Subgroups — C₂₃×#2298 #15669917

Order 2944.

524 conjugacy classes of subgroups.

The subgroup lattice has 152 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	C₂₃×#388	C₂	1	4	yes	maximal
3	C₂₃×#387	C₂	1	4	yes	maximal, complemented
4	C₄×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal, complemented
5	C₄×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal
6	C₂₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄³×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#117	C₄	1	8	yes	complemented
10	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
11	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
13	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₃	C₂²	1	4	yes	—
15	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#95	C₂²	1	4	yes	—
20	C₂₃×#95	C₂²	1	4	yes	—
21	C₂₃×#95	C₂²	1	4	yes	—
22	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₂₃	—	2	4	no	—
24	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
26	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
27	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
28	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
30	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₂₃	—	2	8	no	—
32	C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
33	C₂²⋊C₄×C₂₃	—	2	8	no	—
34	C₂²⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
35	C₄²×C₂₃	—	2	8	no	—
36	C₄²×C₂₃	—	2	4	no	—
37	C₄²×C₂₃	—	2	4	no	—
38	C₄²×C₂₃	C₂×C₄	1	4	yes	—
39	C₄²×C₂₃	C₂×C₄	1	4	yes	—
40	C₂³×C₂₃	—	2	4	no	—
41	C₂³×C₂₃	—	2	2	no	—
42	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
43	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
44	C₂×C₄×C₂₃	—	2	4	no	—
45	C₂×C₄×C₂₃	—	2	4	no	—
46	C₂×C₄×C₂₃	—	2	4	no	—
47	C₂×C₄×C₂₃	—	2	4	no	—
48	C₂×C₄×C₂₃	—	2	2	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	4	no	—
53	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
54	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
55	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₂₃	#41	1	8	yes	—
58	C₂×C₄×C₂₃	#41	1	4	yes	—
59	#2298	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂²×C₂₃	—	2	2	no	—
61	C₂²×C₂₃	—	2	4	no	—
62	C₂²×C₂₃	—	4	4	no	—
63	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
64	C₂²×C₂₃	C₂×#41	1	2	yes	—
65	C₂²×C₂₃	C₂×#41	1	4	yes	—
66	C₄×C₂₃	—	2	2	no	—
67	C₄×C₂₃	—	2	4	no	—
68	C₄×C₂₃	—	2	4	no	—
69	C₄×C₂₃	—	2	8	no	—
70	C₄×C₂₃	—	4	4	no	—
71	C₄×C₂₃	#117	1	4	yes	—
72	C₂×#117	C₂×C₂₃	1	1	yes	characteristic
73	#388	C₂×C₂₃	1	4	yes	—
74	#387	C₂×C₂₃	1	4	yes	complemented
75	C₄×C₄⋊C₄	C₂×C₂₃	1	2	yes	complemented
76	C₄×C₂²⋊C₄	C₂×C₂₃	1	2	yes	—
77	#376	C₂×C₂₃	1	1	yes	characteristic, complemented
78	C₄³	C₂×C₂₃	1	1	yes	characteristic, complemented
79	C₂×C₂₃	—	4	2	no	—
80	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
81	C₂×C₂₃	C₄×#41	1	2	yes	—
82	C₂×C₂₃	#529	1	4	yes	—
83	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
84	#117	C₄×C₂₃	1	8	yes	complemented
85	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
86	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
87	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
88	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
89	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
90	C₂×C₄²	C₂²×C₂₃	1	4	yes	—
91	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
92	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
93	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
94	#95	C₂²×C₂₃	1	4	yes	—
95	#95	C₂²×C₂₃	1	4	yes	—
96	#95	C₂²×C₂₃	1	4	yes	—
97	C₂₃	#2298	1	1	yes	characteristic, direct product factor, Sylow
98	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
99	C₂²×C₄	—	2	4	no	—
100	C₂²×C₄	C₂×C₄×C₂₃	1	4	yes	—
101	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
102	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
103	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
104	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
105	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
106	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
107	C₄⋊C₄	—	2	8	no	—
108	C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
109	C₂²⋊C₄	—	2	8	no	—
110	C₂²⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
111	C₄²	—	2	8	no	—
112	C₄²	—	2	4	no	—
113	C₄²	—	2	4	no	—
114	C₄²	C₂×C₄×C₂₃	1	4	yes	—
115	C₄²	C₂×C₄×C₂₃	1	4	yes	—
116	C₂³	—	2	4	no	—
117	C₂³	—	2	2	no	—
118	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
119	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	8	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
130	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
131	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
132	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
133	C₂×C₄	#41×C₂₃	1	8	yes	—
134	C₂×C₄	#41×C₂₃	1	4	yes	—
135	C₂²	—	2	2	no	—
136	C₂²	—	2	4	no	—
137	C₂²	—	4	4	no	—
138	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
139	C₂²	C₂×#41×C₂₃	1	2	yes	—
140	C₂²	C₂×#41×C₂₃	1	4	yes	—
141	C₄	—	2	2	no	—
142	C₄	—	2	4	no	—
143	C₄	—	2	4	no	—
144	C₄	—	2	8	no	—
145	C₄	—	4	4	no	—
146	C₄	C₂₃×#117	1	4	yes	—
147	C₂	—	4	2	no	—
148	C₂	C₂×C₂₃×#117	1	1	yes	characteristic
149	C₂	C₄×#41×C₂₃	1	2	yes	—
150	C₂	C₂₃×#529	1	4	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
3. C₂₃×#387	C₂	semidirect
4. C₄×C₄⋊C₄×C₂₃	C₂	semidirect
6. C₂₃×#376	C₂	semidirect
7. C₄³×C₂₃	C₂	semidirect
9. C₂₃×#117	C₄	semidirect
59. #2298	C₂₃	direct (N × H)
74. #387	C₂×C₂₃	semidirect
75. C₄×C₄⋊C₄	C₂×C₂₃	semidirect
77. #376	C₂×C₂₃	semidirect
78. C₄³	C₂×C₂₃	semidirect
84. #117	C₄×C₂₃	semidirect
97. C₂₃	#2298	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×#117	2
C₂₃×#388	2
C₂₃×#387	2
C₄×C₄⋊C₄×C₂₃	2
C₄×C₂²⋊C₄×C₂₃	2
C₂₃×#376	2
C₄³×C₂₃	2
#2298	23

Subgroups — C23×#2298 #15669917

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2298 #15669917