Subgroups — C₁₇×#21430 #16879174

Order 4352.

858 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₇×#117	C₂	1	1	yes	maximal, characteristic
2	C₁₇×#1923	C₂	1	4	yes	maximal
3	C₁₇×#1916	C₂	1	4	yes	maximal, complemented
4	C₄×C₁₇×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₇×C₂²⋊C₈	C₂	1	2	yes	maximal
6	C₁₇×#1744	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄²×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₇×#117	C₄	1	1	yes	characteristic
9	C₂×C₁₇×#117	C₄	1	1	yes	characteristic
10	C₂²×C₄²×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₁₇×C₄⋊C₈	C₂²	1	4	yes	—
12	C₂×C₁₇×C₂²⋊C₈	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₁₇	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₁₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₇	C₂²	1	2	yes	—
16	C₄×C₄⋊C₄×C₁₇	C₄	1	2	yes	—
17	C₄×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
18	C₄×C₂²⋊C₄×C₁₇	C₄	1	2	yes	—
19	C₄×C₂²⋊C₄×C₁₇	C₂²	1	2	yes	—
20	C₁₇×#376	C₄	1	1	yes	characteristic
21	C₁₇×#376	C₄	1	1	yes	characteristic
22	C₄³×C₁₇	C₂²	1	1	yes	characteristic
23	C₄³×C₁₇	C₂²	1	1	yes	characteristic
24	C₁₇×#336	C₂²	1	4	yes	—
25	C₁₇×#336	C₂²	1	4	yes	—
26	C₁₇×#336	C₂²	1	4	yes	—
27	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₇	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₁₇	C₂³	1	4	yes	—
31	C₂²×C₈×C₁₇	C₂³	1	4	yes	—
32	C₁₇×#117	—	2	4	no	—
33	C₁₇×#117	C₈	1	8	yes	complemented
34	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₇	—	2	4	no	—
39	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₁₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
51	C₁₇×C₄⋊C₈	—	2	8	no	—
52	C₁₇×C₂²⋊C₈	—	2	8	no	—
53	C₄×C₈×C₁₇	—	2	8	no	—
54	C₄×C₈×C₁₇	—	2	4	no	—
55	C₄×C₈×C₁₇	—	2	4	no	—
56	C₁₇×#95	C₂×C₄	1	2	yes	—
57	C₁₇×#95	C₂×C₄	1	2	yes	—
58	C₁₇×#95	C₂×C₄	1	2	yes	—
59	C₁₇×#95	C₂×C₄	1	2	yes	—
60	C₂⁴×C₁₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	—	2	2	no	—
62	C₂²×C₄×C₁₇	—	2	2	no	—
63	C₂²×C₄×C₁₇	—	2	2	no	—
64	C₂²×C₄×C₁₇	—	2	4	no	—
65	C₂²×C₄×C₁₇	—	2	4	no	—
66	C₂²×C₄×C₁₇	—	2	2	no	—
67	C₂²×C₄×C₁₇	C₂×C₈	1	4	yes	—
68	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₁₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₁₇	—	2	4	no	—
79	C₂×C₈×C₁₇	—	2	4	no	—
80	C₂×C₈×C₁₇	—	2	4	no	—
81	C₂×C₈×C₁₇	—	2	4	no	—
82	C₂×C₈×C₁₇	—	2	4	no	—
83	C₂×C₈×C₁₇	#41	1	8	yes	—
84	C₄⋊C₄×C₁₇	—	2	4	no	—
85	C₄⋊C₄×C₁₇	C₂×C₈	1	8	yes	—
86	C₂²⋊C₄×C₁₇	—	2	4	no	—
87	C₂²⋊C₄×C₁₇	C₂×C₈	1	8	yes	—
88	C₄²×C₁₇	—	2	2	no	—
89	C₄²×C₁₇	—	2	2	no	—
90	C₄²×C₁₇	—	2	2	no	—
91	C₄²×C₁₇	—	2	4	no	—
92	C₄²×C₁₇	—	2	4	no	—
93	C₄²×C₁₇	—	2	4	no	—
94	C₄²×C₁₇	—	2	4	no	—
95	C₄²×C₁₇	—	2	2	no	—
96	C₄²×C₁₇	C₂×C₈	1	4	yes	—
97	C₄²×C₁₇	C₂×C₈	1	4	yes	—
98	C₄²×C₁₇	M₁₆	1	4	yes	—
99	C₄²×C₁₇	#41	1	4	yes	—
100	#21430	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₁₇	—	2	1	no	—
102	C₂³×C₁₇	—	2	2	no	—
103	C₂³×C₁₇	—	2	1	no	—
104	C₂³×C₁₇	—	2	2	no	—
105	C₂³×C₁₇	C₂²×C₈	1	2	yes	—
106	C₂³×C₁₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₇	—	2	1	no	—
108	C₂×C₄×C₁₇	—	2	2	no	—
109	C₂×C₄×C₁₇	—	2	2	no	—
110	C₂×C₄×C₁₇	—	2	2	no	—
111	C₂×C₄×C₁₇	—	2	1	no	—
112	C₂×C₄×C₁₇	—	2	2	no	—
113	C₂×C₄×C₁₇	—	2	2	no	—
114	C₂×C₄×C₁₇	—	2	2	no	—
115	C₂×C₄×C₁₇	—	2	8	no	—
116	C₂×C₄×C₁₇	—	2	1	no	—
117	C₂×C₄×C₁₇	—	2	1	no	—
118	C₂×C₄×C₁₇	—	2	2	no	—
119	C₂×C₄×C₁₇	—	2	2	no	—
120	C₂×C₄×C₁₇	—	2	2	no	—
121	C₂×C₄×C₁₇	—	2	2	no	—
122	C₂×C₄×C₁₇	—	2	2	no	—
123	C₂×C₄×C₁₇	—	4	4	no	—
124	C₂×C₄×C₁₇	—	4	4	no	—
125	C₂×C₄×C₁₇	#117	1	2	yes	—
126	C₂×C₄×C₁₇	#117	1	2	yes	—
127	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
128	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
129	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₁₇	C₂×M₁₆	1	4	yes	—
132	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
133	C₂×C₄×C₁₇	#131	1	4	yes	—
134	C₂×C₄×C₁₇	C₂×#41	1	2	yes	—
135	C₂×C₄×C₁₇	C₂×#41	1	4	yes	—
136	C₈×C₁₇	—	2	8	no	—
137	C₈×C₁₇	—	4	4	no	—
138	C₄×#117	C₂×C₁₇	1	1	yes	characteristic
139	#1923	C₂×C₁₇	1	4	yes	—
140	#1916	C₂×C₁₇	1	4	yes	complemented
141	C₄×C₄⋊C₈	C₂×C₁₇	1	2	yes	complemented
142	C₄×C₂²⋊C₈	C₂×C₁₇	1	2	yes	—
143	#1744	C₂×C₁₇	1	1	yes	characteristic, complemented
144	C₄²×C₈	C₂×C₁₇	1	1	yes	characteristic, complemented
145	C₂²×C₁₇	—	2	2	no	—
146	C₂²×C₁₇	—	2	4	no	—
147	C₂²×C₁₇	—	4	1	no	—
148	C₂²×C₁₇	—	4	2	no	—
149	C₂²×C₁₇	—	4	1	no	—
150	C₂²×C₁₇	C₂×#117	1	1	yes	characteristic
151	C₂²×C₁₇	C₄×#41	1	2	yes	—
152	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
153	C₂²×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
154	C₂²×C₁₇	C₂×#131	1	2	yes	—
155	C₄×C₁₇	—	2	2	no	—
156	C₄×C₁₇	—	2	4	no	—
157	C₄×C₁₇	—	2	4	no	—
158	C₄×C₁₇	—	2	4	no	—
159	C₄×C₁₇	—	4	1	no	—
160	C₄×C₁₇	—	4	2	no	—
161	C₄×C₁₇	—	4	1	no	—
162	C₄×C₁₇	#431	1	4	yes	—
163	C₄×C₁₇	#433	1	4	yes	—
164	C₄×C₁₇	#529	1	4	yes	—
165	C₂×#117	C₄×C₁₇	1	1	yes	characteristic
166	C₂×#117	C₄×C₁₇	1	1	yes	characteristic
167	C₂²×C₄²	C₂²×C₁₇	1	1	yes	characteristic
168	C₂×C₄⋊C₈	C₂²×C₁₇	1	4	yes	—
169	C₂×C₂²⋊C₈	C₂²×C₁₇	1	4	yes	—
170	C₂×C₄×C₈	C₂²×C₁₇	1	4	yes	—
171	C₂×C₄×C₈	C₂²×C₁₇	1	2	yes	—
172	C₂×C₄×C₈	C₂²×C₁₇	1	2	yes	—
173	C₄×C₄⋊C₄	C₄×C₁₇	1	2	yes	—
174	C₄×C₄⋊C₄	C₂²×C₁₇	1	2	yes	—
175	C₄×C₂²⋊C₄	C₄×C₁₇	1	2	yes	—
176	C₄×C₂²⋊C₄	C₂²×C₁₇	1	2	yes	—
177	#376	C₄×C₁₇	1	1	yes	characteristic
178	#376	C₄×C₁₇	1	1	yes	characteristic
179	C₄³	C₂²×C₁₇	1	1	yes	characteristic
180	C₄³	C₂²×C₁₇	1	1	yes	characteristic
181	#336	C₂²×C₁₇	1	4	yes	—
182	#336	C₂²×C₁₇	1	4	yes	—
183	#336	C₂²×C₁₇	1	4	yes	—
184	C₂×C₁₇	—	4	2	no	—
185	C₂×C₁₇	#2298	1	1	yes	characteristic
186	C₂×C₁₇	C₂×#431	1	1	yes	characteristic
187	C₂×C₁₇	C₂×#433	1	1	yes	characteristic
188	C₂×C₁₇	C₈×#41	1	2	yes	—
189	C₂×C₁₇	#2965	1	2	yes	—
190	C₂³×C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
191	C₂³×C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
192	C₂³×C₄	C₂³×C₁₇	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₁₇	1	4	yes	—
194	C₂²×C₈	C₂³×C₁₇	1	4	yes	—
195	#117	—	2	4	no	—
196	#117	C₈×C₁₇	1	8	yes	complemented
197	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
198	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
199	C₂×C₂²⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
200	C₂×C₂²⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
201	C₂×C₄²	—	2	4	no	—
202	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
203	C₂×C₄²	C₂×C₄×C₁₇	1	2	yes	—
204	C₂×C₄²	C₂×C₄×C₁₇	1	2	yes	—
205	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
206	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
207	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
208	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
209	C₂×C₄²	C₂×C₄×C₁₇	1	1	yes	characteristic
210	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
211	C₂×C₄²	C₂³×C₁₇	1	2	yes	—
212	C₂×C₄²	C₂³×C₁₇	1	2	yes	—
213	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
214	C₄⋊C₈	—	2	8	no	—
215	C₂²⋊C₈	—	2	8	no	—
216	C₄×C₈	—	2	8	no	—
217	C₄×C₈	—	2	4	no	—
218	C₄×C₈	—	2	4	no	—
219	#95	C₂×C₄×C₁₇	1	2	yes	—
220	#95	C₂×C₄×C₁₇	1	2	yes	—
221	#95	C₂×C₄×C₁₇	1	2	yes	—
222	#95	C₂×C₄×C₁₇	1	2	yes	—
223	C₁₇	#21430	1	1	yes	characteristic, direct product factor, Sylow
224	C₂⁴	C₂²×C₄×C₁₇	1	1	yes	characteristic
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	C₂×C₈×C₁₇	1	4	yes	—
232	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
233	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
234	C₂²×C₄	C₂²×C₄×C₁₇	1	2	yes	—
235	C₂²×C₄	C₂²×C₄×C₁₇	1	2	yes	—
236	C₂²×C₄	C₂²×C₄×C₁₇	1	2	yes	—
237	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
238	C₂²×C₄	C₂²×C₄×C₁₇	1	2	yes	—
239	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
240	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
241	C₂²×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	#41×C₁₇	1	8	yes	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	C₂×C₈×C₁₇	1	8	yes	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	C₂×C₈×C₁₇	1	8	yes	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	4	no	—
255	C₄²	—	2	4	no	—
256	C₄²	—	2	4	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	C₂×C₈×C₁₇	1	4	yes	—
261	C₄²	C₂×C₈×C₁₇	1	4	yes	—
262	C₄²	M₁₆×C₁₇	1	4	yes	—
263	C₄²	#41×C₁₇	1	4	yes	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	2	no	—
268	C₂³	C₂²×C₈×C₁₇	1	2	yes	—
269	C₂³	C₂³×C₄×C₁₇	1	1	yes	characteristic
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	C₁₇×#117	1	2	yes	—
289	C₂×C₄	C₁₇×#117	1	2	yes	—
290	C₂×C₄	C₂²×C₈×C₁₇	1	4	yes	—
291	C₂×C₄	C₂²×C₈×C₁₇	1	4	yes	—
292	C₂×C₄	C₂²×C₈×C₁₇	1	2	yes	—
293	C₂×C₄	C₂²×C₈×C₁₇	1	2	yes	—
294	C₂×C₄	C₂×M₁₆×C₁₇	1	4	yes	—
295	C₂×C₄	C₂×M₁₆×C₁₇	1	2	yes	—
296	C₂×C₄	C₁₇×#131	1	4	yes	—
297	C₂×C₄	C₂×#41×C₁₇	1	2	yes	—
298	C₂×C₄	C₂×#41×C₁₇	1	4	yes	—
299	C₈	—	2	8	no	—
300	C₈	—	4	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	C₂×C₁₇×#117	1	1	yes	characteristic
307	C₂²	C₄×#41×C₁₇	1	2	yes	—
308	C₂²	C₂³×C₈×C₁₇	1	1	yes	characteristic
309	C₂²	C₂²×M₁₆×C₁₇	1	1	yes	characteristic
310	C₂²	C₂×C₁₇×#131	1	2	yes	—
311	C₄	—	2	4	no	—
312	C₄	—	2	4	no	—
313	C₄	—	2	4	no	—
314	C₄	—	2	2	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	C₁₇×#431	1	4	yes	—
319	C₄	C₁₇×#433	1	4	yes	—
320	C₄	C₁₇×#529	1	4	yes	—
321	C₂	—	4	2	no	—
322	C₂	C₁₇×#2298	1	1	yes	characteristic
323	C₂	C₂×C₁₇×#431	1	1	yes	characteristic
324	C₂	C₂×C₁₇×#433	1	1	yes	characteristic
325	C₂	C₈×#41×C₁₇	1	2	yes	—
326	C₂	C₁₇×#2965	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²×C₁₇	C₂³
commutator	C₂²	C₂³×C₈×C₁₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₇
socle	C₂³×C₁₇	C₂³×C₄
prime radical	C₂⁴×C₁₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₇×#1916	C₂	semidirect
4. C₄×C₁₇×C₄⋊C₈	C₂	semidirect
6. C₁₇×#1744	C₂	semidirect
7. C₄²×C₈×C₁₇	C₂	semidirect
33. C₁₇×#117	C₈	semidirect
100. #21430	C₁₇	direct (N × H)
140. #1916	C₂×C₁₇	semidirect
141. C₄×C₄⋊C₈	C₂×C₁₇	semidirect
143. #1744	C₂×C₁₇	semidirect
144. C₄²×C₈	C₂×C₁₇	semidirect
196. #117	C₈×C₁₇	semidirect
223. C₁₇	#21430	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₁₇×#117	2
C₁₇×#1923	2
C₁₇×#1916	2
C₄×C₁₇×C₄⋊C₈	2
C₄×C₁₇×C₂²⋊C₈	2
C₁₇×#1744	2
C₄²×C₈×C₁₇	2
#21430	17

Subgroups — C17×#21430 #16879174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#21430 #16879174