Subgroups — C₁₇×SD₁₆⋊C₁₆ #16869799

Order 4352.

232 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
3	C₈×SD₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1370	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₇×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#1312	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₆×C₁₇	—	2	1	no	—
9	C₈×Q₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₄×SD₁₆×C₁₇	C₄	1	1	yes	characteristic
11	C₈×D₄×C₁₇	C₂²	1	1	yes	characteristic
12	C₁₇×C₄⋊C₁₆	—	2	1	no	—
13	C₁₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₁₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₁₇×C₂²⋊C₁₆	—	2	1	no	—
16	C₄×C₁₆×C₁₇	—	2	1	no	—
17	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
19	C₁₇×#334	C₄	1	1	yes	characteristic
20	C₁₇×Q₈⋊C₈	C₄	1	1	yes	characteristic
21	C₁₇×D₄⋊C₈	C₄	1	1	yes	characteristic
22	C₈²×C₁₇	C₂²	1	1	yes	characteristic
23	C₂×SD₁₆×C₁₇	C₈	1	1	yes	characteristic
24	C₂²×C₈×C₁₇	—	2	1	no	—
25	C₄×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₄×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₁₇	—	2	1	no	—
28	C₂×C₁₆×C₁₇	—	2	1	no	—
29	C₂×C₁₆×C₁₇	—	2	1	no	—
30	C₂×C₁₆×C₁₇	—	4	2	no	—
31	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
33	C₁₇×#106	C₈	1	1	yes	characteristic
34	C₁₇×C₄⋊C₈	—	2	1	no	—
35	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₁₇×#103	C₈	1	1	yes	characteristic
38	C₁₇×#102	C₈	1	1	yes	characteristic
39	C₁₇×C₂²⋊C₈	—	2	1	no	—
40	C₄×C₈×C₁₇	—	2	1	no	—
41	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
45	C₂×D₄×C₁₇	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	—	2	1	no	—
47	SD₁₆×C₁₇	C₁₆	1	4	yes	complemented
48	C₂×C₈×C₁₇	—	2	1	no	—
49	C₂×C₈×C₁₇	—	2	1	no	—
50	C₂×C₈×C₁₇	—	2	1	no	—
51	C₂×C₈×C₁₇	—	4	1	no	—
52	C₂×C₈×C₁₇	—	4	1	no	—
53	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
54	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
55	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
57	C₄⋊C₄×C₁₇	—	2	1	no	—
58	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
59	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
60	C₂²⋊C₄×C₁₇	—	2	1	no	—
61	C₄²×C₁₇	—	2	1	no	—
62	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
63	C₁₆×C₁₇	—	2	1	no	—
64	C₁₆×C₁₇	—	2	1	no	—
65	C₁₆×C₁₇	—	4	1	no	—
66	C₁₆×C₁₇	—	4	1	no	—
67	C₁₆×C₁₇	—	4	1	no	—
68	SD₁₆⋊C₁₆	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₁₇	—	2	1	no	—
70	Q₈×C₁₇	—	2	1	no	—
71	Q₈×C₁₇	C₂×C₁₆	1	2	yes	complemented
72	D₄×C₁₇	—	2	1	no	—
73	D₄×C₁₇	C₂×C₁₆	1	2	yes	—
74	C₂×C₄×C₁₇	—	2	1	no	—
75	C₂×C₄×C₁₇	—	2	1	no	—
76	C₂×C₄×C₁₇	—	2	1	no	—
77	C₂×C₄×C₁₇	—	4	1	no	—
78	C₂×C₄×C₁₇	—	4	1	no	—
79	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
82	C₈×C₁₇	—	2	1	no	—
83	C₈×C₁₇	—	2	1	no	—
84	C₈×C₁₇	—	4	1	no	—
85	C₈×C₁₇	—	4	1	no	—
86	C₈×C₁₇	—	4	1	no	—
87	C₈×C₁₇	C₂×C₁₆	1	2	yes	complemented
88	C₈×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₈×C₁₇	#137	1	1	yes	characteristic
90	Q₈×C₁₆	C₂×C₁₇	1	1	yes	characteristic, complemented
91	D₄×C₁₆	C₂×C₁₇	1	1	yes	characteristic
92	C₈×SD₁₆	C₂×C₁₇	1	1	yes	characteristic
93	#1370	C₂×C₁₇	1	1	yes	characteristic, complemented
94	Q₈⋊C₁₆	C₂×C₁₇	1	1	yes	characteristic, complemented
95	D₄⋊C₁₆	C₂×C₁₇	1	1	yes	characteristic
96	#1312	C₂×C₁₇	1	1	yes	characteristic, complemented
97	C₂²×C₁₇	—	2	2	no	—
98	C₂²×C₁₇	—	4	1	no	—
99	C₂²×C₁₇	—	4	1	no	—
100	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
101	C₄×C₁₇	—	2	2	no	—
102	C₄×C₁₇	—	2	1	no	—
103	C₄×C₁₇	—	4	1	no	—
104	C₄×C₁₇	—	4	1	no	—
105	C₄×C₁₇	#440	1	1	yes	characteristic
106	C₄×C₁₇	#444	1	1	yes	characteristic
107	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
108	C₄×C₁₇	#504	1	1	yes	characteristic
109	C₂²×C₁₆	—	2	1	no	—
110	C₈×Q₈	C₂²×C₁₇	1	1	yes	characteristic
111	C₄×SD₁₆	C₄×C₁₇	1	1	yes	characteristic
112	C₈×D₄	C₂²×C₁₇	1	1	yes	characteristic
113	C₄⋊C₁₆	—	2	1	no	—
114	C₄⋊C₁₆	C₂²×C₁₇	1	1	yes	characteristic
115	C₄⋊C₁₆	C₂²×C₁₇	1	1	yes	characteristic
116	C₂²⋊C₁₆	—	2	1	no	—
117	C₄×C₁₆	—	2	1	no	—
118	C₄×C₁₆	C₂²×C₁₇	1	1	yes	characteristic
119	C₄×C₁₆	C₂²×C₁₇	1	1	yes	characteristic
120	#334	C₄×C₁₇	1	1	yes	characteristic
121	Q₈⋊C₈	C₄×C₁₇	1	1	yes	characteristic
122	D₄⋊C₈	C₄×C₁₇	1	1	yes	characteristic
123	C₈²	C₂²×C₁₇	1	1	yes	characteristic
124	C₂×C₁₇	—	4	2	no	—
125	C₂×C₁₇	SD₁₆⋊C₈	1	1	yes	characteristic
126	C₂×C₁₇	D₄×C₁₆	1	1	yes	characteristic
127	C₂×C₁₇	#2171	1	1	yes	characteristic
128	C₂×SD₁₆	C₈×C₁₇	1	1	yes	characteristic
129	C₂²×C₈	—	2	1	no	—
130	C₄×Q₈	C₂×C₄×C₁₇	1	1	yes	characteristic
131	C₄×D₄	C₂×C₄×C₁₇	1	1	yes	characteristic
132	C₂×C₁₆	—	2	1	no	—
133	C₂×C₁₆	—	2	1	no	—
134	C₂×C₁₆	—	2	1	no	—
135	C₂×C₁₆	—	4	2	no	—
136	C₂×C₁₆	D₄×C₁₇	1	1	yes	characteristic
137	C₂×C₁₆	D₄×C₁₇	1	1	yes	characteristic
138	#106	C₈×C₁₇	1	1	yes	characteristic
139	C₄⋊C₈	—	2	1	no	—
140	C₄⋊C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
141	C₄⋊C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
142	#103	C₈×C₁₇	1	1	yes	characteristic
143	#102	C₈×C₁₇	1	1	yes	characteristic
144	C₂²⋊C₈	—	2	1	no	—
145	C₄×C₈	—	2	1	no	—
146	C₄×C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
147	C₄×C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
148	C₄×C₈	C₂³×C₁₇	1	1	yes	characteristic
149	C₁₇	SD₁₆⋊C₁₆	1	1	yes	characteristic, direct product factor, Sylow
150	C₂×Q₈	C₂×C₈×C₁₇	1	1	yes	characteristic
151	C₂×D₄	C₂×C₈×C₁₇	1	1	yes	characteristic
152	C₂²×C₄	—	2	1	no	—
153	SD₁₆	C₁₆×C₁₇	1	4	yes	complemented
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	4	1	no	—
158	C₂×C₈	—	4	1	no	—
159	C₂×C₈	C₂×C₈×C₁₇	1	1	yes	characteristic
160	C₂×C₈	C₂×C₈×C₁₇	1	1	yes	characteristic
161	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
162	C₂×C₈	#41×C₁₇	1	1	yes	characteristic
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	C₂×C₈×C₁₇	1	1	yes	characteristic
165	C₄⋊C₄	C₂×C₈×C₁₇	1	1	yes	characteristic
166	C₂²⋊C₄	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	C₂²×C₄×C₁₇	1	1	yes	characteristic
169	C₁₆	—	2	1	no	—
170	C₁₆	—	2	1	no	—
171	C₁₆	—	4	1	no	—
172	C₁₆	—	4	1	no	—
173	C₁₆	—	4	1	no	—
174	C₂³	—	2	1	no	—
175	Q₈	—	2	1	no	—
176	Q₈	C₂×C₁₆×C₁₇	1	2	yes	complemented
177	D₄	—	2	1	no	—
178	D₄	C₂×C₁₆×C₁₇	1	2	yes	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	C₄×D₄×C₁₇	1	1	yes	characteristic
185	C₂×C₄	C₂²×C₈×C₁₇	1	1	yes	characteristic
186	C₂×C₄	C₁₇×#131	1	1	yes	characteristic
187	C₈	—	2	1	no	—
188	C₈	—	2	1	no	—
189	C₈	—	4	1	no	—
190	C₈	—	4	1	no	—
191	C₈	—	4	1	no	—
192	C₈	C₂×C₁₆×C₁₇	1	2	yes	complemented
193	C₈	C₁₇×AGL(1,ℤ/8)	1	1	yes	characteristic
194	C₈	C₁₇×#137	1	1	yes	characteristic
195	C₂²	—	2	2	no	—
196	C₂²	—	4	1	no	—
197	C₂²	—	4	1	no	—
198	C₂²	C₈×D₄×C₁₇	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	1	no	—
201	C₄	—	4	1	no	—
202	C₄	—	4	1	no	—
203	C₄	C₁₇×#440	1	1	yes	characteristic
204	C₄	C₁₇×#444	1	1	yes	characteristic
205	C₄	C₂²×C₁₆×C₁₇	1	1	yes	characteristic
206	C₄	C₁₇×#504	1	1	yes	characteristic
207	C₂	—	4	2	no	—
208	C₂	C₁₇×SD₁₆⋊C₈	1	1	yes	characteristic
209	C₂	D₄×C₁₆×C₁₇	1	1	yes	characteristic
210	C₂	C₁₇×#2171	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	C₈×D₄
prime radical	C₂×D₄×C₁₇	C₂×C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₇ ◁ C₄×C₁₆×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×C₁₆×C₁₇	C₂	semidirect
4. C₁₇×#1370	C₂	semidirect
5. C₁₇×Q₈⋊C₁₆	C₂	semidirect
7. C₁₇×#1312	C₂	semidirect
47. SD₁₆×C₁₇	C₁₆	semidirect
68. SD₁₆⋊C₁₆	C₁₇	direct (N × H)
71. Q₈×C₁₇	C₂×C₁₆	semidirect
87. C₈×C₁₇	C₂×C₁₆	semidirect
90. Q₈×C₁₆	C₂×C₁₇	semidirect
93. #1370	C₂×C₁₇	semidirect
94. Q₈⋊C₁₆	C₂×C₁₇	semidirect
96. #1312	C₂×C₁₇	semidirect
149. C₁₇	SD₁₆⋊C₁₆	direct (N × H)
153. SD₁₆	C₁₆×C₁₇	semidirect
176. Q₈	C₂×C₁₆×C₁₇	semidirect
192. 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
Q₈×C₁₆×C₁₇	2
D₄×C₁₆×C₁₇	2
C₈×SD₁₆×C₁₇	2
C₁₇×#1370	2
C₁₇×Q₈⋊C₁₆	2
C₁₇×D₄⋊C₁₆	2
C₁₇×#1312	2
SD₁₆⋊C₁₆	17

Subgroups — C17×SD16⋊C16 #16869799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×SD₁₆⋊C₁₆ #16869799