Subgroups — #16922947

Order 4352.

352 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14528927	C₂	1	1	yes	maximal, characteristic
2	#14528901	C₂	1	1	yes	maximal, characteristic
3	#14526607	C₂	1	2	yes	maximal
4	#14526607	C₂	1	2	yes	maximal
5	C₁₇×#2864	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#3622	—	2	1	no	—
7	#11762564	—	2	1	no	—
8	#11762564	—	2	1	no	—
9	#11762564	—	2	1	no	—
10	#11762564	C₂²	1	2	yes	—
11	#11762564	C₂²	1	2	yes	—
12	C₁₇×#389	C₄	1	1	yes	characteristic
13	C₁₇×#389	C₄	1	1	yes	characteristic
14	C₁₇×#389	C₂²	1	1	yes	characteristic
15	C₁₇×#389	C₂²	1	1	yes	characteristic
16	C₁₇×#387	—	2	1	no	—
17	C₁₇×#387	—	2	1	no	—
18	C₁₇×#387	—	2	1	no	—
19	C₁₇×#387	—	2	1	no	—
20	C₁₇×#382	C₄	1	1	yes	characteristic
21	C₁₇×#382	C₄	1	1	yes	characteristic
22	C₄³×C₁₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
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×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₇	C₂×C₄	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	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
38	C₂²×#3622	—	2	2	no	—
39	C₂²×#3622	—	2	2	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	—	4	2	no	—
49	C₂²×C₄×C₁₇	—	2	1	no	—
50	C₂²×C₄×C₁₇	—	2	1	no	—
51	C₂²×C₄×C₁₇	—	2	1	no	—
52	C₂²×C₄×C₁₇	—	2	1	no	—
53	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₁₇	—	2	2	no	—
61	C₄⋊C₄×C₁₇	—	2	2	no	—
62	C₄⋊C₄×C₁₇	—	4	1	no	—
63	C₄⋊C₄×C₁₇	—	4	1	no	—
64	C₄⋊C₄×C₁₇	—	4	2	no	—
65	C₄⋊C₄×C₁₇	—	4	2	no	—
66	C₄⋊C₄×C₁₇	—	4	2	no	—
67	C₄⋊C₄×C₁₇	—	4	1	no	—
68	C₄⋊C₄×C₁₇	—	4	1	no	—
69	C₄²×C₁₇	—	2	2	no	—
70	C₄²×C₁₇	—	2	2	no	—
71	C₄²×C₁₇	—	2	2	no	—
72	C₄²×C₁₇	—	2	1	no	—
73	C₄²×C₁₇	—	2	1	no	—
74	C₄²×C₁₇	—	4	1	no	—
75	C₄²×C₁₇	—	4	1	no	—
76	C₄²×C₁₇	—	4	1	no	—
77	C₂×#3622	—	4	2	no	—
78	C₂×#3622	—	4	2	no	—
79	C₂×#3622	—	4	2	no	—
80	C₂×#3622	—	4	2	no	—
81	C₂×#3622	—	4	2	no	—
82	C₂×#3622	—	4	2	no	—
83	#7717	—	17	1	no	maximal, Sylow
84	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₇	—	2	1	no	—
86	C₂×C₄×C₁₇	—	2	1	no	—
87	C₂×C₄×C₁₇	—	2	1	no	—
88	C₂×C₄×C₁₇	—	2	1	no	—
89	C₂×C₄×C₁₇	—	2	1	no	—
90	C₂×C₄×C₁₇	—	2	1	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	—	4	1	no	—
103	C₂×C₄×C₁₇	—	4	1	no	—
104	C₂×C₄×C₁₇	—	4	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	#101	1	2	yes	—
115	C₂×C₄×C₁₇	#131	1	2	yes	—
116	C₂×C₄×C₁₇	#135	1	2	yes	—
117	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
119	#3622	—	8	2	no	—
120	#3622	—	8	2	no	—
121	#2864	D₁₇	1	1	yes	characteristic
122	#1839	—	17	1	no	—
123	#1831	—	17	1	no	—
124	#1271	—	17	2	no	—
125	#1271	—	17	2	no	—
126	C₂²×C₁₇	#408	1	1	yes	characteristic
127	C₂²×C₁₇	#410	1	1	yes	characteristic
128	C₂²×C₁₇	C₂×#101	1	1	yes	characteristic
129	C₂²×C₁₇	#416	1	1	yes	characteristic
130	C₂²×C₁₇	#417	1	1	yes	characteristic
131	C₂²×C₁₇	#421	1	1	yes	characteristic
132	C₂²×C₁₇	#421	1	1	yes	characteristic
133	C₄×C₁₇	—	2	2	no	—
134	C₄×C₁₇	—	2	2	no	—
135	C₄×C₁₇	—	2	1	no	—
136	C₄×C₁₇	—	2	1	no	—
137	C₄×C₁₇	—	2	2	no	—
138	C₄×C₁₇	—	4	1	no	—
139	C₄×C₁₇	—	4	1	no	—
140	C₄×C₁₇	—	4	1	no	—
141	C₄×C₁₇	—	8	1	no	—
142	C₄×C₁₇	—	8	1	no	—
143	C₄×C₁₇	—	8	1	no	—
144	C₄×C₁₇	—	8	1	no	—
145	C₂×C₄⋊C₈	—	34	1	no	—
146	#389	Dic₁₇	1	1	yes	characteristic
147	#389	Dic₁₇	1	1	yes	characteristic
148	#389	C₂×D₁₇	1	1	yes	characteristic
149	#389	C₂×D₁₇	1	1	yes	characteristic
150	#387	—	2	1	no	—
151	#387	—	2	1	no	—
152	#387	—	2	1	no	—
153	#387	—	2	1	no	—
154	#382	Dic₁₇	1	1	yes	characteristic
155	#382	Dic₁₇	1	1	yes	characteristic
156	C₄³	C₂×D₁₇	1	1	yes	characteristic
157	#336	—	17	2	no	—
158	#336	—	17	2	no	—
159	#336	—	34	1	no	—
160	#336	—	34	1	no	—
161	#336	—	34	1	no	—
162	C₂×C₁₇	#1470	1	1	yes	characteristic
163	C₂×C₁₇	#1482	1	1	yes	characteristic
164	C₂×C₁₇	#1495	1	1	yes	characteristic
165	C₂×C₁₇	#1512	1	1	yes	characteristic
166	C₂×C₁₇	#1521	1	1	yes	characteristic
167	C₂×C₁₇	#1537	1	1	yes	characteristic
168	C₂×C₁₇	#1552	1	1	yes	characteristic
169	C₂²×C₈	—	34	2	no	—
170	C₂²×C₈	—	34	2	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	C₂×Dic₁₇	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂×Dic₁₇	1	1	yes	characteristic
177	C₂×C₄⋊C₄	C₂×Dic₁₇	1	1	yes	characteristic
178	C₂×C₄⋊C₄	C₂×Dic₁₇	1	1	yes	characteristic
179	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
180	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
181	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
182	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
183	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
184	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
185	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
186	C₄⋊C₈	—	68	2	no	—
187	#95	—	2	1	no	—
188	#95	—	2	1	no	—
189	#95	—	2	1	no	—
190	#95	—	2	1	no	—
191	#95	—	2	1	no	—
192	#95	—	2	1	no	—
193	#95	—	2	1	no	—
194	#95	—	2	1	no	—
195	C₁₇	#7717	1	1	yes	characteristic, complemented, Sylow
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
202	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
203	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
204	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
207	C₂×C₈	—	68	2	no	—
208	C₂×C₈	—	68	2	no	—
209	C₂×C₈	—	68	2	no	—
210	C₂×C₈	—	68	2	no	—
211	C₂×C₈	—	68	2	no	—
212	C₂×C₈	—	68	2	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄⋊C₄	—	4	1	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	4	1	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	4	1	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	4	1	no	—
228	C₄²	—	4	1	no	—
229	C₄²	—	4	1	no	—
230	C₂³	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	#10587507	1	2	yes	—
261	C₂×C₄	#10587634	1	2	yes	—
262	C₂×C₄	#10587635	1	1	yes	characteristic
263	C₂×C₄	#10587636	1	2	yes	—
264	C₂×C₄	#10587637	1	1	yes	characteristic
265	C₈	—	136	2	no	—
266	C₈	—	136	2	no	—
267	C₂²	#11762776	1	1	yes	characteristic
268	C₂²	#11762783	1	1	yes	characteristic
269	C₂²	C₂×#10587507	1	1	yes	characteristic
270	C₂²	#11762805	1	1	yes	characteristic
271	C₂²	#11762810	1	1	yes	characteristic
272	C₂²	#11762826	1	1	yes	characteristic
273	C₂²	#11762826	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	1	no	—
281	C₄	—	4	1	no	—
282	C₄	—	8	1	no	—
283	C₄	—	8	1	no	—
284	C₄	—	8	1	no	—
285	C₄	—	8	1	no	—
286	C₂	#14527140	1	1	yes	characteristic
287	C₂	#14527192	1	1	yes	characteristic
288	C₂	#14527249	1	1	yes	characteristic
289	C₂	#14527322	1	1	yes	characteristic
290	C₂	#14527361	1	1	yes	characteristic
291	C₂	#14527422	1	1	yes	characteristic
292	C₂	#14527484	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×#2864	C₂
socle, prime radical	C₂³×C₁₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#2864	D₁₇
hypocenter	C₁₇	#7717
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2864

Complements

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

N	H ≅ G/N	type
195. C₁₇	#7717	semidirect

Maximal subgroups

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

subgroup	index
#14528927	2
#14528901	2
#14526607	2
C₁₇×#2864	2
#7717	17