Subgroups — C₁₇×#22674 #16880418

Order 4352.

914 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₇×#441	C₂	1	2	yes	maximal
2	C₂×C₄×Q₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₄⋊C₄×C₁₇	C₂	1	2	yes	maximal
4	C₁₇×#1943	C₂	1	2	yes	maximal
5	C₁₇×#1865	C₂	1	4	yes	maximal
6	C₁₇×#1844	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#1803	C₂	1	2	yes	maximal
8	C₁₇×#1770	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₇	C₂²	1	4	yes	—
11	C₂×C₄×Q₈×C₁₇	C₂²	1	2	yes	—
12	C₁₇×#441	—	2	8	no	—
13	C₄×Q₁₆×C₁₇	C₄	1	8	yes	complemented
14	C₂×C₁₇×#107	C₂²	1	1	yes	characteristic
15	C₂×C₁₇×#106	C₂²	1	2	yes	—
16	C₂×C₁₇×#106	C₂²	1	2	yes	—
17	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₇×#103	C₂²	1	4	yes	—
19	C₂×C₁₇×#103	C₂²	1	2	yes	—
20	C₂×C₁₇×#97	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
22	C₁₇×#389	—	2	4	no	—
23	C₁₇×#389	C₂²	1	4	yes	—
24	C₁₇×#384	—	2	2	no	—
25	C₁₇×#384	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₁₇	—	2	2	no	—
27	C₄×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
28	C₁₇×#340	C₂²	1	4	yes	—
29	C₁₇×#336	C₂²	1	2	yes	—
30	C₂²×Q₈×C₁₇	C₂³	1	2	yes	—
31	C₂×Q₁₆×C₁₇	C₂×C₄	1	4	yes	—
32	C₂×Q₁₆×C₁₇	D₄	1	4	yes	—
33	C₂×Q₁₆×C₁₇	Q₈	1	4	yes	—
34	C₂²×C₈×C₁₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₁₇	—	2	4	no	—
38	C₄×Q₈×C₁₇	—	2	8	no	—
39	C₄×Q₈×C₁₇	—	4	4	no	—
40	C₄×Q₈×C₁₇	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₇	D₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₇	C₂³	1	4	yes	—
48	C₂×C₄⋊C₄×C₁₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₇	—	2	2	no	—
50	C₂×C₄²×C₁₇	—	2	4	no	—
51	C₂×C₄²×C₁₇	C₂³	1	2	yes	—
52	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
53	C₁₇×#107	C₂×C₄	1	4	yes	—
54	C₁₇×#106	—	2	4	no	—
55	C₁₇×#106	—	4	2	no	—
56	C₁₇×C₄⋊C₈	—	2	2	no	—
57	C₁₇×#103	—	2	8	no	—
58	C₁₇×#103	C₂×C₄	1	8	yes	—
59	C₁₇×#97	—	2	4	no	—
60	C₄×C₈×C₁₇	C₂×C₄	1	4	yes	—
61	C₁₇×#95	—	2	4	no	—
62	C₁₇×#95	D₄	1	2	yes	—
63	C₂×Q₈×C₁₇	—	2	2	no	—
64	C₂×Q₈×C₁₇	—	2	2	no	—
65	C₂×Q₈×C₁₇	—	2	2	no	—
66	C₂×Q₈×C₁₇	C₂²×C₄	1	4	yes	—
67	C₂×Q₈×C₁₇	C₂×D₄	1	4	yes	—
68	C₂×Q₈×C₁₇	C₂×Q₈	1	4	yes	—
69	C₂²×C₄×C₁₇	—	2	2	no	—
70	C₂²×C₄×C₁₇	—	2	2	no	—
71	C₂²×C₄×C₁₇	—	2	4	no	—
72	C₂²×C₄×C₁₇	C₂×D₄	1	4	yes	—
73	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
76	Q₁₆×C₁₇	C₄⋊C₄	1	16	yes	complemented
77	C₂×C₈×C₁₇	—	2	1	no	—
78	C₂×C₈×C₁₇	—	2	1	no	—
79	C₂×C₈×C₁₇	—	2	2	no	—
80	C₂×C₈×C₁₇	—	2	2	no	—
81	C₂×C₈×C₁₇	—	2	2	no	—
82	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
86	C₄⋊C₄×C₁₇	—	2	8	no	—
87	C₄⋊C₄×C₁₇	—	2	4	no	—
88	C₄⋊C₄×C₁₇	—	2	2	no	—
89	C₄⋊C₄×C₁₇	—	2	4	no	—
90	C₄⋊C₄×C₁₇	—	2	4	no	—
91	C₄⋊C₄×C₁₇	—	4	8	no	—
92	C₄⋊C₄×C₁₇	—	4	4	no	—
93	C₄⋊C₄×C₁₇	—	8	2	no	—
94	C₄⋊C₄×C₁₇	—	8	4	no	—
95	C₄⋊C₄×C₁₇	C₂²×C₄	1	4	yes	—
96	C₄²×C₁₇	—	2	8	no	—
97	C₄²×C₁₇	—	2	2	no	—
98	C₄²×C₁₇	—	2	2	no	—
99	C₄²×C₁₇	—	4	8	no	—
100	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
101	C₄²×C₁₇	#41	1	2	yes	—
102	#22674	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₁₇	C₂²×D₄	1	1	yes	characteristic
104	Q₈×C₁₇	—	2	4	no	—
105	Q₈×C₁₇	—	2	4	no	—
106	Q₈×C₁₇	—	2	4	no	—
107	Q₈×C₁₇	C₂×C₄⋊C₄	1	8	yes	—
108	C₂×C₄×C₁₇	—	2	4	no	—
109	C₂×C₄×C₁₇	—	2	4	no	—
110	C₂×C₄×C₁₇	—	2	4	no	—
111	C₂×C₄×C₁₇	—	2	2	no	—
112	C₂×C₄×C₁₇	—	2	4	no	—
113	C₂×C₄×C₁₇	—	2	4	no	—
114	C₂×C₄×C₁₇	—	2	4	no	—
115	C₂×C₄×C₁₇	—	2	4	no	—
116	C₂×C₄×C₁₇	—	4	2	no	—
117	C₂×C₄×C₁₇	—	4	2	no	—
118	C₂×C₄×C₁₇	—	4	4	no	—
119	C₂×C₄×C₁₇	—	4	4	no	—
120	C₂×C₄×C₁₇	—	4	4	no	—
121	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₇	#137	1	4	yes	—
124	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂×C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
130	C₈×C₁₇	—	2	2	no	—
131	C₈×C₁₇	—	4	2	no	—
132	C₈×C₁₇	C₂×C₄⋊C₄	1	4	yes	—
133	C₂×#441	C₂×C₁₇	1	2	yes	—
134	C₂×C₄×Q₁₆	C₂×C₁₇	1	1	yes	characteristic
135	Q₈×C₄⋊C₄	C₂×C₁₇	1	2	yes	—
136	#1943	C₂×C₁₇	1	2	yes	—
137	#1865	C₂×C₁₇	1	4	yes	—
138	#1844	C₂×C₁₇	1	1	yes	characteristic
139	#1803	C₂×C₁₇	1	2	yes	—
140	#1770	C₂×C₁₇	1	1	yes	characteristic
141	C₂²×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂²×C₁₇	D₄²	1	1	yes	characteristic
143	C₂²×C₁₇	D₄×Q₈	1	1	yes	characteristic
144	C₂²×C₁₇	C₂×#137	1	2	yes	—
145	C₂²×C₁₇	#577	1	2	yes	—
146	C₄×C₁₇	—	2	8	no	—
147	C₄×C₁₇	—	2	2	no	—
148	C₄×C₁₇	—	4	4	no	—
149	C₄×C₁₇	—	4	4	no	—
150	C₄×C₁₇	—	8	4	no	—
151	C₄×C₁₇	#441	1	4	yes	complemented
152	C₄×C₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
153	C₄×C₁₇	#520	1	1	yes	characteristic
154	C₄×C₁₇	#547	1	1	yes	characteristic
155	C₄×C₁₇	#554	1	1	yes	characteristic
156	C₂²×Q₁₆	C₂²×C₁₇	1	1	yes	characteristic
157	C₂×C₄×Q₈	C₂²×C₁₇	1	4	yes	—
158	C₂×C₄×Q₈	C₂²×C₁₇	1	2	yes	—
159	#441	—	2	8	no	—
160	C₄×Q₁₆	C₄×C₁₇	1	8	yes	complemented
161	C₂×#107	C₂²×C₁₇	1	1	yes	characteristic
162	C₂×#106	C₂²×C₁₇	1	2	yes	—
163	C₂×#106	C₂²×C₁₇	1	2	yes	—
164	C₂×C₄⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
165	C₂×#103	C₂²×C₁₇	1	4	yes	—
166	C₂×#103	C₂²×C₁₇	1	2	yes	—
167	C₂×#97	C₂²×C₁₇	1	2	yes	—
168	C₂×C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
169	#389	—	2	4	no	—
170	#389	C₂²×C₁₇	1	4	yes	—
171	#384	—	2	2	no	—
172	#384	C₂²×C₁₇	1	2	yes	—
173	C₄×C₄⋊C₄	—	2	2	no	—
174	C₄×C₄⋊C₄	C₂²×C₁₇	1	1	yes	characteristic
175	#340	C₂²×C₁₇	1	4	yes	—
176	#336	C₂²×C₁₇	1	2	yes	—
177	C₂×C₁₇	D₄×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₁₇	C₂×#441	1	1	yes	characteristic
179	C₂×C₁₇	#2952	1	1	yes	characteristic
180	C₂×C₁₇	#3278	1	2	yes	—
181	C₂×C₁₇	#3387	1	2	yes	—
182	C₂²×Q₈	C₂³×C₁₇	1	2	yes	—
183	C₂×Q₁₆	C₂×C₄×C₁₇	1	4	yes	—
184	C₂×Q₁₆	D₄×C₁₇	1	4	yes	—
185	C₂×Q₁₆	Q₈×C₁₇	1	4	yes	—
186	C₂²×C₈	C₂³×C₁₇	1	2	yes	—
187	C₂²×C₈	C₂³×C₁₇	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₁₇	1	1	yes	characteristic
189	C₄×Q₈	—	2	4	no	—
190	C₄×Q₈	—	2	8	no	—
191	C₄×Q₈	—	4	4	no	—
192	C₄×Q₈	C₂×C₄×C₁₇	1	8	yes	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	4	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	4	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	D₄×C₁₇	1	2	yes	—
199	C₂×C₄⋊C₄	C₂³×C₁₇	1	4	yes	—
200	C₂×C₄⋊C₄	C₂³×C₁₇	1	2	yes	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	—	2	4	no	—
203	C₂×C₄²	C₂³×C₁₇	1	2	yes	—
204	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
205	#107	C₂×C₄×C₁₇	1	4	yes	—
206	#106	—	2	4	no	—
207	#106	—	4	2	no	—
208	C₄⋊C₈	—	2	2	no	—
209	#103	—	2	8	no	—
210	#103	C₂×C₄×C₁₇	1	8	yes	—
211	#97	—	2	4	no	—
212	C₄×C₈	C₂×C₄×C₁₇	1	4	yes	—
213	#95	—	2	4	no	—
214	#95	D₄×C₁₇	1	2	yes	—
215	C₁₇	#22674	1	1	yes	characteristic, direct product factor, Sylow
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	C₂²×C₄×C₁₇	1	4	yes	—
220	C₂×Q₈	C₂×D₄×C₁₇	1	4	yes	—
221	C₂×Q₈	C₂×Q₈×C₁₇	1	4	yes	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	C₂×D₄×C₁₇	1	4	yes	—
226	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
228	C₂²×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
229	Q₁₆	C₄⋊C₄×C₁₇	1	16	yes	complemented
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	C₂²×C₄×C₁₇	1	2	yes	—
236	C₂×C₈	C₂²×C₄×C₁₇	1	2	yes	—
237	C₂×C₈	C₂×D₄×C₁₇	1	2	yes	—
238	C₂×C₈	C₂×Q₈×C₁₇	1	2	yes	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	8	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	4	8	no	—
245	C₄⋊C₄	—	4	4	no	—
246	C₄⋊C₄	—	8	2	no	—
247	C₄⋊C₄	—	8	4	no	—
248	C₄⋊C₄	C₂²×C₄×C₁₇	1	4	yes	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	8	no	—
252	C₄²	—	4	8	no	—
253	C₄²	C₂²×C₄×C₁₇	1	2	yes	—
254	C₄²	#41×C₁₇	1	2	yes	—
255	C₂³	C₂²×D₄×C₁₇	1	1	yes	characteristic
256	Q₈	—	2	4	no	—
257	Q₈	—	2	4	no	—
258	Q₈	—	2	4	no	—
259	Q₈	C₂×C₄⋊C₄×C₁₇	1	8	yes	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	C₄×D₄×C₁₇	1	2	yes	—
274	C₂×C₄	C₄×D₄×C₁₇	1	2	yes	—
275	C₂×C₄	C₁₇×#137	1	4	yes	—
276	C₂×C₄	C₂³×C₄×C₁₇	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₄×C₁₇	1	1	yes	characteristic
278	C₂×C₄	C₂²×Q₈×C₁₇	1	1	yes	characteristic
279	C₂×C₄	C₂×#41×C₁₇	1	1	yes	characteristic
280	C₂×C₄	C₁₇×H(5,𝔽₂)	1	1	yes	characteristic
281	C₂×C₄	C₁₇×2₋¹⁺⁴	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	4	2	no	—
284	C₈	C₂×C₄⋊C₄×C₁₇	1	4	yes	—
285	C₂²	C₂×C₄×D₄×C₁₇	1	1	yes	characteristic
286	C₂²	D₄²×C₁₇	1	1	yes	characteristic
287	C₂²	D₄×Q₈×C₁₇	1	1	yes	characteristic
288	C₂²	C₂×C₁₇×#137	1	2	yes	—
289	C₂²	C₁₇×#577	1	2	yes	—
290	C₄	—	2	2	no	—
291	C₄	—	2	8	no	—
292	C₄	—	4	4	no	—
293	C₄	—	4	4	no	—
294	C₄	—	8	4	no	—
295	C₄	C₁₇×#441	1	4	yes	complemented
296	C₄	C₂²×C₄⋊C₄×C₁₇	1	1	yes	characteristic
297	C₄	C₁₇×#520	1	1	yes	characteristic
298	C₄	C₁₇×#547	1	1	yes	characteristic
299	C₄	C₁₇×#554	1	1	yes	characteristic
300	C₂	D₄×C₄⋊C₄×C₁₇	1	1	yes	characteristic
301	C₂	C₂×C₁₇×#441	1	1	yes	characteristic
302	C₂	C₁₇×#2952	1	1	yes	characteristic
303	C₂	C₁₇×#3278	1	2	yes	—
304	C₂	C₁₇×#3387	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
13. C₄×Q₁₆×C₁₇	C₄	semidirect
76. Q₁₆×C₁₇	C₄⋊C₄	semidirect
102. #22674	C₁₇	direct (N × H)
151. C₄×C₁₇	#441	semidirect
160. C₄×Q₁₆	C₄×C₁₇	semidirect
215. C₁₇	#22674	direct (N × H)
229. Q₁₆	C₄⋊C₄×C₁₇	semidirect
295. C₄	C₁₇×#441	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₇×#441	2
C₂×C₄×Q₁₆×C₁₇	2
Q₈×C₄⋊C₄×C₁₇	2
C₁₇×#1943	2
C₁₇×#1865	2
C₁₇×#1844	2
C₁₇×#1803	2
C₁₇×#1770	2
#22674	17

Subgroups — C17×#22674 #16880418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#22674 #16880418