Subgroups — #16927445

Order 4352.

482 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₆₈	C₂	1	1	yes	maximal, characteristic
2	#14527532	C₂	1	1	yes	maximal, characteristic, complemented
3	#14527208	C₂	1	2	yes	maximal
4	#14526679	C₂	1	2	yes	maximal, complemented
5	C₁₇×#1848	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₁₇	—	2	1	no	—
7	C₂×C₄×D₆₈	C₂²	1	1	yes	characteristic
8	C₈×D₆₈	—	2	1	no	—
9	C₈×D₆₈	—	2	1	no	—
10	C₈×D₆₈	C₄	1	2	yes	—
11	C₈×D₆₈	C₄	1	2	yes	—
12	#11762862	C₂²	1	2	yes	—
13	C₂×C₄⋊#3622	C₂²	1	1	yes	characteristic
14	C₂×#10587491	—	2	1	no	—
15	#11762562	—	2	1	no	—
16	#11762560	—	2	1	no	—
17	#11762536	—	2	4	no	—
18	C₂×C₁₇×#108	—	2	1	no	—
19	C₂×C₁₇×#108	—	2	1	no	—
20	C₁₇×#423	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
22	C₂²×D₆₈	Q₈	1	1	yes	characteristic
23	C₂²×C₄×D₁₇	—	2	1	no	—
24	C₂×M₁₆×C₁₇	—	2	2	no	—
25	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
28	C₂×#63227	—	2	1	no	—
29	C₂×C₄⋊Dic₁₇	Q₈	1	1	yes	characteristic
30	C₂²×#3622	—	2	1	no	—
31	C₂×C₈×D₁₇	—	2	2	no	—
32	C₂×C₈×D₁₇	—	2	2	no	—
33	C₂×C₈×D₁₇	—	4	2	no	—
34	C₂×C₈×D₁₇	—	4	2	no	—
35	C₄×D₆₈	—	2	1	no	—
36	C₄×D₆₈	—	2	1	no	—
37	C₄×D₆₈	C₂×C₄	1	2	yes	—
38	C₄×D₆₈	D₄	1	2	yes	—
39	C₁₇×#108	—	4	2	no	—
40	C₁₇×#108	—	4	2	no	—
41	C₁₇×C₄⋊C₈	—	2	2	no	—
42	C₁₇×C₄⋊C₈	—	2	2	no	—
43	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
47	C₂²⋊#3622	—	2	2	no	—
48	#10587491	—	2	2	no	—
49	#10587491	—	2	2	no	—
50	C₄⋊#3622	—	2	1	no	—
51	C₄⋊#3622	—	2	2	no	—
52	C₄⋊#3622	C₂×C₄	1	2	yes	—
53	C₄×#3622	—	2	2	no	—
54	C₂³×D₁₇	—	2	1	no	—
55	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
58	C₂²×Dic₁₇	—	2	1	no	—
59	C₂×D₆₈	—	2	1	no	—
60	C₂×D₆₈	—	2	4	no	—
61	C₂×D₆₈	C₄⋊C₄	1	2	yes	—
62	C₂×C₄×D₁₇	—	2	2	no	—
63	C₂×C₄×D₁₇	—	2	2	no	—
64	C₂×C₄×D₁₇	—	4	1	no	—
65	C₂×C₄×D₁₇	—	4	1	no	—
66	C₂×C₄×D₁₇	—	4	1	no	—
67	C₂×C₄×D₁₇	—	4	1	no	—
68	M₁₆×C₁₇	—	4	2	no	—
69	M₁₆×C₁₇	—	4	2	no	—
70	C₂×C₈×C₁₇	—	2	1	no	—
71	C₂×C₈×C₁₇	—	2	1	no	—
72	C₂×C₈×C₁₇	—	4	2	no	—
73	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₇	D₈	1	2	yes	—
78	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
79	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₁₇	#41	1	1	yes	characteristic
83	#63227	—	2	2	no	—
84	#63227	—	2	2	no	—
85	C₄⋊Dic₁₇	—	2	1	no	—
86	C₄⋊Dic₁₇	C₄⋊C₄	1	2	yes	—
87	C₂×#3622	—	2	2	no	—
88	C₂×#3622	—	4	1	no	—
89	C₂×#3622	—	4	1	no	—
90	C₂×#3622	—	4	2	no	—
91	C₈×D₁₇	—	4	4	no	—
92	C₈×D₁₇	—	4	4	no	—
93	#8658	—	17	1	no	maximal, Sylow
94	C₂³×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂²×D₁₇	—	2	4	no	—
96	C₂²×D₁₇	—	2	2	no	—
97	C₂²×D₁₇	—	4	2	no	—
98	C₂²×D₁₇	—	4	1	no	—
99	C₂²×D₁₇	—	4	1	no	—
100	C₂×C₄×C₁₇	—	2	2	no	—
101	C₂×C₄×C₁₇	—	4	1	no	—
102	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
105	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
106	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
107	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
112	C₂×Dic₁₇	—	2	2	no	—
113	C₂×Dic₁₇	—	4	1	no	—
114	C₂×Dic₁₇	—	4	1	no	—
115	D₆₈	—	2	4	no	—
116	D₆₈	—	2	2	no	—
117	D₆₈	#108	1	4	yes	complemented
118	C₄×D₁₇	—	4	2	no	—
119	C₄×D₁₇	—	4	2	no	—
120	C₄×D₁₇	—	4	2	no	—
121	C₄×D₁₇	—	4	2	no	—
122	C₈×C₁₇	—	2	1	no	—
123	C₈×C₁₇	—	2	1	no	—
124	C₈×C₁₇	—	8	2	no	—
125	C₈×C₁₇	#102	1	2	yes	—
126	C₈×C₁₇	#102	1	2	yes	—
127	#3622	—	4	2	no	—
128	#3622	—	8	2	no	—
129	C₂×C₈×D₄	—	17	1	no	—
130	#1848	D₁₇	1	1	yes	characteristic, complemented
131	#1564	—	17	1	no	—
132	#1486	—	17	2	no	—
133	#1301	—	17	2	no	—
134	C₂²×C₁₇	—	2	1	no	—
135	C₂²×C₁₇	—	2	1	no	—
136	C₂²×C₁₇	#380	1	1	yes	characteristic
137	C₂²×C₁₇	#474	1	1	yes	characteristic
138	C₂²×C₁₇	#476	1	1	yes	characteristic
139	C₂×D₁₇	—	2	4	no	—
140	C₂×D₁₇	—	4	4	no	—
141	C₂×D₁₇	—	4	4	no	—
142	C₂×D₁₇	—	4	2	no	—
143	C₂×D₁₇	—	4	2	no	—
144	C₄×C₁₇	—	2	1	no	—
145	C₄×C₁₇	—	2	1	no	—
146	C₄×C₁₇	—	2	1	no	—
147	C₄×C₁₇	C₂×#102	1	1	yes	characteristic
148	C₄×C₁₇	#417	1	1	yes	characteristic
149	C₄×C₁₇	C₂×#108	1	2	yes	—
150	C₄×C₁₇	#459	1	1	yes	characteristic
151	C₄×C₁₇	#461	1	1	yes	characteristic
152	Dic₁₇	—	4	2	no	—
153	C₂³×C₈	—	34	1	no	—
154	C₂×C₄×D₄	—	17	1	no	—
155	C₈×D₄	—	17	2	no	—
156	C₈×D₄	—	17	2	no	—
157	C₈×D₄	—	34	1	no	—
158	C₈×D₄	—	34	1	no	—
159	#431	—	17	2	no	—
160	C₂×#108	—	2	1	no	—
161	C₂×#108	—	2	1	no	—
162	#423	C₂×D₁₇	1	2	yes	—
163	C₂×C₄⋊C₈	—	17	1	no	—
164	C₂×C₂²⋊C₈	—	34	1	no	—
165	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
166	#335	—	34	1	no	—
167	#334	—	34	1	no	—
168	D₄⋊C₈	—	34	4	no	—
169	C₂×C₁₇	—	2	1	no	—
170	C₂×C₁₇	—	2	1	no	—
171	C₂×C₁₇	#1802	1	1	yes	characteristic
172	C₂×C₁₇	#1810	1	1	yes	characteristic
173	C₂×C₁₇	#1814	1	1	yes	characteristic
174	D₁₇	—	4	4	no	—
175	C₂²×D₄	—	17	1	no	—
176	C₂³×C₄	—	34	1	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂²×C₈	—	34	1	no	—
179	C₂²×C₈	—	34	2	no	—
180	C₂²×C₈	—	34	2	no	—
181	C₂²×C₈	—	68	2	no	—
182	C₂²×C₈	—	68	2	no	—
183	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
184	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
185	C₄×D₄	—	17	2	no	—
186	C₄×D₄	—	17	2	no	—
187	C₄×D₄	—	34	1	no	—
188	C₄×D₄	—	34	1	no	—
189	C₂×C₄⋊C₄	—	17	1	no	—
190	C₂×C₂²⋊C₄	—	34	1	no	—
191	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
192	#108	—	4	2	no	—
193	#108	—	4	2	no	—
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	17	2	no	—
197	C₄⋊C₈	—	34	1	no	—
198	C₄⋊C₈	—	34	2	no	—
199	C₂²⋊C₈	—	34	2	no	—
200	C₂²⋊C₈	—	34	2	no	—
201	C₂²⋊C₈	—	34	2	no	—
202	C₄×C₈	—	34	2	no	—
203	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
204	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
205	C₄×C₈	D₆₈	1	1	yes	characteristic
206	C₄×C₈	D₆₈	1	1	yes	characteristic
207	C₁₇	#8658	1	1	yes	characteristic, complemented, Sylow
208	C₂⁴	—	34	1	no	—
209	C₂×D₄	—	17	2	no	—
210	C₂×D₄	—	34	4	no	—
211	C₂×D₄	—	34	1	no	—
212	C₂²×C₄	—	34	1	no	—
213	C₂²×C₄	—	34	2	no	—
214	C₂²×C₄	—	34	2	no	—
215	C₂²×C₄	—	68	1	no	—
216	C₂²×C₄	—	68	1	no	—
217	C₂²×C₄	—	68	1	no	—
218	C₂²×C₄	—	68	1	no	—
219	C₂²×C₄	Q₈×D₁₇	1	1	yes	characteristic
220	C₂²×C₄	#63256	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
222	M₁₆	—	4	2	no	—
223	M₁₆	—	4	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	34	2	no	—
228	C₂×C₈	—	68	1	no	—
229	C₂×C₈	—	68	1	no	—
230	C₂×C₈	—	68	4	no	—
231	C₂×C₈	—	68	4	no	—
232	C₂×C₈	—	68	2	no	—
233	C₂×C₈	#63227	1	1	yes	characteristic
234	C₂×C₈	#63227	1	1	yes	characteristic
235	C₂×C₈	#63227	1	1	yes	characteristic
236	C₂×C₈	#63227	1	1	yes	characteristic
237	C₂×C₈	#63228	1	2	yes	—
238	C₂×C₈	#63230	1	2	yes	—
239	C₄⋊C₄	—	17	2	no	—
240	C₄⋊C₄	—	34	1	no	—
241	C₂²⋊C₄	—	34	2	no	—
242	C₂²⋊C₄	—	34	2	no	—
243	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
244	C₄²	C₂×D₆₈	1	1	yes	characteristic
245	C₄²	D₄×D₁₇	1	1	yes	characteristic
246	C₄²	#63254	1	1	yes	characteristic
247	C₂³	—	34	4	no	—
248	C₂³	—	34	2	no	—
249	C₂³	—	68	2	no	—
250	C₂³	—	68	1	no	—
251	C₂³	—	68	1	no	—
252	C₂³	#10587632	1	1	yes	characteristic
253	D₄	—	17	4	no	—
254	D₄	—	34	4	no	—
255	D₄	—	34	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	34	2	no	—
259	C₂×C₄	—	68	1	no	—
260	C₂×C₄	—	68	1	no	—
261	C₂×C₄	—	68	2	no	—
262	C₂×C₄	—	68	2	no	—
263	C₂×C₄	—	68	2	no	—
264	C₂×C₄	—	68	2	no	—
265	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
266	C₂×C₄	#10587578	1	1	yes	characteristic
267	C₂×C₄	#10587581	1	1	yes	characteristic
268	C₂×C₄	#10587583	1	1	yes	characteristic
269	C₂×C₄	C₂×#63227	1	1	yes	characteristic
270	C₂×C₄	C₂×#63228	1	1	yes	characteristic
271	C₂×C₄	#10587618	1	1	yes	characteristic
272	C₂×C₄	#10587624	1	1	yes	characteristic
273	C₂×C₄	C₂×#63230	1	1	yes	characteristic
274	C₂×C₄	#10587628	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	8	2	no	—
278	C₈	—	68	2	no	—
279	C₈	—	136	2	no	—
280	C₈	#10587480	1	2	yes	—
281	C₈	#10587480	1	2	yes	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	34	4	no	—
285	C₂²	—	68	4	no	—
286	C₂²	—	68	4	no	—
287	C₂²	—	68	2	no	—
288	C₂²	—	68	2	no	—
289	C₂²	#11762666	1	1	yes	characteristic
290	C₂²	#11763086	1	1	yes	characteristic
291	C₂²	#11763096	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	68	2	no	—
296	C₄	C₂×#10587480	1	1	yes	characteristic
297	C₄	#11762808	1	1	yes	characteristic
298	C₄	#108×D₁₇	1	2	yes	—
299	C₄	#11763001	1	1	yes	characteristic
300	C₄	#11763011	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	68	4	no	—
304	C₂	#14528781	1	1	yes	characteristic
305	C₂	#14528819	1	1	yes	characteristic
306	C₂	#14528839	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10587624
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1848	C₂
socle	C₂²×C₁₇	#380
prime radical	C₂²×D₆₈	Q₈
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1848	D₁₇
hypocenter	C₁₇	#8658
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
2. #14527532	C₂	semidirect
4. #14526679	C₂	semidirect
5. C₁₇×#1848	C₂	semidirect
117. D₆₈	#108	semidirect
130. #1848	D₁₇	semidirect
207. C₁₇	#8658	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₆₈	2
#14527532	2
#14527208	2
#14526679	2
C₁₇×#1848	2
#8658	17