Subgroups — #17189816

Order 4352.

490 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	#14539298	C₂	1	2	yes	maximal
2	#14531066	C₂	1	2	yes	maximal, complemented
3	#14531063	C₂	1	4	yes	maximal, complemented
4	#14531059	C₂	1	2	yes	maximal, complemented
5	#14530568	C₂	1	4	yes	maximal
6	C₂×C₄×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
7	#11763366	—	2	2	no	—
8	#11763214	C₂²	1	4	yes	—
9	#11763211	—	2	2	no	—
10	#11763211	C₂²	1	4	yes	—
11	#11763207	—	2	2	no	—
12	#11762847	—	2	2	no	—
13	#11762847	C₂²	1	4	yes	—
14	#11762637	C₂²	1	8	yes	—
15	#11762635	C₂²	1	8	yes	—
16	C₂²×C₁₆×C₁₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
19	C₄×C₁₆×C₁₇	C₂²	1	2	yes	—
20	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
24	#10587612	—	2	4	no	—
25	#10587610	—	4	4	no	—
26	#10587556	—	2	4	no	—
27	#10587555	—	2	4	no	—
28	C₄×Dic₃₄	—	4	4	no	—
29	C₂×C₁₆×C₁₇	Q₈	1	8	yes	complemented
30	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
31	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
32	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
33	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
34	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
35	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
36	#10587489	—	2	4	no	—
37	#10587489	C₂³	1	8	yes	—
38	#10587488	—	2	4	no	—
39	#10587488	—	2	4	no	—
40	C₂²×C₄×C₁₇	D₈	1	2	yes	—
41	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₃₄	—	4	4	no	—
43	C₂×C₈×C₁₇	Q₁₆	1	4	yes	—
44	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₁₇	C₂×Q₈	1	4	yes	—
48	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
49	C₄²×C₁₇	D₈	1	2	yes	—
50	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₂²⋊Dic₁₇	—	4	4	no	—
53	C₄⋊Dic₁₇	—	2	8	no	—
54	C₄⋊Dic₁₇	—	4	4	no	—
55	#63225	—	4	8	no	—
56	C₄×Dic₁₇	—	4	4	no	—
57	C₁₆×C₁₇	C₂×Q₈	1	4	yes	—
58	C₁₆×C₁₇	C₂×Q₈	1	4	yes	—
59	#33726	—	17	1	no	maximal, Sylow
60	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₁₇	—	2	4	no	—
62	C₂×C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
63	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
64	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
65	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
67	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
68	C₂×Dic₁₇	—	4	8	no	—
69	Dic₃₄	—	8	8	no	—
70	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
71	C₈×C₁₇	C₂×Q₁₆	1	2	yes	—
72	C₈×C₁₇	C₂×Q₁₆	1	2	yes	—
73	C₈×C₁₇	C₂²×Q₈	1	2	yes	—
74	#3160	—	17	2	no	—
75	#2254	—	17	2	no	—
76	#2253	—	17	4	no	—
77	#2252	—	17	2	no	—
78	#2158	—	17	4	no	—
79	C₂×C₄×C₁₆	D₁₇	1	1	yes	characteristic
80	C₂²×C₁₇	—	2	2	no	—
81	C₂²×C₁₇	#500	1	2	yes	—
82	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
83	C₄×C₁₇	—	2	2	no	—
84	C₄×C₁₇	#500	1	2	yes	—
85	C₄×C₁₇	#508	1	4	yes	—
86	C₄×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
87	C₄×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
88	Dic₁₇	—	8	8	no	—
89	#533	—	34	2	no	—
90	C₂²×C₁₆	C₂×D₁₇	1	2	yes	—
91	#500	—	17	4	no	—
92	#499	—	17	4	no	—
93	#499	—	34	2	no	—
94	#498	—	34	2	no	—
95	#427	—	17	4	no	—
96	#427	—	34	2	no	—
97	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
98	#367	—	17	8	no	—
99	#366	—	17	8	no	—
100	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
101	C₄×C₁₆	C₂×D₁₇	1	2	yes	—
102	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
103	C₂×C₁₇	—	2	2	no	—
104	C₂×C₁₇	C₂×#500	1	1	yes	characteristic
105	C₂×C₁₇	C₂×#508	1	2	yes	—
106	C₂²×C₈	D₆₈	1	1	yes	characteristic
107	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
108	C₄⋊Q₈	—	34	4	no	—
109	#125	—	34	4	no	—
110	C₂²⋊Q₈	—	68	4	no	—
111	C₄×Q₈	—	68	4	no	—
112	#117	—	34	4	no	—
113	C₂×C₄²	D₆₈	1	1	yes	characteristic
114	C₂×C₁₆	Dic₃₄	1	8	yes	complemented
115	C₂×C₁₆	C₂²×D₁₇	1	2	yes	—
116	C₂×C₁₆	C₂²×D₁₇	1	2	yes	—
117	#107	—	17	8	no	—
118	#107	—	34	4	no	—
119	#106	—	34	4	no	—
120	#106	—	34	4	no	—
121	C₄×C₈	D₆₈	1	1	yes	characteristic
122	C₄×C₈	D₆₈	1	1	yes	characteristic
123	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
124	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
125	C₁₇	#33726	1	1	yes	characteristic, complemented, Sylow
126	C₂×Q₈	—	68	4	no	—
127	C₂²×C₄	D₁₃₆	1	2	yes	—
128	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
129	C₂×C₈	Dic₆₈	1	4	yes	—
130	C₂×C₈	C₂×Dic₃₄	1	4	yes	—
131	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
132	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
133	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
134	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
135	C₄⋊C₄	—	34	8	no	—
136	C₄⋊C₄	—	68	8	no	—
137	C₄⋊C₄	—	68	4	no	—
138	C₂²⋊C₄	—	68	4	no	—
139	C₄²	—	68	4	no	—
140	C₄²	D₁₃₆	1	2	yes	—
141	C₄²	C₂×D₆₈	1	1	yes	characteristic
142	C₄²	C₂×D₆₈	1	1	yes	characteristic
143	C₁₆	C₂×Dic₃₄	1	4	yes	—
144	C₁₆	C₂×Dic₃₄	1	4	yes	—
145	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
146	Q₈	—	136	8	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	68	8	no	—
149	C₂×C₄	#10587555	1	2	yes	—
150	C₂×C₄	C₂×D₁₃₆	1	2	yes	—
151	C₂×C₄	C₂×D₁₃₆	1	2	yes	—
152	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
153	C₂×C₄	C₂×Dic₆₈	1	2	yes	—
154	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
155	C₈	#10587555	1	2	yes	—
156	C₈	C₂×Dic₆₈	1	2	yes	—
157	C₈	C₂×Dic₆₈	1	2	yes	—
158	C₈	C₂²×Dic₃₄	1	2	yes	—
159	C₂²	—	2	2	no	—
160	C₂²	#11763214	1	2	yes	—
161	C₂²	C₂²×D₁₃₆	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	136	8	no	—
164	C₄	#11763214	1	2	yes	—
165	C₄	#11763241	1	4	yes	—
166	C₄	C₂×#10587555	1	1	yes	characteristic
167	C₄	C₂²×Dic₆₈	1	1	yes	characteristic
168	C₂	—	2	2	no	—
169	C₂	C₂×#11763214	1	1	yes	characteristic
170	C₂	C₂×#11763241	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₃₆
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₂×C₄×C₁₆×C₁₇	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	C₂³×C₁₇	C₂×D₈
hypercenter	C₂×C₄×C₁₆	D₁₇
hypocenter	C₁₇	#33726
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #14531066	C₂	semidirect
3. #14531063	C₂	semidirect
4. #14531059	C₂	semidirect
29. C₂×C₁₆×C₁₇	Q₈	semidirect
114. C₂×C₁₆	Dic₃₄	semidirect
125. C₁₇	#33726	semidirect

Maximal subgroups

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

subgroup	index
#14539298	2
#14531066	2
#14531063	2
#14531059	2
#14530568	2
C₂×C₄×C₁₆×C₁₇	2
#33726	17