Subgroups — #16927640

Order 4352.

384 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14537759	C₂	1	1	yes	maximal, characteristic
2	C₄×#10587489	C₂	1	1	yes	maximal, characteristic, complemented
3	#14527310	C₂	1	2	yes	maximal
4	#14526639	C₂	1	2	yes	maximal, complemented
5	C₁₇×#1568	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763343	C₂²	1	1	yes	characteristic
7	#11762939	C₄	1	1	yes	characteristic, complemented
8	C₈×Dic₃₄	C₄	1	1	yes	characteristic, complemented
9	#11762888	C₄	1	1	yes	characteristic, complemented
10	#11762880	—	2	1	no	—
11	C₈×D₆₈	C₄	1	1	yes	characteristic, complemented
12	C₈×C₂²⋊D₁₇	—	2	1	no	—
13	#11762868	—	2	1	no	—
14	#11762866	C₂²	1	1	yes	characteristic
15	C₂×#10587489	—	2	1	no	—
16	#11762763	—	2	1	no	—
17	C₄×C₄⋊Dic₁₇	C₂²	1	2	yes	—
18	#11762572	—	2	1	no	—
19	#11762547	—	2	2	no	—
20	#11762544	—	2	2	no	—
21	C₁₇×#432	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
23	C₁₇×#335	—	2	1	no	—
24	C₁₇×#334	—	2	1	no	—
25	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
28	#10587616	D₄	1	1	yes	characteristic
29	#10587615	—	2	1	no	—
30	C₄×C₂²⋊D₁₇	—	2	1	no	—
31	C₂×C₄⋊Dic₁₇	—	2	2	no	—
32	#10587610	D₄	1	1	yes	characteristic
33	C₂×C₄×Dic₁₇	—	2	2	no	—
34	C₂×#63218	—	2	1	no	—
35	C₂×C₈×D₁₇	—	2	1	no	—
36	#10587562	—	2	1	no	—
37	#10587561	Q₈	1	1	yes	characteristic
38	C₄×D₆₈	C₂×C₄	1	1	yes	characteristic
39	#10587558	—	2	1	no	—
40	#10587556	Q₈	1	1	yes	characteristic
41	C₄×Dic₃₄	C₂×C₄	1	1	yes	characteristic
42	C₁₇×C₄⋊C₈	—	2	2	no	—
43	C₁₇×C₂²⋊C₈	—	2	2	no	—
44	C₁₇×#97	—	2	2	no	—
45	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
49	#10587502	—	2	2	no	—
50	C₂²⋊#3622	—	2	1	no	—
51	#10587491	—	2	1	no	—
52	#10587491	—	2	1	no	—
53	#10587489	—	2	4	no	—
54	#10587486	—	2	1	no	—
55	#10587485	—	2	1	no	—
56	#10587485	—	2	1	no	—
57	C₈×Dic₁₇	—	2	1	no	—
58	C₄⋊#3622	C₂×C₄	1	1	yes	characteristic
59	C₄⋊#3622	C₂×C₄	1	1	yes	characteristic
60	#10587474	—	2	1	no	—
61	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₇	—	2	1	no	—
65	C₂²×Dic₁₇	—	4	2	no	—
66	C₂×D₆₈	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×D₁₇	—	2	1	no	—
68	C₂×Dic₃₄	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₇	—	2	2	no	—
70	C₂×C₈×C₁₇	—	4	1	no	—
71	C₂×C₈×C₁₇	—	4	2	no	—
72	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
79	C₄²×C₁₇	#41	1	1	yes	characteristic
80	C₂²⋊Dic₁₇	—	2	1	no	—
81	#63227	—	2	1	no	—
82	#63227	—	2	1	no	—
83	#63227	—	2	1	no	—
84	#63227	—	2	1	no	—
85	C₄⋊Dic₁₇	—	2	4	no	—
86	C₄⋊Dic₁₇	—	4	2	no	—
87	C₄⋊Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
88	C₄⋊Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
89	#63225	—	2	1	no	—
90	#63225	—	2	1	no	—
91	#63225	—	2	1	no	—
92	#63225	—	2	1	no	—
93	C₄×Dic₁₇	—	2	1	no	—
94	C₄×Dic₁₇	—	8	2	no	—
95	C₂×#3622	—	2	1	no	—
96	C₂×#3622	—	2	1	no	—
97	#63218	—	4	2	no	—
98	C₈×D₁₇	—	4	2	no	—
99	#8697	—	17	1	no	maximal, Sylow
100	C₂³×C₁₇	#121	1	1	yes	characteristic
101	C₂²×D₁₇	—	2	1	no	—
102	C₂×C₄×C₁₇	—	2	1	no	—
103	C₂×C₄×C₁₇	—	2	1	no	—
104	C₂×C₄×C₁₇	—	2	1	no	—
105	C₂×C₄×C₁₇	—	2	1	no	—
106	C₂×C₄×C₁₇	—	4	1	no	—
107	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
110	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
111	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂²⋊D₁₇	—	4	1	no	—
114	C₂²⋊D₁₇	—	4	1	no	—
115	C₂×Dic₁₇	—	2	1	no	—
116	C₂×Dic₁₇	—	2	1	no	—
117	C₂×Dic₁₇	—	2	1	no	—
118	C₂×Dic₁₇	—	4	4	no	—
119	C₂×Dic₁₇	—	8	2	no	—
120	C₂×Dic₁₇	—	8	2	no	—
121	D₆₈	—	4	1	no	—
122	C₄×D₁₇	—	4	1	no	—
123	C₄×D₁₇	—	4	1	no	—
124	Dic₃₄	—	4	1	no	—
125	C₈×C₁₇	—	2	1	no	—
126	C₈×C₁₇	—	2	1	no	—
127	C₈×C₁₇	—	2	1	no	—
128	C₈×C₁₇	—	8	2	no	—
129	C₈×C₁₇	#104	1	2	yes	—
130	#3622	—	4	1	no	—
131	#3622	—	4	1	no	—
132	#2966	—	17	1	no	—
133	C₄×#107	—	17	1	no	—
134	#1568	D₁₇	1	1	yes	characteristic, complemented
135	#1509	—	17	2	no	—
136	#1285	—	17	2	no	—
137	C₂²×C₁₇	—	2	1	no	—
138	C₂²×C₁₇	—	2	1	no	—
139	C₂²×C₁₇	#380	1	1	yes	characteristic
140	C₂²×C₁₇	#472	1	1	yes	characteristic
141	C₂²×C₁₇	#473	1	1	yes	characteristic
142	C₂×D₁₇	—	4	1	no	—
143	C₂×D₁₇	—	4	1	no	—
144	C₂×D₁₇	—	4	1	no	—
145	C₄×C₁₇	—	2	1	no	—
146	C₄×C₁₇	—	2	1	no	—
147	C₄×C₁₇	—	2	1	no	—
148	C₄×C₁₇	—	2	1	no	—
149	C₄×C₁₇	—	2	1	no	—
150	C₄×C₁₇	C₂×#104	1	1	yes	characteristic
151	C₄×C₁₇	#421	1	1	yes	characteristic
152	Dic₁₇	—	4	1	no	—
153	Dic₁₇	—	4	1	no	—
154	Dic₁₇	—	4	1	no	—
155	Dic₁₇	—	8	4	no	—
156	#529	—	17	1	no	—
157	#446	—	17	1	no	—
158	C₈×Q₈	—	17	1	no	—
159	#436	—	17	1	no	—
160	#435	—	34	1	no	—
161	C₈×D₄	—	17	1	no	—
162	C₈×D₄	—	34	1	no	—
163	#433	—	34	1	no	—
164	#432	—	17	1	no	—
165	#432	C₂×D₁₇	1	2	yes	—
166	C₂×#107	—	34	1	no	—
167	#405	—	34	1	no	—
168	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
169	C₄×C₄⋊C₄	—	17	2	no	—
170	#340	—	34	1	no	—
171	#335	—	2	1	no	—
172	#334	—	2	1	no	—
173	#328	—	34	2	no	—
174	#327	—	34	2	no	—
175	C₂×C₁₇	—	2	1	no	—
176	C₂×C₁₇	—	2	1	no	—
177	C₂×C₁₇	#1806	1	1	yes	characteristic
178	C₂×C₁₇	#1812	1	1	yes	characteristic
179	C₂×C₁₇	#1816	1	1	yes	characteristic
180	D₁₇	—	8	1	no	—
181	C₂×M₁₆	—	34	1	no	—
182	C₂²×C₈	—	34	1	no	—
183	C₂²×C₈	D₆₈	1	1	yes	characteristic
184	C₂²×C₈	D₆₈	1	1	yes	characteristic
185	#126	—	34	1	no	—
186	#125	—	17	1	no	—
187	#124	—	17	1	no	—
188	#123	—	34	1	no	—
189	C₂²⋊Q₈	—	17	1	no	—
190	#121	—	17	1	no	—
191	C₄×Q₈	—	17	1	no	—
192	C₄×D₄	—	17	1	no	—
193	C₄×D₄	—	34	1	no	—
194	#117	—	34	1	no	—
195	C₂×C₄⋊C₄	—	34	2	no	—
196	C₂×C₄²	—	34	2	no	—
197	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
198	#107	—	34	4	no	—
199	C₄⋊C₈	—	2	2	no	—
200	C₄⋊C₈	—	17	1	no	—
201	C₄⋊C₈	—	17	1	no	—
202	C₄⋊C₈	—	34	1	no	—
203	C₄⋊C₈	—	34	1	no	—
204	C₂²⋊C₈	—	2	2	no	—
205	C₂²⋊C₈	—	34	1	no	—
206	C₂²⋊C₈	—	34	1	no	—
207	C₂²⋊C₈	—	34	1	no	—
208	#97	—	2	2	no	—
209	#97	—	34	1	no	—
210	#97	—	34	1	no	—
211	C₄×C₈	—	34	1	no	—
212	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
213	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
214	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
215	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
216	#95	—	34	2	no	—
217	C₁₇	#8697	1	1	yes	characteristic, complemented, Sylow
218	C₂×Q₈	—	17	1	no	—
219	C₂×D₄	—	17	1	no	—
220	C₂×D₄	—	34	1	no	—
221	C₂²×C₄	—	34	1	no	—
222	C₂²×C₄	—	68	2	no	—
223	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
224	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
225	C₂²×C₄	#63256	1	1	yes	characteristic
226	M₁₆	—	68	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	34	1	no	—
231	C₂×C₈	—	34	1	no	—
232	C₂×C₈	—	68	2	no	—
233	C₂×C₈	#63227	1	1	yes	characteristic
234	C₂×C₈	#63227	1	1	yes	characteristic
235	C₂×C₈	#63227	1	1	yes	characteristic
236	C₂×C₈	#63227	1	1	yes	characteristic
237	C₄⋊C₄	—	17	1	no	—
238	C₄⋊C₄	—	17	1	no	—
239	C₄⋊C₄	—	34	4	no	—
240	C₄⋊C₄	—	34	1	no	—
241	C₄⋊C₄	—	34	1	no	—
242	C₄⋊C₄	—	34	1	no	—
243	C₄⋊C₄	—	34	1	no	—
244	C₄⋊C₄	—	68	2	no	—
245	C₂²⋊C₄	—	34	1	no	—
246	C₂²⋊C₄	—	34	1	no	—
247	C₂²⋊C₄	—	34	1	no	—
248	C₂²⋊C₄	—	34	1	no	—
249	C₂²⋊C₄	—	34	1	no	—
250	C₄²	—	34	1	no	—
251	C₄²	—	136	2	no	—
252	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
253	C₄²	#63254	1	1	yes	characteristic
254	C₄²	Q₈×D₁₇	1	1	yes	characteristic
255	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
256	C₂³	—	34	1	no	—
257	C₂³	#10587581	1	1	yes	characteristic
258	Q₈	—	68	1	no	—
259	D₄	—	68	1	no	—
260	D₄	—	68	1	no	—
261	D₄	—	68	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	34	1	no	—
268	C₂×C₄	—	34	1	no	—
269	C₂×C₄	—	34	1	no	—
270	C₂×C₄	—	68	1	no	—
271	C₂×C₄	—	68	1	no	—
272	C₂×C₄	—	68	4	no	—
273	C₂×C₄	—	136	2	no	—
274	C₂×C₄	—	136	2	no	—
275	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
276	C₂×C₄	#10587578	1	1	yes	characteristic
277	C₂×C₄	#10587583	1	1	yes	characteristic
278	C₂×C₄	C₂×#63227	1	1	yes	characteristic
279	C₂×C₄	#10587624	1	1	yes	characteristic
280	C₂×C₄	#10587632	1	1	yes	characteristic
281	C₈	—	2	1	no	—
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	8	2	no	—
285	C₈	—	68	1	no	—
286	C₈	—	68	1	no	—
287	C₈	#10587496	1	2	yes	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	68	1	no	—
291	C₂²	—	68	1	no	—
292	C₂²	—	68	1	no	—
293	C₂²	#11762666	1	1	yes	characteristic
294	C₂²	#11763080	1	1	yes	characteristic
295	C₂²	#11763085	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	68	1	no	—
302	C₄	—	68	1	no	—
303	C₄	—	68	1	no	—
304	C₄	—	136	4	no	—
305	C₄	C₂×#10587496	1	1	yes	characteristic
306	C₄	#11762825	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	136	1	no	—
310	C₂	#14528801	1	1	yes	characteristic
311	C₂	#14528827	1	1	yes	characteristic
312	C₂	#14528845	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11762666
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1568	C₂
socle	C₂²×C₁₇	#380
prime radical	#10587616	D₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1568	D₁₇
hypocenter	C₁₇	#8697
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
2. C₄×#10587489	C₂	semidirect
4. #14526639	C₂	semidirect
5. C₁₇×#1568	C₂	semidirect
7. #11762939	C₄	semidirect
8. C₈×Dic₃₄	C₄	semidirect
9. #11762888	C₄	semidirect
11. C₈×D₆₈	C₄	semidirect
134. #1568	D₁₇	semidirect
217. C₁₇	#8697	semidirect

Maximal subgroups

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

subgroup	index
#14537759	2
C₄×#10587489	2
#14527310	2
#14526639	2
C₁₇×#1568	2
#8697	17