Subgroups — #16949331

Order 4352.

338 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	M₃₂×Dic₁₇	C₂	1	2	yes	maximal
2	#14528472	C₂	1	1	yes	maximal, characteristic
3	#14526864	C₂	1	1	yes	maximal, characteristic
4	#14526863	C₂	1	2	yes	maximal
5	C₁₇×#1375	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587483	C₂²	1	2	yes	—
7	C₂×#10587486	C₄	1	1	yes	characteristic
8	C₂×#10587486	C₂²	1	1	yes	characteristic
9	C₂×C₄×#3622	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
12	#11762655	C₄	1	1	yes	characteristic
13	#11762584	C₄	1	4	yes	complemented
14	C₁₆×Dic₁₇	C₄	1	4	yes	complemented
15	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
16	C₂×C₁₇×#97	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
23	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
24	C₁₇×M₃₂	—	2	2	no	—
25	C₁₇×M₃₂	—	2	2	no	—
26	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
27	C₁₇×#97	—	2	2	no	—
28	#10587502	—	2	2	no	—
29	#10587486	—	2	2	no	—
30	#10587486	—	2	1	no	—
31	#10587486	—	2	1	no	—
32	C₈×Dic₁₇	C₈	1	4	yes	—
33	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
34	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
35	#10587483	—	2	2	no	—
36	#10587483	—	2	2	no	—
37	C₂×#63214	C₂×C₄	1	4	yes	—
38	C₄×#3622	C₈	1	4	yes	—
39	C₂²×C₄×C₁₇	—	2	1	no	—
40	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₁₇	—	2	1	no	—
42	C₂²×Dic₁₇	M₁₆	1	2	yes	—
43	C₂×C₈×C₁₇	—	2	1	no	—
44	C₂×C₈×C₁₇	—	2	1	no	—
45	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
47	C₂×C₈×C₁₇	#41	1	2	yes	—
48	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
49	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
50	C₄²×C₁₇	—	2	1	no	—
51	C₄²×C₁₇	C₂×C₈	1	2	yes	—
52	C₄×Dic₁₇	—	2	1	no	—
53	C₄×Dic₁₇	C₄²	1	2	yes	—
54	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
55	C₄×Dic₁₇	M₁₆	1	2	yes	—
56	C₂×#3622	—	2	1	no	—
57	C₂×#3622	—	2	1	no	—
58	C₂×#3622	C₄²	1	1	yes	characteristic
59	C₂×#3622	C₄²	1	1	yes	characteristic
60	C₂×#3622	C₂×C₈	1	2	yes	—
61	C₂×#3622	C₂×C₈	1	2	yes	—
62	C₂×#3622	C₂×C₈	1	2	yes	—
63	C₁₆×C₁₇	—	2	4	no	—
64	#63214	—	2	4	no	—
65	#12593	—	17	1	no	maximal, Sylow
66	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
67	C₂×C₄×C₁₇	—	2	2	no	—
68	C₂×C₄×C₁₇	—	4	2	no	—
69	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
71	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
75	C₂×Dic₁₇	—	2	2	no	—
76	C₂×Dic₁₇	—	4	2	no	—
77	C₂×Dic₁₇	—	4	2	no	—
78	C₂×Dic₁₇	#97	1	4	yes	—
79	C₈×C₁₇	—	2	2	no	—
80	C₈×C₁₇	#117	1	2	yes	—
81	C₈×C₁₇	#117	1	2	yes	—
82	#3622	—	2	2	no	—
83	#3622	C₄×C₈	1	2	yes	—
84	#3622	C₄×C₈	1	2	yes	—
85	C₄×M₃₂	—	17	2	no	—
86	#1726	—	17	1	no	—
87	#1375	—	17	2	no	—
88	#1375	—	17	1	no	—
89	#1375	D₁₇	1	1	yes	characteristic
90	C₂²×C₁₇	—	2	2	no	—
91	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
92	C₂²×C₁₇	#431	1	1	yes	characteristic
93	C₂²×C₁₇	#431	1	1	yes	characteristic
94	C₄×C₁₇	—	4	2	no	—
95	C₄×C₁₇	#376	1	1	yes	characteristic
96	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
97	C₄×C₁₇	#431	1	1	yes	characteristic
98	C₄×C₁₇	#431	1	1	yes	characteristic
99	Dic₁₇	—	2	4	no	—
100	Dic₁₇	—	4	2	no	—
101	C₂×M₃₂	—	17	2	no	—
102	C₂×M₃₂	C₂×D₁₇	1	2	yes	—
103	C₂×#97	—	17	1	no	—
104	C₂×#97	—	17	1	no	—
105	C₂×#97	C₂×D₁₇	1	1	yes	characteristic
106	C₂×C₄×C₈	—	17	1	no	—
107	C₂×C₄×C₈	—	17	1	no	—
108	C₂×C₄×C₈	—	17	1	no	—
109	#376	—	17	1	no	—
110	#346	—	17	4	no	—
111	C₄×C₁₆	—	17	4	no	—
112	C₂×C₁₇	—	2	1	no	—
113	C₂×C₁₇	—	2	1	no	—
114	C₂×C₁₇	#1744	1	1	yes	characteristic
115	C₂×C₁₇	#2110	1	2	yes	—
116	C₂²×C₈	—	17	1	no	—
117	C₂²×C₈	—	17	1	no	—
118	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
120	C₂×C₄²	—	17	1	no	—
121	C₂×C₄²	—	17	1	no	—
122	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
123	M₃₂	—	2	2	no	—
124	M₃₂	—	2	2	no	—
125	M₃₂	—	34	2	no	—
126	M₃₂	—	34	2	no	—
127	C₂×C₁₆	—	17	4	no	—
128	C₂×C₁₆	C₄×D₁₇	1	4	yes	—
129	#97	—	2	2	no	—
130	#97	—	34	2	no	—
131	#97	—	34	1	no	—
132	#97	—	34	1	no	—
133	C₄×C₈	—	17	2	no	—
134	C₄×C₈	—	17	2	no	—
135	C₄×C₈	—	17	4	no	—
136	C₄×C₈	—	17	4	no	—
137	#95	—	34	2	no	—
138	C₁₇	#12593	1	1	yes	characteristic, complemented, Sylow
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	17	2	no	—
141	C₂²×C₄	—	34	1	no	—
142	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	17	1	no	—
146	C₂×C₈	—	17	1	no	—
147	C₂×C₈	—	17	2	no	—
148	C₂×C₈	—	17	2	no	—
149	C₂×C₈	—	17	2	no	—
150	C₂×C₈	—	34	1	no	—
151	C₂×C₈	—	34	1	no	—
152	C₂×C₈	C₈×D₁₇	1	2	yes	—
153	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
154	C₂×C₈	#63254	1	1	yes	characteristic
155	C₂×C₈	#63254	1	2	yes	—
156	C₂×C₈	#63256	1	1	yes	characteristic
157	C₄²	—	2	1	no	—
158	C₄²	—	17	2	no	—
159	C₄²	—	17	2	no	—
160	C₄²	—	17	2	no	—
161	C₄²	—	34	1	no	—
162	C₄²	C₈×D₁₇	1	2	yes	—
163	C₁₆	—	2	4	no	—
164	C₁₆	—	34	4	no	—
165	C₂³	M₁₆×D₁₇	1	1	yes	characteristic
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	17	4	no	—
169	C₂×C₄	—	34	2	no	—
170	C₂×C₄	—	68	2	no	—
171	C₂×C₄	—	68	2	no	—
172	C₂×C₄	C₄²×D₁₇	1	1	yes	characteristic
173	C₂×C₄	#10587563	1	1	yes	characteristic
174	C₂×C₄	#10587578	1	1	yes	characteristic
175	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
176	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
177	C₂×C₄	M₁₆×D₁₇	1	1	yes	characteristic
178	C₈	—	2	2	no	—
179	C₈	—	17	2	no	—
180	C₈	—	17	2	no	—
181	C₈	—	34	2	no	—
182	C₈	#10587563	1	2	yes	—
183	C₈	#10587578	1	2	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	#97×D₁₇	1	1	yes	characteristic
186	C₂²	#11762859	1	1	yes	characteristic
187	C₂²	#11762861	1	1	yes	characteristic
188	C₄	—	4	2	no	—
189	C₄	—	34	4	no	—
190	C₄	—	68	2	no	—
191	C₄	#11762654	1	1	yes	characteristic
192	C₄	C₄×C₈×D₁₇	1	1	yes	characteristic
193	C₄	#11762859	1	1	yes	characteristic
194	C₄	#11762861	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	#14528528	1	1	yes	characteristic
198	C₂	#14530387	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄²×D₁₇
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1375	C₂
socle	C₂²×C₁₇	C₂×#97
prime radical	C₂³×C₁₇	C₂×M₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1375	D₁₇
hypocenter	C₁₇	#12593
LCS	G ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #1375

Complements

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

N	H ≅ G/N	type
13. #11762584	C₄	semidirect
14. C₁₆×Dic₁₇	C₄	semidirect
138. C₁₇	#12593	semidirect

Maximal subgroups

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

subgroup	index
M₃₂×Dic₁₇	2
#14528472	2
#14526864	2
#14526863	2
C₁₇×#1375	2
#12593	17