Subgroups — #16950492

Order 4352.

438 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×#11762635	C₂	1	2	yes	maximal, complemented
2	#14528813	C₂	1	2	yes	maximal
3	#14526876	C₂	1	2	yes	maximal, complemented
4	C₂³×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587489	C₂²	1	4	yes	—
6	#11762669	—	2	2	no	—
7	#11762635	—	2	4	no	—
8	#11762572	—	2	2	no	—
9	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₁₇	—	2	4	no	—
11	C₂²×C₁₆×C₁₇	C₄	1	4	yes	complemented
12	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₇	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₇	—	2	2	no	—
16	C₂²×C₈×C₁₇	—	2	2	no	—
17	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₇	Q₈	1	2	yes	—
20	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
21	C₂×C₂²⋊Dic₁₇	—	4	2	no	—
22	C₂×C₄⋊Dic₁₇	—	2	4	no	—
23	C₂×C₁₆×C₁₇	—	2	8	no	—
24	C₂×C₁₆×C₁₇	—	2	2	no	—
25	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
26	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₁₇	D₄	1	4	yes	—
28	#10587502	—	4	2	no	—
29	#10587489	—	2	4	no	—
30	#10587489	—	2	4	no	—
31	C₂⁴×C₁₇	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₁₇	—	2	2	no	—
33	C₂²×C₄×C₁₇	—	2	2	no	—
34	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₇	Q₁₆	1	2	yes	—
37	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₁₇	—	4	4	no	—
39	C₂×C₈×C₁₇	—	2	2	no	—
40	C₂×C₈×C₁₇	—	2	2	no	—
41	C₂×C₈×C₁₇	—	2	2	no	—
42	C₂×C₈×C₁₇	—	2	4	no	—
43	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₁₇	#41	1	2	yes	—
48	C₂²⋊Dic₁₇	—	8	4	no	—
49	C₄⋊Dic₁₇	—	4	4	no	—
50	C₄⋊Dic₁₇	—	4	4	no	—
51	C₁₆×C₁₇	—	2	2	no	—
52	C₁₆×C₁₇	C₂²⋊C₄	1	4	yes	complemented
53	#12834	—	17	1	no	maximal, Sylow
54	C₂³×C₁₇	—	2	2	no	—
55	C₂³×C₁₇	—	2	2	no	—
56	C₂³×C₁₇	#107	1	2	yes	—
57	C₂³×C₁₇	D₁₆	1	2	yes	—
58	C₂³×C₁₇	Q₃₂	1	2	yes	—
59	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₁₇	—	2	2	no	—
61	C₂×C₄×C₁₇	—	2	2	no	—
62	C₂×C₄×C₁₇	—	2	2	no	—
63	C₂×C₄×C₁₇	—	2	4	no	—
64	C₂×C₄×C₁₇	#107	1	2	yes	—
65	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₇	#121	1	2	yes	—
67	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×C₄×C₁₇	#135	1	2	yes	—
69	C₂×Dic₁₇	—	8	4	no	—
70	C₂×Dic₁₇	—	8	4	no	—
71	C₂×Dic₁₇	—	8	4	no	—
72	C₈×C₁₇	—	2	2	no	—
73	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₈×C₁₇	#117	1	1	yes	characteristic
75	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
76	C₂³×C₁₆	D₁₇	1	1	yes	characteristic
77	C₂×#366	—	17	2	no	—
78	#1808	—	17	2	no	—
79	#1380	—	17	2	no	—
80	C₂²×C₁₇	—	2	2	no	—
81	C₂²×C₁₇	—	2	2	no	—
82	C₂²×C₁₇	—	2	4	no	—
83	C₂²×C₁₇	—	2	4	no	—
84	C₂²×C₁₇	#366	1	4	yes	complemented
85	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
86	C₂²×C₁₇	#466	1	2	yes	—
87	C₂²×C₁₇	C₂×D₁₆	1	1	yes	characteristic
88	C₂²×C₁₇	C₂×Q₃₂	1	1	yes	characteristic
89	C₂²×C₁₇	#508	1	2	yes	—
90	C₄×C₁₇	—	2	2	no	—
91	C₄×C₁₇	#380	1	1	yes	characteristic
92	C₄×C₁₇	#427	1	1	yes	characteristic
93	C₄×C₁₇	#467	1	2	yes	—
94	Dic₁₇	—	16	4	no	—
95	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
96	C₂²×C₁₆	—	2	4	no	—
97	C₂²×C₁₆	Dic₁₇	1	4	yes	complemented
98	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
99	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
100	C₂×#107	—	17	4	no	—
101	#380	—	34	2	no	—
102	#366	—	34	4	no	—
103	#340	—	34	2	no	—
104	C₂×C₁₇	—	2	4	no	—
105	C₂×C₁₇	#1808	1	1	yes	characteristic
106	C₂×C₁₇	C₂×#366	1	1	yes	characteristic
107	C₂×C₁₇	#2158	1	1	yes	characteristic
108	C₂×C₁₇	#2215	1	2	yes	—
109	C₂×C₁₇	#2216	1	2	yes	—
110	C₂³×C₄	D₆₈	1	1	yes	characteristic
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	Dic₃₄	1	2	yes	—
114	C₂²×C₈	D₆₈	1	1	yes	characteristic
115	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
116	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	34	4	no	—
118	C₂×C₂²⋊C₄	—	68	2	no	—
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	2	8	no	—
121	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
122	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
123	C₂×C₁₆	C₂²⋊D₁₇	1	4	yes	—
124	#107	—	34	4	no	—
125	#107	—	34	4	no	—
126	#95	—	68	2	no	—
127	C₁₇	#12834	1	1	yes	characteristic, complemented, Sylow
128	C₂⁴	D₁₃₆	1	1	yes	characteristic
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	68	4	no	—
132	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
133	C₂²×C₄	Dic₆₈	1	2	yes	—
134	C₂²×C₄	C₄⋊Dic₁₇	1	2	yes	—
135	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
141	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
142	C₂×C₈	#63252	1	2	yes	—
143	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
144	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
145	C₄⋊C₄	—	68	4	no	—
146	C₄⋊C₄	—	68	4	no	—
147	C₂²⋊C₄	—	136	4	no	—
148	C₁₆	—	2	2	no	—
149	C₁₆	C₂²⋊Dic₁₇	1	4	yes	complemented
150	C₂³	—	2	2	no	—
151	C₂³	—	2	2	no	—
152	C₂³	D₂₇₂	1	2	yes	—
153	C₂³	Dic₁₃₆	1	2	yes	—
154	C₂³	#10587489	1	2	yes	—
155	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	136	4	no	—
161	C₂×C₄	—	136	4	no	—
162	C₂×C₄	—	136	4	no	—
163	C₂×C₄	#10587489	1	2	yes	—
164	C₂×C₄	#10587590	1	2	yes	—
165	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
166	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
167	C₂×C₄	#10587616	1	2	yes	—
168	C₈	—	2	2	no	—
169	C₈	#10587610	1	2	yes	—
170	C₈	#10587612	1	1	yes	characteristic
171	C₈	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
172	C₂²	—	2	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	2	4	no	—
175	C₂²	—	2	4	no	—
176	C₂²	#11762635	1	4	yes	complemented
177	C₂²	C₂×#10587489	1	1	yes	characteristic
178	C₂²	#11763045	1	2	yes	—
179	C₂²	C₂×D₂₇₂	1	1	yes	characteristic
180	C₂²	C₂×Dic₁₃₆	1	1	yes	characteristic
181	C₂²	#11763241	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	272	4	no	—
184	C₄	#11762669	1	1	yes	characteristic
185	C₄	#11762847	1	1	yes	characteristic
186	C₄	#11763050	1	2	yes	—
187	C₂	—	2	4	no	—
188	C₂	#14528813	1	1	yes	characteristic
189	C₂	C₂×#11762635	1	1	yes	characteristic
190	C₂	#14530568	1	1	yes	characteristic
191	C₂	#14530870	1	2	yes	—
192	C₂	#14530875	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₁₃₆
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂³×C₁₆×C₁₇	C₂
socle	C₂³×C₁₇	C₂×D₈
prime radical	C₂⁴×C₁₇	D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂³×C₁₆	D₁₇
hypocenter	C₁₇	#12834
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#11762635	C₂	semidirect
3. #14526876	C₂	semidirect
11. C₂²×C₁₆×C₁₇	C₄	semidirect
52. C₁₆×C₁₇	C₂²⋊C₄	semidirect
84. C₂²×C₁₇	#366	semidirect
97. C₂²×C₁₆	Dic₁₇	semidirect
127. C₁₇	#12834	semidirect
149. C₁₆	C₂²⋊Dic₁₇	semidirect
176. C₂²	#11762635	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762635	2
#14528813	2
#14526876	2
C₂³×C₁₆×C₁₇	2
#12834	17