Subgroups — #17018202

Order 4352.

606 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539370	C₂	1	1	yes	maximal, characteristic
2	#14538914	C₂	1	2	yes	maximal
3	#14528214	C₂	1	4	yes	maximal, complemented
4	#14528178	C₂	1	2	yes	maximal, complemented
5	#14528154	C₂	1	2	yes	maximal, complemented
6	#14527547	C₂	1	1	yes	maximal, characteristic
7	#14527482	C₂	1	2	yes	maximal
8	C₂×C₈×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic
9	#11763366	C₂²	1	2	yes	—
10	#11763214	C₂²	1	1	yes	characteristic
11	#11763189	C₂²	1	1	yes	characteristic
12	#11763160	C₂²	1	2	yes	—
13	#11763139	—	2	2	no	—
14	#11763094	C₂²	1	4	yes	—
15	#11763050	—	2	1	no	—
16	#11763050	—	2	1	no	—
17	#11763019	C₂²	1	4	yes	—
18	#11762965	—	2	2	no	—
19	#11762914	—	2	4	no	—
20	C₄×Dic₆₈	—	2	1	no	—
21	C₄×Dic₆₈	—	2	1	no	—
22	#11762866	C₂²	1	2	yes	—
23	#11762562	C₂²	1	2	yes	—
24	#11762560	C₂²	1	2	yes	—
25	#11762556	C₂²	1	4	yes	—
26	#11762554	C₂²	1	4	yes	—
27	C₂×C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
28	C₈×Q₈×C₁₇	—	2	1	no	—
29	C₈×Q₈×C₁₇	—	2	1	no	—
30	C₈×Q₈×C₁₇	C₂²	1	2	yes	—
31	C₈×Q₈×C₁₇	C₂²	1	2	yes	—
32	C₂×C₁₇×C₄⋊C₈	—	2	1	no	—
33	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₁₇	—	2	1	no	—
35	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
36	C₂²×Q₈×C₁₇	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₇	—	2	1	no	—
38	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
40	C₄×Q₈×C₁₇	—	2	1	no	—
41	C₄×Q₈×C₁₇	—	2	1	no	—
42	C₄×Q₈×C₁₇	D₄	1	2	yes	—
43	C₄×Q₈×C₁₇	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
45	C₂×C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₁₇	—	2	1	no	—
47	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
48	C₂×#63231	—	2	4	no	—
49	#10587612	—	2	2	no	—
50	#10587610	—	2	2	no	—
51	#10587610	—	2	2	no	—
52	C₂×Dic₆₈	—	2	1	no	—
53	C₂×Dic₆₈	—	2	1	no	—
54	#10587556	—	2	2	no	—
55	#10587555	C₂³	1	2	yes	—
56	#10587555	C₂³	1	2	yes	—
57	C₄×Dic₃₄	—	2	2	no	—
58	C₄×Dic₃₄	—	2	2	no	—
59	C₁₇×C₄⋊C₈	—	2	2	no	—
60	C₁₇×C₄⋊C₈	—	2	2	no	—
61	C₁₇×C₄⋊C₈	—	2	1	no	—
62	C₁₇×C₄⋊C₈	C₂³	1	2	yes	—
63	C₄×C₈×C₁₇	—	2	2	no	—
64	C₄×C₈×C₁₇	—	2	2	no	—
65	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
66	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
67	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
69	#10587506	—	2	4	no	—
70	C₂²⋊#3622	—	2	2	no	—
71	#10587489	—	2	1	no	—
72	#10587489	—	2	1	no	—
73	#10587487	—	2	2	no	—
74	#10587487	—	2	2	no	—
75	#10587481	—	2	4	no	—
76	#10587479	—	2	4	no	—
77	C₄⋊#3622	C₂³	1	4	yes	—
78	#10587474	—	2	2	no	—
79	C₂×Q₈×C₁₇	—	2	4	no	—
80	C₂×Q₈×C₁₇	—	2	1	no	—
81	C₂×Q₈×C₁₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₁₇	—	2	1	no	—
83	C₂²×C₄×C₁₇	—	2	1	no	—
84	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
87	C₂×Dic₃₄	—	2	2	no	—
88	C₂×Dic₃₄	—	2	2	no	—
89	C₂×C₈×C₁₇	—	2	2	no	—
90	C₂×C₈×C₁₇	—	2	2	no	—
91	C₂×C₈×C₁₇	—	4	1	no	—
92	C₂×C₈×C₁₇	—	4	2	no	—
93	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
97	C₄⋊C₄×C₁₇	—	2	2	no	—
98	C₄⋊C₄×C₁₇	—	2	1	no	—
99	C₄⋊C₄×C₁₇	—	2	2	no	—
100	C₄⋊C₄×C₁₇	C₂×D₄	1	2	yes	—
101	C₄²×C₁₇	—	2	2	no	—
102	C₄²×C₁₇	—	2	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	C₂²⋊Dic₁₇	—	4	2	no	—
108	#63231	—	4	8	no	—
109	C₄⋊Dic₁₇	—	2	2	no	—
110	C₄⋊Dic₁₇	—	2	2	no	—
111	C₄⋊Dic₁₇	—	2	2	no	—
112	C₄⋊Dic₁₇	—	2	2	no	—
113	#63225	—	4	2	no	—
114	#63225	—	4	2	no	—
115	C₄×Dic₁₇	—	4	2	no	—
116	C₂×#3622	—	2	4	no	—
117	Dic₆₈	—	4	2	no	—
118	Dic₆₈	—	4	2	no	—
119	#19265	—	17	1	no	maximal, Sylow
120	C₂³×C₁₇	#121	1	1	yes	characteristic
121	Q₈×C₁₇	—	2	4	no	—
122	Q₈×C₁₇	—	2	2	no	—
123	Q₈×C₁₇	#135	1	4	yes	—
124	C₂×C₄×C₁₇	—	2	4	no	—
125	C₂×C₄×C₁₇	—	2	2	no	—
126	C₂×C₄×C₁₇	—	2	2	no	—
127	C₂×C₄×C₁₇	—	2	2	no	—
128	C₂×C₄×C₁₇	—	2	1	no	—
129	C₂×C₄×C₁₇	—	2	1	no	—
130	C₂×C₄×C₁₇	—	2	2	no	—
131	C₂×C₄×C₁₇	—	4	2	no	—
132	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
133	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
134	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
135	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
138	C₂×Dic₁₇	—	4	2	no	—
139	C₂×Dic₁₇	—	4	2	no	—
140	Dic₃₄	—	4	4	no	—
141	Dic₃₄	—	4	2	no	—
142	Dic₃₄	—	4	2	no	—
143	C₈×C₁₇	—	2	1	no	—
144	C₈×C₁₇	—	2	1	no	—
145	C₈×C₁₇	—	2	1	no	—
146	C₈×C₁₇	—	4	2	no	—
147	C₈×C₁₇	#137	1	2	yes	—
148	#3622	—	4	4	no	—
149	#3170	—	17	1	no	—
150	#3113	—	17	2	no	—
151	C₂×C₈×Q₈	D₁₇	1	1	yes	characteristic
152	#1680	—	17	4	no	—
153	#1674	—	17	2	no	—
154	#1670	—	17	2	no	—
155	#1568	—	17	1	no	—
156	#1552	—	17	2	no	—
157	C₂²×C₁₇	—	2	1	no	—
158	C₂²×C₁₇	—	2	1	no	—
159	C₂²×C₁₇	#473	1	2	yes	—
160	C₂²×C₁₇	C₂×#121	1	1	yes	characteristic
161	C₄×C₁₇	—	2	1	no	—
162	C₄×C₁₇	—	2	4	no	—
163	C₄×C₁₇	—	2	1	no	—
164	C₄×C₁₇	—	2	1	no	—
165	C₄×C₁₇	—	2	1	no	—
166	C₄×C₁₇	—	4	2	no	—
167	C₄×C₁₇	C₂×#135	1	2	yes	—
168	C₄×C₁₇	C₂×#137	1	1	yes	characteristic
169	C₄×C₁₇	#575	1	1	yes	characteristic
170	Dic₁₇	—	8	2	no	—
171	Dic₁₇	—	8	2	no	—
172	#533	—	17	2	no	—
173	C₂×C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
174	#500	—	17	1	no	—
175	#494	—	17	1	no	—
176	#487	—	17	2	no	—
177	#483	—	34	2	no	—
178	#475	—	17	4	no	—
179	#467	—	34	1	no	—
180	#467	—	34	1	no	—
181	#462	—	17	4	no	—
182	#451	—	34	2	no	—
183	C₈×Q₈	—	2	1	no	—
184	C₈×Q₈	—	2	1	no	—
185	C₈×Q₈	C₂×D₁₇	1	2	yes	—
186	C₈×Q₈	C₂×D₁₇	1	2	yes	—
187	#441	—	34	4	no	—
188	C₄×Q₁₆	—	34	1	no	—
189	C₄×Q₁₆	—	34	1	no	—
190	#432	—	17	2	no	—
191	C₂×C₄⋊C₈	—	2	1	no	—
192	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
193	C₂×C₄×C₈	—	2	1	no	—
194	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
195	#335	—	17	2	no	—
196	#334	—	17	2	no	—
197	#332	—	17	4	no	—
198	#332	—	17	4	no	—
199	C₂×C₁₇	—	2	2	no	—
200	C₂×C₁₇	#3037	1	1	yes	characteristic
201	C₂×C₁₇	#3050	1	1	yes	characteristic
202	C₂×C₁₇	C₂×#473	1	1	yes	characteristic
203	C₂²×Q₈	D₆₈	1	1	yes	characteristic
204	C₂×Q₁₆	—	34	4	no	—
205	C₂×Q₁₆	—	34	1	no	—
206	C₂×Q₁₆	—	34	1	no	—
207	C₂²×C₈	—	2	1	no	—
208	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
209	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
210	C₄⋊Q₈	—	17	2	no	—
211	C₄⋊Q₈	—	17	2	no	—
212	#125	—	34	2	no	—
213	C₂²⋊Q₈	—	34	2	no	—
214	C₂²⋊Q₈	—	34	2	no	—
215	C₄×Q₈	—	2	1	no	—
216	C₄×Q₈	—	2	1	no	—
217	C₄×Q₈	—	34	2	no	—
218	C₄×Q₈	—	34	2	no	—
219	C₄×Q₈	D₆₈	1	2	yes	—
220	C₄×Q₈	C₂²×D₁₇	1	2	yes	—
221	#117	—	34	2	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	D₆₈	1	1	yes	characteristic
224	C₂×C₄²	—	2	1	no	—
225	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
226	#107	—	34	1	no	—
227	#107	—	34	1	no	—
228	#106	—	34	4	no	—
229	C₄⋊C₈	—	2	2	no	—
230	C₄⋊C₈	—	2	2	no	—
231	C₄⋊C₈	—	2	1	no	—
232	C₄⋊C₈	—	17	4	no	—
233	C₄⋊C₈	C₂²×D₁₇	1	2	yes	—
234	#103	—	34	4	no	—
235	#103	—	34	2	no	—
236	#103	—	34	2	no	—
237	#103	—	34	4	no	—
238	C₂²⋊C₈	—	34	2	no	—
239	#97	—	34	2	no	—
240	C₄×C₈	—	2	2	no	—
241	C₄×C₈	—	2	2	no	—
242	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
243	C₄×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
246	C₁₇	#19265	1	1	yes	characteristic, complemented, Sylow
247	C₂×Q₈	—	2	4	no	—
248	C₂×Q₈	—	2	1	no	—
249	C₂×Q₈	—	34	2	no	—
250	C₂×Q₈	—	34	2	no	—
251	C₂×Q₈	C₂×D₆₈	1	2	yes	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
255	C₂²×C₄	#63252	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
257	Q₁₆	—	68	2	no	—
258	Q₁₆	—	68	8	no	—
259	Q₁₆	—	68	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	—	34	4	no	—
265	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
266	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
267	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
268	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	34	2	no	—
273	C₄⋊C₄	—	34	2	no	—
274	C₄⋊C₄	—	34	2	no	—
275	C₄⋊C₄	—	34	2	no	—
276	C₄⋊C₄	—	68	2	no	—
277	C₄⋊C₄	—	68	2	no	—
278	C₄⋊C₄	C₂×D₆₈	1	2	yes	—
279	C₂²⋊C₄	—	68	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	68	2	no	—
283	C₄²	C₂×D₆₈	1	1	yes	characteristic
284	C₄²	#63252	1	1	yes	characteristic
285	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
286	C₄²	C₂³×D₁₇	1	1	yes	characteristic
287	C₂³	#10587616	1	1	yes	characteristic
288	Q₈	—	2	4	no	—
289	Q₈	—	2	2	no	—
290	Q₈	—	68	4	no	—
291	Q₈	—	68	2	no	—
292	Q₈	—	68	2	no	—
293	Q₈	#10587590	1	4	yes	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	68	2	no	—
303	C₂×C₄	—	68	2	no	—
304	C₂×C₄	#10587616	1	1	yes	characteristic
305	C₂×C₄	#10587616	1	1	yes	characteristic
306	C₂×C₄	#10587616	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
308	C₂×C₄	C₂×#63252	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	—	2	1	no	—
313	C₈	—	4	2	no	—
314	C₈	—	68	4	no	—
315	C₈	#10587628	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	#11763084	1	2	yes	—
319	C₂²	C₂×#10587616	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	4	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	136	2	no	—
327	C₄	—	136	2	no	—
328	C₄	C₂×#10587590	1	2	yes	—
329	C₄	C₂×#10587628	1	1	yes	characteristic
330	C₄	#11763607	1	1	yes	characteristic
331	C₂	—	2	2	no	—
332	C₂	#14538256	1	1	yes	characteristic
333	C₂	#14538365	1	1	yes	characteristic
334	C₂	C₂×#11763084	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#10587616
commutator, cosocle	C₄²×C₁₇	C₂⁴
Frattini	C₄²	C₂³×D₁₇
Fitting	C₂×C₈×Q₈×C₁₇	C₂
socle	C₂²×C₁₇	C₂×#121
prime radical	C₂³×C₁₇	#121
hypercenter	C₂×C₈×Q₈	D₁₇
hypocenter	C₁₇	#19265
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. #14528214	C₂	semidirect
4. #14528178	C₂	semidirect
5. #14528154	C₂	semidirect
246. C₁₇	#19265	semidirect

Maximal subgroups

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

subgroup	index
#14539370	2
#14538914	2
#14528214	2
#14528178	2
#14528154	2
#14527547	2
#14527482	2
C₂×C₈×Q₈×C₁₇	2
#19265	17