Subgroups — #17017398

Order 4352.

714 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	#14539905	C₂	1	2	yes	maximal
2	#14537497	C₂	1	1	yes	maximal, characteristic
3	#14528270	C₂	1	4	yes	maximal, complemented
4	#14528146	C₂	1	4	yes	maximal, complemented
5	#14527532	C₂	1	1	yes	maximal, characteristic, complemented
6	#14527456	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10587587	—	2	1	no	—
9	#11763359	C₂²	1	1	yes	characteristic
10	#11763212	—	2	2	no	—
11	#11763157	—	2	2	no	—
12	#11763122	—	2	4	no	—
13	#11763086	C₂²	1	8	yes	—
14	#11762888	—	2	1	no	—
15	#11762888	C₂²	1	2	yes	—
16	C₈×D₆₈	—	2	1	no	—
17	C₈×D₆₈	C₂²	1	2	yes	—
18	#11762862	C₂²	1	2	yes	—
19	#11762833	C₂²	1	1	yes	characteristic
20	#11762775	—	2	1	no	—
21	#11762562	C₂²	1	2	yes	—
22	#11762560	C₂²	1	2	yes	—
23	#11762555	C₂²	1	8	yes	—
24	C₂²×Q₁₆×C₁₇	—	2	1	no	—
25	C₂²×Q₁₆×C₁₇	—	2	1	no	—
26	C₂×C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
27	C₁₇×#494	—	2	1	no	—
28	C₁₇×#494	—	2	1	no	—
29	C₁₇×#494	C₂²	1	2	yes	—
30	C₁₇×#494	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
32	C₂²×Q₈×C₁₇	—	2	2	no	—
33	C₂×#63252	—	2	1	no	—
34	C₂×Q₁₆×C₁₇	—	2	2	no	—
35	C₂×Q₁₆×C₁₇	—	2	2	no	—
36	C₂×Q₁₆×C₁₇	—	2	2	no	—
37	C₂×Q₁₆×C₁₇	—	2	2	no	—
38	C₂×Q₁₆×C₁₇	—	4	4	no	—
39	C₂×Q₁₆×C₁₇	—	4	4	no	—
40	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
42	C₁₇×C₄⋊Q₈	—	2	2	no	—
43	C₁₇×C₄⋊Q₈	—	2	2	no	—
44	C₁₇×C₄⋊Q₈	—	2	2	no	—
45	C₁₇×C₄⋊Q₈	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
48	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
49	#10587616	—	2	1	no	—
50	#10587616	D₄	1	2	yes	—
51	C₂×#63223	—	2	1	no	—
52	#10587587	—	4	4	no	—
53	C₂×#63218	—	2	2	no	—
54	C₂×C₈×D₁₇	—	2	2	no	—
55	#10587561	—	2	1	no	—
56	#10587560	D₄	1	1	yes	characteristic
57	C₄×D₆₈	—	2	1	no	—
58	C₄×D₆₈	C₂³	1	2	yes	—
59	#10587555	D₄	1	1	yes	characteristic
60	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
61	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
62	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
64	C₂²⋊#3622	—	2	2	no	—
65	#10587491	—	2	2	no	—
66	#10587491	—	2	2	no	—
67	#10587480	—	2	8	no	—
68	#10587478	—	2	8	no	—
69	C₄⋊#3622	—	2	1	no	—
70	C₄⋊#3622	C₂³	1	2	yes	—
71	C₄⋊#3622	C₂³	1	4	yes	—
72	C₄×#3622	—	2	2	no	—
73	C₂×Q₈×C₁₇	—	2	4	no	—
74	C₂×Q₈×C₁₇	—	4	2	no	—
75	C₂×Q₈×C₁₇	—	4	4	no	—
76	C₂×Q₈×C₁₇	—	4	4	no	—
77	C₂²×C₄×C₁₇	—	4	2	no	—
78	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
81	C₂×C₂²⋊D₁₇	—	2	2	no	—
82	#63252	—	4	2	no	—
83	#63252	—	4	2	no	—
84	C₂×D₆₈	—	2	1	no	—
85	C₂×D₆₈	—	2	1	no	—
86	C₂×D₆₈	C₂×D₄	1	2	yes	—
87	C₂×C₄×D₁₇	—	2	2	no	—
88	C₂×Dic₃₄	—	2	1	no	—
89	Q₁₆×C₁₇	—	4	4	no	—
90	Q₁₆×C₁₇	—	4	4	no	—
91	Q₁₆×C₁₇	—	4	4	no	—
92	Q₁₆×C₁₇	—	4	4	no	—
93	C₂×C₈×C₁₇	—	2	2	no	—
94	C₂×C₈×C₁₇	D₈	1	4	yes	—
95	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
99	C₄⋊C₄×C₁₇	—	2	4	no	—
100	C₄⋊C₄×C₁₇	—	2	4	no	—
101	C₄⋊C₄×C₁₇	—	4	2	no	—
102	C₄⋊C₄×C₁₇	—	4	2	no	—
103	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₇	#41	1	1	yes	characteristic
106	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
107	#63227	—	2	2	no	—
108	#63227	—	2	2	no	—
109	C₄⋊Dic₁₇	—	2	1	no	—
110	C₄⋊Dic₁₇	C₂×D₄	1	2	yes	—
111	#63223	—	4	2	no	—
112	C₂×#3622	—	2	4	no	—
113	C₂×#3622	—	2	2	no	—
114	#63218	—	4	4	no	—
115	C₈×D₁₇	—	4	4	no	—
116	#19188	—	17	1	no	maximal, Sylow
117	C₂³×C₁₇	#121	1	1	yes	characteristic
118	C₂²×D₁₇	—	2	2	no	—
119	Q₈×C₁₇	—	4	4	no	—
120	Q₈×C₁₇	—	4	4	no	—
121	Q₈×C₁₇	—	8	4	no	—
122	C₂×C₄×C₁₇	—	2	2	no	—
123	C₂×C₄×C₁₇	—	2	2	no	—
124	C₂×C₄×C₁₇	—	4	4	no	—
125	C₂×C₄×C₁₇	—	8	4	no	—
126	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
127	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
128	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
129	C₂×C₄×C₁₇	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
134	C₂²⋊D₁₇	—	4	4	no	—
135	C₂×Dic₁₇	—	2	2	no	—
136	D₆₈	—	2	4	no	—
137	D₆₈	—	4	2	no	—
138	D₆₈	—	4	2	no	—
139	C₄×D₁₇	—	4	2	no	—
140	C₄×D₁₇	—	4	2	no	—
141	Dic₃₄	—	4	2	no	—
142	C₈×C₁₇	—	2	1	no	—
143	C₈×C₁₇	—	2	1	no	—
144	C₈×C₁₇	C₂×D₈	1	2	yes	—
145	C₈×C₁₇	C₂×D₈	1	2	yes	—
146	#3622	—	4	2	no	—
147	#3622	—	4	4	no	—
148	#3226	—	17	2	no	—
149	C₂×#494	D₁₇	1	1	yes	characteristic, complemented
150	#2931	—	17	1	no	—
151	#1689	—	17	4	no	—
152	#1669	—	17	4	no	—
153	#1564	—	17	1	no	—
154	#1546	—	17	2	no	—
155	C₂²×C₁₇	—	2	1	no	—
156	C₂²×C₁₇	—	2	1	no	—
157	C₂²×C₁₇	#459	1	2	yes	—
158	C₂²×C₁₇	C₂×#121	1	1	yes	characteristic
159	C₂×D₁₇	—	4	2	no	—
160	C₂×D₁₇	—	4	2	no	—
161	C₂×D₁₇	—	4	2	no	—
162	C₄×C₁₇	—	2	1	no	—
163	C₄×C₁₇	—	2	1	no	—
164	C₄×C₁₇	—	8	4	no	—
165	C₄×C₁₇	#459	1	2	yes	—
166	C₄×C₁₇	#473	1	2	yes	—
167	C₄×C₁₇	C₂²×D₈	1	1	yes	characteristic
168	C₄×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₄×C₁₇	#578	1	2	yes	—
170	Dic₁₇	—	4	2	no	—
171	C₂²×Q₁₆	—	2	1	no	—
172	C₂²×Q₁₆	—	2	1	no	—
173	C₂×#131	—	34	1	no	—
174	#532	—	17	1	no	—
175	C₂×C₄⋊Q₈	C₂×D₁₇	1	2	yes	—
176	#500	—	34	2	no	—
177	#494	—	2	1	no	—
178	#494	—	2	1	no	—
179	#494	C₂×D₁₇	1	2	yes	—
180	#494	C₂×D₁₇	1	2	yes	—
181	#486	—	34	2	no	—
182	#480	—	34	4	no	—
183	#474	—	17	8	no	—
184	#436	—	17	2	no	—
185	#436	—	34	1	no	—
186	C₈×D₄	—	17	2	no	—
187	C₈×D₄	—	34	1	no	—
188	#431	—	17	2	no	—
189	#423	—	17	1	no	—
190	#408	—	34	1	no	—
191	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
192	#335	—	17	2	no	—
193	#334	—	17	2	no	—
194	#332	—	17	8	no	—
195	C₂×C₁₇	—	2	2	no	—
196	C₂×C₁₇	C₂×#459	1	1	yes	characteristic
197	C₂×C₁₇	#3059	1	1	yes	characteristic
198	C₂×C₁₇	C₂×#473	1	1	yes	characteristic
199	D₁₇	—	8	2	no	—
200	C₂×#41	—	34	1	no	—
201	C₂²×Q₈	—	2	2	no	—
202	C₂×Q₁₆	—	2	2	no	—
203	C₂×Q₁₆	—	2	2	no	—
204	C₂×Q₁₆	—	2	2	no	—
205	C₂×Q₁₆	—	2	2	no	—
206	C₂×Q₁₆	—	4	4	no	—
207	C₂×Q₁₆	—	4	4	no	—
208	#131	—	68	4	no	—
209	C₂×M₁₆	—	34	1	no	—
210	C₂×M₁₆	—	34	2	no	—
211	C₂²×C₈	—	34	2	no	—
212	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
213	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
214	C₄⋊Q₈	—	2	2	no	—
215	C₄⋊Q₈	—	2	2	no	—
216	C₄⋊Q₈	—	2	2	no	—
217	C₄⋊Q₈	—	17	1	no	—
218	C₄⋊Q₈	C₂²×D₁₇	1	4	yes	—
219	#127	—	17	1	no	—
220	#124	—	34	1	no	—
221	#121	—	17	2	no	—
222	#121	—	34	1	no	—
223	C₄×D₄	—	17	2	no	—
224	C₄×D₄	—	34	1	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
228	#107	—	34	8	no	—
229	C₄⋊C₈	—	17	2	no	—
230	C₄⋊C₈	—	17	4	no	—
231	C₄⋊C₈	—	34	1	no	—
232	#102	—	34	8	no	—
233	C₂²⋊C₈	—	34	2	no	—
234	C₂²⋊C₈	—	34	2	no	—
235	C₂²⋊C₈	—	34	2	no	—
236	C₄×C₈	—	34	2	no	—
237	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
238	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
241	C₁₇	#19188	1	1	yes	characteristic, complemented, Sylow
242	#41	—	68	2	no	—
243	#41	—	68	2	no	—
244	C₂×Q₈	—	2	4	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×Q₈	—	4	4	no	—
247	C₂×Q₈	—	4	4	no	—
248	C₂×Q₈	—	34	1	no	—
249	C₂×D₄	—	17	2	no	—
250	C₂×D₄	—	34	2	no	—
251	C₂×D₄	—	34	1	no	—
252	C₂×D₄	—	34	1	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	—	34	2	no	—
255	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
256	C₂²×C₄	#63254	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
258	Q₁₆	—	4	4	no	—
259	Q₁₆	—	4	4	no	—
260	Q₁₆	—	4	4	no	—
261	Q₁₆	—	4	4	no	—
262	M₁₆	—	68	2	no	—
263	M₁₆	—	68	4	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	34	4	no	—
266	C₂×C₈	—	34	2	no	—
267	C₂×C₈	—	68	4	no	—
268	C₂×C₈	#63228	1	4	yes	—
269	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
270	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
271	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
272	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
273	C₄⋊C₄	—	2	4	no	—
274	C₄⋊C₄	—	2	4	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	17	2	no	—
278	C₄⋊C₄	—	34	1	no	—
279	C₂²⋊C₄	—	34	2	no	—
280	C₂²⋊C₄	—	34	2	no	—
281	C₄²	D₄×D₁₇	1	1	yes	characteristic
282	C₄²	#63254	1	1	yes	characteristic
283	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
284	C₄²	C₂³×D₁₇	1	1	yes	characteristic
285	C₂³	—	34	2	no	—
286	C₂³	#10587624	1	1	yes	characteristic
287	Q₈	—	4	4	no	—
288	Q₈	—	4	4	no	—
289	Q₈	—	8	4	no	—
290	Q₈	—	68	2	no	—
291	D₄	—	34	4	no	—
292	D₄	—	68	4	no	—
293	D₄	—	68	2	no	—
294	D₄	—	68	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	8	4	no	—
299	C₂×C₄	—	34	2	no	—
300	C₂×C₄	—	68	2	no	—
301	C₂×C₄	—	68	2	no	—
302	C₂×C₄	C₂×#63228	1	2	yes	—
303	C₂×C₄	#10587618	1	2	yes	—
304	C₂×C₄	#10587624	1	1	yes	characteristic
305	C₂×C₄	#10587624	1	1	yes	characteristic
306	C₂×C₄	#10587624	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
308	C₂×C₄	C₂×#63254	1	1	yes	characteristic
309	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	68	2	no	—
313	C₈	—	68	4	no	—
314	C₈	C₂×#63228	1	2	yes	—
315	C₈	C₂×#63228	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	68	2	no	—
319	C₂²	—	68	2	no	—
320	C₂²	—	68	2	no	—
321	C₂²	#11763001	1	2	yes	—
322	C₂²	C₂×#10587624	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	8	4	no	—
326	C₄	—	68	2	no	—
327	C₄	#11763001	1	2	yes	—
328	C₄	#11763081	1	2	yes	—
329	C₄	C₂²×#63228	1	1	yes	characteristic
330	C₄	C₂×#10587618	1	1	yes	characteristic
331	C₄	#11763622	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	136	2	no	—
334	C₂	C₂×#11763001	1	1	yes	characteristic
335	C₂	#14538435	1	1	yes	characteristic
336	C₂	C₂×#11763081	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#10587624
commutator, cosocle	C₄²×C₁₇	C₂⁴
Frattini	C₄²	C₂³×D₁₇
Fitting	C₂×C₁₇×#494	C₂
socle	C₂²×C₁₇	C₂×#121
prime radical	#11763359	C₂²
hypercenter	C₂×#494	D₁₇
hypocenter	C₁₇	#19188
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #14528270	C₂	semidirect
4. #14528146	C₂	semidirect
5. #14527532	C₂	semidirect
6. #14527456	C₂	semidirect
7. C₂×C₁₇×#494	C₂	semidirect
149. C₂×#494	D₁₇	semidirect
241. C₁₇	#19188	semidirect

Maximal subgroups

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

subgroup	index
#14539905	2
#14537497	2
#14528270	2
#14528146	2
#14527532	2
#14527456	2
C₂×C₁₇×#494	2
#19188	17