Subgroups — #17043187

Order 4352.

678 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539298	C₂	1	1	yes	maximal, characteristic
2	#14537718	C₂	1	2	yes	maximal
3	#14529761	C₂	1	2	yes	maximal
4	#14529537	C₂	1	4	yes	maximal
5	#14528956	C₂	1	1	yes	maximal, characteristic
6	#14528668	C₂	1	4	yes	maximal
7	C₄×C₁₇×#97	C₂	1	1	yes	maximal, characteristic
8	#11763366	C₂²	1	2	yes	—
9	#11763214	C₄	1	2	yes	complemented
10	#11763211	C₄	1	4	yes	complemented
11	#11763207	C₄	1	2	yes	complemented
12	#11762939	—	2	4	no	—
13	C₈×Dic₃₄	—	2	4	no	—
14	C₂×#10587490	C₂²	1	4	yes	—
15	#11762847	C₂²	1	4	yes	—
16	#11762835	C₂²	1	4	yes	—
17	#11762833	C₂²	1	2	yes	—
18	#11762764	C₂²	1	4	yes	—
19	#11762569	C₂²	1	8	yes	—
20	C₂×C₁₇×#97	C₂²	1	2	yes	—
21	C₂×C₁₇×#97	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
24	C₄³×C₁₇	C₂²	1	1	yes	characteristic
25	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₁₇	C₂³	1	2	yes	—
27	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₇	—	2	2	no	—
29	C₂×C₄²×C₁₇	C₂³	1	2	yes	—
30	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
31	#10587612	C₂³	1	4	yes	—
32	#10587610	—	2	4	no	—
33	C₂×#63223	C₂³	1	4	yes	—
34	#10587556	C₂×C₄	1	4	yes	—
35	#10587555	C₂×C₄	1	4	yes	—
36	C₄×Dic₃₄	—	2	4	no	—
37	C₁₇×#97	—	2	4	no	—
38	C₁₇×#97	Q₈	1	8	yes	complemented
39	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₇	D₄	1	4	yes	—
43	#10587490	—	2	8	no	—
44	#10587489	C₂×C₄	1	8	yes	—
45	#10587488	C₂×C₄	1	8	yes	—
46	#10587486	—	2	4	no	—
47	#10587485	—	2	8	no	—
48	C₈×Dic₁₇	—	2	4	no	—
49	C₄⋊#3622	—	2	4	no	—
50	C₂²×C₄×C₁₇	—	2	2	no	—
51	C₂²×C₄×C₁₇	#41	1	2	yes	—
52	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₃₄	—	2	4	no	—
54	C₂×C₈×C₁₇	—	2	2	no	—
55	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₁₇	C₂×Q₈	1	4	yes	—
60	C₂×C₈×C₁₇	C₂×Q₈	1	4	yes	—
61	C₂×C₈×C₁₇	#41	1	4	yes	—
62	C₄²×C₁₇	—	2	2	no	—
63	C₄²×C₁₇	—	4	4	no	—
64	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₁₇	C₂×D₄	1	2	yes	—
67	C₄²×C₁₇	C₂×Q₈	1	4	yes	—
68	C₂²⋊Dic₁₇	—	2	4	no	—
69	C₄⋊Dic₁₇	—	2	4	no	—
70	C₄⋊Dic₁₇	C₂²×C₄	1	8	yes	—
71	#63225	—	2	8	no	—
72	C₄×Dic₁₇	—	2	4	no	—
73	#63223	—	2	8	no	—
74	C₂×#3622	—	2	4	no	—
75	#21734	—	17	1	no	maximal, Sylow
76	C₂³×C₁₇	C₂×#41	1	1	yes	characteristic
77	C₂×C₄×C₁₇	—	2	4	no	—
78	C₂×C₄×C₁₇	—	2	4	no	—
79	C₂×C₄×C₁₇	—	4	4	no	—
80	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₂²×Q₈	1	2	yes	—
85	C₂×C₄×C₁₇	C₂×#41	1	2	yes	—
86	C₂×Dic₁₇	—	2	8	no	—
87	Dic₃₄	—	4	8	no	—
88	C₈×C₁₇	—	2	2	no	—
89	C₈×C₁₇	C₄×Q₈	1	4	yes	—
90	C₈×C₁₇	C₄×Q₈	1	4	yes	—
91	C₈×C₁₇	C₄⋊Q₈	1	4	yes	—
92	#3622	—	4	4	no	—
93	#3160	—	17	1	no	—
94	#2960	—	17	2	no	—
95	#1994	—	17	2	no	—
96	#1946	—	17	4	no	—
97	#1848	—	17	1	no	—
98	#1773	—	17	4	no	—
99	C₄×#97	D₁₇	1	1	yes	characteristic
100	C₂²×C₁₇	—	2	2	no	—
101	C₂²×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
102	C₂²×C₁₇	#533	1	2	yes	—
103	C₄×C₁₇	—	2	2	no	—
104	C₄×C₁₇	—	4	4	no	—
105	C₄×C₁₇	#444	1	4	yes	—
106	C₄×C₁₇	C₂×C₄×Q₈	1	2	yes	—
107	C₄×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
108	C₄×C₁₇	#532	1	1	yes	characteristic
109	Dic₁₇	—	4	8	no	—
110	#533	—	17	2	no	—
111	#500	—	17	2	no	—
112	#499	—	17	4	no	—
113	#498	—	17	2	no	—
114	#446	—	34	4	no	—
115	C₈×Q₈	—	34	4	no	—
116	C₂×#108	—	17	4	no	—
117	#427	—	17	4	no	—
118	#424	—	17	4	no	—
119	#423	—	17	2	no	—
120	#405	—	17	4	no	—
121	C₂×#97	C₂×D₁₇	1	2	yes	—
122	C₂×#97	C₂×D₁₇	1	2	yes	—
123	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
124	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
125	C₄³	C₂×D₁₇	1	1	yes	characteristic
126	#339	—	17	8	no	—
127	C₂×C₁₇	—	2	2	no	—
128	C₂×C₁₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
129	C₂×C₁₇	C₂×#444	1	2	yes	—
130	C₂×M₁₆	—	17	4	no	—
131	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
132	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
133	C₂²×C₈	C₂²×D₁₇	1	2	yes	—
134	C₄⋊Q₈	—	17	4	no	—
135	#125	—	17	4	no	—
136	C₂²⋊Q₈	—	34	4	no	—
137	C₄×Q₈	—	34	4	no	—
138	#117	—	17	4	no	—
139	C₂×C₄²	—	2	2	no	—
140	C₂×C₄²	C₂²×D₁₇	1	2	yes	—
141	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
142	#108	—	34	8	no	—
143	#107	—	17	8	no	—
144	#106	—	17	8	no	—
145	C₄⋊C₈	—	34	8	no	—
146	C₄⋊C₈	—	34	4	no	—
147	#97	—	2	4	no	—
148	#97	—	34	4	no	—
149	#97	Dic₃₄	1	8	yes	complemented
150	C₄×C₈	—	34	4	no	—
151	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
152	C₄×C₈	C₄×D₁₇	1	2	yes	—
153	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
154	C₄×C₈	D₆₈	1	4	yes	—
155	C₁₇	#21734	1	1	yes	characteristic, complemented, Sylow
156	C₂×Q₈	—	34	4	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	#63252	1	2	yes	—
159	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
160	M₁₆	—	34	8	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	34	4	no	—
163	C₂×C₈	C₂×Dic₃₄	1	4	yes	—
164	C₂×C₈	C₂×Dic₃₄	1	4	yes	—
165	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
166	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
167	C₂×C₈	C₂×D₆₈	1	2	yes	—
168	C₂×C₈	C₂×D₆₈	1	2	yes	—
169	C₂×C₈	#63252	1	4	yes	—
170	C₄⋊C₄	—	17	8	no	—
171	C₄⋊C₄	—	34	8	no	—
172	C₄⋊C₄	—	34	4	no	—
173	C₂²⋊C₄	—	34	4	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	4	4	no	—
176	C₄²	—	34	4	no	—
177	C₄²	C₂×Dic₃₄	1	4	yes	—
178	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
179	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
180	C₄²	C₂×D₆₈	1	2	yes	—
181	C₂³	C₂×#63252	1	1	yes	characteristic
182	Q₈	—	68	8	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	34	8	no	—
187	C₂×C₄	C₄×D₆₈	1	2	yes	—
188	C₂×C₄	C₄×D₆₈	1	2	yes	—
189	C₂×C₄	C₂²×Dic₃₄	1	2	yes	—
190	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
191	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
192	C₂×C₄	C₂×#63252	1	2	yes	—
193	C₈	—	2	2	no	—
194	C₈	—	68	4	no	—
195	C₈	C₄×Dic₃₄	1	4	yes	—
196	C₈	C₄×Dic₃₄	1	4	yes	—
197	C₈	#10587555	1	4	yes	—
198	C₂²	—	2	2	no	—
199	C₂²	C₂×C₄×D₆₈	1	1	yes	characteristic
200	C₂²	#11763366	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	4	4	no	—
203	C₄	—	68	8	no	—
204	C₄	#11762932	1	4	yes	—
205	C₄	C₂×C₄×Dic₃₄	1	2	yes	—
206	C₄	C₂×#10587555	1	1	yes	characteristic
207	C₄	#11763359	1	1	yes	characteristic
208	C₂	—	2	2	no	—
209	C₂	C₄×#10587555	1	1	yes	characteristic
210	C₂	C₂×#11762932	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₆₈
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	C₄×C₁₇×#97	C₂
socle	C₂²×C₁₇	C₂×C₄×D₄
prime radical	C₂³×C₁₇	C₂×#41
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	C₄×#97	D₁₇
hypocenter	C₁₇	#21734
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
9. #11763214	C₄	semidirect
10. #11763211	C₄	semidirect
11. #11763207	C₄	semidirect
38. C₁₇×#97	Q₈	semidirect
149. #97	Dic₃₄	semidirect
155. C₁₇	#21734	semidirect

Maximal subgroups

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

subgroup	index
#14539298	2
#14537718	2
#14529761	2
#14529537	2
#14528956	2
#14528668	2
C₄×C₁₇×#97	2
#21734	17