Subgroups — #17951685

Order 4352.

574 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×#3624	C₂	1	2	yes	maximal
2	#14543606	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#3624	C₄	1	4	yes	—
5	C₂×C₄×#3624	C₄	1	4	yes	—
6	C₂×C₄×#3624	C₂²	1	4	yes	—
7	C₂×C₄×#3622	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₁₇	C₄	1	2	yes	—
9	C₂×C₈×Dic₁₇	C₂²	1	2	yes	—
10	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
12	C₂²×#3624	C₂×C₄	1	8	yes	—
13	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
17	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
19	C₂²×#3622	C₂×C₄	1	2	yes	—
20	C₄×#3624	—	2	8	no	—
21	C₄×#3624	—	2	8	no	—
22	C₄×#3624	C₈	1	16	yes	complemented
23	C₄×C₈×C₁₇	C₈	1	4	yes	complemented
24	C₈×Dic₁₇	—	2	4	no	—
25	C₈×Dic₁₇	—	2	4	no	—
26	C₄×#3622	C₈	1	4	yes	complemented
27	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
28	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
29	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
30	C₂²×Dic₁₇	C₄²	1	2	yes	—
31	C₂²×Dic₁₇	C₂²×C₄	1	2	yes	—
32	C₂×#3624	—	2	8	no	—
33	C₂×#3624	—	2	8	no	—
34	C₂×#3624	C₂×C₈	1	16	yes	—
35	C₂×C₈×C₁₇	—	2	2	no	—
36	C₂×C₈×C₁₇	—	2	2	no	—
37	C₂×C₈×C₁₇	C₂×C₈	1	4	yes	—
38	C₄²×C₁₇	C₂×C₈	1	2	yes	—
39	C₄²×C₁₇	C₂×C₈	1	2	yes	—
40	C₄×Dic₁₇	C₂×C₈	1	8	yes	—
41	C₄×Dic₁₇	M₁₆	1	4	yes	—
42	C₄×Dic₁₇	M₁₆	1	8	yes	—
43	C₄×Dic₁₇	#41	1	4	yes	—
44	C₂×#3622	—	2	2	no	—
45	C₂×#3622	—	2	2	no	—
46	C₂×#3622	C₂×C₈	1	4	yes	—
47	#8131	—	17	1	no	maximal, Sylow
48	C₂³×C₁₇	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄×C₁₇	C₄×C₈	1	4	yes	—
50	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
51	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
52	C₂×C₄×C₁₇	#97	1	4	yes	—
53	C₂×C₄×C₁₇	#117	1	2	yes	—
54	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
57	C₂×Dic₁₇	C₄×C₈	1	4	yes	—
58	C₂×Dic₁₇	#97	1	4	yes	—
59	C₂×Dic₁₇	#97	1	4	yes	—
60	C₂×Dic₁₇	#117	1	4	yes	—
61	C₂×Dic₁₇	C₂²×C₈	1	4	yes	—
62	C₂×Dic₁₇	C₂×M₁₆	1	4	yes	—
63	#3624	—	2	16	no	—
64	C₈×C₁₇	—	2	4	no	—
65	#3622	—	2	4	no	—
66	C₄²×C₈	—	17	2	no	—
67	C₄²×C₈	—	17	1	no	—
68	#1273	—	17	4	no	—
69	C₂²×C₁₇	#376	1	1	yes	characteristic
70	C₂²×C₁₇	C₂×C₄×C₈	1	2	yes	—
71	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
72	C₂²×C₁₇	C₂×#97	1	2	yes	—
73	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
74	C₄×C₁₇	C₈²	1	4	yes	—
75	C₄×C₁₇	#322	1	4	yes	—
76	C₄×C₁₇	#431	1	2	yes	—
77	C₄×C₁₇	#431	1	2	yes	—
78	Dic₁₇	#322	1	8	yes	—
79	Dic₁₇	#431	1	8	yes	—
80	C₂×C₄×C₈	—	17	4	no	—
81	C₂×C₄×C₈	—	17	1	no	—
82	C₂×C₄×C₈	—	17	4	no	—
83	C₂×C₄×C₈	—	17	4	no	—
84	C₂×C₄×C₈	—	17	2	no	—
85	C₂×C₄×C₈	—	17	2	no	—
86	C₂×C₄×C₈	#595	1	1	yes	characteristic
87	C₄³	—	17	1	no	—
88	C₂×C₁₇	C₂×C₈²	1	1	yes	characteristic
89	C₂×C₁₇	C₂×#322	1	1	yes	characteristic
90	C₂×C₁₇	C₂×#322	1	2	yes	—
91	C₂×C₁₇	#1744	1	2	yes	—
92	C₂×C₁₇	#1744	1	1	yes	characteristic
93	C₂²×C₈	—	17	8	no	—
94	C₂²×C₈	—	17	2	no	—
95	C₂²×C₈	C₂×#595	1	2	yes	—
96	C₂×C₄²	—	17	2	no	—
97	C₂×C₄²	—	17	2	no	—
98	C₂×C₄²	—	17	1	no	—
99	C₂×C₄²	—	17	1	no	—
100	C₂×C₄²	C₂×#595	1	1	yes	characteristic
101	C₄×C₈	—	17	16	no	—
102	C₄×C₈	—	17	4	no	—
103	C₄×C₈	—	34	4	no	—
104	C₄×C₈	—	34	4	no	—
105	C₄×C₈	—	34	8	no	—
106	C₄×C₈	—	34	8	no	—
107	C₄×C₈	#3624	1	4	yes	complemented
108	C₁₇	#8131	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₄	—	17	2	no	—
110	C₂²×C₄	—	17	2	no	—
111	C₂²×C₄	C₄×#595	1	1	yes	characteristic
112	C₂²×C₄	C₄×#595	1	1	yes	characteristic
113	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	17	16	no	—
117	C₂×C₈	—	17	4	no	—
118	C₂×C₈	—	34	8	no	—
119	C₂×C₈	—	34	8	no	—
120	C₂×C₈	—	34	2	no	—
121	C₂×C₈	—	34	2	no	—
122	C₂×C₈	C₂×#3624	1	4	yes	—
123	C₄²	—	17	4	no	—
124	C₄²	—	17	4	no	—
125	C₄²	—	17	8	no	—
126	C₄²	—	17	8	no	—
127	C₄²	#63242	1	2	yes	—
128	C₄²	C₂×#3624	1	2	yes	—
129	C₂³	C₂×C₄×#595	1	1	yes	characteristic
130	C₂×C₄	—	17	4	no	—
131	C₂×C₄	—	17	4	no	—
132	C₂×C₄	—	17	4	no	—
133	C₂×C₄	—	17	4	no	—
134	C₂×C₄	—	17	4	no	—
135	C₂×C₄	—	17	4	no	—
136	C₂×C₄	C₈×#595	1	4	yes	—
137	C₂×C₄	#10587532	1	4	yes	—
138	C₂×C₄	C₄×#3624	1	2	yes	—
139	C₂×C₄	C₄×#3624	1	2	yes	—
140	C₂×C₄	C₂×#63242	1	1	yes	characteristic
141	C₂×C₄	D₁₇⋊M₁₆	1	2	yes	—
142	C₂×C₄	#10587677	1	2	yes	—
143	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₈	—	34	16	no	—
146	C₈	—	34	4	no	—
147	C₂²	#11763745	1	1	yes	characteristic
148	C₂²	C₂×C₈×#595	1	2	yes	—
149	C₂²	C₂×C₄×#3624	1	1	yes	characteristic
150	C₂²	C₂×#10587532	1	2	yes	—
151	C₂²	#11763768	1	1	yes	characteristic
152	C₄	—	17	8	no	—
153	C₄	—	17	8	no	—
154	C₄	C₈×#3624	1	4	yes	—
155	C₄	#11763650	1	4	yes	—
156	C₄	#11763825	1	2	yes	—
157	C₄	#11763827	1	2	yes	—
158	C₂	C₂×C₈×#3624	1	1	yes	characteristic
159	C₂	C₂×#11763650	1	1	yes	characteristic
160	C₂	#14543980	1	2	yes	—
161	C₂	#14544497	1	2	yes	—
162	C₂	#14544498	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂×#595
commutator	C₂×C₁₇	C₂×C₈²
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂×C₄²
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#8131
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
22. C₄×#3624	C₈	semidirect
23. C₄×C₈×C₁₇	C₈	semidirect
26. C₄×#3622	C₈	semidirect
107. C₄×C₈	#3624	semidirect
108. C₁₇	#8131	semidirect

Maximal subgroups

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

subgroup	index
C₄²×#3624	2
#14543606	2
C₄×C₈×Dic₁₇	2
#8131	17