Subgroups — #16952952

Order 4352.

342 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763211	C₂	1	1	yes	maximal, characteristic
2	C₄×#10587483	C₂	1	1	yes	maximal, characteristic
3	#14526894	C₂	1	4	yes	maximal
4	C₁₇×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587556	—	2	1	no	—
6	C₂×#10587483	C₂²	1	2	yes	—
7	#11763211	—	2	1	no	—
8	#11763211	—	2	1	no	—
9	#11763211	C₄	1	2	yes	—
10	#11763211	C₄	1	2	yes	—
11	C₂×#10587489	C₂²	1	2	yes	—
12	C₂×#10587488	—	2	1	no	—
13	#11762585	—	2	1	no	—
14	C₄×#63214	—	2	1	no	—
15	C₂×C₁₇×#108	—	2	1	no	—
16	C₂×C₁₇×#108	C₂²	1	2	yes	—
17	C₁₇×#423	—	2	1	no	—
18	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₁₇	—	2	2	no	—
20	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₇	—	2	2	no	—
24	C₂×#63225	—	4	2	no	—
25	#10587556	—	2	2	no	—
26	#10587556	—	2	2	no	—
27	#10587556	—	4	2	no	—
28	C₁₇×#108	—	2	4	no	—
29	C₁₇×#108	—	4	2	no	—
30	C₁₇×C₄⋊C₈	—	4	2	no	—
31	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
34	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
35	#10587489	—	2	2	no	—
36	#10587489	C₂×C₄	1	4	yes	—
37	#10587488	—	2	2	no	—
38	#10587488	—	2	2	no	—
39	#10587483	—	2	2	no	—
40	#10587483	Q₈	1	4	yes	—
41	C₂×#63214	—	2	2	no	—
42	C₂²×C₄×C₁₇	SD₁₆	1	2	yes	—
43	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₁₇	—	4	2	no	—
45	M₁₆×C₁₇	—	4	2	no	—
46	M₁₆×C₁₇	—	4	2	no	—
47	C₂×C₈×C₁₇	—	4	2	no	—
48	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
51	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
52	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₁₇	#41	1	2	yes	—
56	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₄²×C₁₇	D₈	1	2	yes	—
59	C₄⋊Dic₁₇	—	2	4	no	—
60	C₄⋊Dic₁₇	—	4	2	no	—
61	#63225	—	4	4	no	—
62	#63225	—	4	4	no	—
63	#63214	—	2	4	no	—
64	#13289	—	17	1	no	maximal, Sylow
65	C₂³×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₁₇	—	4	2	no	—
67	C₂×C₄×C₁₇	#102	1	2	yes	—
68	C₂×C₄×C₁₇	#102	1	2	yes	—
69	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×Dic₁₇	—	4	4	no	—
76	C₂×Dic₁₇	—	8	4	no	—
77	C₈×C₁₇	—	8	2	no	—
78	C₈×C₁₇	#103	1	2	yes	—
79	C₈×C₁₇	#103	1	2	yes	—
80	C₈×C₁₇	C₄×Q₈	1	2	yes	—
81	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
82	C₂×#499	—	17	1	no	—
83	C₄×M₃₂	—	17	1	no	—
84	#1848	D₁₇	1	1	yes	characteristic
85	#1387	—	17	4	no	—
86	C₂²×C₁₇	—	2	2	no	—
87	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
88	C₂²×C₁₇	#487	1	1	yes	characteristic
89	C₂²×C₁₇	#498	1	1	yes	characteristic
90	C₄×C₁₇	—	2	2	no	—
91	C₄×C₁₇	—	2	2	no	—
92	C₄×C₁₇	#391	1	1	yes	characteristic
93	C₄×C₁₇	C₂×#103	1	1	yes	characteristic
94	C₄×C₁₇	#486	1	1	yes	characteristic
95	C₄×C₁₇	#500	1	1	yes	characteristic
96	Dic₁₇	—	8	4	no	—
97	C₂×#125	—	34	1	no	—
98	C₂×M₃₂	—	17	2	no	—
99	#499	—	17	2	no	—
100	#499	—	17	2	no	—
101	#499	—	34	1	no	—
102	#499	—	34	1	no	—
103	C₂×#108	—	2	1	no	—
104	C₂×#108	C₂×D₁₇	1	2	yes	—
105	C₂×#107	—	17	2	no	—
106	C₂×#106	—	34	1	no	—
107	#423	—	2	1	no	—
108	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
109	#346	—	34	1	no	—
110	C₄×C₁₆	—	34	1	no	—
111	C₂×C₁₇	—	2	1	no	—
112	C₂×C₁₇	—	2	1	no	—
113	C₂×C₁₇	#1990	1	1	yes	characteristic
114	C₂×C₁₇	#2148	1	2	yes	—
115	C₂×M₁₆	—	2	2	no	—
116	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
117	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
118	#125	—	34	2	no	—
119	#125	—	34	2	no	—
120	#125	—	68	2	no	—
121	C₂×C₄⋊C₄	—	34	2	no	—
122	C₂×C₄⋊C₄	—	68	2	no	—
123	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
124	M₃₂	—	17	4	no	—
125	M₃₂	—	34	2	no	—
126	C₂×C₁₆	—	34	2	no	—
127	#108	—	2	4	no	—
128	#108	—	4	2	no	—
129	#107	—	17	4	no	—
130	#107	—	34	2	no	—
131	#106	—	34	2	no	—
132	#106	—	34	2	no	—
133	C₄⋊C₈	—	4	2	no	—
134	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
135	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
136	C₄×C₈	D₆₈	1	1	yes	characteristic
137	C₄×C₈	D₆₈	1	1	yes	characteristic
138	C₁₇	#13289	1	1	yes	characteristic, complemented, Sylow
139	C₂²×C₄	—	68	2	no	—
140	C₂²×C₄	#63230	1	2	yes	—
141	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	#63227	1	1	yes	characteristic
146	C₂×C₈	#63227	1	1	yes	characteristic
147	C₂×C₈	#63229	1	2	yes	—
148	C₂×C₈	#63231	1	2	yes	—
149	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
150	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
151	C₂×C₈	#63254	1	2	yes	—
152	C₂×C₈	Q₈×D₁₇	1	2	yes	—
153	C₄⋊C₄	—	34	4	no	—
154	C₄⋊C₄	—	68	4	no	—
155	C₄⋊C₄	—	68	4	no	—
156	C₄⋊C₄	—	68	2	no	—
157	C₄²	#63227	1	1	yes	characteristic
158	C₄²	#63227	1	1	yes	characteristic
159	C₄²	#63228	1	2	yes	—
160	C₁₆	—	34	4	no	—
161	C₂³	C₂×#63230	1	1	yes	characteristic
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	68	4	no	—
164	C₂×C₄	—	136	4	no	—
165	C₂×C₄	#10587480	1	2	yes	—
166	C₂×C₄	#10587480	1	2	yes	—
167	C₂×C₄	C₂×#63227	1	1	yes	characteristic
168	C₂×C₄	C₂×#63228	1	1	yes	characteristic
169	C₂×C₄	C₂×#63229	1	1	yes	characteristic
170	C₂×C₄	#10587622	1	1	yes	characteristic
171	C₂×C₄	C₂×#63231	1	1	yes	characteristic
172	C₂×C₄	#10587630	1	1	yes	characteristic
173	C₈	—	8	2	no	—
174	C₈	#10587481	1	2	yes	—
175	C₈	#10587481	1	2	yes	—
176	C₈	#10587573	1	2	yes	—
177	C₈	#10587583	1	2	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	C₂×#10587480	1	1	yes	characteristic
180	C₂²	#11763159	1	1	yes	characteristic
181	C₂²	#11763205	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	136	4	no	—
185	C₄	#11762714	1	1	yes	characteristic
186	C₄	C₂×#10587481	1	1	yes	characteristic
187	C₄	#11763155	1	1	yes	characteristic
188	C₄	#11763212	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	#14529744	1	1	yes	characteristic
192	C₂	#14530531	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#10587480
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1848	C₂
socle	C₂²×C₁₇	C₂×#102
prime radical	C₂³×C₁₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1848	D₁₇
hypocenter	C₁₇	#13289
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
138. C₁₇	#13289	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763211	2
C₄×#10587483	2
#14526894	2
C₁₇×#1848	2
#13289	17