Subgroups — #16927367

Order 4352.

542 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11762874	C₂	1	1	yes	maximal, characteristic
2	#14528940	C₂	1	1	yes	maximal, characteristic, complemented
3	#14528785	C₂	1	1	yes	maximal, characteristic
4	#14528781	C₂	1	1	yes	maximal, characteristic
5	C₂×#11762559	C₂	1	1	yes	maximal, characteristic, complemented
6	#14526668	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#1296	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₁₇	—	2	1	no	—
9	C₂×C₄×D₆₈	C₂²	1	1	yes	characteristic
10	#11762874	—	2	2	no	—
11	#11762874	C₄	1	4	yes	complemented
12	C₂×#10587479	—	2	1	no	—
13	C₂×#10587479	—	2	1	no	—
14	C₂×C₄⋊#3622	C₂²	1	1	yes	characteristic
15	C₂×#10587492	—	2	1	no	—
16	C₂×#10587480	—	2	1	no	—
17	C₂×#10587491	—	2	1	no	—
18	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
19	#11762679	C₂²	1	1	yes	characteristic
20	#11762572	—	2	1	no	—
21	#11762571	—	2	1	no	—
22	#11762559	—	2	2	no	—
23	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₁₇×#384	C₂²	1	1	yes	characteristic
26	C₂²×D₆₈	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₁₇	—	2	1	no	—
28	C₂²×C₈×C₁₇	—	2	1	no	—
29	C₂²×C₈×C₁₇	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
33	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
34	C₂×#63227	—	2	1	no	—
35	C₂×C₄⋊Dic₁₇	—	2	1	no	—
36	C₂×C₄⋊Dic₁₇	—	2	1	no	—
37	C₂×C₄⋊Dic₁₇	—	2	1	no	—
38	C₂×C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
39	C₂²×#3622	—	2	1	no	—
40	C₂²×#3622	D₄	1	1	yes	characteristic
41	C₂²×#3622	D₄	1	1	yes	characteristic
42	C₂×C₈×D₁₇	—	2	4	no	—
43	C₂×C₈×D₁₇	—	4	2	no	—
44	C₂×C₈×D₁₇	—	4	2	no	—
45	C₄×D₆₈	—	2	1	no	—
46	C₄×D₆₈	—	2	1	no	—
47	C₄×D₆₈	C₂×C₄	1	2	yes	—
48	C₄×D₆₈	Q₈	1	2	yes	—
49	C₁₇×C₄⋊C₈	—	2	1	no	—
50	C₁₇×C₄⋊C₈	—	2	1	no	—
51	C₁₇×C₄⋊C₈	—	2	1	no	—
52	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
53	#10587492	—	4	2	no	—
54	#10587491	—	2	4	no	—
55	#10587480	—	4	2	no	—
56	#10587479	—	4	2	no	—
57	#10587479	—	4	2	no	—
58	C₄⋊#3622	—	2	1	no	—
59	C₄⋊#3622	—	2	1	no	—
60	C₄×#3622	C₂×C₄	1	2	yes	—
61	C₄×#3622	D₄	1	2	yes	—
62	C₂³×D₁₇	—	2	1	no	—
63	C₂²×C₄×C₁₇	—	4	1	no	—
64	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
67	C₂²×Dic₁₇	—	2	1	no	—
68	C₂²×Dic₁₇	—	4	1	no	—
69	C₂×D₆₈	—	2	1	no	—
70	C₂×D₆₈	—	2	1	no	—
71	C₂×D₆₈	—	2	1	no	—
72	C₂×D₆₈	C₄⋊C₄	1	2	yes	—
73	C₂×D₆₈	SD₁₆	1	2	yes	—
74	C₂×D₆₈	SD₁₆	1	2	yes	—
75	C₂×C₄×D₁₇	—	2	2	no	—
76	C₂×C₄×D₁₇	—	2	2	no	—
77	C₂×C₄×D₁₇	—	4	1	no	—
78	C₂×C₄×D₁₇	—	4	1	no	—
79	C₂×C₄×D₁₇	—	4	1	no	—
80	C₂×C₄×D₁₇	—	4	1	no	—
81	C₂×C₈×C₁₇	—	2	2	no	—
82	C₂×C₈×C₁₇	—	4	1	no	—
83	C₂×C₈×C₁₇	—	4	1	no	—
84	C₂×C₈×C₁₇	—	4	1	no	—
85	C₂×C₈×C₁₇	—	4	1	no	—
86	C₂×C₈×C₁₇	—	4	1	no	—
87	C₄⋊C₄×C₁₇	—	4	1	no	—
88	C₄⋊C₄×C₁₇	—	4	1	no	—
89	C₄⋊C₄×C₁₇	—	4	1	no	—
90	C₄⋊C₄×C₁₇	—	4	1	no	—
91	C₄⋊C₄×C₁₇	—	8	1	no	—
92	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
95	C₄²×C₁₇	#41	1	1	yes	characteristic
96	#63227	—	2	2	no	—
97	#63227	—	2	2	no	—
98	C₄⋊Dic₁₇	—	2	1	no	—
99	C₄⋊Dic₁₇	—	4	1	no	—
100	C₄⋊Dic₁₇	—	4	1	no	—
101	C₄⋊Dic₁₇	—	4	1	no	—
102	C₄⋊Dic₁₇	—	4	1	no	—
103	C₄⋊Dic₁₇	—	8	1	no	—
104	C₄⋊Dic₁₇	C₄⋊C₄	1	2	yes	—
105	C₂×#3622	—	2	1	no	—
106	C₂×#3622	—	2	1	no	—
107	C₂×#3622	—	4	1	no	—
108	C₂×#3622	—	4	1	no	—
109	C₂×#3622	—	4	1	no	—
110	C₂×#3622	C₂²⋊C₄	1	2	yes	—
111	C₂×#3622	C₂²⋊C₄	1	2	yes	—
112	C₂×#3622	D₈	1	2	yes	—
113	C₂×#3622	SD₁₆	1	2	yes	—
114	C₈×D₁₇	—	4	8	no	—
115	#8642	—	17	1	no	maximal, Sylow
116	C₂³×C₁₇	#121	1	1	yes	characteristic
117	C₂²×D₁₇	—	2	2	no	—
118	C₂²×D₁₇	—	2	2	no	—
119	C₂²×D₁₇	—	2	2	no	—
120	C₂²×D₁₇	—	4	1	no	—
121	C₂²×D₁₇	—	4	1	no	—
122	C₂²×D₁₇	—	4	1	no	—
123	C₂²×D₁₇	—	4	1	no	—
124	C₂×C₄×C₁₇	—	2	1	no	—
125	C₂×C₄×C₁₇	—	2	1	no	—
126	C₂×C₄×C₁₇	—	8	1	no	—
127	C₂×C₄×C₁₇	—	8	1	no	—
128	C₂×C₄×C₁₇	—	8	1	no	—
129	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
132	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
133	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
140	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
141	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
143	C₂×Dic₁₇	—	2	2	no	—
144	C₂×Dic₁₇	—	4	1	no	—
145	C₂×Dic₁₇	—	4	1	no	—
146	C₂×Dic₁₇	—	8	1	no	—
147	C₂×Dic₁₇	—	8	1	no	—
148	C₂×Dic₁₇	—	8	1	no	—
149	D₆₈	—	2	2	no	—
150	D₆₈	—	2	2	no	—
151	D₆₈	—	2	2	no	—
152	D₆₈	#106	1	4	yes	complemented
153	C₄×D₁₇	—	4	2	no	—
154	C₄×D₁₇	—	4	2	no	—
155	C₄×D₁₇	—	4	2	no	—
156	C₄×D₁₇	—	4	2	no	—
157	C₈×C₁₇	—	4	2	no	—
158	C₈×C₁₇	—	8	1	no	—
159	#3622	—	2	2	no	—
160	#3622	—	8	1	no	—
161	#3622	#102	1	4	yes	complemented
162	C₂×C₈×D₄	—	17	1	no	—
163	#1844	—	17	1	no	—
164	#1802	—	17	1	no	—
165	#1802	—	17	1	no	—
166	C₂×#334	—	17	1	no	—
167	#1296	—	17	1	no	—
168	#1296	D₁₇	1	1	yes	characteristic, complemented
169	C₂²×C₁₇	#380	1	1	yes	characteristic
170	C₂²×C₁₇	#459	1	1	yes	characteristic
171	C₂²×C₁₇	#461	1	1	yes	characteristic
172	C₂²×C₁₇	#465	1	1	yes	characteristic
173	C₂²×C₁₇	#465	1	1	yes	characteristic
174	C₂²×C₁₇	#471	1	1	yes	characteristic
175	C₂²×C₁₇	#471	1	1	yes	characteristic
176	C₂×D₁₇	—	2	4	no	—
177	C₂×D₁₇	—	4	2	no	—
178	C₂×D₁₇	—	4	2	no	—
179	C₂×D₁₇	—	4	2	no	—
180	C₂×D₁₇	—	4	2	no	—
181	C₂×D₁₇	—	4	2	no	—
182	C₂×D₁₇	—	4	2	no	—
183	C₄×C₁₇	—	2	1	no	—
184	C₄×C₁₇	—	2	1	no	—
185	C₄×C₁₇	—	16	1	no	—
186	C₄×C₁₇	C₂×#102	1	1	yes	characteristic
187	C₄×C₁₇	#417	1	1	yes	characteristic
188	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
189	C₄×C₁₇	#427	1	1	yes	characteristic
190	C₄×C₁₇	#465	1	1	yes	characteristic
191	C₄×C₁₇	#465	1	1	yes	characteristic
192	C₄×C₁₇	#474	1	1	yes	characteristic
193	C₄×C₁₇	#476	1	1	yes	characteristic
194	Dic₁₇	—	4	2	no	—
195	Dic₁₇	—	16	1	no	—
196	C₂³×C₈	—	34	1	no	—
197	C₂×C₄×D₄	—	17	1	no	—
198	C₈×D₄	—	17	4	no	—
199	C₈×D₄	—	34	2	no	—
200	C₂×#106	—	34	1	no	—
201	C₂×#106	—	34	1	no	—
202	C₂×C₄⋊C₈	—	17	1	no	—
203	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
204	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
205	C₂×#102	—	34	1	no	—
206	C₂×#102	—	34	1	no	—
207	C₂×C₂²⋊C₈	—	34	1	no	—
208	C₂×C₄×C₈	—	17	1	no	—
209	#384	—	17	1	no	—
210	#384	C₂×D₁₇	1	1	yes	characteristic
211	#340	—	34	1	no	—
212	#340	—	34	1	no	—
213	#334	—	34	2	no	—
214	C₂×C₁₇	#1665	1	1	yes	characteristic
215	C₂×C₁₇	#1665	1	1	yes	characteristic
216	C₂×C₁₇	#1671	1	1	yes	characteristic
217	C₂×C₁₇	#1671	1	1	yes	characteristic
218	C₂×C₁₇	#1802	1	1	yes	characteristic
219	C₂×C₁₇	#1807	1	1	yes	characteristic
220	C₂×C₁₇	#1815	1	1	yes	characteristic
221	D₁₇	—	4	4	no	—
222	C₂²×D₄	—	17	1	no	—
223	C₂³×C₄	—	34	1	no	—
224	C₂²×C₈	—	2	1	no	—
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	17	1	no	—
227	C₂²×C₈	—	17	1	no	—
228	C₂²×C₈	—	34	4	no	—
229	C₂²×C₈	—	34	1	no	—
230	C₂²×C₈	—	68	2	no	—
231	C₂²×C₈	—	68	2	no	—
232	C₄×D₄	—	17	2	no	—
233	C₄×D₄	—	17	2	no	—
234	C₄×D₄	—	34	1	no	—
235	C₄×D₄	—	34	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	17	1	no	—
240	C₂×C₄⋊C₄	—	34	1	no	—
241	C₂×C₄⋊C₄	—	34	1	no	—
242	C₂×C₄⋊C₄	—	34	1	no	—
243	C₂×C₂²⋊C₄	—	34	1	no	—
244	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
245	#106	—	68	2	no	—
246	#106	—	68	2	no	—
247	C₄⋊C₈	—	2	1	no	—
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	—	34	1	no	—
251	C₄⋊C₈	—	34	1	no	—
252	C₄⋊C₈	C₄×D₁₇	1	2	yes	—
253	#102	—	68	2	no	—
254	#102	—	68	2	no	—
255	C₂²⋊C₈	—	34	4	no	—
256	C₄×C₈	—	17	2	no	—
257	C₄×C₈	—	17	2	no	—
258	C₁₇	#8642	1	1	yes	characteristic, complemented, Sylow
259	C₂⁴	—	34	1	no	—
260	C₂×D₄	—	17	2	no	—
261	C₂×D₄	—	17	2	no	—
262	C₂×D₄	—	17	2	no	—
263	C₂×D₄	—	34	1	no	—
264	C₂×D₄	—	34	1	no	—
265	C₂×D₄	—	34	1	no	—
266	C₂²×C₄	—	4	1	no	—
267	C₂²×C₄	—	34	1	no	—
268	C₂²×C₄	—	34	2	no	—
269	C₂²×C₄	—	34	2	no	—
270	C₂²×C₄	—	68	1	no	—
271	C₂²×C₄	—	68	1	no	—
272	C₂²×C₄	—	68	1	no	—
273	C₂²×C₄	—	68	1	no	—
274	C₂²×C₄	—	68	1	no	—
275	C₂²×C₄	#63252	1	1	yes	characteristic
276	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
277	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	17	2	no	—
285	C₂×C₈	—	17	2	no	—
286	C₂×C₈	—	17	2	no	—
287	C₂×C₈	—	17	2	no	—
288	C₂×C₈	—	34	1	no	—
289	C₂×C₈	—	34	1	no	—
290	C₂×C₈	—	68	8	no	—
291	C₂×C₈	—	68	1	no	—
292	C₂×C₈	—	68	1	no	—
293	C₂×C₈	—	68	1	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₄⋊C₄	—	8	1	no	—
299	C₄⋊C₄	—	17	2	no	—
300	C₄⋊C₄	—	34	1	no	—
301	C₄⋊C₄	—	68	1	no	—
302	C₄⋊C₄	—	68	1	no	—
303	C₄⋊C₄	—	68	1	no	—
304	C₄⋊C₄	—	68	1	no	—
305	C₄⋊C₄	—	136	1	no	—
306	C₂²⋊C₄	—	34	2	no	—
307	C₂²⋊C₄	—	34	2	no	—
308	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
309	C₄²	#63252	1	1	yes	characteristic
310	C₄²	D₄×D₁₇	1	1	yes	characteristic
311	C₄²	Q₈×D₁₇	1	1	yes	characteristic
312	C₂³	—	34	2	no	—
313	C₂³	—	34	2	no	—
314	C₂³	—	34	2	no	—
315	C₂³	—	68	1	no	—
316	C₂³	—	68	1	no	—
317	C₂³	—	68	1	no	—
318	C₂³	—	68	1	no	—
319	C₂³	#10587570	1	1	yes	characteristic
320	D₄	—	17	4	no	—
321	D₄	—	34	2	no	—
322	D₄	—	34	2	no	—
323	D₄	—	34	2	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	8	1	no	—
327	C₂×C₄	—	8	1	no	—
328	C₂×C₄	—	8	1	no	—
329	C₂×C₄	—	34	2	no	—
330	C₂×C₄	—	68	1	no	—
331	C₂×C₄	—	68	1	no	—
332	C₂×C₄	—	68	2	no	—
333	C₂×C₄	—	68	2	no	—
334	C₂×C₄	—	68	2	no	—
335	C₂×C₄	—	68	2	no	—
336	C₂×C₄	—	136	1	no	—
337	C₂×C₄	—	136	1	no	—
338	C₂×C₄	—	136	1	no	—
339	C₂×C₄	#10587558	1	1	yes	characteristic
340	C₂×C₄	C₂²⋊C₄×D₁₇	1	1	yes	characteristic
341	C₂×C₄	#10587570	1	1	yes	characteristic
342	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
343	C₂×C₄	#10587582	1	1	yes	characteristic
344	C₂×C₄	#10587582	1	1	yes	characteristic
345	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
346	C₂×C₄	#10587598	1	1	yes	characteristic
347	C₂×C₄	SD₁₆×D₁₇	1	1	yes	characteristic
348	C₂×C₄	SD₁₆×D₁₇	1	1	yes	characteristic
349	C₂×C₄	SD₁₆×D₁₇	1	1	yes	characteristic
350	C₂×C₄	#10587600	1	1	yes	characteristic
351	C₂×C₄	#10587604	1	1	yes	characteristic
352	C₂×C₄	#10587605	1	1	yes	characteristic
353	C₈	—	4	2	no	—
354	C₈	—	8	1	no	—
355	C₈	—	17	4	no	—
356	C₈	—	34	2	no	—
357	C₈	—	136	1	no	—
358	C₂²	—	34	4	no	—
359	C₂²	—	68	2	no	—
360	C₂²	—	68	2	no	—
361	C₂²	—	68	2	no	—
362	C₂²	—	68	2	no	—
363	C₂²	—	68	2	no	—
364	C₂²	—	68	2	no	—
365	C₂²	#11762668	1	1	yes	characteristic
366	C₂²	#11763003	1	1	yes	characteristic
367	C₂²	#11763013	1	1	yes	characteristic
368	C₂²	#11763037	1	1	yes	characteristic
369	C₂²	#11763041	1	1	yes	characteristic
370	C₂²	#11763071	1	1	yes	characteristic
371	C₂²	#11763075	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	16	1	no	—
375	C₄	—	272	1	no	—
376	C₄	—	68	2	no	—
377	C₄	#102×D₁₇	1	1	yes	characteristic
378	C₄	#11762809	1	1	yes	characteristic
379	C₄	#106×D₁₇	1	1	yes	characteristic
380	C₄	#11762846	1	1	yes	characteristic
381	C₄	#11763037	1	1	yes	characteristic
382	C₄	#11763041	1	1	yes	characteristic
383	C₄	#11763088	1	1	yes	characteristic
384	C₄	#11763098	1	1	yes	characteristic
385	C₂	—	68	4	no	—
386	C₂	#14528124	1	1	yes	characteristic
387	C₂	#14528128	1	1	yes	characteristic
388	C₂	#14528158	1	1	yes	characteristic
389	C₂	#14528162	1	1	yes	characteristic
390	C₂	#14528783	1	1	yes	characteristic
391	C₂	#14528808	1	1	yes	characteristic
392	C₂	#14528842	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587570
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1296	C₂
socle	C₂³×C₁₇	#121
prime radical	C₂²×D₆₈	D₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1296	D₁₇
hypocenter	C₁₇	#8642
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

Complements

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

N	H ≅ G/N	type
2. #14528940	C₂	semidirect
5. C₂×#11762559	C₂	semidirect
6. #14526668	C₂	semidirect
7. C₁₇×#1296	C₂	semidirect
11. #11762874	C₄	semidirect
152. D₆₈	#106	semidirect
161. #3622	#102	semidirect
168. #1296	D₁₇	semidirect
258. C₁₇	#8642	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11762874	2
#14528940	2
#14528785	2
#14528781	2
C₂×#11762559	2
#14526668	2
C₁₇×#1296	2
#8642	17