Subgroups — #17950283

Order 4352.

518 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14544498	C₂	1	1	yes	maximal, characteristic
2	C₄×#10587542	C₂	1	1	yes	maximal, characteristic
3	C₂×#11763651	C₂	1	1	yes	maximal, characteristic
4	C₂×#11763651	C₂	1	1	yes	maximal, characteristic
5	C₂×#11763650	C₂	1	1	yes	maximal, characteristic
6	C₂×C₈×#3624	C₂	1	1	yes	maximal, characteristic
7	#14528528	C₂	1	1	yes	maximal, characteristic
8	C₂×#10587542	C₄	1	4	yes	—
9	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
10	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
11	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
12	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
13	#11763675	C₄	1	4	yes	—
14	#11763651	C₄	1	4	yes	complemented
15	#11763651	C₄	1	4	yes	complemented
16	#11763650	C₄	1	4	yes	complemented
17	C₈×#3624	C₄	1	4	yes	complemented
18	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
20	C₄²×Dic₁₇	C₂²	1	1	yes	characteristic
21	#11762564	C₄	1	1	yes	characteristic
22	#11762563	C₄	1	1	yes	characteristic
23	#11762563	C₄	1	1	yes	characteristic
24	C₁₇×#336	C₄	1	1	yes	characteristic
25	C₂²×#3624	C₂×C₄	1	2	yes	—
26	C₂²×#3624	C₂×C₄	1	2	yes	—
27	C₂²×#3624	C₂×C₄	1	2	yes	—
28	C₂²×#3624	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
34	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
35	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
36	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
37	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
38	#10587542	—	2	8	no	—
39	C₄×#3624	C₂×C₄	1	2	yes	—
40	C₄×#3624	C₂×C₄	1	2	yes	—
41	C₄×#3624	C₂×C₄	1	2	yes	—
42	C₄×#3624	C₂×C₄	1	2	yes	—
43	C₄×#3624	C₂×C₄	1	2	yes	—
44	C₄×#3624	C₂×C₄	1	2	yes	—
45	C₄×#3624	C₂×C₄	1	2	yes	—
46	C₄×#3624	C₂×C₄	1	2	yes	—
47	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
48	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
49	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
50	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
51	C₂²×C₄×C₁₇	C₄²	1	2	yes	—
52	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×Dic₁₇	C₄²	1	2	yes	—
54	C₂²×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×#3624	—	2	4	no	—
57	C₂×#3624	—	2	4	no	—
58	C₂×#3624	—	2	4	no	—
59	C₂×#3624	C₄²	1	4	yes	—
60	C₂×#3624	C₄²	1	4	yes	—
61	C₂×#3624	C₄²	1	4	yes	—
62	C₂×#3624	C₄²	1	4	yes	—
63	C₂×#3624	M₁₆	1	8	yes	—
64	C₂×C₈×C₁₇	—	2	1	no	—
65	C₂×C₈×C₁₇	—	2	1	no	—
66	C₂×C₈×C₁₇	—	2	1	no	—
67	C₂×C₈×C₁₇	—	2	1	no	—
68	C₂×C₈×C₁₇	C₄²	1	2	yes	—
69	C₂×C₈×C₁₇	C₄²	1	2	yes	—
70	C₄²×C₁₇	—	2	1	no	—
71	C₄²×C₁₇	—	2	1	no	—
72	C₄×Dic₁₇	—	2	1	no	—
73	C₄×Dic₁₇	—	2	1	no	—
74	C₄×Dic₁₇	—	2	4	no	—
75	C₄×Dic₁₇	—	2	4	no	—
76	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
77	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
78	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
79	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×#3622	—	2	1	no	—
81	C₂×#3622	—	2	1	no	—
82	C₂×#3622	—	2	1	no	—
83	C₂×#3622	—	2	1	no	—
84	C₂×#3622	C₄²	1	2	yes	—
85	C₂×#3622	C₄²	1	2	yes	—
86	#7844	—	17	1	no	maximal, Sylow
87	C₂³×C₁₇	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₁₇	—	2	2	no	—
89	C₂×C₄×C₁₇	—	2	2	no	—
90	C₂×C₄×C₁₇	—	2	2	no	—
91	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
92	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
94	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
95	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
96	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
97	C₂×Dic₁₇	—	2	2	no	—
98	C₂×Dic₁₇	—	2	2	no	—
99	C₂×Dic₁₇	—	2	2	no	—
100	C₂×Dic₁₇	C₂×C₄²	1	1	yes	characteristic
101	C₂×Dic₁₇	C₂×C₄²	1	1	yes	characteristic
102	C₂×Dic₁₇	C₂×C₄²	1	1	yes	characteristic
103	C₂×Dic₁₇	C₂×C₄²	1	1	yes	characteristic
104	C₂×Dic₁₇	C₂×M₁₆	1	4	yes	—
105	C₂×Dic₁₇	#131	1	4	yes	—
106	#3624	—	2	4	no	—
107	#3624	—	2	4	no	—
108	#3624	—	2	4	no	—
109	#3624	#97	1	8	yes	—
110	C₈×C₁₇	—	2	2	no	—
111	C₈×C₁₇	—	2	2	no	—
112	#3622	—	2	2	no	—
113	#3622	—	2	2	no	—
114	#1744	—	17	1	no	—
115	#1744	—	17	1	no	—
116	C₄×#97	—	17	1	no	—
117	C₂×#322	—	17	1	no	—
118	C₂×#322	—	17	1	no	—
119	C₂×#322	—	17	1	no	—
120	C₂×C₈²	—	17	1	no	—
121	C₂²×C₁₇	C₄³	1	1	yes	characteristic
122	C₂²×C₁₇	C₄×M₁₆	1	2	yes	—
123	C₂²×C₁₇	#405	1	2	yes	—
124	C₂²×C₁₇	#405	1	1	yes	characteristic
125	C₂²×C₁₇	#405	1	1	yes	characteristic
126	C₄×C₁₇	—	4	2	no	—
127	C₄×C₁₇	#405	1	1	yes	characteristic
128	C₄×C₁₇	#405	1	1	yes	characteristic
129	C₄×C₁₇	#405	1	1	yes	characteristic
130	C₄×C₁₇	#405	1	1	yes	characteristic
131	Dic₁₇	—	4	2	no	—
132	Dic₁₇	C₂×#97	1	2	yes	—
133	Dic₁₇	C₄×M₁₆	1	2	yes	—
134	Dic₁₇	#405	1	2	yes	—
135	Dic₁₇	#405	1	2	yes	—
136	C₂×#97	—	17	4	no	—
137	C₂×C₄×C₈	—	17	1	no	—
138	C₂×C₄×C₈	—	17	1	no	—
139	C₂×C₄×C₈	—	17	1	no	—
140	C₂×C₄×C₈	—	17	1	no	—
141	C₂×C₄×C₈	—	17	1	no	—
142	C₂×C₄×C₈	—	17	1	no	—
143	C₄³	—	17	1	no	—
144	#336	—	17	4	no	—
145	#336	—	17	1	no	—
146	#336	—	17	1	no	—
147	#336	—	17	1	no	—
148	#336	#595	1	1	yes	characteristic
149	#322	—	17	4	no	—
150	#322	—	17	4	no	—
151	#322	—	17	4	no	—
152	C₈²	—	17	4	no	—
153	C₂×C₁₇	#1453	1	2	yes	—
154	C₂×C₁₇	#1453	1	2	yes	—
155	C₂×C₁₇	C₄×#97	1	1	yes	characteristic
156	C₂×C₁₇	#1726	1	1	yes	characteristic
157	C₂×C₁₇	#1726	1	1	yes	characteristic
158	C₂²×C₈	—	17	2	no	—
159	C₂²×C₈	—	17	2	no	—
160	C₂²×C₈	—	17	2	no	—
161	C₂²×C₈	—	17	2	no	—
162	C₂²×C₈	—	17	1	no	—
163	C₂²×C₈	—	17	1	no	—
164	C₂²×C₈	C₂×#595	1	1	yes	characteristic
165	C₂²×C₈	C₂×#595	1	1	yes	characteristic
166	C₂×C₄²	—	17	1	no	—
167	C₂×C₄²	—	17	2	no	—
168	C₂×C₄²	—	17	2	no	—
169	C₂×C₄²	—	17	1	no	—
170	C₂×C₄²	C₂×#595	1	1	yes	characteristic
171	#97	—	34	8	no	—
172	C₄×C₈	—	17	2	no	—
173	C₄×C₈	—	17	2	no	—
174	C₄×C₈	—	17	2	no	—
175	C₄×C₈	—	17	2	no	—
176	C₄×C₈	—	17	2	no	—
177	C₄×C₈	—	17	2	no	—
178	C₄×C₈	—	17	2	no	—
179	C₄×C₈	—	17	2	no	—
180	C₄×C₈	—	17	2	no	—
181	C₄×C₈	—	17	2	no	—
182	C₄×C₈	—	17	2	no	—
183	C₄×C₈	—	17	2	no	—
184	C₁₇	#7844	1	1	yes	characteristic, complemented, Sylow
185	C₂²×C₄	—	17	1	no	—
186	C₂²×C₄	—	17	1	no	—
187	C₂²×C₄	—	17	2	no	—
188	C₂²×C₄	C₄×#595	1	2	yes	—
189	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	17	4	no	—
195	C₂×C₈	—	17	8	no	—
196	C₂×C₈	—	17	4	no	—
197	C₂×C₈	—	17	4	no	—
198	C₂×C₈	—	17	4	no	—
199	C₂×C₈	—	17	2	no	—
200	C₂×C₈	—	17	2	no	—
201	C₂×C₈	—	34	4	no	—
202	C₂×C₈	—	34	4	no	—
203	C₂×C₈	—	34	4	no	—
204	C₂×C₈	—	34	1	no	—
205	C₂×C₈	—	34	1	no	—
206	C₂×C₈	—	34	1	no	—
207	C₂×C₈	—	34	1	no	—
208	C₂×C₈	C₄×#595	1	2	yes	—
209	C₂×C₈	C₄×#595	1	2	yes	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	17	1	no	—
213	C₄²	—	17	1	no	—
214	C₄²	—	17	1	no	—
215	C₄²	—	17	1	no	—
216	C₄²	—	34	4	no	—
217	C₄²	—	34	4	no	—
218	C₄²	—	34	1	no	—
219	C₄²	—	34	1	no	—
220	C₂³	C₂×C₄×#595	1	1	yes	characteristic
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	17	1	no	—
225	C₂×C₄	—	17	1	no	—
226	C₂×C₄	—	17	1	no	—
227	C₂×C₄	—	17	1	no	—
228	C₂×C₄	—	17	4	no	—
229	C₂×C₄	—	17	4	no	—
230	C₂×C₄	—	34	2	no	—
231	C₂×C₄	—	34	2	no	—
232	C₂×C₄	—	34	2	no	—
233	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
235	C₂×C₄	#10587678	1	1	yes	characteristic
236	C₂×C₄	#10587678	1	1	yes	characteristic
237	C₂×C₄	#10587678	1	1	yes	characteristic
238	C₂×C₄	#10587680	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	2	2	no	—
241	C₈	—	17	8	no	—
242	C₈	—	34	4	no	—
243	C₈	—	34	4	no	—
244	C₈	—	34	4	no	—
245	C₈	—	34	2	no	—
246	C₈	—	34	2	no	—
247	C₂²	C₄²×#595	1	1	yes	characteristic
248	C₂²	M₁₆×#595	1	2	yes	—
249	C₂²	#11763772	1	2	yes	—
250	C₂²	#11763773	1	1	yes	characteristic
251	C₂²	#11763774	1	1	yes	characteristic
252	C₄	—	4	2	no	—
253	C₄	—	17	2	no	—
254	C₄	—	17	2	no	—
255	C₄	—	17	2	no	—
256	C₄	—	17	2	no	—
257	C₄	—	68	2	no	—
258	C₄	#11763773	1	1	yes	characteristic
259	C₄	#11763773	1	1	yes	characteristic
260	C₄	#11763774	1	1	yes	characteristic
261	C₄	#11763774	1	1	yes	characteristic
262	C₂	#14544001	1	2	yes	—
263	C₂	#14544002	1	2	yes	—
264	C₂	#97×#595	1	1	yes	characteristic
265	C₂	#14544440	1	1	yes	characteristic
266	C₂	#14544441	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#595
commutator	C₂²×C₁₇	C₄³
Fitting	C₁₇×#336	C₄
socle, prime radical	C₂³×C₁₇	C₂×C₄²
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	#336	#595
hypocenter	C₁₇	#7844
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
14. #11763651	C₄	semidirect
15. #11763651	C₄	semidirect
16. #11763650	C₄	semidirect
17. C₈×#3624	C₄	semidirect
184. C₁₇	#7844	semidirect

Maximal subgroups

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

subgroup	index
#14544498	2
C₄×#10587542	2
C₂×#11763651	2
C₂×#11763650	2
C₂×C₈×#3624	2
#14528528	2
#7844	17