Subgroups — #16950700

Order 4352.

286 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528956	C₂	1	1	yes	maximal, characteristic
2	#14528948	C₂	1	1	yes	maximal, characteristic
3	#14526883	C₂	1	4	yes	maximal
4	C₄×C₁₇×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587490	—	2	1	no	—
6	C₂×#10587490	C₂²	1	2	yes	—
7	C₂×#10587489	—	2	1	no	—
8	C₂×#10587488	C₂²	1	2	yes	—
9	#11762833	—	2	1	no	—
10	#11762675	—	2	1	no	—
11	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₁₇×#346	C₄	1	2	yes	complemented
14	C₄×C₁₆×C₁₇	C₄	1	2	yes	complemented
15	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₇	—	2	2	no	—
19	C₂×#63223	—	2	2	no	—
20	C₁₇×M₃₂	—	2	2	no	—
21	C₁₇×M₃₂	—	2	2	no	—
22	C₂×C₁₆×C₁₇	C₂×C₄	1	4	yes	—
23	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
26	C₄×C₈×C₁₇	Q₈	1	1	yes	characteristic
27	#10587502	—	4	2	no	—
28	#10587490	—	2	4	no	—
29	#10587490	—	4	2	no	—
30	#10587489	—	4	2	no	—
31	#10587488	—	2	4	no	—
32	C₄⋊#3622	—	4	2	no	—
33	C₂²×C₄×C₁₇	SD₁₆	1	2	yes	—
34	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×Dic₁₇	—	4	2	no	—
36	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₇	D₈	1	2	yes	—
39	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
40	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₁₇	#41	1	2	yes	—
43	C₂×C₈×C₁₇	#41	1	2	yes	—
44	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₁₇	SD₁₆	1	2	yes	—
47	C₄⋊Dic₁₇	—	4	2	no	—
48	C₄⋊Dic₁₇	—	4	2	no	—
49	#63223	—	4	2	no	—
50	#63223	—	4	2	no	—
51	C₂×#3622	—	4	2	no	—
52	C₁₆×C₁₇	—	2	4	no	—
53	#12887	—	17	1	no	maximal, Sylow
54	C₂³×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
55	C₂×C₄×C₁₇	—	4	2	no	—
56	C₂×C₄×C₁₇	#106	1	2	yes	—
57	C₂×C₄×C₁₇	#106	1	2	yes	—
58	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
60	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
61	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
64	C₂×Dic₁₇	—	8	4	no	—
65	C₂×Dic₁₇	—	8	2	no	—
66	C₈×C₁₇	#107	1	2	yes	—
67	C₈×C₁₇	#107	1	2	yes	—
68	C₈×C₁₇	#117	1	2	yes	—
69	C₈×C₁₇	#125	1	2	yes	—
70	#3622	—	8	2	no	—
71	C₄×M₃₂	D₁₇	1	1	yes	characteristic
72	#1848	—	17	1	no	—
73	#1846	—	17	1	no	—
74	#1383	—	17	4	no	—
75	C₂²×C₁₇	—	2	2	no	—
76	C₂²×C₁₇	C₂×#106	1	1	yes	characteristic
77	C₂²×C₁₇	#486	1	1	yes	characteristic
78	C₂²×C₁₇	#487	1	1	yes	characteristic
79	C₄×C₁₇	—	2	2	no	—
80	C₄×C₁₇	—	2	2	no	—
81	C₄×C₁₇	#382	1	1	yes	characteristic
82	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
83	C₄×C₁₇	#486	1	1	yes	characteristic
84	C₄×C₁₇	#487	1	1	yes	characteristic
85	Dic₁₇	—	16	2	no	—
86	C₂×M₃₂	C₂×D₁₇	1	2	yes	—
87	C₂×#108	—	17	2	no	—
88	C₂×#108	—	34	1	no	—
89	C₂×#107	—	34	1	no	—
90	C₂×#106	—	17	2	no	—
91	#423	—	34	1	no	—
92	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
93	#382	—	34	1	no	—
94	#346	Dic₁₇	1	2	yes	complemented
95	C₄×C₁₆	Dic₁₇	1	2	yes	complemented
96	C₂×C₁₇	—	2	1	no	—
97	C₂×C₁₇	—	2	1	no	—
98	C₂×C₁₇	#1834	1	1	yes	characteristic
99	C₂×C₁₇	#2155	1	2	yes	—
100	C₂×M₁₆	—	34	2	no	—
101	C₂²×C₈	D₆₈	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
103	C₂×C₄⋊C₄	—	34	2	no	—
104	C₂×C₄²	D₆₈	1	1	yes	characteristic
105	M₃₂	—	2	2	no	—
106	M₃₂	—	2	2	no	—
107	C₂×C₁₆	C₂×Dic₁₇	1	4	yes	—
108	#108	—	34	4	no	—
109	#108	—	68	2	no	—
110	#107	—	68	2	no	—
111	#106	—	34	4	no	—
112	C₄⋊C₈	—	68	2	no	—
113	C₄×C₈	Dic₃₄	1	1	yes	characteristic
114	C₄×C₈	Dic₃₄	1	1	yes	characteristic
115	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
116	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
117	#95	—	68	2	no	—
118	C₁₇	#12887	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₄	—	68	2	no	—
120	C₂²×C₄	#63219	1	2	yes	—
121	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
122	M₁₆	—	68	2	no	—
123	M₁₆	—	68	2	no	—
124	C₂×C₈	—	68	2	no	—
125	C₂×C₈	D₁₃₆	1	2	yes	—
126	C₂×C₈	Dic₆₈	1	2	yes	—
127	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
128	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
129	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
130	C₂×C₈	#63252	1	2	yes	—
131	C₂×C₈	#63252	1	2	yes	—
132	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
133	C₄⋊C₄	—	68	2	no	—
134	C₄⋊C₄	—	68	2	no	—
135	C₄²	#63219	1	2	yes	—
136	C₄²	C₄⋊Dic₁₇	1	1	yes	characteristic
137	C₄²	C₄⋊Dic₁₇	1	1	yes	characteristic
138	C₁₆	—	2	4	no	—
139	C₂³	C₂×#63219	1	1	yes	characteristic
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	136	4	no	—
142	C₂×C₄	—	136	2	no	—
143	C₂×C₄	#10587488	1	2	yes	—
144	C₂×C₄	#10587488	1	2	yes	—
145	C₂×C₄	#10587561	1	1	yes	characteristic
146	C₂×C₄	#10587561	1	1	yes	characteristic
147	C₂×C₄	C₂×#63219	1	1	yes	characteristic
148	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
149	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
151	C₈	—	136	2	no	—
152	C₈	#10587489	1	2	yes	—
153	C₈	#10587489	1	2	yes	—
154	C₈	#10587556	1	2	yes	—
155	C₈	#10587612	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	C₂×#10587488	1	1	yes	characteristic
158	C₂²	#11763156	1	1	yes	characteristic
159	C₂²	#11763160	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	272	2	no	—
163	C₄	#11762675	1	1	yes	characteristic
164	C₄	C₂×#10587489	1	1	yes	characteristic
165	C₄	#11763156	1	1	yes	characteristic
166	C₄	#11763160	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#14528912	1	1	yes	characteristic
170	C₂	#14530557	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#10587488
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₄×C₁₇×M₃₂	C₂
socle	C₂²×C₁₇	C₂×#106
prime radical	C₂³×C₁₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₄×M₃₂	D₁₇
hypocenter	C₁₇	#12887
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
13. C₁₇×#346	C₄	semidirect
14. C₄×C₁₆×C₁₇	C₄	semidirect
94. #346	Dic₁₇	semidirect
95. C₄×C₁₆	Dic₁₇	semidirect
118. C₁₇	#12887	semidirect

Maximal subgroups

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

subgroup	index
#14528956	2
#14528948	2
#14526883	2
C₄×C₁₇×M₃₂	2
#12887	17