Subgroups — #17951693

Order 4352.

446 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	#14544499	C₂	1	1	yes	maximal, characteristic
2	C₄²×#3624	C₂	1	1	yes	maximal, characteristic
3	#14543606	C₂	1	4	yes	maximal
4	#14528528	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×#3624	C₄	1	2	yes	—
6	C₂×C₄×#3624	C₄	1	2	yes	—
7	C₂×C₄×#3624	C₂²	1	2	yes	—
8	C₂×C₄×#3624	C₂²	1	2	yes	—
9	#11763675	C₄	1	2	yes	—
10	#11763675	C₄	1	2	yes	—
11	C₂×C₈×Dic₁₇	C₂²	1	2	yes	—
12	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
13	#11762564	C₄	1	1	yes	characteristic
14	#11762563	C₄	1	2	yes	—
15	C₁₇×#336	C₄	1	1	yes	characteristic
16	C₂²×#3624	C₂×C₄	1	4	yes	—
17	C₂²×#3624	C₂×C₄	1	4	yes	—
18	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
26	C₂²×#3622	C₂×C₄	1	2	yes	—
27	C₄×#3624	—	2	4	no	—
28	C₄×#3624	—	2	4	no	—
29	C₄×#3624	—	2	4	no	—
30	C₄×#3624	C₈	1	8	yes	complemented
31	C₈×Dic₁₇	—	2	4	no	—
32	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
34	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
35	C₂²×Dic₁₇	C₄²	1	1	yes	characteristic
36	C₂²×Dic₁₇	C₄²	1	1	yes	characteristic
37	C₂²×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×#3624	—	2	4	no	—
40	C₂×#3624	—	2	4	no	—
41	C₂×#3624	—	2	4	no	—
42	C₂×#3624	—	2	4	no	—
43	C₂×#3624	—	2	4	no	—
44	C₂×#3624	C₂×C₈	1	8	yes	—
45	C₂×C₈×C₁₇	—	2	2	no	—
46	C₂×C₈×C₁₇	—	2	2	no	—
47	C₂×C₈×C₁₇	—	2	2	no	—
48	C₄²×C₁₇	—	2	2	no	—
49	C₄×Dic₁₇	—	2	1	no	—
50	C₄×Dic₁₇	—	2	1	no	—
51	C₄×Dic₁₇	—	2	2	no	—
52	C₄×Dic₁₇	—	2	2	no	—
53	C₄×Dic₁₇	C₂×C₈	1	4	yes	—
54	C₄×Dic₁₇	M₁₆	1	4	yes	—
55	C₄×Dic₁₇	#41	1	4	yes	—
56	C₂×#3622	—	2	2	no	—
57	C₂×#3622	—	2	2	no	—
58	C₂×#3622	—	2	2	no	—
59	#8133	—	17	1	no	maximal, Sylow
60	C₂³×C₁₇	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₁₇	—	2	1	no	—
62	C₂×C₄×C₁₇	—	2	1	no	—
63	C₂×C₄×C₁₇	—	2	1	no	—
64	C₂×C₄×C₁₇	—	2	1	no	—
65	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
66	C₂×C₄×C₁₇	#97	1	2	yes	—
67	C₂×C₄×C₁₇	#117	1	2	yes	—
68	C₂×C₄×C₁₇	#131	1	2	yes	—
69	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
70	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
71	C₂×Dic₁₇	—	2	1	no	—
72	C₂×Dic₁₇	—	2	1	no	—
73	C₂×Dic₁₇	—	2	1	no	—
74	C₂×Dic₁₇	—	2	1	no	—
75	C₂×Dic₁₇	C₄×C₈	1	2	yes	—
76	C₂×Dic₁₇	#97	1	2	yes	—
77	C₂×Dic₁₇	#117	1	2	yes	—
78	C₂×Dic₁₇	#117	1	2	yes	—
79	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
80	C₂×Dic₁₇	C₂×M₁₆	1	2	yes	—
81	C₂×Dic₁₇	#131	1	2	yes	—
82	C₂×Dic₁₇	#131	1	2	yes	—
83	#3624	—	2	8	no	—
84	#3624	—	4	4	no	—
85	C₈×C₁₇	—	4	2	no	—
86	#3622	—	4	2	no	—
87	#1744	—	17	1	no	—
88	#1744	—	17	1	no	—
89	C₄²×C₈	—	17	1	no	—
90	#1273	—	17	4	no	—
91	C₂²×C₁₇	#376	1	1	yes	characteristic
92	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
93	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
94	C₂²×C₁₇	#405	1	1	yes	characteristic
95	C₂²×C₁₇	#405	1	1	yes	characteristic
96	C₂²×C₁₇	#405	1	1	yes	characteristic
97	C₂²×C₁₇	#405	1	1	yes	characteristic
98	C₄×C₁₇	—	2	2	no	—
99	C₄×C₁₇	—	2	2	no	—
100	C₄×C₁₇	#433	1	2	yes	—
101	C₄×C₁₇	#433	1	2	yes	—
102	Dic₁₇	—	2	2	no	—
103	Dic₁₇	—	2	2	no	—
104	Dic₁₇	#431	1	4	yes	—
105	Dic₁₇	#433	1	4	yes	—
106	C₂×C₄×C₈	—	17	2	no	—
107	C₂×C₄×C₈	—	17	2	no	—
108	C₂×C₄×C₈	—	17	2	no	—
109	C₂×C₄×C₈	—	17	2	no	—
110	C₂×C₄×C₈	—	17	2	no	—
111	C₄³	—	17	1	no	—
112	#336	—	17	2	no	—
113	#336	—	17	2	no	—
114	#336	—	17	2	no	—
115	#336	—	17	1	no	—
116	#336	#595	1	1	yes	characteristic
117	C₂×C₁₇	#1450	1	1	yes	characteristic
118	C₂×C₁₇	#1450	1	1	yes	characteristic
119	C₂×C₁₇	#1452	1	1	yes	characteristic
120	C₂×C₁₇	#1452	1	1	yes	characteristic
121	C₂×C₁₇	#1744	1	1	yes	characteristic
122	C₂×C₁₇	#1746	1	1	yes	characteristic
123	C₂×C₁₇	#1746	1	1	yes	characteristic
124	C₂²×C₈	—	17	4	no	—
125	C₂²×C₈	—	17	4	no	—
126	C₂²×C₈	—	17	2	no	—
127	C₂²×C₈	C₂×#595	1	2	yes	—
128	C₂×C₄²	—	17	1	no	—
129	C₂×C₄²	—	17	1	no	—
130	C₂×C₄²	—	17	1	no	—
131	C₂×C₄²	—	17	1	no	—
132	C₂×C₄²	—	17	1	no	—
133	C₂×C₄²	—	17	1	no	—
134	C₂×C₄²	C₂×#595	1	1	yes	characteristic
135	C₄×C₈	—	17	8	no	—
136	C₄×C₈	—	34	4	no	—
137	C₄×C₈	—	34	4	no	—
138	C₄×C₈	—	34	4	no	—
139	C₄×C₈	—	34	4	no	—
140	C₁₇	#8133	1	1	yes	characteristic, complemented, Sylow
141	C₂²×C₄	—	17	1	no	—
142	C₂²×C₄	—	17	1	no	—
143	C₂²×C₄	—	17	1	no	—
144	C₂²×C₄	—	17	1	no	—
145	C₂²×C₄	C₄×#595	1	1	yes	characteristic
146	C₂²×C₄	C₄×#595	1	1	yes	characteristic
147	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	17	8	no	—
152	C₂×C₈	—	34	4	no	—
153	C₂×C₈	—	34	4	no	—
154	C₂×C₈	—	34	4	no	—
155	C₂×C₈	—	34	4	no	—
156	C₂×C₈	—	34	4	no	—
157	C₂×C₈	—	34	2	no	—
158	C₂×C₈	—	34	2	no	—
159	C₂×C₈	—	34	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	17	4	no	—
162	C₄²	—	17	4	no	—
163	C₄²	—	17	4	no	—
164	C₄²	—	34	2	no	—
165	C₄²	—	34	2	no	—
166	C₄²	—	34	1	no	—
167	C₄²	—	34	1	no	—
168	C₂³	C₂×C₄×#595	1	1	yes	characteristic
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	17	2	no	—
174	C₂×C₄	—	17	2	no	—
175	C₂×C₄	—	17	2	no	—
176	C₂×C₄	—	17	2	no	—
177	C₂×C₄	—	17	2	no	—
178	C₂×C₄	—	17	2	no	—
179	C₂×C₄	—	17	2	no	—
180	C₂×C₄	—	17	2	no	—
181	C₂×C₄	—	34	1	no	—
182	C₂×C₄	—	34	1	no	—
183	C₂×C₄	—	34	1	no	—
184	C₂×C₄	—	34	1	no	—
185	C₂×C₄	C₈×#595	1	2	yes	—
186	C₂×C₄	#10587532	1	2	yes	—
187	C₂×C₄	#10587677	1	2	yes	—
188	C₂×C₄	#10587678	1	1	yes	characteristic
189	C₂×C₄	#10587678	1	2	yes	—
190	C₂×C₄	#10587680	1	1	yes	characteristic
191	C₈	—	4	2	no	—
192	C₈	—	34	8	no	—
193	C₈	—	68	4	no	—
194	C₈	—	68	2	no	—
195	C₂²	#11763745	1	1	yes	characteristic
196	C₂²	C₂×C₈×#595	1	1	yes	characteristic
197	C₂²	C₂×#10587532	1	1	yes	characteristic
198	C₂²	#11763773	1	1	yes	characteristic
199	C₂²	#11763774	1	1	yes	characteristic
200	C₂²	#11763775	1	1	yes	characteristic
201	C₂²	#11763775	1	1	yes	characteristic
202	C₄	—	2	2	no	—
203	C₄	—	2	2	no	—
204	C₄	—	17	4	no	—
205	C₄	—	17	4	no	—
206	C₄	—	34	2	no	—
207	C₄	—	34	2	no	—
208	C₄	#11763833	1	2	yes	—
209	C₄	#11763834	1	2	yes	—
210	C₂	#14543989	1	1	yes	characteristic
211	C₂	#14543990	1	1	yes	characteristic
212	C₂	#14543999	1	1	yes	characteristic
213	C₂	#14544000	1	1	yes	characteristic
214	C₂	#14544497	1	1	yes	characteristic
215	C₂	#14544503	1	1	yes	characteristic
216	C₂	#14544504	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#595
commutator	C₂²×C₁₇	C₂×C₄×C₈
Fitting	C₁₇×#336	C₄
socle, prime radical	C₂³×C₁₇	C₂×C₄²
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	#336	#595
hypocenter	C₁₇	#8133
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ #336

Complements

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

N	H ≅ G/N	type
30. C₄×#3624	C₈	semidirect
140. C₁₇	#8133	semidirect

Maximal subgroups

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

subgroup	index
#14544499	2
C₄²×#3624	2
#14543606	2
#14528528	2
#8133	17