Subgroups — C₂×C₁₇×#2974 #16917966

Order 4352.

2654 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₁₇	C₂	1	1	yes	maximal, characteristic
2	C₁₇×#2974	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₁₇×#436	C₂	1	2	yes	maximal, complemented
4	C₂×C₁₇×#435	C₂	1	4	yes	maximal, complemented
5	C₂×C₈×D₄×C₁₇	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₇×#432	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₇×#431	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₇×#407	C₂	1	2	yes	maximal, complemented
9	C₂²×C₁₇×C₂²⋊C₈	C₂	1	2	yes	maximal, complemented
10	C₂³×D₄×C₁₇	C₄	1	1	yes	characteristic
11	C₂⁴×C₄×C₁₇	C₂²	1	2	yes	—
12	C₂²×M₁₆×C₁₇	C₂²	1	4	yes	complemented
13	C₂³×C₈×C₁₇	C₂²	1	4	yes	complemented
14	C₂×C₄×D₄×C₁₇	C₄	1	8	yes	—
15	C₂×C₄×D₄×C₁₇	C₄	1	1	yes	characteristic
16	C₂×C₄×D₄×C₁₇	C₄	1	2	yes	—
17	C₂×C₄×D₄×C₁₇	C₄	1	1	yes	characteristic
18	C₂×C₄×D₄×C₁₇	C₂²	1	8	yes	—
19	C₂×C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₁₇	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₁₇	C₄	1	1	yes	characteristic
23	C₂²×C₂²⋊C₄×C₁₇	C₄	1	2	yes	—
24	C₂²×C₄²×C₁₇	C₂²	1	1	yes	characteristic
25	C₁₇×#436	C₂²	1	16	yes	complemented
26	C₁₇×#435	C₂²	1	32	yes	complemented
27	C₈×D₄×C₁₇	C₂²	1	16	yes	complemented
28	C₁₇×#432	C₂²	1	8	yes	complemented
29	C₁₇×#431	C₂²	1	8	yes	complemented
30	C₂×C₁₇×C₄⋊C₈	C₂²	1	2	yes	complemented
31	C₂×C₁₇×C₄⋊C₈	C₂²	1	2	yes	complemented
32	C₁₇×#407	C₂²	1	16	yes	complemented
33	C₂×C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₁₇×C₂²⋊C₈	C₂²	1	2	yes	—
35	C₂×C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
36	C₂×C₁₇×C₂²⋊C₈	C₂²	1	4	yes	complemented
37	C₂×C₁₇×C₂²⋊C₈	C₂²	1	4	yes	complemented
38	C₂×C₁₇×C₂²⋊C₈	C₂²	1	16	yes	complemented
39	C₂×C₁₇×#97	C₂²	1	2	yes	complemented
40	C₂×C₄×C₈×C₁₇	C₂²	1	2	yes	complemented
41	C₂⁵×C₁₇	C₂×C₄	1	2	yes	—
42	C₂²×D₄×C₁₇	—	2	8	no	—
43	C₂²×D₄×C₁₇	C₂×C₄	1	2	yes	—
44	C₂²×D₄×C₁₇	C₂×C₄	1	8	yes	—
45	C₂²×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
46	C₂²×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₁₇	—	4	8	no	—
48	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₁₇	C₂×C₄	1	2	yes	—
51	C₂³×C₄×C₁₇	C₂×C₄	1	8	yes	—
52	C₂³×C₄×C₁₇	C₂×C₄	1	2	yes	—
53	C₂³×C₄×C₁₇	C₂×C₄	1	2	yes	—
54	C₂³×C₄×C₁₇	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₁₇	C₂³	1	2	yes	—
56	C₂³×C₄×C₁₇	C₂³	1	2	yes	—
57	C₂³×C₄×C₁₇	C₂³	1	8	yes	—
58	C₂×M₁₆×C₁₇	—	4	8	no	—
59	C₂×M₁₆×C₁₇	C₂³	1	16	yes	complemented
60	C₂²×C₈×C₁₇	—	4	8	no	—
61	C₂²×C₈×C₁₇	C₂³	1	2	yes	—
62	C₂²×C₈×C₁₇	C₂³	1	4	yes	—
63	C₂²×C₈×C₁₇	C₂³	1	2	yes	—
64	C₂²×C₈×C₁₇	C₂³	1	16	yes	complemented
65	C₄×D₄×C₁₇	C₂×C₄	1	8	yes	—
66	C₄×D₄×C₁₇	C₂×C₄	1	4	yes	—
67	C₄×D₄×C₁₇	C₂×C₄	1	4	yes	—
68	C₄×D₄×C₁₇	C₂×C₄	1	16	yes	—
69	C₄×D₄×C₁₇	C₂×C₄	1	8	yes	—
70	C₄×D₄×C₁₇	C₂×C₄	1	8	yes	—
71	C₄×D₄×C₁₇	C₂³	1	4	yes	—
72	C₄×D₄×C₁₇	C₂³	1	8	yes	—
73	C₄×D₄×C₁₇	C₂³	1	4	yes	—
74	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
75	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	8	yes	—
78	C₂×C₂²⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
79	C₂×C₂²⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
80	C₂×C₂²⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
81	C₂×C₂²⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
82	C₂×C₂²⋊C₄×C₁₇	C₂×C₄	1	16	yes	—
83	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄²×C₁₇	C₂×C₄	1	4	yes	—
85	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄²×C₁₇	C₂³	1	4	yes	—
87	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
89	C₁₇×C₄⋊C₈	C₂³	1	8	yes	complemented
90	C₁₇×C₄⋊C₈	C₂³	1	8	yes	complemented
91	C₁₇×C₂²⋊C₈	C₂³	1	4	yes	—
92	C₁₇×C₂²⋊C₈	C₂³	1	8	yes	—
93	C₁₇×C₂²⋊C₈	C₂³	1	4	yes	—
94	C₁₇×C₂²⋊C₈	C₂³	1	16	yes	complemented
95	C₁₇×C₂²⋊C₈	C₂³	1	16	yes	complemented
96	C₁₇×#97	C₂³	1	8	yes	complemented
97	C₄×C₈×C₁₇	C₂³	1	8	yes	complemented
98	C₂⁴×C₁₇	—	2	8	no	—
99	C₂⁴×C₁₇	—	2	8	no	—
100	C₂⁴×C₁₇	—	4	4	no	—
101	C₂⁴×C₁₇	C₂²×C₄	1	2	yes	—
102	C₂⁴×C₁₇	C₂²×C₄	1	8	yes	—
103	C₂⁴×C₁₇	C₂²×C₄	1	2	yes	—
104	C₂⁴×C₁₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×D₄×C₁₇	—	2	8	no	—
106	C₂×D₄×C₁₇	—	2	32	no	—
107	C₂×D₄×C₁₇	—	2	4	no	—
108	C₂×D₄×C₁₇	M₁₆	1	8	yes	complemented
109	C₂×D₄×C₁₇	C₂²×C₄	1	8	yes	—
110	C₂×D₄×C₁₇	C₂²×C₄	1	4	yes	—
111	C₂×D₄×C₁₇	C₂²×C₄	1	4	yes	—
112	C₂²×C₄×C₁₇	—	2	4	no	—
113	C₂²×C₄×C₁₇	—	2	4	no	—
114	C₂²×C₄×C₁₇	—	2	8	no	—
115	C₂²×C₄×C₁₇	—	2	8	no	—
116	C₂²×C₄×C₁₇	—	2	4	no	—
117	C₂²×C₄×C₁₇	—	4	16	no	—
118	C₂²×C₄×C₁₇	—	4	16	no	—
119	C₂²×C₄×C₁₇	—	4	4	no	—
120	C₂²×C₄×C₁₇	—	4	4	no	—
121	C₂²×C₄×C₁₇	—	4	4	no	—
122	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
124	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
125	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
126	C₂²×C₄×C₁₇	C₂²×C₄	1	4	yes	—
127	C₂²×C₄×C₁₇	C₂²×C₄	1	4	yes	—
128	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₁₇	C₂²×C₄	1	4	yes	—
130	C₂²×C₄×C₁₇	C₂²×C₄	1	4	yes	—
131	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₁₇	C₂²×C₄	1	8	yes	—
134	C₂²×C₄×C₁₇	C₂²×C₄	1	8	yes	—
135	C₂²×C₄×C₁₇	C₂²×C₄	1	8	yes	—
136	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₁₇	C₂⁴	1	4	yes	—
138	C₂²×C₄×C₁₇	C₂⁴	1	4	yes	—
139	C₂²×C₄×C₁₇	C₂⁴	1	2	yes	—
140	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
141	M₁₆×C₁₇	—	4	16	no	—
142	C₂×C₈×C₁₇	—	2	4	no	—
143	C₂×C₈×C₁₇	—	2	8	no	—
144	C₂×C₈×C₁₇	—	2	4	no	—
145	C₂×C₈×C₁₇	—	4	16	no	—
146	C₂×C₈×C₁₇	C₂⁴	1	4	yes	—
147	C₂×C₈×C₁₇	C₂⁴	1	8	yes	—
148	C₂×C₈×C₁₇	C₂⁴	1	4	yes	—
149	C₄⋊C₄×C₁₇	C₂²×C₄	1	8	yes	—
150	C₄⋊C₄×C₁₇	C₂²×C₄	1	4	yes	—
151	C₄⋊C₄×C₁₇	C₂²×C₄	1	4	yes	—
152	C₂²⋊C₄×C₁₇	C₂²×C₄	1	8	yes	—
153	C₂²⋊C₄×C₁₇	C₂²×C₄	1	8	yes	—
154	C₂²⋊C₄×C₁₇	C₂²×C₄	1	8	yes	—
155	C₂²⋊C₄×C₁₇	C₂²×C₄	1	8	yes	—
156	C₄²×C₁₇	C₂²×C₄	1	4	yes	—
157	C₄²×C₁₇	C₂²×C₄	1	4	yes	—
158	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
159	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
160	C₄²×C₁₇	C₂⁴	1	2	yes	—
161	C₄²×C₁₇	C₂⁴	1	2	yes	—
162	C₂×#2974	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂³×C₁₇	—	2	2	no	—
164	C₂³×C₁₇	—	2	4	no	—
165	C₂³×C₁₇	—	2	2	no	—
166	C₂³×C₁₇	—	2	4	no	—
167	C₂³×C₁₇	—	2	4	no	—
168	C₂³×C₁₇	—	2	16	no	—
169	C₂³×C₁₇	—	2	4	no	—
170	C₂³×C₁₇	—	2	16	no	—
171	C₂³×C₁₇	—	2	4	no	—
172	C₂³×C₁₇	—	4	8	no	—
173	C₂³×C₁₇	—	4	32	no	—
174	C₂³×C₁₇	C₂×M₁₆	1	8	yes	—
175	C₂³×C₁₇	#131	1	4	yes	—
176	C₂³×C₁₇	C₂³×C₄	1	4	yes	—
177	C₂³×C₁₇	C₂³×C₄	1	4	yes	—
178	C₂³×C₁₇	C₂³×C₄	1	2	yes	—
179	C₂³×C₁₇	C₂³×C₄	1	1	yes	characteristic
180	D₄×C₁₇	—	2	16	no	—
181	D₄×C₁₇	—	2	8	no	—
182	D₄×C₁₇	C₂×M₁₆	1	16	yes	complemented
183	C₂×C₄×C₁₇	—	2	2	no	—
184	C₂×C₄×C₁₇	—	2	4	no	—
185	C₂×C₄×C₁₇	—	2	4	no	—
186	C₂×C₄×C₁₇	—	2	2	no	—
187	C₂×C₄×C₁₇	—	2	8	no	—
188	C₂×C₄×C₁₇	—	2	2	no	—
189	C₂×C₄×C₁₇	—	2	2	no	—
190	C₂×C₄×C₁₇	—	2	4	no	—
191	C₂×C₄×C₁₇	—	2	8	no	—
192	C₂×C₄×C₁₇	—	2	8	no	—
193	C₂×C₄×C₁₇	—	2	8	no	—
194	C₂×C₄×C₁₇	—	2	8	no	—
195	C₂×C₄×C₁₇	—	2	8	no	—
196	C₂×C₄×C₁₇	—	2	4	no	—
197	C₂×C₄×C₁₇	—	4	8	no	—
198	C₂×C₄×C₁₇	—	4	8	no	—
199	C₂×C₄×C₁₇	—	4	8	no	—
200	C₂×C₄×C₁₇	C₂×M₁₆	1	4	yes	—
201	C₂×C₄×C₁₇	C₂³×C₄	1	2	yes	—
202	C₂×C₄×C₁₇	C₂³×C₄	1	2	yes	—
203	C₂×C₄×C₁₇	C₂³×C₄	1	2	yes	—
204	C₂×C₄×C₁₇	C₂³×C₄	1	2	yes	—
205	C₂×C₄×C₁₇	C₂³×C₄	1	4	yes	—
206	C₂×C₄×C₁₇	C₂³×C₄	1	4	yes	—
207	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
208	C₂×C₄×C₁₇	C₂³×C₄	1	2	yes	—
209	C₂×C₄×C₁₇	H(5,𝔽₂)	1	4	yes	—
210	C₂×C₄×C₁₇	C₂⁵	1	1	yes	characteristic
211	C₈×C₁₇	—	2	4	no	—
212	C₈×C₁₇	—	2	8	no	—
213	C₈×C₁₇	—	2	4	no	—
214	C₂²×C₄×D₄	C₂×C₁₇	1	1	yes	characteristic
215	#2974	C₂×C₁₇	1	16	yes	direct product factor
216	C₂×#436	C₂×C₁₇	1	2	yes	complemented
217	C₂×#435	C₂×C₁₇	1	4	yes	complemented
218	C₂×C₈×D₄	C₂×C₁₇	1	2	yes	complemented
219	C₂×#432	C₂×C₁₇	1	1	yes	characteristic, complemented
220	C₂×#431	C₂×C₁₇	1	1	yes	characteristic, complemented
221	C₂×#407	C₂×C₁₇	1	2	yes	complemented
222	C₂²×C₂²⋊C₈	C₂×C₁₇	1	2	yes	complemented
223	C₂²×C₁₇	—	2	2	no	—
224	C₂²×C₁₇	—	2	4	no	—
225	C₂²×C₁₇	—	2	8	no	—
226	C₂²×C₁₇	—	2	16	no	—
227	C₂²×C₁₇	—	2	2	no	—
228	C₂²×C₁₇	—	2	4	no	—
229	C₂²×C₁₇	—	2	4	no	—
230	C₂²×C₁₇	—	2	4	no	—
231	C₂²×C₁₇	—	2	4	no	—
232	C₂²×C₁₇	—	4	32	no	—
233	C₂²×C₁₇	—	4	8	no	—
234	C₂²×C₁₇	#518	1	2	yes	—
235	C₂²×C₁₇	C₂²×M₁₆	1	8	yes	—
236	C₂²×C₁₇	C₂²×M₁₆	1	2	yes	—
237	C₂²×C₁₇	C₂×#131	1	4	yes	—
238	C₂²×C₁₇	C₂×#131	1	2	yes	—
239	C₂²×C₁₇	C₂⁴×C₄	1	1	yes	characteristic
240	C₄×C₁₇	—	2	2	no	—
241	C₄×C₁₇	—	2	4	no	—
242	C₄×C₁₇	—	2	4	no	—
243	C₄×C₁₇	—	2	2	no	—
244	C₄×C₁₇	—	2	2	no	—
245	C₄×C₁₇	—	2	2	no	—
246	C₄×C₁₇	#518	1	2	yes	—
247	C₄×C₁₇	C₂²×M₁₆	1	4	yes	—
248	C₄×C₁₇	C₂×H(5,𝔽₂)	1	2	yes	—
249	C₂³×D₄	C₄×C₁₇	1	1	yes	characteristic
250	C₂⁴×C₄	C₂²×C₁₇	1	2	yes	—
251	C₂²×M₁₆	C₂²×C₁₇	1	4	yes	complemented
252	C₂³×C₈	C₂²×C₁₇	1	4	yes	complemented
253	C₂×C₄×D₄	C₄×C₁₇	1	2	yes	—
254	C₂×C₄×D₄	C₄×C₁₇	1	8	yes	—
255	C₂×C₄×D₄	C₄×C₁₇	1	1	yes	characteristic
256	C₂×C₄×D₄	C₄×C₁₇	1	1	yes	characteristic
257	C₂×C₄×D₄	C₂²×C₁₇	1	1	yes	characteristic
258	C₂×C₄×D₄	C₂²×C₁₇	1	2	yes	—
259	C₂×C₄×D₄	C₂²×C₁₇	1	1	yes	characteristic
260	C₂×C₄×D₄	C₂²×C₁₇	1	8	yes	—
261	C₂²×C₄⋊C₄	C₄×C₁₇	1	1	yes	characteristic
262	C₂²×C₂²⋊C₄	C₄×C₁₇	1	2	yes	—
263	C₂²×C₄²	C₂²×C₁₇	1	1	yes	characteristic
264	#436	C₂²×C₁₇	1	16	yes	complemented
265	#435	C₂²×C₁₇	1	32	yes	complemented
266	C₈×D₄	C₂²×C₁₇	1	16	yes	complemented
267	#432	C₂²×C₁₇	1	8	yes	complemented
268	#431	C₂²×C₁₇	1	8	yes	complemented
269	C₂×C₄⋊C₈	C₂²×C₁₇	1	2	yes	complemented
270	C₂×C₄⋊C₈	C₂²×C₁₇	1	2	yes	complemented
271	#407	C₂²×C₁₇	1	16	yes	complemented
272	C₂×C₂²⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
273	C₂×C₂²⋊C₈	C₂²×C₁₇	1	2	yes	—
274	C₂×C₂²⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
275	C₂×C₂²⋊C₈	C₂²×C₁₇	1	16	yes	complemented
276	C₂×C₂²⋊C₈	C₂²×C₁₇	1	4	yes	complemented
277	C₂×C₂²⋊C₈	C₂²×C₁₇	1	4	yes	complemented
278	C₂×#97	C₂²×C₁₇	1	2	yes	complemented
279	C₂×C₄×C₈	C₂²×C₁₇	1	2	yes	complemented
280	C₂×C₁₇	—	2	4	no	—
281	C₂×C₁₇	—	2	8	no	—
282	C₂×C₁₇	—	4	4	no	—
283	C₂×C₁₇	#2974	1	4	yes	direct product factor
284	C₂×C₁₇	C₂×#518	1	1	yes	characteristic
285	C₂×C₁₇	C₂³×M₁₆	1	1	yes	characteristic
286	C₂×C₁₇	C₂²×#131	1	1	yes	characteristic
287	C₂⁵	C₂×C₄×C₁₇	1	2	yes	—
288	C₂²×D₄	—	2	8	no	—
289	C₂²×D₄	C₂×C₄×C₁₇	1	2	yes	—
290	C₂²×D₄	C₂×C₄×C₁₇	1	8	yes	—
291	C₂²×D₄	C₂×C₄×C₁₇	1	1	yes	characteristic
292	C₂²×D₄	C₂×C₄×C₁₇	1	1	yes	characteristic
293	C₂³×C₄	—	4	8	no	—
294	C₂³×C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
295	C₂³×C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
296	C₂³×C₄	C₂×C₄×C₁₇	1	2	yes	—
297	C₂³×C₄	C₂×C₄×C₁₇	1	8	yes	—
298	C₂³×C₄	C₂×C₄×C₁₇	1	2	yes	—
299	C₂³×C₄	C₂×C₄×C₁₇	1	2	yes	—
300	C₂³×C₄	C₂³×C₁₇	1	2	yes	—
301	C₂³×C₄	C₂³×C₁₇	1	2	yes	—
302	C₂³×C₄	C₂³×C₁₇	1	1	yes	characteristic
303	C₂³×C₄	C₂³×C₁₇	1	8	yes	—
304	C₂×M₁₆	—	4	8	no	—
305	C₂×M₁₆	C₂³×C₁₇	1	16	yes	complemented
306	C₂²×C₈	—	4	8	no	—
307	C₂²×C₈	C₂³×C₁₇	1	2	yes	—
308	C₂²×C₈	C₂³×C₁₇	1	4	yes	—
309	C₂²×C₈	C₂³×C₁₇	1	2	yes	—
310	C₂²×C₈	C₂³×C₁₇	1	16	yes	complemented
311	C₄×D₄	C₂×C₄×C₁₇	1	8	yes	—
312	C₄×D₄	C₂×C₄×C₁₇	1	4	yes	—
313	C₄×D₄	C₂×C₄×C₁₇	1	4	yes	—
314	C₄×D₄	C₂×C₄×C₁₇	1	16	yes	—
315	C₄×D₄	C₂×C₄×C₁₇	1	8	yes	—
316	C₄×D₄	C₂×C₄×C₁₇	1	8	yes	—
317	C₄×D₄	C₂³×C₁₇	1	4	yes	—
318	C₄×D₄	C₂³×C₁₇	1	8	yes	—
319	C₄×D₄	C₂³×C₁₇	1	4	yes	—
320	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
321	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	8	yes	—
322	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
323	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
324	C₂×C₂²⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
325	C₂×C₂²⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
326	C₂×C₂²⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
327	C₂×C₂²⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
328	C₂×C₂²⋊C₄	C₂×C₄×C₁₇	1	16	yes	—
329	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
330	C₂×C₄²	C₂×C₄×C₁₇	1	4	yes	—
331	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
332	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
333	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
334	C₂×C₄²	C₂³×C₁₇	1	4	yes	—
335	C₄⋊C₈	C₂³×C₁₇	1	8	yes	complemented
336	C₄⋊C₈	C₂³×C₁₇	1	8	yes	complemented
337	C₂²⋊C₈	C₂³×C₁₇	1	4	yes	—
338	C₂²⋊C₈	C₂³×C₁₇	1	8	yes	—
339	C₂²⋊C₈	C₂³×C₁₇	1	4	yes	—
340	C₂²⋊C₈	C₂³×C₁₇	1	16	yes	complemented
341	C₂²⋊C₈	C₂³×C₁₇	1	16	yes	complemented
342	#97	C₂³×C₁₇	1	8	yes	complemented
343	C₄×C₈	C₂³×C₁₇	1	8	yes	complemented
344	C₁₇	C₂×#2974	1	1	yes	characteristic, direct product factor, Sylow
345	C₂⁴	—	2	8	no	—
346	C₂⁴	—	2	8	no	—
347	C₂⁴	—	4	4	no	—
348	C₂⁴	C₂²×C₄×C₁₇	1	2	yes	—
349	C₂⁴	C₂²×C₄×C₁₇	1	8	yes	—
350	C₂⁴	C₂²×C₄×C₁₇	1	2	yes	—
351	C₂⁴	C₂²×C₄×C₁₇	1	1	yes	characteristic
352	C₂×D₄	—	2	8	no	—
353	C₂×D₄	—	2	32	no	—
354	C₂×D₄	—	2	4	no	—
355	C₂×D₄	M₁₆×C₁₇	1	8	yes	complemented
356	C₂×D₄	C₂²×C₄×C₁₇	1	8	yes	—
357	C₂×D₄	C₂²×C₄×C₁₇	1	4	yes	—
358	C₂×D₄	C₂²×C₄×C₁₇	1	4	yes	—
359	C₂²×C₄	—	2	4	no	—
360	C₂²×C₄	—	2	4	no	—
361	C₂²×C₄	—	2	8	no	—
362	C₂²×C₄	—	2	8	no	—
363	C₂²×C₄	—	2	4	no	—
364	C₂²×C₄	—	4	16	no	—
365	C₂²×C₄	—	4	16	no	—
366	C₂²×C₄	—	4	4	no	—
367	C₂²×C₄	—	4	4	no	—
368	C₂²×C₄	—	4	4	no	—
369	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
370	C₂²×C₄	C₂²×C₄×C₁₇	1	2	yes	—
371	C₂²×C₄	C₂²×C₄×C₁₇	1	4	yes	—
372	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
373	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
374	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
375	C₂²×C₄	C₂²×C₄×C₁₇	1	4	yes	—
376	C₂²×C₄	C₂²×C₄×C₁₇	1	2	yes	—
377	C₂²×C₄	C₂²×C₄×C₁₇	1	2	yes	—
378	C₂²×C₄	C₂²×C₄×C₁₇	1	4	yes	—
379	C₂²×C₄	C₂²×C₄×C₁₇	1	4	yes	—
380	C₂²×C₄	C₂²×C₄×C₁₇	1	8	yes	—
381	C₂²×C₄	C₂²×C₄×C₁₇	1	8	yes	—
382	C₂²×C₄	C₂²×C₄×C₁₇	1	8	yes	—
383	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
384	C₂²×C₄	C₂⁴×C₁₇	1	2	yes	—
385	C₂²×C₄	C₂⁴×C₁₇	1	4	yes	—
386	C₂²×C₄	C₂⁴×C₁₇	1	4	yes	—
387	C₂²×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
388	M₁₆	—	4	16	no	—
389	C₂×C₈	—	2	4	no	—
390	C₂×C₈	—	2	8	no	—
391	C₂×C₈	—	2	4	no	—
392	C₂×C₈	—	4	16	no	—
393	C₂×C₈	C₂⁴×C₁₇	1	4	yes	—
394	C₂×C₈	C₂⁴×C₁₇	1	4	yes	—
395	C₂×C₈	C₂⁴×C₁₇	1	8	yes	—
396	C₄⋊C₄	C₂²×C₄×C₁₇	1	8	yes	—
397	C₄⋊C₄	C₂²×C₄×C₁₇	1	4	yes	—
398	C₄⋊C₄	C₂²×C₄×C₁₇	1	4	yes	—
399	C₂²⋊C₄	C₂²×C₄×C₁₇	1	8	yes	—
400	C₂²⋊C₄	C₂²×C₄×C₁₇	1	8	yes	—
401	C₂²⋊C₄	C₂²×C₄×C₁₇	1	8	yes	—
402	C₂²⋊C₄	C₂²×C₄×C₁₇	1	8	yes	—
403	C₄²	C₂²×C₄×C₁₇	1	4	yes	—
404	C₄²	C₂²×C₄×C₁₇	1	4	yes	—
405	C₄²	C₂²×C₄×C₁₇	1	2	yes	—
406	C₄²	C₂²×C₄×C₁₇	1	2	yes	—
407	C₄²	C₂⁴×C₁₇	1	2	yes	—
408	C₄²	C₂⁴×C₁₇	1	2	yes	—
409	C₂³	—	2	2	no	—
410	C₂³	—	2	4	no	—
411	C₂³	—	2	2	no	—
412	C₂³	—	2	4	no	—
413	C₂³	—	2	4	no	—
414	C₂³	—	2	16	no	—
415	C₂³	—	2	4	no	—
416	C₂³	—	2	16	no	—
417	C₂³	—	2	4	no	—
418	C₂³	—	4	8	no	—
419	C₂³	—	4	32	no	—
420	C₂³	C₂×M₁₆×C₁₇	1	8	yes	—
421	C₂³	C₁₇×#131	1	4	yes	—
422	C₂³	C₂³×C₄×C₁₇	1	4	yes	—
423	C₂³	C₂³×C₄×C₁₇	1	4	yes	—
424	C₂³	C₂³×C₄×C₁₇	1	2	yes	—
425	C₂³	C₂³×C₄×C₁₇	1	1	yes	characteristic
426	D₄	—	2	16	no	—
427	D₄	—	2	8	no	—
428	D₄	C₂×M₁₆×C₁₇	1	16	yes	complemented
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	4	no	—
431	C₂×C₄	—	2	4	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	8	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	4	no	—
437	C₂×C₄	—	2	8	no	—
438	C₂×C₄	—	2	8	no	—
439	C₂×C₄	—	2	8	no	—
440	C₂×C₄	—	2	8	no	—
441	C₂×C₄	—	2	8	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	4	8	no	—
444	C₂×C₄	—	4	8	no	—
445	C₂×C₄	—	4	8	no	—
446	C₂×C₄	C₂×M₁₆×C₁₇	1	4	yes	—
447	C₂×C₄	C₂³×C₄×C₁₇	1	2	yes	—
448	C₂×C₄	C₂³×C₄×C₁₇	1	2	yes	—
449	C₂×C₄	C₂³×C₄×C₁₇	1	2	yes	—
450	C₂×C₄	C₂³×C₄×C₁₇	1	2	yes	—
451	C₂×C₄	C₂³×C₄×C₁₇	1	4	yes	—
452	C₂×C₄	C₂³×C₄×C₁₇	1	4	yes	—
453	C₂×C₄	C₂³×C₄×C₁₇	1	1	yes	characteristic
454	C₂×C₄	C₂³×C₄×C₁₇	1	2	yes	—
455	C₂×C₄	C₁₇×H(5,𝔽₂)	1	4	yes	—
456	C₂×C₄	C₂⁵×C₁₇	1	1	yes	characteristic
457	C₈	—	2	4	no	—
458	C₈	—	2	8	no	—
459	C₈	—	2	4	no	—
460	C₂²	—	2	8	no	—
461	C₂²	—	2	16	no	—
462	C₂²	—	2	2	no	—
463	C₂²	—	2	4	no	—
464	C₂²	—	2	2	no	—
465	C₂²	—	2	4	no	—
466	C₂²	—	2	4	no	—
467	C₂²	—	2	4	no	—
468	C₂²	—	2	4	no	—
469	C₂²	—	4	32	no	—
470	C₂²	—	4	8	no	—
471	C₂²	C₁₇×#518	1	2	yes	—
472	C₂²	C₂²×M₁₆×C₁₇	1	8	yes	—
473	C₂²	C₂²×M₁₆×C₁₇	1	2	yes	—
474	C₂²	C₂×C₁₇×#131	1	4	yes	—
475	C₂²	C₂×C₁₇×#131	1	2	yes	—
476	C₂²	C₂⁴×C₄×C₁₇	1	1	yes	characteristic
477	C₄	—	2	2	no	—
478	C₄	—	2	2	no	—
479	C₄	—	2	2	no	—
480	C₄	—	2	2	no	—
481	C₄	—	2	4	no	—
482	C₄	—	2	4	no	—
483	C₄	C₁₇×#518	1	2	yes	—
484	C₄	C₂²×M₁₆×C₁₇	1	4	yes	—
485	C₄	C₂×C₁₇×H(5,𝔽₂)	1	2	yes	—
486	C₂	—	2	4	no	—
487	C₂	—	2	8	no	—
488	C₂	—	4	4	no	—
489	C₂	C₁₇×#2974	1	4	yes	direct product factor
490	C₂	C₂×C₁₇×#518	1	1	yes	characteristic
491	C₂	C₂³×M₁₆×C₁₇	1	1	yes	characteristic
492	C₂	C₂²×C₁₇×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₇	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₁₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₁₇
socle	C₂³×C₁₇	C₂³×C₄
prime radical	C₂³×D₄×C₁₇	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₇×#2974	C₂	direct (N × H)
3. C₂×C₁₇×#436	C₂	semidirect
4. C₂×C₁₇×#435	C₂	semidirect
5. C₂×C₈×D₄×C₁₇	C₂	semidirect
6. C₂×C₁₇×#432	C₂	semidirect
7. C₂×C₁₇×#431	C₂	semidirect
8. C₂×C₁₇×#407	C₂	semidirect
9. C₂²×C₁₇×C₂²⋊C₈	C₂	semidirect
12. C₂²×M₁₆×C₁₇	C₂²	semidirect
13. C₂³×C₈×C₁₇	C₂²	semidirect
25. C₁₇×#436	C₂²	semidirect
26. C₁₇×#435	C₂²	semidirect
27. C₈×D₄×C₁₇	C₂²	semidirect
28. C₁₇×#432	C₂²	semidirect
29. C₁₇×#431	C₂²	semidirect
30. C₂×C₁₇×C₄⋊C₈	C₂²	semidirect
31. C₂×C₁₇×C₄⋊C₈	C₂²	semidirect
32. C₁₇×#407	C₂²	semidirect
36. C₂×C₁₇×C₂²⋊C₈	C₂²	semidirect
37. C₂×C₁₇×C₂²⋊C₈	C₂²	semidirect
38. C₂×C₁₇×C₂²⋊C₈	C₂²	semidirect
39. C₂×C₁₇×#97	C₂²	semidirect
40. C₂×C₄×C₈×C₁₇	C₂²	semidirect
59. C₂×M₁₆×C₁₇	C₂³	semidirect
64. C₂²×C₈×C₁₇	C₂³	semidirect
89. C₁₇×C₄⋊C₈	C₂³	semidirect
90. C₁₇×C₄⋊C₈	C₂³	semidirect
94. C₁₇×C₂²⋊C₈	C₂³	semidirect
95. C₁₇×C₂²⋊C₈	C₂³	semidirect
96. C₁₇×#97	C₂³	semidirect
97. C₄×C₈×C₁₇	C₂³	semidirect
108. C₂×D₄×C₁₇	M₁₆	semidirect
162. C₂×#2974	C₁₇	direct (N × H)
182. D₄×C₁₇	C₂×M₁₆	semidirect
215. #2974	C₂×C₁₇	direct (N × H)
216. C₂×#436	C₂×C₁₇	semidirect
217. C₂×#435	C₂×C₁₇	semidirect
218. C₂×C₈×D₄	C₂×C₁₇	semidirect
219. C₂×#432	C₂×C₁₇	semidirect
220. C₂×#431	C₂×C₁₇	semidirect
221. C₂×#407	C₂×C₁₇	semidirect
222. C₂²×C₂²⋊C₈	C₂×C₁₇	semidirect
251. C₂²×M₁₆	C₂²×C₁₇	semidirect
252. C₂³×C₈	C₂²×C₁₇	semidirect
264. #436	C₂²×C₁₇	semidirect
265. #435	C₂²×C₁₇	semidirect
266. C₈×D₄	C₂²×C₁₇	semidirect
267. #432	C₂²×C₁₇	semidirect
268. #431	C₂²×C₁₇	semidirect
269. C₂×C₄⋊C₈	C₂²×C₁₇	semidirect
270. C₂×C₄⋊C₈	C₂²×C₁₇	semidirect
271. #407	C₂²×C₁₇	semidirect
275. C₂×C₂²⋊C₈	C₂²×C₁₇	semidirect
276. C₂×C₂²⋊C₈	C₂²×C₁₇	semidirect
277. C₂×C₂²⋊C₈	C₂²×C₁₇	semidirect
278. C₂×#97	C₂²×C₁₇	semidirect
279. C₂×C₄×C₈	C₂²×C₁₇	semidirect
283. C₂×C₁₇	#2974	direct (N × H)
305. C₂×M₁₆	C₂³×C₁₇	semidirect
310. C₂²×C₈	C₂³×C₁₇	semidirect
335. C₄⋊C₈	C₂³×C₁₇	semidirect
336. C₄⋊C₈	C₂³×C₁₇	semidirect
340. C₂²⋊C₈	C₂³×C₁₇	semidirect
341. C₂²⋊C₈	C₂³×C₁₇	semidirect
342. #97	C₂³×C₁₇	semidirect
343. C₄×C₈	C₂³×C₁₇	semidirect
344. C₁₇	C₂×#2974	direct (N × H)
355. C₂×D₄	M₁₆×C₁₇	semidirect
428. D₄	C₂×M₁₆×C₁₇	semidirect
489. C₂	C₁₇×#2974	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₄×C₁₇	2
C₁₇×#2974	2
C₂×C₁₇×#436	2
C₂×C₁₇×#435	2
C₂×C₈×D₄×C₁₇	2
C₂×C₁₇×#432	2
C₂×C₁₇×#431	2
C₂×C₁₇×#407	2
C₂²×C₁₇×C₂²⋊C₈	2
C₂×#2974	17

Subgroups — C2×C17×#2974 #16917966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₇×#2974 #16917966