Subgroups — C₁₇×#3376 #14526382

Order 2176.

346 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₇×#554	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₇×#491	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₇×#489	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#488	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₇×#479	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#478	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₇×#475	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₇×#463	C₂	1	1	yes	maximal, characteristic
10	C₁₇×#463	C₂	1	1	yes	maximal, characteristic
11	C₁₇×#461	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
13	C₁₇×#442	C₂	1	1	yes	maximal, characteristic
14	C₁₇×#440	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₁₇	C₂²	1	1	yes	characteristic
19	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₁₇×#126	—	2	1	no	—
22	C₁₇×#126	—	2	1	no	—
23	C₁₇×#125	C₂²	1	1	yes	characteristic
24	C₁₇×#125	C₂²	1	1	yes	characteristic
25	C₁₇×#124	C₂²	1	1	yes	characteristic
26	C₁₇×#124	C₂²	1	1	yes	characteristic
27	C₁₇×C₂²⋊Q₈	—	2	1	no	—
28	C₁₇×C₂²⋊Q₈	—	2	1	no	—
29	C₁₇×C₂²⋊Q₈	—	2	1	no	—
30	C₁₇×C₂²⋊Q₈	—	2	1	no	—
31	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
32	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
33	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
34	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
35	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
37	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
38	C₁₇×#117	—	2	1	no	—
39	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
40	C₁₇×#107	C₂²	1	1	yes	characteristic
41	C₁₇×#106	C₂²	1	1	yes	characteristic
42	C₁₇×#106	C₂²	1	1	yes	characteristic
43	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
45	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₁₇×#103	C₂²	1	1	yes	characteristic
47	C₁₇×#103	C₂²	1	1	yes	characteristic
48	C₁₇×#103	C₂²	1	1	yes	characteristic
49	C₁₇×#103	C₂²	1	1	yes	characteristic
50	C₁₇×#103	C₂²	1	1	yes	characteristic
51	C₁₇×#103	C₂²	1	1	yes	characteristic
52	C₁₇×#102	C₂²	1	1	yes	characteristic
53	C₁₇×#102	C₂²	1	1	yes	characteristic
54	C₁₇×#102	C₂²	1	1	yes	characteristic
55	C₁₇×#102	C₂²	1	1	yes	characteristic
56	C₁₇×#97	C₂²	1	1	yes	characteristic
57	C₁₇×#97	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	—	2	1	no	—
60	C₂×Q₈×C₁₇	D₄	1	1	yes	characteristic
61	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
62	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
64	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	—	2	1	no	—
66	C₂²×C₄×C₁₇	—	2	1	no	—
67	C₂²×C₄×C₁₇	—	2	1	no	—
68	C₂²×C₄×C₁₇	—	2	1	no	—
69	SD₁₆×C₁₇	—	2	2	no	—
70	SD₁₆×C₁₇	—	2	2	no	—
71	D₈×C₁₇	—	2	2	no	—
72	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₇	—	2	1	no	—
77	C₄⋊C₄×C₁₇	—	2	1	no	—
78	C₄⋊C₄×C₁₇	—	2	1	no	—
79	C₄⋊C₄×C₁₇	—	2	1	no	—
80	C₄⋊C₄×C₁₇	—	2	1	no	—
81	C₄⋊C₄×C₁₇	—	2	1	no	—
82	C₄⋊C₄×C₁₇	—	2	1	no	—
83	C₄⋊C₄×C₁₇	—	2	1	no	—
84	C₄⋊C₄×C₁₇	—	2	1	no	—
85	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
86	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
87	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
88	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
89	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
90	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
91	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
92	C₂²⋊C₄×C₁₇	—	2	1	no	—
93	C₂²⋊C₄×C₁₇	—	2	1	no	—
94	C₂²⋊C₄×C₁₇	—	2	1	no	—
95	C₂²⋊C₄×C₁₇	—	2	1	no	—
96	C₂²⋊C₄×C₁₇	—	2	1	no	—
97	C₂²⋊C₄×C₁₇	—	2	1	no	—
98	C₂²⋊C₄×C₁₇	—	2	1	no	—
99	C₂²⋊C₄×C₁₇	—	2	1	no	—
100	C₄²×C₁₇	—	2	1	no	—
101	C₄²×C₁₇	—	2	1	no	—
102	C₄²×C₁₇	C₂³	1	1	yes	characteristic
103	C₄²×C₁₇	C₂³	1	1	yes	characteristic
104	C₄²×C₁₇	C₂³	1	1	yes	characteristic
105	C₂³×C₁₇	—	2	1	no	—
106	C₂³×C₁₇	—	2	1	no	—
107	Q₈×C₁₇	—	2	1	no	—
108	Q₈×C₁₇	—	2	1	no	—
109	Q₈×C₁₇	—	2	1	no	—
110	Q₈×C₁₇	—	2	1	no	—
111	Q₈×C₁₇	—	4	1	no	—
112	Q₈×C₁₇	—	8	1	no	—
113	D₄×C₁₇	—	2	1	no	—
114	D₄×C₁₇	—	2	1	no	—
115	D₄×C₁₇	—	2	1	no	—
116	D₄×C₁₇	#41	1	2	yes	—
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	—	2	1	no	—
123	C₂×C₄×C₁₇	—	4	2	no	—
124	C₂×C₄×C₁₇	—	4	2	no	—
125	C₂×C₄×C₁₇	—	8	1	no	—
126	C₂×C₄×C₁₇	—	8	1	no	—
127	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
134	C₈×C₁₇	—	2	1	no	—
135	C₈×C₁₇	—	2	1	no	—
136	C₈×C₁₇	—	2	1	no	—
137	C₈×C₁₇	—	2	1	no	—
138	#3376	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂²×C₁₇	—	2	2	no	—
140	C₂²×C₁₇	—	4	1	no	—
141	C₂²×C₁₇	—	4	1	no	—
142	C₂²×C₁₇	—	4	1	no	—
143	C₂²×C₁₇	—	4	1	no	—
144	C₂²×C₁₇	—	4	1	no	—
145	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
146	C₄×C₁₇	—	2	1	no	—
147	C₄×C₁₇	—	2	1	no	—
148	C₄×C₁₇	—	2	1	no	—
149	C₄×C₁₇	—	2	1	no	—
150	C₄×C₁₇	—	2	1	no	—
151	C₄×C₁₇	—	2	1	no	—
152	C₄×C₁₇	—	4	1	no	—
153	C₄×C₁₇	—	4	1	no	—
154	C₄×C₁₇	—	4	1	no	—
155	C₄×C₁₇	—	4	1	no	—
156	C₄×C₁₇	—	4	1	no	—
157	C₄×C₁₇	—	4	1	no	—
158	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
159	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
160	#556	C₂×C₁₇	1	1	yes	characteristic, complemented
161	#554	C₂×C₁₇	1	1	yes	characteristic, complemented
162	#491	C₂×C₁₇	1	1	yes	characteristic, complemented
163	#489	C₂×C₁₇	1	1	yes	characteristic, complemented
164	#488	C₂×C₁₇	1	1	yes	characteristic, complemented
165	#479	C₂×C₁₇	1	1	yes	characteristic, complemented
166	#478	C₂×C₁₇	1	1	yes	characteristic, complemented
167	#475	C₂×C₁₇	1	1	yes	characteristic, complemented
168	#463	C₂×C₁₇	1	1	yes	characteristic
169	#463	C₂×C₁₇	1	1	yes	characteristic
170	#461	C₂×C₁₇	1	1	yes	characteristic, complemented
171	#446	C₂×C₁₇	1	1	yes	characteristic, complemented
172	#442	C₂×C₁₇	1	1	yes	characteristic
173	#440	C₂×C₁₇	1	1	yes	characteristic, complemented
174	C₄×SD₁₆	C₂×C₁₇	1	1	yes	characteristic, complemented
175	C₂×C₁₇	—	4	2	no	—
176	C₂×C₁₇	—	8	1	no	—
177	C₂×C₁₇	#548	1	1	yes	characteristic
178	C₂×C₁₇	#575	1	1	yes	characteristic
179	C₂×C₁₇	#577	1	1	yes	characteristic
180	C₂×SD₁₆	C₂²×C₁₇	1	1	yes	characteristic
181	C₂×SD₁₆	C₂²×C₁₇	1	1	yes	characteristic
182	C₂×D₈	C₂²×C₁₇	1	1	yes	characteristic
183	C₄⋊Q₈	C₂²×C₁₇	1	1	yes	characteristic
184	C₄⋊Q₈	C₂²×C₁₇	1	1	yes	characteristic
185	#126	—	2	1	no	—
186	#126	—	2	1	no	—
187	#125	C₂²×C₁₇	1	1	yes	characteristic
188	#125	C₂²×C₁₇	1	1	yes	characteristic
189	#124	C₂²×C₁₇	1	1	yes	characteristic
190	#124	C₂²×C₁₇	1	1	yes	characteristic
191	C₂²⋊Q₈	—	2	1	no	—
192	C₂²⋊Q₈	—	2	1	no	—
193	C₂²⋊Q₈	—	2	1	no	—
194	C₂²⋊Q₈	—	2	1	no	—
195	C₄×Q₈	C₂²×C₁₇	1	1	yes	characteristic
196	C₄×Q₈	C₂²×C₁₇	1	1	yes	characteristic
197	C₄×Q₈	C₂²×C₁₇	1	1	yes	characteristic
198	C₄×D₄	C₂²×C₁₇	1	1	yes	characteristic
199	C₄×D₄	C₂²×C₁₇	1	1	yes	characteristic
200	C₄×D₄	C₂²×C₁₇	1	1	yes	characteristic
201	C₄×D₄	C₂²×C₁₇	1	1	yes	characteristic
202	#117	—	2	1	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	#107	C₂²×C₁₇	1	1	yes	characteristic
205	#106	C₂²×C₁₇	1	1	yes	characteristic
206	#106	C₂²×C₁₇	1	1	yes	characteristic
207	C₄⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
208	C₄⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
209	C₄⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
210	#103	C₂²×C₁₇	1	1	yes	characteristic
211	#103	C₂²×C₁₇	1	1	yes	characteristic
212	#103	C₂²×C₁₇	1	1	yes	characteristic
213	#103	C₂²×C₁₇	1	1	yes	characteristic
214	#103	C₂²×C₁₇	1	1	yes	characteristic
215	#103	C₂²×C₁₇	1	1	yes	characteristic
216	#102	C₂²×C₁₇	1	1	yes	characteristic
217	#102	C₂²×C₁₇	1	1	yes	characteristic
218	#102	C₂²×C₁₇	1	1	yes	characteristic
219	#102	C₂²×C₁₇	1	1	yes	characteristic
220	#97	C₂²×C₁₇	1	1	yes	characteristic
221	#97	C₂²×C₁₇	1	1	yes	characteristic
222	C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
223	C₁₇	#3376	1	1	yes	characteristic, direct product factor, Sylow
224	C₂×Q₈	—	2	1	no	—
225	C₂×Q₈	D₄×C₁₇	1	1	yes	characteristic
226	C₂×Q₈	C₂³×C₁₇	1	1	yes	characteristic
227	C₂×Q₈	C₂³×C₁₇	1	1	yes	characteristic
228	C₂×D₄	C₂³×C₁₇	1	1	yes	characteristic
229	C₂×D₄	C₂³×C₁₇	1	1	yes	characteristic
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	SD₁₆	—	2	2	no	—
235	SD₁₆	—	2	2	no	—
236	D₈	—	2	2	no	—
237	C₂×C₈	C₂³×C₁₇	1	1	yes	characteristic
238	C₂×C₈	C₂³×C₁₇	1	1	yes	characteristic
239	C₂×C₈	C₂³×C₁₇	1	1	yes	characteristic
240	C₂×C₈	C₂³×C₁₇	1	1	yes	characteristic
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	D₄×C₁₇	1	1	yes	characteristic
251	C₄⋊C₄	D₄×C₁₇	1	1	yes	characteristic
252	C₄⋊C₄	D₄×C₁₇	1	1	yes	characteristic
253	C₄⋊C₄	C₂³×C₁₇	1	1	yes	characteristic
254	C₄⋊C₄	C₂³×C₁₇	1	1	yes	characteristic
255	C₄⋊C₄	C₂³×C₁₇	1	1	yes	characteristic
256	C₄⋊C₄	C₂³×C₁₇	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	C₂³×C₁₇	1	1	yes	characteristic
268	C₄²	C₂³×C₁₇	1	1	yes	characteristic
269	C₄²	C₂³×C₁₇	1	1	yes	characteristic
270	C₂³	—	2	1	no	—
271	C₂³	—	2	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	2	1	no	—
276	Q₈	—	4	1	no	—
277	Q₈	—	8	1	no	—
278	D₄	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	2	1	no	—
281	D₄	#41×C₁₇	1	2	yes	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
293	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
294	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
295	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
296	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
297	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
298	C₂×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	C₂²×D₄×C₁₇	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	C₂×#41×C₁₇	1	1	yes	characteristic
323	C₄	C₁₇×2₋¹⁺⁴	1	1	yes	characteristic
324	C₂	—	4	2	no	—
325	C₂	—	8	1	no	—
326	C₂	C₁₇×#548	1	1	yes	characteristic
327	C₂	C₁₇×#575	1	1	yes	characteristic
328	C₂	C₁₇×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₇
prime radical	C₂×D₈×C₁₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ C₄×Q₈×C₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₇×#556	C₂	semidirect
2. C₁₇×#554	C₂	semidirect
3. C₁₇×#491	C₂	semidirect
4. C₁₇×#489	C₂	semidirect
5. C₁₇×#488	C₂	semidirect
6. C₁₇×#479	C₂	semidirect
7. C₁₇×#478	C₂	semidirect
8. C₁₇×#475	C₂	semidirect
11. C₁₇×#461	C₂	semidirect
12. C₁₇×#446	C₂	semidirect
14. C₁₇×#440	C₂	semidirect
15. C₄×SD₁₆×C₁₇	C₂	semidirect
138. #3376	C₁₇	direct (N × H)
160. #556	C₂×C₁₇	semidirect
161. #554	C₂×C₁₇	semidirect
162. #491	C₂×C₁₇	semidirect
163. #489	C₂×C₁₇	semidirect
164. #488	C₂×C₁₇	semidirect
165. #479	C₂×C₁₇	semidirect
166. #478	C₂×C₁₇	semidirect
167. #475	C₂×C₁₇	semidirect
170. #461	C₂×C₁₇	semidirect
171. #446	C₂×C₁₇	semidirect
173. #440	C₂×C₁₇	semidirect
174. C₄×SD₁₆	C₂×C₁₇	semidirect
223. C₁₇	#3376	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₇×#556	2
C₁₇×#554	2
C₁₇×#491	2
C₁₇×#489	2
C₁₇×#488	2
C₁₇×#479	2
C₁₇×#478	2
C₁₇×#475	2
C₁₇×#463	2
C₁₇×#461	2
C₁₇×#446	2
C₁₇×#442	2
C₁₇×#440	2
C₄×SD₁₆×C₁₇	2
#3376	17

Subgroups — C17×#3376 #14526382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#3376 #14526382