Subgroups — #17043171

Order 4352.

702 conjugacy classes of subgroups.

The subgroup lattice has 218 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	#14529532	C₂	1	4	yes	maximal
5	#14528956	C₂	1	1	yes	maximal, characteristic
6	#14528663	C₂	1	4	yes	maximal
7	C₄²×C₈×C₁₇	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×C₄×C₈×C₁₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₇	C₂²	1	2	yes	—
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	C₄×C₈×C₁₇	—	2	4	no	—
38	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₁₇	D₄	1	2	yes	—
42	C₄×C₈×C₁₇	D₄	1	2	yes	—
43	C₄×C₈×C₁₇	Q₈	1	8	yes	complemented
44	#10587490	—	2	8	no	—
45	#10587489	C₂×C₄	1	8	yes	—
46	#10587488	C₂×C₄	1	8	yes	—
47	#10587486	—	2	4	no	—
48	#10587485	—	2	8	no	—
49	C₈×Dic₁₇	—	2	4	no	—
50	C₄⋊#3622	—	2	4	no	—
51	C₂²×C₄×C₁₇	—	2	2	no	—
52	C₂²×C₄×C₁₇	#41	1	2	yes	—
53	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₃₄	—	2	4	no	—
55	C₂×C₈×C₁₇	—	2	2	no	—
56	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
58	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₇	C₂×Q₈	1	4	yes	—
62	C₂×C₈×C₁₇	C₂×Q₈	1	4	yes	—
63	C₂×C₈×C₁₇	#41	1	4	yes	—
64	C₄²×C₁₇	—	2	8	no	—
65	C₄²×C₁₇	—	2	2	no	—
66	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
67	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₁₇	C₂×D₄	1	2	yes	—
69	C₄²×C₁₇	C₂×Q₈	1	4	yes	—
70	C₂²⋊Dic₁₇	—	2	4	no	—
71	C₄⋊Dic₁₇	—	2	4	no	—
72	C₄⋊Dic₁₇	C₂²×C₄	1	8	yes	—
73	#63225	—	2	8	no	—
74	C₄×Dic₁₇	—	2	4	no	—
75	#63223	—	2	8	no	—
76	C₂×#3622	—	2	4	no	—
77	#21732	—	17	1	no	maximal, Sylow
78	C₂³×C₁₇	C₂×#41	1	1	yes	characteristic
79	C₂×C₄×C₁₇	—	2	8	no	—
80	C₂×C₄×C₁₇	—	2	4	no	—
81	C₂×C₄×C₁₇	—	2	4	no	—
82	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
83	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₇	C₂²×Q₈	1	2	yes	—
87	C₂×C₄×C₁₇	C₂×#41	1	2	yes	—
88	C₂×Dic₁₇	—	2	8	no	—
89	Dic₃₄	—	4	8	no	—
90	C₈×C₁₇	—	2	2	no	—
91	C₈×C₁₇	C₄×Q₈	1	4	yes	—
92	C₈×C₁₇	C₄×Q₈	1	4	yes	—
93	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
94	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
95	#3622	—	4	4	no	—
96	#3160	—	17	1	no	—
97	#2960	—	17	2	no	—
98	#1994	—	17	2	no	—
99	#1945	—	17	4	no	—
100	#1848	—	17	1	no	—
101	#1772	—	17	4	no	—
102	C₄²×C₈	D₁₇	1	1	yes	characteristic
103	C₂²×C₁₇	—	2	2	no	—
104	C₂²×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
105	C₂²×C₁₇	#533	1	2	yes	—
106	C₄×C₁₇	—	2	8	no	—
107	C₄×C₁₇	—	2	2	no	—
108	C₄×C₁₇	#443	1	4	yes	—
109	C₄×C₁₇	C₂×C₄×Q₈	1	2	yes	—
110	C₄×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
111	C₄×C₁₇	#532	1	1	yes	characteristic
112	Dic₁₇	—	4	8	no	—
113	#533	—	17	2	no	—
114	#500	—	17	2	no	—
115	#499	—	17	4	no	—
116	#498	—	17	2	no	—
117	#446	—	34	4	no	—
118	C₈×Q₈	—	34	4	no	—
119	C₂×#108	—	17	4	no	—
120	#427	—	17	4	no	—
121	#424	—	17	4	no	—
122	#423	—	17	2	no	—
123	#405	—	17	4	no	—
124	C₂×C₄×C₈	C₂×D₁₇	1	2	yes	—
125	C₂×C₄×C₈	C₂×D₁₇	1	2	yes	—
126	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
127	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
128	C₄³	C₂×D₁₇	1	1	yes	characteristic
129	#339	—	17	8	no	—
130	C₂×C₁₇	—	2	2	no	—
131	C₂×C₁₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
132	C₂×C₁₇	C₂×#443	1	2	yes	—
133	C₂×M₁₆	—	17	4	no	—
134	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₁₇	1	2	yes	—
137	C₄⋊Q₈	—	17	4	no	—
138	#125	—	17	4	no	—
139	C₂²⋊Q₈	—	34	4	no	—
140	C₄×Q₈	—	34	4	no	—
141	#117	—	17	4	no	—
142	C₂×C₄²	—	2	2	no	—
143	C₂×C₄²	C₂²×D₁₇	1	2	yes	—
144	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
145	#108	—	34	8	no	—
146	#107	—	17	8	no	—
147	#106	—	17	8	no	—
148	C₄⋊C₈	—	34	8	no	—
149	C₄⋊C₈	—	34	4	no	—
150	#97	—	34	4	no	—
151	C₄×C₈	—	2	4	no	—
152	C₄×C₈	—	34	4	no	—
153	C₄×C₈	Dic₃₄	1	8	yes	complemented
154	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
155	C₄×C₈	C₄×D₁₇	1	2	yes	—
156	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
157	C₄×C₈	D₆₈	1	2	yes	—
158	C₄×C₈	D₆₈	1	2	yes	—
159	C₁₇	#21732	1	1	yes	characteristic, complemented, Sylow
160	C₂×Q₈	—	34	4	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	#63252	1	2	yes	—
163	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
164	M₁₆	—	34	8	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	34	4	no	—
167	C₂×C₈	C₂×Dic₃₄	1	4	yes	—
168	C₂×C₈	C₂×Dic₃₄	1	4	yes	—
169	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
170	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
171	C₂×C₈	C₂×D₆₈	1	2	yes	—
172	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
173	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
174	C₂×C₈	#63252	1	4	yes	—
175	C₄⋊C₄	—	17	8	no	—
176	C₄⋊C₄	—	34	8	no	—
177	C₄⋊C₄	—	34	4	no	—
178	C₂²⋊C₄	—	34	4	no	—
179	C₄²	—	2	8	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	34	4	no	—
182	C₄²	C₂×Dic₃₄	1	4	yes	—
183	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
184	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
185	C₄²	C₂×D₆₈	1	2	yes	—
186	C₂³	C₂×#63252	1	1	yes	characteristic
187	Q₈	—	68	8	no	—
188	C₂×C₄	—	2	8	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	34	8	no	—
192	C₂×C₄	C₄×D₆₈	1	2	yes	—
193	C₂×C₄	C₄×D₆₈	1	2	yes	—
194	C₂×C₄	C₂²×Dic₃₄	1	2	yes	—
195	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
196	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
197	C₂×C₄	C₂×#63252	1	2	yes	—
198	C₈	—	2	2	no	—
199	C₈	—	68	4	no	—
200	C₈	C₄×Dic₃₄	1	4	yes	—
201	C₈	C₄×Dic₃₄	1	4	yes	—
202	C₈	#10587555	1	2	yes	—
203	C₈	#10587555	1	2	yes	—
204	C₂²	—	2	2	no	—
205	C₂²	C₂×C₄×D₆₈	1	1	yes	characteristic
206	C₂²	#11763366	1	2	yes	—
207	C₄	—	2	8	no	—
208	C₄	—	2	2	no	—
209	C₄	—	68	8	no	—
210	C₄	#11762927	1	4	yes	—
211	C₄	C₂×C₄×Dic₃₄	1	2	yes	—
212	C₄	C₂×#10587555	1	1	yes	characteristic
213	C₄	#11763359	1	1	yes	characteristic
214	C₂	—	2	2	no	—
215	C₂	C₄×#10587555	1	1	yes	characteristic
216	C₂	C₂×#11762927	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₆₈
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	C₄²×C₈×C₁₇	C₂
socle	C₂²×C₁₇	C₂×C₄×D₄
prime radical	C₂³×C₁₇	C₂×#41
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	C₄²×C₈	D₁₇
hypocenter	C₁₇	#21732
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

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
43. C₄×C₈×C₁₇	Q₈	semidirect
153. C₄×C₈	Dic₃₄	semidirect
159. C₁₇	#21732	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
#14529532	2
#14528956	2
#14528663	2
C₄²×C₈×C₁₇	2
#21732	17