Subgroups — #17957987

Order 4352.

330 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544990	C₂	1	1	yes	maximal, characteristic
2	#14543987	C₂	1	1	yes	maximal, characteristic
3	#14543695	C₂	1	2	yes	maximal, complemented
4	C₁₆×#3624	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587538	C₄	1	2	yes	—
7	#11763825	C₄	1	2	yes	—
8	#11763698	C₄	1	2	yes	—
9	#11763695	C₂²	1	2	yes	—
10	#11763650	C₄	1	2	yes	—
11	C₈×#3624	C₂²	1	2	yes	—
12	C₂²×#63214	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
14	C₁₆×Dic₁₇	C₄	1	1	yes	characteristic
15	C₁₆×Dic₁₇	C₄	1	1	yes	characteristic
16	C₁₆×Dic₁₇	C₂²	1	1	yes	characteristic
17	C₁₆×Dic₁₇	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₁₇	C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
22	C₂²⋊#3624	C₈	1	4	yes	—
23	C₄⋊#3624	C₈	1	4	yes	—
24	C₄×#3624	C₂×C₄	1	4	yes	—
25	#10587538	C₈	1	8	yes	complemented
26	C₂×#63216	C₂×C₄	1	4	yes	—
27	C₂×C₁₆×C₁₇	C₈	1	4	yes	complemented
28	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
33	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
34	C₂×#63214	C₈	1	4	yes	complemented
35	C₂×#63214	C₂×C₄	1	1	yes	characteristic
36	C₂×#63214	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
38	C₂²×Dic₁₇	C₂×C₈	1	2	yes	—
39	C₂×#3624	C₂×C₈	1	8	yes	—
40	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
42	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₄×Dic₁₇	C₄²	1	1	yes	characteristic
45	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
46	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×#3622	C₄²	1	1	yes	characteristic
48	C₂×#3622	C₂×C₈	1	4	yes	—
49	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
50	#63216	C₂×C₈	1	8	yes	—
51	C₁₆×C₁₇	C₂×C₈	1	2	yes	—
52	C₁₆×C₁₇	C₂×C₈	1	2	yes	—
53	#63214	C₂×C₈	1	2	yes	—
54	#63214	C₂×C₈	1	2	yes	—
55	#11399	—	17	1	no	maximal, Sylow
56	C₂³×C₁₇	C₄×C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
58	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
59	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
60	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
61	C₂×Dic₁₇	—	2	4	no	—
62	C₂×Dic₁₇	C₄×C₈	1	2	yes	—
63	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
64	#3624	—	2	8	no	—
65	C₈×C₁₇	C₄×C₈	1	2	yes	—
66	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
67	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
68	#3622	C₄×C₈	1	2	yes	—
69	#3622	C₂²×C₈	1	2	yes	—
70	C₄×M₃₂	—	17	1	no	—
71	C₂×C₄×C₁₆	—	17	1	no	—
72	#1450	—	17	1	no	—
73	#1311	—	17	2	no	—
74	C₈×C₁₆	—	17	2	no	—
75	C₂²×C₁₇	—	2	2	no	—
76	C₂²×C₁₇	C₈²	1	2	yes	—
77	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
78	C₄×C₁₇	C₈²	1	2	yes	—
79	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
80	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
81	Dic₁₇	—	2	2	no	—
82	Dic₁₇	#504	1	4	yes	—
83	C₂×M₃₂	—	17	2	no	—
84	C₂²×C₁₆	—	17	1	no	—
85	C₂²×C₁₆	#595	1	1	yes	characteristic
86	#431	—	17	2	no	—
87	C₂×C₄×C₈	—	17	1	no	—
88	#346	—	17	2	no	—
89	C₄×C₁₆	—	17	2	no	—
90	C₄×C₁₆	—	17	1	no	—
91	C₄×C₁₆	—	17	1	no	—
92	C₄×C₁₆	—	17	1	no	—
93	C₄×C₁₆	—	17	1	no	—
94	#322	—	17	2	no	—
95	C₈²	—	17	2	no	—
96	C₂×C₁₇	—	2	2	no	—
97	C₂×C₁₇	C₂×C₈²	1	1	yes	characteristic
98	C₂×C₁₇	#2108	1	2	yes	—
99	C₂²×C₈	—	17	1	no	—
100	C₂²×C₈	C₂×#595	1	1	yes	characteristic
101	C₂×C₄²	—	17	1	no	—
102	M₃₂	—	17	8	no	—
103	C₂×C₁₆	—	17	4	no	—
104	C₂×C₁₆	—	17	4	no	—
105	C₂×C₁₆	—	17	1	no	—
106	C₂×C₁₆	—	17	1	no	—
107	C₂×C₁₆	#3624	1	4	yes	complemented
108	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
109	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
110	C₄⋊C₈	—	17	4	no	—
111	C₂²⋊C₈	—	17	4	no	—
112	C₄×C₈	—	17	4	no	—
113	C₄×C₈	—	17	1	no	—
114	C₄×C₈	—	17	1	no	—
115	C₄×C₈	—	17	1	no	—
116	C₄×C₈	—	17	1	no	—
117	C₁₇	#11399	1	1	yes	characteristic, complemented, Sylow
118	C₂²×C₄	—	17	2	no	—
119	C₂²×C₄	C₄×#595	1	1	yes	characteristic
120	C₂×C₈	—	17	1	no	—
121	C₂×C₈	—	17	1	no	—
122	C₂×C₈	—	17	4	no	—
123	C₂×C₈	—	17	8	no	—
124	C₂×C₈	#63242	1	2	yes	—
125	C₂×C₈	C₄×#595	1	1	yes	characteristic
126	C₂×C₈	C₂×#3624	1	2	yes	—
127	C₂×C₈	C₂²×#595	1	1	yes	characteristic
128	C₄²	—	17	1	no	—
129	C₄²	—	17	1	no	—
130	C₄²	—	17	2	no	—
131	C₁₆	—	17	8	no	—
132	C₁₆	—	17	2	no	—
133	C₁₆	—	17	2	no	—
134	C₁₆	C₂×#3624	1	2	yes	—
135	C₁₆	C₂×#3624	1	2	yes	—
136	C₂³	C₈×#595	1	1	yes	characteristic
137	C₂×C₄	—	17	2	no	—
138	C₂×C₄	—	17	2	no	—
139	C₂×C₄	—	34	4	no	—
140	C₂×C₄	C₈×#595	1	2	yes	—
141	C₂×C₄	C₈×#595	1	1	yes	characteristic
142	C₂×C₄	C₄×#3624	1	2	yes	—
143	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
144	C₈	—	17	2	no	—
145	C₈	—	17	2	no	—
146	C₈	—	34	8	no	—
147	C₈	C₄×#3624	1	2	yes	—
148	C₈	C₂×#63242	1	1	yes	characteristic
149	C₈	C₂²×#3624	1	1	yes	characteristic
150	C₂²	—	2	2	no	—
151	C₂²	C₈×#3624	1	2	yes	—
152	C₂²	C₂×C₈×#595	1	1	yes	characteristic
153	C₄	—	17	4	no	—
154	C₄	—	34	2	no	—
155	C₄	C₈×#3624	1	2	yes	—
156	C₄	C₂×C₈×#595	1	1	yes	characteristic
157	C₄	C₂×C₄×#3624	1	1	yes	characteristic
158	C₂	—	2	2	no	—
159	C₂	C₂×C₈×#3624	1	1	yes	characteristic
160	C₂	#14544992	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂×#595
commutator	C₂×C₁₇	C₂×C₈²
Frattini	C₂×C₈	C₂²×#595
Fitting	C₂²×C₁₆×C₁₇	C₄
socle	C₂²×C₁₇	C₂×C₄×C₈
prime radical	C₂³×C₁₇	C₄×C₈
cosocle	C₈×Dic₁₇	C₂³
hypercenter	C₂²×C₁₆	#595
hypocenter	C₁₇	#11399
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #14543695	C₂	semidirect
4. C₁₆×#3624	C₂	semidirect
25. #10587538	C₈	semidirect
27. C₂×C₁₆×C₁₇	C₈	semidirect
34. C₂×#63214	C₈	semidirect
107. C₂×C₁₆	#3624	semidirect
117. C₁₇	#11399	semidirect

Maximal subgroups

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

subgroup	index
#14544990	2
#14543987	2
#14543695	2
C₁₆×#3624	2
C₂×C₁₆×Dic₁₇	2
#11399	17