Subgroups — C₂×C₁₇×#1370 #16869747

Order 4352.

286 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇×C₄⋊C₁₆	C₂	1	2	yes	maximal
2	C₂×C₈²×C₁₇	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#1370	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₁₇	—	2	2	no	—
5	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
8	C₁₇×C₄⋊C₁₆	—	2	2	no	—
9	C₁₇×C₄⋊C₁₆	C₂²	1	4	yes	—
10	C₈²×C₁₇	C₄	1	2	yes	—
11	C₈²×C₁₇	C₂²	1	2	yes	—
12	C₂²×C₈×C₁₇	—	2	1	no	—
13	C₂²×C₈×C₁₇	C₈	1	1	yes	characteristic
14	C₂²×C₈×C₁₇	C₈	1	1	yes	characteristic
15	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆×C₁₇	—	2	4	no	—
19	C₂×C₁₆×C₁₇	—	4	4	no	—
20	C₄×C₈×C₁₇	—	2	1	no	—
21	C₄×C₈×C₁₇	C₈	1	2	yes	—
22	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
26	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
31	C₂×C₈×C₁₇	—	2	4	no	—
32	C₂×C₈×C₁₇	—	2	2	no	—
33	C₂×C₈×C₁₇	—	2	2	no	—
34	C₂×C₈×C₁₇	—	2	2	no	—
35	C₂×C₈×C₁₇	C₁₆	1	4	yes	complemented
36	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₁₇	D₈	1	2	yes	—
41	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
42	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
46	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
47	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
48	C₁₆×C₁₇	—	4	4	no	—
49	C₂×#1370	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂³×C₁₇	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇	—	2	2	no	—
52	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₇	#107	1	2	yes	—
56	C₂×C₄×C₁₇	#108	1	2	yes	—
57	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
58	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
59	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
62	C₈×C₁₇	—	2	4	no	—
63	C₈×C₁₇	—	2	1	no	—
64	C₈×C₁₇	—	2	1	no	—
65	C₈×C₁₇	—	2	2	no	—
66	C₈×C₁₇	#107	1	2	yes	—
67	C₈×C₁₇	C₂×C₁₆	1	4	yes	complemented
68	C₈×C₁₇	C₂×D₈	1	1	yes	characteristic
69	C₈×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄⋊C₁₆	C₂×C₁₇	1	2	yes	—
71	C₂×C₈²	C₂×C₁₇	1	1	yes	characteristic
72	#1370	C₂×C₁₇	1	4	yes	direct product factor
73	C₂²×C₁₇	#335	1	2	yes	—
74	C₂²×C₁₇	C₄⋊C₁₆	1	2	yes	—
75	C₂²×C₁₇	#364	1	2	yes	—
76	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
77	C₄×C₁₇	—	2	1	no	—
78	C₄×C₁₇	—	2	1	no	—
79	C₄×C₁₇	#335	1	2	yes	—
80	C₄×C₁₇	C₄⋊C₁₆	1	2	yes	—
81	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
82	C₄×C₁₇	C₂×#108	1	1	yes	characteristic
83	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
84	C₄×C₁₇	C₂×M₃₂	1	1	yes	characteristic
85	C₂²×C₁₆	—	2	2	no	—
86	C₂×C₄×C₈	C₄×C₁₇	1	1	yes	characteristic
87	C₂×C₄×C₈	C₄×C₁₇	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
89	C₄⋊C₁₆	—	2	2	no	—
90	C₄⋊C₁₆	C₂²×C₁₇	1	4	yes	—
91	C₈²	C₄×C₁₇	1	2	yes	—
92	C₈²	C₂²×C₁₇	1	2	yes	—
93	C₂×C₁₇	#1370	1	4	yes	direct product factor
94	C₂×C₁₇	C₂×#335	1	1	yes	characteristic
95	C₂×C₁₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
96	C₂×C₁₇	C₂×#364	1	1	yes	characteristic
97	C₂²×C₈	—	2	1	no	—
98	C₂²×C₈	C₈×C₁₇	1	1	yes	characteristic
99	C₂²×C₈	C₈×C₁₇	1	1	yes	characteristic
100	C₂²×C₈	D₄×C₁₇	1	1	yes	characteristic
101	C₂²×C₈	Q₈×C₁₇	1	1	yes	characteristic
102	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
103	C₂×C₁₆	—	2	4	no	—
104	C₂×C₁₆	—	4	4	no	—
105	C₄×C₈	—	2	1	no	—
106	C₄×C₈	C₈×C₁₇	1	2	yes	—
107	C₄×C₈	C₂×C₄×C₁₇	1	2	yes	—
108	C₄×C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
109	C₄×C₈	C₂×C₄×C₁₇	1	2	yes	—
110	C₄×C₈	D₄×C₁₇	1	1	yes	characteristic
111	C₄×C₈	Q₈×C₁₇	1	1	yes	characteristic
112	C₄×C₈	C₂³×C₁₇	1	1	yes	characteristic
113	C₁₇	C₂×#1370	1	1	yes	characteristic, direct product factor, Sylow
114	C₂²×C₄	C₄⋊C₄×C₁₇	1	1	yes	characteristic
115	C₂²×C₄	C₂×C₈×C₁₇	1	1	yes	characteristic
116	C₂²×C₄	M₁₆×C₁₇	1	1	yes	characteristic
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	C₁₆×C₁₇	1	4	yes	complemented
122	C₂×C₈	C₄⋊C₄×C₁₇	1	1	yes	characteristic
123	C₂×C₈	C₄⋊C₄×C₁₇	1	1	yes	characteristic
124	C₂×C₈	C₂×C₈×C₁₇	1	2	yes	—
125	C₂×C₈	C₂×C₈×C₁₇	1	2	yes	—
126	C₂×C₈	D₈×C₁₇	1	2	yes	—
127	C₂×C₈	Q₁₆×C₁₇	1	2	yes	—
128	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
129	C₂×C₈	C₂×Q₈×C₁₇	1	1	yes	characteristic
130	C₄²	C₄⋊C₄×C₁₇	1	1	yes	characteristic
131	C₄²	C₂×C₈×C₁₇	1	1	yes	characteristic
132	C₄²	M₁₆×C₁₇	1	1	yes	characteristic
133	C₄²	C₂²×C₄×C₁₇	1	1	yes	characteristic
134	C₁₆	—	4	4	no	—
135	C₂³	C₁₇×C₄⋊C₈	1	1	yes	characteristic
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	C₁₇×C₄⋊C₈	1	1	yes	characteristic
138	C₂×C₄	C₁₇×C₄⋊C₈	1	1	yes	characteristic
139	C₂×C₄	C₁₇×C₄⋊C₈	1	1	yes	characteristic
140	C₂×C₄	C₁₇×#107	1	2	yes	—
141	C₂×C₄	C₁₇×#108	1	2	yes	—
142	C₂×C₄	C₂×C₁₆×C₁₇	1	2	yes	—
143	C₂×C₄	C₁₇×M₃₂	1	2	yes	—
144	C₂×C₄	C₂×C₄⋊C₄×C₁₇	1	1	yes	characteristic
145	C₂×C₄	C₂²×C₈×C₁₇	1	1	yes	characteristic
146	C₂×C₄	C₂×M₁₆×C₁₇	1	1	yes	characteristic
147	C₈	—	2	4	no	—
148	C₈	—	2	2	no	—
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	C₁₇×#107	1	2	yes	—
152	C₈	C₂×C₁₆×C₁₇	1	4	yes	complemented
153	C₈	C₂×D₈×C₁₇	1	1	yes	characteristic
154	C₈	C₂×Q₁₆×C₁₇	1	1	yes	characteristic
155	C₂²	C₁₇×#335	1	2	yes	—
156	C₂²	C₁₇×C₄⋊C₁₆	1	2	yes	—
157	C₂²	C₁₇×#364	1	2	yes	—
158	C₂²	C₂×C₁₇×C₄⋊C₈	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	C₁₇×#335	1	2	yes	—
162	C₄	C₁₇×C₄⋊C₁₆	1	2	yes	—
163	C₄	C₂×C₁₇×#107	1	1	yes	characteristic
164	C₄	C₂×C₁₇×#108	1	1	yes	characteristic
165	C₄	C₂²×C₁₆×C₁₇	1	1	yes	characteristic
166	C₄	C₂×C₁₇×M₃₂	1	1	yes	characteristic
167	C₂	C₁₇×#1370	1	4	yes	direct product factor
168	C₂	C₂×C₁₇×#335	1	1	yes	characteristic
169	C₂	C₂×C₁₇×C₄⋊C₁₆	1	1	yes	characteristic
170	C₂	C₂×C₁₇×#364	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₁₇×#1370	C₂	direct (N × H)
35. C₂×C₈×C₁₇	C₁₆	semidirect
49. C₂×#1370	C₁₇	direct (N × H)
67. C₈×C₁₇	C₂×C₁₆	semidirect
72. #1370	C₂×C₁₇	direct (N × H)
93. C₂×C₁₇	#1370	direct (N × H)
113. C₁₇	C₂×#1370	direct (N × H)
121. C₂×C₈	C₁₆×C₁₇	semidirect
152. C₈	C₂×C₁₆×C₁₇	semidirect
167. C₂	C₁₇×#1370	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₇×C₄⋊C₁₆	2
C₂×C₈²×C₁₇	2
C₁₇×#1370	2
C₂×#1370	17

Subgroups — C2×C17×#1370 #16869747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₇×#1370 #16869747