Subgroups — #16925366

Order 4352.

444 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14537760	C₂	1	1	yes	maximal, characteristic
2	#14528472	C₂	1	1	yes	maximal, characteristic, complemented
3	#14527270	C₂	1	1	yes	maximal, characteristic
4	M₁₆×#3622	C₂	1	1	yes	maximal, characteristic, complemented
5	#14526683	C₂	1	1	yes	maximal, characteristic, complemented
6	#14526678	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#1524	C₂	1	1	yes	maximal, characteristic
8	#11762940	C₄	1	1	yes	characteristic, complemented
9	Q₈×#3622	C₄	1	1	yes	characteristic, complemented
10	#11762887	C₄	1	1	yes	characteristic, complemented
11	#11762881	—	2	1	no	—
12	D₄×#3622	—	2	1	no	—
13	D₄×#3622	C₄	1	1	yes	characteristic, complemented
14	#11762870	—	2	1	no	—
15	#11762866	C₂²	1	1	yes	characteristic
16	#11762862	C₂²	1	1	yes	characteristic
17	#11762765	—	2	1	no	—
18	C₂×#10587486	—	2	1	no	—
19	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₁₇	—	2	1	no	—
21	#11762655	C₂²	1	1	yes	characteristic
22	#11762546	C₄	1	2	yes	—
23	#11762543	C₄	1	2	yes	—
24	#11762530	—	2	1	no	—
25	C₈×#3622	—	2	1	no	—
26	C₁₇×#529	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
28	C₂×C₁₇×#97	C₂²	1	1	yes	characteristic
29	C₁₇×#330	C₄	1	2	yes	complemented
30	C₁₇×#329	C₄	1	2	yes	complemented
31	C₂×M₁₆×C₁₇	—	2	1	no	—
32	C₂²×C₈×C₁₇	—	2	1	no	—
33	C₁₇×#126	—	2	1	no	—
34	C₁₇×#125	C₂×C₄	1	1	yes	characteristic
35	C₁₇×#124	C₂×C₄	1	1	yes	characteristic
36	C₁₇×#123	—	2	1	no	—
37	C₁₇×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
38	C₁₇×#121	C₂×C₄	1	1	yes	characteristic
39	C₄×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
40	C₄×D₄×C₁₇	—	2	1	no	—
41	C₄×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
42	C₁₇×#117	—	2	1	no	—
43	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
44	C₂×C₄×Dic₁₇	—	2	1	no	—
45	C₂×#63223	—	2	1	no	—
46	C₂²×#3622	—	2	1	no	—
47	C₂²×#3622	D₄	1	1	yes	characteristic
48	C₂²×#3622	D₄	1	1	yes	characteristic
49	C₁₇×#97	—	2	1	no	—
50	C₁₇×#97	C₂×C₄	1	2	yes	—
51	C₁₇×#97	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₁₇	—	2	1	no	—
53	#10587502	—	2	1	no	—
54	#10587502	C₂×C₄	1	2	yes	—
55	C₂²⋊#3622	—	2	1	no	—
56	C₂²⋊#3622	—	2	1	no	—
57	C₂²⋊#3622	—	2	1	no	—
58	C₂²⋊#3622	C₂×C₄	1	2	yes	—
59	#10587486	—	2	4	no	—
60	C₈×Dic₁₇	—	2	4	no	—
61	C₄⋊#3622	—	2	1	no	—
62	C₄⋊#3622	—	2	1	no	—
63	C₄⋊#3622	—	2	1	no	—
64	C₄⋊#3622	C₂×C₄	1	1	yes	characteristic
65	C₄⋊#3622	C₂×C₄	1	1	yes	characteristic
66	#10587474	—	2	1	no	—
67	#10587474	—	2	1	no	—
68	C₄×#3622	—	2	1	no	—
69	C₄×#3622	—	2	1	no	—
70	C₄×#3622	C₂×C₄	1	1	yes	characteristic
71	C₄×#3622	C₂×C₄	1	1	yes	characteristic
72	C₄×#3622	D₄	1	1	yes	characteristic
73	C₄×#3622	D₄	1	1	yes	characteristic
74	C₂×Q₈×C₁₇	C₄²	1	1	yes	characteristic
75	C₂×D₄×C₁₇	—	2	1	no	—
76	C₂×D₄×C₁₇	C₄²	1	1	yes	characteristic
77	C₂²×C₄×C₁₇	—	2	1	no	—
78	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
81	C₂²×Dic₁₇	—	2	2	no	—
82	M₁₆×C₁₇	—	2	4	no	—
83	C₂×C₈×C₁₇	—	2	2	no	—
84	C₂×C₈×C₁₇	—	2	2	no	—
85	C₂×C₈×C₁₇	—	2	2	no	—
86	C₂×C₈×C₁₇	—	2	2	no	—
87	C₄⋊C₄×C₁₇	—	2	1	no	—
88	C₄⋊C₄×C₁₇	—	2	1	no	—
89	C₄⋊C₄×C₁₇	—	2	1	no	—
90	C₄⋊C₄×C₁₇	—	2	1	no	—
91	C₄⋊C₄×C₁₇	C₄²	1	1	yes	characteristic
92	C₄⋊C₄×C₁₇	C₄²	1	1	yes	characteristic
93	C₂²⋊C₄×C₁₇	—	2	1	no	—
94	C₂²⋊C₄×C₁₇	—	2	1	no	—
95	C₂²⋊C₄×C₁₇	—	2	1	no	—
96	C₂²⋊C₄×C₁₇	—	2	1	no	—
97	C₂²⋊C₄×C₁₇	—	2	1	no	—
98	C₄²×C₁₇	—	2	1	no	—
99	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₇	#41	1	1	yes	characteristic
103	C₄×Dic₁₇	—	2	2	no	—
104	C₄×Dic₁₇	—	2	2	no	—
105	#63223	—	4	2	no	—
106	C₂×#3622	—	2	2	no	—
107	C₂×#3622	—	2	2	no	—
108	C₂×#3622	—	2	1	no	—
109	C₂×#3622	—	2	1	no	—
110	C₂×#3622	—	2	2	no	—
111	C₂×#3622	—	4	2	no	—
112	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
113	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
114	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
115	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
116	#8247	—	17	1	no	maximal, Sylow
117	C₂³×C₁₇	—	2	1	no	—
118	C₂³×C₁₇	C₄×D₄	1	1	yes	characteristic
119	Q₈×C₁₇	—	4	1	no	—
120	D₄×C₁₇	—	4	1	no	—
121	D₄×C₁₇	—	4	2	no	—
122	C₂×C₄×C₁₇	—	2	2	no	—
123	C₂×C₄×C₁₇	—	2	1	no	—
124	C₂×C₄×C₁₇	—	2	1	no	—
125	C₂×C₄×C₁₇	—	2	1	no	—
126	C₂×C₄×C₁₇	—	2	1	no	—
127	C₂×C₄×C₁₇	—	2	1	no	—
128	C₂×C₄×C₁₇	—	4	1	no	—
129	C₂×C₄×C₁₇	—	4	1	no	—
130	C₂×C₄×C₁₇	—	4	1	no	—
131	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
132	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
134	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
137	C₂×Dic₁₇	—	2	4	no	—
138	C₂×Dic₁₇	—	4	2	no	—
139	C₂×Dic₁₇	—	4	2	no	—
140	C₈×C₁₇	—	2	4	no	—
141	C₈×C₁₇	—	2	4	no	—
142	#3622	—	2	1	no	—
143	#3622	—	2	1	no	—
144	#3622	—	4	2	no	—
145	#3622	—	4	1	no	—
146	#3622	—	4	1	no	—
147	#3622	#104	1	2	yes	—
148	#3622	#104	1	2	yes	—
149	#2966	—	17	1	no	—
150	#1726	—	17	1	no	—
151	#1524	D₁₇	1	1	yes	characteristic
152	#1500	—	17	1	no	—
153	C₈×M₁₆	—	17	1	no	—
154	#1302	—	17	1	no	—
155	#1300	—	17	1	no	—
156	C₂²×C₁₇	—	2	1	no	—
157	C₂²×C₁₇	—	2	1	no	—
158	C₂²×C₁₇	—	4	1	no	—
159	C₂²×C₁₇	—	4	1	no	—
160	C₂²×C₁₇	—	4	1	no	—
161	C₂²×C₁₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
162	C₂²×C₁₇	#443	1	2	yes	—
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	—	2	1	no	—
168	C₄×C₁₇	—	4	1	no	—
169	C₄×C₁₇	—	4	1	no	—
170	C₄×C₁₇	—	4	1	no	—
171	C₄×C₁₇	C₂×#104	1	1	yes	characteristic
172	C₄×C₁₇	C₂×#104	1	1	yes	characteristic
173	Dic₁₇	—	4	4	no	—
174	#529	C₂×D₁₇	1	1	yes	characteristic
175	#446	—	17	1	no	—
176	C₈×Q₈	—	17	1	no	—
177	#436	—	17	1	no	—
178	#435	—	34	1	no	—
179	C₈×D₄	—	17	1	no	—
180	C₈×D₄	—	34	1	no	—
181	#433	—	34	1	no	—
182	#432	—	17	1	no	—
183	#431	—	17	1	no	—
184	#405	—	34	1	no	—
185	C₄×M₁₆	C₂×D₁₇	1	1	yes	characteristic
186	C₂×#97	—	34	1	no	—
187	C₂×#97	C₂×D₁₇	1	1	yes	characteristic
188	C₂×C₄×C₈	—	17	1	no	—
189	C₂×C₄×C₈	—	34	1	no	—
190	#376	—	17	1	no	—
191	#330	Dic₁₇	1	2	yes	complemented
192	#329	Dic₁₇	1	2	yes	complemented
193	#328	—	17	2	no	—
194	#327	—	17	2	no	—
195	#322	—	34	1	no	—
196	C₈²	—	34	1	no	—
197	C₂×C₁₇	—	2	2	no	—
198	C₂×C₁₇	—	8	1	no	—
199	C₂×C₁₇	#1757	1	1	yes	characteristic
200	C₂×C₁₇	C₄×#104	1	1	yes	characteristic
201	C₂×C₁₇	#1764	1	1	yes	characteristic
202	C₂×M₁₆	—	2	1	no	—
203	C₂×M₁₆	—	34	1	no	—
204	C₂²×C₈	—	2	1	no	—
205	C₂²×C₈	—	17	1	no	—
206	C₂²×C₈	—	17	1	no	—
207	C₂²×C₈	—	34	1	no	—
208	#126	—	2	1	no	—
209	#125	C₂×Dic₁₇	1	1	yes	characteristic
210	#124	C₂×Dic₁₇	1	1	yes	characteristic
211	#123	—	2	1	no	—
212	C₂²⋊Q₈	C₄×D₁₇	1	1	yes	characteristic
213	#121	C₄×D₁₇	1	1	yes	characteristic
214	C₄×Q₈	C₄×D₁₇	1	1	yes	characteristic
215	C₄×D₄	—	2	1	no	—
216	C₄×D₄	C₄×D₁₇	1	1	yes	characteristic
217	#117	—	2	1	no	—
218	C₂×C₄²	—	34	1	no	—
219	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
220	C₄⋊C₈	—	17	1	no	—
221	C₄⋊C₈	—	17	1	no	—
222	C₄⋊C₈	—	34	1	no	—
223	C₄⋊C₈	—	34	1	no	—
224	C₄⋊C₈	—	34	1	no	—
225	C₂²⋊C₈	—	17	2	no	—
226	C₂²⋊C₈	—	34	1	no	—
227	C₂²⋊C₈	—	34	1	no	—
228	C₂²⋊C₈	—	34	1	no	—
229	#97	—	2	1	no	—
230	#97	—	34	1	no	—
231	#97	—	34	4	no	—
232	#97	—	34	1	no	—
233	#97	C₂×Dic₁₇	1	2	yes	—
234	#97	C₂×Dic₁₇	1	2	yes	—
235	C₄×C₈	—	2	1	no	—
236	C₄×C₈	—	17	1	no	—
237	C₄×C₈	—	17	1	no	—
238	C₄×C₈	—	17	1	no	—
239	C₄×C₈	—	17	1	no	—
240	C₄×C₈	—	34	1	no	—
241	C₄×C₈	—	34	4	no	—
242	C₄×C₈	—	34	1	no	—
243	#95	—	17	2	no	—
244	#95	—	34	1	no	—
245	C₁₇	#8247	1	1	yes	characteristic, complemented, Sylow
246	C₂×Q₈	C₄×Dic₁₇	1	1	yes	characteristic
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	C₄×Dic₁₇	1	1	yes	characteristic
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	34	2	no	—
251	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
252	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
253	C₂²×C₄	#63254	1	1	yes	characteristic
254	M₁₆	—	2	4	no	—
255	M₁₆	—	68	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	17	1	no	—
261	C₂×C₈	—	17	1	no	—
262	C₂×C₈	—	17	1	no	—
263	C₂×C₈	—	17	1	no	—
264	C₂×C₈	—	34	2	no	—
265	C₂×C₈	—	34	2	no	—
266	C₂×C₈	—	34	1	no	—
267	C₂×C₈	—	34	1	no	—
268	C₂×C₈	—	34	2	no	—
269	C₂×C₈	—	68	2	no	—
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	C₄×Dic₁₇	1	1	yes	characteristic
275	C₄⋊C₄	C₄×Dic₁₇	1	1	yes	characteristic
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	34	2	no	—
283	C₄²	—	34	2	no	—
284	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
285	C₄²	D₄×D₁₇	1	1	yes	characteristic
286	C₄²	#63254	1	1	yes	characteristic
287	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
288	C₂³	—	2	1	no	—
289	C₂³	#10587567	1	1	yes	characteristic
290	Q₈	—	4	1	no	—
291	D₄	—	4	1	no	—
292	D₄	—	4	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	34	4	no	—
303	C₂×C₄	—	68	2	no	—
304	C₂×C₄	—	68	2	no	—
305	C₂×C₄	#10587563	1	1	yes	characteristic
306	C₂×C₄	C₂²⋊C₄×D₁₇	1	1	yes	characteristic
307	C₂×C₄	#10587567	1	1	yes	characteristic
308	C₂×C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
309	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
310	C₂×C₄	D₄×Dic₁₇	1	1	yes	characteristic
311	C₈	—	2	4	no	—
312	C₈	—	2	4	no	—
313	C₈	—	17	2	no	—
314	C₈	—	17	2	no	—
315	C₈	—	34	1	no	—
316	C₈	—	34	1	no	—
317	C₈	—	68	2	no	—
318	C₈	—	68	1	no	—
319	C₈	—	68	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	C₂²⋊C₄×Dic₁₇	1	1	yes	characteristic
326	C₂²	#11762923	1	2	yes	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	68	4	no	—
336	C₄	#104×D₁₇	1	1	yes	characteristic
337	C₄	#104×D₁₇	1	1	yes	characteristic
338	C₂	—	2	2	no	—
339	C₂	—	8	1	no	—
340	C₂	#14528587	1	1	yes	characteristic
341	C₂	#104×Dic₁₇	1	1	yes	characteristic
342	C₂	#14528622	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1524	C₂
socle	C₂²×C₁₇	C₄×C₂²⋊C₄
prime radical	C₁₇×#121	C₂×C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1524	D₁₇
hypocenter	C₁₇	#8247
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1524

Complements

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

N	H ≅ G/N	type
2. #14528472	C₂	semidirect
4. M₁₆×#3622	C₂	semidirect
5. #14526683	C₂	semidirect
6. #14526678	C₂	semidirect
8. #11762940	C₄	semidirect
9. Q₈×#3622	C₄	semidirect
10. #11762887	C₄	semidirect
13. D₄×#3622	C₄	semidirect
29. C₁₇×#330	C₄	semidirect
30. C₁₇×#329	C₄	semidirect
191. #330	Dic₁₇	semidirect
192. #329	Dic₁₇	semidirect
245. C₁₇	#8247	semidirect

Maximal subgroups

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

subgroup	index
#14537760	2
#14528472	2
#14527270	2
M₁₆×#3622	2
#14526683	2
#14526678	2
C₁₇×#1524	2
#8247	17