Subgroups — #16950773

Order 4352.

314 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762639	C₂	1	4	yes	maximal
2	#14528956	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587490	—	2	2	no	—
5	C₂×#10587490	C₂²	1	4	yes	—
6	#11762833	—	2	2	no	—
7	#11762639	—	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	2	yes	—
11	C₄×C₁₆×C₁₇	C₄	1	2	yes	—
12	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
15	C₂×#63223	—	2	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	1	yes	characteristic
20	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
22	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
23	#10587490	—	2	8	no	—
24	#10587490	—	4	4	no	—
25	C₄⋊#3622	—	4	4	no	—
26	C₂²×C₄×C₁₇	Q₁₆	1	2	yes	—
27	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
28	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₇	D₈	1	2	yes	—
31	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
32	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
34	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
35	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
36	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
38	C₄²×C₁₇	D₈	1	2	yes	—
39	#63223	—	4	4	no	—
40	#63223	—	4	4	no	—
41	C₂×#3622	—	4	4	no	—
42	C₁₆×C₁₇	C₄⋊C₄	1	8	yes	—
43	#12909	—	17	1	no	maximal, Sylow
44	C₂³×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
45	C₂×C₄×C₁₇	—	2	4	no	—
46	C₂×C₄×C₁₇	#107	1	2	yes	—
47	C₂×C₄×C₁₇	#107	1	2	yes	—
48	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
50	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
54	C₈×C₁₇	#107	1	2	yes	—
55	C₈×C₁₇	#107	1	2	yes	—
56	C₈×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
57	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
58	#3622	—	8	4	no	—
59	C₂×#368	—	17	4	no	—
60	C₂×C₄×C₁₆	D₁₇	1	1	yes	characteristic
61	#1848	—	17	2	no	—
62	C₂²×C₁₇	—	2	2	no	—
63	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
64	C₂²×C₁₇	#494	1	1	yes	characteristic
65	C₂²×C₁₇	#500	1	1	yes	characteristic
66	C₄×C₁₇	—	2	2	no	—
67	C₄×C₁₇	#368	1	4	yes	complemented
68	C₄×C₁₇	#384	1	1	yes	characteristic
69	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
70	C₄×C₁₇	#493	1	1	yes	characteristic
71	C₄×C₁₇	#500	1	1	yes	characteristic
72	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
73	C₂×#108	—	17	4	no	—
74	C₂×#108	—	34	2	no	—
75	#423	—	34	2	no	—
76	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
77	#368	—	34	8	no	—
78	C₄×C₁₆	Dic₁₇	1	2	yes	—
79	C₄×C₁₆	Dic₁₇	1	2	yes	—
80	C₂×C₁₇	—	2	1	no	—
81	C₂×C₁₇	—	2	1	no	—
82	C₂×C₁₇	#1845	1	1	yes	characteristic
83	C₂×C₁₇	C₂×#368	1	2	yes	—
84	C₂×M₁₆	—	34	4	no	—
85	C₂²×C₈	D₆₈	1	1	yes	characteristic
86	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
87	C₂×C₄²	D₆₈	1	1	yes	characteristic
88	C₂×C₁₆	Dic₃₄	1	4	yes	—
89	C₂×C₁₆	D₆₈	1	4	yes	—
90	C₂×C₁₆	C₂×Dic₁₇	1	4	yes	—
91	#108	—	34	8	no	—
92	#108	—	68	4	no	—
93	C₄⋊C₈	—	68	4	no	—
94	C₄×C₈	Dic₃₄	1	1	yes	characteristic
95	C₄×C₈	Dic₃₄	1	1	yes	characteristic
96	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
97	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
98	C₁₇	#12909	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₄	Dic₆₈	1	2	yes	—
100	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
101	M₁₆	—	68	4	no	—
102	M₁₆	—	68	4	no	—
103	C₂×C₈	—	68	4	no	—
104	C₂×C₈	D₁₃₆	1	2	yes	—
105	C₂×C₈	Dic₆₈	1	2	yes	—
106	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
107	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
108	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
109	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
110	C₂×C₈	C₂×D₆₈	1	2	yes	—
111	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
112	C₄²	D₁₃₆	1	2	yes	—
113	C₄²	C₄⋊Dic₁₇	1	1	yes	characteristic
114	C₄²	C₄⋊Dic₁₇	1	1	yes	characteristic
115	C₁₆	C₄⋊Dic₁₇	1	8	yes	—
116	C₂³	C₂×Dic₆₈	1	1	yes	characteristic
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	#10587489	1	2	yes	—
119	C₂×C₄	#10587489	1	2	yes	—
120	C₂×C₄	#10587555	1	1	yes	characteristic
121	C₂×C₄	#10587560	1	1	yes	characteristic
122	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
123	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
124	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
125	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
126	C₈	—	136	4	no	—
127	C₈	#10587489	1	2	yes	—
128	C₈	#10587489	1	2	yes	—
129	C₈	#10587555	1	2	yes	—
130	C₈	C₂×C₄⋊Dic₁₇	1	2	yes	—
131	C₂²	—	2	2	no	—
132	C₂²	C₂×#10587489	1	1	yes	characteristic
133	C₂²	#11763189	1	1	yes	characteristic
134	C₂²	#11763214	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	#11762639	1	4	yes	complemented
137	C₄	#11762679	1	1	yes	characteristic
138	C₄	C₂×#10587489	1	1	yes	characteristic
139	C₄	#11763186	1	1	yes	characteristic
140	C₄	#11763214	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	#14528944	1	1	yes	characteristic
144	C₂	C₂×#11762639	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₃₆
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₄×C₁₆×C₁₇	C₂
socle	C₂²×C₁₇	C₂×#107
prime radical	C₂³×C₁₇	C₂×Q₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₇
hypocenter	C₁₇	#12909
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
67. C₄×C₁₇	#368	semidirect
98. C₁₇	#12909	semidirect
136. C₄	#11762639	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762639	2
#14528956	2
C₂×C₄×C₁₆×C₁₇	2
#12909	17