Subgroups — C₁₇×#25412 #16883156

Order 4352.

1038 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇×#495	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₇×#467	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₇×#466	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×Q₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#2426	C₂	1	2	yes	maximal, complemented
6	C₁₇×#2042	C₂	1	2	yes	maximal, complemented
7	C₁₇×#2000	C₂	1	2	yes	maximal, complemented
8	C₁₇×#1970	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₇×#1942	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₇×#1880	C₂	1	2	yes	maximal, complemented
11	C₈×C₂²⋊C₄×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₁₇	C₂²	1	2	yes	—
14	C₂²×D₈×C₁₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₁₇×#124	C₂²	1	2	yes	—
17	C₂×C₁₇×C₂²⋊Q₈	C₂²	1	2	yes	—
18	C₂×C₁₇×#121	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₁₇	C₂²	1	2	yes	—
20	C₁₇×#495	—	2	4	no	—
21	C₁₇×#467	—	2	4	no	—
22	C₁₇×#466	—	2	4	no	—
23	C₄×Q₁₆×C₁₇	—	2	4	no	—
24	C₂×C₁₇×#107	C₂²	1	1	yes	characteristic
25	C₂×C₁₇×#107	C₂²	1	1	yes	characteristic
26	C₂×C₁₇×#103	C₂²	1	2	yes	—
27	C₂×C₁₇×#103	C₂²	1	2	yes	—
28	C₂×C₁₇×#102	C₂²	1	2	yes	—
29	C₂×C₁₇×#102	C₂²	1	2	yes	—
30	C₂×C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
33	C₁₇×#396	—	2	2	no	—
34	C₁₇×#394	—	2	2	no	—
35	C₁₇×#390	—	2	2	no	—
36	C₁₇×#390	C₂²	1	2	yes	—
37	C₁₇×#389	C₂²	1	2	yes	—
38	C₁₇×#388	—	2	2	no	—
39	C₄×C₂²⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
40	C₁₇×#340	C₂²	1	2	yes	—
41	C₁₇×#336	C₂²	1	1	yes	characteristic
42	C₁₇×#336	C₂²	1	1	yes	characteristic
43	C₂²×Q₈×C₁₇	C₂³	1	2	yes	—
44	C₂²×D₄×C₁₇	—	2	2	no	—
45	C₂²×D₄×C₁₇	C₂³	1	2	yes	—
46	C₂³×C₄×C₁₇	C₂³	1	1	yes	characteristic
47	C₂×Q₁₆×C₁₇	—	2	2	no	—
48	C₂×Q₁₆×C₁₇	—	2	2	no	—
49	C₂×Q₁₆×C₁₇	D₄	1	4	yes	complemented
50	C₂×SD₁₆×C₁₇	—	2	4	no	—
51	C₂×SD₁₆×C₁₇	—	2	4	no	—
52	C₂×SD₁₆×C₁₇	—	4	2	no	—
53	C₂×D₈×C₁₇	—	2	2	no	—
54	C₂×D₈×C₁₇	—	2	2	no	—
55	C₂×D₈×C₁₇	—	2	2	no	—
56	C₂²×C₈×C₁₇	—	2	2	no	—
57	C₂²×C₈×C₁₇	—	2	2	no	—
58	C₂²×C₈×C₁₇	—	2	2	no	—
59	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
63	C₁₇×#124	—	2	4	no	—
64	C₁₇×#124	—	4	2	no	—
65	C₁₇×C₂²⋊Q₈	—	2	4	no	—
66	C₁₇×C₂²⋊Q₈	—	4	2	no	—
67	C₁₇×#121	—	2	4	no	—
68	C₁₇×#121	—	4	2	no	—
69	C₄×Q₈×C₁₇	—	2	4	no	—
70	C₄×Q₈×C₁₇	—	4	2	no	—
71	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
72	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
73	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
74	C₂×C₄⋊C₄×C₁₇	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₁₇	C₂³	1	2	yes	—
76	C₂×C₂²⋊C₄×C₁₇	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₁₇	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₁₇	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₁₇	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₁₇	—	2	2	no	—
81	C₂×C₂²⋊C₄×C₁₇	—	2	2	no	—
82	C₂×C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
83	C₂×C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
84	C₂×C₄²×C₁₇	—	2	2	no	—
85	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
87	C₁₇×#107	—	2	2	no	—
88	C₁₇×#107	D₄	1	4	yes	—
89	C₁₇×#103	—	2	4	no	—
90	C₁₇×#103	—	2	4	no	—
91	C₁₇×#102	—	2	4	no	—
92	C₁₇×#102	—	4	2	no	—
93	C₁₇×C₂²⋊C₈	—	2	2	no	—
94	C₄×C₈×C₁₇	—	2	2	no	—
95	C₄×C₈×C₁₇	—	2	2	no	—
96	C₁₇×#95	—	2	2	no	—
97	C₁₇×#95	D₄	1	1	yes	characteristic
98	C₁₇×#95	D₄	1	1	yes	characteristic
99	C₂⁴×C₁₇	—	2	2	no	—
100	C₂⁴×C₁₇	C₂×D₄	1	1	yes	characteristic
101	C₂×Q₈×C₁₇	—	2	2	no	—
102	C₂×Q₈×C₁₇	—	2	2	no	—
103	C₂×Q₈×C₁₇	—	2	2	no	—
104	C₂×Q₈×C₁₇	—	2	2	no	—
105	C₂×Q₈×C₁₇	—	2	2	no	—
106	C₂×Q₈×C₁₇	C₂×D₄	1	4	yes	—
107	C₂×D₄×C₁₇	—	2	2	no	—
108	C₂×D₄×C₁₇	—	2	2	no	—
109	C₂×D₄×C₁₇	—	2	2	no	—
110	C₂×D₄×C₁₇	—	2	2	no	—
111	C₂×D₄×C₁₇	—	2	2	no	—
112	C₂×D₄×C₁₇	—	2	2	no	—
113	C₂×D₄×C₁₇	—	4	4	no	—
114	C₂×D₄×C₁₇	—	4	4	no	—
115	C₂×D₄×C₁₇	—	8	2	no	—
116	C₂²×C₄×C₁₇	—	2	2	no	—
117	C₂²×C₄×C₁₇	—	2	2	no	—
118	C₂²×C₄×C₁₇	—	2	1	no	—
119	C₂²×C₄×C₁₇	—	2	1	no	—
120	C₂²×C₄×C₁₇	—	2	1	no	—
121	C₂²×C₄×C₁₇	—	2	1	no	—
122	C₂²×C₄×C₁₇	—	2	1	no	—
123	C₂²×C₄×C₁₇	—	2	1	no	—
124	C₂²×C₄×C₁₇	—	2	2	no	—
125	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
131	Q₁₆×C₁₇	—	2	8	no	—
132	SD₁₆×C₁₇	—	4	8	no	—
133	D₈×C₁₇	—	4	4	no	—
134	C₂×C₈×C₁₇	—	2	1	no	—
135	C₂×C₈×C₁₇	—	2	1	no	—
136	C₂×C₈×C₁₇	—	2	1	no	—
137	C₂×C₈×C₁₇	—	2	1	no	—
138	C₂×C₈×C₁₇	—	2	1	no	—
139	C₂×C₈×C₁₇	—	2	1	no	—
140	C₂×C₈×C₁₇	—	2	1	no	—
141	C₂×C₈×C₁₇	—	2	1	no	—
142	C₂×C₈×C₁₇	—	4	4	no	—
143	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
144	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
145	C₂×C₈×C₁₇	#41	1	2	yes	—
146	C₂×C₈×C₁₇	#41	1	2	yes	—
147	C₄⋊C₄×C₁₇	—	2	2	no	—
148	C₄⋊C₄×C₁₇	—	2	2	no	—
149	C₄⋊C₄×C₁₇	—	2	2	no	—
150	C₄⋊C₄×C₁₇	—	4	4	no	—
151	C₄⋊C₄×C₁₇	—	4	4	no	—
152	C₄⋊C₄×C₁₇	—	8	2	no	—
153	C₄⋊C₄×C₁₇	C₂×D₄	1	4	yes	—
154	C₂²⋊C₄×C₁₇	—	2	2	no	—
155	C₂²⋊C₄×C₁₇	—	2	2	no	—
156	C₂²⋊C₄×C₁₇	—	4	4	no	—
157	C₂²⋊C₄×C₁₇	—	4	4	no	—
158	C₂²⋊C₄×C₁₇	—	4	4	no	—
159	C₂²⋊C₄×C₁₇	—	4	4	no	—
160	C₂²⋊C₄×C₁₇	—	4	4	no	—
161	C₂²⋊C₄×C₁₇	—	8	2	no	—
162	C₄²×C₁₇	—	2	1	no	—
163	C₄²×C₁₇	—	2	1	no	—
164	C₄²×C₁₇	—	2	1	no	—
165	C₄²×C₁₇	—	2	1	no	—
166	C₄²×C₁₇	—	4	4	no	—
167	#25412	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂³×C₁₇	—	2	1	no	—
169	C₂³×C₁₇	—	2	1	no	—
170	C₂³×C₁₇	—	2	1	no	—
171	C₂³×C₁₇	—	2	1	no	—
172	C₂³×C₁₇	—	2	1	no	—
173	C₂³×C₁₇	—	2	1	no	—
174	C₂³×C₁₇	—	4	2	no	—
175	C₂³×C₁₇	—	4	2	no	—
176	C₂³×C₁₇	—	4	2	no	—
177	C₂³×C₁₇	—	4	2	no	—
178	C₂³×C₁₇	—	4	2	no	—
179	C₂³×C₁₇	—	4	2	no	—
180	C₂³×C₁₇	—	4	2	no	—
181	C₂³×C₁₇	#135	1	2	yes	—
182	C₂³×C₁₇	C₂²×D₄	1	1	yes	characteristic
183	Q₈×C₁₇	—	2	4	no	—
184	Q₈×C₁₇	—	2	4	no	—
185	Q₈×C₁₇	—	4	2	no	—
186	Q₈×C₁₇	—	4	2	no	—
187	D₄×C₁₇	—	4	2	no	—
188	D₄×C₁₇	—	4	2	no	—
189	D₄×C₁₇	—	4	2	no	—
190	D₄×C₁₇	—	4	2	no	—
191	D₄×C₁₇	—	8	8	no	—
192	C₂×C₄×C₁₇	—	2	1	no	—
193	C₂×C₄×C₁₇	—	2	1	no	—
194	C₂×C₄×C₁₇	—	2	1	no	—
195	C₂×C₄×C₁₇	—	2	1	no	—
196	C₂×C₄×C₁₇	—	2	1	no	—
197	C₂×C₄×C₁₇	—	2	1	no	—
198	C₂×C₄×C₁₇	—	2	4	no	—
199	C₂×C₄×C₁₇	—	2	1	no	—
200	C₂×C₄×C₁₇	—	2	4	no	—
201	C₂×C₄×C₁₇	—	2	1	no	—
202	C₂×C₄×C₁₇	—	2	1	no	—
203	C₂×C₄×C₁₇	—	2	1	no	—
204	C₂×C₄×C₁₇	—	2	1	no	—
205	C₂×C₄×C₁₇	—	2	1	no	—
206	C₂×C₄×C₁₇	—	2	1	no	—
207	C₂×C₄×C₁₇	—	2	1	no	—
208	C₂×C₄×C₁₇	—	4	2	no	—
209	C₂×C₄×C₁₇	—	4	2	no	—
210	C₂×C₄×C₁₇	—	4	2	no	—
211	C₂×C₄×C₁₇	—	4	2	no	—
212	C₂×C₄×C₁₇	—	4	1	no	—
213	C₂×C₄×C₁₇	—	4	1	no	—
214	C₂×C₄×C₁₇	—	4	1	no	—
215	C₂×C₄×C₁₇	—	4	1	no	—
216	C₂×C₄×C₁₇	—	4	2	no	—
217	C₂×C₄×C₁₇	—	4	2	no	—
218	C₂×C₄×C₁₇	—	4	2	no	—
219	C₂×C₄×C₁₇	#135	1	2	yes	—
220	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
223	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
224	C₂×C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
225	C₂×C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
226	C₈×C₁₇	—	2	2	no	—
227	C₈×C₁₇	—	4	1	no	—
228	C₈×C₁₇	—	4	1	no	—
229	C₈×C₁₇	#121	1	4	yes	—
230	C₂×#495	C₂×C₁₇	1	1	yes	characteristic, complemented
231	C₂×#467	C₂×C₁₇	1	1	yes	characteristic, complemented
232	C₂×#466	C₂×C₁₇	1	1	yes	characteristic
233	C₂×C₄×Q₁₆	C₂×C₁₇	1	1	yes	characteristic, complemented
234	#2426	C₂×C₁₇	1	2	yes	complemented
235	#2042	C₂×C₁₇	1	2	yes	complemented
236	#2000	C₂×C₁₇	1	2	yes	complemented
237	#1970	C₂×C₁₇	1	1	yes	characteristic, complemented
238	#1942	C₂×C₁₇	1	1	yes	characteristic, complemented
239	#1880	C₂×C₁₇	1	2	yes	complemented
240	C₈×C₂²⋊C₄	C₂×C₁₇	1	1	yes	characteristic, complemented
241	C₂²×C₁₇	—	2	2	no	—
242	C₂²×C₁₇	—	2	2	no	—
243	C₂²×C₁₇	—	2	2	no	—
244	C₂²×C₁₇	—	4	1	no	—
245	C₂²×C₁₇	—	4	1	no	—
246	C₂²×C₁₇	—	4	1	no	—
247	C₂²×C₁₇	—	4	1	no	—
248	C₂²×C₁₇	—	8	2	no	—
249	C₂²×C₁₇	—	8	2	no	—
250	C₂²×C₁₇	—	8	2	no	—
251	C₂²×C₁₇	—	8	2	no	—
252	C₂²×C₁₇	—	8	2	no	—
253	C₂²×C₁₇	—	8	2	no	—
254	C₂²×C₁₇	—	8	2	no	—
255	C₂²×C₁₇	#532	1	1	yes	characteristic
256	C₂²×C₁₇	D₄²	1	1	yes	characteristic
257	C₂²×C₁₇	D₄²	1	1	yes	characteristic
258	C₂²×C₁₇	C₂×#135	1	1	yes	characteristic
259	C₂²×C₁₇	C₂×#135	1	1	yes	characteristic
260	C₂²×C₁₇	#576	1	1	yes	characteristic
261	C₂²×C₁₇	#578	1	1	yes	characteristic
262	C₄×C₁₇	—	2	2	no	—
263	C₄×C₁₇	—	4	1	no	—
264	C₄×C₁₇	—	4	4	no	—
265	C₄×C₁₇	—	4	1	no	—
266	C₄×C₁₇	—	4	4	no	—
267	C₄×C₁₇	—	4	1	no	—
268	C₄×C₁₇	—	4	1	no	—
269	C₄×C₁₇	—	8	2	no	—
270	C₄×C₁₇	C₂×#121	1	1	yes	characteristic
271	C₄×C₁₇	#540	1	1	yes	characteristic
272	C₄×C₁₇	#546	1	1	yes	characteristic
273	C₄×C₁₇	#550	1	1	yes	characteristic
274	C₂²×Q₁₆	C₂²×C₁₇	1	1	yes	characteristic
275	C₂²×SD₁₆	C₂²×C₁₇	1	2	yes	—
276	C₂²×D₈	C₂²×C₁₇	1	1	yes	characteristic
277	C₂³×C₈	C₂²×C₁₇	1	1	yes	characteristic
278	C₂×#124	C₂²×C₁₇	1	2	yes	—
279	C₂×C₂²⋊Q₈	C₂²×C₁₇	1	2	yes	—
280	C₂×#121	C₂²×C₁₇	1	2	yes	—
281	C₂×C₄×Q₈	C₂²×C₁₇	1	2	yes	—
282	#495	—	2	4	no	—
283	#467	—	2	4	no	—
284	#466	—	2	4	no	—
285	C₄×Q₁₆	—	2	4	no	—
286	C₂×#107	C₂²×C₁₇	1	1	yes	characteristic
287	C₂×#107	C₂²×C₁₇	1	1	yes	characteristic
288	C₂×#103	C₂²×C₁₇	1	2	yes	—
289	C₂×#103	C₂²×C₁₇	1	2	yes	—
290	C₂×#102	C₂²×C₁₇	1	2	yes	—
291	C₂×#102	C₂²×C₁₇	1	2	yes	—
292	C₂×C₂²⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
293	C₂×C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
294	C₂×C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
295	#396	—	2	2	no	—
296	#394	—	2	2	no	—
297	#390	—	2	2	no	—
298	#390	C₂²×C₁₇	1	2	yes	—
299	#389	C₂²×C₁₇	1	2	yes	—
300	#388	—	2	2	no	—
301	C₄×C₂²⋊C₄	C₂²×C₁₇	1	1	yes	characteristic
302	#340	C₂²×C₁₇	1	2	yes	—
303	#336	C₂²×C₁₇	1	1	yes	characteristic
304	#336	C₂²×C₁₇	1	1	yes	characteristic
305	C₂×C₁₇	—	4	2	no	—
306	C₂×C₁₇	—	16	2	no	—
307	C₂×C₁₇	#2428	1	1	yes	characteristic
308	C₂×C₁₇	#3168	1	1	yes	characteristic
309	C₂×C₁₇	#3175	1	1	yes	characteristic
310	C₂×C₁₇	#3280	1	1	yes	characteristic
311	C₂×C₁₇	#3285	1	1	yes	characteristic
312	C₂×C₁₇	#3285	1	1	yes	characteristic
313	C₂×C₁₇	#3287	1	1	yes	characteristic
314	C₂²×Q₈	C₂³×C₁₇	1	2	yes	—
315	C₂²×D₄	—	2	2	no	—
316	C₂²×D₄	C₂³×C₁₇	1	2	yes	—
317	C₂³×C₄	C₂³×C₁₇	1	1	yes	characteristic
318	C₂×Q₁₆	—	2	2	no	—
319	C₂×Q₁₆	—	2	2	no	—
320	C₂×Q₁₆	D₄×C₁₇	1	4	yes	complemented
321	C₂×SD₁₆	—	2	4	no	—
322	C₂×SD₁₆	—	2	4	no	—
323	C₂×SD₁₆	—	4	2	no	—
324	C₂×D₈	—	2	2	no	—
325	C₂×D₈	—	2	2	no	—
326	C₂×D₈	—	2	2	no	—
327	C₂²×C₈	—	2	2	no	—
328	C₂²×C₈	—	2	2	no	—
329	C₂²×C₈	—	2	2	no	—
330	C₂²×C₈	C₂³×C₁₇	1	1	yes	characteristic
331	C₂²×C₈	C₂³×C₁₇	1	1	yes	characteristic
332	C₂²×C₈	C₂³×C₁₇	1	1	yes	characteristic
333	C₂²×C₈	C₂³×C₁₇	1	1	yes	characteristic
334	#124	—	2	4	no	—
335	#124	—	4	2	no	—
336	C₂²⋊Q₈	—	2	4	no	—
337	C₂²⋊Q₈	—	4	2	no	—
338	#121	—	2	4	no	—
339	#121	—	4	2	no	—
340	C₄×Q₈	—	2	4	no	—
341	C₄×Q₈	—	4	2	no	—
342	C₂×C₄⋊C₄	—	2	2	no	—
343	C₂×C₄⋊C₄	—	2	2	no	—
344	C₂×C₄⋊C₄	—	2	2	no	—
345	C₂×C₄⋊C₄	C₂³×C₁₇	1	2	yes	—
346	C₂×C₄⋊C₄	C₂³×C₁₇	1	2	yes	—
347	C₂×C₂²⋊C₄	—	2	2	no	—
348	C₂×C₂²⋊C₄	—	2	2	no	—
349	C₂×C₂²⋊C₄	—	2	2	no	—
350	C₂×C₂²⋊C₄	—	2	2	no	—
351	C₂×C₂²⋊C₄	—	2	2	no	—
352	C₂×C₂²⋊C₄	—	2	2	no	—
353	C₂×C₂²⋊C₄	D₄×C₁₇	1	1	yes	characteristic
354	C₂×C₂²⋊C₄	D₄×C₁₇	1	1	yes	characteristic
355	C₂×C₄²	—	2	2	no	—
356	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
357	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
358	#107	—	2	2	no	—
359	#107	D₄×C₁₇	1	4	yes	—
360	#103	—	2	4	no	—
361	#103	—	2	4	no	—
362	#102	—	2	4	no	—
363	#102	—	4	2	no	—
364	C₂²⋊C₈	—	2	2	no	—
365	C₄×C₈	—	2	2	no	—
366	C₄×C₈	—	2	2	no	—
367	#95	—	2	2	no	—
368	#95	D₄×C₁₇	1	1	yes	characteristic
369	#95	D₄×C₁₇	1	1	yes	characteristic
370	C₁₇	#25412	1	1	yes	characteristic, direct product factor, Sylow
371	C₂⁴	—	2	2	no	—
372	C₂⁴	C₂×D₄×C₁₇	1	1	yes	characteristic
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	—	2	2	no	—
375	C₂×Q₈	—	2	2	no	—
376	C₂×Q₈	—	2	2	no	—
377	C₂×Q₈	—	2	2	no	—
378	C₂×Q₈	C₂×D₄×C₁₇	1	4	yes	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	2	2	no	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	—	2	2	no	—
384	C₂×D₄	—	2	2	no	—
385	C₂×D₄	—	4	4	no	—
386	C₂×D₄	—	4	4	no	—
387	C₂×D₄	—	8	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	2	1	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
398	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
399	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
400	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
401	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
402	C₂²×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
403	Q₁₆	—	2	8	no	—
404	SD₁₆	—	4	8	no	—
405	D₈	—	4	4	no	—
406	C₂×C₈	—	2	1	no	—
407	C₂×C₈	—	2	1	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	1	no	—
412	C₂×C₈	—	2	1	no	—
413	C₂×C₈	—	2	1	no	—
414	C₂×C₈	—	4	4	no	—
415	C₂×C₈	C₂×D₄×C₁₇	1	2	yes	—
416	C₂×C₈	C₂×D₄×C₁₇	1	2	yes	—
417	C₂×C₈	#41×C₁₇	1	2	yes	—
418	C₂×C₈	#41×C₁₇	1	2	yes	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	—	2	2	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	4	4	no	—
423	C₄⋊C₄	—	4	4	no	—
424	C₄⋊C₄	—	8	2	no	—
425	C₄⋊C₄	C₂×D₄×C₁₇	1	4	yes	—
426	C₂²⋊C₄	—	2	2	no	—
427	C₂²⋊C₄	—	2	2	no	—
428	C₂²⋊C₄	—	4	4	no	—
429	C₂²⋊C₄	—	4	4	no	—
430	C₂²⋊C₄	—	4	4	no	—
431	C₂²⋊C₄	—	4	4	no	—
432	C₂²⋊C₄	—	4	4	no	—
433	C₂²⋊C₄	—	8	2	no	—
434	C₄²	—	2	1	no	—
435	C₄²	—	2	1	no	—
436	C₄²	—	2	1	no	—
437	C₄²	—	2	1	no	—
438	C₄²	—	4	4	no	—
439	C₂³	—	2	1	no	—
440	C₂³	—	2	1	no	—
441	C₂³	—	2	1	no	—
442	C₂³	—	2	1	no	—
443	C₂³	—	2	1	no	—
444	C₂³	—	2	1	no	—
445	C₂³	—	4	2	no	—
446	C₂³	—	4	2	no	—
447	C₂³	—	4	2	no	—
448	C₂³	—	4	2	no	—
449	C₂³	—	4	2	no	—
450	C₂³	—	4	2	no	—
451	C₂³	—	4	2	no	—
452	C₂³	C₁₇×#135	1	2	yes	—
453	C₂³	C₂²×D₄×C₁₇	1	1	yes	characteristic
454	Q₈	—	2	4	no	—
455	Q₈	—	2	4	no	—
456	Q₈	—	4	2	no	—
457	Q₈	—	4	2	no	—
458	D₄	—	4	2	no	—
459	D₄	—	4	2	no	—
460	D₄	—	4	2	no	—
461	D₄	—	4	2	no	—
462	D₄	—	8	8	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	2	1	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	2	no	—
482	C₂×C₄	—	4	2	no	—
483	C₂×C₄	—	4	1	no	—
484	C₂×C₄	—	4	1	no	—
485	C₂×C₄	—	4	1	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	C₁₇×#135	1	2	yes	—
491	C₂×C₄	C₂²×D₄×C₁₇	1	1	yes	characteristic
492	C₂×C₄	C₂²×D₄×C₁₇	1	1	yes	characteristic
493	C₂×C₄	C₂×#41×C₁₇	1	1	yes	characteristic
494	C₂×C₄	C₂×#41×C₁₇	1	1	yes	characteristic
495	C₂×C₄	C₁₇×H(5,𝔽₂)	1	1	yes	characteristic
496	C₂×C₄	C₁₇×H(5,𝔽₂)	1	1	yes	characteristic
497	C₈	—	2	2	no	—
498	C₈	—	4	1	no	—
499	C₈	—	4	1	no	—
500	C₈	C₁₇×#121	1	4	yes	—
501	C₂²	—	2	2	no	—
502	C₂²	—	2	2	no	—
503	C₂²	—	2	2	no	—
504	C₂²	—	4	1	no	—
505	C₂²	—	4	1	no	—
506	C₂²	—	4	1	no	—
507	C₂²	—	4	1	no	—
508	C₂²	—	8	2	no	—
509	C₂²	—	8	2	no	—
510	C₂²	—	8	2	no	—
511	C₂²	—	8	2	no	—
512	C₂²	—	8	2	no	—
513	C₂²	—	8	2	no	—
514	C₂²	—	8	2	no	—
515	C₂²	C₁₇×#532	1	1	yes	characteristic
516	C₂²	D₄²×C₁₇	1	1	yes	characteristic
517	C₂²	D₄²×C₁₇	1	1	yes	characteristic
518	C₂²	C₂×C₁₇×#135	1	1	yes	characteristic
519	C₂²	C₂×C₁₇×#135	1	1	yes	characteristic
520	C₂²	C₁₇×#576	1	1	yes	characteristic
521	C₂²	C₁₇×#578	1	1	yes	characteristic
522	C₄	—	2	2	no	—
523	C₄	—	4	1	no	—
524	C₄	—	4	4	no	—
525	C₄	—	4	1	no	—
526	C₄	—	4	1	no	—
527	C₄	—	4	1	no	—
528	C₄	—	4	4	no	—
529	C₄	—	8	2	no	—
530	C₄	C₂×C₁₇×#121	1	1	yes	characteristic
531	C₄	C₁₇×#540	1	1	yes	characteristic
532	C₄	C₁₇×#546	1	1	yes	characteristic
533	C₄	C₁₇×#550	1	1	yes	characteristic
534	C₂	—	4	2	no	—
535	C₂	—	16	2	no	—
536	C₂	C₁₇×#2428	1	1	yes	characteristic
537	C₂	C₁₇×#3168	1	1	yes	characteristic
538	C₂	C₁₇×#3175	1	1	yes	characteristic
539	C₂	C₁₇×#3280	1	1	yes	characteristic
540	C₂	C₁₇×#3285	1	1	yes	characteristic
541	C₂	C₁₇×#3285	1	1	yes	characteristic
542	C₂	C₁₇×#3287	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₇
prime radical	C₂×C₁₇×#466	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ C₄×C₂²⋊C₄×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₇×#495	C₂	semidirect
2. C₂×C₁₇×#467	C₂	semidirect
4. C₂×C₄×Q₁₆×C₁₇	C₂	semidirect
5. C₁₇×#2426	C₂	semidirect
6. C₁₇×#2042	C₂	semidirect
7. C₁₇×#2000	C₂	semidirect
8. C₁₇×#1970	C₂	semidirect
9. C₁₇×#1942	C₂	semidirect
10. C₁₇×#1880	C₂	semidirect
11. C₈×C₂²⋊C₄×C₁₇	C₂	semidirect
49. C₂×Q₁₆×C₁₇	D₄	semidirect
167. #25412	C₁₇	direct (N × H)
230. C₂×#495	C₂×C₁₇	semidirect
231. C₂×#467	C₂×C₁₇	semidirect
233. C₂×C₄×Q₁₆	C₂×C₁₇	semidirect
234. #2426	C₂×C₁₇	semidirect
235. #2042	C₂×C₁₇	semidirect
236. #2000	C₂×C₁₇	semidirect
237. #1970	C₂×C₁₇	semidirect
238. #1942	C₂×C₁₇	semidirect
239. #1880	C₂×C₁₇	semidirect
240. C₈×C₂²⋊C₄	C₂×C₁₇	semidirect
320. C₂×Q₁₆	D₄×C₁₇	semidirect
370. C₁₇	#25412	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₇×#495	2
C₂×C₁₇×#467	2
C₂×C₁₇×#466	2
C₂×C₄×Q₁₆×C₁₇	2
C₁₇×#2426	2
C₁₇×#2042	2
C₁₇×#2000	2
C₁₇×#1970	2
C₁₇×#1942	2
C₁₇×#1880	2
C₈×C₂²⋊C₄×C₁₇	2
#25412	17

Subgroups — C17×#25412 #16883156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#25412 #16883156