Subgroups — #16950281

Order 4352.

440 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14537206	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762629	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11762629	C₂	1	1	yes	maximal, characteristic, complemented
4	#14530413	C₂	1	1	yes	maximal, characteristic
5	#14530411	C₂	1	1	yes	maximal, characteristic
6	#14526870	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#1378	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10587593	C₄	1	1	yes	characteristic
9	C₂×#10587587	C₂²	1	1	yes	characteristic
10	#11763277	C₄	1	1	yes	characteristic
11	C₂×#10587483	C₂²	1	1	yes	characteristic
12	C₂×#10587483	C₂²	1	1	yes	characteristic
13	#11762766	—	2	1	no	—
14	#11762765	C₄	1	1	yes	characteristic
15	#11762763	—	2	1	no	—
16	#11762761	—	2	1	no	—
17	M₁₆×Dic₁₇	C₄	1	1	yes	characteristic
18	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
19	C₄×C₈×D₁₇	—	2	1	no	—
20	#11762629	—	2	1	no	—
21	#11762629	—	2	1	no	—
22	#11762629	—	2	1	no	—
23	#11762629	—	2	1	no	—
24	#11762594	—	2	2	no	—
25	#11762592	—	2	2	no	—
26	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
27	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
28	C₁₇×#405	C₂²	1	1	yes	characteristic
29	C₂×#63252	C₂×C₄	1	1	yes	characteristic
30	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
32	C₁₇×#117	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
34	C₂×#63223	C₂×C₄	1	1	yes	characteristic
35	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
36	#10587593	—	2	2	no	—
37	#10587593	C₈	1	4	yes	—
38	#10587587	—	2	1	no	—
39	#10587587	—	2	1	no	—
40	#10587587	D₄	1	2	yes	—
41	#10587587	Q₈	1	2	yes	—
42	C₂×#63218	—	2	1	no	—
43	C₂×#63218	—	2	1	no	—
44	C₂×C₈×D₁₇	—	2	1	no	—
45	C₂×C₈×D₁₇	—	2	1	no	—
46	#10587579	—	2	1	no	—
47	#10587573	—	2	1	no	—
48	#10587567	—	2	2	no	—
49	#10587558	—	2	1	no	—
50	C₄²×D₁₇	—	2	1	no	—
51	C₁₇×M₃₂	—	2	1	no	—
52	C₁₇×M₃₂	—	2	1	no	—
53	C₁₇×M₃₂	—	2	1	no	—
54	C₁₇×M₃₂	—	2	1	no	—
55	C₂×C₁₆×C₁₇	—	2	1	no	—
56	C₂×C₁₆×C₁₇	—	2	1	no	—
57	C₁₇×#97	—	2	1	no	—
58	C₄×C₈×C₁₇	—	2	1	no	—
59	#10587486	—	2	1	no	—
60	C₈×Dic₁₇	—	2	1	no	—
61	C₈×Dic₁₇	D₄	1	1	yes	characteristic
62	C₈×Dic₁₇	D₄	1	1	yes	characteristic
63	C₈×Dic₁₇	D₄	1	1	yes	characteristic
64	C₈×Dic₁₇	D₄	1	1	yes	characteristic
65	#10587483	—	2	1	no	—
66	#10587483	—	2	1	no	—
67	#10587483	—	2	1	no	—
68	#10587483	—	2	1	no	—
69	C₂×#63214	—	2	1	no	—
70	C₂×#63214	—	2	1	no	—
71	#10587474	—	2	1	no	—
72	C₄×#3622	—	2	1	no	—
73	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₂²⋊D₁₇	—	2	1	no	—
75	C₂²×Dic₁₇	—	2	1	no	—
76	#63252	—	2	1	no	—
77	#63252	—	2	1	no	—
78	#63252	C₄⋊C₄	1	2	yes	—
79	#63252	C₂×C₈	1	2	yes	—
80	C₂×D₆₈	M₁₆	1	1	yes	characteristic
81	C₂×C₄×D₁₇	—	2	1	no	—
82	C₂×C₄×D₁₇	—	2	2	no	—
83	C₂×Dic₃₄	M₁₆	1	1	yes	characteristic
84	M₁₆×C₁₇	—	2	1	no	—
85	M₁₆×C₁₇	C₂×C₈	1	2	yes	—
86	C₂×C₈×C₁₇	—	2	1	no	—
87	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
91	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
92	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
93	C₄⋊C₄×C₁₇	—	2	1	no	—
94	C₂²⋊C₄×C₁₇	M₁₆	1	2	yes	—
95	C₄²×C₁₇	—	2	1	no	—
96	#63227	—	2	2	no	—
97	#63225	—	2	2	no	—
98	C₄×Dic₁₇	—	2	2	no	—
99	C₄×Dic₁₇	—	2	1	no	—
100	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
101	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
102	#63223	—	2	1	no	—
103	#63223	C₄⋊C₄	1	2	yes	—
104	C₂×#3622	—	2	1	no	—
105	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×#3622	C₂×C₈	1	2	yes	—
108	C₂×#3622	M₁₆	1	2	yes	—
109	#63218	—	2	2	no	—
110	#63218	—	2	2	no	—
111	#63218	—	2	4	no	—
112	C₈×D₁₇	—	2	2	no	—
113	C₈×D₁₇	—	2	2	no	—
114	C₈×D₁₇	—	2	4	no	—
115	C₁₆×C₁₇	—	4	1	no	—
116	C₁₆×C₁₇	—	4	1	no	—
117	#63214	—	4	1	no	—
118	#63214	—	4	1	no	—
119	#12792	—	17	1	no	maximal, Sylow
120	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
121	C₂²×D₁₇	—	2	1	no	—
122	C₂×C₄×C₁₇	—	2	2	no	—
123	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
126	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
128	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
129	C₂²⋊D₁₇	—	2	2	no	—
130	C₂²⋊D₁₇	—	2	2	no	—
131	C₂×Dic₁₇	—	2	1	no	—
132	C₂×Dic₁₇	—	2	1	no	—
133	C₂×Dic₁₇	—	2	1	no	—
134	C₂×Dic₁₇	—	4	2	no	—
135	D₆₈	—	2	1	no	—
136	D₆₈	C₄⋊C₈	1	2	yes	—
137	C₄×D₁₇	—	2	2	no	—
138	C₄×D₁₇	—	2	2	no	—
139	C₄×D₁₇	—	4	4	no	—
140	Dic₃₄	—	2	1	no	—
141	Dic₃₄	C₄⋊C₈	1	2	yes	—
142	C₈×C₁₇	—	2	2	no	—
143	C₈×C₁₇	#121	1	1	yes	characteristic
144	C₈×C₁₇	#121	1	1	yes	characteristic
145	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
146	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
147	#3622	—	2	2	no	—
148	#3622	C₂²⋊C₈	1	2	yes	—
149	#3622	C₂²⋊C₈	1	2	yes	—
150	C₄×#131	—	17	1	no	—
151	C₂×#364	—	17	1	no	—
152	C₂×#364	—	17	1	no	—
153	#2117	—	17	1	no	—
154	#2117	—	17	1	no	—
155	#1378	—	17	1	no	—
156	#1378	D₁₇	1	1	yes	characteristic, complemented
157	C₂²×C₁₇	—	4	1	no	—
158	C₂²×C₁₇	#407	1	1	yes	characteristic
159	C₂²×C₁₇	#423	1	1	yes	characteristic
160	C₂²×C₁₇	#431	1	1	yes	characteristic
161	C₂×D₁₇	—	2	2	no	—
162	C₂×D₁₇	—	4	2	no	—
163	C₄×C₁₇	—	4	2	no	—
164	C₄×C₁₇	#380	1	1	yes	characteristic
165	C₄×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
166	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
167	C₄×C₁₇	#432	1	1	yes	characteristic
168	Dic₁₇	—	2	2	no	—
169	Dic₁₇	—	2	2	no	—
170	Dic₁₇	—	4	1	no	—
171	C₂×#131	—	17	1	no	—
172	C₂×#131	—	17	1	no	—
173	C₄×#41	—	17	1	no	—
174	C₂×M₃₂	—	17	1	no	—
175	C₂×M₃₂	—	17	1	no	—
176	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
177	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
178	#405	—	17	1	no	—
179	#405	—	34	1	no	—
180	#405	—	34	1	no	—
181	#405	C₂×D₁₇	1	1	yes	characteristic
182	C₄×M₁₆	—	17	1	no	—
183	C₄×M₁₆	—	34	1	no	—
184	C₂×C₄×C₈	—	17	1	no	—
185	C₂×C₄×C₈	—	34	1	no	—
186	#364	—	34	1	no	—
187	#364	—	34	1	no	—
188	#364	—	34	1	no	—
189	#364	—	34	1	no	—
190	#350	—	34	2	no	—
191	#350	—	34	2	no	—
192	C₂×C₁₇	—	2	1	no	—
193	C₂×C₁₇	—	2	1	no	—
194	C₂×C₁₇	—	2	1	no	—
195	C₂×C₁₇	#1797	1	1	yes	characteristic
196	D₁₇	—	4	2	no	—
197	C₂×#41	—	17	1	no	—
198	#131	—	17	4	no	—
199	#131	—	17	2	no	—
200	#131	—	17	2	no	—
201	#131	—	34	2	no	—
202	#131	—	34	1	no	—
203	#131	—	34	1	no	—
204	C₂×M₁₆	—	17	1	no	—
205	C₂×M₁₆	—	34	1	no	—
206	C₂×M₁₆	—	34	1	no	—
207	C₂×M₁₆	C₄×D₁₇	1	1	yes	characteristic
208	C₂²×C₈	—	17	1	no	—
209	C₂²×C₈	—	34	1	no	—
210	C₂²×C₈	—	34	1	no	—
211	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
212	C₄×Q₈	—	34	1	no	—
213	C₄×D₄	—	34	1	no	—
214	C₄×D₄	—	34	2	no	—
215	#117	—	34	1	no	—
216	#117	C₄×D₁₇	1	1	yes	characteristic
217	C₂×C₄²	—	17	1	no	—
218	C₂×C₄²	—	34	1	no	—
219	M₃₂	—	2	1	no	—
220	M₃₂	—	2	1	no	—
221	M₃₂	—	2	1	no	—
222	M₃₂	—	2	1	no	—
223	M₃₂	—	34	1	no	—
224	M₃₂	—	34	1	no	—
225	M₃₂	—	34	1	no	—
226	M₃₂	—	34	1	no	—
227	C₂×C₁₆	—	2	1	no	—
228	C₂×C₁₆	—	2	1	no	—
229	C₂×C₁₆	—	34	1	no	—
230	C₂×C₁₆	—	34	1	no	—
231	#97	—	2	1	no	—
232	#97	—	34	1	no	—
233	#97	—	34	1	no	—
234	C₄×C₈	—	2	1	no	—
235	C₄×C₈	—	17	1	no	—
236	C₄×C₈	—	17	1	no	—
237	C₄×C₈	—	17	1	no	—
238	C₄×C₈	—	17	1	no	—
239	C₄×C₈	—	34	1	no	—
240	C₄×C₈	—	34	1	no	—
241	C₁₇	#12792	1	1	yes	characteristic, complemented, Sylow
242	#41	—	17	2	no	—
243	#41	—	17	2	no	—
244	#41	—	34	1	no	—
245	#41	—	34	1	no	—
246	C₂×Q₈	—	17	1	no	—
247	C₂×D₄	—	17	1	no	—
248	C₂×D₄	—	34	1	no	—
249	C₂²×C₄	—	34	2	no	—
250	C₂²×C₄	—	34	1	no	—
251	C₂²×C₄	—	34	1	no	—
252	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
253	M₁₆	—	2	1	no	—
254	M₁₆	—	17	2	no	—
255	M₁₆	—	34	4	no	—
256	M₁₆	—	34	1	no	—
257	M₁₆	—	34	2	no	—
258	M₁₆	—	34	2	no	—
259	M₁₆	C₈×D₁₇	1	2	yes	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	17	1	no	—
262	C₂×C₈	—	17	1	no	—
263	C₂×C₈	—	17	2	no	—
264	C₂×C₈	—	17	2	no	—
265	C₂×C₈	—	34	4	no	—
266	C₂×C₈	—	34	1	no	—
267	C₂×C₈	—	34	2	no	—
268	C₂×C₈	—	34	2	no	—
269	C₂×C₈	#63252	1	1	yes	characteristic
270	C₂×C₈	#63252	1	1	yes	characteristic
271	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
272	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
273	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
274	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	34	2	no	—
277	C₂²⋊C₄	—	34	2	no	—
278	C₂²⋊C₄	#63218	1	2	yes	—
279	C₄²	—	2	1	no	—
280	C₄²	—	17	1	no	—
281	C₄²	—	17	1	no	—
282	C₄²	—	34	2	no	—
283	C₄²	—	34	1	no	—
284	C₁₆	—	4	1	no	—
285	C₁₆	—	4	1	no	—
286	C₁₆	—	68	1	no	—
287	C₁₆	—	68	1	no	—
288	C₂³	—	34	1	no	—
289	C₂³	C₂×#63218	1	1	yes	characteristic
290	Q₈	—	17	2	no	—
291	Q₈	—	34	1	no	—
292	D₄	—	17	2	no	—
293	D₄	—	34	2	no	—
294	D₄	—	34	2	no	—
295	D₄	—	34	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	34	1	no	—
298	C₂×C₄	—	34	1	no	—
299	C₂×C₄	—	34	2	no	—
300	C₂×C₄	—	34	2	no	—
301	C₂×C₄	—	34	1	no	—
302	C₂×C₄	—	68	4	no	—
303	C₂×C₄	—	68	2	no	—
304	C₂×C₄	#10587558	1	1	yes	characteristic
305	C₂×C₄	C₂²⋊C₄×D₁₇	1	1	yes	characteristic
306	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
307	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
308	C₂×C₄	M₁₆×D₁₇	1	1	yes	characteristic
309	C₂×C₄	M₁₆×D₁₇	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	17	2	no	—
312	C₈	—	17	2	no	—
313	C₈	—	34	2	no	—
314	C₈	#10587570	1	1	yes	characteristic
315	C₈	#10587570	1	1	yes	characteristic
316	C₈	#10587582	1	1	yes	characteristic
317	C₈	#10587582	1	1	yes	characteristic
318	C₂²	—	4	1	no	—
319	C₂²	—	34	2	no	—
320	C₂²	—	68	2	no	—
321	C₂²	#11762771	1	1	yes	characteristic
322	C₂²	#11762831	1	1	yes	characteristic
323	C₂²	#11762860	1	1	yes	characteristic
324	C₄	—	4	2	no	—
325	C₄	—	34	2	no	—
326	C₄	—	34	2	no	—
327	C₄	—	68	1	no	—
328	C₄	#11762668	1	1	yes	characteristic
329	C₄	C₂²⋊C₈×D₁₇	1	1	yes	characteristic
330	C₄	C₄⋊C₈×D₁₇	1	1	yes	characteristic
331	C₄	#11762864	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	68	2	no	—
336	C₂	#14528763	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#10587570
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1378	C₂
socle	C₂×C₁₇	#1797
prime radical	C₂×#63252	C₂×C₄
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1378	D₁₇
hypocenter	C₁₇	#12792
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1378

Complements

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

N	H ≅ G/N	type
2. C₂×#11762629	C₂	semidirect
3. C₂×#11762629	C₂	semidirect
6. #14526870	C₂	semidirect
7. C₁₇×#1378	C₂	semidirect
156. #1378	D₁₇	semidirect
241. C₁₇	#12792	semidirect

Maximal subgroups

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

subgroup	index
#14537206	2
C₂×#11762629	2
#14530413	2
#14530411	2
#14526870	2
C₁₇×#1378	2
#12792	17