Subgroups — #16922927

Order 4352.

372 conjugacy classes of subgroups.

The subgroup lattice has 300 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	#14528637	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×#10587474	—	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×#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	#10587474	—	4	2	no	—
50	C₂×Q₈×C₁₇	—	2	2	no	—
51	C₂×Q₈×C₁₇	—	4	1	no	—
52	C₂×Q₈×C₁₇	—	4	1	no	—
53	C₂²×C₄×C₁₇	—	2	1	no	—
54	C₂²×C₄×C₁₇	—	2	1	no	—
55	C₂²×C₄×C₁₇	—	2	1	no	—
56	C₂²×C₄×C₁₇	—	2	1	no	—
57	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
64	C₄⋊C₄×C₁₇	—	2	2	no	—
65	C₄⋊C₄×C₁₇	—	4	1	no	—
66	C₄⋊C₄×C₁₇	—	4	1	no	—
67	C₄⋊C₄×C₁₇	—	4	2	no	—
68	C₄⋊C₄×C₁₇	—	4	2	no	—
69	C₄⋊C₄×C₁₇	—	4	1	no	—
70	C₄⋊C₄×C₁₇	—	4	1	no	—
71	C₄²×C₁₇	—	2	2	no	—
72	C₄²×C₁₇	—	2	2	no	—
73	C₄²×C₁₇	—	2	1	no	—
74	C₄²×C₁₇	—	2	2	no	—
75	C₄²×C₁₇	—	2	1	no	—
76	C₄²×C₁₇	—	4	1	no	—
77	C₄²×C₁₇	—	4	1	no	—
78	C₄²×C₁₇	—	4	1	no	—
79	C₂×#3622	—	2	4	no	—
80	C₂×#3622	—	4	2	no	—
81	C₂×#3622	—	4	2	no	—
82	C₂×#3622	—	4	2	no	—
83	C₂×#3622	—	4	2	no	—
84	C₂×#3622	—	4	2	no	—
85	#7713	—	17	1	no	maximal, Sylow
86	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	Q₈×C₁₇	—	8	2	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	—	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	#104	1	2	yes	—
118	C₂×C₄×C₁₇	#131	1	2	yes	—
119	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₁₇	#137	1	2	yes	—
121	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
122	#3622	—	4	4	no	—
123	#3622	—	8	2	no	—
124	#2863	D₁₇	1	1	yes	characteristic
125	#1839	—	17	1	no	—
126	#1767	—	17	1	no	—
127	#1272	—	17	2	no	—
128	#1271	—	17	2	no	—
129	C₂²×C₁₇	#408	1	1	yes	characteristic
130	C₂²×C₁₇	#410	1	1	yes	characteristic
131	C₂²×C₁₇	#413	1	1	yes	characteristic
132	C₂²×C₁₇	#417	1	1	yes	characteristic
133	C₂²×C₁₇	#419	1	1	yes	characteristic
134	C₂²×C₁₇	C₂×#104	1	1	yes	characteristic
135	C₂²×C₁₇	#421	1	1	yes	characteristic
136	C₄×C₁₇	—	2	2	no	—
137	C₄×C₁₇	—	2	1	no	—
138	C₄×C₁₇	—	2	1	no	—
139	C₄×C₁₇	—	2	2	no	—
140	C₄×C₁₇	—	2	2	no	—
141	C₄×C₁₇	—	4	1	no	—
142	C₄×C₁₇	—	4	1	no	—
143	C₄×C₁₇	—	4	1	no	—
144	C₄×C₁₇	—	8	1	no	—
145	C₄×C₁₇	—	8	1	no	—
146	C₄×C₁₇	—	8	1	no	—
147	C₄×C₁₇	—	8	1	no	—
148	C₂×C₄⋊C₈	—	34	1	no	—
149	C₂×#97	—	34	1	no	—
150	#391	Dic₁₇	1	1	yes	characteristic
151	#391	Dic₁₇	1	1	yes	characteristic
152	#391	C₂×D₁₇	1	1	yes	characteristic
153	#389	Dic₁₇	1	1	yes	characteristic
154	#389	Dic₁₇	1	1	yes	characteristic
155	#389	C₂×D₁₇	1	1	yes	characteristic
156	#387	—	2	1	no	—
157	#387	—	2	1	no	—
158	#387	—	2	1	no	—
159	#383	—	2	1	no	—
160	C₄³	C₂×D₁₇	1	1	yes	characteristic
161	#336	—	17	2	no	—
162	#336	—	17	2	no	—
163	#336	—	34	1	no	—
164	#336	—	34	1	no	—
165	C₂×C₁₇	#1473	1	1	yes	characteristic
166	C₂×C₁₇	#1485	1	1	yes	characteristic
167	C₂×C₁₇	#1495	1	1	yes	characteristic
168	C₂×C₁₇	#1509	1	1	yes	characteristic
169	C₂×C₁₇	#1515	1	1	yes	characteristic
170	C₂×C₁₇	#1532	1	1	yes	characteristic
171	C₂×C₁₇	#1559	1	1	yes	characteristic
172	C₂²×Q₈	C₂×Dic₁₇	1	1	yes	characteristic
173	C₂²×C₈	—	34	2	no	—
174	C₂²×C₈	—	34	2	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	C₂×Dic₁₇	1	1	yes	characteristic
179	C₂×C₄⋊C₄	C₂×Dic₁₇	1	1	yes	characteristic
180	C₂×C₄⋊C₄	C₂×Dic₁₇	1	1	yes	characteristic
181	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
182	C₂×C₄²	C₂×Dic₁₇	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	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
187	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
188	C₄⋊C₈	—	34	4	no	—
189	#97	—	68	2	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	#95	—	2	1	no	—
196	#95	—	2	1	no	—
197	#95	—	2	1	no	—
198	#95	—	2	1	no	—
199	C₁₇	#7713	1	1	yes	characteristic, complemented, Sylow
200	C₂×Q₈	—	2	2	no	—
201	C₂×Q₈	—	4	1	no	—
202	C₂×Q₈	—	4	1	no	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
209	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
210	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
211	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
214	C₂×C₈	—	34	4	no	—
215	C₂×C₈	—	68	2	no	—
216	C₂×C₈	—	68	2	no	—
217	C₂×C₈	—	68	2	no	—
218	C₂×C₈	—	68	2	no	—
219	C₂×C₈	—	68	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	4	1	no	—
222	C₄⋊C₄	—	4	1	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	4	1	no	—
226	C₄⋊C₄	—	4	1	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	1	no	—
231	C₄²	—	2	1	no	—
232	C₄²	—	4	1	no	—
233	C₄²	—	4	1	no	—
234	C₄²	—	4	1	no	—
235	C₂³	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
236	Q₈	—	8	2	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	—	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	—	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	—	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	#10587508	1	2	yes	—
267	C₂×C₄	#10587628	1	2	yes	—
268	C₂×C₄	#10587634	1	2	yes	—
269	C₂×C₄	#10587635	1	1	yes	characteristic
270	C₂×C₄	#10587637	1	1	yes	characteristic
271	C₈	—	68	4	no	—
272	C₈	—	136	2	no	—
273	C₂²	#11762776	1	1	yes	characteristic
274	C₂²	#11762783	1	1	yes	characteristic
275	C₂²	#11762793	1	1	yes	characteristic
276	C₂²	#11762810	1	1	yes	characteristic
277	C₂²	#11762818	1	1	yes	characteristic
278	C₂²	C₂×#10587508	1	1	yes	characteristic
279	C₂²	#11762826	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	4	1	no	—
286	C₄	—	4	1	no	—
287	C₄	—	4	1	no	—
288	C₄	—	8	1	no	—
289	C₄	—	8	1	no	—
290	C₄	—	8	1	no	—
291	C₄	—	8	1	no	—
292	C₂	#14527153	1	1	yes	characteristic
293	C₂	#14527205	1	1	yes	characteristic
294	C₂	#14527249	1	1	yes	characteristic
295	C₂	#14527309	1	1	yes	characteristic
296	C₂	#14527335	1	1	yes	characteristic
297	C₂	#14527404	1	1	yes	characteristic
298	C₂	#14527514	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	#7713
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
199. C₁₇	#7713	semidirect

Maximal subgroups

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

subgroup	index
#14528927	2
#14528637	2
#14526610	2
#14526607	2
C₁₇×#2863	2
#7713	17