Subgroups — #17189066

Order 4352.

524 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14539296	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11762925	C₂	1	1	yes	maximal, characteristic
3	#14530959	C₂	1	4	yes	maximal, complemented
4	#14530954	C₂	1	4	yes	maximal, complemented
5	#14530558	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11762629	C₂	1	1	yes	maximal, characteristic, complemented
7	#14530521	C₂	1	1	yes	maximal, characteristic
8	#14530519	C₂	1	1	yes	maximal, characteristic
9	C₁₇×#2155	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10587590	C₂²	1	1	yes	characteristic
11	C₂×#10587593	—	2	1	no	—
12	#11763362	—	2	1	no	—
13	C₂×#10587483	C₂²	1	1	yes	characteristic
14	#11763213	—	2	1	no	—
15	#11763210	—	2	1	no	—
16	#11763209	C₂²	1	2	yes	complemented
17	#11763206	C₂²	1	2	yes	complemented
18	#11762925	—	2	1	no	—
19	#11762925	—	2	1	no	—
20	#11762925	C₂²	1	2	yes	—
21	#11762925	C₂²	1	2	yes	—
22	#11762847	C₂²	1	1	yes	characteristic
23	#11762845	—	2	1	no	—
24	C₂×#10587496	—	2	1	no	—
25	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
26	#11762633	C₂²	1	4	yes	complemented
27	#11762629	C₂²	1	2	yes	complemented
28	#11762629	C₂²	1	2	yes	complemented
29	#11762620	C₂²	1	4	yes	—
30	#11762618	C₂²	1	4	yes	—
31	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
32	C₂×C₁₇×#108	C₂²	1	1	yes	characteristic
33	C₁₇×#427	C₂²	1	1	yes	characteristic
34	C₁₇×#365	C₂²	1	4	yes	complemented
35	C₂×#63252	—	2	1	no	—
36	C₂×M₁₆×C₁₇	—	2	1	no	—
37	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
38	C₁₇×#117	—	2	1	no	—
39	#10587612	—	2	1	no	—
40	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
41	C₂²×#3622	D₄	1	1	yes	characteristic
42	#10587593	—	2	2	no	—
43	#10587593	—	2	2	no	—
44	#10587593	—	4	1	no	—
45	#10587593	—	4	1	no	—
46	C₂×Dic₆₈	Q₈	1	1	yes	characteristic, complemented
47	#10587590	—	2	1	no	—
48	#10587590	—	2	1	no	—
49	#10587590	Q₈	1	2	yes	complemented
50	#10587590	C₂³	1	2	yes	—
51	C₂×D₁₃₆	Q₈	1	1	yes	characteristic, complemented
52	C₂×#63219	—	2	1	no	—
53	C₂×#63218	—	4	1	no	—
54	C₂×C₈×D₁₇	—	4	1	no	—
55	#10587576	—	2	2	no	—
56	#10587574	—	2	2	no	—
57	#10587564	—	4	2	no	—
58	C₄×Dic₃₄	—	4	2	no	—
59	C₁₇×M₃₂	—	2	1	no	—
60	C₁₇×M₃₂	C₂³	1	2	yes	—
61	C₂×C₁₆×C₁₇	—	2	1	no	—
62	C₁₇×#108	—	2	1	no	—
63	C₁₇×#108	C₂³	1	2	yes	—
64	C₁₇×#107	—	2	1	no	—
65	C₁₇×#106	C₂³	1	2	yes	—
66	#10587496	—	2	2	no	—
67	#10587496	—	2	2	no	—
68	#10587489	—	2	1	no	—
69	#10587488	C₂³	1	2	yes	—
70	C₈×Dic₁₇	D₄	1	1	yes	characteristic
71	C₈×Dic₁₇	D₄	1	1	yes	characteristic
72	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
73	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
74	#10587483	—	2	1	no	—
75	#10587483	C₂³	1	2	yes	—
76	C₂×#63214	—	2	1	no	—
77	#10587479	—	2	2	no	—
78	#10587478	—	2	2	no	—
79	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₂²⋊D₁₇	—	4	1	no	—
81	C₂²×Dic₁₇	—	2	1	no	—
82	#63252	—	2	2	no	—
83	#63252	—	2	2	no	—
84	#63252	—	4	1	no	—
85	#63252	—	4	1	no	—
86	C₂×D₆₈	—	2	1	no	—
87	C₂×C₄×D₁₇	—	4	1	no	—
88	C₂×Dic₃₄	—	2	1	no	—
89	C₂×Dic₃₄	—	4	2	no	—
90	M₁₆×C₁₇	—	2	2	no	—
91	M₁₆×C₁₇	—	4	1	no	—
92	C₂×C₈×C₁₇	—	4	1	no	—
93	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
95	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
96	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
97	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
98	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₁₇	—	2	2	no	—
100	C₂²⋊C₄×C₁₇	—	4	1	no	—
101	C₄²×C₁₇	—	4	1	no	—
102	C₂²⋊Dic₁₇	—	4	1	no	—
103	C₄⋊Dic₁₇	—	2	2	no	—
104	C₄⋊Dic₁₇	—	4	2	no	—
105	#63225	—	4	2	no	—
106	#63225	—	4	2	no	—
107	C₄×Dic₁₇	—	2	1	no	—
108	C₄×Dic₁₇	—	4	1	no	—
109	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
110	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
111	C₂×#3622	SD₁₆	1	1	yes	characteristic
112	C₂×#3622	SD₁₆	1	1	yes	characteristic
113	C₂×#3622	SD₁₆	1	1	yes	characteristic
114	C₂×#3622	SD₁₆	1	1	yes	characteristic
115	C₂×#3622	C₂×D₄	1	1	yes	characteristic
116	C₂×#3622	C₂×D₄	1	1	yes	characteristic
117	Dic₆₈	—	2	1	no	—
118	Dic₆₈	C₂×Q₈	1	2	yes	—
119	D₁₃₆	—	2	1	no	—
120	D₁₃₆	C₂×Q₈	1	2	yes	—
121	#63219	—	2	2	no	—
122	#63219	—	2	2	no	—
123	#63218	—	4	2	no	—
124	#63218	—	4	2	no	—
125	C₈×D₁₇	—	4	2	no	—
126	C₈×D₁₇	—	4	2	no	—
127	C₁₆×C₁₇	—	2	2	no	—
128	#63214	—	2	2	no	—
129	#33637	—	17	1	no	maximal, Sylow
130	C₂³×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂²×D₁₇	—	4	1	no	—
132	C₂×C₄×C₁₇	—	4	2	no	—
133	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
139	C₂²⋊D₁₇	—	4	2	no	—
140	C₂²⋊D₁₇	—	4	2	no	—
141	C₂×Dic₁₇	—	2	1	no	—
142	C₂×Dic₁₇	—	2	1	no	—
143	C₂×Dic₁₇	—	4	2	no	—
144	C₂×Dic₁₇	—	4	1	no	—
145	C₂×Dic₁₇	—	4	2	no	—
146	D₆₈	—	2	2	no	—
147	D₆₈	—	4	1	no	—
148	C₄×D₁₇	—	4	2	no	—
149	C₄×D₁₇	—	4	2	no	—
150	Dic₃₄	—	2	2	no	—
151	Dic₃₄	—	4	1	no	—
152	Dic₃₄	—	8	4	no	—
153	C₈×C₁₇	—	4	2	no	—
154	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
155	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
156	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
157	C₈×C₁₇	C₂×#41	1	1	yes	characteristic
158	#3622	C₂×SD₁₆	1	1	yes	characteristic
159	#3622	C₂×SD₁₆	1	1	yes	characteristic
160	#3622	C₂×SD₁₆	1	1	yes	characteristic
161	#3622	C₂×SD₁₆	1	1	yes	characteristic
162	#3160	—	17	1	no	—
163	C₂×#443	—	17	1	no	—
164	#2231	—	17	4	no	—
165	#2230	—	17	4	no	—
166	#2155	—	17	1	no	—
167	#2155	D₁₇	1	1	yes	characteristic, complemented
168	C₂×#364	—	17	1	no	—
169	#2145	—	17	1	no	—
170	#2145	—	17	1	no	—
171	C₂²×C₁₇	—	4	1	no	—
172	C₂²×C₁₇	#476	1	1	yes	characteristic
173	C₂²×C₁₇	#476	1	1	yes	characteristic
174	C₂²×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₂×D₁₇	—	4	2	no	—
176	C₂×D₁₇	—	8	2	no	—
177	C₄×C₁₇	—	8	2	no	—
178	C₄×C₁₇	#476	1	1	yes	characteristic
179	C₄×C₁₇	#476	1	1	yes	characteristic
180	C₄×C₁₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
181	C₄×C₁₇	C₂²×SD₁₆	1	1	yes	characteristic
182	Dic₁₇	—	2	2	no	—
183	Dic₁₇	—	4	1	no	—
184	Dic₁₇	—	4	2	no	—
185	Dic₁₇	—	8	2	no	—
186	C₂×#135	—	17	1	no	—
187	C₂×#131	—	34	1	no	—
188	#533	—	34	1	no	—
189	C₂×M₃₂	—	17	1	no	—
190	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
191	#500	—	34	1	no	—
192	#499	—	17	2	no	—
193	#499	—	34	1	no	—
194	#498	—	17	2	no	—
195	#443	—	17	2	no	—
196	#443	—	17	2	no	—
197	#443	—	34	1	no	—
198	#443	—	34	1	no	—
199	C₂×#108	C₂×D₁₇	1	1	yes	characteristic
200	#427	—	17	1	no	—
201	#427	—	34	1	no	—
202	#427	C₂×D₁₇	1	1	yes	characteristic
203	C₂×#104	—	34	1	no	—
204	C₂×C₄×C₈	—	17	1	no	—
205	#365	—	17	4	no	—
206	#365	C₂×D₁₇	1	4	yes	complemented
207	#364	—	17	2	no	—
208	#364	—	17	2	no	—
209	#360	—	17	4	no	—
210	#360	—	17	4	no	—
211	C₂×C₁₇	—	2	1	no	—
212	C₂×C₁₇	—	2	1	no	—
213	C₂×C₁₇	—	2	1	no	—
214	C₂×C₁₇	C₂×#476	1	1	yes	characteristic
215	D₁₇	—	8	2	no	—
216	C₂×#41	—	34	1	no	—
217	#135	—	17	2	no	—
218	#135	—	17	2	no	—
219	#135	—	34	1	no	—
220	#135	—	34	1	no	—
221	C₂×Q₁₆	—	17	1	no	—
222	C₂×SD₁₆	—	34	1	no	—
223	C₂×D₈	—	17	1	no	—
224	#131	—	34	2	no	—
225	#131	—	34	2	no	—
226	#131	—	68	1	no	—
227	#131	—	68	1	no	—
228	C₂×M₁₆	—	2	1	no	—
229	C₂×M₁₆	—	68	1	no	—
230	C₂²×C₈	—	17	1	no	—
231	C₂²×C₈	—	68	1	no	—
232	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
233	C₄⋊Q₈	—	34	2	no	—
234	#125	—	34	2	no	—
235	C₂²⋊Q₈	—	68	2	no	—
236	C₄×Q₈	—	68	2	no	—
237	#117	—	2	1	no	—
238	#117	—	34	1	no	—
239	C₂×C₄²	—	17	1	no	—
240	M₃₂	—	2	1	no	—
241	M₃₂	—	17	2	no	—
242	M₃₂	—	34	1	no	—
243	M₃₂	C₂²×D₁₇	1	2	yes	—
244	C₂×C₁₆	—	2	1	no	—
245	C₂×C₁₆	—	34	1	no	—
246	#108	—	2	1	no	—
247	#108	C₂²×D₁₇	1	2	yes	—
248	#107	—	2	1	no	—
249	#107	—	34	2	no	—
250	#107	—	34	1	no	—
251	#106	—	17	2	no	—
252	#106	—	34	2	no	—
253	#106	C₂²×D₁₇	1	2	yes	—
254	#104	—	34	2	no	—
255	#104	—	34	2	no	—
256	C₄×C₈	—	17	1	no	—
257	C₄×C₈	—	17	1	no	—
258	C₄×C₈	—	17	1	no	—
259	C₄×C₈	—	17	1	no	—
260	C₁₇	#33637	1	1	yes	characteristic, complemented, Sylow
261	#41	—	34	2	no	—
262	#41	—	34	2	no	—
263	#41	—	68	1	no	—
264	#41	—	68	1	no	—
265	C₂×Q₈	—	34	1	no	—
266	C₂×Q₈	—	68	2	no	—
267	C₂×D₄	—	34	1	no	—
268	C₂×D₄	—	68	1	no	—
269	C₂²×C₄	—	34	1	no	—
270	C₂²×C₄	—	68	1	no	—
271	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
272	Q₁₆	—	17	2	no	—
273	Q₁₆	—	34	1	no	—
274	SD₁₆	—	34	2	no	—
275	SD₁₆	—	34	2	no	—
276	D₈	—	17	2	no	—
277	D₈	—	34	1	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	4	1	no	—
280	M₁₆	—	68	2	no	—
281	M₁₆	—	68	2	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	—	17	1	no	—
284	C₂×C₈	—	17	1	no	—
285	C₂×C₈	—	17	1	no	—
286	C₂×C₈	—	17	1	no	—
287	C₂×C₈	—	17	1	no	—
288	C₂×C₈	—	17	1	no	—
289	C₂×C₈	—	68	2	no	—
290	C₂×C₈	—	68	2	no	—
291	C₂×C₈	#63252	1	1	yes	characteristic
292	C₂×C₈	#63252	1	1	yes	characteristic
293	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
294	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
295	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
296	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	34	2	no	—
299	C₄⋊C₄	—	68	2	no	—
300	C₄⋊C₄	—	68	2	no	—
301	C₄⋊C₄	—	68	2	no	—
302	C₂²⋊C₄	—	4	1	no	—
303	C₂²⋊C₄	—	68	1	no	—
304	C₄²	—	4	1	no	—
305	C₄²	—	17	1	no	—
306	C₄²	—	17	1	no	—
307	C₄²	—	34	1	no	—
308	C₄²	—	68	1	no	—
309	C₁₆	—	2	2	no	—
310	C₁₆	—	34	2	no	—
311	C₂³	—	68	1	no	—
312	C₂³	#10587600	1	1	yes	characteristic
313	Q₈	—	34	2	no	—
314	Q₈	—	68	1	no	—
315	Q₈	—	136	4	no	—
316	D₄	—	34	2	no	—
317	D₄	—	68	2	no	—
318	D₄	—	68	2	no	—
319	D₄	—	68	1	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	34	1	no	—
322	C₂×C₄	—	34	1	no	—
323	C₂×C₄	—	68	2	no	—
324	C₂×C₄	—	68	1	no	—
325	C₂×C₄	—	68	2	no	—
326	C₂×C₄	—	68	2	no	—
327	C₂×C₄	—	68	2	no	—
328	C₂×C₄	#10587582	1	1	yes	characteristic
329	C₂×C₄	#10587582	1	1	yes	characteristic
330	C₂×C₄	SD₁₆×D₁₇	1	1	yes	characteristic
331	C₂×C₄	SD₁₆×D₁₇	1	1	yes	characteristic
332	C₂×C₄	#10587600	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
334	C₈	—	4	2	no	—
335	C₈	—	17	1	no	—
336	C₈	—	17	1	no	—
337	C₈	—	17	1	no	—
338	C₈	—	17	1	no	—
339	C₈	#10587582	1	1	yes	characteristic
340	C₈	#10587582	1	1	yes	characteristic
341	C₈	C₂×#63252	1	1	yes	characteristic
342	C₈	C₂×Q₈×D₁₇	1	1	yes	characteristic
343	C₂²	—	4	1	no	—
344	C₂²	—	68	2	no	—
345	C₂²	—	136	2	no	—
346	C₂²	#11763098	1	1	yes	characteristic
347	C₂²	#11763098	1	1	yes	characteristic
348	C₂²	C₂×#10587600	1	1	yes	characteristic
349	C₄	—	8	2	no	—
350	C₄	—	34	2	no	—
351	C₄	—	68	1	no	—
352	C₄	—	68	2	no	—
353	C₄	—	136	2	no	—
354	C₄	#11763098	1	1	yes	characteristic
355	C₄	#11763098	1	1	yes	characteristic
356	C₄	C₂×#10587582	1	1	yes	characteristic
357	C₄	C₂×SD₁₆×D₁₇	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	136	2	no	—
362	C₂	C₂×#11763098	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11763098
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#2155	C₂
socle	C₂×C₁₇	C₂×#476
prime radical	C₂×#10587590	C₂²
hypercenter	#2155	D₁₇
hypocenter	C₁₇	#33637
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2155

Complements

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

N	H ≅ G/N	type
1. #14539296	C₂	semidirect
3. #14530959	C₂	semidirect
4. #14530954	C₂	semidirect
5. #14530558	C₂	semidirect
6. C₂×#11762629	C₂	semidirect
9. C₁₇×#2155	C₂	semidirect
16. #11763209	C₂²	semidirect
17. #11763206	C₂²	semidirect
26. #11762633	C₂²	semidirect
27. #11762629	C₂²	semidirect
28. #11762629	C₂²	semidirect
34. C₁₇×#365	C₂²	semidirect
46. C₂×Dic₆₈	Q₈	semidirect
49. #10587590	Q₈	semidirect
51. C₂×D₁₃₆	Q₈	semidirect
167. #2155	D₁₇	semidirect
206. #365	C₂×D₁₇	semidirect
260. C₁₇	#33637	semidirect

Maximal subgroups

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

subgroup	index
#14539296	2
C₂×#11762925	2
#14530959	2
#14530954	2
#14530558	2
C₂×#11762629	2
#14530521	2
#14530519	2
C₁₇×#2155	2
#33637	17