Subgroups — #16952950

Order 4352.

354 conjugacy classes of subgroups.

The subgroup lattice has 200 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	#14528948	C₂	1	1	yes	maximal, characteristic
3	#14526889	C₂	1	4	yes	maximal
4	C₁₇×#3160	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587483	C₂²	1	2	yes	—
6	C₂×#10587489	—	2	1	no	—
7	C₂×#10587488	C₂²	1	2	yes	—
8	#11762675	—	2	1	no	—
9	#11762585	—	2	1	no	—
10	C₄×#63214	—	2	1	no	—
11	C₁₇×#533	—	2	1	no	—
12	C₁₇×#500	C₄	1	2	yes	complemented
13	C₁₇×#499	—	2	1	no	—
14	C₁₇×#499	C₄	1	2	yes	complemented
15	C₁₇×#498	—	2	1	no	—
16	C₁₇×#427	—	2	1	no	—
17	C₁₇×#427	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
21	C₁₇×C₄⋊Q₈	—	2	2	no	—
22	C₁₇×#125	—	2	2	no	—
23	C₁₇×C₂²⋊Q₈	—	4	2	no	—
24	C₄×Q₈×C₁₇	—	4	2	no	—
25	C₁₇×#117	—	2	2	no	—
26	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₇	—	2	2	no	—
28	C₁₇×#107	—	2	2	no	—
29	C₁₇×#107	C₂×C₄	1	4	yes	—
30	C₁₇×#106	—	2	2	no	—
31	C₁₇×#106	—	2	2	no	—
32	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
35	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
36	#10587502	—	4	2	no	—
37	#10587489	—	4	2	no	—
38	#10587488	—	2	4	no	—
39	#10587483	—	2	2	no	—
40	#10587483	Q₈	1	4	yes	complemented
41	C₂×#63214	—	2	2	no	—
42	C₂×Q₈×C₁₇	—	4	2	no	—
43	C₂²×C₄×C₁₇	D₈	1	2	yes	—
44	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₁₇	—	4	2	no	—
46	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
49	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
50	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₁₇	#41	1	2	yes	—
54	C₄⋊C₄×C₁₇	—	2	4	no	—
55	C₄⋊C₄×C₁₇	—	4	4	no	—
56	C₄⋊C₄×C₁₇	—	4	2	no	—
57	C₂²⋊C₄×C₁₇	—	4	2	no	—
58	C₄²×C₁₇	—	4	2	no	—
59	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₁₇	SD₁₆	1	2	yes	—
62	C₄⋊Dic₁₇	—	4	2	no	—
63	C₄⋊Dic₁₇	—	4	2	no	—
64	#63214	—	2	4	no	—
65	#13288	—	17	1	no	maximal, Sylow
66	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
67	Q₈×C₁₇	—	8	4	no	—
68	C₂×C₄×C₁₇	—	4	2	no	—
69	C₂×C₄×C₁₇	—	4	4	no	—
70	C₂×C₄×C₁₇	#102	1	2	yes	—
71	C₂×C₄×C₁₇	#102	1	2	yes	—
72	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×Dic₁₇	—	8	4	no	—
79	C₂×Dic₁₇	—	8	2	no	—
80	C₈×C₁₇	#103	1	2	yes	—
81	C₈×C₁₇	#103	1	2	yes	—
82	C₈×C₁₇	C₄×Q₈	1	2	yes	—
83	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
84	#3160	D₁₇	1	1	yes	characteristic
85	C₄×M₃₂	—	17	1	no	—
86	#1846	—	17	1	no	—
87	#1386	—	17	4	no	—
88	C₂²×C₁₇	—	2	2	no	—
89	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
90	C₂²×C₁₇	#486	1	1	yes	characteristic
91	C₂²×C₁₇	#500	1	1	yes	characteristic
92	C₄×C₁₇	—	2	2	no	—
93	C₄×C₁₇	—	2	2	no	—
94	C₄×C₁₇	—	8	4	no	—
95	C₄×C₁₇	#391	1	1	yes	characteristic
96	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
97	C₄×C₁₇	#487	1	1	yes	characteristic
98	C₄×C₁₇	#498	1	1	yes	characteristic
99	Dic₁₇	—	16	2	no	—
100	#533	—	2	1	no	—
101	C₂×M₃₂	—	17	2	no	—
102	#500	Dic₁₇	1	2	yes	complemented
103	#499	—	2	1	no	—
104	#499	Dic₁₇	1	2	yes	complemented
105	#498	—	2	1	no	—
106	#427	—	2	1	no	—
107	#427	C₂×D₁₇	1	2	yes	—
108	C₂×#107	—	34	1	no	—
109	C₂×#106	—	17	2	no	—
110	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
111	#382	—	34	1	no	—
112	#346	—	34	1	no	—
113	C₄×C₁₆	—	34	1	no	—
114	C₂×C₁₇	—	2	1	no	—
115	C₂×C₁₇	—	2	1	no	—
116	C₂×C₁₇	#1990	1	1	yes	characteristic
117	C₂×C₁₇	#2145	1	2	yes	—
118	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
120	C₄⋊Q₈	—	2	2	no	—
121	#125	—	2	2	no	—
122	C₂²⋊Q₈	—	4	2	no	—
123	C₄×Q₈	—	4	2	no	—
124	#117	—	2	2	no	—
125	C₂×C₄⋊C₄	—	34	2	no	—
126	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
127	M₃₂	—	17	4	no	—
128	M₃₂	—	34	2	no	—
129	C₂×C₁₆	—	34	2	no	—
130	#107	—	2	2	no	—
131	#107	—	68	2	no	—
132	#107	C₂×Dic₁₇	1	4	yes	—
133	#106	—	2	2	no	—
134	#106	—	2	2	no	—
135	#106	—	34	4	no	—
136	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
137	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
140	#95	—	68	2	no	—
141	C₁₇	#13288	1	1	yes	characteristic, complemented, Sylow
142	C₂×Q₈	—	4	2	no	—
143	C₂²×C₄	—	68	2	no	—
144	C₂²×C₄	#63228	1	2	yes	—
145	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
146	C₂×C₈	#63230	1	2	yes	—
147	C₂×C₈	#63231	1	2	yes	—
148	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
149	C₂×C₈	C₂²⋊Dic₁₇	1	1	yes	characteristic
150	C₂×C₈	Q₈×D₁₇	1	2	yes	—
151	C₂×C₈	#63256	1	2	yes	—
152	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
153	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	68	2	no	—
158	C₄⋊C₄	—	68	2	no	—
159	C₂²⋊C₄	—	4	2	no	—
160	C₄²	—	4	2	no	—
161	C₄²	#63229	1	2	yes	—
162	C₄²	C₂²⋊Dic₁₇	1	1	yes	characteristic
163	C₄²	C₂²⋊Dic₁₇	1	1	yes	characteristic
164	C₁₆	—	34	4	no	—
165	C₂³	C₂×#63228	1	1	yes	characteristic
166	Q₈	—	8	4	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	136	4	no	—
170	C₂×C₄	—	136	2	no	—
171	C₂×C₄	#10587503	1	2	yes	—
172	C₂×C₄	#10587503	1	2	yes	—
173	C₂×C₄	C₂×#63229	1	1	yes	characteristic
174	C₂×C₄	C₂×#63230	1	1	yes	characteristic
175	C₂×C₄	C₂×#63231	1	1	yes	characteristic
176	C₂×C₄	#10587630	1	1	yes	characteristic
177	C₂×C₄	#10587633	1	1	yes	characteristic
178	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
179	C₈	#10587506	1	2	yes	—
180	C₈	#10587506	1	2	yes	—
181	C₈	Q₈×Dic₁₇	1	2	yes	—
182	C₈	#10587632	1	2	yes	—
183	C₂²	—	2	2	no	—
184	C₂²	C₂×#10587503	1	1	yes	characteristic
185	C₂²	#11763157	1	1	yes	characteristic
186	C₂²	#11763212	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	2	2	no	—
189	C₄	—	8	4	no	—
190	C₄	—	272	2	no	—
191	C₄	#11762716	1	1	yes	characteristic
192	C₄	C₂×#10587506	1	1	yes	characteristic
193	C₄	#11763161	1	1	yes	characteristic
194	C₄	#11763205	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	#14529746	1	1	yes	characteristic
198	C₂	#14530520	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#10587503
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#3160	C₂
socle	C₂²×C₁₇	C₂×#102
prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#3160	D₁₇
hypocenter	C₁₇	#13288
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
12. C₁₇×#500	C₄	semidirect
14. C₁₇×#499	C₄	semidirect
40. #10587483	Q₈	semidirect
102. #500	Dic₁₇	semidirect
104. #499	Dic₁₇	semidirect
141. C₁₇	#13288	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
#14528948	2
#14526889	2
C₁₇×#3160	2
#13288	17