Subgroups — C₄×C₇×C₂₇×C₃₂ #72201459

Order 24192.

288 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×C₁₆×C₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₂₇×C₃₂	C₂	1	2	yes	maximal
3	C₄×C₇×C₉×C₃₂	C₃	1	1	yes	maximal, characteristic
4	C₄×C₇×C₈×C₂₇	C₄	1	1	yes	characteristic
5	C₂×C₇×C₁₆×C₂₇	C₄	1	1	yes	characteristic
6	C₂×C₇×C₁₆×C₂₇	C₂²	1	1	yes	characteristic
7	C₇×C₂₇×C₃₂	C₄	1	4	yes	direct product factor
8	C₄×C₇×C₉×C₁₆	C₂×C₃	1	1	yes	characteristic
9	C₂×C₇×C₉×C₃₂	C₂×C₃	1	2	yes	—
10	C₄×C₂₇×C₃₂	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₄²×C₇×C₂₇	C₈	1	1	yes	characteristic
12	C₂×C₇×C₈×C₂₇	C₈	1	1	yes	characteristic
13	C₂×C₇×C₈×C₂₇	C₂×C₄	1	1	yes	characteristic
14	C₇×C₁₆×C₂₇	C₈	1	2	yes	—
15	C₇×C₁₆×C₂₇	C₂×C₄	1	2	yes	—
16	C₃×C₄×C₇×C₃₂	C₉	1	1	yes	characteristic
17	C₄×C₇×C₈×C₉	C₃×C₄	1	1	yes	characteristic
18	C₂×C₇×C₉×C₁₆	C₃×C₄	1	1	yes	characteristic
19	C₂×C₇×C₉×C₁₆	C₂²×C₃	1	1	yes	characteristic
20	C₇×C₉×C₃₂	C₃×C₄	1	4	yes	—
21	C₂×C₂₇×C₃₂	C₂×C₇	1	2	yes	—
22	C₄×C₁₆×C₂₇	C₂×C₇	1	1	yes	characteristic
23	C₂×C₄×C₇×C₂₇	C₁₆	1	2	yes	—
24	C₂×C₄×C₇×C₂₇	C₂×C₈	1	1	yes	characteristic
25	C₇×C₈×C₂₇	C₁₆	1	2	yes	—
26	C₇×C₈×C₂₇	C₄²	1	1	yes	characteristic
27	C₇×C₈×C₂₇	C₂×C₈	1	1	yes	characteristic
28	C₂×C₃×C₇×C₃₂	C₂×C₉	1	2	yes	—
29	C₃×C₄×C₇×C₁₆	C₂×C₉	1	1	yes	characteristic
30	C₄×C₉×C₃₂	C₃×C₇	1	1	yes	characteristic
31	C₂×C₇×C₈×C₉	C₃×C₈	1	1	yes	characteristic
32	C₂×C₇×C₈×C₉	C₂×C₃×C₄	1	1	yes	characteristic
33	C₄²×C₇×C₉	C₃×C₈	1	1	yes	characteristic
34	C₇×C₉×C₁₆	C₃×C₈	1	2	yes	—
35	C₇×C₉×C₁₆	C₂×C₃×C₄	1	2	yes	—
36	C₄×C₇×C₃₂	C₂₇	1	1	yes	characteristic, direct product factor, Hall
37	C₂×C₁₆×C₂₇	C₄×C₇	1	1	yes	characteristic
38	C₂×C₁₆×C₂₇	C₂²×C₇	1	1	yes	characteristic
39	C₄×C₈×C₂₇	C₄×C₇	1	1	yes	characteristic
40	C₂₇×C₃₂	C₄×C₇	1	4	yes	direct product factor
41	C₂²×C₇×C₂₇	C₂×C₁₆	1	1	yes	characteristic
42	C₄×C₇×C₂₇	C₃₂	1	4	yes	direct product factor
43	C₄×C₇×C₂₇	C₄×C₈	1	1	yes	characteristic
44	C₄×C₇×C₂₇	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₃×C₇×C₁₆	C₄×C₉	1	1	yes	characteristic
46	C₂×C₃×C₇×C₁₆	C₂²×C₉	1	1	yes	characteristic
47	C₃×C₄×C₇×C₈	C₄×C₉	1	1	yes	characteristic
48	C₃×C₇×C₃₂	C₄×C₉	1	4	yes	—
49	C₂×C₉×C₃₂	C₂×C₃×C₇	1	2	yes	—
50	C₄×C₉×C₁₆	C₂×C₃×C₇	1	1	yes	characteristic
51	C₂×C₄×C₇×C₉	C₃×C₁₆	1	2	yes	—
52	C₂×C₄×C₇×C₉	C₂×C₃×C₈	1	1	yes	characteristic
53	C₇×C₈×C₉	C₃×C₁₆	1	2	yes	—
54	C₇×C₈×C₉	C₃×C₄²	1	1	yes	characteristic
55	C₇×C₈×C₉	C₂×C₃×C₈	1	1	yes	characteristic
56	C₂×C₇×C₃₂	C₂×C₂₇	1	2	yes	—
57	C₄×C₇×C₁₆	C₂×C₂₇	1	1	yes	characteristic
58	C₂×C₈×C₂₇	C₇×C₈	1	1	yes	characteristic
59	C₂×C₈×C₂₇	C₂×C₄×C₇	1	1	yes	characteristic
60	C₄²×C₂₇	C₇×C₈	1	1	yes	characteristic
61	C₁₆×C₂₇	C₇×C₈	1	2	yes	—
62	C₁₆×C₂₇	C₂×C₄×C₇	1	2	yes	—
63	C₃×C₄×C₃₂	C₇×C₉	1	1	yes	characteristic
64	C₂×C₇×C₂₇	C₄×C₁₆	1	1	yes	characteristic
65	C₂×C₇×C₂₇	C₂×C₃₂	1	2	yes	—
66	C₂×C₃×C₇×C₈	C₈×C₉	1	1	yes	characteristic
67	C₂×C₃×C₇×C₈	C₂×C₄×C₉	1	1	yes	characteristic
68	C₃×C₄²×C₇	C₈×C₉	1	1	yes	characteristic
69	C₃×C₇×C₁₆	C₈×C₉	1	2	yes	—
70	C₃×C₇×C₁₆	C₂×C₄×C₉	1	2	yes	—
71	C₂×C₉×C₁₆	C₃×C₄×C₇	1	1	yes	characteristic
72	C₂×C₉×C₁₆	C₂²×C₃×C₇	1	1	yes	characteristic
73	C₄×C₈×C₉	C₃×C₄×C₇	1	1	yes	characteristic
74	C₉×C₃₂	C₃×C₄×C₇	1	4	yes	—
75	C₂²×C₇×C₉	C₂×C₃×C₁₆	1	1	yes	characteristic
76	C₄×C₇×C₉	C₃×C₃₂	1	4	yes	—
77	C₄×C₇×C₉	C₃×C₄×C₈	1	1	yes	characteristic
78	C₄×C₇×C₉	C₂×C₃×C₁₆	1	1	yes	characteristic
79	C₂×C₇×C₁₆	C₄×C₂₇	1	1	yes	characteristic
80	C₂×C₇×C₁₆	C₂²×C₂₇	1	1	yes	characteristic
81	C₄×C₇×C₈	C₄×C₂₇	1	1	yes	characteristic
82	C₇×C₃₂	C₄×C₂₇	1	4	yes	direct product factor
83	C₂×C₄×C₂₇	C₇×C₁₆	1	2	yes	—
84	C₂×C₄×C₂₇	C₂×C₇×C₈	1	1	yes	characteristic
85	C₈×C₂₇	C₇×C₁₆	1	2	yes	—
86	C₈×C₂₇	C₄²×C₇	1	1	yes	characteristic
87	C₈×C₂₇	C₂×C₇×C₈	1	1	yes	characteristic
88	C₂×C₃×C₃₂	C₂×C₇×C₉	1	2	yes	—
89	C₃×C₄×C₁₆	C₂×C₇×C₉	1	1	yes	characteristic
90	C₇×C₂₇	C₄×C₃₂	1	1	yes	characteristic, direct product factor, Hall
91	C₂×C₃×C₄×C₇	C₉×C₁₆	1	2	yes	—
92	C₂×C₃×C₄×C₇	C₂×C₈×C₉	1	1	yes	characteristic
93	C₃×C₇×C₈	C₉×C₁₆	1	2	yes	—
94	C₃×C₇×C₈	C₄²×C₉	1	1	yes	characteristic
95	C₃×C₇×C₈	C₂×C₈×C₉	1	1	yes	characteristic
96	C₂×C₈×C₉	C₃×C₇×C₈	1	1	yes	characteristic
97	C₂×C₈×C₉	C₂×C₃×C₄×C₇	1	1	yes	characteristic
98	C₄²×C₉	C₃×C₇×C₈	1	1	yes	characteristic
99	C₉×C₁₆	C₃×C₇×C₈	1	2	yes	—
100	C₉×C₁₆	C₂×C₃×C₄×C₇	1	2	yes	—
101	C₄×C₃₂	C₇×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
102	C₂×C₇×C₉	C₃×C₄×C₁₆	1	1	yes	characteristic
103	C₂×C₇×C₉	C₂×C₃×C₃₂	1	2	yes	—
104	C₂×C₇×C₈	C₈×C₂₇	1	1	yes	characteristic
105	C₂×C₇×C₈	C₂×C₄×C₂₇	1	1	yes	characteristic
106	C₄²×C₇	C₈×C₂₇	1	1	yes	characteristic
107	C₇×C₁₆	C₈×C₂₇	1	2	yes	—
108	C₇×C₁₆	C₂×C₄×C₂₇	1	2	yes	—
109	C₂²×C₂₇	C₂×C₇×C₁₆	1	1	yes	characteristic
110	C₄×C₂₇	C₇×C₃₂	1	4	yes	direct product factor
111	C₄×C₂₇	C₄×C₇×C₈	1	1	yes	characteristic
112	C₄×C₂₇	C₂×C₇×C₁₆	1	1	yes	characteristic
113	C₂×C₃×C₁₆	C₄×C₇×C₉	1	1	yes	characteristic
114	C₂×C₃×C₁₆	C₂²×C₇×C₉	1	1	yes	characteristic
115	C₃×C₄×C₈	C₄×C₇×C₉	1	1	yes	characteristic
116	C₃×C₃₂	C₄×C₇×C₉	1	4	yes	—
117	C₂²×C₃×C₇	C₂×C₉×C₁₆	1	1	yes	characteristic
118	C₃×C₄×C₇	C₉×C₃₂	1	4	yes	—
119	C₃×C₄×C₇	C₄×C₈×C₉	1	1	yes	characteristic
120	C₃×C₄×C₇	C₂×C₉×C₁₆	1	1	yes	characteristic
121	C₂×C₄×C₉	C₃×C₇×C₁₆	1	2	yes	—
122	C₂×C₄×C₉	C₂×C₃×C₇×C₈	1	1	yes	characteristic
123	C₈×C₉	C₃×C₇×C₁₆	1	2	yes	—
124	C₈×C₉	C₃×C₄²×C₇	1	1	yes	characteristic
125	C₈×C₉	C₂×C₃×C₇×C₈	1	1	yes	characteristic
126	C₂×C₃₂	C₂×C₇×C₂₇	1	2	yes	—
127	C₄×C₁₆	C₂×C₇×C₂₇	1	1	yes	characteristic
128	C₇×C₉	C₃×C₄×C₃₂	1	1	yes	characteristic
129	C₂×C₄×C₇	C₁₆×C₂₇	1	2	yes	—
130	C₂×C₄×C₇	C₂×C₈×C₂₇	1	1	yes	characteristic
131	C₇×C₈	C₁₆×C₂₇	1	2	yes	—
132	C₇×C₈	C₄²×C₂₇	1	1	yes	characteristic
133	C₇×C₈	C₂×C₈×C₂₇	1	1	yes	characteristic
134	C₂×C₂₇	C₄×C₇×C₁₆	1	1	yes	characteristic
135	C₂×C₂₇	C₂×C₇×C₃₂	1	2	yes	—
136	C₂×C₃×C₈	C₇×C₈×C₉	1	1	yes	characteristic
137	C₂×C₃×C₈	C₂×C₄×C₇×C₉	1	1	yes	characteristic
138	C₃×C₄²	C₇×C₈×C₉	1	1	yes	characteristic
139	C₃×C₁₆	C₇×C₈×C₉	1	2	yes	—
140	C₃×C₁₆	C₂×C₄×C₇×C₉	1	2	yes	—
141	C₂×C₃×C₇	C₄×C₉×C₁₆	1	1	yes	characteristic
142	C₂×C₃×C₇	C₂×C₉×C₃₂	1	2	yes	—
143	C₂²×C₉	C₂×C₃×C₇×C₁₆	1	1	yes	characteristic
144	C₄×C₉	C₃×C₇×C₃₂	1	4	yes	—
145	C₄×C₉	C₃×C₄×C₇×C₈	1	1	yes	characteristic
146	C₄×C₉	C₂×C₃×C₇×C₁₆	1	1	yes	characteristic
147	C₂×C₁₆	C₄×C₇×C₂₇	1	1	yes	characteristic
148	C₂×C₁₆	C₂²×C₇×C₂₇	1	1	yes	characteristic
149	C₄×C₈	C₄×C₇×C₂₇	1	1	yes	characteristic
150	C₃₂	C₄×C₇×C₂₇	1	4	yes	direct product factor
151	C₂²×C₇	C₂×C₁₆×C₂₇	1	1	yes	characteristic
152	C₄×C₇	C₂₇×C₃₂	1	4	yes	direct product factor
153	C₄×C₇	C₄×C₈×C₂₇	1	1	yes	characteristic
154	C₄×C₇	C₂×C₁₆×C₂₇	1	1	yes	characteristic
155	C₂₇	C₄×C₇×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
156	C₂×C₃×C₄	C₇×C₉×C₁₆	1	2	yes	—
157	C₂×C₃×C₄	C₂×C₇×C₈×C₉	1	1	yes	characteristic
158	C₃×C₈	C₇×C₉×C₁₆	1	2	yes	—
159	C₃×C₈	C₄²×C₇×C₉	1	1	yes	characteristic
160	C₃×C₈	C₂×C₇×C₈×C₉	1	1	yes	characteristic
161	C₃×C₇	C₄×C₉×C₃₂	1	1	yes	characteristic
162	C₂×C₉	C₃×C₄×C₇×C₁₆	1	1	yes	characteristic
163	C₂×C₉	C₂×C₃×C₇×C₃₂	1	2	yes	—
164	C₂×C₈	C₇×C₈×C₂₇	1	1	yes	characteristic
165	C₂×C₈	C₂×C₄×C₇×C₂₇	1	1	yes	characteristic
166	C₄²	C₇×C₈×C₂₇	1	1	yes	characteristic
167	C₁₆	C₇×C₈×C₂₇	1	2	yes	—
168	C₁₆	C₂×C₄×C₇×C₂₇	1	2	yes	—
169	C₂×C₇	C₄×C₁₆×C₂₇	1	1	yes	characteristic
170	C₂×C₇	C₂×C₂₇×C₃₂	1	2	yes	—
171	C₂²×C₃	C₂×C₇×C₉×C₁₆	1	1	yes	characteristic
172	C₃×C₄	C₇×C₉×C₃₂	1	4	yes	—
173	C₃×C₄	C₂×C₇×C₉×C₁₆	1	1	yes	characteristic
174	C₃×C₄	C₄×C₇×C₈×C₉	1	1	yes	characteristic
175	C₉	C₃×C₄×C₇×C₃₂	1	1	yes	characteristic
176	C₂×C₄	C₇×C₁₆×C₂₇	1	2	yes	—
177	C₂×C₄	C₂×C₇×C₈×C₂₇	1	1	yes	characteristic
178	C₈	C₇×C₁₆×C₂₇	1	2	yes	—
179	C₈	C₂×C₇×C₈×C₂₇	1	1	yes	characteristic
180	C₈	C₄²×C₇×C₂₇	1	1	yes	characteristic
181	C₇	C₄×C₂₇×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
182	C₂×C₃	C₂×C₇×C₉×C₃₂	1	2	yes	—
183	C₂×C₃	C₄×C₇×C₉×C₁₆	1	1	yes	characteristic
184	C₂²	C₂×C₇×C₁₆×C₂₇	1	1	yes	characteristic
185	C₄	C₇×C₂₇×C₃₂	1	4	yes	direct product factor
186	C₄	C₂×C₇×C₁₆×C₂₇	1	1	yes	characteristic
187	C₄	C₄×C₇×C₈×C₂₇	1	1	yes	characteristic
188	C₃	C₄×C₇×C₉×C₃₂	1	1	yes	characteristic
189	C₂	C₂×C₇×C₂₇×C₃₂	1	2	yes	—
190	C₂	C₄×C₇×C₁₆×C₂₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₉×C₁₆	C₂²×C₃×C₇
socle, prime radical	C₂²×C₃×C₇	C₂×C₉×C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₇×C₂₇×C₃₂	C₄	direct (N × H)
10. C₄×C₂₇×C₃₂	C₇	direct (N × H)
36. C₄×C₇×C₃₂	C₂₇	direct (N × H)
40. C₂₇×C₃₂	C₄×C₇	direct (N × H)
42. C₄×C₇×C₂₇	C₃₂	direct (N × H)
82. C₇×C₃₂	C₄×C₂₇	direct (N × H)
90. C₇×C₂₇	C₄×C₃₂	direct (N × H)
101. C₄×C₃₂	C₇×C₂₇	direct (N × H)
110. C₄×C₂₇	C₇×C₃₂	direct (N × H)
150. C₃₂	C₄×C₇×C₂₇	direct (N × H)
152. C₄×C₇	C₂₇×C₃₂	direct (N × H)
155. C₂₇	C₄×C₇×C₃₂	direct (N × H)
181. C₇	C₄×C₂₇×C₃₂	direct (N × H)
185. C₄	C₇×C₂₇×C₃₂	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×C₇×C₁₆×C₂₇	2
C₂×C₇×C₂₇×C₃₂	2
C₄×C₇×C₉×C₃₂	3
C₄×C₂₇×C₃₂	7

Subgroups — C4×C7×C27×C32 #72201459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₇×C₂₇×C₃₂ #72201459