Subgroups — #17187709

Order 4352.

674 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763242	C₂	1	1	yes	maximal, characteristic
2	#14537659	C₂	1	2	yes	maximal, complemented
3	#14530651	C₂	1	2	yes	maximal, complemented
4	#14530646	C₂	1	2	yes	maximal, complemented
5	Q₃₂×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₂₇₂	C₂	1	1	yes	maximal, characteristic, complemented
7	#14530630	C₂	1	1	yes	maximal, characteristic, complemented
8	D₁₇×#366	C₂	1	1	yes	maximal, characteristic, complemented
9	#14530495	C₂	1	2	yes	maximal, complemented
10	#14530379	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₁₇×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10587605	C₂²	1	2	yes	—
13	#11763291	—	2	2	no	—
14	#11763242	C₄	1	8	yes	complemented
15	C₂×#10587499	C₂²	1	2	yes	—
16	C₂×D₂₇₂	C₂²	1	1	yes	characteristic
17	C₂×#10587469	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₁₇	C₂²	1	1	yes	characteristic
19	#11762912	—	2	2	no	—
20	#11762907	—	2	2	no	—
21	Q₁₆×Dic₁₇	C₂²	1	2	yes	—
22	C₄×D₁₃₆	C₂²	1	2	yes	—
23	#11762891	C₂²	1	2	yes	—
24	#107×D₁₇	C₂²	1	2	yes	—
25	#11762809	—	2	2	no	—
26	#11762763	C₂²	1	1	yes	characteristic
27	#11762635	C₂²	1	1	yes	characteristic
28	#11762634	C₂²	1	2	yes	—
29	#11762613	C₂²	1	2	yes	—
30	#11762611	C₂²	1	2	yes	—
31	#11762609	C₂²	1	2	yes	—
32	#11762585	C₂²	1	1	yes	characteristic
33	#11762584	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₁₇	C₂²	1	1	yes	characteristic
35	C₂×C₁₇×Q₃₂	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₁₇	C₂²	1	2	yes	—
37	C₁₇×#366	C₂²	1	1	yes	characteristic
38	C₁₇×#358	C₂²	1	2	yes	—
39	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
40	C₂×#63256	—	2	2	no	—
41	C₂×Q₁₆×C₁₇	C₂³	1	2	yes	—
42	C₄×Q₈×C₁₇	—	2	2	no	—
43	Q₈×Dic₁₇	—	2	2	no	—
44	C₂×#63230	—	2	2	no	—
45	#10587605	—	2	4	no	—
46	#10587605	C₂×C₄	1	8	yes	complemented
47	C₂×D₁₃₆	C₂³	1	2	yes	—
48	C₂×#63218	D₄	1	1	yes	characteristic
49	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
50	#10587579	—	2	2	no	—
51	#10587579	—	4	2	no	—
52	C₄⋊C₄×D₁₇	—	2	2	no	—
53	C₄⋊C₄×D₁₇	—	4	2	no	—
54	C₄×D₆₈	—	2	2	no	—
55	C₄×D₆₈	—	4	2	no	—
56	#10587558	—	4	2	no	—
57	#10587558	D₄	1	1	yes	characteristic
58	C₁₇×Q₃₂	C₂×C₄	1	4	yes	complemented
59	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
61	C₁₇×#107	C₂³	1	2	yes	—
62	C₁₇×#103	—	2	2	no	—
63	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
64	#10587506	—	2	2	no	—
65	#10587499	C₂×C₄	1	8	yes	complemented
66	#10587492	—	2	2	no	—
67	#10587489	C₂³	1	2	yes	—
68	#10587486	D₄	1	1	yes	characteristic
69	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
70	C₂×#63214	C₂³	1	1	yes	characteristic
71	C₂×#63214	C₂³	1	1	yes	characteristic
72	#10587480	—	2	2	no	—
73	#10587478	—	2	2	no	—
74	#10587474	D₄	1	1	yes	characteristic
75	D₂₇₂	C₂×C₄	1	4	yes	complemented
76	#10587469	—	2	2	no	—
77	C₁₆×D₁₇	C₂×C₄	1	4	yes	complemented
78	C₂×Q₈×C₁₇	—	2	2	no	—
79	#63256	—	2	8	no	—
80	#63256	—	4	4	no	—
81	C₂×D₆₈	—	2	2	no	—
82	C₂×D₆₈	—	4	2	no	—
83	C₂×C₄×D₁₇	—	4	2	no	—
84	C₂×C₄×D₁₇	—	4	2	no	—
85	C₂×C₄×D₁₇	—	4	2	no	—
86	C₂×C₄×D₁₇	—	4	2	no	—
87	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
88	Q₁₆×C₁₇	—	2	2	no	—
89	Q₁₆×C₁₇	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₁₇	—	2	2	no	—
93	C₄⋊C₄×C₁₇	—	4	2	no	—
94	C₄²×C₁₇	—	4	2	no	—
95	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
96	#63230	—	2	8	no	—
97	#63227	—	4	2	no	—
98	#63227	—	4	2	no	—
99	#63227	D₈	1	1	yes	characteristic
100	#63227	D₈	1	1	yes	characteristic
101	C₄⋊Dic₁₇	—	2	2	no	—
102	C₄⋊Dic₁₇	—	4	2	no	—
103	#63225	—	4	2	no	—
104	#63225	D₈	1	1	yes	characteristic
105	#63225	D₈	1	1	yes	characteristic
106	C₄×Dic₁₇	—	4	2	no	—
107	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
108	C₂×#3622	C₂×D₄	1	1	yes	characteristic
109	C₂×#3622	C₂×D₄	1	1	yes	characteristic
110	D₁₃₆	—	2	2	no	—
111	D₁₃₆	C₂²×C₄	1	4	yes	—
112	#63218	—	2	2	no	—
113	C₈×D₁₇	C₂²×C₄	1	2	yes	—
114	C₈×D₁₇	#41	1	2	yes	—
115	C₁₆×C₁₇	—	2	1	no	—
116	C₁₆×C₁₇	C₂²×C₄	1	2	yes	—
117	#63214	—	2	1	no	—
118	#63214	C₂²×C₄	1	2	yes	—
119	#33459	—	17	1	no	maximal, Sylow
120	C₂²×D₁₇	—	4	2	no	—
121	C₂²×D₁₇	C₂×D₈	1	1	yes	characteristic
122	Q₈×C₁₇	—	2	4	no	—
123	Q₈×C₁₇	—	4	2	no	—
124	C₂×C₄×C₁₇	—	4	2	no	—
125	C₂×C₄×C₁₇	—	4	2	no	—
126	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
129	C₂×Dic₁₇	—	4	2	no	—
130	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
132	C₂×Dic₁₇	C₂×D₈	1	1	yes	characteristic
133	D₆₈	—	2	4	no	—
134	D₆₈	—	4	8	no	—
135	D₆₈	—	4	2	no	—
136	C₄×D₁₇	—	4	8	no	—
137	C₄×D₁₇	—	16	2	no	—
138	C₄×D₁₇	—	16	2	no	—
139	C₄×D₁₇	—	16	2	no	—
140	C₄×D₁₇	C₄×D₄	1	2	yes	—
141	C₄×D₁₇	#135	1	2	yes	—
142	C₈×C₁₇	—	2	1	no	—
143	C₈×C₁₇	C₂³×C₄	1	1	yes	characteristic
144	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
145	#3622	—	2	1	no	—
146	#3622	C₄×D₄	1	2	yes	—
147	C₂×#508	—	17	1	no	—
148	#2951	—	17	2	no	—
149	#2175	—	17	2	no	—
150	#2175	—	17	2	no	—
151	C₄×Q₃₂	—	17	1	no	—
152	C₄×Q₃₂	D₁₇	1	1	yes	characteristic, complemented
153	C₄×D₁₆	—	17	1	no	—
154	C₄×D₁₆	—	17	1	no	—
155	C₂×#366	—	17	1	no	—
156	#2139	—	17	2	no	—
157	#2108	—	17	1	no	—
158	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
159	C₂×D₁₇	—	2	1	no	—
160	C₂×D₁₇	—	2	1	no	—
161	C₂×D₁₇	—	4	4	no	—
162	C₂×D₁₇	—	8	2	no	—
163	C₂×D₁₇	—	8	2	no	—
164	C₂×D₁₇	C₄×D₈	1	2	yes	—
165	C₄×C₁₇	—	2	1	no	—
166	C₄×C₁₇	—	2	1	no	—
167	C₄×C₁₇	—	4	4	no	—
168	C₄×C₁₇	—	8	2	no	—
169	C₄×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
170	C₄×C₁₇	C₂×#135	1	1	yes	characteristic
171	Dic₁₇	—	2	1	no	—
172	Dic₁₇	—	2	1	no	—
173	Dic₁₇	—	8	2	no	—
174	Dic₁₇	C₄×D₈	1	2	yes	—
175	C₂×#135	—	17	2	no	—
176	#520	—	34	2	no	—
177	#508	—	17	8	no	—
178	C₂×Q₃₂	C₂×D₁₇	1	1	yes	characteristic
179	C₂×SD₃₂	—	17	2	no	—
180	C₂×D₁₆	—	17	1	no	—
181	C₂×M₃₂	—	17	1	no	—
182	C₂²×C₁₆	—	17	1	no	—
183	#440	—	34	2	no	—
184	#440	—	34	2	no	—
185	C₄×Q₁₆	—	17	2	no	—
186	C₄×Q₁₆	C₂×D₁₇	1	2	yes	—
187	C₄×D₈	—	17	2	no	—
188	C₄×D₈	—	17	2	no	—
189	C₂×#107	—	17	2	no	—
190	#417	—	34	2	no	—
191	#405	—	17	1	no	—
192	#366	—	17	1	no	—
193	#366	—	17	2	no	—
194	#366	C₂×D₁₇	1	1	yes	characteristic
195	#358	—	17	2	no	—
196	#358	C₂×D₁₇	1	2	yes	—
197	#357	—	17	2	no	—
198	#357	—	17	2	no	—
199	#346	—	17	1	no	—
200	#346	—	17	1	no	—
201	C₄×C₁₆	—	17	1	no	—
202	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
203	C₂×C₁₇	C₂×C₄×D₈	1	1	yes	characteristic
204	C₂×C₁₇	#3415	1	1	yes	characteristic
205	C₂×C₁₇	#3417	1	1	yes	characteristic
206	D₁₇	—	2	2	no	—
207	D₁₇	—	8	4	no	—
208	C₂×#41	—	34	2	no	—
209	#135	—	17	8	no	—
210	#135	—	34	4	no	—
211	C₂×Q₁₆	C₂²×D₁₇	1	2	yes	—
212	C₂×SD₁₆	—	34	2	no	—
213	C₂×D₈	—	17	2	no	—
214	C₂×M₁₆	—	17	1	no	—
215	C₂²×C₈	—	17	1	no	—
216	C₄×Q₈	—	2	2	no	—
217	C₄×Q₈	—	34	2	no	—
218	C₄×D₄	—	34	2	no	—
219	C₄×D₄	—	34	2	no	—
220	C₄×D₄	—	68	2	no	—
221	C₄×D₄	—	68	2	no	—
222	#117	—	17	1	no	—
223	#117	—	68	2	no	—
224	C₂×C₄⋊C₄	—	34	2	no	—
225	C₂×C₄⋊C₄	—	68	2	no	—
226	Q₃₂	C₄×D₁₇	1	4	yes	complemented
227	SD₃₂	—	17	8	no	—
228	D₁₆	—	17	4	no	—
229	M₃₂	—	34	2	no	—
230	C₂×C₁₆	—	17	1	no	—
231	C₂×C₁₆	—	17	4	no	—
232	C₂×C₁₆	—	17	1	no	—
233	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
234	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
235	#107	—	17	2	no	—
236	#107	—	34	2	no	—
237	#107	C₂²×D₁₇	1	2	yes	—
238	#103	—	2	2	no	—
239	#103	—	34	2	no	—
240	#102	—	34	2	no	—
241	#102	—	34	2	no	—
242	#97	—	17	1	no	—
243	#97	—	17	1	no	—
244	C₄×C₈	—	17	1	no	—
245	C₄×C₈	C₂²×D₁₇	1	1	yes	characteristic
246	C₁₇	#33459	1	1	yes	characteristic, complemented, Sylow
247	#41	—	34	8	no	—
248	#41	—	68	4	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×D₄	—	34	2	no	—
251	C₂×D₄	—	68	2	no	—
252	C₂²×C₄	—	17	1	no	—
253	C₂²×C₄	—	68	2	no	—
254	C₂²×C₄	—	68	2	no	—
255	C₂²×C₄	—	68	2	no	—
256	C₂²×C₄	—	68	2	no	—
257	Q₁₆	—	2	2	no	—
258	Q₁₆	C₂×C₄×D₁₇	1	4	yes	—
259	SD₁₆	—	34	8	no	—
260	D₈	—	17	4	no	—
261	D₈	—	34	2	no	—
262	M₁₆	—	34	2	no	—
263	C₂×C₈	—	17	1	no	—
264	C₂×C₈	—	17	2	no	—
265	C₂×C₈	—	17	2	no	—
266	C₂×C₈	—	17	1	no	—
267	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
268	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	17	1	no	—
272	C₄⋊C₄	—	17	1	no	—
273	C₄⋊C₄	—	34	2	no	—
274	C₄⋊C₄	—	68	2	no	—
275	C₄⋊C₄	—	68	2	no	—
276	C₂²⋊C₄	—	17	1	no	—
277	C₂²⋊C₄	—	17	1	no	—
278	C₂²⋊C₄	—	68	2	no	—
279	C₂²⋊C₄	—	68	2	no	—
280	C₄²	—	4	2	no	—
281	C₄²	—	17	1	no	—
282	C₄²	—	68	2	no	—
283	C₄²	D₄×D₁₇	1	1	yes	characteristic
284	C₁₆	—	2	1	no	—
285	C₁₆	—	17	2	no	—
286	C₁₆	—	34	1	no	—
287	C₁₆	C₂×C₄×D₁₇	1	2	yes	—
288	C₂³	—	17	1	no	—
289	C₂³	—	68	2	no	—
290	Q₈	—	2	4	no	—
291	Q₈	—	4	2	no	—
292	D₄	—	34	4	no	—
293	D₄	—	68	8	no	—
294	D₄	—	68	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	272	2	no	—
298	C₂×C₄	—	272	2	no	—
299	C₂×C₄	—	272	2	no	—
300	C₂×C₄	—	17	1	no	—
301	C₂×C₄	—	17	2	no	—
302	C₂×C₄	—	17	2	no	—
303	C₂×C₄	—	17	1	no	—
304	C₂×C₄	—	17	1	no	—
305	C₂×C₄	—	68	8	no	—
306	C₂×C₄	—	68	2	no	—
307	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
308	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	17	2	no	—
312	C₈	—	34	1	no	—
313	C₈	C₂²×C₄×D₁₇	1	1	yes	characteristic
314	C₈	#41×D₁₇	1	1	yes	characteristic
315	C₂²	—	17	2	no	—
316	C₂²	—	34	1	no	—
317	C₂²	—	34	1	no	—
318	C₂²	—	68	4	no	—
319	C₂²	—	136	2	no	—
320	C₂²	—	136	2	no	—
321	C₂²	C₂×D₈×D₁₇	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	4	no	—
325	C₄	—	8	2	no	—
326	C₄	—	17	2	no	—
327	C₄	—	34	1	no	—
328	C₄	—	34	1	no	—
329	C₄	—	136	2	no	—
330	C₄	C₄×D₄×D₁₇	1	1	yes	characteristic
331	C₄	#135×D₁₇	1	1	yes	characteristic
332	C₂	—	34	2	no	—
333	C₂	—	136	4	no	—
334	C₂	C₄×D₈×D₁₇	1	1	yes	characteristic
335	C₂	#14541922	1	1	yes	characteristic
336	C₂	#14541933	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₇
commutator	C₈×C₁₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₄×C₁₇×Q₃₂	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	C₂×#11763242	C₂
cosocle	C₂×C₈×C₁₇	C₂⁴
hypercenter	C₄×Q₃₂	D₁₇
hypocenter	C₁₇	#33459
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #14537659	C₂	semidirect
3. #14530651	C₂	semidirect
4. #14530646	C₂	semidirect
5. Q₃₂×Dic₁₇	C₂	semidirect
6. C₄×D₂₇₂	C₂	semidirect
7. #14530630	C₂	semidirect
8. D₁₇×#366	C₂	semidirect
9. #14530495	C₂	semidirect
10. #14530379	C₂	semidirect
11. C₄×C₁₇×Q₃₂	C₂	semidirect
14. #11763242	C₄	semidirect
46. #10587605	C₂×C₄	semidirect
58. C₁₇×Q₃₂	C₂×C₄	semidirect
65. #10587499	C₂×C₄	semidirect
75. D₂₇₂	C₂×C₄	semidirect
77. C₁₆×D₁₇	C₂×C₄	semidirect
152. C₄×Q₃₂	D₁₇	semidirect
226. Q₃₂	C₄×D₁₇	semidirect
246. C₁₇	#33459	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763242	2
#14537659	2
#14530651	2
#14530646	2
Q₃₂×Dic₁₇	2
C₄×D₂₇₂	2
#14530630	2
D₁₇×#366	2
#14530495	2
#14530379	2
C₄×C₁₇×Q₃₂	2
#33459	17