Subgroups — #16954505

Order 4352.

238 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	C₂×#11762642	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₂×D₁₇	C₂	1	1	yes	maximal, characteristic
3	#14530379	C₂	1	1	yes	maximal, characteristic
4	#14526917	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#10587465	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₂×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#1397	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10587469	C₄	1	1	yes	characteristic
9	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
10	#11762763	C₄	1	1	yes	characteristic
11	#11762642	C₄	1	4	yes	complemented
12	C₂×#10587465	C₂²	1	1	yes	characteristic
13	C₂×#10587465	C₂²	1	1	yes	characteristic
14	C₃₂×D₁₇	C₄	1	4	yes	complemented
15	#11762585	C₄	1	1	yes	characteristic
16	#11762584	C₄	1	1	yes	characteristic
17	C₁₆×Dic₁₇	C₂²	1	1	yes	characteristic
18	C₂×C₁₇×C₃₂	C₂²	1	1	yes	characteristic
19	C₂×C₁₇×C₃₂	C₂²	1	1	yes	characteristic
20	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
21	C₂×#63218	C₈	1	1	yes	characteristic
22	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
23	#10587558	C₈	1	1	yes	characteristic
24	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
26	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	#10587486	C₈	1	1	yes	characteristic
28	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂×#63214	C₂×C₄	1	1	yes	characteristic
30	C₂×#63214	C₂×C₄	1	1	yes	characteristic
31	#10587474	C₈	1	1	yes	characteristic
32	#10587469	C₈	1	4	yes	—
33	C₁₆×D₁₇	C₂×C₄	1	2	yes	—
34	C₁₆×D₁₇	C₂×C₄	1	2	yes	—
35	C₁₇×C₃₂	C₂×C₄	1	2	yes	—
36	C₁₇×C₃₂	C₂×C₄	1	2	yes	—
37	#10587465	C₂×C₄	1	2	yes	—
38	#10587465	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
42	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
43	#63227	C₁₆	1	2	yes	—
44	#63225	C₁₆	1	2	yes	—
45	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
46	C₂×#3622	C₂×C₈	1	1	yes	characteristic
47	C₂×#3622	C₂×C₈	1	1	yes	characteristic
48	#63218	C₁₆	1	4	yes	—
49	C₈×D₁₇	C₄²	1	2	yes	—
50	C₈×D₁₇	C₂×C₈	1	2	yes	—
51	C₁₆×C₁₇	C₂×C₈	1	2	yes	—
52	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
54	#63214	C₄²	1	2	yes	—
55	#63214	C₂×C₈	1	2	yes	—
56	#13549	—	17	1	no	maximal, Sylow
57	C₂²×D₁₇	C₂×C₁₆	1	1	yes	characteristic
58	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
59	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
60	C₂×Dic₁₇	C₂×C₁₆	1	2	yes	—
61	C₂×Dic₁₇	C₂×C₁₆	1	1	yes	characteristic
62	C₄×D₁₇	C₄×C₈	1	2	yes	—
63	C₄×D₁₇	C₂×C₁₆	1	2	yes	—
64	C₈×C₁₇	C₂×C₁₆	1	2	yes	—
65	C₈×C₁₇	C₂×C₄²	1	1	yes	characteristic
66	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
67	#3622	C₄×C₈	1	2	yes	—
68	#3622	C₂×C₁₆	1	2	yes	—
69	C₂×M₆₄	—	17	1	no	—
70	C₂²×C₃₂	—	17	1	no	—
71	#2108	—	17	1	no	—
72	#1397	—	17	1	no	—
73	#1397	D₁₇	1	1	yes	characteristic, complemented
74	C₄×C₃₂	—	17	1	no	—
75	C₄×C₃₂	—	17	1	no	—
76	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
77	C₂×D₁₇	—	2	2	no	—
78	C₂×D₁₇	C₄×C₁₆	1	2	yes	—
79	C₄×C₁₇	—	2	2	no	—
80	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
81	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
82	Dic₁₇	—	2	2	no	—
83	Dic₁₇	C₄×C₁₆	1	2	yes	—
84	C₂×M₃₂	—	17	1	no	—
85	C₂²×C₁₆	—	17	1	no	—
86	#405	—	17	1	no	—
87	M₆₄	—	17	4	no	—
88	C₂×C₃₂	—	17	1	no	—
89	C₂×C₃₂	—	17	1	no	—
90	C₂×C₃₂	—	17	4	no	—
91	C₂×C₃₂	C₂×D₁₇	1	1	yes	characteristic
92	C₂×C₃₂	C₂×D₁₇	1	1	yes	characteristic
93	#346	—	17	1	no	—
94	#346	—	17	1	no	—
95	C₄×C₁₆	—	17	1	no	—
96	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
97	C₂×C₁₇	C₂×C₄×C₁₆	1	1	yes	characteristic
98	C₂×C₁₇	#2258	1	2	yes	—
99	D₁₇	—	2	2	no	—
100	C₂×M₁₆	—	17	1	no	—
101	C₂²×C₈	—	17	1	no	—
102	#117	—	17	1	no	—
103	M₃₂	—	17	4	no	—
104	C₂×C₁₆	—	17	1	no	—
105	C₂×C₁₆	—	17	1	no	—
106	C₂×C₁₆	—	17	2	no	—
107	C₂×C₁₆	—	17	2	no	—
108	C₂×C₁₆	C₄×D₁₇	1	1	yes	characteristic
109	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
110	#97	—	17	1	no	—
111	#97	—	17	1	no	—
112	C₄×C₈	—	17	1	no	—
113	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
114	C₃₂	—	17	2	no	—
115	C₃₂	—	17	2	no	—
116	C₃₂	C₄×D₁₇	1	2	yes	—
117	C₃₂	C₄×D₁₇	1	2	yes	—
118	C₁₇	#13549	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₄	—	17	1	no	—
120	M₁₆	—	17	4	no	—
121	C₂×C₈	—	17	1	no	—
122	C₂×C₈	—	17	1	no	—
123	C₂×C₈	—	17	2	no	—
124	C₂×C₈	—	17	2	no	—
125	C₂×C₈	C₈×D₁₇	1	1	yes	characteristic
126	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
127	C₄⋊C₄	—	17	2	no	—
128	C₂²⋊C₄	—	17	2	no	—
129	C₄²	—	17	1	no	—
130	C₄²	C₈×D₁₇	1	1	yes	characteristic
131	C₁₆	—	17	2	no	—
132	C₁₆	—	17	2	no	—
133	C₁₆	C₈×D₁₇	1	2	yes	—
134	C₁₆	C₂×C₄×D₁₇	1	1	yes	characteristic
135	C₁₆	C₂×C₄×D₁₇	1	1	yes	characteristic
136	C₂³	—	17	1	no	—
137	C₂×C₄	—	17	1	no	—
138	C₂×C₄	—	17	2	no	—
139	C₂×C₄	—	17	2	no	—
140	C₂×C₄	—	17	2	no	—
141	C₂×C₄	C₁₆×D₁₇	1	2	yes	—
142	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
143	C₈	—	17	2	no	—
144	C₈	—	17	2	no	—
145	C₈	C₁₆×D₁₇	1	2	yes	—
146	C₈	C₄²×D₁₇	1	1	yes	characteristic
147	C₈	C₂×C₈×D₁₇	1	1	yes	characteristic
148	C₂²	—	17	2	no	—
149	C₂²	—	34	2	no	—
150	C₂²	C₂×C₁₆×D₁₇	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	17	2	no	—
153	C₄	—	34	2	no	—
154	C₄	C₄×C₈×D₁₇	1	1	yes	characteristic
155	C₄	C₂×C₁₆×D₁₇	1	1	yes	characteristic
156	C₂	—	34	2	no	—
157	C₂	C₄×C₁₆×D₁₇	1	1	yes	characteristic
158	C₂	#14531079	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₁₇
commutator	C₂×C₁₇	C₂×C₄×C₁₆
Fitting	C₁₇×#1397	C₂
socle	C₂²×C₁₇	C₂²×C₁₆
prime radical	C₂²×D₁₇	C₂×C₁₆
cosocle	C₂×C₁₆×C₁₇	C₂³
hypercenter	#1397	D₁₇
hypocenter	C₁₇	#13549
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₁₆ ◁ #1397

Complements

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

N	H ≅ G/N	type
4. #14526917	C₂	semidirect
5. C₄×#10587465	C₂	semidirect
6. C₃₂×Dic₁₇	C₂	semidirect
7. C₁₇×#1397	C₂	semidirect
11. #11762642	C₄	semidirect
14. C₃₂×D₁₇	C₄	semidirect
73. #1397	D₁₇	semidirect
118. C₁₇	#13549	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762642	2
C₂×C₃₂×D₁₇	2
#14530379	2
#14526917	2
C₄×#10587465	2
C₃₂×Dic₁₇	2
C₁₇×#1397	2
#13549	17