Subgroups — #16950761

Order 4352.

374 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762637	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	#11762637	—	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	#12906	—	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	SD₃₂	1	8	yes	—
43	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
45	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
48	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
49	C₂×Dic₁₇	—	8	4	no	—
50	C₂×Dic₁₇	—	8	8	no	—
51	C₈×C₁₇	#107	1	4	yes	—
52	C₈×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
53	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
54	C₂×#367	—	17	4	no	—
55	C₂×C₄×C₁₆	D₁₇	1	1	yes	characteristic
56	#1845	—	17	2	no	—
57	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
58	C₂²×C₁₇	#493	1	1	yes	characteristic
59	C₂²×C₁₇	#500	1	1	yes	characteristic
60	C₂²×C₁₇	C₂×SD₃₂	1	4	yes	—
61	C₄×C₁₇	#367	1	8	yes	complemented
62	C₄×C₁₇	#384	1	1	yes	characteristic
63	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
64	C₄×C₁₇	#494	1	1	yes	characteristic
65	C₄×C₁₇	#500	1	1	yes	characteristic
66	Dic₁₇	—	16	4	no	—
67	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
68	C₂×#107	—	17	4	no	—
69	C₂×#107	—	34	2	no	—
70	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
71	#384	—	34	2	no	—
72	#367	—	34	8	no	—
73	C₄×C₁₆	Dic₁₇	1	4	yes	complemented
74	C₂×C₁₇	#1845	1	1	yes	characteristic
75	C₂×C₁₇	C₂×#367	1	2	yes	—
76	C₂×C₁₇	#2248	1	2	yes	—
77	C₂×C₁₇	#2254	1	2	yes	—
78	C₂²×C₈	D₆₈	1	1	yes	characteristic
79	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
80	C₂×C₄⋊C₄	—	34	4	no	—
81	C₂×C₄⋊C₄	—	68	4	no	—
82	C₂×C₄²	D₆₈	1	1	yes	characteristic
83	C₂×C₁₆	Dic₃₄	1	4	yes	—
84	C₂×C₁₆	D₆₈	1	4	yes	—
85	C₂×C₁₆	C₂×Dic₁₇	1	4	yes	—
86	#107	—	34	8	no	—
87	#107	—	68	4	no	—
88	C₄×C₈	Dic₃₄	1	2	yes	—
89	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
90	C₁₇	#12906	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₄	—	68	4	no	—
92	C₂²×C₄	D₁₃₆	1	2	yes	—
93	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
94	C₂×C₈	D₁₃₆	1	2	yes	—
95	C₂×C₈	Dic₆₈	1	2	yes	—
96	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
97	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
98	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
99	C₂×C₈	C₂×D₆₈	1	2	yes	—
100	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
101	C₄⋊C₄	—	68	4	no	—
102	C₄⋊C₄	—	68	4	no	—
103	C₄⋊C₄	—	136	8	no	—
104	C₄²	Dic₆₈	1	2	yes	—
105	C₄²	C₄⋊Dic₁₇	1	2	yes	—
106	C₁₆	C₄⋊Dic₁₇	1	8	yes	complemented
107	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
108	C₂×C₄	—	136	8	no	—
109	C₂×C₄	—	136	4	no	—
110	C₂×C₄	#10587471	1	8	yes	—
111	C₂×C₄	#10587489	1	4	yes	—
112	C₂×C₄	#10587555	1	1	yes	characteristic
113	C₂×C₄	#10587560	1	1	yes	characteristic
114	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
115	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
116	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
118	C₈	#10587489	1	4	yes	—
119	C₈	#10587555	1	2	yes	—
120	C₈	C₂×C₄⋊Dic₁₇	1	2	yes	—
121	C₂²	C₂×#10587489	1	1	yes	characteristic
122	C₂²	#11763186	1	1	yes	characteristic
123	C₂²	#11763214	1	1	yes	characteristic
124	C₂²	C₂×#10587471	1	4	yes	—
125	C₄	—	272	4	no	—
126	C₄	#11762637	1	8	yes	complemented
127	C₄	#11762679	1	1	yes	characteristic
128	C₄	C₂×#10587489	1	1	yes	characteristic
129	C₄	#11763189	1	1	yes	characteristic
130	C₄	#11763214	1	1	yes	characteristic
131	C₂	#14528944	1	1	yes	characteristic
132	C₂	C₂×#11762637	1	2	yes	—
133	C₂	#14531040	1	2	yes	—
134	C₂	#14531066	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	#12906
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
61. C₄×C₁₇	#367	semidirect
73. C₄×C₁₆	Dic₁₇	semidirect
90. C₁₇	#12906	semidirect
106. C₁₆	C₄⋊Dic₁₇	semidirect
126. C₄	#11762637	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762637	2
#14528944	2
C₂×C₄×C₁₆×C₁₇	2
#12906	17