Subgroups — #16950758

Order 4352.

374 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762635	C₂	1	4	yes	maximal
2	#14528944	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587489	—	2	2	no	—
5	C₂×#10587489	C₂²	1	4	yes	—
6	#11762679	—	2	2	no	—
7	#11762635	—	2	8	no	—
8	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₁₇	C₄	1	4	yes	complemented
11	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
12	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
13	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₇	—	2	4	no	—
15	C₂×C₄⋊Dic₁₇	—	4	4	no	—
16	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
17	C₂×C₁₆×C₁₇	D₄	1	4	yes	—
18	C₂×C₁₆×C₁₇	Q₈	1	4	yes	—
19	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₁₇	Q₈	1	2	yes	—
21	#10587489	—	2	8	no	—
22	#10587489	—	4	4	no	—
23	C₂²×C₄×C₁₇	D₈	1	2	yes	—
24	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
25	C₂²×Dic₁₇	—	4	4	no	—
26	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₁₇	D₈	1	2	yes	—
28	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
29	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
31	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
32	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
33	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
34	C₄²×C₁₇	Q₁₆	1	2	yes	—
35	C₄⋊Dic₁₇	—	4	4	no	—
36	C₄⋊Dic₁₇	—	4	4	no	—
37	C₄⋊Dic₁₇	—	8	8	no	—
38	C₁₆×C₁₇	C₄⋊C₄	1	8	yes	complemented
39	#12905	—	17	1	no	maximal, Sylow
40	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
41	C₂×C₄×C₁₇	#107	1	4	yes	—
42	C₂×C₄×C₁₇	D₁₆	1	4	yes	—
43	C₂×C₄×C₁₇	Q₃₂	1	4	yes	—
44	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
50	C₂×Dic₁₇	—	8	4	no	—
51	C₂×Dic₁₇	—	8	8	no	—
52	C₈×C₁₇	#107	1	4	yes	—
53	C₈×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
54	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
55	C₂×#366	—	17	4	no	—
56	C₂×C₄×C₁₆	D₁₇	1	1	yes	characteristic
57	#1845	—	17	2	no	—
58	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
59	C₂²×C₁₇	#493	1	1	yes	characteristic
60	C₂²×C₁₇	#500	1	1	yes	characteristic
61	C₂²×C₁₇	C₂×D₁₆	1	2	yes	—
62	C₂²×C₁₇	C₂×Q₃₂	1	2	yes	—
63	C₄×C₁₇	#366	1	8	yes	complemented
64	C₄×C₁₇	#384	1	1	yes	characteristic
65	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
66	C₄×C₁₇	#494	1	1	yes	characteristic
67	C₄×C₁₇	#500	1	1	yes	characteristic
68	Dic₁₇	—	16	4	no	—
69	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
70	C₂×#107	—	17	4	no	—
71	C₂×#107	—	34	2	no	—
72	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
73	#384	—	34	2	no	—
74	#366	—	34	8	no	—
75	C₄×C₁₆	Dic₁₇	1	4	yes	complemented
76	C₂×C₁₇	#1845	1	1	yes	characteristic
77	C₂×C₁₇	C₂×#366	1	2	yes	—
78	C₂×C₁₇	#2246	1	1	yes	characteristic
79	C₂×C₁₇	#2247	1	1	yes	characteristic
80	C₂×C₁₇	#2252	1	2	yes	—
81	C₂²×C₈	D₆₈	1	1	yes	characteristic
82	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
83	C₂×C₄⋊C₄	—	34	4	no	—
84	C₂×C₄⋊C₄	—	68	4	no	—
85	C₂×C₄²	D₆₈	1	1	yes	characteristic
86	C₂×C₁₆	Dic₃₄	1	4	yes	—
87	C₂×C₁₆	D₆₈	1	4	yes	—
88	C₂×C₁₆	C₂×Dic₁₇	1	4	yes	—
89	#107	—	34	8	no	—
90	#107	—	68	4	no	—
91	C₄×C₈	Dic₃₄	1	2	yes	—
92	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
93	C₁₇	#12905	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₄	—	68	4	no	—
95	C₂²×C₄	D₁₃₆	1	2	yes	—
96	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
97	C₂×C₈	D₁₃₆	1	2	yes	—
98	C₂×C₈	Dic₆₈	1	2	yes	—
99	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
100	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
101	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
102	C₂×C₈	C₂×D₆₈	1	2	yes	—
103	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
104	C₄⋊C₄	—	68	4	no	—
105	C₄⋊C₄	—	68	4	no	—
106	C₄⋊C₄	—	136	8	no	—
107	C₄²	Dic₆₈	1	2	yes	—
108	C₄²	C₄⋊Dic₁₇	1	2	yes	—
109	C₁₆	C₄⋊Dic₁₇	1	8	yes	complemented
110	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
111	C₂×C₄	—	136	8	no	—
112	C₂×C₄	—	136	4	no	—
113	C₂×C₄	D₂₇₂	1	4	yes	—
114	C₂×C₄	Dic₁₃₆	1	4	yes	—
115	C₂×C₄	#10587489	1	4	yes	—
116	C₂×C₄	#10587555	1	1	yes	characteristic
117	C₂×C₄	#10587560	1	1	yes	characteristic
118	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
119	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
120	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
121	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
122	C₈	#10587489	1	4	yes	—
123	C₈	#10587555	1	2	yes	—
124	C₈	C₂×C₄⋊Dic₁₇	1	2	yes	—
125	C₂²	C₂×#10587489	1	1	yes	characteristic
126	C₂²	#11763186	1	1	yes	characteristic
127	C₂²	#11763214	1	1	yes	characteristic
128	C₂²	C₂×D₂₇₂	1	2	yes	—
129	C₂²	C₂×Dic₁₃₆	1	2	yes	—
130	C₄	—	272	4	no	—
131	C₄	#11762635	1	8	yes	complemented
132	C₄	#11762679	1	1	yes	characteristic
133	C₄	C₂×#10587489	1	1	yes	characteristic
134	C₄	#11763189	1	1	yes	characteristic
135	C₄	#11763214	1	1	yes	characteristic
136	C₂	#14528944	1	1	yes	characteristic
137	C₂	C₂×#11762635	1	2	yes	—
138	C₂	#14531034	1	1	yes	characteristic
139	C₂	#14531037	1	1	yes	characteristic
140	C₂	#14531059	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₁₃₆
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₄×C₁₆×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₇
hypocenter	C₁₇	#12905
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
10. C₄×C₁₆×C₁₇	C₄	semidirect
38. C₁₆×C₁₇	C₄⋊C₄	semidirect
63. C₄×C₁₇	#366	semidirect
75. C₄×C₁₆	Dic₁₇	semidirect
93. C₁₇	#12905	semidirect
109. C₁₆	C₄⋊Dic₁₇	semidirect
131. C₄	#11762635	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762635	2
#14528944	2
C₂×C₄×C₁₆×C₁₇	2
#12905	17