Subgroups — #16922011

Order 4352.

202 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10587465	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#10587465	—	2	2	no	—
4	C₂×#10587465	—	2	2	no	—
5	C₂×#10587465	C₄	1	4	yes	complemented
6	C₂²×C₁₆×C₁₇	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
9	C₄×C₁₆×C₁₇	—	2	2	no	—
10	C₂²×C₈×C₁₇	C₈	1	1	yes	characteristic
11	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄²×C₁₇	C₈	1	1	yes	characteristic
13	C₂×C₁₆×C₁₇	—	2	1	no	—
14	C₂×C₁₆×C₁₇	—	2	1	no	—
15	C₂×C₁₆×C₁₇	C₈	1	2	yes	—
16	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
17	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
18	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
19	C₂×C₁₆×C₁₇	Q₈	1	1	yes	characteristic
20	C₄×C₈×C₁₇	—	2	1	no	—
21	C₄×C₈×C₁₇	—	2	1	no	—
22	#10587465	—	2	4	no	—
23	C₂²×C₄×C₁₇	C₁₆	1	2	yes	—
24	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
25	C₂×C₈×C₁₇	—	2	1	no	—
26	C₂×C₈×C₁₇	—	2	1	no	—
27	C₂×C₈×C₁₇	C₁₆	1	2	yes	—
28	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
29	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₁₇	M₁₆	1	1	yes	characteristic
33	C₂×C₈×C₁₇	M₁₆	1	1	yes	characteristic
34	C₄²×C₁₇	—	2	1	no	—
35	C₄²×C₁₇	—	2	1	no	—
36	C₁₆×C₁₇	—	2	2	no	—
37	C₁₆×C₁₇	C₂²⋊C₄	1	2	yes	—
38	C₁₆×C₁₇	C₄⋊C₄	1	2	yes	—
39	#7465	—	17	1	no	maximal, Sylow
40	C₂³×C₁₇	C₂×C₁₆	1	1	yes	characteristic
41	C₂×C₄×C₁₇	—	2	2	no	—
42	C₂×C₄×C₁₇	—	2	2	no	—
43	C₂×C₄×C₁₇	C₃₂	1	4	yes	complemented
44	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
45	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
47	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
48	C₂×C₄×C₁₇	M₃₂	1	1	yes	characteristic
49	C₂×C₄×C₁₇	M₃₂	1	1	yes	characteristic
50	C₈×C₁₇	—	2	2	no	—
51	C₈×C₁₇	#95	1	1	yes	characteristic
52	C₈×C₁₇	#97	1	1	yes	characteristic
53	C₈×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
54	C₈×C₁₇	C₄⋊C₈	1	1	yes	characteristic
55	C₂²×C₃₂	—	17	2	no	—
56	C₂×C₄×C₁₆	D₁₇	1	1	yes	characteristic
57	C₂²×C₁₇	C₄×C₁₆	1	1	yes	characteristic
58	C₂²×C₁₇	C₂²⋊C₁₆	1	1	yes	characteristic
59	C₂²×C₁₇	C₄⋊C₁₆	1	1	yes	characteristic
60	C₂²×C₁₇	C₂×C₃₂	1	2	yes	—
61	C₂²×C₁₇	M₆₄	1	2	yes	—
62	C₄×C₁₇	—	2	4	no	—
63	C₄×C₁₇	#336	1	1	yes	characteristic
64	C₄×C₁₇	#346	1	1	yes	characteristic
65	C₄×C₁₇	C₂²⋊C₁₆	1	1	yes	characteristic
66	C₄×C₁₇	C₄⋊C₁₆	1	1	yes	characteristic
67	C₂²×C₁₆	Dic₁₇	1	1	yes	characteristic
68	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
69	C₂×C₄×C₈	Dic₁₇	1	1	yes	characteristic
70	C₂×C₃₂	—	17	4	no	—
71	C₂×C₃₂	—	34	2	no	—
72	C₂×C₃₂	—	34	2	no	—
73	C₄×C₁₆	—	2	2	no	—
74	C₂×C₁₇	#1374	1	1	yes	characteristic
75	C₂×C₁₇	C₄×C₃₂	1	1	yes	characteristic
76	C₂×C₁₇	#1397	1	1	yes	characteristic
77	C₂×C₁₇	C₂²⋊C₃₂	1	2	yes	—
78	C₂×C₁₇	C₄⋊C₃₂	1	2	yes	—
79	C₂²×C₈	#3622	1	1	yes	characteristic
80	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
81	C₂×C₄²	#3622	1	1	yes	characteristic
82	C₂×C₁₆	—	2	1	no	—
83	C₂×C₁₆	—	2	1	no	—
84	C₂×C₁₆	#3622	1	2	yes	—
85	C₂×C₁₆	Dic₃₄	1	1	yes	characteristic
86	C₂×C₁₆	C₄×D₁₇	1	2	yes	—
87	C₂×C₁₆	D₆₈	1	1	yes	characteristic
88	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
89	C₄×C₈	—	2	1	no	—
90	C₄×C₈	—	2	1	no	—
91	C₃₂	—	34	4	no	—
92	C₁₇	#7465	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₄	#63214	1	2	yes	—
94	C₂²×C₄	C₂×#3622	1	1	yes	characteristic
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	#63214	1	2	yes	—
98	C₂×C₈	C₂×#3622	1	1	yes	characteristic
99	C₂×C₈	#63223	1	1	yes	characteristic
100	C₂×C₈	#63223	1	1	yes	characteristic
101	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
102	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
103	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
104	C₄²	—	2	1	no	—
105	C₄²	—	2	1	no	—
106	C₁₆	—	2	2	no	—
107	C₁₆	#63225	1	2	yes	—
108	C₁₆	#63227	1	2	yes	—
109	C₂³	C₂×#63214	1	1	yes	characteristic
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	#10587465	1	4	yes	complemented
113	C₂×C₄	C₄×#3622	1	1	yes	characteristic
114	C₂×C₄	C₄⋊#3622	1	1	yes	characteristic
115	C₂×C₄	C₂×#63214	1	1	yes	characteristic
116	C₂×C₄	#10587483	1	1	yes	characteristic
117	C₂×C₄	#10587483	1	1	yes	characteristic
118	C₂×C₄	C₂²⋊#3622	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	#10587474	1	1	yes	characteristic
121	C₈	C₄⋊#3622	1	1	yes	characteristic
122	C₈	C₂²⋊#3622	1	1	yes	characteristic
123	C₈	#10587502	1	1	yes	characteristic
124	C₂²	C₄×#63214	1	1	yes	characteristic
125	C₂²	C₂²⋊#63214	1	1	yes	characteristic
126	C₂²	C₄⋊#63214	1	1	yes	characteristic
127	C₂²	C₂×#10587465	1	2	yes	—
128	C₂²	#11762643	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	#11762564	1	1	yes	characteristic
131	C₄	#11762585	1	1	yes	characteristic
132	C₄	C₂²⋊#63214	1	1	yes	characteristic
133	C₄	C₄⋊#63214	1	1	yes	characteristic
134	C₂	#14526862	1	1	yes	characteristic
135	C₂	C₄×#10587465	1	1	yes	characteristic
136	C₂	#14526918	1	1	yes	characteristic
137	C₂	C₂²⋊#10587465	1	2	yes	—
138	C₂	C₄⋊#10587465	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₁₆	C₂×D₁₇
commutator	C₂×C₁₇	C₄×C₃₂
Fitting	C₂×C₄×C₁₆×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂×C₁₆
cosocle	C₂²×C₁₆×C₁₇	C₂²
hypercenter	C₂×C₄×C₁₆	D₁₇
hypocenter	C₁₇	#7465
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
5. C₂×#10587465	C₄	semidirect
43. C₂×C₄×C₁₇	C₃₂	semidirect
92. C₁₇	#7465	semidirect
112. C₂×C₄	#10587465	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#10587465	2
C₂×C₄×C₁₆×C₁₇	2
#7465	17