Subgroups — #16950820

Order 4352.

376 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	#14541886	C₂	1	1	yes	maximal, characteristic
2	#14530550	C₂	1	1	yes	maximal, characteristic, complemented
3	#14530539	C₂	1	1	yes	maximal, characteristic, complemented
4	#14530411	C₂	1	1	yes	maximal, characteristic
5	#14528623	C₂	1	1	yes	maximal, characteristic
6	#14526865	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#1378	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10587587	C₂²	1	1	yes	characteristic
9	#11763227	—	2	2	no	—
10	#11763225	—	2	2	no	—
11	C₂×#10587483	C₂²	1	1	yes	characteristic
12	C₂×#10587483	C₂²	1	1	yes	characteristic
13	M₃₂×D₁₇	—	2	2	no	—
14	C₂×#10587496	C₄	1	1	yes	characteristic
15	#11762803	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
17	#11762630	—	2	2	no	—
18	#11762627	—	2	2	no	—
19	#11762592	—	2	2	no	—
20	#11762573	C₄	1	1	yes	characteristic
21	#11762568	C₄	1	1	yes	characteristic
22	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
23	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
24	C₁₇×#405	C₂²	1	1	yes	characteristic
25	C₂×#63252	C₂×C₄	1	1	yes	characteristic
26	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
28	C₁₇×#117	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂×#63223	C₂×C₄	1	1	yes	characteristic
31	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
32	#10587587	—	2	1	no	—
33	#10587587	—	2	1	no	—
34	#10587587	D₄	1	2	yes	—
35	#10587587	Q₈	1	2	yes	—
36	C₂×#63218	—	2	1	no	—
37	C₂×C₈×D₁₇	—	2	1	no	—
38	C₁₇×M₃₂	—	2	1	no	—
39	C₁₇×M₃₂	—	2	1	no	—
40	C₁₇×M₃₂	—	2	1	no	—
41	C₁₇×M₃₂	—	2	1	no	—
42	C₂×C₁₆×C₁₇	—	2	1	no	—
43	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
44	C₁₇×#97	—	2	1	no	—
45	C₄×C₈×C₁₇	—	2	1	no	—
46	#10587496	C₈	1	2	yes	—
47	#10587496	C₈	1	2	yes	—
48	C₈×Dic₁₇	—	2	1	no	—
49	C₈×Dic₁₇	—	2	1	no	—
50	#10587483	—	2	1	no	—
51	#10587483	—	2	1	no	—
52	#10587483	—	2	1	no	—
53	#10587483	D₄	1	2	yes	—
54	C₂×#63214	—	2	1	no	—
55	C₂×#63214	—	2	1	no	—
56	#10587481	—	2	1	no	—
57	#10587480	—	2	1	no	—
58	#10587469	—	2	4	no	—
59	C₁₆×D₁₇	—	2	4	no	—
60	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₇	—	2	1	no	—
62	C₂²×Dic₁₇	—	2	1	no	—
63	#63252	—	2	1	no	—
64	#63252	—	2	1	no	—
65	#63252	C₄⋊C₄	1	2	yes	—
66	#63252	C₂×C₈	1	2	yes	—
67	C₂×D₆₈	M₁₆	1	1	yes	characteristic
68	C₂×C₄×D₁₇	—	2	1	no	—
69	C₂×Dic₃₄	M₁₆	1	1	yes	characteristic
70	M₁₆×C₁₇	—	2	1	no	—
71	M₁₆×C₁₇	C₂×C₈	1	2	yes	—
72	C₂×C₈×C₁₇	—	2	1	no	—
73	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
78	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
79	C₄⋊C₄×C₁₇	—	2	1	no	—
80	C₂²⋊C₄×C₁₇	—	2	1	no	—
81	C₄²×C₁₇	—	2	1	no	—
82	C₄×Dic₁₇	—	2	1	no	—
83	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
84	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
85	#63223	—	2	1	no	—
86	#63223	C₄⋊C₄	1	2	yes	—
87	C₂×#3622	—	2	1	no	—
88	C₂×#3622	—	2	1	no	—
89	C₂×#3622	—	2	1	no	—
90	C₂×#3622	—	2	1	no	—
91	#63218	—	2	2	no	—
92	#63218	—	2	2	no	—
93	C₈×D₁₇	—	2	2	no	—
94	C₈×D₁₇	—	2	2	no	—
95	C₁₆×C₁₇	—	2	2	no	—
96	C₁₆×C₁₇	—	4	1	no	—
97	#63214	—	2	2	no	—
98	#63214	—	4	1	no	—
99	#12918	—	17	1	no	maximal, Sylow
100	C₂³×C₁₇	#131	1	1	yes	characteristic
101	C₂²×D₁₇	—	2	1	no	—
102	C₂×C₄×C₁₇	—	4	1	no	—
103	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
109	C₂²⋊D₁₇	—	2	2	no	—
110	C₂²⋊D₁₇	—	2	2	no	—
111	C₂×Dic₁₇	—	2	1	no	—
112	C₂×Dic₁₇	—	2	1	no	—
113	C₂×Dic₁₇	—	2	1	no	—
114	C₂×Dic₁₇	—	4	2	no	—
115	D₆₈	—	2	1	no	—
116	D₆₈	C₄⋊C₈	1	2	yes	—
117	C₄×D₁₇	—	2	2	no	—
118	C₄×D₁₇	—	2	2	no	—
119	Dic₃₄	—	2	1	no	—
120	Dic₃₄	C₄⋊C₈	1	2	yes	—
121	C₈×C₁₇	—	2	2	no	—
122	C₈×C₁₇	#120	1	1	yes	characteristic
123	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
124	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
125	C₈×C₁₇	#123	1	1	yes	characteristic
126	#3622	—	2	2	no	—
127	#3622	—	4	1	no	—
128	#3407	—	17	1	no	—
129	#2153	—	17	1	no	—
130	#2150	—	17	1	no	—
131	#2117	—	17	1	no	—
132	#1764	—	17	1	no	—
133	#1378	D₁₇	1	1	yes	characteristic, complemented
134	#1376	—	17	1	no	—
135	C₂²×C₁₇	—	4	1	no	—
136	C₂²×C₁₇	#424	1	1	yes	characteristic
137	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
138	C₂²×C₁₇	#435	1	1	yes	characteristic
139	C₂×D₁₇	—	2	2	no	—
140	C₂×D₁₇	—	8	1	no	—
141	C₄×C₁₇	—	8	1	no	—
142	C₄×C₁₇	#385	1	1	yes	characteristic
143	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
144	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
145	C₄×C₁₇	#435	1	1	yes	characteristic
146	Dic₁₇	—	2	2	no	—
147	Dic₁₇	—	2	2	no	—
148	Dic₁₇	—	4	1	no	—
149	C₂×#131	—	17	1	no	—
150	#504	—	34	2	no	—
151	#504	—	34	2	no	—
152	C₂×M₃₂	—	17	1	no	—
153	C₂×M₃₂	—	17	1	no	—
154	C₂×M₃₂	—	34	2	no	—
155	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
156	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
157	C₂×#104	—	17	1	no	—
158	#416	—	17	1	no	—
159	#405	C₂×D₁₇	1	1	yes	characteristic
160	C₂×C₄×C₈	—	17	1	no	—
161	#364	—	34	2	no	—
162	C₄⋊C₁₆	—	34	2	no	—
163	#350	—	34	2	no	—
164	#341	—	17	1	no	—
165	#339	—	17	1	no	—
166	C₂×C₁₇	—	2	1	no	—
167	C₂×C₁₇	—	2	1	no	—
168	C₂×C₁₇	—	2	1	no	—
169	C₂×C₁₇	#1850	1	1	yes	characteristic
170	D₁₇	—	4	2	no	—
171	C₂×#41	—	17	1	no	—
172	#131	—	17	2	no	—
173	#131	—	17	2	no	—
174	#131	—	34	1	no	—
175	#131	—	34	1	no	—
176	C₂×M₁₆	—	17	1	no	—
177	C₂×M₁₆	—	34	1	no	—
178	C₂×M₁₆	C₄×D₁₇	1	1	yes	characteristic
179	C₂²×C₈	—	17	1	no	—
180	C₂²×C₈	—	34	1	no	—
181	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
182	#117	C₄×D₁₇	1	1	yes	characteristic
183	C₂×C₄²	—	17	1	no	—
184	M₃₂	—	2	1	no	—
185	M₃₂	—	2	1	no	—
186	M₃₂	—	2	1	no	—
187	M₃₂	—	2	1	no	—
188	M₃₂	—	17	2	no	—
189	M₃₂	—	34	4	no	—
190	M₃₂	—	34	1	no	—
191	M₃₂	—	34	1	no	—
192	M₃₂	—	34	1	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	—	34	4	no	—
195	C₂×C₁₆	—	34	1	no	—
196	C₂×C₁₆	—	34	1	no	—
197	C₂×C₁₆	D₆₈	1	2	yes	—
198	#104	—	17	2	no	—
199	#104	—	17	2	no	—
200	#103	—	34	1	no	—
201	#102	—	34	1	no	—
202	#97	—	2	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	34	1	no	—
205	C₄×C₈	—	34	1	no	—
206	C₁₇	#12918	1	1	yes	characteristic, complemented, Sylow
207	#41	—	17	2	no	—
208	#41	—	17	2	no	—
209	#41	—	34	1	no	—
210	#41	—	34	1	no	—
211	C₂×Q₈	—	17	1	no	—
212	C₂×D₄	—	17	1	no	—
213	C₂×D₄	—	34	1	no	—
214	C₂²×C₄	—	34	1	no	—
215	C₂²×C₄	—	34	1	no	—
216	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
217	M₁₆	—	2	1	no	—
218	M₁₆	—	17	2	no	—
219	M₁₆	—	34	1	no	—
220	M₁₆	—	34	2	no	—
221	M₁₆	—	34	2	no	—
222	M₁₆	C₈×D₁₇	1	2	yes	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	34	1	no	—
225	C₂×C₈	—	34	2	no	—
226	C₂×C₈	—	34	2	no	—
227	C₂×C₈	—	34	1	no	—
228	C₂×C₈	—	34	1	no	—
229	C₂×C₈	—	34	1	no	—
230	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
231	C₂×C₈	#63252	1	1	yes	characteristic
232	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
233	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
234	C₂×C₈	#63254	1	1	yes	characteristic
235	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
236	C₄⋊C₄	—	2	1	no	—
237	C₂²⋊C₄	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	17	1	no	—
240	C₄²	—	17	1	no	—
241	C₄²	—	34	1	no	—
242	C₁₆	—	2	2	no	—
243	C₁₆	—	4	1	no	—
244	C₁₆	—	34	2	no	—
245	C₁₆	—	68	1	no	—
246	C₂³	—	34	1	no	—
247	C₂³	#10587587	1	1	yes	characteristic
248	Q₈	—	17	2	no	—
249	Q₈	—	34	1	no	—
250	D₄	—	17	2	no	—
251	D₄	—	34	2	no	—
252	D₄	—	34	2	no	—
253	D₄	—	34	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	34	1	no	—
256	C₂×C₄	—	34	1	no	—
257	C₂×C₄	—	34	2	no	—
258	C₂×C₄	—	34	2	no	—
259	C₂×C₄	—	34	1	no	—
260	C₂×C₄	—	68	2	no	—
261	C₂×C₄	C₄×D₆₈	1	1	yes	characteristic
262	C₂×C₄	#10587567	1	1	yes	characteristic
263	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
264	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
265	C₂×C₄	M₁₆×D₁₇	1	1	yes	characteristic
266	C₂×C₄	#10587593	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	34	2	no	—
269	C₈	—	68	1	no	—
270	C₈	#10587568	1	1	yes	characteristic
271	C₈	#10587569	1	1	yes	characteristic
272	C₈	#10587582	1	1	yes	characteristic
273	C₈	#10587583	1	1	yes	characteristic
274	C₂²	—	4	1	no	—
275	C₂²	—	34	2	no	—
276	C₂²	—	136	1	no	—
277	C₂²	#11762834	1	1	yes	characteristic
278	C₂²	C₈×D₆₈	1	1	yes	characteristic
279	C₂²	#11762883	1	1	yes	characteristic
280	C₄	—	8	1	no	—
281	C₄	—	34	2	no	—
282	C₄	—	34	2	no	—
283	C₄	—	68	1	no	—
284	C₄	#11762683	1	1	yes	characteristic
285	C₄	C₄⋊C₈×D₁₇	1	1	yes	characteristic
286	C₄	#11762872	1	1	yes	characteristic
287	C₄	#11762882	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	68	2	no	—
292	C₂	#14528963	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#10587568
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1378	C₂
socle	C₂×C₁₇	#1850
prime radical	C₂×#63252	C₂×C₄
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1378	D₁₇
hypocenter	C₁₇	#12918
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1378

Complements

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

N	H ≅ G/N	type
2. #14530550	C₂	semidirect
3. #14530539	C₂	semidirect
6. #14526865	C₂	semidirect
7. C₁₇×#1378	C₂	semidirect
133. #1378	D₁₇	semidirect
206. C₁₇	#12918	semidirect

Maximal subgroups

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

subgroup	index
#14541886	2
#14530550	2
#14530539	2
#14530411	2
#14528623	2
#14526865	2
C₁₇×#1378	2
#12918	17