Subgroups — C₂₃×#2922 #15670541

Order 2944.

400 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#433	C₂	1	4	yes	maximal, complemented
3	C₂₃×#431	C₂	1	4	yes	maximal, complemented
4	C₂₃×#424	C₂	1	2	yes	maximal, complemented
5	C₂₃×#408	C₂	1	2	yes	maximal
6	C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
10	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
11	C₄×Q₈×C₂₃	C₄	1	2	yes	—
12	C₄×D₄×C₂₃	C₄	1	4	yes	—
13	C₄×D₄×C₂₃	C₄	1	2	yes	—
14	C₂₃×#117	C₂²	1	1	yes	characteristic
15	C₂₃×#117	C₂²	1	2	yes	—
16	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
18	C₂₃×C₄⋊C₈	C₂²	1	8	yes	—
19	C₂₃×C₂²⋊C₈	C₂²	1	8	yes	—
20	C₂₃×#97	C₂²	1	2	yes	—
21	C₄×C₈×C₂₃	C₂²	1	2	yes	—
22	C₄×C₈×C₂₃	C₂²	1	2	yes	—
23	C₄×C₈×C₂₃	C₂²	1	2	yes	—
24	#41×C₂₃	—	2	4	no	—
25	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
27	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
31	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
32	M₁₆×C₂₃	—	2	4	no	—
33	C₂×C₈×C₂₃	—	2	4	no	—
34	C₂×C₈×C₂₃	C₂³	1	4	yes	—
35	C₂×C₈×C₂₃	C₂³	1	4	yes	—
36	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
37	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
38	C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
39	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
40	C₄²×C₂₃	C₂×C₄	1	4	yes	—
41	C₄²×C₂₃	C₂×C₄	1	2	yes	—
42	C₄²×C₂₃	C₂³	1	2	yes	—
43	C₄²×C₂₃	C₂³	1	1	yes	characteristic
44	C₄²×C₂₃	C₂³	1	1	yes	characteristic
45	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
46	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₂₃	—	2	2	no	—
48	D₄×C₂₃	—	2	4	no	—
49	D₄×C₂₃	—	2	2	no	—
50	C₂×C₄×C₂₃	—	2	4	no	—
51	C₂×C₄×C₂₃	—	2	8	no	—
52	C₂×C₄×C₂₃	—	2	4	no	—
53	C₂×C₄×C₂₃	C₂²×C₄	1	2	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	1	yes	characteristic
57	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₂₃	#41	1	4	yes	—
59	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
60	C₈×C₂₃	—	2	4	no	—
61	C₈×C₂₃	—	2	4	no	—
62	#2922	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂²×C₂₃	—	2	2	no	—
64	C₂²×C₂₃	—	2	2	no	—
65	C₂²×C₂₃	—	2	4	no	—
66	C₂²×C₂₃	#117	1	2	yes	—
67	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
68	C₄×C₂₃	—	2	2	no	—
69	C₄×C₂₃	—	2	2	no	—
70	C₄×C₂₃	—	2	4	no	—
71	C₄×C₂₃	#117	1	2	yes	—
72	C₄×C₂₃	#131	1	4	yes	—
73	C₄×C₂₃	C₂×#41	1	2	yes	—
74	C₄×#41	C₂×C₂₃	1	1	yes	characteristic
75	#433	C₂×C₂₃	1	4	yes	complemented
76	#431	C₂×C₂₃	1	4	yes	complemented
77	#424	C₂×C₂₃	1	2	yes	complemented
78	#408	C₂×C₂₃	1	2	yes	—
79	C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
80	C₂×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
81	C₂×C₂₃	—	2	2	no	—
82	C₂×C₂₃	—	4	2	no	—
83	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
84	C₂×C₂₃	C₂×#131	1	2	yes	—
85	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
86	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
87	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
88	C₄×Q₈	C₄×C₂₃	1	2	yes	—
89	C₄×D₄	C₄×C₂₃	1	4	yes	—
90	C₄×D₄	C₄×C₂₃	1	2	yes	—
91	#117	C₂²×C₂₃	1	1	yes	characteristic
92	#117	C₂²×C₂₃	1	2	yes	—
93	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
94	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
95	C₄⋊C₈	C₂²×C₂₃	1	8	yes	—
96	C₂²⋊C₈	C₂²×C₂₃	1	8	yes	—
97	#97	C₂²×C₂₃	1	2	yes	—
98	C₄×C₈	C₂²×C₂₃	1	2	yes	—
99	C₄×C₈	C₂²×C₂₃	1	2	yes	—
100	C₄×C₈	C₂²×C₂₃	1	2	yes	—
101	C₂₃	#2922	1	1	yes	characteristic, direct product factor, Sylow
102	#41	—	2	4	no	—
103	C₂×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
104	C₂×D₄	C₂×C₄×C₂₃	1	2	yes	—
105	C₂×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
106	C₂²×C₄	C₂×C₄×C₂₃	1	4	yes	—
107	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
108	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
109	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
110	M₁₆	—	2	4	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	C₂³×C₂₃	1	4	yes	—
113	C₂×C₈	C₂³×C₂₃	1	4	yes	—
114	C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
115	C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
116	C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
117	C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
118	C₄²	C₂×C₄×C₂₃	1	4	yes	—
119	C₄²	C₂×C₄×C₂₃	1	2	yes	—
120	C₄²	C₂³×C₂₃	1	1	yes	characteristic
121	C₄²	C₂³×C₂₃	1	1	yes	characteristic
122	C₄²	C₂³×C₂₃	1	2	yes	—
123	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
124	C₂³	C₂²×C₄×C₂₃	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	D₄	—	2	4	no	—
127	D₄	—	2	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	8	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
132	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
133	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
134	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
135	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
136	C₂×C₄	#41×C₂₃	1	4	yes	—
137	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₈	—	2	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	2	4	no	—
142	C₂²	—	2	2	no	—
143	C₂²	C₂₃×#117	1	2	yes	—
144	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	4	no	—
147	C₄	—	2	2	no	—
148	C₄	C₂₃×#117	1	2	yes	—
149	C₄	C₂₃×#131	1	4	yes	—
150	C₄	C₂×#41×C₂₃	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	4	2	no	—
153	C₂	C₂×C₂₃×#117	1	1	yes	characteristic
154	C₂	C₂×C₂₃×#131	1	2	yes	—

	subgroup	G/H
center	C₄²×C₂₃	C₂³
commutator	C₂²	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂³×C₄
prime radical	C₂×#41×C₂₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#433	C₂	semidirect
3. C₂₃×#431	C₂	semidirect
4. C₂₃×#424	C₂	semidirect
6. C₄×M₁₆×C₂₃	C₂	semidirect
7. C₂×C₄×C₈×C₂₃	C₂	semidirect
62. #2922	C₂₃	direct (N × H)
75. #433	C₂×C₂₃	semidirect
76. #431	C₂×C₂₃	semidirect
77. #424	C₂×C₂₃	semidirect
79. C₄×M₁₆	C₂×C₂₃	semidirect
80. C₂×C₄×C₈	C₂×C₂₃	semidirect
101. C₂₃	#2922	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#41×C₂₃	2
C₂₃×#433	2
C₂₃×#431	2
C₂₃×#424	2
C₂₃×#408	2
C₄×M₁₆×C₂₃	2
C₂×C₄×C₈×C₂₃	2
#2922	23

Subgroups — C23×#2922 #15670541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2922 #15670541