Subgroups — #16949332

Order 4352.

338 conjugacy classes of subgroups.

The subgroup lattice has 160 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	#14528470	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×#10587474	C₄	1	1	yes	characteristic
7	C₂×#10587486	C₂²	1	2	yes	—
8	C₂×C₈×Dic₁₇	C₄	1	2	yes	—
9	#11762655	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	C₂×C₄	1	2	yes	—
20	C₂²×#3622	C₂×C₄	1	2	yes	—
21	C₁₇×M₃₂	—	2	4	no	—
22	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
23	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
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	#10587502	—	2	2	no	—
28	#10587486	—	2	4	no	—
29	C₈×Dic₁₇	C₈	1	8	yes	—
30	#10587483	—	2	4	no	—
31	C₂×#63214	C₂×C₄	1	4	yes	—
32	#10587474	—	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	C₂×C₈	1	4	yes	—
47	C₂×#3622	—	2	2	no	—
48	C₂×#3622	—	2	2	no	—
49	C₂×#3622	C₂×C₈	1	4	yes	—
50	C₁₆×C₁₇	—	2	4	no	—
51	#63214	—	2	4	no	—
52	#12593	—	17	1	no	maximal, Sylow
53	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
54	C₂×C₄×C₁₇	—	4	2	no	—
55	C₂×C₄×C₁₇	#97	1	2	yes	—
56	C₂×C₄×C₁₇	#97	1	2	yes	—
57	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₄×C₁₇	#117	1	2	yes	—
59	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
60	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
61	C₂×Dic₁₇	—	2	4	no	—
62	C₂×Dic₁₇	—	4	4	no	—
63	C₈×C₁₇	C₄×C₈	1	2	yes	—
64	C₈×C₁₇	C₄×C₈	1	2	yes	—
65	C₈×C₁₇	#117	1	2	yes	—
66	C₈×C₁₇	#117	1	2	yes	—
67	#3622	—	2	4	no	—
68	C₄×M₃₂	—	17	1	no	—
69	C₄×M₃₂	D₁₇	1	1	yes	characteristic
70	#1726	—	17	1	no	—
71	#1375	—	17	4	no	—
72	C₂²×C₁₇	—	2	2	no	—
73	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
74	C₂²×C₁₇	#431	1	2	yes	—
75	C₄×C₁₇	—	2	2	no	—
76	C₄×C₁₇	—	2	2	no	—
77	C₄×C₁₇	#376	1	1	yes	characteristic
78	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
79	C₄×C₁₇	#431	1	2	yes	—
80	Dic₁₇	—	4	4	no	—
81	C₂×M₃₂	—	17	2	no	—
82	C₂×M₃₂	C₂×D₁₇	1	2	yes	—
83	C₂×#97	—	17	2	no	—
84	C₂×#97	—	17	1	no	—
85	C₂×C₄×C₈	—	17	2	no	—
86	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
87	#376	—	17	1	no	—
88	#346	—	17	2	no	—
89	#346	Dic₁₇	1	2	yes	complemented
90	C₄×C₁₆	—	17	2	no	—
91	C₄×C₁₆	Dic₁₇	1	2	yes	complemented
92	C₂×C₁₇	—	2	2	no	—
93	C₂×C₁₇	#1744	1	1	yes	characteristic
94	C₂×C₁₇	#2110	1	2	yes	—
95	C₂²×C₈	—	17	2	no	—
96	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
97	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
98	C₂×C₄²	—	17	2	no	—
99	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
100	M₃₂	—	2	4	no	—
101	M₃₂	—	34	4	no	—
102	C₂×C₁₆	—	17	4	no	—
103	C₂×C₁₆	C₂×Dic₁₇	1	4	yes	—
104	#97	—	34	4	no	—
105	#97	—	34	2	no	—
106	C₄×C₈	—	17	8	no	—
107	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
108	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
109	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
110	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
111	#95	—	34	2	no	—
112	C₁₇	#12593	1	1	yes	characteristic, complemented, Sylow
113	C₂²×C₄	—	34	2	no	—
114	C₂²×C₄	#63218	1	2	yes	—
115	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
116	C₂×C₈	—	17	4	no	—
117	C₂×C₈	—	34	2	no	—
118	C₂×C₈	—	34	2	no	—
119	C₂×C₈	C₈×D₁₇	1	4	yes	—
120	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
121	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
122	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
123	C₂×C₈	#63252	1	4	yes	—
124	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
125	C₄²	—	17	4	no	—
126	C₄²	—	34	2	no	—
127	C₄²	#63218	1	2	yes	—
128	C₄²	C₄×Dic₁₇	1	1	yes	characteristic
129	C₄²	C₄×Dic₁₇	1	1	yes	characteristic
130	C₁₆	—	2	4	no	—
131	C₁₆	—	34	4	no	—
132	C₂³	C₂×#63218	1	1	yes	characteristic
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	34	4	no	—
135	C₂×C₄	—	68	4	no	—
136	C₂×C₄	#10587486	1	2	yes	—
137	C₂×C₄	#10587486	1	2	yes	—
138	C₂×C₄	#10587558	1	2	yes	—
139	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
140	C₂×C₄	C₂×#63218	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
142	C₈	—	34	4	no	—
143	C₈	C₈×Dic₁₇	1	2	yes	—
144	C₈	C₈×Dic₁₇	1	2	yes	—
145	C₈	#10587558	1	2	yes	—
146	C₈	#10587612	1	2	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	C₂×#10587486	1	1	yes	characteristic
149	C₂²	#11762860	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	68	4	no	—
153	C₄	#11762655	1	1	yes	characteristic
154	C₄	C₂×C₈×Dic₁₇	1	1	yes	characteristic
155	C₄	#11762860	1	2	yes	—
156	C₂	—	2	2	no	—
157	C₂	#14528529	1	1	yes	characteristic
158	C₂	#14530388	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	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	#12593
LCS	G ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ 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
89. #346	Dic₁₇	semidirect
91. C₄×C₁₆	Dic₁₇	semidirect
112. C₁₇	#12593	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
#14528470	2
#14526863	2
C₄×C₁₇×M₃₂	2
#12593	17