Subgroups — #17005178

Order 4352.

584 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539343	C₂	1	1	yes	maximal, characteristic
2	#14537878	C₂	1	1	yes	maximal, characteristic, complemented
3	#14537859	C₂	1	1	yes	maximal, characteristic
4	#14527664	C₂	1	2	yes	maximal, complemented
5	#14527643	C₂	1	2	yes	maximal, complemented
6	#14527629	C₂	1	2	yes	maximal, complemented
7	#14527622	C₂	1	2	yes	maximal, complemented
8	#14527547	C₂	1	1	yes	maximal, characteristic, complemented
9	#14527258	C₂	1	1	yes	maximal, characteristic, complemented
10	#14527254	C₂	1	1	yes	maximal, characteristic
11	C₁₇×#1452	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763366	C₂²	1	1	yes	characteristic
13	#11763359	C₂²	1	1	yes	characteristic
14	#11763207	C₄	1	1	yes	characteristic
15	#11763182	C₄	1	1	yes	characteristic
16	#11763160	C₄	1	1	yes	characteristic
17	#11763156	C₄	1	1	yes	characteristic
18	#11763038	C₄	1	2	yes	—
19	#11763038	C₄	1	2	yes	—
20	#11762939	C₂²	1	2	yes	—
21	#11762938	—	2	2	no	—
22	C₈×Dic₃₄	C₂²	1	2	yes	—
23	C₄×#63219	C₂²	1	2	yes	—
24	C₄×#63219	C₂²	1	2	yes	—
25	#11762888	C₂²	1	2	yes	—
26	#11762879	—	2	2	no	—
27	C₈×D₆₈	C₂²	1	2	yes	—
28	#11762867	—	2	2	no	—
29	#11762866	C₂²	1	1	yes	characteristic
30	#11762866	C₂²	1	1	yes	characteristic
31	#11762774	—	2	2	no	—
32	#11762562	C₂²	1	2	yes	—
33	#11762560	C₂²	1	2	yes	—
34	#11762541	C₂²	1	2	yes	—
35	#11762541	C₂²	1	2	yes	—
36	#11762538	C₂²	1	2	yes	—
37	#11762538	C₂²	1	2	yes	—
38	C₁₇×#431	C₂²	1	1	yes	characteristic
39	C₁₇×#431	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
41	C₁₇×#322	C₂²	1	1	yes	characteristic
42	C₁₇×#322	C₂²	1	1	yes	characteristic
43	C₁₇×#322	C₂²	1	1	yes	characteristic
44	C₁₇×#322	C₂²	1	1	yes	characteristic
45	C₂×#63252	—	2	1	no	—
46	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
49	#10587616	—	2	1	no	—
50	#10587616	C₂×C₄	1	2	yes	—
51	#10587612	—	2	1	no	—
52	#10587610	—	2	1	no	—
53	#10587610	C₂×C₄	1	2	yes	—
54	C₂×#63219	C₂×C₄	1	2	yes	—
55	C₂×#63219	C₂×C₄	1	2	yes	—
56	C₂×#63218	—	2	2	no	—
57	C₂×C₈×D₁₇	—	2	2	no	—
58	#10587561	—	2	1	no	—
59	#10587560	C₂×C₄	1	1	yes	characteristic
60	C₄×D₆₈	—	2	1	no	—
61	C₄×D₆₈	C₂³	1	2	yes	—
62	#10587556	—	2	1	no	—
63	#10587555	C₂×C₄	1	1	yes	characteristic
64	#10587555	C₂×C₄	1	1	yes	characteristic
65	#10587555	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₃₄	—	2	1	no	—
67	C₄×Dic₃₄	C₂³	1	2	yes	—
68	C₁₇×C₄⋊C₈	—	2	1	no	—
69	C₁₇×C₄⋊C₈	D₄	1	2	yes	—
70	C₁₇×C₂²⋊C₈	—	2	1	no	—
71	C₁₇×C₂²⋊C₈	D₄	1	2	yes	—
72	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
76	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
80	C₂²⋊#3622	—	2	1	no	—
81	C₂²⋊#3622	—	2	1	no	—
82	#10587492	C₂×C₄	1	2	yes	—
83	#10587492	C₂×C₄	1	2	yes	—
84	#10587491	—	2	2	no	—
85	#10587491	—	2	2	no	—
86	#10587488	C₂×C₄	1	2	yes	—
87	#10587488	C₂×C₄	1	2	yes	—
88	#10587487	C₂×C₄	1	2	yes	—
89	#10587487	C₂×C₄	1	2	yes	—
90	#10587486	—	2	2	no	—
91	#10587485	—	2	2	no	—
92	#10587485	—	2	2	no	—
93	C₈×Dic₁₇	—	2	2	no	—
94	C₄⋊#3622	C₂³	1	2	yes	—
95	C₄⋊#3622	C₂³	1	2	yes	—
96	#10587474	—	2	1	no	—
97	#10587474	—	2	1	no	—
98	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
101	C₂×C₂²⋊D₁₇	—	2	2	no	—
102	#63252	—	8	2	no	—
103	C₂×D₆₈	—	2	1	no	—
104	C₂×D₆₈	—	2	1	no	—
105	C₂×D₆₈	C₂²×C₄	1	2	yes	—
106	C₂×C₄×D₁₇	—	2	2	no	—
107	C₂×Dic₃₄	—	2	1	no	—
108	C₂×Dic₃₄	—	2	1	no	—
109	C₂×Dic₃₄	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₁₇	—	2	1	no	—
111	C₂×C₈×C₁₇	—	2	1	no	—
112	C₂×C₈×C₁₇	—	4	1	no	—
113	C₂×C₈×C₁₇	—	4	1	no	—
114	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
119	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
120	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₁₇	#41	1	1	yes	characteristic
123	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
124	C₂²⋊Dic₁₇	—	2	2	no	—
125	#63227	—	2	2	no	—
126	#63227	—	2	2	no	—
127	C₄⋊Dic₁₇	—	2	1	no	—
128	C₄⋊Dic₁₇	—	2	1	no	—
129	C₄⋊Dic₁₇	—	2	1	no	—
130	C₄⋊Dic₁₇	—	2	1	no	—
131	C₄⋊Dic₁₇	C₂²×C₄	1	2	yes	—
132	C₄⋊Dic₁₇	C₂²×C₄	1	2	yes	—
133	#63225	—	2	2	no	—
134	#63225	—	2	2	no	—
135	C₄×Dic₁₇	—	2	2	no	—
136	C₂×#3622	—	2	2	no	—
137	C₂×#3622	—	2	2	no	—
138	#63219	—	4	2	no	—
139	#63219	—	4	2	no	—
140	#63218	—	8	2	no	—
141	C₈×D₁₇	—	8	2	no	—
142	#17990	—	17	1	no	maximal, Sylow
143	C₂³×C₁₇	C₄×D₄	1	1	yes	characteristic
144	C₂²×D₁₇	—	2	2	no	—
145	C₂×C₄×C₁₇	—	2	1	no	—
146	C₂×C₄×C₁₇	—	2	1	no	—
147	C₂×C₄×C₁₇	—	2	2	no	—
148	C₂×C₄×C₁₇	—	4	1	no	—
149	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
155	C₂²⋊D₁₇	—	8	2	no	—
156	C₂×Dic₁₇	—	2	2	no	—
157	C₂×Dic₁₇	—	2	2	no	—
158	C₂×Dic₁₇	—	2	2	no	—
159	D₆₈	—	2	2	no	—
160	D₆₈	—	2	2	no	—
161	D₆₈	—	8	1	no	—
162	D₆₈	—	8	1	no	—
163	C₄×D₁₇	—	4	2	no	—
164	C₄×D₁₇	—	4	2	no	—
165	Dic₃₄	—	2	2	no	—
166	Dic₃₄	—	2	2	no	—
167	Dic₃₄	—	8	1	no	—
168	Dic₃₄	—	8	1	no	—
169	C₈×C₁₇	—	2	1	no	—
170	C₈×C₁₇	—	2	1	no	—
171	C₈×C₁₇	—	2	1	no	—
172	C₈×C₁₇	—	2	1	no	—
173	C₈×C₁₇	—	2	2	no	—
174	C₈×C₁₇	—	2	2	no	—
175	C₈×C₁₇	—	4	1	no	—
176	C₈×C₁₇	—	4	1	no	—
177	#3622	—	4	2	no	—
178	#3622	—	4	2	no	—
179	#3167	—	17	1	no	—
180	#2985	—	17	1	no	—
181	#2981	—	17	1	no	—
182	#1588	—	17	2	no	—
183	#1585	—	17	2	no	—
184	#1583	—	17	2	no	—
185	#1582	—	17	2	no	—
186	#1568	—	17	1	no	—
187	#1497	—	17	1	no	—
188	#1496	—	17	1	no	—
189	#1452	D₁₇	1	1	yes	characteristic, complemented
190	C₂²×C₁₇	—	2	1	no	—
191	C₂²×C₁₇	—	2	1	no	—
192	C₂²×C₁₇	—	2	1	no	—
193	C₂²×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
194	C₂×D₁₇	—	4	2	no	—
195	C₂×D₁₇	—	4	2	no	—
196	C₂×D₁₇	—	4	2	no	—
197	C₄×C₁₇	—	2	1	no	—
198	C₄×C₁₇	—	2	1	no	—
199	C₄×C₁₇	—	2	1	no	—
200	C₄×C₁₇	—	2	1	no	—
201	C₄×C₁₇	#568	1	1	yes	characteristic
202	C₄×C₁₇	#568	1	1	yes	characteristic
203	C₄×C₁₇	#577	1	2	yes	—
204	Dic₁₇	—	4	2	no	—
205	Dic₁₇	—	4	2	no	—
206	Dic₁₇	—	4	2	no	—
207	#533	—	17	1	no	—
208	#532	—	17	1	no	—
209	#498	—	17	1	no	—
210	#492	—	17	1	no	—
211	#487	—	17	1	no	—
212	#486	—	17	1	no	—
213	#465	—	17	2	no	—
214	#465	—	17	2	no	—
215	#446	—	17	2	no	—
216	#446	—	34	2	no	—
217	C₈×Q₈	—	17	2	no	—
218	C₄×SD₁₆	—	17	2	no	—
219	C₄×SD₁₆	—	17	2	no	—
220	#436	—	17	2	no	—
221	#435	—	34	2	no	—
222	C₈×D₄	—	17	2	no	—
223	#433	—	34	2	no	—
224	#432	—	17	1	no	—
225	#432	—	17	1	no	—
226	#431	C₂×D₁₇	1	1	yes	characteristic
227	#431	C₂×D₁₇	1	1	yes	characteristic
228	#408	—	34	2	no	—
229	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
230	#335	—	17	2	no	—
231	#334	—	17	2	no	—
232	Q₈⋊C₈	—	17	2	no	—
233	Q₈⋊C₈	—	17	2	no	—
234	D₄⋊C₈	—	17	2	no	—
235	D₄⋊C₈	—	17	2	no	—
236	#322	C₂×D₁₇	1	1	yes	characteristic
237	#322	C₂×D₁₇	1	1	yes	characteristic
238	#322	C₂×D₁₇	1	1	yes	characteristic
239	#322	C₂×D₁₇	1	1	yes	characteristic
240	C₂×C₁₇	—	4	1	no	—
241	C₂×C₁₇	#2932	1	1	yes	characteristic
242	C₂×C₁₇	#2949	1	1	yes	characteristic
243	C₂×C₁₇	#2957	1	1	yes	characteristic
244	D₁₇	—	8	2	no	—
245	C₂×#41	—	34	1	no	—
246	C₂×SD₁₆	—	17	2	no	—
247	C₂×SD₁₆	—	17	2	no	—
248	C₂×M₁₆	—	34	2	no	—
249	C₂²×C₈	—	34	2	no	—
250	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
251	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
252	C₄⋊Q₈	—	17	1	no	—
253	C₄⋊Q₈	—	17	1	no	—
254	C₄⋊Q₈	—	17	1	no	—
255	#127	—	17	1	no	—
256	#125	—	34	1	no	—
257	#124	—	34	1	no	—
258	C₂²⋊Q₈	—	17	2	no	—
259	C₂²⋊Q₈	—	34	1	no	—
260	#121	—	17	2	no	—
261	#121	—	34	1	no	—
262	C₄×Q₈	—	17	2	no	—
263	C₄×Q₈	—	34	1	no	—
264	C₄×D₄	—	17	2	no	—
265	C₄×D₄	—	34	1	no	—
266	#117	—	34	1	no	—
267	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
268	#106	—	17	2	no	—
269	#106	—	17	2	no	—
270	C₄⋊C₈	—	2	1	no	—
271	C₄⋊C₈	—	17	2	no	—
272	C₄⋊C₈	—	17	2	no	—
273	C₄⋊C₈	—	34	2	no	—
274	C₄⋊C₈	—	34	2	no	—
275	C₄⋊C₈	D₆₈	1	2	yes	—
276	#103	—	17	2	no	—
277	#103	—	17	2	no	—
278	#102	—	17	2	no	—
279	#102	—	17	2	no	—
280	C₂²⋊C₈	—	2	1	no	—
281	C₂²⋊C₈	—	34	1	no	—
282	C₂²⋊C₈	—	34	1	no	—
283	C₂²⋊C₈	—	34	2	no	—
284	C₂²⋊C₈	—	34	2	no	—
285	C₂²⋊C₈	D₆₈	1	2	yes	—
286	#97	—	34	2	no	—
287	#97	—	34	1	no	—
288	#97	—	34	1	no	—
289	C₄×C₈	—	34	2	no	—
290	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
291	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
292	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
293	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
294	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
295	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
296	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
297	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
298	C₁₇	#17990	1	1	yes	characteristic, complemented, Sylow
299	#41	—	136	2	no	—
300	C₂×Q₈	—	17	2	no	—
301	C₂×Q₈	—	34	1	no	—
302	C₂×Q₈	—	34	1	no	—
303	C₂×D₄	—	17	2	no	—
304	C₂×D₄	—	34	2	no	—
305	C₂×D₄	—	34	1	no	—
306	C₂×D₄	—	34	1	no	—
307	C₂²×C₄	—	34	2	no	—
308	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
309	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
310	C₂²×C₄	#63252	1	1	yes	characteristic
311	SD₁₆	—	68	2	no	—
312	SD₁₆	—	68	2	no	—
313	M₁₆	—	136	2	no	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	4	1	no	—
317	C₂×C₈	—	4	1	no	—
318	C₂×C₈	—	34	2	no	—
319	C₂×C₈	—	34	2	no	—
320	C₂×C₈	—	136	2	no	—
321	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
322	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
323	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
324	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
325	C₂×C₈	C₂×D₆₈	1	2	yes	—
326	C₂×C₈	C₂×D₆₈	1	2	yes	—
327	C₄⋊C₄	—	17	2	no	—
328	C₄⋊C₄	—	17	2	no	—
329	C₄⋊C₄	—	34	2	no	—
330	C₄⋊C₄	—	34	1	no	—
331	C₄⋊C₄	—	34	1	no	—
332	C₄⋊C₄	—	34	1	no	—
333	C₄⋊C₄	—	34	2	no	—
334	C₄⋊C₄	—	34	1	no	—
335	C₂²⋊C₄	—	34	2	no	—
336	C₂²⋊C₄	—	34	2	no	—
337	C₂²⋊C₄	—	34	2	no	—
338	C₄²	—	34	2	no	—
339	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
340	C₄²	C₂×D₆₈	1	1	yes	characteristic
341	C₄²	#63252	1	1	yes	characteristic
342	C₄²	C₂³×D₁₇	1	1	yes	characteristic
343	C₂³	—	34	2	no	—
344	C₂³	C₄×D₆₈	1	1	yes	characteristic
345	Q₈	—	34	2	no	—
346	Q₈	—	34	2	no	—
347	Q₈	—	136	1	no	—
348	Q₈	—	136	1	no	—
349	D₄	—	34	2	no	—
350	D₄	—	34	2	no	—
351	D₄	—	136	1	no	—
352	D₄	—	136	1	no	—
353	D₄	—	136	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	34	2	no	—
359	C₂×C₄	—	34	2	no	—
360	C₂×C₄	—	34	2	no	—
361	C₂×C₄	—	68	2	no	—
362	C₂×C₄	—	68	2	no	—
363	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
364	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
365	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
366	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
367	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
368	C₂×C₄	C₂×#63252	1	1	yes	characteristic
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	—	2	2	no	—
374	C₈	—	2	2	no	—
375	C₈	—	4	1	no	—
376	C₈	—	4	1	no	—
377	C₈	—	68	2	no	—
378	C₈	—	68	2	no	—
379	C₂²	—	2	1	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	68	2	no	—
383	C₂²	—	68	2	no	—
384	C₂²	—	68	2	no	—
385	C₂²	C₂×C₄×D₆₈	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	2	1	no	—
388	C₄	—	2	1	no	—
389	C₄	—	2	1	no	—
390	C₄	—	68	2	no	—
391	C₄	—	68	2	no	—
392	C₄	—	68	2	no	—
393	C₄	#11763575	1	1	yes	characteristic
394	C₄	#11763575	1	1	yes	characteristic
395	C₄	#11763621	1	2	yes	—
396	C₂	—	4	1	no	—
397	C₂	—	136	2	no	—
398	C₂	#14537503	1	1	yes	characteristic
399	C₂	#14537645	1	1	yes	characteristic
400	C₂	#14537702	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₆₈
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₄²	C₂³×D₁₇
Fitting	C₁₇×#1452	C₂
socle	C₂²×C₁₇	C₂×C₄×D₄
prime radical	#11763359	C₂²
cosocle	C₄²×C₁₇	C₂⁴
hypercenter	#1452	D₁₇
hypocenter	C₁₇	#17990
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1452

Complements

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

N	H ≅ G/N	type
2. #14537878	C₂	semidirect
4. #14527664	C₂	semidirect
5. #14527643	C₂	semidirect
6. #14527629	C₂	semidirect
7. #14527622	C₂	semidirect
8. #14527547	C₂	semidirect
9. #14527258	C₂	semidirect
11. C₁₇×#1452	C₂	semidirect
189. #1452	D₁₇	semidirect
298. C₁₇	#17990	semidirect

Maximal subgroups

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

subgroup	index
#14539343	2
#14537878	2
#14537859	2
#14527664	2
#14527643	2
#14527629	2
#14527622	2
#14527547	2
#14527258	2
#14527254	2
C₁₇×#1452	2
#17990	17