Subgroups — C₁₇×#12015 #16869759

Order 4352.

246 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#2162	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₈×M₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1370	C₂	1	2	yes	maximal, complemented
5	C₁₇×#1367	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₆×C₁₇	—	2	1	no	—
7	C₁₇×#431	C₄	1	1	yes	characteristic
8	C₄×M₁₆×C₁₇	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₁₇×C₄⋊C₁₆	—	2	1	no	—
11	C₁₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₁₇×C₄⋊C₁₆	C₂²	1	2	yes	—
13	C₁₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₁₇×C₂²⋊C₁₆	—	2	1	no	—
15	C₄×C₁₆×C₁₇	—	2	1	no	—
16	C₁₇×#322	C₄	1	2	yes	—
17	C₈²×C₁₇	C₂²	1	2	yes	—
18	C₂×M₁₆×C₁₇	C₈	1	1	yes	characteristic
19	C₂×M₁₆×C₁₇	C₈	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₁₇	—	2	2	no	—
24	C₂×C₁₆×C₁₇	—	2	1	no	—
25	C₂×C₁₆×C₁₇	—	2	1	no	—
26	C₂×C₁₆×C₁₇	—	4	2	no	—
27	C₁₇×C₄⋊C₈	—	2	1	no	—
28	C₁₇×C₂²⋊C₈	—	2	1	no	—
29	C₁₇×#97	C₈	1	2	yes	—
30	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
34	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
35	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
39	M₁₆×C₁₇	—	2	2	no	—
40	M₁₆×C₁₇	C₁₆	1	4	yes	complemented
41	C₂×C₈×C₁₇	—	2	1	no	—
42	C₂×C₈×C₁₇	—	2	1	no	—
43	C₂×C₈×C₁₇	—	2	2	no	—
44	C₂×C₈×C₁₇	—	4	1	no	—
45	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
54	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
55	C₁₆×C₁₇	—	4	1	no	—
56	C₁₆×C₁₇	—	4	2	no	—
57	C₁₆×C₁₇	—	4	1	no	—
58	#12015	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₁₇	C₄⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₇	—	2	1	no	—
61	C₂×C₄×C₁₇	—	2	1	no	—
62	C₂×C₄×C₁₇	—	4	1	no	—
63	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
67	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
68	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
71	C₈×C₁₇	—	2	1	no	—
72	C₈×C₁₇	—	2	1	no	—
73	C₈×C₁₇	—	2	1	no	—
74	C₈×C₁₇	—	2	2	no	—
75	C₈×C₁₇	—	4	2	no	—
76	C₈×C₁₇	C₂×C₁₆	1	4	yes	complemented
77	C₈×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₈×C₁₇	#137	1	1	yes	characteristic
79	#2162	C₂×C₁₇	1	1	yes	characteristic
80	C₂×C₄⋊C₁₆	C₂×C₁₇	1	1	yes	characteristic
81	C₈×M₁₆	C₂×C₁₇	1	1	yes	characteristic
82	#1370	C₂×C₁₇	1	2	yes	complemented
83	#1367	C₂×C₁₇	1	2	yes	complemented
84	C₂²×C₁₇	—	2	1	no	—
85	C₂²×C₁₇	—	2	1	no	—
86	C₂²×C₁₇	C₄⋊C₁₆	1	2	yes	—
87	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
88	C₄×C₁₇	—	2	1	no	—
89	C₄×C₁₇	—	2	1	no	—
90	C₄×C₁₇	—	2	1	no	—
91	C₄×C₁₇	C₄⋊C₁₆	1	2	yes	—
92	C₄×C₁₇	#428	1	1	yes	characteristic
93	C₄×C₁₇	#430	1	1	yes	characteristic
94	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
95	C₄×C₁₇	C₂×M₃₂	1	1	yes	characteristic
96	C₂²×C₁₆	—	2	1	no	—
97	#431	C₄×C₁₇	1	1	yes	characteristic
98	C₄×M₁₆	C₄×C₁₇	1	1	yes	characteristic
99	C₂×C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
100	C₄⋊C₁₆	—	2	1	no	—
101	C₄⋊C₁₆	C₂²×C₁₇	1	1	yes	characteristic
102	C₄⋊C₁₆	C₂²×C₁₇	1	2	yes	—
103	C₄⋊C₁₆	C₂²×C₁₇	1	1	yes	characteristic
104	C₂²⋊C₁₆	—	2	1	no	—
105	C₄×C₁₆	—	2	1	no	—
106	#322	C₄×C₁₇	1	2	yes	—
107	C₈²	C₂²×C₁₇	1	2	yes	—
108	C₂×C₁₇	—	2	2	no	—
109	C₂×C₁₇	#1565	1	1	yes	characteristic
110	C₂×C₁₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
111	C₂×C₁₇	#2153	1	1	yes	characteristic
112	C₂×M₁₆	C₈×C₁₇	1	1	yes	characteristic
113	C₂×M₁₆	C₈×C₁₇	1	1	yes	characteristic
114	C₂²×C₈	D₄×C₁₇	1	1	yes	characteristic
115	C₂²×C₈	Q₈×C₁₇	1	1	yes	characteristic
116	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
117	C₂×C₁₆	—	2	1	no	—
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	4	2	no	—
121	C₄⋊C₈	—	2	1	no	—
122	C₂²⋊C₈	—	2	1	no	—
123	#97	C₈×C₁₇	1	2	yes	—
124	C₄×C₈	C₂×C₄×C₁₇	1	2	yes	—
125	C₄×C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
126	C₄×C₈	C₂×C₄×C₁₇	1	2	yes	—
127	C₄×C₈	D₄×C₁₇	1	1	yes	characteristic
128	C₄×C₈	Q₈×C₁₇	1	1	yes	characteristic
129	C₄×C₈	C₂³×C₁₇	1	1	yes	characteristic
130	C₁₇	#12015	1	1	yes	characteristic, direct product factor, Sylow
131	C₂²×C₄	C₄⋊C₄×C₁₇	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₈×C₁₇	1	1	yes	characteristic
133	C₂²×C₄	M₁₆×C₁₇	1	1	yes	characteristic
134	M₁₆	—	2	2	no	—
135	M₁₆	C₁₆×C₁₇	1	4	yes	complemented
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	C₄⋊C₄×C₁₇	1	1	yes	characteristic
141	C₂×C₈	C₄⋊C₄×C₁₇	1	1	yes	characteristic
142	C₂×C₈	C₂×C₈×C₁₇	1	2	yes	—
143	C₂×C₈	C₂×C₈×C₁₇	1	2	yes	—
144	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
145	C₂×C₈	C₂×Q₈×C₁₇	1	1	yes	characteristic
146	C₄²	C₄⋊C₄×C₁₇	1	1	yes	characteristic
147	C₄²	C₂×C₈×C₁₇	1	1	yes	characteristic
148	C₄²	M₁₆×C₁₇	1	1	yes	characteristic
149	C₄²	C₂²×C₄×C₁₇	1	1	yes	characteristic
150	C₁₆	—	4	2	no	—
151	C₁₆	—	4	1	no	—
152	C₁₆	—	4	1	no	—
153	C₂³	C₁₇×C₄⋊C₈	1	1	yes	characteristic
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	C₁₇×C₄⋊C₈	1	1	yes	characteristic
158	C₂×C₄	C₁₇×C₄⋊C₈	1	1	yes	characteristic
159	C₂×C₄	C₁₇×C₄⋊C₈	1	1	yes	characteristic
160	C₂×C₄	C₂×C₁₆×C₁₇	1	2	yes	—
161	C₂×C₄	C₁₇×M₃₂	1	2	yes	—
162	C₂×C₄	C₂×C₄⋊C₄×C₁₇	1	1	yes	characteristic
163	C₂×C₄	C₂²×C₈×C₁₇	1	1	yes	characteristic
164	C₂×C₄	C₂×M₁₆×C₁₇	1	1	yes	characteristic
165	C₈	—	2	2	no	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	4	2	no	—
170	C₈	C₂×C₁₆×C₁₇	1	4	yes	complemented
171	C₈	C₁₇×AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₈	C₁₇×#137	1	1	yes	characteristic
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	C₁₇×C₄⋊C₁₆	1	2	yes	—
176	C₂²	C₂×C₁₇×C₄⋊C₈	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	C₁₇×C₄⋊C₁₆	1	2	yes	—
181	C₄	C₁₇×#428	1	1	yes	characteristic
182	C₄	C₁₇×#430	1	1	yes	characteristic
183	C₄	C₂²×C₁₆×C₁₇	1	1	yes	characteristic
184	C₄	C₂×C₁₇×M₃₂	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	C₁₇×#1565	1	1	yes	characteristic
187	C₂	C₂×C₁₇×C₄⋊C₁₆	1	1	yes	characteristic
188	C₂	C₁₇×#2153	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₇	C₂×D₄
commutator	C₄	C₂²×C₁₆×C₁₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₇
socle	C₂²×C₁₇	C₂×C₄⋊C₈
prime radical	C₂³×C₁₇	C₄⋊C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₇ ◁ C₂×C₄×C₈×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₇×#1370	C₂	semidirect
5. C₁₇×#1367	C₂	semidirect
40. M₁₆×C₁₇	C₁₆	semidirect
58. #12015	C₁₇	direct (N × H)
76. C₈×C₁₇	C₂×C₁₆	semidirect
82. #1370	C₂×C₁₇	semidirect
83. #1367	C₂×C₁₇	semidirect
130. C₁₇	#12015	direct (N × H)
135. M₁₆	C₁₆×C₁₇	semidirect
170. C₈	C₂×C₁₆×C₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₇×#2162	2
C₂×C₁₇×C₄⋊C₁₆	2
C₈×M₁₆×C₁₇	2
C₁₇×#1370	2
C₁₇×#1367	2
#12015	17

Subgroups — C17×#12015 #16869759

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#12015 #16869759