Subgroups — #17967310

Order 4352.

560 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545689	C₂	1	2	yes	maximal
2	#14544431	C₂	1	2	yes	maximal, complemented
3	#14544424	C₂	1	2	yes	maximal, complemented
4	#14544377	C₂	1	2	yes	maximal, complemented
5	Q₁₆×#3624	C₂	1	2	yes	maximal, complemented
6	#14544345	C₂	1	1	yes	maximal, characteristic
7	#14544147	C₂	1	2	yes	maximal
8	#14543988	C₂	1	1	yes	maximal, characteristic
9	#14539368	C₂	1	1	yes	maximal, characteristic
10	#11763861	—	2	4	no	—
11	#11763860	C₂²	1	4	yes	—
12	Q₈×#3624	C₂²	1	4	yes	—
13	#11763834	—	2	4	no	—
14	#11763831	C₂²	1	2	yes	—
15	#11763827	C₂²	1	1	yes	characteristic
16	#11763827	C₂²	1	1	yes	characteristic
17	#11763672	C₂²	1	2	yes	—
18	#11763670	C₂²	1	2	yes	—
19	Q₈⋊#3624	C₂²	1	4	yes	—
20	Q₈⋊#3624	C₂²	1	4	yes	—
21	#11763650	C₂²	1	1	yes	characteristic
22	#11763650	C₂²	1	1	yes	characteristic
23	C₈×#3624	C₂²	1	1	yes	characteristic
24	C₈×#3624	C₂²	1	1	yes	characteristic
25	#11763364	C₂²	1	2	yes	—
26	#11763214	C₄	1	1	yes	characteristic
27	#11763159	C₄	1	2	yes	—
28	#11763047	C₄	1	2	yes	—
29	#11763047	C₄	1	2	yes	—
30	Q₁₆×Dic₁₇	C₂²	1	2	yes	—
31	Q₁₆×Dic₁₇	C₂²	1	2	yes	—
32	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
33	C₁₇×#494	C₄	1	1	yes	characteristic
34	C₂×Q₁₆×C₁₇	C₂×C₄	1	2	yes	—
35	C₂×Q₁₆×C₁₇	C₂×C₄	1	2	yes	—
36	C₁₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
37	#10587632	C₂×C₄	1	4	yes	—
38	Q₈×Dic₁₇	C₂³	1	4	yes	—
39	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
41	#10587583	—	2	2	no	—
42	#10587578	—	2	2	no	—
43	#10587576	—	2	2	no	—
44	#10587573	—	2	2	no	—
45	C₄²×D₁₇	C₂³	1	1	yes	characteristic
46	#10587555	C₂×C₄	1	2	yes	—
47	#10587544	—	2	1	no	—
48	#10587544	—	2	4	no	—
49	#10587544	D₄	1	2	yes	—
50	#10587543	—	2	2	no	—
51	#10587543	—	2	1	no	—
52	#10587543	D₄	1	2	yes	—
53	#10587542	—	2	2	no	—
54	#10587542	—	2	4	no	—
55	C₄⋊#3624	—	2	4	no	—
56	C₄⋊#3624	C₂³	1	4	yes	—
57	C₄×#3624	—	2	4	no	—
58	C₄×#3624	C₂³	1	1	yes	characteristic
59	C₄×#3624	C₂³	1	1	yes	characteristic
60	C₄×#3624	C₂³	1	1	yes	characteristic
61	C₄×#3624	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
63	#10587506	C₂×C₄	1	4	yes	—
64	#10587506	C₂×C₄	1	4	yes	—
65	#10587489	C₂×C₄	1	2	yes	—
66	#10587489	C₂×C₄	1	2	yes	—
67	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
68	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
69	C₄×#3622	C₂×C₄	1	1	yes	characteristic
70	C₂×Q₈×C₁₇	C₂²×C₄	1	4	yes	—
71	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×D₁₇	#41	1	1	yes	characteristic
74	C₂×Dic₃₄	—	2	2	no	—
75	C₂×#3624	—	2	4	no	—
76	C₂×#3624	—	2	1	no	—
77	C₂×#3624	—	2	1	no	—
78	C₂×#3624	C₂×D₄	1	2	yes	—
79	C₂×#3624	C₂×D₄	1	2	yes	—
80	Q₁₆×C₁₇	—	2	4	no	—
81	Q₁₆×C₁₇	—	2	4	no	—
82	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
84	C₄⋊C₄×C₁₇	—	2	2	no	—
85	C₄⋊C₄×C₁₇	—	2	2	no	—
86	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
87	#63227	—	2	4	no	—
88	C₄⋊Dic₁₇	—	2	2	no	—
89	C₄⋊Dic₁₇	—	2	4	no	—
90	C₄⋊Dic₁₇	C₂²×C₄	1	4	yes	—
91	#63225	—	2	4	no	—
92	C₄×Dic₁₇	—	2	4	no	—
93	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
94	C₄×Dic₁₇	#41	1	1	yes	characteristic
95	C₄×Dic₁₇	C₂⁴	1	1	yes	characteristic
96	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
97	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
98	C₈×D₁₇	—	2	2	no	—
99	C₈×D₁₇	—	2	2	no	—
100	#17972	—	17	1	no	maximal, Sylow
101	C₂²×D₁₇	C₄×D₄	1	1	yes	characteristic
102	Q₈×C₁₇	—	2	4	no	—
103	Q₈×C₁₇	—	2	4	no	—
104	C₂×C₄×C₁₇	—	2	4	no	—
105	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
108	C₂×Dic₁₇	—	2	4	no	—
109	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
110	C₂×Dic₁₇	C₂²×D₄	1	1	yes	characteristic
111	C₂×Dic₁₇	C₂×#41	1	1	yes	characteristic
112	C₄×D₁₇	—	2	1	no	—
113	C₄×D₁₇	—	2	1	no	—
114	C₄×D₁₇	—	2	2	no	—
115	C₄×D₁₇	—	4	1	no	—
116	Dic₃₄	—	8	2	no	—
117	#3624	—	2	2	no	—
118	#3624	—	2	2	no	—
119	#3624	—	4	1	no	—
120	#3624	—	4	1	no	—
121	#3624	—	4	4	no	—
122	C₈×C₁₇	#131	1	2	yes	—
123	C₈×C₁₇	#131	1	2	yes	—
124	#3622	—	2	1	no	—
125	#3622	—	2	1	no	—
126	#3170	—	17	1	no	—
127	#2985	—	17	2	no	—
128	#1593	—	17	2	no	—
129	#1591	—	17	2	no	—
130	#1579	—	17	2	no	—
131	C₈×Q₁₆	—	17	2	no	—
132	#1568	—	17	1	no	—
133	#1497	—	17	2	no	—
134	#1450	—	17	1	no	—
135	C₂²×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂×D₁₇	—	2	1	no	—
137	C₂×D₁₇	—	2	1	no	—
138	C₂×D₁₇	—	2	1	no	—
139	C₄×C₁₇	—	2	1	no	—
140	C₄×C₁₇	—	2	1	no	—
141	C₄×C₁₇	—	4	4	no	—
142	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
143	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
144	Dic₁₇	—	2	1	no	—
145	Dic₁₇	—	2	1	no	—
146	Dic₁₇	—	4	4	no	—
147	Dic₁₇	#578	1	2	yes	—
148	#533	—	17	2	no	—
149	#500	—	17	1	no	—
150	#494	#595	1	1	yes	characteristic
151	#487	—	17	2	no	—
152	#467	—	17	2	no	—
153	#467	—	17	2	no	—
154	#446	—	17	4	no	—
155	#446	—	34	4	no	—
156	C₈×Q₈	—	17	4	no	—
157	C₄×Q₁₆	—	17	2	no	—
158	C₄×Q₁₆	—	17	2	no	—
159	#433	—	34	4	no	—
160	#432	—	17	2	no	—
161	#431	—	17	1	no	—
162	#431	—	17	1	no	—
163	C₂×C₄×C₈	—	17	1	no	—
164	#335	—	17	2	no	—
165	#334	—	17	2	no	—
166	Q₈⋊C₈	—	17	4	no	—
167	Q₈⋊C₈	—	17	4	no	—
168	#322	—	17	1	no	—
169	#322	—	17	1	no	—
170	C₈²	—	17	1	no	—
171	C₈²	—	17	1	no	—
172	C₂×C₁₇	#2931	1	1	yes	characteristic
173	C₂×C₁₇	#2950	1	1	yes	characteristic
174	C₂×C₁₇	#2957	1	1	yes	characteristic
175	D₁₇	—	4	1	no	—
176	C₂×Q₁₆	C₂×#595	1	2	yes	—
177	C₂×Q₁₆	C₂×#595	1	2	yes	—
178	C₂²×C₈	—	17	1	no	—
179	C₂²×C₈	—	17	1	no	—
180	C₄⋊Q₈	—	17	2	no	—
181	C₄⋊Q₈	C₂×#595	1	2	yes	—
182	#125	—	34	2	no	—
183	C₂²⋊Q₈	—	17	4	no	—
184	C₂²⋊Q₈	—	34	2	no	—
185	C₄×Q₈	—	17	4	no	—
186	C₄×Q₈	—	34	2	no	—
187	#117	—	34	2	no	—
188	C₂×C₄²	—	17	1	no	—
189	#107	—	17	2	no	—
190	#107	—	17	2	no	—
191	C₄⋊C₈	—	17	2	no	—
192	C₄⋊C₈	—	17	4	no	—
193	C₄⋊C₈	—	34	1	no	—
194	C₄⋊C₈	—	34	4	no	—
195	C₄⋊C₈	—	34	4	no	—
196	#103	—	17	4	no	—
197	#103	—	17	4	no	—
198	C₂²⋊C₈	—	17	2	no	—
199	C₂²⋊C₈	—	34	1	no	—
200	C₂²⋊C₈	—	34	2	no	—
201	#97	—	34	2	no	—
202	#97	—	34	4	no	—
203	C₄×C₈	—	17	1	no	—
204	C₄×C₈	—	17	1	no	—
205	C₄×C₈	—	17	1	no	—
206	C₄×C₈	—	17	1	no	—
207	C₄×C₈	—	17	1	no	—
208	C₄×C₈	—	17	1	no	—
209	C₄×C₈	—	17	1	no	—
210	C₄×C₈	—	34	4	no	—
211	C₄×C₈	C₂×#595	1	1	yes	characteristic
212	C₁₇	#17972	1	1	yes	characteristic, complemented, Sylow
213	C₂×Q₈	—	34	2	no	—
214	C₂×Q₈	C₂²×#595	1	4	yes	—
215	C₂²×C₄	—	17	1	no	—
216	C₂²×C₄	—	17	1	no	—
217	C₂²×C₄	—	17	1	no	—
218	Q₁₆	—	2	4	no	—
219	Q₁₆	—	2	4	no	—
220	C₂×C₈	—	17	2	no	—
221	C₂×C₈	—	17	2	no	—
222	C₂×C₈	—	17	1	no	—
223	C₂×C₈	—	17	1	no	—
224	C₂×C₈	—	34	2	no	—
225	C₂×C₈	—	34	2	no	—
226	C₂×C₈	—	34	1	no	—
227	C₂×C₈	—	34	1	no	—
228	C₂×C₈	—	34	4	no	—
229	C₂×C₈	C₂²×#595	1	1	yes	characteristic
230	C₂×C₈	C₂²×#595	1	1	yes	characteristic
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	17	4	no	—
234	C₄⋊C₄	—	34	4	no	—
235	C₄⋊C₄	—	34	4	no	—
236	C₄⋊C₄	—	34	2	no	—
237	C₂²⋊C₄	—	34	4	no	—
238	C₄²	—	17	1	no	—
239	C₄²	—	17	1	no	—
240	C₄²	—	17	1	no	—
241	C₄²	—	34	4	no	—
242	C₄²	C₂²×#595	1	1	yes	characteristic
243	C₂³	—	17	1	no	—
244	Q₈	—	2	4	no	—
245	Q₈	—	2	4	no	—
246	Q₈	—	136	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	17	1	no	—
249	C₂×C₄	—	17	1	no	—
250	C₂×C₄	—	17	1	no	—
251	C₂×C₄	—	34	2	no	—
252	C₂×C₄	—	34	1	no	—
253	C₂×C₄	—	34	1	no	—
254	C₂×C₄	—	34	4	no	—
255	C₂×C₄	—	68	1	no	—
256	C₂×C₄	D₄×#595	1	1	yes	characteristic
257	C₂×C₄	D₄×#595	1	1	yes	characteristic
258	C₂×C₄	C₂³×#595	1	1	yes	characteristic
259	C₈	—	34	2	no	—
260	C₈	—	34	2	no	—
261	C₈	—	34	1	no	—
262	C₈	—	34	1	no	—
263	C₈	—	68	4	no	—
264	C₈	—	68	1	no	—
265	C₈	—	68	1	no	—
266	C₈	#10587678	1	2	yes	—
267	C₈	#10587678	1	2	yes	—
268	C₂²	—	34	1	no	—
269	C₂²	—	34	1	no	—
270	C₂²	—	34	1	no	—
271	C₂²	C₂×D₄×#595	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	4	4	no	—
275	C₄	—	17	2	no	—
276	C₄	—	34	1	no	—
277	C₄	—	34	1	no	—
278	C₄	—	68	4	no	—
279	C₄	C₂×#10587678	1	1	yes	characteristic
280	C₄	C₂×#10587678	1	1	yes	characteristic
281	C₂	—	68	1	no	—
282	C₂	#14545506	1	1	yes	characteristic
283	C₂	#14545545	1	1	yes	characteristic
284	C₂	#14545564	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#595
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×#595
Fitting	C₁₇×#494	C₄
socle	C₂²×C₁₇	C₂×C₄×D₄
prime radical	C₂²×D₁₇	C₄×D₄
cosocle	C₄×Dic₁₇	C₂⁴
hypercenter	#494	#595
hypocenter	C₁₇	#17972
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
2. #14544431	C₂	semidirect
3. #14544424	C₂	semidirect
4. #14544377	C₂	semidirect
5. Q₁₆×#3624	C₂	semidirect
212. C₁₇	#17972	semidirect

Maximal subgroups

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

subgroup	index
#14545689	2
#14544431	2
#14544424	2
#14544377	2
Q₁₆×#3624	2
#14544345	2
#14544147	2
#14543988	2
#14539368	2
#17972	17