Subgroups — D₁₇×#2183 #17187807

Order 4352.

542 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Q₈×D₁₇	C₂	1	1	yes	maximal, characteristic
2	#14530693	C₂	1	1	yes	maximal, characteristic, complemented
3	#14530692	C₂	1	1	yes	maximal, characteristic, complemented
4	#14530691	C₂	1	2	yes	maximal, complemented
5	#14530690	C₂	1	2	yes	maximal, complemented
6	#14530689	C₂	1	1	yes	maximal, characteristic, complemented
7	D₁₇×C₄⋊C₁₆	C₂	1	2	yes	maximal
8	D₁₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
9	D₁₇×#346	C₂	1	2	yes	maximal
10	C₄×C₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
11	C₁₇×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×Q₈×D₁₇	C₄	1	1	yes	characteristic
13	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₁₇	C₂²	1	2	yes	—
15	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
16	Q₈×#3622	C₄	1	1	yes	characteristic
17	C₈×Dic₃₄	C₂²	1	1	yes	characteristic
18	C₈×Dic₃₄	C₂²	1	2	yes	—
19	#11762933	C₄	1	1	yes	characteristic
20	#11762933	C₄	1	2	yes	—
21	C₄⋊C₈×D₁₇	C₄	1	1	yes	characteristic
22	C₄⋊C₈×D₁₇	C₄	1	2	yes	—
23	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
24	C₄×C₈×D₁₇	C₂²	1	2	yes	—
25	C₄⋊#63214	C₂²	1	2	yes	—
26	C₄⋊#63214	C₂²	1	1	yes	characteristic
27	#11762627	C₂²	1	2	yes	—
28	#11762627	C₂²	1	2	yes	—
29	#11762627	C₂²	1	2	yes	—
30	#11762585	C₂²	1	2	yes	—
31	#11762584	C₂²	1	2	yes	—
32	#11762584	C₂²	1	2	yes	—
33	C₄×#63214	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₁₇	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₁₇	C₂²	1	1	yes	characteristic
36	C₈×Q₈×C₁₇	C₂²	1	1	yes	characteristic
37	C₁₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₁₇×C₄⋊C₁₆	C₂²	1	2	yes	—
39	C₁₇×#346	C₂²	1	2	yes	—
40	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
41	C₂×Q₈×D₁₇	C₈	1	1	yes	characteristic
42	C₄×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
43	Q₈×Dic₁₇	C₈	1	1	yes	characteristic
44	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×D₁₇	C₈	1	2	yes	—
48	C₄⋊C₄×D₁₇	C₈	1	1	yes	characteristic
49	#10587573	C₈	1	2	yes	—
50	#10587573	C₈	1	1	yes	characteristic
51	C₄²×D₁₇	C₂×C₄	1	2	yes	—
52	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
53	C₄×Dic₃₄	C₂×C₄	1	1	yes	characteristic
54	C₄×Dic₃₄	C₂×C₄	1	2	yes	—
55	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
56	C₂×C₁₆×C₁₇	C₂³	1	2	yes	—
57	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
58	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
59	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₁₇	C₂³	1	2	yes	—
62	#10587485	C₂×C₄	1	2	yes	—
63	#10587485	C₂×C₄	1	2	yes	—
64	#10587485	C₂×C₄	1	2	yes	—
65	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
67	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
68	C₈×Dic₁₇	C₂³	1	2	yes	—
69	C₂×#63214	C₂³	1	1	yes	characteristic
70	C₂×#63214	C₂³	1	2	yes	—
71	C₂×#63214	C₂³	1	1	yes	characteristic
72	C₄⋊#3622	C₂×C₄	1	1	yes	characteristic
73	C₄⋊#3622	C₂×C₄	1	2	yes	—
74	C₄×#3622	C₂×C₄	1	1	yes	characteristic
75	C₄×#3622	C₂×C₄	1	2	yes	—
76	C₁₆×D₁₇	—	2	2	no	—
77	C₁₆×D₁₇	—	2	4	no	—
78	C₁₆×D₁₇	Q₈	1	4	yes	complemented
79	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
80	Q₈×D₁₇	—	2	4	no	—
81	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
82	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
83	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₃₄	C₂×C₈	1	1	yes	characteristic
87	C₂×Dic₃₄	C₂×C₈	1	2	yes	—
88	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₇	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
92	C₄⋊C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
93	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
94	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
95	C₄⋊Dic₁₇	C₂×C₈	1	2	yes	—
96	C₄⋊Dic₁₇	C₂×C₈	1	1	yes	characteristic
97	#63225	C₂×C₈	1	2	yes	—
98	#63225	C₂×C₈	1	2	yes	—
99	#63225	C₂×C₈	1	2	yes	—
100	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
101	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
102	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
103	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
104	C₄×Dic₁₇	C₂²×C₄	1	2	yes	—
105	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
107	C₂×#3622	C₂²×C₄	1	2	yes	—
108	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
109	C₈×D₁₇	—	2	2	no	—
110	C₈×D₁₇	—	2	4	no	—
111	C₈×D₁₇	C₂×Q₈	1	2	yes	—
112	C₈×D₁₇	#41	1	2	yes	—
113	C₁₆×C₁₇	—	2	1	no	—
114	C₁₆×C₁₇	—	2	2	no	—
115	C₁₆×C₁₇	C₂×Q₈	1	2	yes	complemented
116	#63214	—	2	1	no	—
117	#63214	—	2	2	no	—
118	#63214	C₂×Q₈	1	2	yes	complemented
119	C₂×#2183	—	17	1	no	maximal, Sylow
120	C₂²×D₁₇	C₂²×C₈	1	1	yes	characteristic
121	Q₈×C₁₇	—	2	2	no	—
122	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
127	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
128	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
129	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
130	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
131	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
132	C₄×D₁₇	—	2	4	no	—
133	C₄×D₁₇	—	2	4	no	—
134	C₄×D₁₇	—	2	2	no	—
135	C₄×D₁₇	M₃₂	1	4	yes	—
136	C₄×D₁₇	C₄×Q₈	1	2	yes	—
137	C₄×D₁₇	#131	1	2	yes	—
138	Dic₃₄	—	2	2	no	—
139	Dic₃₄	—	2	4	no	—
140	C₈×C₁₇	—	2	1	no	—
141	C₈×C₁₇	—	2	2	no	—
142	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
143	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
144	#3622	—	2	1	no	—
145	#3622	—	2	2	no	—
146	#3622	C₄×Q₈	1	2	yes	—
147	C₂×C₈×Q₈	—	17	1	no	—
148	#2183	—	17	1	no	—
149	#2183	—	17	1	no	—
150	#2183	—	17	2	no	—
151	#2183	—	17	2	no	—
152	#2183	—	17	1	no	—
153	#2183	D₁₇	1	1	yes	characteristic, direct product factor
154	C₂×C₄⋊C₁₆	—	17	1	no	—
155	C₂×C₄⋊C₁₆	—	17	2	no	—
156	C₂×#346	—	17	2	no	—
157	C₂×C₄×C₁₆	—	17	1	no	—
158	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
159	C₂×D₁₇	C₈×Q₈	1	2	yes	—
160	C₂×D₁₇	C₂×M₃₂	1	2	yes	—
161	C₂×D₁₇	#504	1	2	yes	—
162	C₄×C₁₇	—	2	2	no	—
163	C₄×C₁₇	—	2	2	no	—
164	C₄×C₁₇	—	2	1	no	—
165	C₄×C₁₇	C₂×M₃₂	1	2	yes	—
166	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
167	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
168	Dic₁₇	—	2	2	no	—
169	Dic₁₇	—	2	2	no	—
170	Dic₁₇	—	2	1	no	—
171	Dic₁₇	C₈×Q₈	1	2	yes	—
172	Dic₁₇	C₂×M₃₂	1	2	yes	—
173	C₂×C₄×Q₈	—	17	1	no	—
174	C₂²×C₁₆	—	17	1	no	—
175	C₂²×C₁₆	—	17	1	no	—
176	C₂²×C₁₆	—	17	2	no	—
177	C₈×Q₈	—	17	1	no	—
178	C₈×Q₈	—	17	2	no	—
179	C₈×Q₈	—	17	1	no	—
180	C₈×Q₈	—	17	2	no	—
181	C₈×Q₈	—	17	1	no	—
182	C₈×Q₈	C₂×D₁₇	1	1	yes	characteristic
183	C₂×C₄⋊C₈	—	17	1	no	—
184	C₂×C₄⋊C₈	—	17	2	no	—
185	C₂×C₄×C₈	—	17	1	no	—
186	C₂×C₄×C₈	—	17	2	no	—
187	C₄⋊C₁₆	—	17	2	no	—
188	C₄⋊C₁₆	—	17	2	no	—
189	C₄⋊C₁₆	—	17	2	no	—
190	C₄⋊C₁₆	—	17	1	no	—
191	C₄⋊C₁₆	—	17	2	no	—
192	C₄⋊C₁₆	C₂×D₁₇	1	1	yes	characteristic
193	C₄⋊C₁₆	C₂×D₁₇	1	2	yes	—
194	#346	—	17	2	no	—
195	#346	—	17	2	no	—
196	#346	—	17	2	no	—
197	#346	C₂×D₁₇	1	2	yes	—
198	C₄×C₁₆	—	17	1	no	—
199	C₄×C₁₆	—	17	1	no	—
200	C₄×C₁₆	—	17	1	no	—
201	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
202	C₂×C₁₇	C₂×C₈×Q₈	1	1	yes	characteristic
203	C₂×C₁₇	C₂²×M₃₂	1	1	yes	characteristic
204	C₂×C₁₇	C₂×#504	1	1	yes	characteristic
205	D₁₇	#2183	1	4	yes	direct product factor
206	C₂²×Q₈	—	17	1	no	—
207	C₂²×C₈	—	17	1	no	—
208	C₂²×C₈	—	17	1	no	—
209	C₂²×C₈	—	17	2	no	—
210	C₄×Q₈	—	17	1	no	—
211	C₄×Q₈	—	17	2	no	—
212	C₄×Q₈	—	17	1	no	—
213	C₄×Q₈	—	17	2	no	—
214	C₄×Q₈	—	17	1	no	—
215	C₄×Q₈	C₄×D₁₇	1	1	yes	characteristic
216	C₂×C₄⋊C₄	—	17	1	no	—
217	C₂×C₄⋊C₄	—	17	2	no	—
218	C₂×C₄²	—	17	1	no	—
219	C₂×C₄²	—	17	2	no	—
220	C₂×C₁₆	—	17	1	no	—
221	C₂×C₁₆	—	17	4	no	—
222	C₂×C₁₆	—	17	1	no	—
223	C₂×C₁₆	—	17	2	no	—
224	C₂×C₁₆	—	34	2	no	—
225	C₂×C₁₆	—	34	4	no	—
226	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
227	C₂×C₁₆	C₂²×D₁₇	1	2	yes	—
228	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
229	C₄⋊C₈	—	17	2	no	—
230	C₄⋊C₈	—	17	2	no	—
231	C₄⋊C₈	—	17	2	no	—
232	C₄⋊C₈	—	17	2	no	—
233	C₄⋊C₈	—	17	1	no	—
234	C₄⋊C₈	C₄×D₁₇	1	2	yes	—
235	C₄⋊C₈	C₄×D₁₇	1	1	yes	characteristic
236	C₄×C₈	—	17	1	no	—
237	C₄×C₈	—	17	2	no	—
238	C₄×C₈	—	17	1	no	—
239	C₄×C₈	—	17	2	no	—
240	C₄×C₈	—	17	1	no	—
241	C₄×C₈	—	17	2	no	—
242	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₁₇	1	2	yes	—
244	C₁₇	C₂×#2183	1	1	yes	characteristic, complemented, Sylow
245	C₂×Q₈	—	17	2	no	—
246	C₂×Q₈	—	17	1	no	—
247	C₂×Q₈	—	34	4	no	—
248	C₂×Q₈	C₈×D₁₇	1	1	yes	characteristic
249	C₂²×C₄	—	17	1	no	—
250	C₂²×C₄	—	17	1	no	—
251	C₂²×C₄	—	17	1	no	—
252	C₂²×C₄	—	17	2	no	—
253	C₂²×C₄	—	17	2	no	—
254	C₂×C₈	—	17	1	no	—
255	C₂×C₈	—	17	2	no	—
256	C₂×C₈	—	17	2	no	—
257	C₂×C₈	—	17	1	no	—
258	C₂×C₈	—	17	2	no	—
259	C₂×C₈	—	34	2	no	—
260	C₂×C₈	—	34	4	no	—
261	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
262	C₂×C₈	C₂×C₄×D₁₇	1	2	yes	—
263	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
264	C₄⋊C₄	—	17	2	no	—
265	C₄⋊C₄	—	17	2	no	—
266	C₄⋊C₄	—	17	2	no	—
267	C₄⋊C₄	—	17	2	no	—
268	C₄⋊C₄	—	17	1	no	—
269	C₄⋊C₄	C₈×D₁₇	1	2	yes	—
270	C₄⋊C₄	C₈×D₁₇	1	1	yes	characteristic
271	C₄²	—	17	1	no	—
272	C₄²	—	17	2	no	—
273	C₄²	—	17	1	no	—
274	C₄²	—	17	2	no	—
275	C₄²	—	17	1	no	—
276	C₄²	—	17	2	no	—
277	C₄²	C₂×C₄×D₁₇	1	2	yes	—
278	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
279	C₁₆	—	2	1	no	—
280	C₁₆	—	2	2	no	—
281	C₁₆	—	17	2	no	—
282	C₁₆	—	34	1	no	—
283	C₁₆	—	34	2	no	—
284	C₁₆	Q₈×D₁₇	1	2	yes	complemented
285	C₂³	—	17	1	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	34	4	no	—
288	Q₈	—	34	2	no	—
289	C₂×C₄	—	17	1	no	—
290	C₂×C₄	—	17	1	no	—
291	C₂×C₄	—	17	4	no	—
292	C₂×C₄	—	17	2	no	—
293	C₂×C₄	—	17	2	no	—
294	C₂×C₄	—	17	1	no	—
295	C₂×C₄	—	17	2	no	—
296	C₂×C₄	—	17	2	no	—
297	C₂×C₄	—	34	4	no	—
298	C₂×C₄	—	34	2	no	—
299	C₂×C₄	—	34	4	no	—
300	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
301	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
302	C₂×C₄	C₂×C₈×D₁₇	1	2	yes	—
303	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	2	no	—
307	C₈	—	17	2	no	—
308	C₈	—	34	1	no	—
309	C₈	—	34	2	no	—
310	C₈	C₂×Q₈×D₁₇	1	1	yes	characteristic
311	C₈	#41×D₁₇	1	1	yes	characteristic
312	C₂²	—	17	2	no	—
313	C₂²	—	17	2	no	—
314	C₂²	—	17	2	no	—
315	C₂²	C₂²×C₈×D₁₇	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	2	no	—
319	C₄	—	17	2	no	—
320	C₄	—	17	2	no	—
321	C₄	—	34	2	no	—
322	C₄	—	34	1	no	—
323	C₄	—	34	2	no	—
324	C₄	M₃₂×D₁₇	1	2	yes	—
325	C₄	C₄×Q₈×D₁₇	1	1	yes	characteristic
326	C₄	#131×D₁₇	1	1	yes	characteristic
327	C₂	—	17	4	no	—
328	C₂	C₈×Q₈×D₁₇	1	1	yes	characteristic
329	C₂	C₂×M₃₂×D₁₇	1	1	yes	characteristic
330	C₂	D₁₇×#504	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₈
Fitting	C₁₇×#2183	C₂
prime radical	C₂²×D₁₇	C₂²×C₈
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	#2183	D₁₇
hypocenter	C₁₇	C₂×#2183
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
2. #14530693	C₂	semidirect
3. #14530692	C₂	semidirect
4. #14530691	C₂	semidirect
5. #14530690	C₂	semidirect
6. #14530689	C₂	semidirect
11. C₁₇×#2183	C₂	semidirect
78. C₁₆×D₁₇	Q₈	semidirect
115. C₁₆×C₁₇	C₂×Q₈	semidirect
118. #63214	C₂×Q₈	semidirect
153. #2183	D₁₇	direct (N × H)
205. D₁₇	#2183	direct (N × H)
244. C₁₇	C₂×#2183	semidirect
284. C₁₆	Q₈×D₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₈×Q₈×D₁₇	2
#14530693	2
#14530692	2
#14530691	2
#14530690	2
#14530689	2
D₁₇×C₄⋊C₁₆	2
D₁₇×#346	2
C₄×C₁₆×D₁₇	2
C₁₇×#2183	2
C₂×#2183	17

Subgroups — D17×#2183 #17187807

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₇×#2183 #17187807