Subgroups — #16954504

Order 4352.

238 conjugacy classes of subgroups.

The subgroup lattice has 130 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762643	C₂	1	1	yes	maximal, characteristic
2	C₂²×#10587465	C₂	1	1	yes	maximal, characteristic
3	#14526918	C₂	1	2	yes	maximal, complemented
4	C₄×#10587465	C₂	1	2	yes	maximal, complemented
5	C₁₇×#2108	C₂	1	1	yes	maximal, characteristic
6	#11762643	C₄	1	4	yes	complemented
7	C₂×#10587465	C₄	1	4	yes	complemented
8	C₂×#10587465	C₂²	1	1	yes	characteristic
9	C₂×#10587465	C₂²	1	1	yes	characteristic
10	C₂×#10587465	C₂²	1	2	yes	—
11	C₂×C₁₇×M₃₂	C₄	1	1	yes	characteristic
12	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
13	C₁₇×#405	C₄	1	1	yes	characteristic
14	C₁₇×#346	C₄	1	2	yes	—
15	C₄×C₁₆×C₁₇	C₂²	1	2	yes	—
16	C₂×M₁₆×C₁₇	C₈	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₁₇×#117	C₈	1	1	yes	characteristic
19	C₁₇×M₃₂	C₈	1	4	yes	—
20	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
22	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
25	C₁₇×#97	C₈	1	2	yes	—
26	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
27	#10587465	C₂×C₄	1	2	yes	—
28	#10587465	C₂×C₄	1	2	yes	—
29	#10587465	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
31	M₁₆×C₁₇	C₁₆	1	4	yes	—
32	C₂×C₈×C₁₇	C₄²	1	2	yes	—
33	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
36	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
37	C₄⋊C₄×C₁₇	C₁₆	1	2	yes	—
38	C₂²⋊C₄×C₁₇	C₁₆	1	2	yes	—
39	C₄²×C₁₇	C₂×C₈	1	2	yes	—
40	C₁₆×C₁₇	C₄²	1	2	yes	—
41	C₁₆×C₁₇	C₂×C₈	1	4	yes	—
42	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
43	C₁₆×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	#13549	—	17	1	no	maximal, Sylow
45	C₂³×C₁₇	C₂×C₁₆	1	1	yes	characteristic
46	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
47	C₂×C₄×C₁₇	C₂×C₁₆	1	4	yes	—
48	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
50	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
51	C₈×C₁₇	C₄×C₈	1	2	yes	—
52	C₈×C₁₇	C₂×C₁₆	1	4	yes	—
53	C₈×C₁₇	C₂×C₄²	1	1	yes	characteristic
54	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
55	C₂×M₆₄	—	17	1	no	—
56	C₂²×C₃₂	—	17	1	no	—
57	#2108	D₁₇	1	1	yes	characteristic
58	#1397	—	17	2	no	—
59	C₄×C₃₂	—	17	2	no	—
60	C₂²×C₁₇	—	2	2	no	—
61	C₂²×C₁₇	C₄×C₁₆	1	2	yes	—
62	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
63	C₄×C₁₇	—	2	4	no	—
64	C₄×C₁₇	C₄×C₁₆	1	2	yes	—
65	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
66	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
67	C₂×M₃₂	Dic₁₇	1	1	yes	characteristic
68	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
69	#405	Dic₁₇	1	1	yes	characteristic
70	M₆₄	—	17	4	no	—
71	C₂×C₃₂	—	17	4	no	—
72	C₂×C₃₂	—	17	1	no	—
73	C₂×C₃₂	—	17	2	no	—
74	C₂×C₃₂	—	17	1	no	—
75	#346	Dic₁₇	1	2	yes	—
76	C₄×C₁₆	C₂×D₁₇	1	2	yes	—
77	C₂×C₁₇	—	2	2	no	—
78	C₂×C₁₇	C₂×C₄×C₁₆	1	1	yes	characteristic
79	C₂×C₁₇	#2258	1	2	yes	—
80	C₂×M₁₆	#3622	1	1	yes	characteristic
81	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
82	#117	#3622	1	1	yes	characteristic
83	M₃₂	#3622	1	4	yes	—
84	C₂×C₁₆	C₄×D₁₇	1	2	yes	—
85	C₂×C₁₆	C₄×D₁₇	1	2	yes	—
86	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
87	C₂×C₁₆	C₂×Dic₁₇	1	1	yes	characteristic
88	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
89	#97	#3622	1	2	yes	—
90	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
91	C₃₂	—	17	2	no	—
92	C₃₂	—	17	2	no	—
93	C₃₂	—	17	4	no	—
94	C₁₇	#13549	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₄	C₂×#3622	1	1	yes	characteristic
96	M₁₆	#63214	1	4	yes	—
97	C₂×C₈	C₂×#3622	1	1	yes	characteristic
98	C₂×C₈	C₂×#3622	1	2	yes	—
99	C₂×C₈	C₂×#3622	1	2	yes	—
100	C₂×C₈	C₄×Dic₁₇	1	2	yes	—
101	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
102	C₄⋊C₄	#63214	1	2	yes	—
103	C₂²⋊C₄	#63214	1	2	yes	—
104	C₄²	C₂×#3622	1	2	yes	—
105	C₁₆	C₂×#3622	1	4	yes	—
106	C₁₆	C₄×Dic₁₇	1	2	yes	—
107	C₁₆	C₂×C₄×D₁₇	1	1	yes	characteristic
108	C₁₆	C₂×C₄×D₁₇	1	1	yes	characteristic
109	C₂³	C₂×#63214	1	1	yes	characteristic
110	C₂×C₄	C₄×#3622	1	2	yes	—
111	C₂×C₄	C₂×#63214	1	4	yes	—
112	C₂×C₄	C₂×#63214	1	2	yes	—
113	C₂×C₄	C₂×#63214	1	1	yes	characteristic
114	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
115	C₈	C₄×#3622	1	2	yes	—
116	C₈	C₂×#63214	1	4	yes	—
117	C₈	C₂²×#3622	1	1	yes	characteristic
118	C₈	C₂×C₄×Dic₁₇	1	1	yes	characteristic
119	C₂²	—	2	2	no	—
120	C₂²	C₄×#63214	1	2	yes	—
121	C₂²	C₂²×#63214	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	C₄×#63214	1	2	yes	—
124	C₄	C₂×C₄×#3622	1	1	yes	characteristic
125	C₄	C₂²×#63214	1	1	yes	characteristic
126	C₂	—	2	2	no	—
127	C₂	C₂×C₄×#63214	1	1	yes	characteristic
128	C₂	#14531078	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₁₇
commutator	C₂×C₁₇	C₂×C₄×C₁₆
Fitting	C₁₇×#2108	C₂
socle	C₂²×C₁₇	C₂²×C₁₆
prime radical	C₂³×C₁₇	C₂×C₁₆
cosocle	C₂×C₁₆×C₁₇	C₂³
hypercenter	#2108	D₁₇
hypocenter	C₁₇	#13549
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
3. #14526918	C₂	semidirect
4. C₄×#10587465	C₂	semidirect
6. #11762643	C₄	semidirect
7. C₂×#10587465	C₄	semidirect
94. 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×#11762643	2
C₂²×#10587465	2
#14526918	2
C₄×#10587465	2
C₁₇×#2108	2
#13549	17