Subgroups — #17004102

Order 4352.

666 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14537751	C₂	1	2	yes	maximal, complemented
2	#14527698	C₂	1	2	yes	maximal, complemented
3	#14527693	C₂	1	2	yes	maximal, complemented
4	#14527687	C₂	1	4	yes	maximal, complemented
5	#14527532	C₂	1	1	yes	maximal, characteristic, complemented
6	#14527166	C₂	1	2	yes	maximal, complemented
7	C₂×#11762530	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₁₇×#135	C₂	1	1	yes	maximal, characteristic
9	#11762940	C₂²	1	4	yes	—
10	#11762887	C₂²	1	4	yes	—
11	#11762881	—	2	4	no	—
12	#11762862	—	2	2	no	—
13	#11762862	C₂²	1	2	yes	—
14	C₄×#63223	—	2	2	no	—
15	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
16	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
17	#11762562	C₂²	1	2	yes	—
18	#11762560	C₂²	1	2	yes	—
19	Q₈⋊#3622	C₂²	1	4	yes	—
20	D₄⋊#3622	C₂²	1	4	yes	—
21	#11762530	C₂²	1	2	yes	—
22	#11762530	C₂²	1	2	yes	—
23	C₂×C₁₇×#135	C₄	1	1	yes	characteristic
24	C₄×#41×C₁₇	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₁₇	C₄	1	1	yes	characteristic
26	C₄×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
27	C₄×SD₁₆×C₁₇	C₄	1	2	yes	—
28	C₄×SD₁₆×C₁₇	C₂²	1	2	yes	—
29	C₄×D₈×C₁₇	C₄	1	1	yes	characteristic
30	C₄×D₈×C₁₇	C₂²	1	1	yes	characteristic
31	C₁₇×#427	C₄	1	1	yes	characteristic
32	C₁₇×#416	C₄	1	2	yes	—
33	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
34	C₂×#41×C₁₇	C₂×C₄	1	2	yes	—
35	C₁₇×#135	—	2	4	no	—
36	C₂×Q₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
37	C₂×SD₁₆×C₁₇	C₂×C₄	1	2	yes	—
38	C₂×D₈×C₁₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
41	C₄×Q₈×C₁₇	C₂×C₄	1	2	yes	—
42	C₄×Q₈×C₁₇	C₂³	1	2	yes	—
43	C₄×D₄×C₁₇	—	2	2	no	—
44	C₄×D₄×C₁₇	—	2	2	no	—
45	C₄×D₄×C₁₇	C₂×C₄	1	2	yes	—
46	C₄×D₄×C₁₇	C₂³	1	2	yes	—
47	C₁₇×#117	—	2	2	no	—
48	C₁₇×#117	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₁₇	—	2	2	no	—
50	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
51	C₂×#63223	—	2	4	no	—
52	C₂²×#3622	—	2	1	no	—
53	C₂²×#3622	D₄	1	2	yes	—
54	C₁₇×#107	C₂×C₄	1	1	yes	characteristic
55	C₁₇×#107	C₂×C₄	1	1	yes	characteristic
56	C₁₇×#106	C₂×C₄	1	2	yes	—
57	C₁₇×#103	C₂×C₄	1	2	yes	—
58	C₁₇×#103	C₂×C₄	1	2	yes	—
59	C₁₇×#102	C₂×C₄	1	2	yes	—
60	C₁₇×#102	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
65	C₂²⋊#3622	—	2	4	no	—
66	C₂²⋊#3622	—	2	2	no	—
67	C₄⋊#3622	—	2	4	no	—
68	C₄⋊#3622	C₂³	1	4	yes	—
69	#10587474	—	2	4	no	—
70	C₄×#3622	—	2	2	no	—
71	C₄×#3622	—	2	1	no	—
72	C₄×#3622	D₄	1	2	yes	—
73	C₄×#3622	C₂³	1	2	yes	—
74	C₄×#3622	C₂³	1	2	yes	—
75	#41×C₁₇	—	2	4	no	—
76	#41×C₁₇	—	2	4	no	—
77	C₂×Q₈×C₁₇	C₂²×C₄	1	2	yes	—
78	C₂×D₄×C₁₇	—	2	2	no	—
79	C₂×D₄×C₁₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₁₇	—	2	2	no	—
81	C₂²×C₄×C₁₇	—	2	2	no	—
82	C₂²×C₄×C₁₇	—	2	2	no	—
83	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
86	Q₁₆×C₁₇	—	2	2	no	—
87	SD₁₆×C₁₇	—	2	4	no	—
88	D₈×C₁₇	—	2	2	no	—
89	C₂×C₈×C₁₇	—	2	2	no	—
90	C₂×C₈×C₁₇	M₁₆	1	4	yes	complemented
91	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
95	C₄⋊C₄×C₁₇	—	2	2	no	—
96	C₄⋊C₄×C₁₇	—	2	2	no	—
97	C₄⋊C₄×C₁₇	C₂²×C₄	1	2	yes	—
98	C₄⋊C₄×C₁₇	C₂²×C₄	1	2	yes	—
99	C₂²⋊C₄×C₁₇	—	2	2	no	—
100	C₂²⋊C₄×C₁₇	—	2	2	no	—
101	C₂²⋊C₄×C₁₇	—	2	2	no	—
102	C₄²×C₁₇	—	2	2	no	—
103	C₄²×C₁₇	—	2	2	no	—
104	C₄²×C₁₇	—	2	2	no	—
105	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₇	#41	1	1	yes	characteristic
108	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
109	#63223	—	4	8	no	—
110	C₂×#3622	—	2	4	no	—
111	C₂×#3622	—	2	2	no	—
112	C₂×#3622	—	4	2	no	—
113	C₂×#3622	—	4	2	no	—
114	C₂×#3622	C₂×D₄	1	4	yes	—
115	#17890	—	17	1	no	maximal, Sylow
116	C₂³×C₁₇	—	2	2	no	—
117	C₂³×C₁₇	C₄×D₄	1	1	yes	characteristic
118	Q₈×C₁₇	—	2	2	no	—
119	Q₈×C₁₇	—	2	2	no	—
120	D₄×C₁₇	—	2	2	no	—
121	D₄×C₁₇	—	2	2	no	—
122	D₄×C₁₇	—	4	4	no	—
123	C₂×C₄×C₁₇	—	2	2	no	—
124	C₂×C₄×C₁₇	—	2	2	no	—
125	C₂×C₄×C₁₇	—	2	2	no	—
126	C₂×C₄×C₁₇	—	2	2	no	—
127	C₂×C₄×C₁₇	—	2	2	no	—
128	C₂×C₄×C₁₇	—	4	4	no	—
129	C₂×C₄×C₁₇	—	4	4	no	—
130	C₂×C₄×C₁₇	—	4	4	no	—
131	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
135	C₂×C₄×C₁₇	#131	1	2	yes	—
136	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
139	C₈×C₁₇	—	2	1	no	—
140	C₈×C₁₇	—	2	1	no	—
141	C₈×C₁₇	C₂×M₁₆	1	2	yes	—
142	C₈×C₁₇	C₂×M₁₆	1	2	yes	—
143	#3622	—	2	4	no	—
144	#3622	—	4	2	no	—
145	#3622	—	4	4	no	—
146	#2965	—	17	2	no	—
147	C₄×#135	D₁₇	1	1	yes	characteristic
148	#1594	—	17	2	no	—
149	#1593	—	17	2	no	—
150	#1592	—	17	4	no	—
151	#1564	—	17	1	no	—
152	#1476	—	17	2	no	—
153	C₂×#322	—	17	1	no	—
154	C₂²×C₁₇	—	2	1	no	—
155	C₂²×C₁₇	—	2	1	no	—
156	C₂²×C₁₇	—	4	2	no	—
157	C₂²×C₁₇	—	4	2	no	—
158	C₂²×C₁₇	—	4	2	no	—
159	C₂²×C₁₇	#436	1	2	yes	—
160	C₂²×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
161	C₄×C₁₇	—	2	1	no	—
162	C₄×C₁₇	—	2	1	no	—
163	C₄×C₁₇	—	4	2	no	—
164	C₄×C₁₇	—	4	2	no	—
165	C₄×C₁₇	—	4	2	no	—
166	C₄×C₁₇	#436	1	2	yes	—
167	C₄×C₁₇	#444	1	2	yes	—
168	C₄×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
169	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
170	C₄×C₁₇	#575	1	2	yes	—
171	C₂×#135	Dic₁₇	1	1	yes	characteristic
172	C₄×#41	C₂×D₁₇	1	2	yes	—
173	#446	—	17	4	no	—
174	C₄×Q₁₆	Dic₁₇	1	1	yes	characteristic
175	C₄×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
176	C₄×SD₁₆	Dic₁₇	1	2	yes	—
177	C₄×SD₁₆	C₂×D₁₇	1	2	yes	—
178	C₄×D₈	Dic₁₇	1	1	yes	characteristic
179	C₄×D₈	C₂×D₁₇	1	1	yes	characteristic
180	#436	—	17	4	no	—
181	#435	—	34	4	no	—
182	#431	—	17	2	no	—
183	#431	—	34	2	no	—
184	#427	Dic₁₇	1	1	yes	characteristic
185	#416	Dic₁₇	1	2	yes	—
186	C₄×M₁₆	—	34	2	no	—
187	C₂×C₄×C₈	—	17	1	no	—
188	C₂×C₄×C₈	—	17	1	no	—
189	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
190	#335	—	17	2	no	—
191	#334	—	17	2	no	—
192	Q₈⋊C₈	—	17	4	no	—
193	D₄⋊C₈	—	17	4	no	—
194	#322	—	17	2	no	—
195	#322	—	17	2	no	—
196	C₂×C₁₇	—	2	2	no	—
197	C₂×C₁₇	—	8	2	no	—
198	C₂×C₁₇	C₂×#436	1	1	yes	characteristic
199	C₂×C₁₇	#2943	1	1	yes	characteristic
200	C₂×C₁₇	C₂×#444	1	1	yes	characteristic
201	C₂×#41	C₂×Dic₁₇	1	2	yes	—
202	#135	—	2	4	no	—
203	C₂×Q₁₆	C₂×Dic₁₇	1	1	yes	characteristic
204	C₂×SD₁₆	C₂×Dic₁₇	1	2	yes	—
205	C₂×D₈	C₂×Dic₁₇	1	1	yes	characteristic
206	C₂×M₁₆	—	34	4	no	—
207	C₂²×C₈	—	17	2	no	—
208	C₂²×C₈	—	34	1	no	—
209	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
210	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
211	C₄×Q₈	C₂×Dic₁₇	1	2	yes	—
212	C₄×Q₈	C₂²×D₁₇	1	2	yes	—
213	C₄×D₄	—	2	2	no	—
214	C₄×D₄	—	2	2	no	—
215	C₄×D₄	C₂×Dic₁₇	1	2	yes	—
216	C₄×D₄	C₂²×D₁₇	1	2	yes	—
217	#117	—	2	2	no	—
218	#117	C₂×Dic₁₇	1	2	yes	—
219	C₂×C₄²	—	2	2	no	—
220	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
221	#107	C₂×Dic₁₇	1	1	yes	characteristic
222	#107	C₂×Dic₁₇	1	1	yes	characteristic
223	#106	C₂×Dic₁₇	1	2	yes	—
224	C₄⋊C₈	—	17	4	no	—
225	C₄⋊C₈	—	34	4	no	—
226	#103	C₂×Dic₁₇	1	2	yes	—
227	#103	C₂×Dic₁₇	1	2	yes	—
228	#102	C₂×Dic₁₇	1	2	yes	—
229	#102	C₂×Dic₁₇	1	2	yes	—
230	C₂²⋊C₈	—	34	4	no	—
231	C₂²⋊C₈	—	34	2	no	—
232	#97	—	34	4	no	—
233	C₄×C₈	—	17	2	no	—
234	C₄×C₈	—	17	2	no	—
235	C₄×C₈	—	17	2	no	—
236	C₄×C₈	—	34	2	no	—
237	C₄×C₈	—	34	1	no	—
238	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
239	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
242	C₁₇	#17890	1	1	yes	characteristic, complemented, Sylow
243	#41	—	2	4	no	—
244	#41	—	2	4	no	—
245	C₂×Q₈	C₂²×Dic₁₇	1	2	yes	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	C₂²×Dic₁₇	1	2	yes	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
252	C₂²×C₄	#63254	1	1	yes	characteristic
253	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
254	Q₁₆	—	2	2	no	—
255	SD₁₆	—	2	4	no	—
256	D₈	—	2	2	no	—
257	M₁₆	—	68	8	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	17	4	no	—
260	C₂×C₈	—	34	4	no	—
261	C₂×C₈	—	34	2	no	—
262	C₂×C₈	—	68	2	no	—
263	C₂×C₈	—	68	2	no	—
264	C₂×C₈	#63223	1	4	yes	complemented
265	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
266	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
267	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
268	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	C₂²×Dic₁₇	1	2	yes	—
272	C₄⋊C₄	C₂²×Dic₁₇	1	2	yes	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	2	2	no	—
275	C₂²⋊C₄	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	D₄×D₁₇	1	1	yes	characteristic
280	C₄²	#63254	1	1	yes	characteristic
281	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
282	C₄²	C₂³×D₁₇	1	1	yes	characteristic
283	C₂³	—	2	2	no	—
284	C₂³	D₄×Dic₁₇	1	1	yes	characteristic
285	Q₈	—	2	2	no	—
286	Q₈	—	2	2	no	—
287	D₄	—	2	2	no	—
288	D₄	—	2	2	no	—
289	D₄	—	4	4	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	C₂×#63223	1	2	yes	—
299	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
300	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
301	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
302	C₂×C₄	#10587634	1	2	yes	—
303	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
304	C₂×C₄	C₂×#63254	1	1	yes	characteristic
305	C₂×C₄	C₂³×Dic₁₇	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	34	4	no	—
309	C₈	—	68	2	no	—
310	C₈	—	68	4	no	—
311	C₈	C₂×#63223	1	2	yes	—
312	C₈	C₂×#63223	1	2	yes	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	#11762887	1	2	yes	—
319	C₂²	C₂×D₄×Dic₁₇	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	#11762887	1	2	yes	—
326	C₄	#11762931	1	2	yes	—
327	C₄	C₂²×#63223	1	1	yes	characteristic
328	C₄	C₂×#10587634	1	1	yes	characteristic
329	C₄	#11763606	1	2	yes	—
330	C₂	—	2	2	no	—
331	C₂	—	8	2	no	—
332	C₂	C₂×#11762887	1	1	yes	characteristic
333	C₂	#14537585	1	1	yes	characteristic
334	C₂	C₂×#11762931	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₄²	C₂³×D₁₇
Fitting	C₄×C₁₇×#135	C₂
socle	C₂²×C₁₇	C₂×C₄×D₄
prime radical	C₂×C₁₇×#135	C₄
cosocle	C₄²×C₁₇	C₂⁴
hypercenter	C₄×#135	D₁₇
hypocenter	C₁₇	#17890
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #14537751	C₂	semidirect
2. #14527698	C₂	semidirect
3. #14527693	C₂	semidirect
4. #14527687	C₂	semidirect
5. #14527532	C₂	semidirect
6. #14527166	C₂	semidirect
7. C₂×#11762530	C₂	semidirect
90. C₂×C₈×C₁₇	M₁₆	semidirect
242. C₁₇	#17890	semidirect
264. C₂×C₈	#63223	semidirect

Maximal subgroups

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

subgroup	index
#14537751	2
#14527698	2
#14527693	2
#14527687	2
#14527532	2
#14527166	2
C₂×#11762530	2
C₄×C₁₇×#135	2
#17890	17