Subgroups — #17005195

Order 4352.

576 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539354	C₂	1	1	yes	maximal, characteristic
2	#14537935	C₂	1	2	yes	maximal, complemented
3	#14527664	C₂	1	2	yes	maximal, complemented
4	#14527643	C₂	1	2	yes	maximal, complemented
5	#14527636	C₂	1	2	yes	maximal, complemented
6	#14527615	C₂	1	2	yes	maximal, complemented
7	#14527543	C₂	1	1	yes	maximal, characteristic, complemented
8	#14527250	C₂	1	2	yes	maximal, complemented
9	C₁₇×#1452	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763343	C₂²	1	2	yes	—
11	#11763211	C₄	1	1	yes	characteristic
12	#11763193	C₄	1	1	yes	characteristic
13	#11763164	C₄	1	2	yes	—
14	#11763050	C₄	1	1	yes	characteristic
15	#11763045	C₄	1	1	yes	characteristic
16	#11763038	C₄	1	2	yes	—
17	#11762939	C₂²	1	2	yes	—
18	C₈×Dic₃₄	C₂²	1	2	yes	—
19	C₄×Dic₆₈	C₂²	1	1	yes	characteristic
20	C₄×#63219	C₂²	1	2	yes	—
21	C₄×D₁₃₆	C₂²	1	1	yes	characteristic
22	#11762888	C₂²	1	2	yes	—
23	#11762884	—	2	2	no	—
24	#11762883	—	2	2	no	—
25	C₈×D₆₈	C₂²	1	2	yes	—
26	#11762869	—	2	2	no	—
27	#11762834	—	2	2	no	—
28	#11762833	C₂²	1	2	yes	—
29	#11762562	C₂²	1	2	yes	—
30	#11762562	C₂²	1	1	yes	characteristic
31	#11762562	C₂²	1	1	yes	characteristic
32	#11762541	C₂²	1	2	yes	—
33	#11762541	C₂²	1	2	yes	—
34	#11762538	C₂²	1	2	yes	—
35	#11762538	C₂²	1	2	yes	—
36	C₁₇×#431	C₂²	1	1	yes	characteristic
37	C₁₇×#431	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
39	C₁₇×#322	C₂²	1	2	yes	—
40	C₁₇×#322	C₂²	1	2	yes	—
41	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
44	#10587616	C₂×C₄	1	2	yes	—
45	#10587615	—	2	2	no	—
46	C₄×C₂²⋊D₁₇	—	2	2	no	—
47	#10587610	C₂×C₄	1	2	yes	—
48	C₂×#63223	—	2	2	no	—
49	C₂×Dic₆₈	C₂×C₄	1	1	yes	characteristic
50	C₂×D₁₃₆	C₂×C₄	1	1	yes	characteristic
51	C₂×#63219	C₂×C₄	1	2	yes	—
52	C₂×#63218	—	2	2	no	—
53	C₂×C₈×D₁₇	—	2	2	no	—
54	#10587562	—	2	2	no	—
55	#10587561	C₂×C₄	1	2	yes	—
56	C₄×D₆₈	C₂³	1	2	yes	—
57	#10587558	—	2	2	no	—
58	#10587556	C₂×C₄	1	2	yes	—
59	C₄×Dic₃₄	C₂³	1	2	yes	—
60	C₁₇×C₄⋊C₈	—	2	1	no	—
61	C₁₇×C₄⋊C₈	D₄	1	1	yes	characteristic
62	C₁₇×C₄⋊C₈	D₄	1	1	yes	characteristic
63	C₁₇×C₂²⋊C₈	—	2	1	no	—
64	C₁₇×C₂²⋊C₈	D₄	1	2	yes	—
65	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₁₇	C₂³	1	2	yes	—
68	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₇	C₂³	1	2	yes	—
71	#10587492	C₂×C₄	1	2	yes	—
72	#10587492	C₂×C₄	1	2	yes	—
73	#10587491	—	2	2	no	—
74	#10587491	—	2	2	no	—
75	#10587489	C₂×C₄	1	1	yes	characteristic
76	#10587489	C₂×C₄	1	1	yes	characteristic
77	#10587488	C₂×C₄	1	2	yes	—
78	#10587487	C₂×C₄	1	2	yes	—
79	#10587487	C₂×C₄	1	2	yes	—
80	#10587486	—	2	2	no	—
81	#10587485	—	2	2	no	—
82	#10587485	—	2	2	no	—
83	C₈×Dic₁₇	—	2	2	no	—
84	C₄⋊#3622	—	2	2	no	—
85	C₄⋊#3622	C₂³	1	2	yes	—
86	C₄⋊#3622	C₂³	1	2	yes	—
87	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
90	C₂×C₂²⋊D₁₇	—	2	2	no	—
91	C₂×D₆₈	C₂²×C₄	1	2	yes	—
92	C₂×C₄×D₁₇	—	2	2	no	—
93	C₂×Dic₃₄	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₁₇	—	2	2	no	—
95	C₂×C₈×C₁₇	—	4	1	no	—
96	C₂×C₈×C₁₇	—	4	1	no	—
97	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
103	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₇	#41	1	1	yes	characteristic
106	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
107	C₂²⋊Dic₁₇	—	2	2	no	—
108	#63227	—	2	2	no	—
109	#63227	—	2	2	no	—
110	#63227	—	2	2	no	—
111	#63227	—	2	2	no	—
112	C₄⋊Dic₁₇	C₂²×C₄	1	2	yes	—
113	C₄⋊Dic₁₇	C₂²×C₄	1	2	yes	—
114	#63225	—	2	2	no	—
115	#63225	—	2	2	no	—
116	#63225	—	2	2	no	—
117	#63225	—	2	2	no	—
118	C₄×Dic₁₇	—	2	2	no	—
119	#63223	—	8	2	no	—
120	C₂×#3622	—	2	2	no	—
121	C₂×#3622	—	2	2	no	—
122	Dic₆₈	—	4	1	no	—
123	D₁₃₆	—	4	1	no	—
124	#63219	—	4	2	no	—
125	#63218	—	8	2	no	—
126	C₈×D₁₇	—	8	2	no	—
127	#17992	—	17	1	no	maximal, Sylow
128	C₂³×C₁₇	C₄×D₄	1	1	yes	characteristic
129	C₂²×D₁₇	—	2	2	no	—
130	C₂×C₄×C₁₇	—	2	1	no	—
131	C₂×C₄×C₁₇	—	2	1	no	—
132	C₂×C₄×C₁₇	—	2	1	no	—
133	C₂×C₄×C₁₇	—	2	1	no	—
134	C₂×C₄×C₁₇	—	4	1	no	—
135	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
141	C₂²⋊D₁₇	—	8	2	no	—
142	C₂×Dic₁₇	—	2	2	no	—
143	C₂×Dic₁₇	—	2	2	no	—
144	C₂×Dic₁₇	—	2	2	no	—
145	D₆₈	—	2	2	no	—
146	D₆₈	—	2	2	no	—
147	C₄×D₁₇	—	4	2	no	—
148	C₄×D₁₇	—	4	2	no	—
149	Dic₃₄	—	2	2	no	—
150	Dic₃₄	—	2	2	no	—
151	C₈×C₁₇	—	2	1	no	—
152	C₈×C₁₇	—	2	1	no	—
153	C₈×C₁₇	—	2	1	no	—
154	C₈×C₁₇	—	2	1	no	—
155	C₈×C₁₇	—	2	2	no	—
156	C₈×C₁₇	—	2	2	no	—
157	C₈×C₁₇	—	4	2	no	—
158	#3622	—	4	2	no	—
159	#3622	—	4	2	no	—
160	#3168	—	17	1	no	—
161	#2993	—	17	2	no	—
162	#1588	—	17	2	no	—
163	#1585	—	17	2	no	—
164	#1584	—	17	2	no	—
165	#1581	—	17	2	no	—
166	#1567	—	17	1	no	—
167	#1495	—	17	2	no	—
168	#1452	D₁₇	1	1	yes	characteristic, complemented
169	C₂²×C₁₇	—	2	1	no	—
170	C₂²×C₁₇	—	2	1	no	—
171	C₂²×C₁₇	—	2	1	no	—
172	C₂²×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂×D₁₇	—	4	2	no	—
174	C₂×D₁₇	—	4	2	no	—
175	C₂×D₁₇	—	4	2	no	—
176	C₄×C₁₇	—	2	1	no	—
177	C₄×C₁₇	—	2	1	no	—
178	C₄×C₁₇	—	2	1	no	—
179	C₄×C₁₇	—	2	1	no	—
180	C₄×C₁₇	#568	1	1	yes	characteristic
181	C₄×C₁₇	#568	1	1	yes	characteristic
182	C₄×C₁₇	#576	1	1	yes	characteristic
183	C₄×C₁₇	#578	1	1	yes	characteristic
184	Dic₁₇	—	4	2	no	—
185	Dic₁₇	—	4	2	no	—
186	Dic₁₇	—	4	2	no	—
187	#529	—	17	2	no	—
188	#499	—	17	1	no	—
189	#495	—	17	1	no	—
190	#488	—	17	2	no	—
191	#467	—	17	1	no	—
192	#466	—	17	1	no	—
193	#465	—	17	2	no	—
194	#446	—	17	2	no	—
195	C₈×Q₈	—	17	2	no	—
196	C₄×Q₁₆	—	17	1	no	—
197	C₄×SD₁₆	—	17	2	no	—
198	C₄×D₈	—	17	1	no	—
199	#436	—	17	2	no	—
200	#436	—	34	2	no	—
201	#435	—	34	2	no	—
202	C₈×D₄	—	17	2	no	—
203	#433	—	34	2	no	—
204	#431	C₂×D₁₇	1	1	yes	characteristic
205	#431	C₂×D₁₇	1	1	yes	characteristic
206	#424	—	34	2	no	—
207	#423	—	17	2	no	—
208	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
209	#335	—	17	2	no	—
210	#335	—	17	1	no	—
211	#335	—	17	1	no	—
212	Q₈⋊C₈	—	17	2	no	—
213	Q₈⋊C₈	—	17	2	no	—
214	D₄⋊C₈	—	17	2	no	—
215	D₄⋊C₈	—	17	2	no	—
216	#322	C₂×D₁₇	1	2	yes	—
217	#322	C₂×D₁₇	1	2	yes	—
218	C₂×C₁₇	—	4	1	no	—
219	C₂×C₁₇	#2932	1	1	yes	characteristic
220	C₂×C₁₇	#2951	1	1	yes	characteristic
221	C₂×C₁₇	#2957	1	1	yes	characteristic
222	D₁₇	—	8	2	no	—
223	C₂×Q₁₆	—	17	1	no	—
224	C₂×SD₁₆	—	17	2	no	—
225	C₂×D₈	—	17	1	no	—
226	C₂×M₁₆	—	34	2	no	—
227	C₂×M₁₆	—	34	2	no	—
228	C₂²×C₈	—	34	2	no	—
229	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
230	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
231	#126	—	34	2	no	—
232	#125	—	17	2	no	—
233	#124	—	17	2	no	—
234	#123	—	34	2	no	—
235	C₂²⋊Q₈	—	17	2	no	—
236	#121	—	17	2	no	—
237	C₄×Q₈	—	17	2	no	—
238	C₄×D₄	—	17	2	no	—
239	C₄×D₄	—	34	2	no	—
240	#117	—	34	2	no	—
241	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
242	#107	—	17	1	no	—
243	#107	—	17	1	no	—
244	#106	—	17	2	no	—
245	C₄⋊C₈	—	2	1	no	—
246	C₄⋊C₈	—	17	2	no	—
247	C₄⋊C₈	—	17	2	no	—
248	C₄⋊C₈	—	34	2	no	—
249	C₄⋊C₈	—	34	2	no	—
250	C₄⋊C₈	—	34	2	no	—
251	C₄⋊C₈	D₆₈	1	1	yes	characteristic
252	C₄⋊C₈	D₆₈	1	1	yes	characteristic
253	#103	—	17	2	no	—
254	#103	—	17	2	no	—
255	#102	—	17	2	no	—
256	#102	—	17	2	no	—
257	C₂²⋊C₈	—	2	1	no	—
258	C₂²⋊C₈	—	34	2	no	—
259	C₂²⋊C₈	—	34	2	no	—
260	C₂²⋊C₈	D₆₈	1	2	yes	—
261	#97	—	34	2	no	—
262	C₄×C₈	—	34	2	no	—
263	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
264	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
265	C₄×C₈	C₂²×D₁₇	1	2	yes	—
266	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
268	C₄×C₈	C₂²×D₁₇	1	2	yes	—
269	C₁₇	#17992	1	1	yes	characteristic, complemented, Sylow
270	C₂×Q₈	—	17	2	no	—
271	C₂×D₄	—	17	2	no	—
272	C₂×D₄	—	34	2	no	—
273	C₂²×C₄	—	34	2	no	—
274	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
276	C₂²×C₄	#63252	1	1	yes	characteristic
277	Q₁₆	—	68	1	no	—
278	SD₁₆	—	68	2	no	—
279	D₈	—	68	1	no	—
280	M₁₆	—	136	2	no	—
281	M₁₆	—	136	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	34	2	no	—
286	C₂×C₈	—	34	2	no	—
287	C₂×C₈	—	136	2	no	—
288	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
289	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
290	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
292	C₂×C₈	C₂×D₆₈	1	2	yes	—
293	C₂×C₈	C₂×D₆₈	1	2	yes	—
294	C₄⋊C₄	—	17	2	no	—
295	C₄⋊C₄	—	17	2	no	—
296	C₄⋊C₄	—	34	2	no	—
297	C₄⋊C₄	—	34	2	no	—
298	C₄⋊C₄	—	34	2	no	—
299	C₄⋊C₄	—	34	2	no	—
300	C₂²⋊C₄	—	34	2	no	—
301	C₂²⋊C₄	—	34	2	no	—
302	C₂²⋊C₄	—	34	2	no	—
303	C₂²⋊C₄	—	34	2	no	—
304	C₂²⋊C₄	—	34	2	no	—
305	C₄²	—	34	2	no	—
306	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
307	C₄²	C₂×D₆₈	1	1	yes	characteristic
308	C₄²	#63252	1	1	yes	characteristic
309	C₄²	C₂³×D₁₇	1	1	yes	characteristic
310	C₂³	—	34	2	no	—
311	C₂³	C₄×D₆₈	1	1	yes	characteristic
312	Q₈	—	34	2	no	—
313	Q₈	—	34	2	no	—
314	D₄	—	34	2	no	—
315	D₄	—	34	2	no	—
316	D₄	—	136	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	34	2	no	—
323	C₂×C₄	—	34	2	no	—
324	C₂×C₄	—	34	2	no	—
325	C₂×C₄	—	68	2	no	—
326	C₂×C₄	—	68	2	no	—
327	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
328	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
329	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
331	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
332	C₂×C₄	C₂×#63252	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	2	2	no	—
338	C₈	—	2	2	no	—
339	C₈	—	4	2	no	—
340	C₈	—	68	2	no	—
341	C₈	—	68	2	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	68	2	no	—
346	C₂²	—	68	2	no	—
347	C₂²	—	68	2	no	—
348	C₂²	C₂×C₄×D₆₈	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	68	2	no	—
354	C₄	—	68	2	no	—
355	C₄	—	68	2	no	—
356	C₄	#11763575	1	1	yes	characteristic
357	C₄	#11763575	1	1	yes	characteristic
358	C₄	#11763614	1	1	yes	characteristic
359	C₄	#11763626	1	1	yes	characteristic
360	C₂	—	4	1	no	—
361	C₂	—	136	2	no	—
362	C₂	#14537503	1	1	yes	characteristic
363	C₂	#14537661	1	1	yes	characteristic
364	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	#11763045	C₄
cosocle	C₄²×C₁₇	C₂⁴
hypercenter	#1452	D₁₇
hypocenter	C₁₇	#17992
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. #14537935	C₂	semidirect
3. #14527664	C₂	semidirect
4. #14527643	C₂	semidirect
5. #14527636	C₂	semidirect
6. #14527615	C₂	semidirect
7. #14527543	C₂	semidirect
8. #14527250	C₂	semidirect
9. C₁₇×#1452	C₂	semidirect
168. #1452	D₁₇	semidirect
269. C₁₇	#17992	semidirect

Maximal subgroups

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

subgroup	index
#14539354	2
#14537935	2
#14527664	2
#14527643	2
#14527636	2
#14527615	2
#14527543	2
#14527250	2
C₁₇×#1452	2
#17992	17