Subgroups — #17960650

Order 4352.

414 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10587529	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×#595	C₂	1	1	yes	maximal, characteristic
3	#14543821	C₂	1	4	yes	maximal
4	C₄×C₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587529	C₂²	1	2	yes	—
6	C₄×#63242	C₄	1	1	yes	characteristic
7	C₂×C₈×#595	C₄	1	2	yes	—
8	C₂×C₈×#595	C₂²	1	2	yes	—
9	C₄²×#595	C₄	1	1	yes	characteristic
10	#11763695	C₄	1	2	yes	complemented
11	C₄×#63216	C₄	1	2	yes	complemented
12	C₂×C₁₆×D₁₇	C₂²	1	2	yes	—
13	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
14	C₄×#63214	C₄	1	1	yes	characteristic, complemented
15	C₁₆×Dic₁₇	C₄	1	2	yes	complemented
16	C₄×C₁₆×C₁₇	C₄	1	1	yes	characteristic, complemented
17	C₂×C₄×#595	C₈	1	2	yes	—
18	C₂×C₄×#595	C₈	1	2	yes	—
19	C₂×C₄×#595	C₂×C₄	1	2	yes	—
20	C₂×#63242	C₂×C₄	1	2	yes	—
21	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
23	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
24	C₄×#3624	C₈	1	2	yes	—
25	C₄×#3624	C₈	1	2	yes	—
26	C₂×#63216	C₂×C₄	1	4	yes	—
27	C₈×#595	—	2	4	no	—
28	C₈×#595	—	2	4	no	—
29	#10587529	—	2	4	no	—
30	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
33	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
34	C₂×#63214	C₂×C₄	1	2	yes	—
35	C₄×#3622	C₂×C₄	1	1	yes	characteristic
36	C₁₆×D₁₇	—	2	4	no	—
37	C₂²×#595	C₂×C₈	1	4	yes	—
38	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
39	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
40	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×#3624	C₂×C₈	1	4	yes	—
42	C₄×#595	—	2	4	no	—
43	C₄×#595	—	2	2	no	—
44	C₄×#595	—	2	2	no	—
45	C₄×#595	C₁₆	1	8	yes	complemented
46	#63242	—	2	2	no	—
47	#63242	—	2	2	no	—
48	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₁₇	C₄²	1	1	yes	characteristic
51	C₄×Dic₁₇	C₄²	1	1	yes	characteristic
52	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
53	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
54	C₂×#3622	C₄²	1	1	yes	characteristic
55	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
56	C₈×D₁₇	M₁₆	1	4	yes	—
57	C₈×D₁₇	#41	1	4	yes	—
58	#63216	—	2	4	no	—
59	C₁₆×C₁₇	—	2	2	no	—
60	#63214	—	2	2	no	—
61	#12590	—	17	1	no	maximal, Sylow
62	C₂²×D₁₇	C₂²×C₈	1	1	yes	characteristic
63	C₂×#595	—	2	4	no	—
64	C₂×#595	—	2	4	no	—
65	C₂×#595	C₂×C₁₆	1	8	yes	—
66	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
69	C₂×Dic₁₇	C₄×C₈	1	1	yes	characteristic
70	C₂×Dic₁₇	C₄×C₈	1	1	yes	characteristic
71	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
72	C₄×D₁₇	C₂×C₁₆	1	4	yes	—
73	C₄×D₁₇	M₃₂	1	4	yes	—
74	C₄×D₁₇	#117	1	2	yes	—
75	C₄×D₁₇	C₂×M₁₆	1	2	yes	—
76	#3624	—	2	4	no	—
77	C₈×C₁₇	#97	1	2	yes	—
78	C₈×C₁₇	#117	1	2	yes	—
79	#3622	#97	1	2	yes	—
80	#3622	#117	1	2	yes	—
81	C₂×C₄×C₁₆	—	17	1	no	—
82	C₂×C₄×C₁₆	—	17	1	no	—
83	C₄²×C₈	—	17	1	no	—
84	#1374	—	17	4	no	—
85	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
86	C₂×D₁₇	#431	1	2	yes	—
87	C₂×D₁₇	C₂²×C₁₆	1	2	yes	—
88	C₂×D₁₇	C₂×M₃₂	1	2	yes	—
89	#595	—	2	8	no	—
90	C₄×C₁₇	C₄×C₁₆	1	2	yes	—
91	C₄×C₁₇	#346	1	2	yes	—
92	C₄×C₁₇	#376	1	1	yes	characteristic
93	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
94	Dic₁₇	C₄×C₁₆	1	2	yes	—
95	Dic₁₇	#346	1	2	yes	—
96	Dic₁₇	#431	1	2	yes	—
97	C₂²×C₁₆	—	17	2	no	—
98	C₂²×C₁₆	—	17	2	no	—
99	C₂×C₄×C₈	—	17	1	no	—
100	C₂×C₄×C₈	—	17	1	no	—
101	C₂×C₄×C₈	—	17	2	no	—
102	C₂×C₄×C₈	—	17	2	no	—
103	C₄³	—	17	1	no	—
104	C₄×C₁₆	—	17	2	no	—
105	C₄×C₁₆	—	17	2	no	—
106	C₄×C₁₆	—	17	2	no	—
107	C₄×C₁₆	—	17	1	no	—
108	C₄×C₁₆	#595	1	1	yes	characteristic, complemented
109	C₂×C₁₇	#1744	1	1	yes	characteristic
110	C₂×C₁₇	C₂×C₄×C₁₆	1	1	yes	characteristic
111	C₂×C₁₇	C₂×#346	1	1	yes	characteristic
112	D₁₇	#2162	1	4	yes	—
113	C₂²×C₈	—	17	1	no	—
114	C₂²×C₈	—	17	1	no	—
115	C₂²×C₈	—	17	2	no	—
116	C₂×C₄²	—	17	2	no	—
117	C₂×C₄²	—	17	1	no	—
118	C₂×C₄²	—	17	2	no	—
119	C₂×C₄²	—	17	2	no	—
120	C₂×C₁₆	—	17	4	no	—
121	C₂×C₁₆	—	17	2	no	—
122	C₂×C₁₆	—	34	4	no	—
123	C₂×C₁₆	—	34	4	no	—
124	C₂×C₁₆	C₂×#595	1	2	yes	—
125	C₄×C₈	—	17	2	no	—
126	C₄×C₈	—	17	2	no	—
127	C₄×C₈	—	17	1	no	—
128	C₄×C₈	—	17	1	no	—
129	C₄×C₈	—	17	1	no	—
130	C₄×C₈	—	34	4	no	—
131	C₄×C₈	—	34	4	no	—
132	C₄×C₈	C₂×#595	1	1	yes	characteristic
133	C₁₇	#12590	1	1	yes	characteristic, complemented, Sylow
134	C₂²×C₄	—	17	1	no	—
135	C₂²×C₄	—	17	1	no	—
136	C₂²×C₄	—	17	1	no	—
137	C₂²×C₄	—	17	4	no	—
138	C₂×C₈	—	17	1	no	—
139	C₂×C₈	—	17	1	no	—
140	C₂×C₈	—	17	4	no	—
141	C₂×C₈	—	17	4	no	—
142	C₂×C₈	—	17	4	no	—
143	C₂×C₈	—	34	2	no	—
144	C₂×C₈	—	34	2	no	—
145	C₂×C₈	C₄×#595	1	1	yes	characteristic
146	C₂×C₈	C₂²×#595	1	1	yes	characteristic
147	C₄²	—	17	8	no	—
148	C₄²	—	17	1	no	—
149	C₄²	—	17	1	no	—
150	C₄²	—	17	1	no	—
151	C₄²	—	34	2	no	—
152	C₄²	—	34	2	no	—
153	C₄²	—	34	4	no	—
154	C₄²	C₄×#595	1	1	yes	characteristic
155	C₁₆	—	2	2	no	—
156	C₁₆	—	34	4	no	—
157	C₁₆	—	34	2	no	—
158	C₂³	—	17	1	no	—
159	C₂×C₄	—	17	1	no	—
160	C₂×C₄	—	17	1	no	—
161	C₂×C₄	—	17	1	no	—
162	C₂×C₄	—	17	2	no	—
163	C₂×C₄	—	17	2	no	—
164	C₂×C₄	—	17	4	no	—
165	C₂×C₄	—	17	4	no	—
166	C₂×C₄	—	17	8	no	—
167	C₂×C₄	—	34	4	no	—
168	C₂×C₄	—	34	4	no	—
169	C₂×C₄	C₈×#595	1	1	yes	characteristic
170	C₂×C₄	C₈×#595	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
172	C₈	—	17	2	no	—
173	C₈	—	17	2	no	—
174	C₈	—	34	4	no	—
175	C₈	#10587532	1	2	yes	—
176	C₈	#10587677	1	2	yes	—
177	C₂²	—	17	2	no	—
178	C₂²	—	17	2	no	—
179	C₂²	—	17	2	no	—
180	C₂²	C₂×C₈×#595	1	1	yes	characteristic
181	C₄	—	17	2	no	—
182	C₄	—	17	2	no	—
183	C₄	—	17	2	no	—
184	C₄	—	34	8	no	—
185	C₄	C₁₆×#595	1	2	yes	—
186	C₄	#11763696	1	2	yes	—
187	C₄	#11763745	1	1	yes	characteristic
188	C₄	C₂×#10587532	1	1	yes	characteristic
189	C₂	—	17	4	no	—
190	C₂	#14544497	1	1	yes	characteristic
191	C₂	C₂×C₁₆×#595	1	1	yes	characteristic
192	C₂	C₂×#11763696	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂×#595
commutator	C₂×C₁₇	C₂×C₄×C₁₆
Frattini	C₂×C₈	C₂²×#595
Fitting	C₄×C₁₆×C₁₇	C₄
socle	C₂²×C₁₇	C₂×C₄×C₈
prime radical	C₂²×D₁₇	C₂²×C₈
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	C₄×C₁₆	#595
hypocenter	C₁₇	#12590
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
10. #11763695	C₄	semidirect
11. C₄×#63216	C₄	semidirect
14. C₄×#63214	C₄	semidirect
15. C₁₆×Dic₁₇	C₄	semidirect
16. C₄×C₁₆×C₁₇	C₄	semidirect
45. C₄×#595	C₁₆	semidirect
108. C₄×C₁₆	#595	semidirect
133. C₁₇	#12590	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10587529	2
C₄×C₈×#595	2
#14543821	2
C₄×C₁₆×D₁₇	2
#12590	17