Subgroups — Q₈×C₁₆×D₁₇ #17187803

Order 4352.

582 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Q₈×D₁₇	C₂	1	1	yes	maximal, characteristic
2	Q₈×#63214	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₆×Dic₃₄	C₂	1	3	yes	maximal, complemented
4	#14530686	C₂	1	3	yes	maximal, complemented
5	D₁₇×C₄⋊C₁₆	C₂	1	3	yes	maximal
6	C₄×C₁₆×D₁₇	C₂	1	3	yes	maximal
7	Q₈×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Q₈×D₁₇	C₄	1	1	yes	characteristic
9	C₂×C₁₆×D₁₇	C₂²	1	3	yes	—
10	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
11	Q₈×#3622	C₄	1	1	yes	characteristic
12	C₈×Dic₃₄	C₂²	1	3	yes	—
13	#11762933	C₄	1	3	yes	—
14	C₄⋊C₈×D₁₇	C₄	1	3	yes	—
15	C₄×C₈×D₁₇	C₂²	1	3	yes	—
16	C₄⋊#63214	C₂²	1	3	yes	—
17	#11762627	C₂²	1	6	yes	—
18	C₄×#63214	C₂²	1	3	yes	—
19	C₁₆×Dic₁₇	C₂²	1	3	yes	—
20	C₁₆×Dic₁₇	C₂²	1	3	yes	—
21	C₈×Q₈×C₁₇	C₂²	1	1	yes	characteristic
22	C₁₇×C₄⋊C₁₆	C₂²	1	3	yes	—
23	C₄×C₁₆×C₁₇	C₂²	1	3	yes	—
24	C₂×Q₈×D₁₇	C₈	1	1	yes	characteristic
25	C₄×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	Q₈×Dic₁₇	C₈	1	1	yes	characteristic
27	C₂×C₈×D₁₇	C₂×C₄	1	3	yes	—
28	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×D₁₇	C₈	1	3	yes	—
30	#10587573	C₈	1	3	yes	—
31	C₄²×D₁₇	C₂×C₄	1	3	yes	—
32	C₄×Dic₃₄	C₂×C₄	1	3	yes	—
33	C₂×C₁₆×C₁₇	C₂³	1	3	yes	—
34	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
35	C₁₇×C₄⋊C₈	C₂×C₄	1	3	yes	—
36	C₄×C₈×C₁₇	C₂³	1	3	yes	—
37	#10587485	C₂×C₄	1	6	yes	—
38	C₈×Dic₁₇	C₂×C₄	1	3	yes	—
39	C₈×Dic₁₇	C₂³	1	3	yes	—
40	C₂×#63214	C₂³	1	3	yes	—
41	C₂×#63214	C₂³	1	1	yes	characteristic
42	C₄⋊#3622	C₂×C₄	1	3	yes	—
43	C₄×#3622	C₂×C₄	1	3	yes	—
44	C₁₆×D₁₇	—	2	6	no	—
45	C₁₆×D₁₇	Q₈	1	4	yes	direct product factor
46	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
47	Q₈×D₁₇	C₁₆	1	8	yes	direct product factor
48	C₂×C₄×D₁₇	C₂×C₈	1	3	yes	—
49	C₂×C₄×D₁₇	C₂×C₈	1	3	yes	—
50	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×Dic₃₄	C₂×C₈	1	3	yes	—
52	C₂×C₈×C₁₇	C₂²×C₄	1	3	yes	—
53	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
54	C₄⋊C₄×C₁₇	C₂×C₈	1	3	yes	—
55	C₄²×C₁₇	C₂²×C₄	1	3	yes	—
56	C₄⋊Dic₁₇	C₂×C₈	1	3	yes	—
57	#63225	C₂×C₈	1	6	yes	—
58	C₄×Dic₁₇	C₂×C₈	1	3	yes	—
59	C₄×Dic₁₇	C₂×C₈	1	3	yes	—
60	C₄×Dic₁₇	C₂²×C₄	1	3	yes	—
61	C₂×#3622	C₂²×C₄	1	3	yes	—
62	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
63	C₈×D₁₇	—	2	6	no	—
64	C₈×D₁₇	C₂×Q₈	1	2	yes	—
65	C₈×D₁₇	#41	1	2	yes	—
66	C₁₆×C₁₇	—	2	3	no	—
67	C₁₆×C₁₇	C₂×Q₈	1	2	yes	complemented
68	#63214	—	2	3	no	—
69	#63214	C₂×Q₈	1	2	yes	complemented
70	C₂×Q₈×C₁₆	—	17	1	no	maximal, Sylow
71	C₂²×D₁₇	C₂²×C₈	1	1	yes	characteristic
72	Q₈×C₁₇	C₂×C₁₆	1	4	yes	complemented
73	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
74	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
75	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
76	C₂×Dic₁₇	C₂²×C₈	1	3	yes	—
77	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
78	C₂×Dic₁₇	C₂²×C₈	1	3	yes	—
79	C₄×D₁₇	—	2	6	no	—
80	C₄×D₁₇	C₂×C₁₆	1	12	yes	—
81	C₄×D₁₇	C₄×Q₈	1	2	yes	—
82	C₄×D₁₇	#131	1	2	yes	—
83	Dic₃₄	C₂×C₁₆	1	12	yes	complemented
84	C₈×C₁₇	—	2	3	no	—
85	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
86	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
87	#3622	—	2	3	no	—
88	#3622	C₄×Q₈	1	2	yes	—
89	C₂×C₈×Q₈	—	17	1	no	—
90	Q₈×C₁₆	—	17	1	no	—
91	Q₈×C₁₆	—	17	3	no	—
92	Q₈×C₁₆	—	17	3	no	—
93	Q₈×C₁₆	D₁₇	1	1	yes	characteristic, direct product factor
94	C₂×C₄⋊C₁₆	—	17	3	no	—
95	C₂×C₄×C₁₆	—	17	3	no	—
96	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
97	C₂×D₁₇	C₈×Q₈	1	2	yes	—
98	C₂×D₁₇	C₂²×C₁₆	1	2	yes	—
99	C₂×D₁₇	#504	1	2	yes	—
100	C₄×C₁₇	—	2	3	no	—
101	C₄×C₁₇	C₂²×C₁₆	1	6	yes	—
102	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
103	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
104	Dic₁₇	—	2	3	no	—
105	Dic₁₇	C₈×Q₈	1	2	yes	—
106	Dic₁₇	C₂²×C₁₆	1	6	yes	—
107	C₂×C₄×Q₈	—	17	1	no	—
108	C₂²×C₁₆	—	17	1	no	—
109	C₂²×C₁₆	—	17	3	no	—
110	C₈×Q₈	—	17	1	no	—
111	C₈×Q₈	—	17	3	no	—
112	C₈×Q₈	—	17	3	no	—
113	C₈×Q₈	C₂×D₁₇	1	1	yes	characteristic
114	C₂×C₄⋊C₈	—	17	3	no	—
115	C₂×C₄×C₈	—	17	3	no	—
116	C₄⋊C₁₆	—	17	6	no	—
117	C₄⋊C₁₆	—	17	3	no	—
118	C₄⋊C₁₆	C₂×D₁₇	1	3	yes	—
119	C₄×C₁₆	—	17	3	no	—
120	C₄×C₁₆	—	17	3	no	—
121	C₄×C₁₆	—	17	3	no	—
122	C₄×C₁₆	C₂×D₁₇	1	3	yes	—
123	C₂×C₁₇	C₂×C₈×Q₈	1	1	yes	characteristic
124	C₂×C₁₇	C₂³×C₁₆	1	1	yes	characteristic
125	C₂×C₁₇	C₂×#504	1	1	yes	characteristic
126	D₁₇	Q₈×C₁₆	1	4	yes	direct product factor
127	C₂²×Q₈	—	17	1	no	—
128	C₂²×C₈	—	17	1	no	—
129	C₂²×C₈	—	17	3	no	—
130	C₄×Q₈	—	17	1	no	—
131	C₄×Q₈	—	17	3	no	—
132	C₄×Q₈	—	17	3	no	—
133	C₄×Q₈	C₄×D₁₇	1	1	yes	characteristic
134	C₂×C₄⋊C₄	—	17	3	no	—
135	C₂×C₄²	—	17	3	no	—
136	C₂×C₁₆	—	17	1	no	—
137	C₂×C₁₆	—	17	4	no	—
138	C₂×C₁₆	—	17	3	no	—
139	C₂×C₁₆	—	34	6	no	—
140	C₂×C₁₆	C₂²×D₁₇	1	3	yes	—
141	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
142	C₄⋊C₈	—	17	6	no	—
143	C₄⋊C₈	—	17	3	no	—
144	C₄⋊C₈	C₄×D₁₇	1	3	yes	—
145	C₄×C₈	—	17	3	no	—
146	C₄×C₈	—	17	3	no	—
147	C₄×C₈	—	17	3	no	—
148	C₄×C₈	C₂²×D₁₇	1	3	yes	—
149	C₁₇	C₂×Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
150	C₂×Q₈	—	17	8	no	—
151	C₂×Q₈	—	17	3	no	—
152	C₂×Q₈	C₈×D₁₇	1	1	yes	characteristic
153	C₂²×C₄	—	17	1	no	—
154	C₂²×C₄	—	17	3	no	—
155	C₂²×C₄	—	17	3	no	—
156	C₂×C₈	—	17	1	no	—
157	C₂×C₈	—	17	2	no	—
158	C₂×C₈	—	17	2	no	—
159	C₂×C₈	—	17	3	no	—
160	C₂×C₈	—	34	6	no	—
161	C₂×C₈	C₂×C₄×D₁₇	1	3	yes	—
162	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
163	C₄⋊C₄	—	17	6	no	—
164	C₄⋊C₄	—	17	3	no	—
165	C₄⋊C₄	C₈×D₁₇	1	3	yes	—
166	C₄²	—	17	3	no	—
167	C₄²	—	17	3	no	—
168	C₄²	—	17	3	no	—
169	C₄²	C₂×C₄×D₁₇	1	3	yes	—
170	C₁₆	—	2	3	no	—
171	C₁₆	—	17	2	no	—
172	C₁₆	—	34	3	no	—
173	C₁₆	Q₈×D₁₇	1	2	yes	direct product factor
174	C₂³	—	17	1	no	—
175	Q₈	—	17	12	no	—
176	Q₈	C₁₆×D₁₇	1	4	yes	direct product factor
177	C₂×C₄	—	17	1	no	—
178	C₂×C₄	—	17	3	no	—
179	C₂×C₄	—	17	12	no	—
180	C₂×C₄	—	17	2	no	—
181	C₂×C₄	—	17	2	no	—
182	C₂×C₄	—	17	3	no	—
183	C₂×C₄	—	34	6	no	—
184	C₂×C₄	C₂×C₈×D₁₇	1	3	yes	—
185	C₂×C₄	C₂×C₈×D₁₇	1	3	yes	—
186	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
187	C₈	—	2	3	no	—
188	C₈	—	17	2	no	—
189	C₈	—	34	3	no	—
190	C₈	C₂×Q₈×D₁₇	1	1	yes	characteristic
191	C₈	#41×D₁₇	1	1	yes	characteristic
192	C₂²	—	17	2	no	—
193	C₂²	—	17	2	no	—
194	C₂²	—	17	2	no	—
195	C₂²	C₂²×C₈×D₁₇	1	1	yes	characteristic
196	C₄	—	2	3	no	—
197	C₄	—	17	6	no	—
198	C₄	—	17	2	no	—
199	C₄	—	34	3	no	—
200	C₄	C₂×C₁₆×D₁₇	1	6	yes	—
201	C₄	C₄×Q₈×D₁₇	1	1	yes	characteristic
202	C₄	#131×D₁₇	1	1	yes	characteristic
203	C₂	—	17	4	no	—
204	C₂	C₈×Q₈×D₁₇	1	1	yes	characteristic
205	C₂	C₂²×C₁₆×D₁₇	1	1	yes	characteristic
206	C₂	D₁₇×#504	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₁₇
commutator	C₂×C₁₇	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	Q₈×C₁₆×C₁₇	C₂
socle	C₂²×C₁₇	C₂³×C₈
prime radical	C₂²×D₁₇	C₂²×C₈
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	Q₈×C₁₆	D₁₇
hypocenter	C₁₇	C₂×Q₈×C₁₆
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₁₆ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×#63214	C₂	semidirect
3. C₁₆×Dic₃₄	C₂	semidirect
4. #14530686	C₂	semidirect
7. Q₈×C₁₆×C₁₇	C₂	semidirect
45. C₁₆×D₁₇	Q₈	direct (N × H)
47. Q₈×D₁₇	C₁₆	direct (N × H)
67. C₁₆×C₁₇	C₂×Q₈	semidirect
69. #63214	C₂×Q₈	semidirect
72. Q₈×C₁₇	C₂×C₁₆	semidirect
83. Dic₃₄	C₂×C₁₆	semidirect
93. Q₈×C₁₆	D₁₇	direct (N × H)
126. D₁₇	Q₈×C₁₆	direct (N × H)
149. C₁₇	C₂×Q₈×C₁₆	semidirect
173. C₁₆	Q₈×D₁₇	direct (N × H)
176. Q₈	C₁₆×D₁₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈×Q₈×D₁₇	2
Q₈×#63214	2
C₁₆×Dic₃₄	2
#14530686	2
D₁₇×C₄⋊C₁₆	2
C₄×C₁₆×D₁₇	2
Q₈×C₁₆×C₁₇	2
C₂×Q₈×C₁₆	17

Subgroups — Q8×C16×D17 #17187803

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₁₆×D₁₇ #17187803