Subgroups — C₂²×C₃×C₇×C₁₆×C₂₃ #72207666

Order 30912.

392 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	C₂²×C₃×C₇×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₇×C₁₆×C₂₃	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₇×C₁₆×C₂₃	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃×C₄×C₇×C₂₃	C₄	1	1	yes	characteristic
5	C₂×C₃×C₇×C₈×C₂₃	C₄	1	3	yes	—
6	C₂×C₃×C₇×C₈×C₂₃	C₂²	1	3	yes	—
7	C₃×C₇×C₁₆×C₂₃	C₂²	1	4	yes	direct product factor
8	C₂²×C₇×C₈×C₂₃	C₂×C₃	1	1	yes	characteristic
9	C₂×C₇×C₁₆×C₂₃	C₂×C₃	1	6	yes	direct product factor
10	C₂²×C₃×C₁₆×C₂₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂³×C₃×C₇×C₂₃	C₈	1	1	yes	characteristic
12	C₂×C₃×C₄×C₇×C₂₃	C₈	1	3	yes	—
13	C₂×C₃×C₄×C₇×C₂₃	C₂×C₄	1	3	yes	—
14	C₃×C₇×C₈×C₂₃	C₂×C₄	1	3	yes	—
15	C₃×C₇×C₈×C₂₃	C₂³	1	1	yes	characteristic
16	C₂²×C₄×C₇×C₂₃	C₃×C₄	1	1	yes	characteristic
17	C₂×C₇×C₈×C₂₃	C₃×C₄	1	3	yes	—
18	C₂×C₇×C₈×C₂₃	C₂²×C₃	1	3	yes	—
19	C₇×C₁₆×C₂₃	C₂²×C₃	1	4	yes	direct product factor
20	C₂²×C₃×C₈×C₂₃	C₂×C₇	1	1	yes	characteristic
21	C₂×C₃×C₁₆×C₂₃	C₂×C₇	1	6	yes	direct product factor
22	C₂²×C₃×C₇×C₂₃	C₁₆	1	4	yes	direct product factor
23	C₂²×C₃×C₇×C₂₃	C₂×C₈	1	3	yes	—
24	C₃×C₄×C₇×C₂₃	C₂×C₈	1	3	yes	—
25	C₃×C₄×C₇×C₂₃	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₁₆×C₂₃	C₃×C₇	1	1	yes	characteristic, direct product factor, Hall
27	C₂²×C₃×C₇×C₁₆	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
28	C₂³×C₇×C₂₃	C₃×C₈	1	1	yes	characteristic
29	C₂×C₄×C₇×C₂₃	C₃×C₈	1	3	yes	—
30	C₂×C₄×C₇×C₂₃	C₂×C₃×C₄	1	3	yes	—
31	C₇×C₈×C₂₃	C₂×C₃×C₄	1	3	yes	—
32	C₇×C₈×C₂₃	C₂³×C₃	1	1	yes	characteristic
33	C₂²×C₃×C₄×C₂₃	C₄×C₇	1	1	yes	characteristic
34	C₂×C₃×C₈×C₂₃	C₄×C₇	1	3	yes	—
35	C₂×C₃×C₈×C₂₃	C₂²×C₇	1	3	yes	—
36	C₃×C₁₆×C₂₃	C₂²×C₇	1	4	yes	direct product factor
37	C₂×C₃×C₇×C₂₃	C₂×C₁₆	1	6	yes	direct product factor
38	C₂×C₃×C₇×C₂₃	C₂²×C₈	1	1	yes	characteristic
39	C₂²×C₈×C₂₃	C₂×C₃×C₇	1	1	yes	characteristic
40	C₂×C₁₆×C₂₃	C₂×C₃×C₇	1	6	yes	direct product factor
41	C₂²×C₃×C₇×C₈	C₂×C₂₃	1	1	yes	characteristic
42	C₂×C₃×C₇×C₁₆	C₂×C₂₃	1	6	yes	direct product factor
43	C₂²×C₇×C₂₃	C₃×C₁₆	1	4	yes	direct product factor
44	C₂²×C₇×C₂₃	C₂×C₃×C₈	1	3	yes	—
45	C₄×C₇×C₂₃	C₂×C₃×C₈	1	3	yes	—
46	C₄×C₇×C₂₃	C₂²×C₃×C₄	1	1	yes	characteristic
47	C₂³×C₃×C₂₃	C₇×C₈	1	1	yes	characteristic
48	C₂×C₃×C₄×C₂₃	C₇×C₈	1	3	yes	—
49	C₂×C₃×C₄×C₂₃	C₂×C₄×C₇	1	3	yes	—
50	C₃×C₈×C₂₃	C₂×C₄×C₇	1	3	yes	—
51	C₃×C₈×C₂₃	C₂³×C₇	1	1	yes	characteristic
52	C₃×C₇×C₂₃	C₂²×C₁₆	1	1	yes	characteristic, direct product factor, Hall
53	C₂²×C₇×C₁₆	C₃×C₂₃	1	1	yes	characteristic, direct product factor, Hall
54	C₂²×C₄×C₂₃	C₃×C₄×C₇	1	1	yes	characteristic
55	C₂×C₈×C₂₃	C₃×C₄×C₇	1	3	yes	—
56	C₂×C₈×C₂₃	C₂²×C₃×C₇	1	3	yes	—
57	C₁₆×C₂₃	C₂²×C₃×C₇	1	4	yes	direct product factor
58	C₂²×C₃×C₄×C₇	C₄×C₂₃	1	1	yes	characteristic
59	C₂×C₃×C₇×C₈	C₄×C₂₃	1	3	yes	—
60	C₂×C₃×C₇×C₈	C₂²×C₂₃	1	3	yes	—
61	C₃×C₇×C₁₆	C₂²×C₂₃	1	4	yes	direct product factor
62	C₂×C₇×C₂₃	C₂×C₃×C₁₆	1	6	yes	direct product factor
63	C₂×C₇×C₂₃	C₂²×C₃×C₈	1	1	yes	characteristic
64	C₂²×C₃×C₂₃	C₇×C₁₆	1	4	yes	direct product factor
65	C₂²×C₃×C₂₃	C₂×C₇×C₈	1	3	yes	—
66	C₃×C₄×C₂₃	C₂×C₇×C₈	1	3	yes	—
67	C₃×C₄×C₂₃	C₂²×C₄×C₇	1	1	yes	characteristic
68	C₂²×C₇×C₈	C₂×C₃×C₂₃	1	1	yes	characteristic
69	C₂×C₇×C₁₆	C₂×C₃×C₂₃	1	6	yes	direct product factor
70	C₂²×C₃×C₁₆	C₇×C₂₃	1	1	yes	characteristic, direct product factor, Hall
71	C₂³×C₂₃	C₃×C₇×C₈	1	1	yes	characteristic
72	C₂×C₄×C₂₃	C₃×C₇×C₈	1	3	yes	—
73	C₂×C₄×C₂₃	C₂×C₃×C₄×C₇	1	3	yes	—
74	C₈×C₂₃	C₂×C₃×C₄×C₇	1	3	yes	—
75	C₈×C₂₃	C₂³×C₃×C₇	1	1	yes	characteristic
76	C₂³×C₃×C₇	C₈×C₂₃	1	1	yes	characteristic
77	C₂×C₃×C₄×C₇	C₈×C₂₃	1	3	yes	—
78	C₂×C₃×C₄×C₇	C₂×C₄×C₂₃	1	3	yes	—
79	C₃×C₇×C₈	C₂×C₄×C₂₃	1	3	yes	—
80	C₃×C₇×C₈	C₂³×C₂₃	1	1	yes	characteristic
81	C₇×C₂₃	C₂²×C₃×C₁₆	1	1	yes	characteristic, direct product factor, Hall
82	C₂×C₃×C₂₃	C₂×C₇×C₁₆	1	6	yes	direct product factor
83	C₂×C₃×C₂₃	C₂²×C₇×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₇	C₃×C₄×C₂₃	1	1	yes	characteristic
85	C₂×C₇×C₈	C₃×C₄×C₂₃	1	3	yes	—
86	C₂×C₇×C₈	C₂²×C₃×C₂₃	1	3	yes	—
87	C₇×C₁₆	C₂²×C₃×C₂₃	1	4	yes	direct product factor
88	C₂²×C₃×C₈	C₂×C₇×C₂₃	1	1	yes	characteristic
89	C₂×C₃×C₁₆	C₂×C₇×C₂₃	1	6	yes	direct product factor
90	C₂²×C₂₃	C₃×C₇×C₁₆	1	4	yes	direct product factor
91	C₂²×C₂₃	C₂×C₃×C₇×C₈	1	3	yes	—
92	C₄×C₂₃	C₂×C₃×C₇×C₈	1	3	yes	—
93	C₄×C₂₃	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
94	C₂²×C₃×C₇	C₁₆×C₂₃	1	4	yes	direct product factor
95	C₂²×C₃×C₇	C₂×C₈×C₂₃	1	3	yes	—
96	C₃×C₄×C₇	C₂×C₈×C₂₃	1	3	yes	—
97	C₃×C₄×C₇	C₂²×C₄×C₂₃	1	1	yes	characteristic
98	C₃×C₂₃	C₂²×C₇×C₁₆	1	1	yes	characteristic, direct product factor, Hall
99	C₂²×C₁₆	C₃×C₇×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
100	C₂³×C₇	C₃×C₈×C₂₃	1	1	yes	characteristic
101	C₂×C₄×C₇	C₃×C₈×C₂₃	1	3	yes	—
102	C₂×C₄×C₇	C₂×C₃×C₄×C₂₃	1	3	yes	—
103	C₇×C₈	C₂×C₃×C₄×C₂₃	1	3	yes	—
104	C₇×C₈	C₂³×C₃×C₂₃	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₄×C₇×C₂₃	1	1	yes	characteristic
106	C₂×C₃×C₈	C₄×C₇×C₂₃	1	3	yes	—
107	C₂×C₃×C₈	C₂²×C₇×C₂₃	1	3	yes	—
108	C₃×C₁₆	C₂²×C₇×C₂₃	1	4	yes	direct product factor
109	C₂×C₂₃	C₂×C₃×C₇×C₁₆	1	6	yes	direct product factor
110	C₂×C₂₃	C₂²×C₃×C₇×C₈	1	1	yes	characteristic
111	C₂×C₃×C₇	C₂×C₁₆×C₂₃	1	6	yes	direct product factor
112	C₂×C₃×C₇	C₂²×C₈×C₂₃	1	1	yes	characteristic
113	C₂²×C₈	C₂×C₃×C₇×C₂₃	1	1	yes	characteristic
114	C₂×C₁₆	C₂×C₃×C₇×C₂₃	1	6	yes	direct product factor
115	C₂²×C₇	C₃×C₁₆×C₂₃	1	4	yes	direct product factor
116	C₂²×C₇	C₂×C₃×C₈×C₂₃	1	3	yes	—
117	C₄×C₇	C₂×C₃×C₈×C₂₃	1	3	yes	—
118	C₄×C₇	C₂²×C₃×C₄×C₂₃	1	1	yes	characteristic
119	C₂³×C₃	C₇×C₈×C₂₃	1	1	yes	characteristic
120	C₂×C₃×C₄	C₇×C₈×C₂₃	1	3	yes	—
121	C₂×C₃×C₄	C₂×C₄×C₇×C₂₃	1	3	yes	—
122	C₃×C₈	C₂×C₄×C₇×C₂₃	1	3	yes	—
123	C₃×C₈	C₂³×C₇×C₂₃	1	1	yes	characteristic
124	C₂₃	C₂²×C₃×C₇×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
125	C₃×C₇	C₂²×C₁₆×C₂₃	1	1	yes	characteristic, direct product factor, Hall
126	C₂²×C₄	C₃×C₄×C₇×C₂₃	1	1	yes	characteristic
127	C₂×C₈	C₃×C₄×C₇×C₂₃	1	3	yes	—
128	C₂×C₈	C₂²×C₃×C₇×C₂₃	1	3	yes	—
129	C₁₆	C₂²×C₃×C₇×C₂₃	1	4	yes	direct product factor
130	C₂×C₇	C₂×C₃×C₁₆×C₂₃	1	6	yes	direct product factor
131	C₂×C₇	C₂²×C₃×C₈×C₂₃	1	1	yes	characteristic
132	C₂²×C₃	C₇×C₁₆×C₂₃	1	4	yes	direct product factor
133	C₂²×C₃	C₂×C₇×C₈×C₂₃	1	3	yes	—
134	C₃×C₄	C₂×C₇×C₈×C₂₃	1	3	yes	—
135	C₃×C₄	C₂²×C₄×C₇×C₂₃	1	1	yes	characteristic
136	C₂³	C₃×C₇×C₈×C₂₃	1	1	yes	characteristic
137	C₂×C₄	C₃×C₇×C₈×C₂₃	1	3	yes	—
138	C₂×C₄	C₂×C₃×C₄×C₇×C₂₃	1	3	yes	—
139	C₈	C₂×C₃×C₄×C₇×C₂₃	1	3	yes	—
140	C₈	C₂³×C₃×C₇×C₂₃	1	1	yes	characteristic
141	C₇	C₂²×C₃×C₁₆×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
142	C₂×C₃	C₂×C₇×C₁₆×C₂₃	1	6	yes	direct product factor
143	C₂×C₃	C₂²×C₇×C₈×C₂₃	1	1	yes	characteristic
144	C₂²	C₃×C₇×C₁₆×C₂₃	1	4	yes	direct product factor
145	C₂²	C₂×C₃×C₇×C₈×C₂₃	1	3	yes	—
146	C₄	C₂×C₃×C₇×C₈×C₂₃	1	3	yes	—
147	C₄	C₂²×C₃×C₄×C₇×C₂₃	1	1	yes	characteristic
148	C₃	C₂²×C₇×C₁₆×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
149	C₂	C₂×C₃×C₇×C₁₆×C₂₃	1	6	yes	direct product factor
150	C₂	C₂²×C₃×C₇×C₈×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂³×C₃×C₇×C₂₃
socle, prime radical	C₂³×C₃×C₇×C₂₃	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₇×C₁₆×C₂₃	C₂	direct (N × H)
3. C₂²×C₇×C₁₆×C₂₃	C₃	direct (N × H)
7. C₃×C₇×C₁₆×C₂₃	C₂²	direct (N × H)
9. C₂×C₇×C₁₆×C₂₃	C₂×C₃	direct (N × H)
10. C₂²×C₃×C₁₆×C₂₃	C₇	direct (N × H)
19. C₇×C₁₆×C₂₃	C₂²×C₃	direct (N × H)
21. C₂×C₃×C₁₆×C₂₃	C₂×C₇	direct (N × H)
22. C₂²×C₃×C₇×C₂₃	C₁₆	direct (N × H)
26. C₂²×C₁₆×C₂₃	C₃×C₇	direct (N × H)
27. C₂²×C₃×C₇×C₁₆	C₂₃	direct (N × H)
36. C₃×C₁₆×C₂₃	C₂²×C₇	direct (N × H)
37. C₂×C₃×C₇×C₂₃	C₂×C₁₆	direct (N × H)
40. C₂×C₁₆×C₂₃	C₂×C₃×C₇	direct (N × H)
42. C₂×C₃×C₇×C₁₆	C₂×C₂₃	direct (N × H)
43. C₂²×C₇×C₂₃	C₃×C₁₆	direct (N × H)
52. C₃×C₇×C₂₃	C₂²×C₁₆	direct (N × H)
53. C₂²×C₇×C₁₆	C₃×C₂₃	direct (N × H)
57. C₁₆×C₂₃	C₂²×C₃×C₇	direct (N × H)
61. C₃×C₇×C₁₆	C₂²×C₂₃	direct (N × H)
62. C₂×C₇×C₂₃	C₂×C₃×C₁₆	direct (N × H)
64. C₂²×C₃×C₂₃	C₇×C₁₆	direct (N × H)
69. C₂×C₇×C₁₆	C₂×C₃×C₂₃	direct (N × H)
70. C₂²×C₃×C₁₆	C₇×C₂₃	direct (N × H)
81. C₇×C₂₃	C₂²×C₃×C₁₆	direct (N × H)
82. C₂×C₃×C₂₃	C₂×C₇×C₁₆	direct (N × H)
87. C₇×C₁₆	C₂²×C₃×C₂₃	direct (N × H)
89. C₂×C₃×C₁₆	C₂×C₇×C₂₃	direct (N × H)
90. C₂²×C₂₃	C₃×C₇×C₁₆	direct (N × H)
94. C₂²×C₃×C₇	C₁₆×C₂₃	direct (N × H)
98. C₃×C₂₃	C₂²×C₇×C₁₆	direct (N × H)
99. C₂²×C₁₆	C₃×C₇×C₂₃	direct (N × H)
108. C₃×C₁₆	C₂²×C₇×C₂₃	direct (N × H)
109. C₂×C₂₃	C₂×C₃×C₇×C₁₆	direct (N × H)
111. C₂×C₃×C₇	C₂×C₁₆×C₂₃	direct (N × H)
114. C₂×C₁₆	C₂×C₃×C₇×C₂₃	direct (N × H)
115. C₂²×C₇	C₃×C₁₆×C₂₃	direct (N × H)
124. C₂₃	C₂²×C₃×C₇×C₁₆	direct (N × H)
125. C₃×C₇	C₂²×C₁₆×C₂₃	direct (N × H)
129. C₁₆	C₂²×C₃×C₇×C₂₃	direct (N × H)
130. C₂×C₇	C₂×C₃×C₁₆×C₂₃	direct (N × H)
132. C₂²×C₃	C₇×C₁₆×C₂₃	direct (N × H)
141. C₇	C₂²×C₃×C₁₆×C₂₃	direct (N × H)
142. C₂×C₃	C₂×C₇×C₁₆×C₂₃	direct (N × H)
144. C₂²	C₃×C₇×C₁₆×C₂₃	direct (N × H)
148. C₃	C₂²×C₇×C₁₆×C₂₃	direct (N × H)
149. C₂	C₂×C₃×C₇×C₁₆×C₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₇×C₈×C₂₃	2
C₂×C₃×C₇×C₁₆×C₂₃	2
C₂²×C₇×C₁₆×C₂₃	3
C₂²×C₃×C₁₆×C₂₃	7
C₂²×C₃×C₇×C₁₆	23

Subgroups — C22×C3×C7×C16×C23 #72207666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₇×C₁₆×C₂₃ #72207666