Subgroups — #17953769

Order 4352.

362 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544717	C₂	1	1	yes	maximal, characteristic
2	#14544716	C₂	1	1	yes	maximal, characteristic
3	#14543614	C₂	1	4	yes	maximal
4	#14528528	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587544	—	2	1	no	—
6	C₂×#10587544	—	2	1	no	—
7	C₂×C₄⋊#3624	—	2	1	no	—
8	C₂×C₄⋊#3624	—	2	1	no	—
9	C₂×C₄⋊#3624	C₂²	1	2	yes	—
10	C₂×C₄⋊#3624	C₂²	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	—	2	2	no	—
17	C₂²×#3624	—	2	2	no	—
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	D₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₇	Q₈	1	1	yes	characteristic
24	C₂×C₄×Dic₁₇	Q₈	1	1	yes	characteristic
25	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
26	C₂²×#3622	C₂×C₄	1	2	yes	—
27	#10587544	—	4	2	no	—
28	#10587544	—	4	2	no	—
29	C₄⋊#3624	—	2	4	no	—
30	C₄⋊#3624	—	2	4	no	—
31	C₄⋊#3624	—	4	2	no	—
32	C₄⋊#3624	—	4	2	no	—
33	C₈×Dic₁₇	—	2	2	no	—
34	C₈×Dic₁₇	—	2	2	no	—
35	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×Dic₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×#3624	—	4	2	no	—
43	C₂×#3624	—	4	2	no	—
44	C₂×#3624	—	4	2	no	—
45	C₂×#3624	—	4	2	no	—
46	C₂×#3624	—	4	2	no	—
47	C₂×#3624	—	4	2	no	—
48	C₂×C₈×C₁₇	—	2	2	no	—
49	C₂×C₈×C₁₇	—	2	2	no	—
50	C₂×C₈×C₁₇	—	2	2	no	—
51	C₄²×C₁₇	—	2	2	no	—
52	C₄×Dic₁₇	—	2	1	no	—
53	C₄×Dic₁₇	—	2	1	no	—
54	C₄×Dic₁₇	—	4	1	no	—
55	C₄×Dic₁₇	—	4	1	no	—
56	C₄×Dic₁₇	D₈	1	2	yes	—
57	C₄×Dic₁₇	SD₁₆	1	2	yes	—
58	C₄×Dic₁₇	SD₁₆	1	2	yes	—
59	C₄×Dic₁₇	Q₁₆	1	2	yes	—
60	C₄×Dic₁₇	#41	1	2	yes	—
61	C₄×Dic₁₇	#41	1	2	yes	—
62	C₂×#3622	—	2	2	no	—
63	C₂×#3622	—	2	2	no	—
64	C₂×#3622	—	2	2	no	—
65	#8787	—	17	1	no	maximal, Sylow
66	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₇	—	2	1	no	—
68	C₂×C₄×C₁₇	—	2	1	no	—
69	C₂×C₄×C₁₇	—	2	1	no	—
70	C₂×C₄×C₁₇	—	2	1	no	—
71	C₂×C₄×C₁₇	#106	1	2	yes	—
72	C₂×C₄×C₁₇	#107	1	2	yes	—
73	C₂×C₄×C₁₇	#117	1	2	yes	—
74	C₂×C₄×C₁₇	#131	1	2	yes	—
75	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
76	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
77	C₂×Dic₁₇	—	2	1	no	—
78	C₂×Dic₁₇	—	2	1	no	—
79	C₂×Dic₁₇	—	2	1	no	—
80	C₂×Dic₁₇	—	2	1	no	—
81	C₂×Dic₁₇	—	2	1	no	—
82	C₂×Dic₁₇	—	2	1	no	—
83	C₂×Dic₁₇	#106	1	2	yes	—
84	C₂×Dic₁₇	#107	1	2	yes	—
85	C₂×Dic₁₇	#124	1	1	yes	characteristic
86	C₂×Dic₁₇	#124	1	1	yes	characteristic
87	C₂×Dic₁₇	#125	1	2	yes	—
88	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
89	C₂×Dic₁₇	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×Dic₁₇	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×Dic₁₇	C₂×Q₁₆	1	1	yes	characteristic
92	#3624	—	8	2	no	—
93	#3624	—	8	2	no	—
94	C₈×C₁₇	—	4	2	no	—
95	#3622	—	4	2	no	—
96	#1842	—	17	1	no	—
97	#1842	—	17	1	no	—
98	#1744	—	17	1	no	—
99	#1277	—	17	4	no	—
100	C₂²×C₁₇	#382	1	1	yes	characteristic
101	C₂²×C₁₇	#424	1	1	yes	characteristic
102	C₂²×C₁₇	#424	1	1	yes	characteristic
103	C₂²×C₁₇	C₂×#106	1	1	yes	characteristic
104	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
105	C₂²×C₁₇	#430	1	1	yes	characteristic
106	C₂²×C₁₇	#430	1	1	yes	characteristic
107	C₄×C₁₇	—	2	2	no	—
108	C₄×C₁₇	—	2	2	no	—
109	C₄×C₁₇	#433	1	2	yes	—
110	C₄×C₁₇	#433	1	2	yes	—
111	Dic₁₇	—	2	2	no	—
112	Dic₁₇	—	2	2	no	—
113	Dic₁₇	—	2	2	no	—
114	Dic₁₇	#486	1	1	yes	characteristic
115	Dic₁₇	#486	1	1	yes	characteristic
116	Dic₁₇	#487	1	1	yes	characteristic
117	Dic₁₇	#487	1	1	yes	characteristic
118	C₂×C₄⋊C₈	—	17	2	no	—
119	C₂×C₄⋊C₈	—	17	2	no	—
120	C₂×C₄⋊C₈	—	34	1	no	—
121	C₂×C₄⋊C₈	—	34	1	no	—
122	C₂×C₄⋊C₈	—	34	1	no	—
123	C₂×C₄⋊C₈	—	34	1	no	—
124	C₂×C₄×C₈	—	17	2	no	—
125	C₄³	—	17	1	no	—
126	#336	—	17	2	no	—
127	#336	—	17	1	no	—
128	#336	#595	1	1	yes	characteristic
129	C₂×C₁₇	#1570	1	1	yes	characteristic
130	C₂×C₁₇	#1570	1	1	yes	characteristic
131	C₂×C₁₇	#1571	1	1	yes	characteristic
132	C₂×C₁₇	#1571	1	1	yes	characteristic
133	C₂×C₁₇	#1832	1	1	yes	characteristic
134	C₂×C₁₇	#1834	1	1	yes	characteristic
135	C₂×C₁₇	#1838	1	1	yes	characteristic
136	C₂²×C₈	—	17	2	no	—
137	C₂²×C₈	—	34	2	no	—
138	C₂²×C₈	—	34	2	no	—
139	C₂²×C₈	C₂×#595	1	2	yes	—
140	C₂×C₄²	—	17	1	no	—
141	C₂×C₄²	—	17	1	no	—
142	C₂×C₄²	—	17	1	no	—
143	C₂×C₄²	—	17	1	no	—
144	C₂×C₄²	—	17	1	no	—
145	C₂×C₄²	—	17	1	no	—
146	C₂×C₄²	C₂×#595	1	1	yes	characteristic
147	C₄⋊C₈	—	34	4	no	—
148	C₄⋊C₈	—	34	4	no	—
149	C₄⋊C₈	—	68	2	no	—
150	C₄⋊C₈	—	68	2	no	—
151	C₄⋊C₈	—	68	2	no	—
152	C₄⋊C₈	—	68	2	no	—
153	C₄×C₈	—	34	2	no	—
154	C₄×C₈	—	34	2	no	—
155	C₁₇	#8787	1	1	yes	characteristic, complemented, Sylow
156	C₂²×C₄	—	17	1	no	—
157	C₂²×C₄	—	17	1	no	—
158	C₂²×C₄	—	17	1	no	—
159	C₂²×C₄	—	17	1	no	—
160	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
161	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
162	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	34	2	no	—
167	C₂×C₈	—	34	2	no	—
168	C₂×C₈	—	34	2	no	—
169	C₂×C₈	—	68	2	no	—
170	C₂×C₈	—	68	2	no	—
171	C₂×C₈	—	68	2	no	—
172	C₂×C₈	—	68	2	no	—
173	C₂×C₈	—	68	2	no	—
174	C₂×C₈	—	68	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	—	17	2	no	—
177	C₄²	—	17	2	no	—
178	C₄²	—	17	2	no	—
179	C₄²	—	17	2	no	—
180	C₄²	—	17	2	no	—
181	C₄²	—	17	2	no	—
182	C₄²	—	34	1	no	—
183	C₄²	—	34	1	no	—
184	C₄²	—	68	1	no	—
185	C₄²	—	68	1	no	—
186	C₂³	C₂×C₄⋊#595	1	1	yes	characteristic
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	17	1	no	—
192	C₂×C₄	—	17	1	no	—
193	C₂×C₄	—	17	1	no	—
194	C₂×C₄	—	17	1	no	—
195	C₂×C₄	—	17	1	no	—
196	C₂×C₄	—	17	1	no	—
197	C₂×C₄	—	17	2	no	—
198	C₂×C₄	—	17	2	no	—
199	C₂×C₄	—	17	2	no	—
200	C₂×C₄	—	34	1	no	—
201	C₂×C₄	—	34	1	no	—
202	C₂×C₄	—	34	1	no	—
203	C₂×C₄	—	34	1	no	—
204	C₂×C₄	—	34	1	no	—
205	C₂×C₄	—	34	1	no	—
206	C₂×C₄	#10587533	1	2	yes	—
207	C₂×C₄	#10587534	1	2	yes	—
208	C₂×C₄	#10587677	1	2	yes	—
209	C₂×C₄	#10587678	1	1	yes	characteristic
210	C₂×C₄	#10587678	1	2	yes	—
211	C₂×C₄	#10587680	1	1	yes	characteristic
212	C₈	—	4	2	no	—
213	C₈	—	68	2	no	—
214	C₈	—	136	2	no	—
215	C₈	—	136	2	no	—
216	C₂²	#11763753	1	1	yes	characteristic
217	C₂²	#11763815	1	1	yes	characteristic
218	C₂²	#11763816	1	1	yes	characteristic
219	C₂²	C₂×#10587533	1	1	yes	characteristic
220	C₂²	C₂×#10587534	1	1	yes	characteristic
221	C₂²	#11763824	1	1	yes	characteristic
222	C₂²	#11763824	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	17	1	no	—
226	C₄	—	17	1	no	—
227	C₄	—	17	1	no	—
228	C₄	—	17	1	no	—
229	C₄	—	34	2	no	—
230	C₄	—	34	2	no	—
231	C₄	—	34	2	no	—
232	C₄	#11763833	1	2	yes	—
233	C₄	#11763834	1	2	yes	—
234	C₂	#14544351	1	1	yes	characteristic
235	C₂	#14544352	1	1	yes	characteristic
236	C₂	#14544353	1	1	yes	characteristic
237	C₂	#14544354	1	1	yes	characteristic
238	C₂	#14544702	1	1	yes	characteristic
239	C₂	#14544705	1	1	yes	characteristic
240	C₂	#14544710	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#595
commutator	C₂²×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₁₇×#336	C₄
socle, prime radical	C₂³×C₁₇	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	#336	#595
hypocenter	C₁₇	#8787
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

Complements

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

N	H ≅ G/N	type
155. C₁₇	#8787	semidirect

Maximal subgroups

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

subgroup	index
#14544717	2
#14544716	2
#14543614	2
#14528528	2
#8787	17