Subgroups — #16945883

Order 4352.

242 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10587483	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762560	C₂	1	1	yes	maximal, characteristic
3	#14526849	C₂	1	4	yes	maximal, complemented
4	C₁₇×#1564	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587483	C₄	1	1	yes	characteristic
6	C₂×#10587483	C₄	1	1	yes	characteristic
7	C₂×C₄⋊#3622	—	2	1	no	—
8	#11762585	C₂²	1	2	yes	—
9	C₄×#63214	C₄	1	2	yes	—
10	#11762560	—	2	1	no	—
11	#11762560	C₂²	1	2	yes	—
12	C₁₇×#431	—	2	1	no	—
13	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×#335	—	2	1	no	—
15	C₁₇×#334	C₂²	1	2	yes	—
16	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
19	C₂²×#3622	—	4	1	no	—
20	C₁₇×C₄⋊C₈	—	2	2	no	—
21	C₁₇×C₂²⋊C₈	—	4	1	no	—
22	C₄×C₈×C₁₇	—	4	1	no	—
23	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
25	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
26	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
27	#10587483	—	2	2	no	—
28	#10587483	C₈	1	4	yes	complemented
29	C₂×#63214	C₂×C₄	1	2	yes	—
30	C₂×#63214	C₂×C₄	1	2	yes	—
31	C₄⋊#3622	—	2	2	no	—
32	C₄⋊#3622	—	4	1	no	—
33	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
36	C₂×C₈×C₁₇	—	2	2	no	—
37	C₂×C₈×C₁₇	—	4	2	no	—
38	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
42	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₁₇	SD₁₆	1	1	yes	characteristic
46	C₄²×C₁₇	SD₁₆	1	1	yes	characteristic
47	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
48	C₂×#3622	—	4	2	no	—
49	C₂×#3622	—	8	2	no	—
50	#63214	—	2	2	no	—
51	#63214	C₂×C₈	1	4	yes	complemented
52	#12000	—	17	1	no	maximal, Sylow
53	C₂³×C₁₇	#106	1	1	yes	characteristic
54	C₂×C₄×C₁₇	—	2	2	no	—
55	C₂×C₄×C₁₇	—	4	1	no	—
56	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
57	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
58	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
59	C₂×C₄×C₁₇	#106	1	1	yes	characteristic
60	C₂×C₄×C₁₇	#108	1	2	yes	—
61	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₈×C₁₇	—	2	1	no	—
65	C₈×C₁₇	—	2	1	no	—
66	C₈×C₁₇	—	8	2	no	—
67	C₈×C₁₇	C₄⋊C₈	1	2	yes	—
68	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
69	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
70	#3622	—	8	2	no	—
71	C₄×M₃₂	—	17	1	no	—
72	#1564	D₁₇	1	1	yes	characteristic
73	C₂×#334	—	17	1	no	—
74	#1368	—	17	4	no	—
75	C₂²×C₁₇	—	2	1	no	—
76	C₂²×C₁₇	—	2	1	no	—
77	C₂²×C₁₇	#334	1	2	yes	—
78	C₂²×C₁₇	C₂×#106	1	1	yes	characteristic
79	C₄×C₁₇	—	2	1	no	—
80	C₄×C₁₇	—	2	1	no	—
81	C₄×C₁₇	—	2	1	no	—
82	C₄×C₁₇	#334	1	2	yes	—
83	C₄×C₁₇	#365	1	2	yes	—
84	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₄×C₁₇	C₂×#108	1	1	yes	characteristic
86	C₂×M₃₂	—	17	1	no	—
87	C₂×M₃₂	—	17	1	no	—
88	#431	—	2	1	no	—
89	C₂×C₄⋊C₈	—	34	1	no	—
90	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
91	#346	—	17	2	no	—
92	C₄×C₁₆	—	17	2	no	—
93	#335	—	2	1	no	—
94	#334	—	17	2	no	—
95	#334	—	34	1	no	—
96	#334	C₂×D₁₇	1	2	yes	—
97	C₂×C₁₇	—	2	2	no	—
98	C₂×C₁₇	C₂×#334	1	1	yes	characteristic
99	C₂×C₁₇	C₂×#365	1	1	yes	characteristic
100	C₂×C₁₇	#2155	1	1	yes	characteristic
101	C₂²×C₈	—	68	1	no	—
102	C₂²×C₈	Dic₃₄	1	1	yes	characteristic
103	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
104	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
105	M₃₂	—	17	4	no	—
106	M₃₂	—	34	2	no	—
107	C₂×C₁₆	—	17	2	no	—
108	C₂×C₁₆	—	17	2	no	—
109	C₄⋊C₈	—	2	2	no	—
110	C₄⋊C₈	—	34	2	no	—
111	C₄⋊C₈	—	68	1	no	—
112	C₂²⋊C₈	—	4	1	no	—
113	C₄×C₈	—	4	1	no	—
114	C₄×C₈	Dic₃₄	1	1	yes	characteristic
115	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
116	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
117	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
118	C₁₇	#12000	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₄	#63225	1	1	yes	characteristic
120	C₂²×C₄	#63229	1	1	yes	characteristic
121	C₂²×C₄	#63230	1	1	yes	characteristic
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	4	2	no	—
124	C₂×C₈	—	68	2	no	—
125	C₂×C₈	—	136	2	no	—
126	C₂×C₈	C₈×D₁₇	1	2	yes	—
127	C₂×C₈	#63218	1	2	yes	—
128	C₂×C₈	#63225	1	1	yes	characteristic
129	C₂×C₈	#63225	1	1	yes	characteristic
130	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
131	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
132	C₄²	#63225	1	1	yes	characteristic
133	C₄²	#63229	1	1	yes	characteristic
134	C₄²	#63230	1	1	yes	characteristic
135	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
136	C₁₆	—	17	4	no	—
137	C₁₆	—	34	2	no	—
138	C₂³	#10587479	1	1	yes	characteristic
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	#10587479	1	1	yes	characteristic
142	C₂×C₄	#10587479	1	1	yes	characteristic
143	C₂×C₄	#10587479	1	1	yes	characteristic
144	C₂×C₄	#10587485	1	2	yes	—
145	C₂×C₄	#10587493	1	2	yes	—
146	C₂×C₄	C₂×#63225	1	1	yes	characteristic
147	C₂×C₄	C₂×#63229	1	1	yes	characteristic
148	C₂×C₄	C₂×#63230	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	8	2	no	—
152	C₈	—	136	2	no	—
153	C₈	#10587485	1	2	yes	—
154	C₈	C₂×C₈×D₁₇	1	1	yes	characteristic
155	C₈	C₂×#63218	1	1	yes	characteristic
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	#11762559	1	2	yes	—
159	C₂²	C₂×#10587479	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	#11762559	1	2	yes	—
164	C₄	#11762633	1	2	yes	—
165	C₄	C₂×#10587485	1	1	yes	characteristic
166	C₄	C₂×#10587493	1	1	yes	characteristic
167	C₂	—	2	2	no	—
168	C₂	C₂×#11762559	1	1	yes	characteristic
169	C₂	C₂×#11762633	1	1	yes	characteristic
170	C₂	#14530556	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10587479
commutator	C₈×C₁₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₇
Fitting	C₁₇×#1564	C₂
socle	C₂²×C₁₇	C₂×#106
prime radical	C₂³×C₁₇	#106
cosocle	C₄×C₈×C₁₇	C₂³
hypercenter	#1564	D₁₇
hypocenter	C₁₇	#12000
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #14526849	C₂	semidirect
28. #10587483	C₈	semidirect
51. #63214	C₂×C₈	semidirect
118. C₁₇	#12000	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10587483	2
C₂×#11762560	2
#14526849	2
C₁₇×#1564	2
#12000	17