Subgroups — #16950710

Order 4352.

318 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528948	C₂	1	1	yes	maximal, characteristic
2	#14528944	C₂	1	1	yes	maximal, characteristic
3	#14526876	C₂	1	4	yes	maximal
4	C₂×C₁₇×#346	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587489	—	2	1	no	—
6	C₂×#10587489	C₂²	1	2	yes	—
7	C₂×#10587489	C₂²	1	2	yes	—
8	C₂×#10587488	—	2	1	no	—
9	#11762679	—	2	1	no	—
10	#11762675	—	2	1	no	—
11	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₁₇×#346	C₄	1	4	yes	complemented
14	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₇	—	2	2	no	—
18	C₂×C₄⋊Dic₁₇	—	2	2	no	—
19	C₂×C₄⋊Dic₁₇	—	4	2	no	—
20	C₂×C₁₆×C₁₇	—	2	2	no	—
21	C₂×C₁₆×C₁₇	—	2	2	no	—
22	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
23	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₁₇	Q₈	1	2	yes	—
25	#10587502	—	4	2	no	—
26	#10587489	—	2	4	no	—
27	#10587489	—	2	4	no	—
28	#10587489	—	4	2	no	—
29	#10587488	—	4	2	no	—
30	C₂²×C₄×C₁₇	D₈	1	2	yes	—
31	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
32	C₂²×Dic₁₇	—	4	2	no	—
33	C₂²×Dic₁₇	—	4	2	no	—
34	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
36	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
37	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₁₇	#41	1	2	yes	—
40	C₂×C₈×C₁₇	#41	1	2	yes	—
41	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
42	C₄²×C₁₇	Q₁₆	1	2	yes	—
43	C₄⋊Dic₁₇	—	4	2	no	—
44	C₄⋊Dic₁₇	—	4	2	no	—
45	C₄⋊Dic₁₇	—	4	2	no	—
46	C₄⋊Dic₁₇	—	4	2	no	—
47	C₄⋊Dic₁₇	—	8	4	no	—
48	C₁₆×C₁₇	—	2	4	no	—
49	#12890	—	17	1	no	maximal, Sylow
50	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₁₇	—	2	2	no	—
52	C₂×C₄×C₁₇	—	2	2	no	—
53	C₂×C₄×C₁₇	#107	1	4	yes	—
54	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
56	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
57	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
59	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
60	C₂×Dic₁₇	—	8	2	no	—
61	C₂×Dic₁₇	—	8	2	no	—
62	C₂×Dic₁₇	—	8	2	no	—
63	C₂×Dic₁₇	—	8	4	no	—
64	C₂×Dic₁₇	—	8	2	no	—
65	C₈×C₁₇	#106	1	4	yes	—
66	C₈×C₁₇	#117	1	2	yes	—
67	C₈×C₁₇	#125	1	2	yes	—
68	C₂×#346	D₁₇	1	1	yes	characteristic
69	#1846	—	17	1	no	—
70	#1845	—	17	1	no	—
71	#1380	—	17	4	no	—
72	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
73	C₂²×C₁₇	#486	1	1	yes	characteristic
74	C₂²×C₁₇	#486	1	1	yes	characteristic
75	C₂²×C₁₇	#509	1	2	yes	—
76	C₂²×C₁₇	#510	1	2	yes	—
77	C₄×C₁₇	—	2	4	no	—
78	C₄×C₁₇	#382	1	1	yes	characteristic
79	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
80	C₄×C₁₇	#487	1	1	yes	characteristic
81	C₄×C₁₇	#487	1	1	yes	characteristic
82	Dic₁₇	—	16	2	no	—
83	Dic₁₇	—	16	2	no	—
84	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
85	C₂×#107	—	17	2	no	—
86	C₂×#107	—	17	2	no	—
87	C₂×#107	—	34	1	no	—
88	C₂×#106	—	34	1	no	—
89	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
90	#384	—	34	1	no	—
91	#382	—	34	1	no	—
92	#346	Dic₁₇	1	4	yes	complemented
93	C₂×C₁₇	#1834	1	1	yes	characteristic
94	C₂×C₁₇	#2159	1	2	yes	—
95	C₂×C₁₇	#2243	1	2	yes	—
96	C₂×C₁₇	#2244	1	1	yes	characteristic
97	C₂×C₁₇	#2245	1	1	yes	characteristic
98	C₂²×C₈	D₆₈	1	1	yes	characteristic
99	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
100	C₂×C₄⋊C₄	—	34	2	no	—
101	C₂×C₄⋊C₄	—	34	2	no	—
102	C₂×C₄⋊C₄	—	68	2	no	—
103	C₂×C₄²	D₆₈	1	1	yes	characteristic
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	C₂×Dic₁₇	1	4	yes	—
107	#107	—	34	4	no	—
108	#107	—	34	4	no	—
109	#107	—	68	2	no	—
110	#106	—	68	2	no	—
111	C₄×C₈	Dic₃₄	1	2	yes	—
112	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
113	#95	—	68	2	no	—
114	C₁₇	#12890	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₄	—	68	2	no	—
116	C₂²×C₄	—	68	2	no	—
117	C₂²×C₄	D₁₃₆	1	2	yes	—
118	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
119	C₂×C₈	#63219	1	2	yes	—
120	C₂×C₈	#63219	1	2	yes	—
121	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
122	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
123	C₂×C₈	#63252	1	2	yes	—
124	C₂×C₈	#63252	1	2	yes	—
125	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
126	C₄⋊C₄	—	68	2	no	—
127	C₄⋊C₄	—	68	2	no	—
128	C₄⋊C₄	—	68	2	no	—
129	C₄⋊C₄	—	68	2	no	—
130	C₄⋊C₄	—	136	4	no	—
131	C₄²	Dic₆₈	1	2	yes	—
132	C₄²	C₄⋊Dic₁₇	1	2	yes	—
133	C₁₆	—	2	4	no	—
134	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	136	4	no	—
138	C₂×C₄	—	136	2	no	—
139	C₂×C₄	—	136	2	no	—
140	C₂×C₄	—	136	2	no	—
141	C₂×C₄	—	136	2	no	—
142	C₂×C₄	#10587489	1	4	yes	—
143	C₂×C₄	#10587561	1	1	yes	characteristic
144	C₂×C₄	#10587561	1	1	yes	characteristic
145	C₂×C₄	C₂×#63219	1	1	yes	characteristic
146	C₂×C₄	C₂×#63219	1	1	yes	characteristic
147	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
148	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
149	C₈	#10587488	1	4	yes	—
150	C₈	#10587556	1	2	yes	—
151	C₈	#10587612	1	2	yes	—
152	C₂²	C₂×#10587489	1	1	yes	characteristic
153	C₂²	#11763156	1	1	yes	characteristic
154	C₂²	#11763156	1	1	yes	characteristic
155	C₂²	#11763247	1	2	yes	—
156	C₂²	#11763252	1	2	yes	—
157	C₄	—	2	4	no	—
158	C₄	—	272	2	no	—
159	C₄	—	272	2	no	—
160	C₄	#11762675	1	1	yes	characteristic
161	C₄	C₂×#10587488	1	1	yes	characteristic
162	C₄	#11763160	1	1	yes	characteristic
163	C₄	#11763160	1	1	yes	characteristic
164	C₂	#14528912	1	1	yes	characteristic
165	C₂	#14530572	1	2	yes	—
166	C₂	#14531021	1	2	yes	—
167	C₂	#14531026	1	1	yes	characteristic
168	C₂	#14531030	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₃₆
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#346	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#346	D₁₇
hypocenter	C₁₇	#12890
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
13. C₁₇×#346	C₄	semidirect
92. #346	Dic₁₇	semidirect
114. C₁₇	#12890	semidirect

Maximal subgroups

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

subgroup	index
#14528948	2
#14528944	2
#14526876	2
C₂×C₁₇×#346	2
#12890	17