Subgroups — #16952355

Order 4352.

326 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762635	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	#14529518	C₂	1	2	yes	maximal
4	#14526875	C₂	1	2	yes	maximal
5	C₂×C₁₇×#366	C₂	1	1	yes	maximal, characteristic
6	C₂²×#63214	C₂²	1	1	yes	characteristic
7	C₂×#10587489	C₂²	1	2	yes	—
8	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
9	#11762708	—	2	2	no	—
10	#11762635	—	2	2	no	—
11	C₁₆×Dic₁₇	—	2	2	no	—
12	#11762571	—	2	2	no	—
13	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
14	C₂×C₁₇×#107	C₂²	1	2	yes	—
15	C₁₇×#366	C₄	1	4	yes	complemented
16	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
18	C₂×C₄⋊Dic₁₇	—	2	2	no	—
19	C₂×C₄⋊Dic₁₇	—	4	2	no	—
20	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
21	C₂²×#3622	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
24	C₂×C₁₆×C₁₇	Q₈	1	2	yes	—
25	C₁₇×#107	—	2	2	no	—
26	C₁₇×#107	C₂×C₄	1	4	yes	—
27	#10587502	—	4	2	no	—
28	#10587489	—	2	2	no	—
29	#10587489	—	2	2	no	—
30	C₈×Dic₁₇	—	2	1	no	—
31	C₈×Dic₁₇	—	2	1	no	—
32	C₂×#63214	—	2	1	no	—
33	C₂×#63214	—	2	1	no	—
34	C₂×#63214	Q₈	1	2	yes	—
35	C₂²×C₄×C₁₇	—	4	2	no	—
36	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₁₇	—	4	2	no	—
38	C₂²×Dic₁₇	D₈	1	1	yes	characteristic
39	C₂²×Dic₁₇	D₈	1	1	yes	characteristic
40	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
45	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
46	C₄⋊C₄×C₁₇	—	2	4	no	—
47	C₄⋊C₄×C₁₇	—	4	2	no	—
48	C₄⋊Dic₁₇	—	4	2	no	—
49	C₄⋊Dic₁₇	—	4	2	no	—
50	C₄⋊Dic₁₇	—	8	4	no	—
51	C₄×Dic₁₇	—	2	1	no	—
52	C₄×Dic₁₇	—	2	1	no	—
53	C₂×#3622	—	2	1	no	—
54	C₂×#3622	—	2	1	no	—
55	C₂×#3622	Q₁₆	1	2	yes	—
56	C₁₆×C₁₇	C₄⋊C₄	1	4	yes	complemented
57	#63214	—	2	2	no	—
58	#13165	—	17	1	no	maximal, Sylow
59	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₁₇	—	4	4	no	—
61	C₂×C₄×C₁₇	—	8	2	no	—
62	C₂×C₄×C₁₇	—	8	2	no	—
63	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
68	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
69	C₂×Dic₁₇	—	2	1	no	—
70	C₂×Dic₁₇	—	2	1	no	—
71	C₂×Dic₁₇	—	2	1	no	—
72	C₂×Dic₁₇	—	2	1	no	—
73	C₂×Dic₁₇	—	8	2	no	—
74	C₂×Dic₁₇	—	8	2	no	—
75	C₂×Dic₁₇	—	8	2	no	—
76	C₂×Dic₁₇	D₁₆	1	2	yes	—
77	C₂×Dic₁₇	Q₃₂	1	2	yes	—
78	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₈×C₁₇	C₄×Q₈	1	1	yes	characteristic
80	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
81	C₈×C₁₇	#125	1	1	yes	characteristic
82	#3622	—	2	2	no	—
83	C₂×#366	—	17	1	no	—
84	C₂×#366	D₁₇	1	1	yes	characteristic
85	C₂×C₄×C₁₆	—	17	1	no	—
86	#1942	—	17	2	no	—
87	#1380	—	17	2	no	—
88	C₂²×C₁₇	C₄×D₈	1	1	yes	characteristic
89	C₂²×C₁₇	#466	1	1	yes	characteristic
90	C₂²×C₁₇	#500	1	1	yes	characteristic
91	C₂²×C₁₇	C₂×D₁₆	1	1	yes	characteristic
92	C₂²×C₁₇	C₂×Q₃₂	1	1	yes	characteristic
93	C₂²×C₁₇	#508	1	1	yes	characteristic
94	C₂²×C₁₇	#508	1	1	yes	characteristic
95	C₄×C₁₇	—	8	4	no	—
96	C₄×C₁₇	#389	1	1	yes	characteristic
97	C₄×C₁₇	C₄×Q₁₆	1	1	yes	characteristic
98	C₄×C₁₇	#467	1	1	yes	characteristic
99	C₄×C₁₇	#499	1	1	yes	characteristic
100	Dic₁₇	—	2	2	no	—
101	Dic₁₇	—	2	2	no	—
102	Dic₁₇	—	16	2	no	—
103	C₂²×C₁₆	—	17	1	no	—
104	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
105	C₂×#107	—	17	2	no	—
106	C₂×#107	C₂×D₁₇	1	2	yes	—
107	C₂×C₄×C₈	—	17	1	no	—
108	#389	—	34	2	no	—
109	#366	—	34	2	no	—
110	#366	Dic₁₇	1	4	yes	complemented
111	C₄×C₁₆	—	34	2	no	—
112	#340	—	34	2	no	—
113	C₂×C₁₇	#1942	1	1	yes	characteristic
114	C₂×C₁₇	C₄×D₁₆	1	1	yes	characteristic
115	C₂×C₁₇	C₄×Q₃₂	1	1	yes	characteristic
116	C₂×C₁₇	#2215	1	1	yes	characteristic
117	C₂×C₁₇	#2216	1	1	yes	characteristic
118	C₂×C₁₇	#2252	1	1	yes	characteristic
119	C₂×C₁₇	#2253	1	1	yes	characteristic
120	C₂²×C₈	—	17	1	no	—
121	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	34	2	no	—
124	C₂×C₄⋊C₄	—	68	2	no	—
125	C₂×C₄²	—	17	1	no	—
126	C₂×C₁₆	—	17	2	no	—
127	C₂×C₁₆	—	34	1	no	—
128	C₂×C₁₆	—	34	1	no	—
129	C₂×C₁₆	Dic₃₄	1	2	yes	—
130	C₂×C₁₆	D₆₈	1	2	yes	—
131	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
132	#107	—	2	2	no	—
133	#107	—	34	2	no	—
134	#107	—	34	2	no	—
135	#107	C₂×Dic₁₇	1	4	yes	—
136	C₄×C₈	—	34	1	no	—
137	C₄×C₈	—	34	1	no	—
138	#95	—	68	2	no	—
139	C₁₇	#13165	1	1	yes	characteristic, complemented, Sylow
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	—	17	1	no	—
142	C₂²×C₄	—	17	1	no	—
143	C₂²×C₄	—	68	2	no	—
144	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
145	C₂×C₈	—	17	2	no	—
146	C₂×C₈	—	34	1	no	—
147	C₂×C₈	—	34	1	no	—
148	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
149	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
150	C₂×C₈	#63254	1	1	yes	characteristic
151	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
152	C₂×C₈	#63256	1	1	yes	characteristic
153	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	68	2	no	—
157	C₄⋊C₄	—	68	2	no	—
158	C₄⋊C₄	—	136	4	no	—
159	C₄²	—	34	1	no	—
160	C₄²	—	34	1	no	—
161	C₁₆	—	34	2	no	—
162	C₁₆	C₄⋊Dic₁₇	1	4	yes	complemented
163	C₂³	D₈×D₁₇	1	1	yes	characteristic
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	17	2	no	—
168	C₂×C₄	—	17	2	no	—
169	C₂×C₄	—	34	1	no	—
170	C₂×C₄	—	34	1	no	—
171	C₂×C₄	—	34	1	no	—
172	C₂×C₄	—	34	1	no	—
173	C₂×C₄	—	136	2	no	—
174	C₂×C₄	—	136	2	no	—
175	C₂×C₄	—	136	2	no	—
176	C₂×C₄	#10587574	1	1	yes	characteristic
177	C₂×C₄	#10587581	1	1	yes	characteristic
178	C₂×C₄	#10587598	1	1	yes	characteristic
179	C₂×C₄	Q₁₆×D₁₇	1	1	yes	characteristic
180	C₂×C₄	#10587605	1	1	yes	characteristic
181	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
182	C₈	—	34	2	no	—
183	C₈	#10587576	1	1	yes	characteristic
184	C₈	#10587583	1	1	yes	characteristic
185	C₈	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
186	C₈	Q₈×Dic₁₇	1	1	yes	characteristic
187	C₂²	D₈×Dic₁₇	1	1	yes	characteristic
188	C₂²	#11763046	1	1	yes	characteristic
189	C₂²	#11763213	1	1	yes	characteristic
190	C₂²	D₁₆×D₁₇	1	1	yes	characteristic
191	C₂²	Q₃₂×D₁₇	1	1	yes	characteristic
192	C₂²	#11763238	1	1	yes	characteristic
193	C₂²	#11763242	1	1	yes	characteristic
194	C₄	—	8	4	no	—
195	C₄	—	272	2	no	—
196	C₄	—	34	2	no	—
197	C₄	—	34	2	no	—
198	C₄	#11762708	1	1	yes	characteristic
199	C₄	Q₁₆×Dic₁₇	1	1	yes	characteristic
200	C₄	#11763051	1	1	yes	characteristic
201	C₄	#11763210	1	1	yes	characteristic
202	C₂	#14529518	1	1	yes	characteristic
203	C₂	D₁₆×Dic₁₇	1	1	yes	characteristic
204	C₂	Q₃₂×Dic₁₇	1	1	yes	characteristic
205	C₂	#14530871	1	1	yes	characteristic
206	C₂	#14530876	1	1	yes	characteristic
207	C₂	#14531058	1	1	yes	characteristic
208	C₂	#14531061	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₁₇
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#366	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#366	D₁₇
hypocenter	C₁₇	#13165
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
15. C₁₇×#366	C₄	semidirect
56. C₁₆×C₁₇	C₄⋊C₄	semidirect
110. #366	Dic₁₇	semidirect
139. C₁₇	#13165	semidirect
162. C₁₆	C₄⋊Dic₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762635	2
C₂×C₁₆×Dic₁₇	2
#14529518	2
#14526875	2
C₂×C₁₇×#366	2
#13165	17