Subgroups — #17189814

Order 4352.

490 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539296	C₂	1	2	yes	maximal
2	#14531065	C₂	1	2	yes	maximal, complemented
3	#14531062	C₂	1	2	yes	maximal, complemented
4	#14531061	C₂	1	2	yes	maximal, complemented
5	#14531058	C₂	1	2	yes	maximal, complemented
6	#14530568	C₂	1	1	yes	maximal, characteristic
7	#14530566	C₂	1	2	yes	maximal
8	C₂×C₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic
9	C₁₇×#2158	C₂	1	1	yes	maximal, characteristic
10	#11763362	—	2	2	no	—
11	C₂²×#63214	C₂²	1	1	yes	characteristic
12	#11763213	C₂²	1	4	yes	—
13	#11763210	C₂²	1	4	yes	—
14	#11763209	—	2	2	no	—
15	#11763206	—	2	2	no	—
16	#11762847	C₂²	1	2	yes	—
17	#11762845	—	2	2	no	—
18	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
19	#11762637	C₂²	1	2	yes	—
20	#11762636	C₂²	1	4	yes	—
21	#11762635	C₂²	1	2	yes	—
22	#11762634	C₂²	1	4	yes	—
23	C₁₆×Dic₁₇	C₂²	1	1	yes	characteristic
24	C₁₆×Dic₁₇	C₂²	1	1	yes	characteristic
25	C₁₆×Dic₁₇	C₂²	1	1	yes	characteristic
26	C₁₆×Dic₁₇	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
28	C₁₇×#427	C₂²	1	2	yes	—
29	C₁₇×#367	C₂²	1	2	yes	—
30	C₁₇×#366	C₂²	1	2	yes	—
31	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
32	C₁₇×#117	—	2	2	no	—
33	#10587612	—	2	2	no	—
34	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
35	C₂²×#3622	D₄	1	1	yes	characteristic
36	#10587576	—	2	4	no	—
37	#10587574	—	2	4	no	—
38	#10587564	—	4	4	no	—
39	C₄×Dic₃₄	—	4	4	no	—
40	C₂×C₁₆×C₁₇	Q₈	1	4	yes	complemented
41	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
42	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
43	C₁₇×#107	C₂³	1	4	yes	—
44	C₁₇×#106	—	2	2	no	—
45	#10587489	C₂³	1	4	yes	—
46	#10587488	—	2	2	no	—
47	C₈×Dic₁₇	D₄	1	1	yes	characteristic
48	C₈×Dic₁₇	D₄	1	1	yes	characteristic
49	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
50	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
51	C₂×#63214	Q₈	1	4	yes	complemented
52	C₂×#63214	C₂³	1	1	yes	characteristic
53	C₂×#63214	C₂³	1	1	yes	characteristic
54	#10587479	—	2	4	no	—
55	#10587478	—	2	4	no	—
56	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₁₇	D₈	1	2	yes	—
58	C₂×Dic₃₄	—	4	4	no	—
59	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
61	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
62	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
63	C₄⋊C₄×C₁₇	—	2	4	no	—
64	C₂²⋊C₄×C₁₇	—	4	2	no	—
65	C₄²×C₁₇	—	4	2	no	—
66	C₂²⋊Dic₁₇	—	4	2	no	—
67	C₄⋊Dic₁₇	—	2	4	no	—
68	C₄⋊Dic₁₇	—	4	4	no	—
69	#63225	—	4	4	no	—
70	#63225	—	4	4	no	—
71	C₄×Dic₁₇	—	4	2	no	—
72	C₄×Dic₁₇	D₈	1	2	yes	—
73	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
74	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
75	C₂×#3622	Q₁₆	1	4	yes	—
76	C₂×#3622	C₂×D₄	1	1	yes	characteristic
77	C₂×#3622	C₂×D₄	1	1	yes	characteristic
78	C₁₆×C₁₇	C₂×Q₈	1	2	yes	—
79	C₁₆×C₁₇	C₂×Q₈	1	2	yes	—
80	#63214	C₂×Q₈	1	2	yes	—
81	#63214	C₂×Q₈	1	2	yes	—
82	#33726	—	17	1	no	maximal, Sylow
83	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₁₇	—	4	4	no	—
85	C₂×C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
86	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
88	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×Dic₁₇	—	2	4	no	—
90	C₂×Dic₁₇	—	4	4	no	—
91	C₂×Dic₁₇	C₂×D₈	1	2	yes	—
92	C₂×Dic₁₇	C₂×D₈	1	2	yes	—
93	Dic₃₄	—	8	8	no	—
94	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
95	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
96	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
97	#3622	C₂×Q₁₆	1	2	yes	—
98	#3622	C₂×Q₁₆	1	2	yes	—
99	#3160	—	17	2	no	—
100	#2254	—	17	2	no	—
101	#2253	—	17	2	no	—
102	#2253	—	17	2	no	—
103	#2252	—	17	2	no	—
104	#2158	—	17	1	no	—
105	#2158	—	17	2	no	—
106	#2158	D₁₇	1	1	yes	characteristic
107	C₂×C₄×C₁₆	—	17	1	no	—
108	C₂²×C₁₇	—	2	2	no	—
109	C₂²×C₁₇	#500	1	2	yes	—
110	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
111	C₄×C₁₇	—	8	4	no	—
112	C₄×C₁₇	#500	1	2	yes	—
113	C₄×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
114	C₄×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
115	Dic₁₇	—	2	2	no	—
116	Dic₁₇	—	8	4	no	—
117	Dic₁₇	#508	1	4	yes	—
118	#533	—	34	2	no	—
119	C₂²×C₁₆	—	17	1	no	—
120	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
121	#500	—	17	4	no	—
122	#499	—	17	4	no	—
123	#499	—	34	2	no	—
124	#498	—	34	2	no	—
125	#427	—	17	2	no	—
126	#427	—	34	2	no	—
127	#427	C₂×D₁₇	1	2	yes	—
128	C₂×C₄×C₈	—	17	1	no	—
129	#367	—	17	2	no	—
130	#367	—	17	4	no	—
131	#367	C₂×D₁₇	1	2	yes	—
132	#366	—	17	2	no	—
133	#366	—	17	4	no	—
134	#366	C₂×D₁₇	1	2	yes	—
135	C₄×C₁₆	—	17	1	no	—
136	C₄×C₁₆	—	17	1	no	—
137	C₄×C₁₆	—	17	1	no	—
138	C₄×C₁₆	—	17	1	no	—
139	C₂×C₁₇	—	2	2	no	—
140	C₂×C₁₇	C₂×#500	1	1	yes	characteristic
141	C₂×C₁₇	C₂×#508	1	2	yes	—
142	C₂²×C₈	—	17	1	no	—
143	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
144	C₄⋊Q₈	—	34	4	no	—
145	#125	—	34	4	no	—
146	C₂²⋊Q₈	—	68	4	no	—
147	C₄×Q₈	—	68	4	no	—
148	#117	—	2	2	no	—
149	#117	—	34	2	no	—
150	C₂×C₄²	—	17	1	no	—
151	C₂×C₁₆	—	17	4	no	—
152	C₂×C₁₆	—	17	1	no	—
153	C₂×C₁₆	—	17	1	no	—
154	C₂×C₁₆	Dic₃₄	1	4	yes	complemented
155	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
156	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
157	#107	—	17	4	no	—
158	#107	—	34	4	no	—
159	#107	C₂²×D₁₇	1	4	yes	—
160	#106	—	2	2	no	—
161	#106	—	34	4	no	—
162	#106	—	34	2	no	—
163	C₄×C₈	—	17	1	no	—
164	C₄×C₈	—	17	1	no	—
165	C₄×C₈	—	17	1	no	—
166	C₄×C₈	—	17	1	no	—
167	C₁₇	#33726	1	1	yes	characteristic, complemented, Sylow
168	C₂×Q₈	—	68	4	no	—
169	C₂²×C₄	—	17	2	no	—
170	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
171	C₂×C₈	—	17	1	no	—
172	C₂×C₈	—	17	1	no	—
173	C₂×C₈	—	17	4	no	—
174	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
175	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
176	C₂×C₈	Q₈×D₁₇	1	2	yes	—
177	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	34	4	no	—
180	C₄⋊C₄	—	68	4	no	—
181	C₄⋊C₄	—	68	4	no	—
182	C₄⋊C₄	—	68	4	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₂²⋊C₄	—	68	2	no	—
185	C₄²	—	4	2	no	—
186	C₄²	—	17	1	no	—
187	C₄²	—	17	1	no	—
188	C₄²	—	17	2	no	—
189	C₄²	—	68	2	no	—
190	C₁₆	—	17	2	no	—
191	C₁₆	—	17	2	no	—
192	C₁₆	C₂×Dic₃₄	1	2	yes	—
193	C₁₆	C₂×Dic₃₄	1	2	yes	—
194	C₂³	D₈×D₁₇	1	1	yes	characteristic
195	Q₈	—	136	8	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	17	2	no	—
198	C₂×C₄	—	17	2	no	—
199	C₂×C₄	—	34	4	no	—
200	C₂×C₄	—	68	4	no	—
201	C₂×C₄	#10587574	1	2	yes	—
202	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
203	C₂×C₄	Q₁₆×D₁₇	1	2	yes	—
204	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
205	C₈	—	17	2	no	—
206	C₈	—	17	2	no	—
207	C₈	#10587574	1	2	yes	—
208	C₈	C₂²×Dic₃₄	1	1	yes	characteristic
209	C₈	C₂×Q₈×D₁₇	1	1	yes	characteristic
210	C₂²	—	2	2	no	—
211	C₂²	#11763213	1	2	yes	—
212	C₂²	C₂×D₈×D₁₇	1	1	yes	characteristic
213	C₄	—	8	4	no	—
214	C₄	—	17	4	no	—
215	C₄	—	34	2	no	—
216	C₄	—	136	4	no	—
217	C₄	#11763213	1	2	yes	—
218	C₄	C₂×#10587574	1	1	yes	characteristic
219	C₄	C₂×Q₁₆×D₁₇	1	1	yes	characteristic
220	C₂	—	2	2	no	—
221	C₂	C₂×#11763213	1	1	yes	characteristic
222	C₂	D₁₇×#508	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₁₇
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#2158	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	C₂³×C₁₇	C₂×D₈
hypercenter	#2158	D₁₇
hypocenter	C₁₇	#33726
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
2. #14531065	C₂	semidirect
3. #14531062	C₂	semidirect
4. #14531061	C₂	semidirect
5. #14531058	C₂	semidirect
40. C₂×C₁₆×C₁₇	Q₈	semidirect
51. C₂×#63214	Q₈	semidirect
154. C₂×C₁₆	Dic₃₄	semidirect
167. C₁₇	#33726	semidirect

Maximal subgroups

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

subgroup	index
#14539296	2
#14531065	2
#14531062	2
#14531061	2
#14531058	2
#14530568	2
#14530566	2
C₂×C₁₆×Dic₁₇	2
C₁₇×#2158	2
#33726	17