Subgroups — C₂×C₁₇×#3185 #16918436

Order 4352.

1954 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×#533	C₂	1	2	yes	maximal
2	C₁₇×#3185	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₁₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₇×#487	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₇×#470	C₂	1	4	yes	maximal
7	C₂×C₁₇×#441	C₂	1	4	yes	maximal, complemented
8	C₂×C₄×M₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₁₇	C₂²	1	4	yes	—
10	C₂²×M₁₆×C₁₇	C₂²	1	2	yes	—
11	C₁₇×#533	—	2	8	no	—
12	C₁₇×#533	C₂²	1	16	yes	—
13	C₂×C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₂×C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₂×C₁₇×#125	—	2	2	no	—
16	C₂×C₁₇×C₂²⋊Q₈	—	2	2	no	—
17	C₂×C₁₇×C₂²⋊Q₈	C₂²	1	4	yes	—
18	C₂×C₄×Q₈×C₁₇	—	2	2	no	—
19	C₂×C₄×Q₈×C₁₇	C₂²	1	4	yes	—
20	C₂×C₁₇×#117	—	2	2	no	—
21	C₂²×C₄²×C₁₇	C₂²	1	1	yes	characteristic
22	C₁₇×#498	C₂²	1	8	yes	complemented
23	C₁₇×#494	C₂²	1	8	yes	complemented
24	C₁₇×#487	C₂²	1	16	yes	complemented
25	C₁₇×#470	C₂²	1	32	yes	—
26	C₁₇×#441	C₂²	1	32	yes	complemented
27	C₂×C₁₇×#106	C₂²	1	4	yes	—
28	C₂×C₁₇×#103	C₂²	1	8	yes	—
29	C₄×M₁₆×C₁₇	C₂²	1	8	yes	—
30	C₂×C₁₇×#97	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
33	C₂²×Q₈×C₁₇	—	2	2	no	—
34	C₂²×Q₈×C₁₇	C₂³	1	4	yes	—
35	C₂³×C₄×C₁₇	D₄	1	2	yes	—
36	C₂³×C₄×C₁₇	C₂³	1	1	yes	characteristic
37	C₂×Q₁₆×C₁₇	—	2	16	no	—
38	C₂×Q₁₆×C₁₇	C₂³	1	16	yes	—
39	C₂×M₁₆×C₁₇	—	2	8	no	—
40	C₂×M₁₆×C₁₇	C₂³	1	8	yes	—
41	C₂²×C₈×C₁₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₇	C₂³	1	2	yes	—
43	C₁₇×C₄⋊Q₈	—	2	4	no	—
44	C₁₇×C₄⋊Q₈	—	2	4	no	—
45	C₁₇×C₄⋊Q₈	C₂³	1	8	yes	—
46	C₁₇×C₄⋊Q₈	C₂³	1	8	yes	—
47	C₁₇×#125	—	2	16	no	—
48	C₁₇×C₂²⋊Q₈	—	2	16	no	—
49	C₁₇×C₂²⋊Q₈	—	2	8	no	—
50	C₁₇×C₂²⋊Q₈	C₂³	1	16	yes	—
51	C₄×Q₈×C₁₇	—	2	16	no	—
52	C₄×Q₈×C₁₇	—	2	8	no	—
53	C₄×Q₈×C₁₇	C₂³	1	16	yes	—
54	C₁₇×#117	—	2	16	no	—
55	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₁₇	—	2	4	no	—
59	C₂×C₄⋊C₄×C₁₇	—	2	4	no	—
60	C₂×C₄⋊C₄×C₁₇	C₂³	1	4	yes	—
61	C₂×C₂²⋊C₄×C₁₇	—	2	4	no	—
62	C₂×C₄²×C₁₇	—	2	4	no	—
63	C₂×C₄²×C₁₇	D₄	1	2	yes	—
64	C₂×C₄²×C₁₇	D₄	1	4	yes	—
65	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₁₇	C₂³	1	4	yes	—
68	C₁₇×#106	C₂³	1	16	yes	—
69	C₁₇×#103	C₂³	1	32	yes	—
70	C₁₇×#97	C₂³	1	8	yes	—
71	C₄×C₈×C₁₇	C₂³	1	4	yes	—
72	C₄×C₈×C₁₇	C₂³	1	4	yes	—
73	C₂⁴×C₁₇	C₂×D₄	1	1	yes	characteristic
74	C₂×Q₈×C₁₇	—	2	4	no	—
75	C₂×Q₈×C₁₇	—	2	8	no	—
76	C₂×Q₈×C₁₇	—	2	16	no	—
77	C₂×Q₈×C₁₇	—	4	8	no	—
78	C₂×Q₈×C₁₇	C₂⁴	1	8	yes	—
79	C₂²×C₄×C₁₇	—	2	8	no	—
80	C₂²×C₄×C₁₇	—	2	4	no	—
81	C₂²×C₄×C₁₇	—	2	4	no	—
82	C₂²×C₄×C₁₇	—	2	4	no	—
83	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₁₇	C₂×D₄	1	8	yes	—
85	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₁₇	C₂⁴	1	2	yes	—
89	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
90	Q₁₆×C₁₇	—	2	32	no	—
91	M₁₆×C₁₇	—	2	16	no	—
92	C₂×C₈×C₁₇	—	2	4	no	—
93	C₂×C₈×C₁₇	#41	1	8	yes	—
94	C₂×C₈×C₁₇	C₂⁴	1	4	yes	—
95	C₂×C₈×C₁₇	C₂⁴	1	4	yes	—
96	C₄⋊C₄×C₁₇	—	2	16	no	—
97	C₄⋊C₄×C₁₇	—	2	4	no	—
98	C₄⋊C₄×C₁₇	—	2	8	no	—
99	C₄⋊C₄×C₁₇	—	2	16	no	—
100	C₄⋊C₄×C₁₇	—	2	8	no	—
101	C₄⋊C₄×C₁₇	—	2	8	no	—
102	C₄⋊C₄×C₁₇	C₂⁴	1	8	yes	—
103	C₂²⋊C₄×C₁₇	—	2	16	no	—
104	C₄²×C₁₇	—	2	16	no	—
105	C₄²×C₁₇	C₂×D₄	1	8	yes	—
106	C₄²×C₁₇	C₂×D₄	1	2	yes	—
107	C₄²×C₁₇	C₂×D₄	1	2	yes	—
108	C₄²×C₁₇	C₂⁴	1	2	yes	—
109	C₄²×C₁₇	C₂⁴	1	2	yes	—
110	C₂×#3185	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂³×C₁₇	—	2	2	no	—
112	C₂³×C₁₇	—	2	4	no	—
113	C₂³×C₁₇	C₂²×D₄	1	2	yes	—
114	C₂³×C₁₇	C₂²×D₄	1	1	yes	characteristic
115	Q₈×C₁₇	—	2	16	no	—
116	Q₈×C₁₇	—	4	8	no	—
117	Q₈×C₁₇	—	4	16	no	—
118	C₂×C₄×C₁₇	—	2	2	no	—
119	C₂×C₄×C₁₇	—	2	4	no	—
120	C₂×C₄×C₁₇	—	2	4	no	—
121	C₂×C₄×C₁₇	—	2	16	no	—
122	C₂×C₄×C₁₇	—	2	4	no	—
123	C₂×C₄×C₁₇	—	2	8	no	—
124	C₂×C₄×C₁₇	—	2	8	no	—
125	C₂×C₄×C₁₇	—	4	8	no	—
126	C₂×C₄×C₁₇	—	4	8	no	—
127	C₂×C₄×C₁₇	#137	1	8	yes	—
128	C₂×C₄×C₁₇	C₂²×D₄	1	4	yes	—
129	C₂×C₄×C₁₇	C₂²×D₄	1	2	yes	—
130	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₇	C₂²×D₄	1	4	yes	—
132	C₂×C₄×C₁₇	C₂×#41	1	4	yes	—
133	C₂×C₄×C₁₇	C₂⁵	1	1	yes	characteristic
134	C₈×C₁₇	—	2	4	no	—
135	C₈×C₁₇	C₂×#41	1	8	yes	—
136	C₂×#533	C₂×C₁₇	1	2	yes	—
137	#3185	C₂×C₁₇	1	16	yes	direct product factor
138	C₂×#498	C₂×C₁₇	1	1	yes	characteristic, complemented
139	C₂×#494	C₂×C₁₇	1	1	yes	characteristic, complemented
140	C₂×#487	C₂×C₁₇	1	2	yes	complemented
141	C₂×#470	C₂×C₁₇	1	4	yes	—
142	C₂×#441	C₂×C₁₇	1	4	yes	complemented
143	C₂×C₄×M₁₆	C₂×C₁₇	1	1	yes	characteristic
144	C₂²×C₁₇	—	2	2	no	—
145	C₂²×C₁₇	—	2	4	no	—
146	C₂²×C₁₇	—	4	4	no	—
147	C₂²×C₁₇	#532	1	2	yes	—
148	C₂²×C₁₇	C₂×#137	1	4	yes	—
149	C₂²×C₁₇	C₂³×D₄	1	1	yes	characteristic
150	C₄×C₁₇	—	2	2	no	—
151	C₄×C₁₇	—	2	4	no	—
152	C₄×C₁₇	—	4	8	no	—
153	C₄×C₁₇	—	4	8	no	—
154	C₄×C₁₇	#532	1	2	yes	—
155	C₄×C₁₇	C₂×#137	1	8	yes	—
156	C₄×C₁₇	C₂²×#41	1	2	yes	—
157	C₂²×Q₁₆	C₂²×C₁₇	1	4	yes	—
158	C₂²×M₁₆	C₂²×C₁₇	1	2	yes	—
159	#533	—	2	8	no	—
160	#533	C₂²×C₁₇	1	16	yes	—
161	C₂×C₄⋊Q₈	C₂²×C₁₇	1	2	yes	—
162	C₂×C₄⋊Q₈	C₂²×C₁₇	1	2	yes	—
163	C₂×#125	—	2	2	no	—
164	C₂×C₂²⋊Q₈	—	2	2	no	—
165	C₂×C₂²⋊Q₈	C₂²×C₁₇	1	4	yes	—
166	C₂×C₄×Q₈	—	2	2	no	—
167	C₂×C₄×Q₈	C₂²×C₁₇	1	4	yes	—
168	C₂×#117	—	2	2	no	—
169	C₂²×C₄²	C₂²×C₁₇	1	1	yes	characteristic
170	#498	C₂²×C₁₇	1	8	yes	complemented
171	#494	C₂²×C₁₇	1	8	yes	complemented
172	#487	C₂²×C₁₇	1	16	yes	complemented
173	#470	C₂²×C₁₇	1	32	yes	—
174	#441	C₂²×C₁₇	1	32	yes	complemented
175	C₂×#106	C₂²×C₁₇	1	4	yes	—
176	C₂×#103	C₂²×C₁₇	1	8	yes	—
177	C₄×M₁₆	C₂²×C₁₇	1	8	yes	—
178	C₂×#97	C₂²×C₁₇	1	2	yes	—
179	C₂×C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
181	C₂×C₁₇	—	4	2	no	—
182	C₂×C₁₇	#3185	1	4	yes	direct product factor
183	C₂×C₁₇	C₂×#532	1	1	yes	characteristic
184	C₂×C₁₇	C₂²×#137	1	2	yes	—
185	C₂²×Q₈	—	2	2	no	—
186	C₂²×Q₈	C₂³×C₁₇	1	4	yes	—
187	C₂³×C₄	D₄×C₁₇	1	2	yes	—
188	C₂³×C₄	C₂³×C₁₇	1	1	yes	characteristic
189	C₂×Q₁₆	—	2	16	no	—
190	C₂×Q₁₆	C₂³×C₁₇	1	16	yes	—
191	C₂×M₁₆	—	2	8	no	—
192	C₂×M₁₆	C₂³×C₁₇	1	8	yes	—
193	C₂²×C₈	C₂³×C₁₇	1	2	yes	—
194	C₂²×C₈	C₂³×C₁₇	1	2	yes	—
195	C₄⋊Q₈	—	2	4	no	—
196	C₄⋊Q₈	—	2	4	no	—
197	C₄⋊Q₈	C₂³×C₁₇	1	8	yes	—
198	C₄⋊Q₈	C₂³×C₁₇	1	8	yes	—
199	#125	—	2	16	no	—
200	C₂²⋊Q₈	—	2	16	no	—
201	C₂²⋊Q₈	—	2	8	no	—
202	C₂²⋊Q₈	C₂³×C₁₇	1	16	yes	—
203	C₄×Q₈	—	2	16	no	—
204	C₄×Q₈	—	2	8	no	—
205	C₄×Q₈	C₂³×C₁₇	1	16	yes	—
206	#117	—	2	16	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	4	no	—
211	C₂×C₄⋊C₄	—	2	4	no	—
212	C₂×C₄⋊C₄	C₂³×C₁₇	1	4	yes	—
213	C₂×C₂²⋊C₄	—	2	4	no	—
214	C₂×C₄²	—	2	4	no	—
215	C₂×C₄²	D₄×C₁₇	1	2	yes	—
216	C₂×C₄²	D₄×C₁₇	1	4	yes	—
217	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
218	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
219	C₂×C₄²	C₂³×C₁₇	1	4	yes	—
220	#106	C₂³×C₁₇	1	16	yes	—
221	#103	C₂³×C₁₇	1	32	yes	—
222	#97	C₂³×C₁₇	1	8	yes	—
223	C₄×C₈	C₂³×C₁₇	1	4	yes	—
224	C₄×C₈	C₂³×C₁₇	1	4	yes	—
225	C₁₇	C₂×#3185	1	1	yes	characteristic, direct product factor, Sylow
226	C₂⁴	C₂×D₄×C₁₇	1	1	yes	characteristic
227	C₂×Q₈	—	2	4	no	—
228	C₂×Q₈	—	2	8	no	—
229	C₂×Q₈	—	2	16	no	—
230	C₂×Q₈	—	4	8	no	—
231	C₂×Q₈	C₂⁴×C₁₇	1	8	yes	—
232	C₂²×C₄	—	2	8	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	C₂×D₄×C₁₇	1	2	yes	—
237	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₁₇	1	2	yes	—
239	C₂²×C₄	C₂×D₄×C₁₇	1	8	yes	—
240	C₂²×C₄	C₂×D₄×C₁₇	1	2	yes	—
241	C₂²×C₄	C₂⁴×C₁₇	1	2	yes	—
242	C₂²×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
243	Q₁₆	—	2	32	no	—
244	M₁₆	—	2	16	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	#41×C₁₇	1	8	yes	—
247	C₂×C₈	C₂⁴×C₁₇	1	4	yes	—
248	C₂×C₈	C₂⁴×C₁₇	1	4	yes	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄⋊C₄	—	2	8	no	—
251	C₄⋊C₄	—	2	8	no	—
252	C₄⋊C₄	—	2	16	no	—
253	C₄⋊C₄	—	2	16	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	C₂⁴×C₁₇	1	8	yes	—
256	C₂²⋊C₄	—	2	16	no	—
257	C₄²	—	2	16	no	—
258	C₄²	C₂×D₄×C₁₇	1	2	yes	—
259	C₄²	C₂×D₄×C₁₇	1	8	yes	—
260	C₄²	C₂×D₄×C₁₇	1	2	yes	—
261	C₄²	C₂⁴×C₁₇	1	2	yes	—
262	C₄²	C₂⁴×C₁₇	1	2	yes	—
263	C₂³	—	2	2	no	—
264	C₂³	—	2	4	no	—
265	C₂³	C₂²×D₄×C₁₇	1	2	yes	—
266	C₂³	C₂²×D₄×C₁₇	1	1	yes	characteristic
267	Q₈	—	2	16	no	—
268	Q₈	—	4	8	no	—
269	Q₈	—	4	16	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	16	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	4	8	no	—
278	C₂×C₄	—	4	8	no	—
279	C₂×C₄	C₁₇×#137	1	8	yes	—
280	C₂×C₄	C₂²×D₄×C₁₇	1	4	yes	—
281	C₂×C₄	C₂²×D₄×C₁₇	1	2	yes	—
282	C₂×C₄	C₂²×D₄×C₁₇	1	1	yes	characteristic
283	C₂×C₄	C₂²×D₄×C₁₇	1	4	yes	—
284	C₂×C₄	C₂×#41×C₁₇	1	4	yes	—
285	C₂×C₄	C₂⁵×C₁₇	1	1	yes	characteristic
286	C₈	—	2	4	no	—
287	C₈	C₂×#41×C₁₇	1	8	yes	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	4	4	no	—
291	C₂²	C₁₇×#532	1	2	yes	—
292	C₂²	C₂×C₁₇×#137	1	4	yes	—
293	C₂²	C₂³×D₄×C₁₇	1	1	yes	characteristic
294	C₄	—	2	4	no	—
295	C₄	—	2	2	no	—
296	C₄	—	4	8	no	—
297	C₄	—	4	8	no	—
298	C₄	C₁₇×#532	1	2	yes	—
299	C₄	C₂×C₁₇×#137	1	8	yes	—
300	C₄	C₂²×#41×C₁₇	1	2	yes	—
301	C₂	—	4	2	no	—
302	C₂	C₁₇×#3185	1	4	yes	direct product factor
303	C₂	C₂×C₁₇×#532	1	1	yes	characteristic
304	C₂	C₂²×C₁₇×#137	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₇
prime radical	C₂⁴×C₁₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇ ◁ C₂²×C₄²×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₇×#3185	C₂	direct (N × H)
3. C₂×C₁₇×#498	C₂	semidirect
4. C₂×C₁₇×#494	C₂	semidirect
5. C₂×C₁₇×#487	C₂	semidirect
7. C₂×C₁₇×#441	C₂	semidirect
22. C₁₇×#498	C₂²	semidirect
23. C₁₇×#494	C₂²	semidirect
24. C₁₇×#487	C₂²	semidirect
26. C₁₇×#441	C₂²	semidirect
110. C₂×#3185	C₁₇	direct (N × H)
137. #3185	C₂×C₁₇	direct (N × H)
138. C₂×#498	C₂×C₁₇	semidirect
139. C₂×#494	C₂×C₁₇	semidirect
140. C₂×#487	C₂×C₁₇	semidirect
142. C₂×#441	C₂×C₁₇	semidirect
170. #498	C₂²×C₁₇	semidirect
171. #494	C₂²×C₁₇	semidirect
172. #487	C₂²×C₁₇	semidirect
174. #441	C₂²×C₁₇	semidirect
182. C₂×C₁₇	#3185	direct (N × H)
225. C₁₇	C₂×#3185	direct (N × H)
302. C₂	C₁₇×#3185	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×#533	2
C₁₇×#3185	2
C₂×C₁₇×#498	2
C₂×C₁₇×#494	2
C₂×C₁₇×#487	2
C₂×C₁₇×#470	2
C₂×C₁₇×#441	2
C₂×C₄×M₁₆×C₁₇	2
C₂×#3185	17

Subgroups — C2×C17×#3185 #16918436

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₇×#3185 #16918436