Subgroups — #16950496

Order 4352.

438 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762637	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	#11762637	—	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	#12835	—	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	SD₃₂	1	4	yes	—
58	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₁₇	—	2	2	no	—
60	C₂×C₄×C₁₇	—	2	2	no	—
61	C₂×C₄×C₁₇	—	2	2	no	—
62	C₂×C₄×C₁₇	—	2	4	no	—
63	C₂×C₄×C₁₇	#107	1	2	yes	—
64	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₇	#121	1	2	yes	—
66	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×C₄×C₁₇	#135	1	2	yes	—
68	C₂×Dic₁₇	—	8	4	no	—
69	C₂×Dic₁₇	—	8	4	no	—
70	C₂×Dic₁₇	—	8	4	no	—
71	C₈×C₁₇	—	2	2	no	—
72	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₈×C₁₇	#117	1	1	yes	characteristic
74	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
75	C₂³×C₁₆	D₁₇	1	1	yes	characteristic
76	C₂×#367	—	17	2	no	—
77	#1808	—	17	2	no	—
78	#1380	—	17	2	no	—
79	C₂²×C₁₇	—	2	2	no	—
80	C₂²×C₁₇	—	2	2	no	—
81	C₂²×C₁₇	—	2	8	no	—
82	C₂²×C₁₇	#367	1	4	yes	complemented
83	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
84	C₂²×C₁₇	#466	1	2	yes	—
85	C₂²×C₁₇	C₂×SD₃₂	1	2	yes	—
86	C₂²×C₁₇	#508	1	2	yes	—
87	C₄×C₁₇	—	2	2	no	—
88	C₄×C₁₇	#380	1	1	yes	characteristic
89	C₄×C₁₇	#427	1	1	yes	characteristic
90	C₄×C₁₇	#467	1	2	yes	—
91	Dic₁₇	—	16	4	no	—
92	C₂³×C₈	C₂×D₁₇	1	1	yes	characteristic
93	C₂²×C₁₆	—	2	4	no	—
94	C₂²×C₁₆	Dic₁₇	1	4	yes	complemented
95	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
96	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
97	C₂×#107	—	17	4	no	—
98	#380	—	34	2	no	—
99	#367	—	34	4	no	—
100	#340	—	34	2	no	—
101	C₂×C₁₇	—	2	4	no	—
102	C₂×C₁₇	#1808	1	1	yes	characteristic
103	C₂×C₁₇	C₂×#367	1	1	yes	characteristic
104	C₂×C₁₇	#2158	1	1	yes	characteristic
105	C₂×C₁₇	#2217	1	4	yes	—
106	C₂³×C₄	D₆₈	1	1	yes	characteristic
107	C₂²×C₈	—	2	2	no	—
108	C₂²×C₈	—	2	2	no	—
109	C₂²×C₈	Dic₃₄	1	2	yes	—
110	C₂²×C₈	D₆₈	1	1	yes	characteristic
111	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
112	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
113	C₂×C₄⋊C₄	—	34	4	no	—
114	C₂×C₂²⋊C₄	—	68	2	no	—
115	C₂×C₁₆	—	2	2	no	—
116	C₂×C₁₆	—	2	8	no	—
117	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
118	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
119	C₂×C₁₆	C₂²⋊D₁₇	1	4	yes	—
120	#107	—	34	4	no	—
121	#107	—	34	4	no	—
122	#95	—	68	2	no	—
123	C₁₇	#12835	1	1	yes	characteristic, complemented, Sylow
124	C₂⁴	D₁₃₆	1	1	yes	characteristic
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	—	68	4	no	—
128	C₂²×C₄	D₁₃₆	1	1	yes	characteristic
129	C₂²×C₄	Dic₆₈	1	2	yes	—
130	C₂²×C₄	C₄⋊Dic₁₇	1	2	yes	—
131	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
137	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
138	C₂×C₈	#63252	1	2	yes	—
139	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
140	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
141	C₄⋊C₄	—	68	4	no	—
142	C₄⋊C₄	—	68	4	no	—
143	C₂²⋊C₄	—	136	4	no	—
144	C₁₆	—	2	2	no	—
145	C₁₆	C₂²⋊Dic₁₇	1	4	yes	complemented
146	C₂³	—	2	2	no	—
147	C₂³	—	2	2	no	—
148	C₂³	#10587471	1	4	yes	—
149	C₂³	#10587489	1	2	yes	—
150	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	136	4	no	—
156	C₂×C₄	—	136	4	no	—
157	C₂×C₄	—	136	4	no	—
158	C₂×C₄	#10587489	1	2	yes	—
159	C₂×C₄	#10587590	1	2	yes	—
160	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
161	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
162	C₂×C₄	#10587616	1	2	yes	—
163	C₈	—	2	2	no	—
164	C₈	#10587610	1	2	yes	—
165	C₈	#10587612	1	1	yes	characteristic
166	C₈	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
167	C₂²	—	2	2	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	8	no	—
170	C₂²	#11762637	1	4	yes	complemented
171	C₂²	C₂×#10587489	1	1	yes	characteristic
172	C₂²	#11763045	1	2	yes	—
173	C₂²	C₂×#10587471	1	2	yes	—
174	C₂²	#11763241	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	272	4	no	—
177	C₄	#11762669	1	1	yes	characteristic
178	C₄	#11762847	1	1	yes	characteristic
179	C₄	#11763050	1	2	yes	—
180	C₂	—	2	4	no	—
181	C₂	#14528813	1	1	yes	characteristic
182	C₂	C₂×#11762637	1	1	yes	characteristic
183	C₂	#14530568	1	1	yes	characteristic
184	C₂	#14530880	1	4	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	#12835
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×#11762637	C₂	semidirect
3. #14526876	C₂	semidirect
11. C₂²×C₁₆×C₁₇	C₄	semidirect
52. C₁₆×C₁₇	C₂²⋊C₄	semidirect
82. C₂²×C₁₇	#367	semidirect
94. C₂²×C₁₆	Dic₁₇	semidirect
123. C₁₇	#12835	semidirect
145. C₁₆	C₂²⋊Dic₁₇	semidirect
170. C₂²	#11762637	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762637	2
#14528813	2
#14526876	2
C₂³×C₁₆×C₁₇	2
#12835	17