Subgroups — #16922815

Order 4352.

410 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528637	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#3622	C₂	1	1	yes	maximal, characteristic
3	#14526610	C₂	1	2	yes	maximal
4	#14526607	C₂	1	2	yes	maximal
5	C₁₇×#2863	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#3622	—	2	1	no	—
7	C₂×C₄⋊#3622	—	2	1	no	—
8	C₂×#10587474	—	2	1	no	—
9	C₂×C₄×#3622	—	2	1	no	—
10	#11762564	C₂²	1	2	yes	—
11	#11762564	C₂²	1	2	yes	—
12	C₁₇×#391	C₄	1	1	yes	characteristic
13	C₁₇×#391	C₄	1	1	yes	characteristic
14	C₁₇×#391	C₂²	1	1	yes	characteristic
15	C₁₇×#389	C₄	1	1	yes	characteristic
16	C₁₇×#389	C₄	1	1	yes	characteristic
17	C₁₇×#389	C₂²	1	1	yes	characteristic
18	C₁₇×#387	—	2	1	no	—
19	C₁₇×#387	—	2	1	no	—
20	C₁₇×#387	—	2	1	no	—
21	C₁₇×#383	—	2	1	no	—
22	C₄³×C₁₇	C₂²	1	1	yes	characteristic
23	C₂²×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
25	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
26	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
27	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
37	C₂²×#3622	—	2	2	no	—
38	C₂²×#3622	—	2	2	no	—
39	C₁₇×#95	—	2	1	no	—
40	C₁₇×#95	—	2	1	no	—
41	C₁₇×#95	—	2	1	no	—
42	C₁₇×#95	—	2	1	no	—
43	C₁₇×#95	—	2	1	no	—
44	C₁₇×#95	—	2	1	no	—
45	C₁₇×#95	—	2	1	no	—
46	C₁₇×#95	—	2	1	no	—
47	C₁₇×#95	—	2	1	no	—
48	C₄⋊#3622	—	2	4	no	—
49	C₄⋊#3622	—	2	4	no	—
50	#10587474	—	4	2	no	—
51	C₄×#3622	—	4	2	no	—
52	C₂×Q₈×C₁₇	—	2	2	no	—
53	C₂×Q₈×C₁₇	—	4	1	no	—
54	C₂×Q₈×C₁₇	—	4	1	no	—
55	C₂²×C₄×C₁₇	—	2	1	no	—
56	C₂²×C₄×C₁₇	—	2	1	no	—
57	C₂²×C₄×C₁₇	—	2	1	no	—
58	C₂²×C₄×C₁₇	—	2	1	no	—
59	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₁₇	—	2	2	no	—
67	C₄⋊C₄×C₁₇	—	4	1	no	—
68	C₄⋊C₄×C₁₇	—	4	1	no	—
69	C₄⋊C₄×C₁₇	—	4	2	no	—
70	C₄⋊C₄×C₁₇	—	4	2	no	—
71	C₄⋊C₄×C₁₇	—	4	1	no	—
72	C₄⋊C₄×C₁₇	—	4	1	no	—
73	C₄²×C₁₇	—	2	2	no	—
74	C₄²×C₁₇	—	2	2	no	—
75	C₄²×C₁₇	—	2	2	no	—
76	C₄²×C₁₇	—	2	1	no	—
77	C₄²×C₁₇	—	2	1	no	—
78	C₄²×C₁₇	—	4	1	no	—
79	C₄²×C₁₇	—	4	1	no	—
80	C₄²×C₁₇	SD₁₆	1	2	yes	—
81	C₄²×C₁₇	Q₁₆	1	2	yes	—
82	C₂×#3622	—	2	4	no	—
83	C₂×#3622	—	2	4	no	—
84	C₂×#3622	—	4	2	no	—
85	C₂×#3622	—	4	2	no	—
86	C₂×#3622	—	4	2	no	—
87	C₂×#3622	—	4	2	no	—
88	#7689	—	17	1	no	maximal, Sylow
89	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	Q₈×C₁₇	—	8	2	no	—
91	C₂×C₄×C₁₇	—	2	1	no	—
92	C₂×C₄×C₁₇	—	2	1	no	—
93	C₂×C₄×C₁₇	—	2	1	no	—
94	C₂×C₄×C₁₇	—	2	1	no	—
95	C₂×C₄×C₁₇	—	2	1	no	—
96	C₂×C₄×C₁₇	—	2	1	no	—
97	C₂×C₄×C₁₇	—	2	1	no	—
98	C₂×C₄×C₁₇	—	2	1	no	—
99	C₂×C₄×C₁₇	—	2	1	no	—
100	C₂×C₄×C₁₇	—	2	1	no	—
101	C₂×C₄×C₁₇	—	2	1	no	—
102	C₂×C₄×C₁₇	—	2	1	no	—
103	C₂×C₄×C₁₇	—	2	1	no	—
104	C₂×C₄×C₁₇	—	2	1	no	—
105	C₂×C₄×C₁₇	—	4	1	no	—
106	C₂×C₄×C₁₇	—	4	1	no	—
107	C₂×C₄×C₁₇	—	4	1	no	—
108	C₂×C₄×C₁₇	—	4	1	no	—
109	C₂×C₄×C₁₇	—	4	1	no	—
110	C₂×C₄×C₁₇	—	4	1	no	—
111	C₂×C₄×C₁₇	—	4	1	no	—
112	C₂×C₄×C₁₇	—	4	1	no	—
113	C₂×C₄×C₁₇	—	4	1	no	—
114	C₂×C₄×C₁₇	—	4	1	no	—
115	C₂×C₄×C₁₇	—	4	1	no	—
116	C₂×C₄×C₁₇	—	4	1	no	—
117	C₂×C₄×C₁₇	#103	1	2	yes	—
118	C₂×C₄×C₁₇	#103	1	2	yes	—
119	C₂×C₄×C₁₇	#104	1	2	yes	—
120	C₂×C₄×C₁₇	#104	1	2	yes	—
121	C₂×C₄×C₁₇	#131	1	2	yes	—
122	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
124	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
126	#3622	—	4	4	no	—
127	#3622	—	4	4	no	—
128	#2863	D₁₇	1	1	yes	characteristic
129	#1767	—	17	1	no	—
130	C₄×C₄⋊C₈	—	17	1	no	—
131	#1272	—	17	2	no	—
132	#1271	—	17	2	no	—
133	C₂²×C₁₇	#408	1	1	yes	characteristic
134	C₂²×C₁₇	#410	1	1	yes	characteristic
135	C₂²×C₁₇	#413	1	1	yes	characteristic
136	C₂²×C₁₇	C₂×#103	1	1	yes	characteristic
137	C₂²×C₁₇	#417	1	1	yes	characteristic
138	C₂²×C₁₇	C₂×#104	1	1	yes	characteristic
139	C₂²×C₁₇	C₂×#104	1	1	yes	characteristic
140	C₄×C₁₇	—	2	2	no	—
141	C₄×C₁₇	—	2	2	no	—
142	C₄×C₁₇	—	2	2	no	—
143	C₄×C₁₇	—	2	2	no	—
144	C₄×C₁₇	—	2	1	no	—
145	C₄×C₁₇	—	2	1	no	—
146	C₄×C₁₇	—	2	1	no	—
147	C₄×C₁₇	—	2	1	no	—
148	C₄×C₁₇	—	2	2	no	—
149	C₄×C₁₇	—	8	1	no	—
150	C₄×C₁₇	—	8	1	no	—
151	C₄×C₁₇	—	8	1	no	—
152	C₄×C₁₇	—	8	1	no	—
153	C₂×C₄⋊C₈	—	34	1	no	—
154	C₂×C₄⋊C₈	—	34	1	no	—
155	C₂×#97	—	34	1	no	—
156	C₂×C₄×C₈	—	34	1	no	—
157	#391	Dic₁₇	1	1	yes	characteristic
158	#391	Dic₁₇	1	1	yes	characteristic
159	#391	C₂×D₁₇	1	1	yes	characteristic
160	#389	Dic₁₇	1	1	yes	characteristic
161	#389	Dic₁₇	1	1	yes	characteristic
162	#389	C₂×D₁₇	1	1	yes	characteristic
163	#387	—	2	1	no	—
164	#387	—	2	1	no	—
165	#387	—	2	1	no	—
166	#383	—	2	1	no	—
167	C₄³	C₂×D₁₇	1	1	yes	characteristic
168	#336	—	17	2	no	—
169	#336	—	17	2	no	—
170	C₂×C₁₇	#1473	1	1	yes	characteristic
171	C₂×C₁₇	#1483	1	1	yes	characteristic
172	C₂×C₁₇	#1493	1	1	yes	characteristic
173	C₂×C₁₇	#1503	1	1	yes	characteristic
174	C₂×C₁₇	#1509	1	1	yes	characteristic
175	C₂×C₁₇	#1524	1	1	yes	characteristic
176	C₂×C₁₇	#1550	1	1	yes	characteristic
177	C₂²×Q₈	C₂×Dic₁₇	1	1	yes	characteristic
178	C₂²×C₈	—	34	2	no	—
179	C₂²×C₈	—	34	2	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	C₂×Dic₁₇	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂×Dic₁₇	1	1	yes	characteristic
185	C₂×C₄⋊C₄	C₂×Dic₁₇	1	1	yes	characteristic
186	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
187	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
188	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
191	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
192	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
193	C₄⋊C₈	—	34	4	no	—
194	C₄⋊C₈	—	34	4	no	—
195	#97	—	68	2	no	—
196	C₄×C₈	—	68	2	no	—
197	#95	—	2	1	no	—
198	#95	—	2	1	no	—
199	#95	—	2	1	no	—
200	#95	—	2	1	no	—
201	#95	—	2	1	no	—
202	#95	—	2	1	no	—
203	#95	—	2	1	no	—
204	#95	—	2	1	no	—
205	#95	—	2	1	no	—
206	C₁₇	#7689	1	1	yes	characteristic, complemented, Sylow
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	4	1	no	—
209	C₂×Q₈	—	4	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
215	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
216	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
217	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
218	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
221	C₂×C₈	—	34	4	no	—
222	C₂×C₈	—	34	4	no	—
223	C₂×C₈	—	68	2	no	—
224	C₂×C₈	—	68	2	no	—
225	C₂×C₈	—	68	2	no	—
226	C₂×C₈	—	68	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	4	1	no	—
240	C₄²	—	4	1	no	—
241	C₄²	#63230	1	2	yes	—
242	C₄²	#63231	1	2	yes	—
243	C₂³	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
244	Q₈	—	8	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	#10587506	1	2	yes	—
272	C₂×C₄	#10587506	1	2	yes	—
273	C₂×C₄	#10587508	1	2	yes	—
274	C₂×C₄	#10587508	1	2	yes	—
275	C₂×C₄	C₂×#63230	1	1	yes	characteristic
276	C₂×C₄	C₂×#63231	1	1	yes	characteristic
277	C₂×C₄	#10587634	1	2	yes	—
278	C₂×C₄	#10587635	1	1	yes	characteristic
279	C₂×C₄	#10587637	1	1	yes	characteristic
280	C₈	—	68	4	no	—
281	C₈	—	68	4	no	—
282	C₂²	#11762776	1	1	yes	characteristic
283	C₂²	#11762783	1	1	yes	characteristic
284	C₂²	#11762793	1	1	yes	characteristic
285	C₂²	C₂×#10587506	1	1	yes	characteristic
286	C₂²	#11762810	1	1	yes	characteristic
287	C₂²	C₂×#10587508	1	1	yes	characteristic
288	C₂²	C₂×#10587508	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	8	1	no	—
299	C₄	—	8	1	no	—
300	C₄	—	8	1	no	—
301	C₄	—	8	1	no	—
302	C₂	#14527153	1	1	yes	characteristic
303	C₂	#14527197	1	1	yes	characteristic
304	C₂	#14527239	1	1	yes	characteristic
305	C₂	#14527283	1	1	yes	characteristic
306	C₂	#14527309	1	1	yes	characteristic
307	C₂	#14527372	1	1	yes	characteristic
308	C₂	#14527474	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₇
commutator	C₂²×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#2863	C₂
socle, prime radical	C₂³×C₁₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#2863	D₁₇
hypocenter	C₁₇	#7689
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2863

Complements

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

N	H ≅ G/N	type
206. C₁₇	#7689	semidirect

Maximal subgroups

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

subgroup	index
#14528637	2
C₄×C₄⋊#3622	2
#14526610	2
#14526607	2
C₁₇×#2863	2
#7689	17