Subgroups — #17005756

Order 4352.

656 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14537799	C₂	1	2	yes	maximal, complemented
2	C₄×#10587590	C₂	1	1	yes	maximal, characteristic
3	#14527682	C₂	1	2	yes	maximal, complemented
4	#14527622	C₂	1	4	yes	maximal, complemented
5	C₈×D₁₃₆	C₂	1	2	yes	maximal, complemented
6	#14527547	C₂	1	1	yes	maximal, characteristic, complemented
7	#14527219	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10587590	C₄	1	1	yes	characteristic
10	C₄×#63252	C₂²	1	2	yes	—
11	#11762939	C₂²	1	4	yes	—
12	C₄×Dic₆₈	C₄	1	1	yes	characteristic
13	C₄×Dic₆₈	C₂²	1	1	yes	characteristic
14	C₄×#63219	C₄	1	2	yes	—
15	C₄×#63219	C₂²	1	2	yes	—
16	C₄×D₁₃₆	C₄	1	1	yes	characteristic
17	C₄×D₁₃₆	C₂²	1	1	yes	characteristic
18	#11762878	—	2	4	no	—
19	C₈×D₆₈	C₂²	1	4	yes	—
20	#11762866	C₂²	1	2	yes	—
21	#11762864	—	2	2	no	—
22	#11762847	C₄	1	1	yes	characteristic
23	#11762806	C₄	1	2	yes	—
24	#97×D₁₇	—	2	2	no	—
25	#11762562	C₂²	1	2	yes	—
26	#11762560	C₂²	1	2	yes	—
27	#11762541	C₂²	1	4	yes	—
28	#11762538	C₂²	1	4	yes	—
29	C₁₇×#431	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
32	C₁₇×#322	C₂²	1	2	yes	—
33	C₈²×C₁₇	C₂²	1	2	yes	—
34	C₂×#63252	C₂×C₄	1	2	yes	—
35	C₂×M₁₆×C₁₇	D₄	1	2	yes	—
36	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
39	C₄×C₂²⋊D₁₇	—	2	2	no	—
40	C₄×C₂²⋊D₁₇	—	2	2	no	—
41	#10587612	C₂×C₄	1	2	yes	—
42	C₂×Dic₆₈	C₂×C₄	1	1	yes	characteristic
43	#10587590	—	2	4	no	—
44	C₂×D₁₃₆	C₂×C₄	1	1	yes	characteristic
45	C₂×#63219	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₁₇	—	2	4	no	—
47	C₄×D₆₈	C₂×C₄	1	2	yes	—
48	C₄×D₆₈	C₂³	1	2	yes	—
49	#10587558	—	2	2	no	—
50	C₄²×D₁₇	—	2	2	no	—
51	C₄×Dic₃₄	C₂×C₄	1	2	yes	—
52	C₄×Dic₃₄	C₂³	1	2	yes	—
53	C₁₇×C₄⋊C₈	—	2	1	no	—
54	C₁₇×C₂²⋊C₈	—	2	1	no	—
55	C₁₇×#97	D₄	1	1	yes	characteristic
56	C₁₇×#97	D₄	1	1	yes	characteristic
57	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₁₇	C₂³	1	2	yes	—
60	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₁₇	C₂³	1	2	yes	—
63	C₂²⋊#3622	—	2	2	no	—
64	#10587492	C₂×C₄	1	2	yes	—
65	#10587492	C₂×C₄	1	2	yes	—
66	#10587491	—	2	4	no	—
67	#10587489	C₂×C₄	1	1	yes	characteristic
68	#10587489	C₂×C₄	1	1	yes	characteristic
69	#10587488	C₂×C₄	1	2	yes	—
70	#10587487	C₂×C₄	1	2	yes	—
71	#10587487	C₂×C₄	1	2	yes	—
72	#10587486	—	2	4	no	—
73	#10587485	—	2	4	no	—
74	C₄⋊#3622	C₂³	1	4	yes	—
75	#10587474	—	2	2	no	—
76	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
79	C₂×C₂²⋊D₁₇	—	2	2	no	—
80	#63252	—	2	4	no	—
81	#63252	—	4	2	no	—
82	C₂×D₆₈	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₁₇	—	2	2	no	—
84	C₂×C₄×D₁₇	—	2	2	no	—
85	C₂×C₄×D₁₇	—	2	2	no	—
86	C₂×Dic₃₄	C₂²×C₄	1	2	yes	—
87	M₁₆×C₁₇	—	2	4	no	—
88	C₂×C₈×C₁₇	—	2	2	no	—
89	C₂×C₈×C₁₇	—	2	2	no	—
90	C₂×C₈×C₁₇	—	4	1	no	—
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	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
97	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₁₇	#41	1	1	yes	characteristic
100	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
101	C₂²⋊Dic₁₇	—	2	2	no	—
102	#63227	—	2	2	no	—
103	#63227	—	2	2	no	—
104	C₄⋊Dic₁₇	C₂²×C₄	1	2	yes	—
105	C₄⋊Dic₁₇	C₂²×C₄	1	2	yes	—
106	#63225	—	2	2	no	—
107	#63225	—	2	2	no	—
108	C₄×Dic₁₇	—	2	2	no	—
109	C₄×Dic₁₇	—	2	2	no	—
110	C₄×Dic₁₇	—	2	2	no	—
111	C₂×#3622	—	2	4	no	—
112	Dic₆₈	—	2	2	no	—
113	D₁₃₆	M₁₆	1	4	yes	complemented
114	#63219	—	2	4	no	—
115	C₈×D₁₇	—	4	8	no	—
116	#18035	—	17	1	no	maximal, Sylow
117	C₂³×C₁₇	C₄×D₄	1	1	yes	characteristic
118	C₂²×D₁₇	—	2	2	no	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	—	2	1	no	—
121	C₂×C₄×C₁₇	—	2	2	no	—
122	C₂×C₄×C₁₇	—	4	1	no	—
123	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
129	C₂²⋊D₁₇	—	4	4	no	—
130	C₂×Dic₁₇	—	2	2	no	—
131	C₂×Dic₁₇	—	2	2	no	—
132	C₂×Dic₁₇	—	2	2	no	—
133	D₆₈	—	2	2	no	—
134	D₆₈	C₂×M₁₆	1	4	yes	—
135	C₄×D₁₇	—	4	4	no	—
136	C₄×D₁₇	—	4	4	no	—
137	C₄×D₁₇	—	8	2	no	—
138	Dic₃₄	—	2	2	no	—
139	Dic₃₄	—	2	2	no	—
140	C₈×C₁₇	—	2	1	no	—
141	C₈×C₁₇	—	2	1	no	—
142	C₈×C₁₇	—	2	1	no	—
143	C₈×C₁₇	—	2	2	no	—
144	C₈×C₁₇	—	4	2	no	—
145	C₈×C₁₇	C₂×M₁₆	1	2	yes	—
146	C₈×C₁₇	#135	1	4	yes	complemented
147	#3622	—	4	4	no	—
148	#2970	—	17	2	no	—
149	C₄×#135	—	17	1	no	—
150	#1591	—	17	2	no	—
151	#1582	—	17	4	no	—
152	C₈×D₈	—	17	2	no	—
153	#1568	—	17	1	no	—
154	#1488	—	17	2	no	—
155	C₈×M₁₆	D₁₇	1	1	yes	characteristic, complemented
156	C₂²×C₁₇	—	2	1	no	—
157	C₂²×C₁₇	—	2	1	no	—
158	C₂²×C₁₇	—	2	1	no	—
159	C₂²×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
160	C₂×D₁₇	—	2	4	no	—
161	C₂×D₁₇	—	4	2	no	—
162	C₂×D₁₇	—	4	2	no	—
163	C₄×C₁₇	—	2	1	no	—
164	C₄×C₁₇	—	2	1	no	—
165	C₄×C₁₇	—	2	1	no	—
166	C₄×C₁₇	—	2	1	no	—
167	C₄×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
168	C₄×C₁₇	#568	1	1	yes	characteristic
169	C₄×C₁₇	C₂×#135	1	2	yes	—
170	Dic₁₇	—	4	2	no	—
171	Dic₁₇	—	4	2	no	—
172	Dic₁₇	—	4	2	no	—
173	C₂×#135	—	17	1	no	—
174	C₄×#41	—	17	2	no	—
175	#446	—	17	4	no	—
176	C₄×Q₁₆	—	17	1	no	—
177	C₄×Q₁₆	—	17	1	no	—
178	C₄×SD₁₆	—	17	2	no	—
179	C₄×SD₁₆	—	17	2	no	—
180	C₄×D₈	—	17	1	no	—
181	C₄×D₈	—	17	1	no	—
182	#435	—	34	4	no	—
183	C₈×D₄	—	17	4	no	—
184	#432	—	17	2	no	—
185	#432	—	34	2	no	—
186	#431	C₂×D₁₇	1	1	yes	characteristic
187	#427	—	17	1	no	—
188	#416	—	17	2	no	—
189	C₄×M₁₆	C₂×D₁₇	1	1	yes	characteristic
190	C₂×#97	—	34	2	no	—
191	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
192	#335	—	17	2	no	—
193	#334	—	17	2	no	—
194	Q₈⋊C₈	—	17	4	no	—
195	D₄⋊C₈	—	17	4	no	—
196	#322	C₂×D₁₇	1	2	yes	—
197	C₈²	C₂×D₁₇	1	2	yes	—
198	C₂×C₁₇	—	4	1	no	—
199	C₂×C₁₇	D₄×M₁₆	1	1	yes	characteristic
200	C₂×C₁₇	C₄×#135	1	1	yes	characteristic
201	C₂×C₁₇	#2957	1	1	yes	characteristic
202	D₁₇	—	4	4	no	—
203	C₂×#41	—	17	2	no	—
204	#135	—	34	4	no	—
205	C₂×Q₁₆	—	17	1	no	—
206	C₂×SD₁₆	—	17	2	no	—
207	C₂×D₈	—	17	1	no	—
208	C₂×M₁₆	D₆₈	1	2	yes	—
209	C₂²×C₈	—	34	4	no	—
210	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
211	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
212	C₄×Q₈	—	17	2	no	—
213	C₄×Q₈	—	17	2	no	—
214	C₄×D₄	—	17	2	no	—
215	C₄×D₄	—	17	2	no	—
216	C₄×D₄	—	34	2	no	—
217	C₄×D₄	—	34	2	no	—
218	#117	—	17	2	no	—
219	#117	—	34	2	no	—
220	C₂×C₄²	—	34	2	no	—
221	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
222	#107	—	17	1	no	—
223	#107	—	17	1	no	—
224	#106	—	17	2	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	17	4	no	—
227	C₄⋊C₈	—	34	4	no	—
228	#103	—	17	2	no	—
229	#103	—	17	2	no	—
230	#102	—	17	2	no	—
231	#102	—	17	2	no	—
232	C₂²⋊C₈	—	2	1	no	—
233	C₂²⋊C₈	—	34	2	no	—
234	C₂²⋊C₈	—	34	4	no	—
235	#97	—	34	4	no	—
236	#97	—	34	2	no	—
237	#97	D₆₈	1	1	yes	characteristic
238	#97	D₆₈	1	1	yes	characteristic
239	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
240	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₁₇	1	2	yes	—
242	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₁₇	1	2	yes	—
245	C₁₇	#18035	1	1	yes	characteristic, complemented, Sylow
246	#41	—	34	4	no	—
247	#41	—	68	2	no	—
248	C₂×Q₈	—	17	2	no	—
249	C₂×D₄	—	17	2	no	—
250	C₂×D₄	—	34	2	no	—
251	C₂²×C₄	—	34	2	no	—
252	C₂²×C₄	—	34	2	no	—
253	C₂²×C₄	—	34	2	no	—
254	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
256	C₂²×C₄	#63252	1	1	yes	characteristic
257	Q₁₆	—	34	2	no	—
258	SD₁₆	—	34	4	no	—
259	D₈	—	17	4	no	—
260	M₁₆	—	2	4	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	34	4	no	—
265	C₂×C₈	—	68	8	no	—
266	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
270	C₂×C₈	C₂×D₆₈	1	2	yes	—
271	C₂×C₈	C₂×D₆₈	1	2	yes	—
272	C₄⋊C₄	—	17	2	no	—
273	C₄⋊C₄	—	17	2	no	—
274	C₄⋊C₄	—	34	2	no	—
275	C₄⋊C₄	—	34	2	no	—
276	C₂²⋊C₄	—	34	2	no	—
277	C₂²⋊C₄	—	34	2	no	—
278	C₂²⋊C₄	—	34	2	no	—
279	C₄²	—	34	2	no	—
280	C₄²	—	34	2	no	—
281	C₄²	—	34	2	no	—
282	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
283	C₄²	C₂×D₆₈	1	1	yes	characteristic
284	C₄²	#63252	1	1	yes	characteristic
285	C₄²	C₂³×D₁₇	1	1	yes	characteristic
286	C₂³	—	34	2	no	—
287	C₂³	C₄×D₆₈	1	1	yes	characteristic
288	Q₈	—	34	2	no	—
289	Q₈	—	34	2	no	—
290	D₄	—	17	4	no	—
291	D₄	—	34	2	no	—
292	D₄	—	68	4	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	34	2	no	—
298	C₂×C₄	—	34	2	no	—
299	C₂×C₄	—	34	2	no	—
300	C₂×C₄	—	68	4	no	—
301	C₂×C₄	—	68	4	no	—
302	C₂×C₄	—	136	2	no	—
303	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
304	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
305	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
308	C₂×C₄	C₂×#63252	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	2	2	no	—
313	C₈	—	4	2	no	—
314	C₈	—	68	4	no	—
315	C₈	#10587590	1	4	yes	complemented
316	C₈	M₁₆×D₁₇	1	2	yes	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	34	4	no	—
321	C₂²	—	68	2	no	—
322	C₂²	—	68	2	no	—
323	C₂²	C₂×C₄×D₆₈	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	68	2	no	—
329	C₄	—	68	2	no	—
330	C₄	—	68	2	no	—
331	C₄	C₂×M₁₆×D₁₇	1	1	yes	characteristic
332	C₄	#11763575	1	1	yes	characteristic
333	C₄	C₂×#10587590	1	2	yes	—
334	C₂	—	4	1	no	—
335	C₂	—	68	4	no	—
336	C₂	M₁₆×D₆₈	1	1	yes	characteristic
337	C₂	C₄×#10587590	1	1	yes	characteristic
338	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	C₈×M₁₆×C₁₇	C₂
socle	C₂²×C₁₇	C₂×C₄×D₄
prime radical	C₂×#10587590	C₄
cosocle	C₄²×C₁₇	C₂⁴
hypercenter	C₈×M₁₆	D₁₇
hypocenter	C₁₇	#18035
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
1. #14537799	C₂	semidirect
3. #14527682	C₂	semidirect
4. #14527622	C₂	semidirect
5. C₈×D₁₃₆	C₂	semidirect
6. #14527547	C₂	semidirect
7. #14527219	C₂	semidirect
8. C₈×M₁₆×C₁₇	C₂	semidirect
113. D₁₃₆	M₁₆	semidirect
146. C₈×C₁₇	#135	semidirect
155. C₈×M₁₆	D₁₇	semidirect
245. C₁₇	#18035	semidirect
315. C₈	#10587590	semidirect

Maximal subgroups

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

subgroup	index
#14537799	2
C₄×#10587590	2
#14527682	2
#14527622	2
C₈×D₁₃₆	2
#14527547	2
#14527219	2
C₈×M₁₆×C₁₇	2
#18035	17