Subgroups — C₁₇×#13188 #16870932

Order 4352.

280 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	C₁₇×#2155	C₂	1	2	yes	maximal
2	C₂×C₁₇×#364	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#1848	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#1772	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1388	C₂	1	2	yes	maximal
6	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
7	C₂×C₁₇×#108	—	2	1	no	—
8	C₂×C₁₇×#108	C₂²	1	2	yes	—
9	C₂×C₁₇×#108	C₂²	1	1	yes	characteristic
10	C₁₇×#427	C₂²	1	1	yes	characteristic
11	C₁₇×#423	—	2	1	no	—
12	C₁₇×#405	—	2	1	no	—
13	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×#365	—	2	4	no	—
15	C₁₇×#364	C₄	1	2	yes	complemented
16	C₁₇×#364	C₄	1	2	yes	complemented
17	C₁₇×#339	—	2	1	no	—
18	C₂×M₁₆×C₁₇	—	2	1	no	—
19	C₂×M₁₆×C₁₇	—	2	2	no	—
20	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
22	C₁₇×#117	—	2	1	no	—
23	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
24	C₁₇×M₃₂	—	2	2	no	—
25	C₁₇×M₃₂	C₂×C₄	1	4	yes	—
26	C₂×C₁₆×C₁₇	—	2	2	no	—
27	C₁₇×#108	—	2	2	no	—
28	C₁₇×#108	—	2	1	no	—
29	C₁₇×#108	—	2	2	no	—
30	C₁₇×#108	—	4	1	no	—
31	C₁₇×#108	—	4	1	no	—
32	C₁₇×#108	Q₈	1	2	yes	—
33	C₁₇×#107	—	2	1	no	—
34	C₁₇×#106	D₄	1	2	yes	—
35	C₁₇×C₄⋊C₈	—	4	2	no	—
36	C₁₇×#97	—	4	1	no	—
37	C₄×C₈×C₁₇	—	4	1	no	—
38	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	—	2	1	no	—
43	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
44	M₁₆×C₁₇	—	2	2	no	—
45	M₁₆×C₁₇	—	4	1	no	—
46	M₁₆×C₁₇	—	4	2	no	—
47	M₁₆×C₁₇	—	4	2	no	—
48	C₂×C₈×C₁₇	—	4	1	no	—
49	C₂×C₈×C₁₇	—	4	2	no	—
50	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₇	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₁₇	SD₁₆	1	1	yes	characteristic
54	C₂×C₈×C₁₇	SD₁₆	1	1	yes	characteristic
55	C₂×C₈×C₁₇	SD₁₆	1	1	yes	characteristic
56	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
61	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
62	C₄⋊C₄×C₁₇	—	2	2	no	—
63	C₂²⋊C₄×C₁₇	—	4	1	no	—
64	C₄²×C₁₇	—	2	1	no	—
65	C₄²×C₁₇	—	4	1	no	—
66	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
67	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
68	C₁₆×C₁₇	—	2	4	no	—
69	#13188	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₁₇	#137	1	1	yes	characteristic
71	C₂×C₄×C₁₇	—	2	1	no	—
72	C₂×C₄×C₁₇	—	2	1	no	—
73	C₂×C₄×C₁₇	—	4	2	no	—
74	C₂×C₄×C₁₇	—	4	2	no	—
75	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₈×C₁₇	—	4	2	no	—
82	C₈×C₁₇	—	8	2	no	—
83	C₈×C₁₇	#106	1	1	yes	characteristic
84	C₈×C₁₇	#106	1	1	yes	characteristic
85	C₈×C₁₇	#106	1	1	yes	characteristic
86	C₈×C₁₇	#106	1	1	yes	characteristic
87	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
88	C₈×C₁₇	C₄×Q₈	1	1	yes	characteristic
89	C₈×C₁₇	#125	1	1	yes	characteristic
90	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
91	#2155	C₂×C₁₇	1	2	yes	—
92	C₂×#364	C₂×C₁₇	1	1	yes	characteristic
93	#1848	C₂×C₁₇	1	1	yes	characteristic
94	#1772	C₂×C₁₇	1	1	yes	characteristic
95	#1388	C₂×C₁₇	1	2	yes	—
96	C₂²×C₁₇	—	4	1	no	—
97	C₂²×C₁₇	#428	1	1	yes	characteristic
98	C₂²×C₁₇	#461	1	1	yes	characteristic
99	C₂²×C₁₇	#475	1	1	yes	characteristic
100	C₄×C₁₇	—	2	2	no	—
101	C₄×C₁₇	—	4	1	no	—
102	C₄×C₁₇	—	8	2	no	—
103	C₄×C₁₇	#389	1	1	yes	characteristic
104	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
105	C₄×C₁₇	#460	1	1	yes	characteristic
106	C₄×C₁₇	#476	1	1	yes	characteristic
107	C₂×M₃₂	C₂²×C₁₇	1	2	yes	—
108	C₂×#108	—	2	1	no	—
109	C₂×#108	C₂²×C₁₇	1	2	yes	—
110	C₂×#108	C₂²×C₁₇	1	1	yes	characteristic
111	#427	C₂²×C₁₇	1	1	yes	characteristic
112	#423	—	2	1	no	—
113	#405	—	2	1	no	—
114	C₂×C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
115	#365	—	2	4	no	—
116	#364	C₄×C₁₇	1	2	yes	complemented
117	#364	C₄×C₁₇	1	2	yes	complemented
118	#339	—	2	1	no	—
119	C₂×C₁₇	—	2	1	no	—
120	C₂×C₁₇	—	2	1	no	—
121	C₂×C₁₇	—	2	1	no	—
122	C₂×C₁₇	#1948	1	1	yes	characteristic
123	C₂×M₁₆	—	2	1	no	—
124	C₂×M₁₆	—	2	2	no	—
125	C₂²×C₈	D₄×C₁₇	1	1	yes	characteristic
126	C₂²×C₈	C₂³×C₁₇	1	1	yes	characteristic
127	#117	—	2	1	no	—
128	C₂×C₄²	D₄×C₁₇	1	1	yes	characteristic
129	M₃₂	—	2	2	no	—
130	M₃₂	C₂×C₄×C₁₇	1	4	yes	—
131	C₂×C₁₆	—	2	2	no	—
132	#108	—	2	2	no	—
133	#108	—	2	1	no	—
134	#108	—	2	2	no	—
135	#108	—	4	1	no	—
136	#108	—	4	1	no	—
137	#108	Q₈×C₁₇	1	2	yes	—
138	#107	—	2	1	no	—
139	#106	D₄×C₁₇	1	2	yes	—
140	C₄⋊C₈	—	4	2	no	—
141	#97	—	4	1	no	—
142	C₄×C₈	—	4	1	no	—
143	C₄×C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
144	C₄×C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
145	C₄×C₈	Q₈×C₁₇	1	1	yes	characteristic
146	C₄×C₈	Q₈×C₁₇	1	1	yes	characteristic
147	C₁₇	#13188	1	1	yes	characteristic, direct product factor, Sylow
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	—	4	1	no	—
152	M₁₆	—	4	2	no	—
153	M₁₆	—	4	2	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	C₄⋊C₄×C₁₇	1	1	yes	characteristic
157	C₂×C₈	C₄⋊C₄×C₁₇	1	1	yes	characteristic
158	C₂×C₈	SD₁₆×C₁₇	1	1	yes	characteristic
159	C₂×C₈	SD₁₆×C₁₇	1	1	yes	characteristic
160	C₂×C₈	SD₁₆×C₁₇	1	1	yes	characteristic
161	C₂×C₈	SD₁₆×C₁₇	1	1	yes	characteristic
162	C₂×C₈	C₂²×C₄×C₁₇	1	1	yes	characteristic
163	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
164	C₂×C₈	C₂×Q₈×C₁₇	1	1	yes	characteristic
165	C₂×C₈	C₂×Q₈×C₁₇	1	1	yes	characteristic
166	C₂×C₈	#41×C₁₇	1	1	yes	characteristic
167	C₂×C₈	#41×C₁₇	1	1	yes	characteristic
168	C₄⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	4	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	4	1	no	—
172	C₄²	C₄⋊C₄×C₁₇	1	1	yes	characteristic
173	C₄²	C₄⋊C₄×C₁₇	1	1	yes	characteristic
174	C₁₆	—	2	4	no	—
175	C₂³	C₁₇×#137	1	1	yes	characteristic
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	C₂×C₄⋊C₄×C₁₇	1	1	yes	characteristic
181	C₂×C₄	C₁₇×#121	1	1	yes	characteristic
182	C₂×C₄	C₁₇×C₂²⋊Q₈	1	1	yes	characteristic
183	C₂×C₄	C₂×SD₁₆×C₁₇	1	1	yes	characteristic
184	C₂×C₄	C₂×SD₁₆×C₁₇	1	1	yes	characteristic
185	C₂×C₄	C₁₇×AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₈	—	4	2	no	—
187	C₈	—	8	2	no	—
188	C₈	C₁₇×#106	1	1	yes	characteristic
189	C₈	C₁₇×#106	1	1	yes	characteristic
190	C₈	C₁₇×#106	1	1	yes	characteristic
191	C₈	C₁₇×#106	1	1	yes	characteristic
192	C₈	C₄×D₄×C₁₇	1	1	yes	characteristic
193	C₈	C₄×Q₈×C₁₇	1	1	yes	characteristic
194	C₈	C₁₇×#125	1	1	yes	characteristic
195	C₈	C₁₇×C₄⋊Q₈	1	1	yes	characteristic
196	C₂²	—	4	1	no	—
197	C₂²	C₁₇×#428	1	1	yes	characteristic
198	C₂²	C₁₇×#461	1	1	yes	characteristic
199	C₂²	C₁₇×#475	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	4	1	no	—
202	C₄	—	8	2	no	—
203	C₄	C₁₇×#389	1	1	yes	characteristic
204	C₄	C₂×C₁₇×#106	1	1	yes	characteristic
205	C₄	C₁₇×#460	1	1	yes	characteristic
206	C₄	C₁₇×#476	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	C₁₇×#1948	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₇	#476
commutator	C₂×C₈	C₂²×C₄×C₁₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₇
socle	C₂×C₁₇	#1948
prime radical	C₂³×C₁₇	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₇ ◁ C₂²×C₄×C₁₇ ◁ C₂×C₄×C₈×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₁₇×#364	C₄	semidirect
16. C₁₇×#364	C₄	semidirect
69. #13188	C₁₇	direct (N × H)
116. #364	C₄×C₁₇	semidirect
117. #364	C₄×C₁₇	semidirect
147. C₁₇	#13188	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₇×#2155	2
C₂×C₁₇×#364	2
C₁₇×#1848	2
C₁₇×#1772	2
C₁₇×#1388	2
#13188	17

Subgroups — C17×#13188 #16870932

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#13188 #16870932