Subgroups — #17187800

Order 4352.

668 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14537701	C₂	1	1	yes	maximal, characteristic, complemented
2	#14537636	C₂	1	1	yes	maximal, characteristic, complemented
3	#14530683	C₂	1	8	yes	maximal, complemented
4	#14530557	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11762622	C₂	1	1	yes	maximal, characteristic, complemented
6	#14530520	C₂	1	1	yes	maximal, characteristic, complemented
7	#14530388	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₇×#509	C₂	1	1	yes	maximal, characteristic
9	#11763576	—	2	1	no	—
10	#11763284	—	2	1	no	—
11	C₂×#10587483	C₂²	1	1	yes	characteristic
12	#11762931	C₂²	1	4	yes	complemented
13	#11762926	—	2	2	no	—
14	#11762921	C₂²	1	4	yes	complemented
15	D₈×Dic₁₇	—	2	2	no	—
16	C₂×#10587490	C₂²	1	1	yes	characteristic
17	#11762847	C₂²	1	1	yes	characteristic
18	#11762826	—	2	1	no	—
19	#11762814	—	2	1	no	—
20	#11762764	C₂²	1	1	yes	characteristic
21	#11762632	C₂²	1	4	yes	complemented
22	#11762622	C₂²	1	4	yes	complemented
23	#11762619	C₂²	1	4	yes	complemented
24	#11762586	C₂²	1	4	yes	complemented
25	C₂×C₁₇×#135	C₂²	1	1	yes	characteristic
26	C₂²×D₈×C₁₇	C₂²	1	1	yes	characteristic
27	C₁₇×#509	C₄	1	4	yes	complemented
28	C₁₇×#509	C₂²	1	4	yes	—
29	C₂×C₁₇×SD₃₂	C₄	1	2	yes	complemented
30	C₂×C₁₇×D₁₆	C₄	1	2	yes	complemented
31	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
32	C₂×#41×C₁₇	—	2	1	no	—
33	C₂²×D₄×C₁₇	—	2	1	no	—
34	C₁₇×#135	—	2	1	no	—
35	C₁₇×#135	—	2	1	no	—
36	C₁₇×#135	C₂×C₄	1	2	yes	complemented
37	C₁₇×#135	C₂³	1	2	yes	—
38	C₂×Q₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic, complemented
39	C₂×SD₁₆×C₁₇	—	2	1	no	—
40	C₂×D₈×C₁₇	—	2	1	no	—
41	C₂×D₈×C₁₇	—	2	1	no	—
42	C₂×D₈×C₁₇	—	2	1	no	—
43	C₂×D₈×C₁₇	C₂×C₄	1	2	yes	—
44	C₂×D₈×C₁₇	C₂×C₄	1	2	yes	—
45	C₂×D₈×C₁₇	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂×D₈×C₁₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊Dic₁₇	—	4	2	no	—
49	#10587634	—	2	4	no	—
50	#10587634	—	4	2	no	—
51	D₄×Dic₁₇	—	2	4	no	—
52	D₄×Dic₁₇	—	4	2	no	—
53	#10587612	—	2	1	no	—
54	#10587612	D₄	1	1	yes	characteristic
55	C₂×#63223	—	2	1	no	—
56	C₂×#63223	—	4	2	no	—
57	C₂×#63223	D₄	1	1	yes	characteristic
58	C₁₇×SD₃₂	C₂×C₄	1	8	yes	complemented
59	C₁₇×D₁₆	C₂×C₄	1	8	yes	complemented
60	C₁₇×M₃₂	C₂×C₄	1	2	yes	complemented
61	C₁₇×M₃₂	C₂³	1	2	yes	—
62	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	complemented
63	#10587508	—	2	4	no	—
64	#10587503	—	2	4	no	—
65	#10587490	—	2	1	no	—
66	#10587490	C₂³	1	2	yes	—
67	#10587489	—	2	1	no	—
68	#10587488	C₂³	1	2	yes	—
69	#10587486	C₂³	1	2	yes	—
70	C₈×Dic₁₇	D₄	1	2	yes	—
71	#10587483	—	2	1	no	—
72	#10587483	C₂³	1	2	yes	—
73	C₂×#63214	—	2	1	no	—
74	C₂⁴×C₁₇	—	4	1	no	—
75	#41×C₁₇	—	2	2	no	—
76	#41×C₁₇	—	2	2	no	—
77	#41×C₁₇	—	4	1	no	—
78	#41×C₁₇	—	4	1	no	—
79	C₂×Q₈×C₁₇	—	2	1	no	—
80	C₂×D₄×C₁₇	—	2	2	no	—
81	C₂×D₄×C₁₇	—	2	2	no	—
82	C₂×D₄×C₁₇	—	2	1	no	—
83	C₂×D₄×C₁₇	—	2	2	no	—
84	C₂×D₄×C₁₇	—	4	1	no	—
85	C₂×D₄×C₁₇	—	4	1	no	—
86	C₂×D₄×C₁₇	—	4	1	no	—
87	C₂×D₄×C₁₇	—	4	1	no	—
88	C₂²×C₄×C₁₇	—	4	1	no	—
89	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×Dic₁₇	—	4	4	no	—
91	Q₁₆×C₁₇	—	2	1	no	—
92	Q₁₆×C₁₇	C₂²×C₄	1	2	yes	—
93	SD₁₆×C₁₇	—	2	2	no	—
94	SD₁₆×C₁₇	—	2	2	no	—
95	D₈×C₁₇	—	2	2	no	—
96	D₈×C₁₇	—	2	2	no	—
97	D₈×C₁₇	—	2	1	no	—
98	D₈×C₁₇	—	2	2	no	—
99	D₈×C₁₇	C₂²×C₄	1	4	yes	—
100	D₈×C₁₇	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
105	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
106	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
107	C₂²⋊Dic₁₇	—	2	1	no	—
108	C₂²⋊Dic₁₇	—	4	4	no	—
109	C₂²⋊Dic₁₇	—	4	1	no	—
110	C₄⋊Dic₁₇	—	2	1	no	—
111	C₄⋊Dic₁₇	—	2	2	no	—
112	C₄×Dic₁₇	—	4	1	no	—
113	C₄×Dic₁₇	C₂×D₄	1	2	yes	—
114	#63223	—	2	1	no	—
115	#63223	—	2	1	no	—
116	#63223	—	2	2	no	—
117	#63223	—	4	4	no	—
118	#63223	—	4	1	no	—
119	C₂×#3622	—	4	4	no	—
120	C₂×#3622	—	4	1	no	—
121	C₂×#3622	C₂×D₄	1	2	yes	—
122	C₁₆×C₁₇	C₂²×C₄	1	4	yes	—
123	#63214	—	2	2	no	—
124	#33471	—	17	1	no	maximal, Sylow
125	C₂³×C₁₇	—	4	2	no	—
126	C₂³×C₁₇	—	4	1	no	—
127	C₂³×C₁₇	—	4	2	no	—
128	C₂³×C₁₇	—	4	2	no	—
129	C₂³×C₁₇	—	16	2	no	—
130	C₂³×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
131	Q₈×C₁₇	—	2	2	no	—
132	Q₈×C₁₇	—	4	1	no	—
133	D₄×C₁₇	—	2	4	no	—
134	D₄×C₁₇	—	2	2	no	—
135	D₄×C₁₇	—	4	2	no	—
136	D₄×C₁₇	—	4	2	no	—
137	D₄×C₁₇	—	4	2	no	—
138	D₄×C₁₇	—	4	2	no	—
139	D₄×C₁₇	—	4	2	no	—
140	D₄×C₁₇	—	4	1	no	—
141	C₂×C₄×C₁₇	—	4	1	no	—
142	C₂×C₄×C₁₇	—	4	2	no	—
143	C₂×C₄×C₁₇	—	4	2	no	—
144	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×Dic₁₇	—	2	2	no	—
151	C₂×Dic₁₇	—	4	2	no	—
152	C₂×Dic₁₇	—	16	4	no	—
153	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
154	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
155	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
156	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
157	#3622	—	2	2	no	—
158	#3622	—	4	2	no	—
159	C₂×#509	D₁₇	1	1	yes	characteristic
160	#2957	—	17	1	no	—
161	#2948	—	17	1	no	—
162	AGL(1,ℤ/16)	—	17	8	no	—
163	#2155	—	17	1	no	—
164	C₂×#361	—	17	1	no	—
165	#2145	—	17	1	no	—
166	#2110	—	17	1	no	—
167	C₂²×C₁₇	—	4	1	no	—
168	C₂²×C₁₇	—	4	2	no	—
169	C₂²×C₁₇	—	4	4	no	—
170	C₂²×C₁₇	—	8	4	no	—
171	C₂²×C₁₇	—	16	1	no	—
172	C₂²×C₁₇	—	16	2	no	—
173	C₂²×C₁₇	—	16	2	no	—
174	C₂²×C₁₇	#442	1	1	yes	characteristic
175	C₂²×C₁₇	#442	1	1	yes	characteristic
176	C₂²×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₄×C₁₇	—	4	2	no	—
178	C₄×C₁₇	#442	1	1	yes	characteristic
179	C₄×C₁₇	#442	1	1	yes	characteristic
180	C₄×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
181	C₄×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	Dic₁₇	—	4	2	no	—
183	Dic₁₇	—	8	2	no	—
184	C₂×#135	C₂×D₁₇	1	1	yes	characteristic
185	C₂²×D₈	C₂×D₁₇	1	1	yes	characteristic
186	#568	—	34	1	no	—
187	#518	—	34	1	no	—
188	#509	Dic₁₇	1	4	yes	complemented
189	#509	C₂×D₁₇	1	4	yes	—
190	C₂×SD₃₂	Dic₁₇	1	2	yes	complemented
191	C₂×D₁₆	Dic₁₇	1	2	yes	complemented
192	C₂×M₃₂	—	17	1	no	—
193	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
194	#444	—	17	4	no	—
195	#443	—	34	2	no	—
196	#442	—	17	4	no	—
197	C₄×D₈	—	34	2	no	—
198	C₂×#108	—	17	1	no	—
199	#427	—	17	1	no	—
200	#421	—	34	1	no	—
201	#418	—	34	1	no	—
202	#405	—	17	1	no	—
203	#365	—	17	4	no	—
204	#361	—	17	4	no	—
205	#360	—	17	4	no	—
206	#347	—	17	4	no	—
207	C₂×C₁₇	—	2	1	no	—
208	C₂×C₁₇	—	2	1	no	—
209	C₂×C₁₇	—	2	1	no	—
210	C₂×C₁₇	—	8	4	no	—
211	C₂×C₁₇	—	8	2	no	—
212	C₂×C₁₇	C₂×#442	1	1	yes	characteristic
213	C₂×#41	—	2	1	no	—
214	C₂²×D₄	—	2	1	no	—
215	#135	—	2	1	no	—
216	#135	—	2	1	no	—
217	#135	C₂×Dic₁₇	1	2	yes	complemented
218	#135	C₂²×D₁₇	1	2	yes	—
219	C₂×Q₁₆	C₂×Dic₁₇	1	1	yes	characteristic, complemented
220	C₂×SD₁₆	—	2	1	no	—
221	C₂×D₈	—	2	1	no	—
222	C₂×D₈	—	2	1	no	—
223	C₂×D₈	—	2	1	no	—
224	C₂×D₈	C₂×Dic₁₇	1	2	yes	—
225	C₂×D₈	C₂×Dic₁₇	1	2	yes	—
226	C₂×D₈	C₂×Dic₁₇	1	1	yes	characteristic, complemented
227	C₂×D₈	C₂²×D₁₇	1	2	yes	—
228	#131	—	34	4	no	—
229	#131	—	68	2	no	—
230	C₂×M₁₆	—	17	1	no	—
231	C₂×M₁₆	—	34	1	no	—
232	C₂×M₁₆	—	68	2	no	—
233	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
234	C₄×D₄	—	34	4	no	—
235	C₄×D₄	—	68	2	no	—
236	#117	—	17	1	no	—
237	#117	—	34	1	no	—
238	C₂×C₂²⋊C₄	—	68	2	no	—
239	SD₃₂	C₂×Dic₁₇	1	8	yes	complemented
240	D₁₆	C₂×Dic₁₇	1	8	yes	complemented
241	M₃₂	—	17	2	no	—
242	M₃₂	—	34	1	no	—
243	M₃₂	C₂×Dic₁₇	1	2	yes	complemented
244	M₃₂	C₂²×D₁₇	1	2	yes	—
245	C₂×C₁₆	—	34	1	no	—
246	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	complemented
247	#108	—	17	2	no	—
248	#108	—	34	1	no	—
249	#107	—	34	1	no	—
250	#106	—	17	2	no	—
251	#104	—	34	4	no	—
252	#102	—	34	4	no	—
253	#97	—	17	2	no	—
254	C₄×C₈	—	17	2	no	—
255	C₁₇	#33471	1	1	yes	characteristic, complemented, Sylow
256	C₂⁴	—	4	1	no	—
257	#41	—	2	2	no	—
258	#41	—	2	2	no	—
259	#41	—	4	1	no	—
260	#41	—	4	1	no	—
261	C₂×Q₈	—	2	1	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	2	1	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	4	1	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂²×C₄	—	4	1	no	—
271	C₂²×C₄	—	68	4	no	—
272	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
273	Q₁₆	—	2	1	no	—
274	Q₁₆	C₂²×Dic₁₇	1	2	yes	—
275	SD₁₆	—	2	2	no	—
276	SD₁₆	—	2	2	no	—
277	D₈	—	2	2	no	—
278	D₈	—	2	2	no	—
279	D₈	—	2	2	no	—
280	D₈	—	2	1	no	—
281	D₈	C₂²×Dic₁₇	1	2	yes	—
282	D₈	C₂²×Dic₁₇	1	4	yes	—
283	M₁₆	—	34	1	no	—
284	M₁₆	—	34	1	no	—
285	M₁₆	—	34	2	no	—
286	M₁₆	—	68	4	no	—
287	M₁₆	—	68	1	no	—
288	C₂×C₈	—	17	2	no	—
289	C₂×C₈	—	68	4	no	—
290	C₂×C₈	—	68	1	no	—
291	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
292	C₂×C₈	#63254	1	1	yes	characteristic
293	C₂×C₈	#63254	1	1	yes	characteristic
294	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
295	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
296	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
297	C₄⋊C₄	—	34	1	no	—
298	C₄⋊C₄	—	34	2	no	—
299	C₂²⋊C₄	—	34	1	no	—
300	C₂²⋊C₄	—	68	4	no	—
301	C₂²⋊C₄	—	68	1	no	—
302	C₄²	—	17	2	no	—
303	C₄²	—	68	1	no	—
304	C₁₆	—	34	2	no	—
305	C₁₆	C₂²×Dic₁₇	1	4	yes	—
306	C₂³	—	4	2	no	—
307	C₂³	—	4	1	no	—
308	C₂³	—	4	2	no	—
309	C₂³	—	4	2	no	—
310	C₂³	—	16	2	no	—
311	C₂³	#10587597	1	1	yes	characteristic
312	Q₈	—	2	2	no	—
313	Q₈	—	4	1	no	—
314	D₄	—	2	4	no	—
315	D₄	—	2	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	2	no	—
320	D₄	—	4	2	no	—
321	D₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	272	4	no	—
326	C₂×C₄	—	34	2	no	—
327	C₂×C₄	—	68	2	no	—
328	C₂×C₄	#10587597	1	1	yes	characteristic
329	C₂×C₄	#10587597	1	1	yes	characteristic
330	C₂×C₄	#10587597	1	1	yes	characteristic
331	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
332	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
334	C₈	—	34	2	no	—
335	C₈	—	68	2	no	—
336	C₈	D₄×Dic₁₇	1	1	yes	characteristic
337	C₈	D₄×Dic₁₇	1	1	yes	characteristic
338	C₈	C₂×#63254	1	1	yes	characteristic
339	C₈	C₂³×Dic₁₇	1	1	yes	characteristic
340	C₂²	—	4	1	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	4	4	no	—
343	C₂²	—	8	4	no	—
344	C₂²	—	16	2	no	—
345	C₂²	—	16	2	no	—
346	C₂²	—	16	1	no	—
347	C₂²	#11762921	1	1	yes	characteristic
348	C₂²	#11762921	1	1	yes	characteristic
349	C₂²	C₂×#10587597	1	1	yes	characteristic
350	C₄	—	4	2	no	—
351	C₄	—	68	2	no	—
352	C₄	—	136	2	no	—
353	C₄	#11762921	1	1	yes	characteristic
354	C₄	#11762921	1	1	yes	characteristic
355	C₄	C₂×D₄×Dic₁₇	1	1	yes	characteristic
356	C₄	C₂×#10587597	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	8	4	no	—
361	C₂	—	8	2	no	—
362	C₂	C₂×#11762921	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#11762921
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₇
Fitting, prime radical	C₂×C₁₇×#509	C₂
socle	C₂×C₁₇	C₂×#442
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₂×#509	D₁₇
hypocenter	C₁₇	#33471
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#509

Complements

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

N	H ≅ G/N	type
1. #14537701	C₂	semidirect
2. #14537636	C₂	semidirect
3. #14530683	C₂	semidirect
4. #14530557	C₂	semidirect
5. C₂×#11762622	C₂	semidirect
6. #14530520	C₂	semidirect
7. #14530388	C₂	semidirect
12. #11762931	C₂²	semidirect
14. #11762921	C₂²	semidirect
21. #11762632	C₂²	semidirect
22. #11762622	C₂²	semidirect
23. #11762619	C₂²	semidirect
24. #11762586	C₂²	semidirect
27. C₁₇×#509	C₄	semidirect
29. C₂×C₁₇×SD₃₂	C₄	semidirect
30. C₂×C₁₇×D₁₆	C₄	semidirect
36. C₁₇×#135	C₂×C₄	semidirect
38. C₂×Q₁₆×C₁₇	C₂×C₄	semidirect
45. C₂×D₈×C₁₇	C₂×C₄	semidirect
58. C₁₇×SD₃₂	C₂×C₄	semidirect
59. C₁₇×D₁₆	C₂×C₄	semidirect
60. C₁₇×M₃₂	C₂×C₄	semidirect
62. C₂×C₁₆×C₁₇	C₂×C₄	semidirect
188. #509	Dic₁₇	semidirect
190. C₂×SD₃₂	Dic₁₇	semidirect
191. C₂×D₁₆	Dic₁₇	semidirect
217. #135	C₂×Dic₁₇	semidirect
219. C₂×Q₁₆	C₂×Dic₁₇	semidirect
226. C₂×D₈	C₂×Dic₁₇	semidirect
239. SD₃₂	C₂×Dic₁₇	semidirect
240. D₁₆	C₂×Dic₁₇	semidirect
243. M₃₂	C₂×Dic₁₇	semidirect
246. C₂×C₁₆	C₂×Dic₁₇	semidirect
255. C₁₇	#33471	semidirect

Maximal subgroups

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

subgroup	index
#14537701	2
#14537636	2
#14530683	2
#14530557	2
C₂×#11762622	2
#14530520	2
#14530388	2
C₂×C₁₇×#509	2
#33471	17