1	2	yes	maximal
6	C₄²×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
7	C₂×C₂₃×#125	C₂²	1	4	yes	—
8	C₂₃×#499	—	2	16	no	—
9	C₂×C₂₃×#107	C₂²	1	8	yes	—
10	C₂×C₂₃×#106	C₂²	1	8	yes	—
11	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
15	C₂₃×#389	—	2	8	no	—
16	C₂₃×#384	C₂²	1	4	yes	—
17	C₂₃×#382	C₂²	1	4	yes	—
18	C₄³×C₂₃	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
20	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
21	C₂₃×#125	—	2	16	no	—
22	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
23	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
24	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
25	C₂×C₄²×C₂₃	D₄	1	4	yes	—
26	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
29	C₂₃×#107	—	2	16	no	—
30	C₂₃×#106	—	2	16	no	—
31	C₄×C₈×C₂₃	D₄	1	8	yes	—
32	C₄×C₈×C₂₃	Q₈	1	4	yes	—
33	C₄×C₈×C₂₃	Q₈	1	4	yes	—
34	C₄×C₈×C₂₃	Q₈	1	8	yes	—
35	C₂₃×#95	—	2	8	no	—
36	C₂²×C₄×C₂₃	—	2	8	no	—
37	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
39	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
40	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
41	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
42	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
43	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
44	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
45	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
46	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
47	C₄⋊C₄×C₂₃	—	2	16	no	—
48	C₄⋊C₄×C₂₃	—	4	16	no	—
49	C₄⋊C₄×C₂₃	—	4	16	no	—
50	C₄²×C₂₃	—	2	8	no	—
51	C₄²×C₂₃	C₂×D₄	1	4	yes	—
52	C₄²×C₂₃	C₂×Q₈	1	4	yes	—
53	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
54	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
55	#33311	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
57	C₂×C₄×C₂₃	—	2	4	no	—
58	C₂×C₄×C₂₃	—	2	4	no	—
59	C₂×C₄×C₂₃	—	4	8	no	—
60	C₂×C₄×C₂₃	—	4	16	no	—
61	C₂×C₄×C₂₃	#127	1	4	yes	—
62	C₂×C₄×C₂₃	C₄⋊Q₈	1	4	yes	—
63	C₂×C₄×C₂₃	C₄⋊Q₈	1	4	yes	—
64	C₂×C₄×C₂₃	#135	1	8	yes	—
65	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
66	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
69	C₈×C₂₃	C₄⋊Q₈	1	8	yes	—
70	C₈×C₂₃	C₄⋊Q₈	1	8	yes	—
71	C₂×#499	C₂×C₂₃	1	4	yes	—
72	#2859	C₂×C₂₃	1	2	yes	—
73	#1846	C₂×C₂₃	1	4	yes	—
74	#1845	C₂×C₂₃	1	2	yes	—
75	#1844	C₂×C₂₃	1	2	yes	—
76	C₄²×C₈	C₂×C₂₃	1	1	yes	characteristic
77	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
78	C₂²×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
79	C₂²×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
80	C₂²×C₂₃	C₂×#135	1	4	yes	—
81	C₄×C₂₃	—	2	4	no	—
82	C₄×C₂₃	—	2	4	no	—
83	C₄×C₂₃	—	8	8	no	—
84	C₄×C₂₃	#499	1	8	yes	complemented
85	C₄×C₂₃	C₂×C₄⋊Q₈	1	2	yes	—
86	C₄×C₂₃	C₂×C₄⋊Q₈	1	2	yes	—
87	C₂×#125	C₂²×C₂₃	1	4	yes	—
88	#499	—	2	16	no	—
89	C₂×#107	C₂²×C₂₃	1	8	yes	—
90	C₂×#106	C₂²×C₂₃	1	8	yes	—
91	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
92	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
94	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
95	#389	—	2	8	no	—
96	#384	C₂²×C₂₃	1	4	yes	—
97	#382	C₂²×C₂₃	1	4	yes	—
98	C₄³	C₂²×C₂₃	1	1	yes	characteristic
99	C₂×C₂₃	#2868	1	1	yes	characteristic
100	C₂×C₂₃	#3147	1	2	yes	—
101	C₂×C₂₃	C₂×#499	1	2	yes	—
102	C₂×C₂₃	#3160	1	2	yes	—
103	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
104	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
105	#125	—	2	16	no	—
106	C₂×C₄⋊C₄	—	2	8	no	—
107	C₂×C₄⋊C₄	—	2	8	no	—
108	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
109	C₂×C₄²	D₄×C₂₃	1	4	yes	—
110	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
111	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
112	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
113	#107	—	2	16	no	—
114	#106	—	2	16	no	—
115	C₄×C₈	D₄×C₂₃	1	8	yes	—
116	C₄×C₈	Q₈×C₂₃	1	4	yes	—
117	C₄×C₈	Q₈×C₂₃	1	4	yes	—
118	C₄×C₈	Q₈×C₂₃	1	8	yes	—
119	#95	—	2	8	no	—
120	C₂₃	#33311	1	1	yes	characteristic, direct product factor, Sylow
121	C₂²×C₄	—	2	8	no	—
122	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
123	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
124	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
125	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
126	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
127	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
128	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
129	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
130	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
131	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
132	C₄⋊C₄	—	2	16	no	—
133	C₄⋊C₄	—	4	16	no	—
134	C₄⋊C₄	—	4	16	no	—
135	C₄²	—	2	8	no	—
136	C₄²	C₂×D₄×C₂₃	1	4	yes	—
137	C₄²	C₂×Q₈×C₂₃	1	4	yes	—
138	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
139	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
140	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	4	16	no	—
144	C₂×C₄	—	4	8	no	—
145	C₂×C₄	C₂₃×#127	1	4	yes	—
146	C₂×C₄	C₂₃×C₄⋊Q₈	1	4	yes	—
147	C₂×C₄	C₂₃×C₄⋊Q₈	1	4	yes	—
148	C₂×C₄	C₂₃×#135	1	8	yes	—
149	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
150	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
151	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
152	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
153	C₈	C₂₃×C₄⋊Q₈	1	8	yes	—
154	C₈	C₂₃×C₄⋊Q₈	1	8	yes	—
155	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
156	C₂²	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
157	C₂²	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
158	C₂²	C₂×C₂₃×#135	1	4	yes	—
159	C₄	—	2	4	no	—
160	C₄	—	2	4	no	—
161	C₄	—	8	8	no	—
162	C₄	C₂₃×#499	1	8	yes	complemented
163	C₄	C₂×C₂₃×C₄⋊Q₈	1	2	yes	—
164	C₄	C₂×C₂₃×C₄⋊Q₈	1	2	yes	—
165	C₂	C₂₃×#2868	1	1	yes	characteristic
166	C₂	C₂₃×#3147	1	2	yes	—
167	C₂	C₂×C₂₃×#499	1	2	yes	—
168	C₂	C₂₃×#3160	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
55. #33311	C₂₃	direct (N × H)
84. C₄×C₂₃	#499	semidirect
120. C₂₃	#33311	direct (N × H)
162. C₄	C₂₃×#499	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#499	2
C₂₃×#2859	2
C₂₃×#1846	2
C₂₃×#1845	2
C₂₃×#1844	2
C₄²×C₈×C₂₃	2
#33311	23

Subgroups — C23×#33311 #19180937

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33311 #19180937