Subgroups — #16950765

Order 4352.

342 conjugacy classes of subgroups.

The subgroup lattice has 158 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
2	C₂×#11762635	C₂	1	2	yes	maximal
3	#14528948	C₂	1	2	yes	maximal
4	C₂×C₄×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587489	C₂²	1	4	yes	—
6	C₂×#10587488	—	2	2	no	—
7	#11762675	—	2	2	no	—
8	#11762637	—	2	4	no	—
9	#11762635	—	2	4	no	—
10	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₁₇	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	4	no	—
18	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
21	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
22	C₂×C₁₆×C₁₇	Q₈	1	2	yes	—
23	C₂×C₁₆×C₁₇	Q₈	1	2	yes	—
24	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₁₇	Q₈	1	2	yes	—
26	#10587502	—	4	4	no	—
27	#10587489	—	2	4	no	—
28	#10587489	—	2	4	no	—
29	#10587488	—	4	4	no	—
30	C₂²×C₄×C₁₇	D₈	1	2	yes	—
31	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
32	C₂²×Dic₁₇	—	4	4	no	—
33	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
34	C₂×C₈×C₁₇	D₈	1	2	yes	—
35	C₂×C₈×C₁₇	Q₁₆	1	2	yes	—
36	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
43	C₄²×C₁₇	Q₁₆	1	2	yes	—
44	C₄⋊Dic₁₇	—	4	4	no	—
45	C₄⋊Dic₁₇	—	4	4	no	—
46	C₁₆×C₁₇	C₄⋊C₄	1	4	yes	—
47	C₁₆×C₁₇	C₄⋊C₄	1	4	yes	—
48	#12907	—	17	1	no	maximal, Sylow
49	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₁₇	—	2	4	no	—
51	C₂×C₄×C₁₇	#107	1	4	yes	—
52	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
54	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
55	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
57	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
58	C₂×Dic₁₇	—	8	4	no	—
59	C₂×Dic₁₇	—	8	8	no	—
60	C₈×C₁₇	#107	1	4	yes	—
61	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
64	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×#367	—	17	2	no	—
66	C₂×#366	—	17	2	no	—
67	C₂×C₄×C₁₆	D₁₇	1	1	yes	characteristic
68	#1846	—	17	2	no	—
69	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
70	C₂²×C₁₇	#493	1	1	yes	characteristic
71	C₂²×C₁₇	#500	1	1	yes	characteristic
72	C₂²×C₁₇	#508	1	4	yes	—
73	C₄×C₁₇	—	2	4	no	—
74	C₄×C₁₇	#384	1	1	yes	characteristic
75	C₄×C₁₇	C₂×#107	1	1	yes	characteristic
76	C₄×C₁₇	#494	1	1	yes	characteristic
77	C₄×C₁₇	#500	1	1	yes	characteristic
78	Dic₁₇	—	16	4	no	—
79	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
80	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
81	C₂×#107	—	17	4	no	—
82	C₂×#106	—	34	2	no	—
83	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
84	#382	—	34	2	no	—
85	#367	—	34	4	no	—
86	#366	—	34	4	no	—
87	C₄×C₁₆	Dic₁₇	1	4	yes	complemented
88	C₂×C₁₇	#1845	1	1	yes	characteristic
89	C₂×C₁₇	#2158	1	2	yes	—
90	C₂×C₁₇	#2249	1	2	yes	—
91	C₂×C₁₇	#2253	1	2	yes	—
92	C₂²×C₈	D₆₈	1	1	yes	characteristic
93	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
94	C₂×C₄⋊C₄	—	34	4	no	—
95	C₂×C₄²	D₆₈	1	1	yes	characteristic
96	C₂×C₁₆	Dic₃₄	1	2	yes	—
97	C₂×C₁₆	Dic₃₄	1	2	yes	—
98	C₂×C₁₆	D₆₈	1	2	yes	—
99	C₂×C₁₆	D₆₈	1	2	yes	—
100	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
101	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
102	#107	—	34	4	no	—
103	#107	—	34	4	no	—
104	#106	—	68	4	no	—
105	C₄×C₈	Dic₃₄	1	2	yes	—
106	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
107	#95	—	68	4	no	—
108	C₁₇	#12907	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₄	—	68	4	no	—
110	C₂²×C₄	D₁₃₆	1	2	yes	—
111	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
112	C₂×C₈	D₁₃₆	1	2	yes	—
113	C₂×C₈	Dic₆₈	1	2	yes	—
114	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
115	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
116	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
117	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
118	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
119	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
120	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
121	C₄⋊C₄	—	68	4	no	—
122	C₄⋊C₄	—	68	4	no	—
123	C₄²	Dic₆₈	1	2	yes	—
124	C₄²	C₄⋊Dic₁₇	1	2	yes	—
125	C₁₆	C₄⋊Dic₁₇	1	4	yes	—
126	C₁₆	C₄⋊Dic₁₇	1	4	yes	—
127	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	136	8	no	—
130	C₂×C₄	—	136	4	no	—
131	C₂×C₄	#10587489	1	4	yes	—
132	C₂×C₄	#10587555	1	1	yes	characteristic
133	C₂×C₄	#10587560	1	1	yes	characteristic
134	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
135	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
136	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
138	C₈	#10587489	1	4	yes	—
139	C₈	#10587555	1	1	yes	characteristic
140	C₈	#10587555	1	1	yes	characteristic
141	C₈	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
142	C₈	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
143	C₂²	C₂×#10587489	1	1	yes	characteristic
144	C₂²	#11763186	1	1	yes	characteristic
145	C₂²	#11763214	1	1	yes	characteristic
146	C₂²	#11763241	1	4	yes	—
147	C₄	—	2	4	no	—
148	C₄	—	272	4	no	—
149	C₄	#11762679	1	1	yes	characteristic
150	C₄	C₂×#10587489	1	1	yes	characteristic
151	C₄	#11763189	1	1	yes	characteristic
152	C₄	#11763214	1	1	yes	characteristic
153	C₂	#14528944	1	1	yes	characteristic
154	C₂	#14530568	1	2	yes	—
155	C₂	#14531045	1	2	yes	—
156	C₂	#14531063	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₁₃₆
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₄×C₁₆×C₁₇	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₇
hypocenter	C₁₇	#12907
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
13. C₄×C₁₆×C₁₇	C₄	semidirect
87. C₄×C₁₆	Dic₁₇	semidirect
108. C₁₇	#12907	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
C₂×#11762635	2
#14528948	2
C₂×C₄×C₁₆×C₁₇	2
#12907	17