Subgroups — D₁₇×#485 #14538672

Order 2176.

416 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊Q₈×D₁₇	C₂	1	1	yes	maximal, characteristic
2	#117×D₁₇	C₂	1	1	yes	maximal, characteristic
3	#11763153	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763152	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763151	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763150	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763149	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763148	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763147	C₂	1	1	yes	maximal, characteristic, complemented
10	#107×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#106×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#103×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
13	#103×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²⋊C₈×D₁₇	C₂	1	1	yes	maximal, characteristic
15	C₁₇×#485	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
19	C₁₇×#117	C₂²	1	1	yes	characteristic
20	#10587632	C₂²	1	1	yes	characteristic
21	#10587612	C₂²	1	1	yes	characteristic
22	#10587610	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
24	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
25	#10587583	C₂²	1	1	yes	characteristic
26	#10587582	—	2	1	no	—
27	#10587578	C₂²	1	1	yes	characteristic
28	#10587578	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×D₁₇	—	2	1	no	—
30	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
33	C₂²⋊C₄×D₁₇	—	2	1	no	—
34	C₂²⋊C₄×D₁₇	—	2	1	no	—
35	#10587564	—	2	1	no	—
36	#10587563	—	2	1	no	—
37	#10587558	—	2	1	no	—
38	C₄²×D₁₇	—	2	1	no	—
39	C₁₇×#107	C₂²	1	1	yes	characteristic, complemented
40	C₁₇×#106	C₂²	1	1	yes	characteristic, complemented
41	C₁₇×#103	C₂²	1	1	yes	characteristic, complemented
42	C₁₇×#103	C₂²	1	1	yes	characteristic, complemented
43	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	#10587506	C₂²	1	1	yes	characteristic, complemented
45	#10587506	C₂²	1	1	yes	characteristic, complemented
46	C₂²⋊#3622	C₂²	1	1	yes	characteristic
47	#10587491	C₂²	1	1	yes	characteristic
48	#10587491	C₂²	1	1	yes	characteristic
49	#10587489	C₂²	1	1	yes	characteristic, complemented
50	#10587488	C₂²	1	1	yes	characteristic, complemented
51	#10587487	C₂²	1	1	yes	characteristic, complemented
52	#10587487	C₂²	1	1	yes	characteristic, complemented
53	#10587481	C₂²	1	1	yes	characteristic, complemented
54	#10587481	C₂²	1	1	yes	characteristic, complemented
55	#10587479	C₂²	1	1	yes	characteristic, complemented
56	#10587479	C₂²	1	1	yes	characteristic, complemented
57	#10587478	C₂²	1	1	yes	characteristic, complemented
58	#10587478	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₁₇	D₄	1	1	yes	characteristic
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂²×Dic₁₇	D₄	1	1	yes	characteristic
63	Q₈×D₁₇	—	2	2	no	—
64	Q₈×D₁₇	—	2	2	no	—
65	C₂×C₄×D₁₇	—	2	2	no	—
66	C₂×C₄×D₁₇	—	2	2	no	—
67	C₂×C₄×D₁₇	—	2	1	no	—
68	C₂×C₄×D₁₇	—	2	1	no	—
69	C₂×C₄×D₁₇	—	2	1	no	—
70	C₂×C₄×D₁₇	—	2	1	no	—
71	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
72	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
73	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
74	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
75	C₂×Dic₃₄	—	2	1	no	—
76	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₁₇	—	2	1	no	—
80	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₁₇	—	2	1	no	—
84	C₂²⋊C₄×C₁₇	—	2	1	no	—
85	C₄²×C₁₇	—	2	1	no	—
86	C₂²⋊Dic₁₇	—	2	1	no	—
87	C₂²⋊Dic₁₇	—	2	1	no	—
88	#63227	—	2	1	no	—
89	#63227	—	2	1	no	—
90	#63227	—	2	1	no	—
91	#63227	—	2	1	no	—
92	C₄⋊Dic₁₇	—	2	1	no	—
93	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
94	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
96	#63225	—	2	1	no	—
97	#63225	—	2	1	no	—
98	#63225	—	2	1	no	—
99	#63225	—	2	1	no	—
100	#63225	—	2	1	no	—
101	C₄×Dic₁₇	—	2	1	no	—
102	C₄×Dic₁₇	—	2	1	no	—
103	C₄×Dic₁₇	—	2	1	no	—
104	C₂×#3622	C₂³	1	1	yes	characteristic
105	C₂×#3622	C₂³	1	1	yes	characteristic
106	C₈×D₁₇	—	2	2	no	—
107	C₈×D₁₇	—	2	2	no	—
108	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₁₇	—	2	2	no	—
110	C₂²×D₁₇	—	2	2	no	—
111	C₂²×D₁₇	—	2	2	no	—
112	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₁₇	—	2	1	no	—
115	Q₈×C₁₇	—	2	1	no	—
116	C₂×C₄×C₁₇	—	2	1	no	—
117	C₂×C₄×C₁₇	—	2	1	no	—
118	C₂×C₄×C₁₇	—	2	1	no	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	—	2	1	no	—
121	C₂×C₄×C₁₇	—	2	1	no	—
122	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
124	C₂×Dic₁₇	—	2	2	no	—
125	C₂×Dic₁₇	—	2	1	no	—
126	C₂×Dic₁₇	—	2	1	no	—
127	C₂×Dic₁₇	—	2	1	no	—
128	C₂×Dic₁₇	—	2	1	no	—
129	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
131	C₄×D₁₇	—	2	2	no	—
132	C₄×D₁₇	—	2	1	no	—
133	C₄×D₁₇	—	2	1	no	—
134	C₄×D₁₇	—	2	2	no	—
135	C₄×D₁₇	—	4	2	no	—
136	C₄×D₁₇	—	4	2	no	—
137	C₄×D₁₇	—	4	2	no	—
138	C₄×D₁₇	—	4	2	no	—
139	C₄×D₁₇	#41	1	2	yes	—
140	C₄×D₁₇	#41	1	2	yes	—
141	Dic₃₄	—	2	1	no	—
142	Dic₃₄	—	2	1	no	—
143	Dic₃₄	—	4	2	no	—
144	C₈×C₁₇	—	2	1	no	—
145	C₈×C₁₇	—	2	1	no	—
146	#3622	—	2	1	no	—
147	#3622	—	2	1	no	—
148	C₂×#485	—	17	1	no	maximal, Sylow
149	C₂²×C₁₇	—	2	1	no	—
150	C₂²×C₁₇	—	2	1	no	—
151	C₂²×C₁₇	—	2	1	no	—
152	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
153	C₂×D₁₇	—	2	1	no	—
154	C₂×D₁₇	—	2	1	no	—
155	C₂×D₁₇	—	2	1	no	—
156	C₂×D₁₇	—	4	4	no	—
157	C₂×D₁₇	#123	1	2	yes	—
158	C₂×D₁₇	#135	1	2	yes	—
159	C₂×D₁₇	#137	1	2	yes	—
160	C₄×C₁₇	—	2	1	no	—
161	C₄×C₁₇	—	4	1	no	—
162	C₄×C₁₇	—	4	1	no	—
163	C₄×C₁₇	—	4	1	no	—
164	C₄×C₁₇	—	4	1	no	—
165	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
166	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
167	Dic₁₇	—	2	1	no	—
168	Dic₁₇	—	4	1	no	—
169	Dic₁₇	—	4	1	no	—
170	Dic₁₇	—	4	1	no	—
171	Dic₁₇	—	4	1	no	—
172	Dic₁₇	#123	1	2	yes	—
173	C₂×C₂²⋊Q₈	—	17	1	no	—
174	C₂×#117	—	17	1	no	—
175	#485	—	17	1	no	—
176	#485	—	17	1	no	—
177	#485	—	17	1	no	—
178	#485	—	17	1	no	—
179	#485	—	17	1	no	—
180	#485	—	17	1	no	—
181	#485	—	17	1	no	—
182	#485	D₁₇	1	1	yes	characteristic, direct product factor
183	C₂×#107	—	17	1	no	—
184	C₂×#106	—	17	1	no	—
185	C₂×#103	—	17	1	no	—
186	C₂×#103	—	17	1	no	—
187	C₂×C₂²⋊C₈	—	17	1	no	—
188	C₂×C₁₇	—	4	1	no	—
189	C₂×C₁₇	C₂×#123	1	1	yes	characteristic
190	C₂×C₁₇	C₂×#135	1	1	yes	characteristic
191	C₂×C₁₇	C₂×#137	1	1	yes	characteristic
192	D₁₇	—	4	1	no	—
193	D₁₇	#485	1	4	yes	direct product factor
194	C₂²×Q₈	—	17	1	no	—
195	C₂³×C₄	—	17	1	no	—
196	C₂²×C₈	—	17	1	no	—
197	C₂²×C₈	—	17	1	no	—
198	C₂²⋊Q₈	—	17	1	no	—
199	C₂²⋊Q₈	—	17	1	no	—
200	C₂²⋊Q₈	—	17	1	no	—
201	C₂²⋊Q₈	—	34	1	no	—
202	C₂²⋊Q₈	—	34	1	no	—
203	C₂²⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
204	#117	—	17	1	no	—
205	#117	—	17	1	no	—
206	#117	—	17	1	no	—
207	#117	—	34	1	no	—
208	#117	—	34	1	no	—
209	#117	C₂×D₁₇	1	1	yes	characteristic
210	C₂×C₄⋊C₄	—	17	1	no	—
211	C₂×C₄⋊C₄	—	17	1	no	—
212	C₂×C₄⋊C₄	—	17	1	no	—
213	C₂×C₄⋊C₄	—	34	1	no	—
214	C₂×C₂²⋊C₄	—	34	1	no	—
215	C₂×C₂²⋊C₄	—	34	1	no	—
216	C₂×C₄²	—	34	1	no	—
217	#107	—	17	1	no	—
218	#107	—	17	1	no	—
219	#107	—	17	1	no	—
220	#107	C₂×D₁₇	1	1	yes	characteristic, complemented
221	#106	—	17	1	no	—
222	#106	—	17	1	no	—
223	#106	—	17	1	no	—
224	#106	C₂×D₁₇	1	1	yes	characteristic, complemented
225	#103	—	17	1	no	—
226	#103	—	17	1	no	—
227	#103	—	17	1	no	—
228	#103	—	17	1	no	—
229	#103	—	17	1	no	—
230	#103	—	17	1	no	—
231	#103	C₂×D₁₇	1	1	yes	characteristic, complemented
232	#103	C₂×D₁₇	1	1	yes	characteristic, complemented
233	C₂²⋊C₈	—	17	1	no	—
234	C₂²⋊C₈	—	17	1	no	—
235	C₂²⋊C₈	—	17	1	no	—
236	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
237	C₁₇	C₂×#485	1	1	yes	characteristic, complemented, Sylow
238	C₂⁴	—	17	1	no	—
239	C₂×Q₈	—	17	1	no	—
240	C₂×Q₈	—	34	1	no	—
241	C₂×Q₈	—	34	2	no	—
242	C₂×Q₈	—	34	2	no	—
243	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
244	C₂²×C₄	—	17	1	no	—
245	C₂²×C₄	—	17	1	no	—
246	C₂²×C₄	—	17	1	no	—
247	C₂²×C₄	—	17	1	no	—
248	C₂²×C₄	—	17	1	no	—
249	C₂²×C₄	—	34	2	no	—
250	C₂²×C₄	—	34	2	no	—
251	C₂²×C₄	—	34	1	no	—
252	C₂²×C₄	—	34	1	no	—
253	C₂²×C₄	—	34	1	no	—
254	C₂²×C₄	—	34	1	no	—
255	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
256	C₂×C₈	—	17	1	no	—
257	C₂×C₈	—	17	1	no	—
258	C₂×C₈	—	34	2	no	—
259	C₂×C₈	—	34	2	no	—
260	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	17	1	no	—
264	C₄⋊C₄	—	17	1	no	—
265	C₄⋊C₄	—	17	1	no	—
266	C₄⋊C₄	—	34	1	no	—
267	C₄⋊C₄	—	34	1	no	—
268	C₄⋊C₄	—	34	1	no	—
269	C₄⋊C₄	—	34	1	no	—
270	C₄⋊C₄	—	34	1	no	—
271	C₄⋊C₄	—	34	1	no	—
272	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
273	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	34	1	no	—
278	C₂²⋊C₄	—	34	1	no	—
279	C₂²⋊C₄	—	34	1	no	—
280	C₂²⋊C₄	—	34	1	no	—
281	C₂²⋊C₄	—	34	1	no	—
282	C₂²⋊C₄	—	34	1	no	—
283	C₄²	—	2	1	no	—
284	C₄²	—	34	1	no	—
285	C₄²	—	34	1	no	—
286	C₄²	—	34	1	no	—
287	C₂³	—	17	1	no	—
288	C₂³	—	17	1	no	—
289	C₂³	—	34	2	no	—
290	C₂³	—	34	2	no	—
291	C₂³	—	34	2	no	—
292	C₂³	D₄×D₁₇	1	1	yes	characteristic
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	34	1	no	—
296	Q₈	—	34	1	no	—
297	Q₈	—	68	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	17	1	no	—
305	C₂×C₄	—	17	2	no	—
306	C₂×C₄	—	17	2	no	—
307	C₂×C₄	—	17	1	no	—
308	C₂×C₄	—	34	2	no	—
309	C₂×C₄	—	34	2	no	—
310	C₂×C₄	—	34	2	no	—
311	C₂×C₄	—	34	1	no	—
312	C₂×C₄	—	34	1	no	—
313	C₂×C₄	—	34	1	no	—
314	C₂×C₄	—	34	1	no	—
315	C₂×C₄	—	34	1	no	—
316	C₂×C₄	—	34	1	no	—
317	C₂×C₄	—	68	2	no	—
318	C₂×C₄	—	68	2	no	—
319	C₂×C₄	—	68	2	no	—
320	C₂×C₄	—	68	2	no	—
321	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
322	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	34	1	no	—
326	C₈	—	34	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	17	2	no	—
331	C₂²	—	17	2	no	—
332	C₂²	—	17	2	no	—
333	C₂²	—	34	1	no	—
334	C₂²	—	34	1	no	—
335	C₂²	—	34	1	no	—
336	C₂²	—	68	4	no	—
337	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	17	2	no	—
344	C₄	—	34	1	no	—
345	C₄	—	68	1	no	—
346	C₄	—	68	1	no	—
347	C₄	—	68	1	no	—
348	C₄	—	68	1	no	—
349	C₄	#41×D₁₇	1	1	yes	characteristic
350	C₄	#41×D₁₇	1	1	yes	characteristic
351	C₂	—	4	1	no	—
352	C₂	—	17	4	no	—
353	C₂	—	68	1	no	—
354	C₂	#123×D₁₇	1	1	yes	characteristic
355	C₂	#135×D₁₇	1	1	yes	characteristic
356	C₂	#137×D₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#485	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂³×D₁₇	D₄
hypercenter	#485	D₁₇
hypocenter	C₁₇	C₂×#485
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
3. #11763153	C₂	semidirect
4. #11763152	C₂	semidirect
5. #11763151	C₂	semidirect
6. #11763150	C₂	semidirect
7. #11763149	C₂	semidirect
8. #11763148	C₂	semidirect
9. #11763147	C₂	semidirect
10. #107×D₁₇	C₂	semidirect
11. #106×D₁₇	C₂	semidirect
12. #103×D₁₇	C₂	semidirect
13. #103×D₁₇	C₂	semidirect
15. C₁₇×#485	C₂	semidirect
39. C₁₇×#107	C₂²	semidirect
40. C₁₇×#106	C₂²	semidirect
41. C₁₇×#103	C₂²	semidirect
42. C₁₇×#103	C₂²	semidirect
44. #10587506	C₂²	semidirect
45. #10587506	C₂²	semidirect
49. #10587489	C₂²	semidirect
50. #10587488	C₂²	semidirect
51. #10587487	C₂²	semidirect
52. #10587487	C₂²	semidirect
53. #10587481	C₂²	semidirect
54. #10587481	C₂²	semidirect
55. #10587479	C₂²	semidirect
56. #10587479	C₂²	semidirect
57. #10587478	C₂²	semidirect
58. #10587478	C₂²	semidirect
182. #485	D₁₇	direct (N × H)
193. D₁₇	#485	direct (N × H)
220. #107	C₂×D₁₇	semidirect
224. #106	C₂×D₁₇	semidirect
231. #103	C₂×D₁₇	semidirect
232. #103	C₂×D₁₇	semidirect
237. C₁₇	C₂×#485	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊Q₈×D₁₇	2
#117×D₁₇	2
#11763153	2
#11763152	2
#11763151	2
#11763150	2
#11763149	2
#11763148	2
#11763147	2
#107×D₁₇	2
#106×D₁₇	2
#103×D₁₇	2
C₂²⋊C₈×D₁₇	2
C₁₇×#485	2
C₂×#485	17

Subgroups — D17×#485 #14538672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₇×#485 #14538672