Subgroups — #16949335

Order 4352.

354 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10587483	C₂	1	1	yes	maximal, characteristic
2	C₄×#10587486	C₂	1	1	yes	maximal, characteristic
3	#14526863	C₂	1	4	yes	maximal
4	C₄×C₁₇×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587483	C₂²	1	2	yes	—
6	C₂×#10587486	C₄	1	2	yes	—
7	C₂×#10587486	C₂²	1	2	yes	—
8	C₂×C₄×#3622	C₄	1	1	yes	characteristic
9	C₄²×Dic₁₇	C₄	1	1	yes	characteristic
10	#11762585	C₄	1	2	yes	complemented
11	C₄×#63214	C₄	1	2	yes	complemented
12	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×#346	C₄	1	2	yes	complemented
15	C₄×C₁₆×C₁₇	C₄	1	2	yes	complemented
16	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₁₇	—	2	2	no	—
20	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
21	C₂²×#3622	C₂×C₄	1	2	yes	—
22	C₁₇×M₃₂	—	2	4	no	—
23	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
24	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
28	#10587486	—	2	4	no	—
29	#10587486	C₈	1	8	yes	—
30	#10587483	—	2	4	no	—
31	C₂×#63214	C₂×C₄	1	4	yes	—
32	C₄×#3622	—	2	2	no	—
33	C₂²×C₄×C₁₇	M₁₆	1	2	yes	—
34	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
35	C₂²×Dic₁₇	—	2	2	no	—
36	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₁₇	C₂×C₈	1	4	yes	—
39	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₇	#41	1	4	yes	—
42	C₄²×C₁₇	C₄²	1	1	yes	characteristic
43	C₄²×C₁₇	C₄²	1	1	yes	characteristic
44	C₄²×C₁₇	M₁₆	1	2	yes	—
45	C₄×Dic₁₇	—	2	2	no	—
46	C₄×Dic₁₇	—	4	4	no	—
47	C₄×Dic₁₇	C₂×C₈	1	4	yes	—
48	C₂×#3622	—	2	2	no	—
49	C₂×#3622	—	2	2	no	—
50	C₂×#3622	C₂×C₈	1	4	yes	—
51	C₁₆×C₁₇	—	2	4	no	—
52	#63214	—	2	4	no	—
53	#12594	—	17	1	no	maximal, Sylow
54	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
55	C₂×C₄×C₁₇	—	2	4	no	—
56	C₂×C₄×C₁₇	#97	1	2	yes	—
57	C₂×C₄×C₁₇	#97	1	2	yes	—
58	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
59	C₂×C₄×C₁₇	#117	1	2	yes	—
60	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
61	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
62	C₂×Dic₁₇	—	2	4	no	—
63	C₂×Dic₁₇	—	4	4	no	—
64	C₈×C₁₇	C₄×C₈	1	2	yes	—
65	C₈×C₁₇	C₄×C₈	1	2	yes	—
66	C₈×C₁₇	#117	1	2	yes	—
67	C₈×C₁₇	#117	1	2	yes	—
68	#3622	—	2	4	no	—
69	C₄×M₃₂	—	17	1	no	—
70	C₄×M₃₂	D₁₇	1	1	yes	characteristic
71	C₄×#97	—	17	1	no	—
72	#1375	—	17	4	no	—
73	C₂²×C₁₇	—	2	2	no	—
74	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
75	C₂²×C₁₇	#431	1	2	yes	—
76	C₄×C₁₇	—	2	2	no	—
77	C₄×C₁₇	#347	1	4	yes	—
78	C₄×C₁₇	#376	1	1	yes	characteristic
79	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
80	C₄×C₁₇	#431	1	2	yes	—
81	Dic₁₇	—	4	4	no	—
82	C₂×M₃₂	—	17	2	no	—
83	C₂×M₃₂	C₂×D₁₇	1	2	yes	—
84	C₂×#97	—	17	2	no	—
85	C₂×#97	—	17	2	no	—
86	C₂×C₄×C₈	—	17	1	no	—
87	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
88	C₄³	—	17	1	no	—
89	#346	—	17	2	no	—
90	#346	Dic₁₇	1	2	yes	complemented
91	C₄×C₁₆	—	17	2	no	—
92	C₄×C₁₆	Dic₁₇	1	2	yes	complemented
93	C₂×C₁₇	—	2	2	no	—
94	C₂×C₁₇	#1744	1	1	yes	characteristic
95	C₂×C₁₇	C₂×#347	1	2	yes	—
96	C₂²×C₈	—	17	2	no	—
97	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
98	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
99	C₂×C₄²	—	17	2	no	—
100	C₂×C₄²	—	34	2	no	—
101	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
102	M₃₂	—	2	4	no	—
103	M₃₂	—	34	4	no	—
104	C₂×C₁₆	—	17	4	no	—
105	C₂×C₁₆	C₂×Dic₁₇	1	4	yes	—
106	#97	—	17	8	no	—
107	#97	—	34	4	no	—
108	C₄×C₈	—	34	2	no	—
109	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
110	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
111	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
112	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
113	C₁₇	#12594	1	1	yes	characteristic, complemented, Sylow
114	C₂²×C₄	—	34	2	no	—
115	C₂²×C₄	#63218	1	2	yes	—
116	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
117	C₂×C₈	—	17	4	no	—
118	C₂×C₈	—	34	2	no	—
119	C₂×C₈	—	34	2	no	—
120	C₂×C₈	C₈×D₁₇	1	4	yes	—
121	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
122	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
123	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
124	C₂×C₈	#63252	1	4	yes	—
125	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
126	C₄²	—	17	4	no	—
127	C₄²	—	34	2	no	—
128	C₄²	—	68	4	no	—
129	C₄²	#63218	1	2	yes	—
130	C₄²	C₄×Dic₁₇	1	1	yes	characteristic
131	C₄²	C₄×Dic₁₇	1	1	yes	characteristic
132	C₁₆	—	2	4	no	—
133	C₁₆	—	34	4	no	—
134	C₂³	C₂×#63218	1	1	yes	characteristic
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	34	4	no	—
137	C₂×C₄	—	68	4	no	—
138	C₂×C₄	#10587486	1	2	yes	—
139	C₂×C₄	#10587486	1	2	yes	—
140	C₂×C₄	#10587558	1	2	yes	—
141	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
142	C₂×C₄	C₂×#63218	1	1	yes	characteristic
143	C₂×C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
144	C₈	—	34	4	no	—
145	C₈	C₈×Dic₁₇	1	2	yes	—
146	C₈	C₈×Dic₁₇	1	2	yes	—
147	C₈	#10587558	1	2	yes	—
148	C₈	#10587612	1	2	yes	—
149	C₂²	—	2	2	no	—
150	C₂²	C₂×#10587486	1	1	yes	characteristic
151	C₂²	#11762860	1	2	yes	—
152	C₄	—	2	2	no	—
153	C₄	—	68	4	no	—
154	C₄	#11762586	1	4	yes	—
155	C₄	#11762655	1	1	yes	characteristic
156	C₄	C₂×C₈×Dic₁₇	1	1	yes	characteristic
157	C₄	#11762860	1	2	yes	—
158	C₂	—	2	2	no	—
159	C₂	#14528529	1	1	yes	characteristic
160	C₂	C₂×#11762586	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×Dic₁₇
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₄×C₁₇×M₃₂	C₂
socle	C₂²×C₁₇	C₂×#97
prime radical	C₂³×C₁₇	C₂×M₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₄×M₃₂	D₁₇
hypocenter	C₁₇	#12594
LCS	G ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄² ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
10. #11762585	C₄	semidirect
11. C₄×#63214	C₄	semidirect
14. C₁₇×#346	C₄	semidirect
15. C₄×C₁₆×C₁₇	C₄	semidirect
90. #346	Dic₁₇	semidirect
92. C₄×C₁₆	Dic₁₇	semidirect
113. C₁₇	#12594	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10587483	2
C₄×#10587486	2
#14526863	2
C₄×C₁₇×M₃₂	2
#12594	17