Subgroups — C₂²×C₄×C₈×C₁₁×C₁₇ #72201218

Order 23936.

1520 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄²×C₁₁×C₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₈×C₁₁×C₁₇	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₁₁×C₁₇	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₄×C₁₁×C₁₇	C₄	1	2	yes	—
5	C₂³×C₄×C₁₁×C₁₇	C₂²	1	1	yes	characteristic
6	C₂²×C₈×C₁₁×C₁₇	C₄	1	16	yes	direct product factor
7	C₂²×C₈×C₁₁×C₁₇	C₂²	1	12	yes	—
8	C₂×C₄²×C₁₁×C₁₇	C₄	1	6	yes	—
9	C₂×C₄²×C₁₁×C₁₇	C₂²	1	6	yes	—
10	C₄×C₈×C₁₁×C₁₇	C₂²	1	16	yes	direct product factor
11	C₂⁴×C₁₁×C₁₇	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₁₁×C₁₇	C₈	1	16	yes	direct product factor
13	C₂²×C₄×C₁₁×C₁₇	C₂×C₄	1	3	yes	—
14	C₂²×C₄×C₁₁×C₁₇	C₂×C₄	1	8	yes	—
15	C₂²×C₄×C₁₁×C₁₇	C₂×C₄	1	12	yes	—
16	C₂²×C₄×C₁₁×C₁₇	C₂³	1	3	yes	—
17	C₄²×C₁₁×C₁₇	C₂×C₄	1	12	yes	—
18	C₄²×C₁₁×C₁₇	C₂³	1	4	yes	—
19	C₂×C₈×C₁₁×C₁₇	C₂×C₄	1	48	yes	direct product factor
20	C₂×C₈×C₁₁×C₁₇	C₂³	1	8	yes	—
21	C₂²×C₄×C₈×C₁₇	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂³×C₁₁×C₁₇	C₄²	1	4	yes	—
23	C₂³×C₁₁×C₁₇	C₂×C₈	1	8	yes	—
24	C₂³×C₁₁×C₁₇	C₂²×C₄	1	3	yes	—
25	C₂×C₄×C₁₁×C₁₇	C₄²	1	12	yes	—
26	C₂×C₄×C₁₁×C₁₇	C₂×C₈	1	48	yes	direct product factor
27	C₂×C₄×C₁₁×C₁₇	C₂²×C₄	1	3	yes	—
28	C₂×C₄×C₁₁×C₁₇	C₂²×C₄	1	8	yes	—
29	C₂×C₄×C₁₁×C₁₇	C₂²×C₄	1	12	yes	—
30	C₂×C₄×C₁₁×C₁₇	C₂⁴	1	1	yes	characteristic
31	C₈×C₁₁×C₁₇	C₂²×C₄	1	16	yes	direct product factor
32	C₂²×C₄×C₈×C₁₁	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
33	C₂³×C₈×C₁₇	C₂×C₁₁	1	2	yes	—
34	C₂²×C₄²×C₁₇	C₂×C₁₁	1	1	yes	characteristic
35	C₂×C₄×C₈×C₁₇	C₂×C₁₁	1	12	yes	direct product factor
36	C₂²×C₁₁×C₁₇	C₄×C₈	1	16	yes	direct product factor
37	C₂²×C₁₁×C₁₇	C₂×C₄²	1	6	yes	—
38	C₂²×C₁₁×C₁₇	C₂²×C₈	1	12	yes	—
39	C₂²×C₁₁×C₁₇	C₂³×C₄	1	1	yes	characteristic
40	C₄×C₁₁×C₁₇	C₂×C₄²	1	6	yes	—
41	C₄×C₁₁×C₁₇	C₂²×C₈	1	16	yes	direct product factor
42	C₄×C₁₁×C₁₇	C₂³×C₄	1	2	yes	—
43	C₂³×C₈×C₁₁	C₂×C₁₇	1	2	yes	—
44	C₂²×C₄²×C₁₁	C₂×C₁₇	1	1	yes	characteristic
45	C₂×C₄×C₈×C₁₁	C₂×C₁₇	1	12	yes	direct product factor
46	C₂³×C₄×C₁₇	C₄×C₁₁	1	2	yes	—
47	C₂³×C₄×C₁₇	C₂²×C₁₁	1	1	yes	characteristic
48	C₂²×C₈×C₁₇	C₄×C₁₁	1	16	yes	direct product factor
49	C₂²×C₈×C₁₇	C₂²×C₁₁	1	12	yes	—
50	C₂×C₄²×C₁₇	C₄×C₁₁	1	6	yes	—
51	C₂×C₄²×C₁₇	C₂²×C₁₁	1	6	yes	—
52	C₄×C₈×C₁₇	C₂²×C₁₁	1	16	yes	direct product factor
53	C₂×C₁₁×C₁₇	C₂×C₄×C₈	1	12	yes	direct product factor
54	C₂×C₁₁×C₁₇	C₂²×C₄²	1	1	yes	characteristic
55	C₂×C₁₁×C₁₇	C₂³×C₈	1	2	yes	—
56	C₂³×C₄×C₁₁	C₄×C₁₇	1	2	yes	—
57	C₂³×C₄×C₁₁	C₂²×C₁₇	1	1	yes	characteristic
58	C₂²×C₈×C₁₁	C₄×C₁₇	1	16	yes	direct product factor
59	C₂²×C₈×C₁₁	C₂²×C₁₇	1	12	yes	—
60	C₂×C₄²×C₁₁	C₄×C₁₇	1	6	yes	—
61	C₂×C₄²×C₁₁	C₂²×C₁₇	1	6	yes	—
62	C₄×C₈×C₁₁	C₂²×C₁₇	1	16	yes	direct product factor
63	C₂⁴×C₁₇	C₂×C₄×C₁₁	1	1	yes	characteristic
64	C₂²×C₄×C₁₇	C₈×C₁₁	1	16	yes	direct product factor
65	C₂²×C₄×C₁₇	C₂×C₄×C₁₁	1	3	yes	—
66	C₂²×C₄×C₁₇	C₂×C₄×C₁₁	1	8	yes	—
67	C₂²×C₄×C₁₇	C₂×C₄×C₁₁	1	12	yes	—
68	C₂²×C₄×C₁₇	C₂³×C₁₁	1	3	yes	—
69	C₂×C₈×C₁₇	C₂×C₄×C₁₁	1	48	yes	direct product factor
70	C₂×C₈×C₁₇	C₂³×C₁₁	1	8	yes	—
71	C₄²×C₁₇	C₂×C₄×C₁₁	1	12	yes	—
72	C₄²×C₁₇	C₂³×C₁₁	1	4	yes	—
73	C₁₁×C₁₇	C₂²×C₄×C₈	1	1	yes	characteristic, direct product factor, Hall
74	C₂⁴×C₁₁	C₂×C₄×C₁₇	1	1	yes	characteristic
75	C₂²×C₄×C₁₁	C₈×C₁₇	1	16	yes	direct product factor
76	C₂²×C₄×C₁₁	C₂×C₄×C₁₇	1	3	yes	—
77	C₂²×C₄×C₁₁	C₂×C₄×C₁₇	1	8	yes	—
78	C₂²×C₄×C₁₁	C₂×C₄×C₁₇	1	12	yes	—
79	C₂²×C₄×C₁₁	C₂³×C₁₇	1	3	yes	—
80	C₂×C₈×C₁₁	C₂×C₄×C₁₇	1	48	yes	direct product factor
81	C₂×C₈×C₁₁	C₂³×C₁₇	1	8	yes	—
82	C₄²×C₁₁	C₂×C₄×C₁₇	1	12	yes	—
83	C₄²×C₁₁	C₂³×C₁₇	1	4	yes	—
84	C₂³×C₁₇	C₄²×C₁₁	1	4	yes	—
85	C₂³×C₁₇	C₂×C₈×C₁₁	1	8	yes	—
86	C₂³×C₁₇	C₂²×C₄×C₁₁	1	3	yes	—
87	C₂×C₄×C₁₇	C₄²×C₁₁	1	12	yes	—
88	C₂×C₄×C₁₇	C₂×C₈×C₁₁	1	48	yes	direct product factor
89	C₂×C₄×C₁₇	C₂²×C₄×C₁₁	1	3	yes	—
90	C₂×C₄×C₁₇	C₂²×C₄×C₁₁	1	8	yes	—
91	C₂×C₄×C₁₇	C₂²×C₄×C₁₁	1	12	yes	—
92	C₂×C₄×C₁₇	C₂⁴×C₁₁	1	1	yes	characteristic
93	C₈×C₁₇	C₂²×C₄×C₁₁	1	16	yes	direct product factor
94	C₂²×C₄×C₈	C₁₁×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
95	C₂³×C₁₁	C₄²×C₁₇	1	4	yes	—
96	C₂³×C₁₁	C₂×C₈×C₁₇	1	8	yes	—
97	C₂³×C₁₁	C₂²×C₄×C₁₇	1	3	yes	—
98	C₂×C₄×C₁₁	C₄²×C₁₇	1	12	yes	—
99	C₂×C₄×C₁₁	C₂×C₈×C₁₇	1	48	yes	direct product factor
100	C₂×C₄×C₁₁	C₂²×C₄×C₁₇	1	3	yes	—
101	C₂×C₄×C₁₁	C₂²×C₄×C₁₇	1	8	yes	—
102	C₂×C₄×C₁₁	C₂²×C₄×C₁₇	1	12	yes	—
103	C₂×C₄×C₁₁	C₂⁴×C₁₇	1	1	yes	characteristic
104	C₈×C₁₁	C₂²×C₄×C₁₇	1	16	yes	direct product factor
105	C₂²×C₁₇	C₄×C₈×C₁₁	1	16	yes	direct product factor
106	C₂²×C₁₇	C₂×C₄²×C₁₁	1	6	yes	—
107	C₂²×C₁₇	C₂²×C₈×C₁₁	1	12	yes	—
108	C₂²×C₁₇	C₂³×C₄×C₁₁	1	1	yes	characteristic
109	C₄×C₁₇	C₂×C₄²×C₁₁	1	6	yes	—
110	C₄×C₁₇	C₂²×C₈×C₁₁	1	16	yes	direct product factor
111	C₄×C₁₇	C₂³×C₄×C₁₁	1	2	yes	—
112	C₂³×C₈	C₂×C₁₁×C₁₇	1	2	yes	—
113	C₂²×C₄²	C₂×C₁₁×C₁₇	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂×C₁₁×C₁₇	1	12	yes	direct product factor
115	C₂²×C₁₁	C₄×C₈×C₁₇	1	16	yes	direct product factor
116	C₂²×C₁₁	C₂×C₄²×C₁₇	1	6	yes	—
117	C₂²×C₁₁	C₂²×C₈×C₁₇	1	12	yes	—
118	C₂²×C₁₁	C₂³×C₄×C₁₇	1	1	yes	characteristic
119	C₄×C₁₁	C₂×C₄²×C₁₇	1	6	yes	—
120	C₄×C₁₁	C₂²×C₈×C₁₇	1	16	yes	direct product factor
121	C₄×C₁₁	C₂³×C₄×C₁₇	1	2	yes	—
122	C₂×C₁₇	C₂×C₄×C₈×C₁₁	1	12	yes	direct product factor
123	C₂×C₁₇	C₂²×C₄²×C₁₁	1	1	yes	characteristic
124	C₂×C₁₇	C₂³×C₈×C₁₁	1	2	yes	—
125	C₂³×C₄	C₄×C₁₁×C₁₇	1	2	yes	—
126	C₂³×C₄	C₂²×C₁₁×C₁₇	1	1	yes	characteristic
127	C₂²×C₈	C₄×C₁₁×C₁₇	1	16	yes	direct product factor
128	C₂²×C₈	C₂²×C₁₁×C₁₇	1	12	yes	—
129	C₂×C₄²	C₄×C₁₁×C₁₇	1	6	yes	—
130	C₂×C₄²	C₂²×C₁₁×C₁₇	1	6	yes	—
131	C₄×C₈	C₂²×C₁₁×C₁₇	1	16	yes	direct product factor
132	C₂×C₁₁	C₂×C₄×C₈×C₁₇	1	12	yes	direct product factor
133	C₂×C₁₁	C₂²×C₄²×C₁₇	1	1	yes	characteristic
134	C₂×C₁₁	C₂³×C₈×C₁₇	1	2	yes	—
135	C₁₇	C₂²×C₄×C₈×C₁₁	1	1	yes	characteristic, direct product factor, Sylow
136	C₂⁴	C₂×C₄×C₁₁×C₁₇	1	1	yes	characteristic
137	C₂²×C₄	C₈×C₁₁×C₁₇	1	16	yes	direct product factor
138	C₂²×C₄	C₂×C₄×C₁₁×C₁₇	1	3	yes	—
139	C₂²×C₄	C₂×C₄×C₁₁×C₁₇	1	8	yes	—
140	C₂²×C₄	C₂×C₄×C₁₁×C₁₇	1	12	yes	—
141	C₂²×C₄	C₂³×C₁₁×C₁₇	1	3	yes	—
142	C₂×C₈	C₂×C₄×C₁₁×C₁₇	1	48	yes	direct product factor
143	C₂×C₈	C₂³×C₁₁×C₁₇	1	8	yes	—
144	C₄²	C₂×C₄×C₁₁×C₁₇	1	12	yes	—
145	C₄²	C₂³×C₁₁×C₁₇	1	4	yes	—
146	C₁₁	C₂²×C₄×C₈×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
147	C₂³	C₂×C₈×C₁₁×C₁₇	1	8	yes	—
148	C₂³	C₄²×C₁₁×C₁₇	1	4	yes	—
149	C₂³	C₂²×C₄×C₁₁×C₁₇	1	3	yes	—
150	C₂×C₄	C₂×C₈×C₁₁×C₁₇	1	48	yes	direct product factor
151	C₂×C₄	C₄²×C₁₁×C₁₇	1	12	yes	—
152	C₂×C₄	C₂²×C₄×C₁₁×C₁₇	1	3	yes	—
153	C₂×C₄	C₂²×C₄×C₁₁×C₁₇	1	8	yes	—
154	C₂×C₄	C₂²×C₄×C₁₁×C₁₇	1	12	yes	—
155	C₂×C₄	C₂⁴×C₁₁×C₁₇	1	1	yes	characteristic
156	C₈	C₂²×C₄×C₁₁×C₁₇	1	16	yes	direct product factor
157	C₂²	C₄×C₈×C₁₁×C₁₇	1	16	yes	direct product factor
158	C₂²	C₂×C₄²×C₁₁×C₁₇	1	6	yes	—
159	C₂²	C₂²×C₈×C₁₁×C₁₇	1	12	yes	—
160	C₂²	C₂³×C₄×C₁₁×C₁₇	1	1	yes	characteristic
161	C₄	C₂×C₄²×C₁₁×C₁₇	1	6	yes	—
162	C₄	C₂²×C₈×C₁₁×C₁₇	1	16	yes	direct product factor
163	C₄	C₂³×C₄×C₁₁×C₁₇	1	2	yes	—
164	C₂	C₂×C₄×C₈×C₁₁×C₁₇	1	12	yes	direct product factor
165	C₂	C₂³×C₈×C₁₁×C₁₇	1	2	yes	—
166	C₂	C₂²×C₄²×C₁₁×C₁₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₁×C₁₇
socle, prime radical	C₂⁴×C₁₁×C₁₇	C₂×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×C₈×C₁₁×C₁₇	C₂	direct (N × H)
6. C₂²×C₈×C₁₁×C₁₇	C₄	direct (N × H)
10. C₄×C₈×C₁₁×C₁₇	C₂²	direct (N × H)
12. C₂²×C₄×C₁₁×C₁₇	C₈	direct (N × H)
19. C₂×C₈×C₁₁×C₁₇	C₂×C₄	direct (N × H)
21. C₂²×C₄×C₈×C₁₇	C₁₁	direct (N × H)
26. C₂×C₄×C₁₁×C₁₇	C₂×C₈	direct (N × H)
31. C₈×C₁₁×C₁₇	C₂²×C₄	direct (N × H)
32. C₂²×C₄×C₈×C₁₁	C₁₇	direct (N × H)
35. C₂×C₄×C₈×C₁₇	C₂×C₁₁	direct (N × H)
36. C₂²×C₁₁×C₁₇	C₄×C₈	direct (N × H)
41. C₄×C₁₁×C₁₇	C₂²×C₈	direct (N × H)
45. C₂×C₄×C₈×C₁₁	C₂×C₁₇	direct (N × H)
48. C₂²×C₈×C₁₇	C₄×C₁₁	direct (N × H)
52. C₄×C₈×C₁₇	C₂²×C₁₁	direct (N × H)
53. C₂×C₁₁×C₁₇	C₂×C₄×C₈	direct (N × H)
58. C₂²×C₈×C₁₁	C₄×C₁₇	direct (N × H)
62. C₄×C₈×C₁₁	C₂²×C₁₇	direct (N × H)
64. C₂²×C₄×C₁₇	C₈×C₁₁	direct (N × H)
69. C₂×C₈×C₁₇	C₂×C₄×C₁₁	direct (N × H)
73. C₁₁×C₁₇	C₂²×C₄×C₈	direct (N × H)
75. C₂²×C₄×C₁₁	C₈×C₁₇	direct (N × H)
80. C₂×C₈×C₁₁	C₂×C₄×C₁₇	direct (N × H)
88. C₂×C₄×C₁₇	C₂×C₈×C₁₁	direct (N × H)
93. C₈×C₁₇	C₂²×C₄×C₁₁	direct (N × H)
94. C₂²×C₄×C₈	C₁₁×C₁₇	direct (N × H)
99. C₂×C₄×C₁₁	C₂×C₈×C₁₇	direct (N × H)
104. C₈×C₁₁	C₂²×C₄×C₁₇	direct (N × H)
105. C₂²×C₁₇	C₄×C₈×C₁₁	direct (N × H)
110. C₄×C₁₇	C₂²×C₈×C₁₁	direct (N × H)
114. C₂×C₄×C₈	C₂×C₁₁×C₁₇	direct (N × H)
115. C₂²×C₁₁	C₄×C₈×C₁₇	direct (N × H)
120. C₄×C₁₁	C₂²×C₈×C₁₇	direct (N × H)
122. C₂×C₁₇	C₂×C₄×C₈×C₁₁	direct (N × H)
127. C₂²×C₈	C₄×C₁₁×C₁₇	direct (N × H)
131. C₄×C₈	C₂²×C₁₁×C₁₇	direct (N × H)
132. C₂×C₁₁	C₂×C₄×C₈×C₁₇	direct (N × H)
135. C₁₇	C₂²×C₄×C₈×C₁₁	direct (N × H)
137. C₂²×C₄	C₈×C₁₁×C₁₇	direct (N × H)
142. C₂×C₈	C₂×C₄×C₁₁×C₁₇	direct (N × H)
146. C₁₁	C₂²×C₄×C₈×C₁₇	direct (N × H)
150. C₂×C₄	C₂×C₈×C₁₁×C₁₇	direct (N × H)
156. C₈	C₂²×C₄×C₁₁×C₁₇	direct (N × H)
157. C₂²	C₄×C₈×C₁₁×C₁₇	direct (N × H)
162. C₄	C₂²×C₈×C₁₁×C₁₇	direct (N × H)
164. C₂	C₂×C₄×C₈×C₁₁×C₁₇	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×C₄²×C₁₁×C₁₇	2
C₂³×C₈×C₁₁×C₁₇	2
C₂×C₄×C₈×C₁₁×C₁₇	2
C₂²×C₄×C₈×C₁₇	11
C₂²×C₄×C₈×C₁₁	17

Subgroups — C22×C4×C8×C11×C17 #72201218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₈×C₁₁×C₁₇ #72201218