Subgroups — C₁₆×C₂²⋊Dic₁₇ #16949403

Order 4352.

510 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#63214	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₁₇	C₂	1	2	yes	maximal, complemented
3	C₈×C₂²⋊Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	#14526861	C₂	1	2	yes	maximal, complemented
5	C₂³×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#63214	C₂²	1	2	yes	—
7	C₂×C₂²⋊#3622	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₁₇	C₂²	1	2	yes	—
9	C₄×C₂²⋊Dic₁₇	C₄	1	1	yes	characteristic
10	C₂²⋊#63214	C₄	1	4	yes	complemented
11	C₁₆×Dic₁₇	—	2	4	no	—
12	#11762563	C₄	1	2	yes	—
13	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₁₇	—	2	4	no	—
15	C₂²×C₁₆×C₁₇	C₄	1	4	yes	complemented
16	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	—	2	2	no	—
20	C₂²×C₈×C₁₇	—	2	2	no	—
21	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
25	C₂×C₂²⋊Dic₁₇	C₈	1	2	yes	—
26	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
27	C₂²×#3622	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₁₇	—	2	8	no	—
29	C₂×C₁₆×C₁₇	—	2	2	no	—
30	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₁₇	D₄	1	4	yes	—
33	#10587502	C₈	1	2	yes	—
34	C₂²⋊#3622	C₈	1	4	yes	—
35	C₈×Dic₁₇	—	2	2	no	—
36	C₈×Dic₁₇	—	2	2	no	—
37	C₂×#63214	—	2	2	no	—
38	C₂×#63214	—	2	2	no	—
39	C₂×#63214	C₂×C₄	1	4	yes	—
40	C₂⁴×C₁₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	—	2	2	no	—
42	C₂²×C₄×C₁₇	—	2	2	no	—
43	C₂²×C₄×C₁₇	C₄²	1	2	yes	—
44	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×Dic₁₇	C₂×C₈	1	4	yes	—
48	C₂×C₈×C₁₇	—	2	2	no	—
49	C₂×C₈×C₁₇	—	2	2	no	—
50	C₂×C₈×C₁₇	—	2	2	no	—
51	C₂×C₈×C₁₇	—	2	4	no	—
52	C₂×C₈×C₁₇	C₄²	1	2	yes	—
53	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₁₇	#41	1	2	yes	—
57	C₂²⋊Dic₁₇	C₁₆	1	8	yes	direct product factor
58	C₄×Dic₁₇	—	2	2	no	—
59	C₄×Dic₁₇	—	2	2	no	—
60	C₂×#3622	—	2	2	no	—
61	C₂×#3622	—	2	2	no	—
62	C₂×#3622	C₂×C₈	1	4	yes	—
63	C₁₆×C₁₇	—	2	2	no	—
64	C₁₆×C₁₇	C₂²⋊C₄	1	4	yes	complemented
65	#63214	—	2	4	no	—
66	C₁₆×C₂²⋊C₄	—	17	1	no	maximal, Sylow
67	C₂³×C₁₇	—	2	2	no	—
68	C₂³×C₁₇	—	2	2	no	—
69	C₂³×C₁₇	C₄×C₈	1	2	yes	—
70	C₂³×C₁₇	C₂×C₁₆	1	4	yes	—
71	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₁₇	—	2	2	no	—
73	C₂×C₄×C₁₇	—	2	2	no	—
74	C₂×C₄×C₁₇	—	2	2	no	—
75	C₂×C₄×C₁₇	—	2	4	no	—
76	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
77	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₇	#131	1	2	yes	—
81	C₂×Dic₁₇	—	2	4	no	—
82	C₂×Dic₁₇	—	2	4	no	—
83	C₂×Dic₁₇	C₂×C₁₆	1	8	yes	complemented
84	C₈×C₁₇	—	2	2	no	—
85	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₈×C₁₇	#117	1	1	yes	characteristic
87	C₈×C₁₇	C₄×D₄	1	2	yes	—
88	#3622	—	2	4	no	—
89	C₂³×C₁₆	D₁₇	1	1	yes	characteristic
90	C₂×C₂²⋊C₁₆	—	17	1	no	—
91	C₂×C₄×C₁₆	—	17	2	no	—
92	C₈×C₂²⋊C₄	—	17	1	no	—
93	#1374	—	17	2	no	—
94	C₂²×C₁₇	—	2	2	no	—
95	C₂²×C₁₇	—	2	8	no	—
96	C₂²×C₁₇	—	2	2	no	—
97	C₂²×C₁₇	C₄×C₁₆	1	4	yes	complemented
98	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
99	C₂²×C₁₇	C₈×D₄	1	2	yes	—
100	C₂²×C₁₇	C₂²×C₁₆	1	2	yes	—
101	C₂²×C₁₇	#504	1	2	yes	—
102	C₄×C₁₇	—	2	2	no	—
103	C₄×C₁₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
104	C₄×C₁₇	#405	1	1	yes	characteristic
105	C₄×C₁₇	C₈×D₄	1	2	yes	—
106	Dic₁₇	—	2	8	no	—
107	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
108	C₂²×C₁₆	—	2	4	no	—
109	C₂²×C₁₆	—	17	2	no	—
110	C₂²×C₁₆	Dic₁₇	1	4	yes	complemented
111	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
112	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
113	C₂×C₂²⋊C₈	—	17	1	no	—
114	C₂×C₄×C₈	—	17	2	no	—
115	C₄×C₂²⋊C₄	—	17	1	no	—
116	C₂²⋊C₁₆	—	17	4	no	—
117	C₄×C₁₆	—	34	4	no	—
118	#336	—	17	2	no	—
119	C₂×C₁₇	—	2	4	no	—
120	C₂×C₁₇	C₈×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₁₇	C₂×C₄×C₁₆	1	1	yes	characteristic
122	C₂×C₁₇	#2108	1	1	yes	characteristic
123	C₂×C₁₇	D₄×C₁₆	1	4	yes	—
124	C₂³×C₄	C₄×D₁₇	1	1	yes	characteristic
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	17	2	no	—
128	C₂²×C₈	C₄×D₁₇	1	2	yes	—
129	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
130	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
131	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
132	C₂×C₂²⋊C₄	—	17	2	no	—
133	C₂×C₄²	—	17	2	no	—
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	2	8	no	—
136	C₂×C₁₆	—	17	4	no	—
137	C₂×C₁₆	—	34	2	no	—
138	C₂×C₁₆	—	34	2	no	—
139	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
140	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
141	C₂×C₁₆	C₂²⋊D₁₇	1	4	yes	—
142	C₂²⋊C₈	—	17	4	no	—
143	C₄×C₈	—	34	2	no	—
144	C₄×C₈	—	34	2	no	—
145	#95	—	17	2	no	—
146	C₁₇	C₁₆×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
147	C₂⁴	C₈×D₁₇	1	1	yes	characteristic
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	17	4	no	—
151	C₂²×C₄	C₈×D₁₇	1	1	yes	characteristic
152	C₂²×C₄	C₈×D₁₇	1	2	yes	—
153	C₂²×C₄	C₄×Dic₁₇	1	2	yes	—
154	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	17	4	no	—
160	C₂×C₈	—	34	2	no	—
161	C₂×C₈	—	34	2	no	—
162	C₂×C₈	C₄×Dic₁₇	1	2	yes	—
163	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
164	C₂×C₈	#63252	1	2	yes	—
165	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
166	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
167	C₂²⋊C₄	—	17	8	no	—
168	C₄²	—	34	2	no	—
169	C₄²	—	34	2	no	—
170	C₁₆	—	2	2	no	—
171	C₁₆	—	34	4	no	—
172	C₁₆	C₂²⋊Dic₁₇	1	4	yes	direct product factor
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	C₁₆×D₁₇	1	4	yes	—
176	C₂³	C₈×Dic₁₇	1	2	yes	—
177	C₂³	C₂×C₈×D₁₇	1	1	yes	characteristic
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	17	8	no	—
183	C₂×C₄	—	34	4	no	—
184	C₂×C₄	—	34	4	no	—
185	C₂×C₄	C₈×Dic₁₇	1	2	yes	—
186	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
187	C₂×C₄	#10587587	1	2	yes	—
188	C₂×C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
189	C₂×C₄	C₄×C₂²⋊D₁₇	1	2	yes	—
190	C₈	—	2	2	no	—
191	C₈	—	34	4	no	—
192	C₈	#10587612	1	1	yes	characteristic
193	C₈	C₄×C₂²⋊D₁₇	1	2	yes	—
194	C₈	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
195	C₂²	—	2	8	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	C₁₆×Dic₁₇	1	4	yes	complemented
199	C₂²	C₂×C₈×Dic₁₇	1	1	yes	characteristic
200	C₂²	C₈×C₂²⋊D₁₇	1	2	yes	—
201	C₂²	C₂×C₁₆×D₁₇	1	2	yes	—
202	C₂²	#11763225	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	34	8	no	—
205	C₄	C₄×C₂²⋊Dic₁₇	1	1	yes	characteristic
206	C₄	#11762764	1	1	yes	characteristic
207	C₄	C₈×C₂²⋊D₁₇	1	2	yes	—
208	C₂	—	2	4	no	—
209	C₂	C₈×C₂²⋊Dic₁₇	1	1	yes	characteristic
210	C₂	C₂×C₁₆×Dic₁₇	1	1	yes	characteristic
211	C₂	#14530380	1	1	yes	characteristic
212	C₂	C₁₆×C₂²⋊D₁₇	1	4	yes	—

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₁₇
commutator	C₂×C₁₇	C₂×C₄×C₁₆
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂³×C₁₆×C₁₇	C₂
socle	C₂³×C₁₇	C₂²×C₈
prime radical	C₂⁴×C₁₇	C₂×C₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂³×C₁₆	D₁₇
hypocenter	C₁₇	C₁₆×C₂²⋊C₄
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₆×Dic₁₇	C₂	semidirect
4. #14526861	C₂	semidirect
10. C₂²⋊#63214	C₄	semidirect
15. C₂²×C₁₆×C₁₇	C₄	semidirect
57. C₂²⋊Dic₁₇	C₁₆	direct (N × H)
64. C₁₆×C₁₇	C₂²⋊C₄	semidirect
83. C₂×Dic₁₇	C₂×C₁₆	semidirect
97. C₂²×C₁₇	C₄×C₁₆	semidirect
110. C₂²×C₁₆	Dic₁₇	semidirect
146. C₁₇	C₁₆×C₂²⋊C₄	semidirect
172. C₁₆	C₂²⋊Dic₁₇	direct (N × H)
198. C₂²	C₁₆×Dic₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊#63214	2
C₂×C₁₆×Dic₁₇	2
C₈×C₂²⋊Dic₁₇	2
#14526861	2
C₂³×C₁₆×C₁₇	2
C₁₆×C₂²⋊C₄	17

Subgroups — C16×C22⋊Dic17 #16949403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×C₂²⋊Dic₁₇ #16949403