Subgroups — #14540674

Order 2176.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763498	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763469	C₂	1	1	yes	maximal, characteristic
3	#11763165	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763124	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763119	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763048	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763039	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763034	C₂	1	1	yes	maximal, characteristic, complemented
9	#11762973	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762966	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762903	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762872	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762803	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#485	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₇	C₂²	1	1	yes	characteristic
17	C₂×#63252	C₂²	1	1	yes	characteristic
18	C₁₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
19	C₁₇×#117	C₂²	1	1	yes	characteristic
20	#10587633	C₂²	1	1	yes	characteristic
21	C₂×#63231	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₁₇	C₂²	1	1	yes	characteristic
23	#10587610	C₂²	1	1	yes	characteristic
24	C₂²×#3622	C₂²	1	1	yes	characteristic
25	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
26	C₂×#63219	C₂²	1	1	yes	characteristic
27	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
28	#10587583	C₂²	1	1	yes	characteristic
29	#10587582	—	2	1	no	—
30	#10587581	C₂²	1	1	yes	characteristic
31	#10587579	—	2	1	no	—
32	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
33	#10587576	C₂²	1	1	yes	characteristic
34	#10587573	C₂²	1	1	yes	characteristic
35	#10587573	C₂²	1	1	yes	characteristic
36	#10587571	—	2	1	no	—
37	#10587570	—	2	1	no	—
38	#10587569	—	2	1	no	—
39	#10587565	—	2	1	no	—
40	#10587558	—	2	1	no	—
41	#10587556	—	2	1	no	—
42	C₁₇×#107	C₂²	1	1	yes	characteristic
43	C₁₇×#106	C₂²	1	1	yes	characteristic
44	C₁₇×#103	C₂²	1	1	yes	characteristic
45	C₁₇×#103	C₂²	1	1	yes	characteristic
46	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
47	#10587506	C₂²	1	1	yes	characteristic
48	#10587492	C₂²	1	1	yes	characteristic
49	#10587491	C₂²	1	1	yes	characteristic
50	#10587487	C₂²	1	1	yes	characteristic
51	#10587485	C₂²	1	1	yes	characteristic
52	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
53	#10587481	C₂²	1	1	yes	characteristic
54	#10587481	C₂²	1	1	yes	characteristic
55	#10587481	C₂²	1	1	yes	characteristic
56	#10587480	C₂²	1	1	yes	characteristic
57	#10587480	C₂²	1	1	yes	characteristic
58	#10587478	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₇	—	2	1	no	—
62	C₂×C₂²⋊D₁₇	D₄	1	1	yes	characteristic
63	Q₈×D₁₇	—	2	2	no	—
64	Q₈×D₁₇	—	4	1	no	—
65	#63252	—	2	2	no	—
66	#63252	—	4	1	no	—
67	C₂×D₆₈	—	2	1	no	—
68	C₂×D₆₈	C₂³	1	1	yes	characteristic
69	C₂×C₄×D₁₇	—	2	1	no	—
70	C₂×C₄×D₁₇	—	2	1	no	—
71	C₂×C₄×D₁₇	—	2	1	no	—
72	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
73	C₂×Dic₃₄	—	2	1	no	—
74	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
75	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₇	—	2	1	no	—
79	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
82	C₂²⋊C₄×C₁₇	—	2	1	no	—
83	C₂²⋊C₄×C₁₇	—	2	1	no	—
84	C₄²×C₁₇	—	2	1	no	—
85	C₂²⋊Dic₁₇	—	2	1	no	—
86	C₂²⋊Dic₁₇	D₄	1	1	yes	characteristic
87	#63231	—	4	1	no	—
88	#63227	—	2	1	no	—
89	#63227	—	2	1	no	—
90	#63227	—	2	1	no	—
91	#63227	—	2	1	no	—
92	#63227	D₄	1	1	yes	characteristic
93	C₄⋊Dic₁₇	—	2	1	no	—
94	C₄⋊Dic₁₇	—	2	1	no	—
95	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
96	#63225	—	2	1	no	—
97	#63225	—	2	1	no	—
98	#63225	—	2	1	no	—
99	#63225	—	2	1	no	—
100	#63225	—	2	1	no	—
101	#63225	D₄	1	1	yes	characteristic
102	C₄×Dic₁₇	—	2	1	no	—
103	C₄×Dic₁₇	—	2	1	no	—
104	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
105	C₂×#3622	—	4	1	no	—
106	C₂×#3622	C₂³	1	1	yes	characteristic
107	C₂×#3622	C₂³	1	1	yes	characteristic
108	Dic₆₈	—	2	2	no	—
109	#63219	—	4	1	no	—
110	C₈×D₁₇	—	4	1	no	—
111	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₁₇	—	2	1	no	—
113	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₁₇	—	2	1	no	—
115	Q₈×C₁₇	—	2	1	no	—
116	C₂×C₄×C₁₇	—	2	1	no	—
117	C₂×C₄×C₁₇	—	2	1	no	—
118	C₂×C₄×C₁₇	—	2	1	no	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	—	2	1	no	—
121	C₂×C₄×C₁₇	—	2	1	no	—
122	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
124	C₂²⋊D₁₇	—	2	2	no	—
125	C₂²⋊D₁₇	—	4	2	no	—
126	C₂×Dic₁₇	—	2	1	no	—
127	C₂×Dic₁₇	—	2	1	no	—
128	C₂×Dic₁₇	—	2	1	no	—
129	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
132	D₆₈	—	2	1	no	—
133	D₆₈	—	2	1	no	—
134	D₆₈	—	4	2	no	—
135	C₄×D₁₇	—	2	1	no	—
136	C₄×D₁₇	—	2	1	no	—
137	C₄×D₁₇	—	4	2	no	—
138	C₄×D₁₇	—	4	2	no	—
139	C₄×D₁₇	—	4	2	no	—
140	Dic₃₄	—	2	1	no	—
141	Dic₃₄	—	2	1	no	—
142	Dic₃₄	—	2	1	no	—
143	Dic₃₄	—	4	2	no	—
144	Dic₃₄	#41	1	2	yes	—
145	C₈×C₁₇	—	2	1	no	—
146	C₈×C₁₇	—	2	1	no	—
147	#3622	—	2	1	no	—
148	#3622	—	2	1	no	—
149	#3300	—	17	1	no	maximal, Sylow
150	C₂²×C₁₇	—	2	1	no	—
151	C₂²×C₁₇	—	2	1	no	—
152	C₂²×C₁₇	—	2	1	no	—
153	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₁₇	—	2	1	no	—
155	C₂×D₁₇	—	2	1	no	—
156	C₂×D₁₇	—	4	1	no	—
157	C₂×D₁₇	—	4	1	no	—
158	C₂×D₁₇	—	4	1	no	—
159	C₂×D₁₇	#135	1	2	yes	—
160	C₄×C₁₇	—	2	1	no	—
161	C₄×C₁₇	—	4	1	no	—
162	C₄×C₁₇	—	4	1	no	—
163	C₄×C₁₇	—	4	1	no	—
164	C₄×C₁₇	—	4	1	no	—
165	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
166	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
167	Dic₁₇	—	2	2	no	—
168	Dic₁₇	—	2	1	no	—
169	Dic₁₇	—	2	1	no	—
170	Dic₁₇	—	2	1	no	—
171	Dic₁₇	—	4	1	no	—
172	Dic₁₇	—	4	1	no	—
173	#552	—	17	1	no	—
174	#548	—	17	1	no	—
175	#488	—	17	1	no	—
176	#485	D₁₇	1	1	yes	characteristic, complemented
177	#480	—	17	1	no	—
178	#479	—	17	1	no	—
179	#467	—	17	1	no	—
180	#465	—	17	1	no	—
181	#464	—	17	1	no	—
182	#452	—	17	1	no	—
183	#451	—	17	1	no	—
184	C₄×Q₁₆	—	17	1	no	—
185	C₈×D₄	—	17	1	no	—
186	#416	—	17	1	no	—
187	C₂×#103	—	17	1	no	—
188	C₂×C₁₇	—	4	1	no	—
189	C₂×C₁₇	#546	1	1	yes	characteristic
190	C₂×C₁₇	C₂×#135	1	1	yes	characteristic
191	C₂×C₁₇	#578	1	1	yes	characteristic
192	D₁₇	—	2	2	no	—
193	D₁₇	—	8	1	no	—
194	C₂×#41	—	17	1	no	—
195	C₂²×Q₈	—	17	1	no	—
196	C₂×Q₁₆	—	17	1	no	—
197	C₂×Q₁₆	—	17	1	no	—
198	C₂×SD₁₆	—	17	1	no	—
199	C₂²×C₈	—	17	1	no	—
200	C₂²×C₈	—	17	1	no	—
201	#126	—	34	1	no	—
202	#125	—	17	1	no	—
203	#125	—	34	1	no	—
204	#124	—	17	1	no	—
205	#124	—	34	1	no	—
206	#123	—	34	1	no	—
207	C₂²⋊Q₈	—	17	1	no	—
208	C₂²⋊Q₈	—	17	1	no	—
209	C₂²⋊Q₈	—	34	1	no	—
210	C₂²⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
211	#121	—	17	1	no	—
212	#121	—	34	1	no	—
213	C₄×Q₈	—	17	1	no	—
214	C₄×Q₈	—	17	1	no	—
215	C₄×D₄	—	17	1	no	—
216	C₄×D₄	—	34	1	no	—
217	#117	—	34	1	no	—
218	#117	C₂×D₁₇	1	1	yes	characteristic
219	C₂×C₄⋊C₄	—	17	1	no	—
220	#107	—	17	1	no	—
221	#107	C₂×D₁₇	1	1	yes	characteristic
222	#106	C₂×D₁₇	1	1	yes	characteristic
223	C₄⋊C₈	—	17	1	no	—
224	#103	—	17	1	no	—
225	#103	—	17	1	no	—
226	#103	—	17	1	no	—
227	#103	—	17	1	no	—
228	#103	—	17	1	no	—
229	#103	C₂×D₁₇	1	1	yes	characteristic
230	#103	C₂×D₁₇	1	1	yes	characteristic
231	#102	—	17	1	no	—
232	#102	—	17	1	no	—
233	#102	—	17	1	no	—
234	C₂²⋊C₈	—	17	1	no	—
235	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
236	C₄×C₈	—	17	1	no	—
237	C₁₇	#3300	1	1	yes	characteristic, complemented, Sylow
238	#41	—	34	2	no	—
239	#41	—	68	1	no	—
240	C₂×Q₈	—	17	1	no	—
241	C₂×Q₈	—	17	1	no	—
242	C₂×Q₈	—	34	1	no	—
243	C₂×Q₈	—	34	2	no	—
244	C₂×Q₈	—	68	1	no	—
245	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
246	C₂×D₄	—	17	1	no	—
247	C₂×D₄	—	17	1	no	—
248	C₂×D₄	—	34	1	no	—
249	C₂×D₄	—	34	1	no	—
250	C₂²×C₄	—	17	1	no	—
251	C₂²×C₄	—	34	1	no	—
252	C₂²×C₄	—	34	1	no	—
253	C₂²×C₄	—	34	1	no	—
254	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
255	Q₁₆	—	34	2	no	—
256	Q₁₆	—	68	1	no	—
257	SD₁₆	—	68	1	no	—
258	C₂×C₈	—	17	1	no	—
259	C₂×C₈	—	17	1	no	—
260	C₂×C₈	—	68	1	no	—
261	C₂×C₈	—	68	1	no	—
262	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	17	1	no	—
266	C₄⋊C₄	—	17	1	no	—
267	C₄⋊C₄	—	34	1	no	—
268	C₄⋊C₄	—	34	1	no	—
269	C₄⋊C₄	—	34	1	no	—
270	C₄⋊C₄	—	34	1	no	—
271	C₄⋊C₄	—	34	1	no	—
272	C₄⋊C₄	—	34	1	no	—
273	C₄⋊C₄	—	34	1	no	—
274	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
275	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	17	1	no	—
280	C₂²⋊C₄	—	17	1	no	—
281	C₂²⋊C₄	—	34	1	no	—
282	C₂²⋊C₄	—	34	1	no	—
283	C₂²⋊C₄	—	34	1	no	—
284	C₂²⋊C₄	—	34	1	no	—
285	C₂²⋊C₄	—	34	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	17	1	no	—
288	C₄²	—	34	1	no	—
289	C₄²	—	34	1	no	—
290	C₂³	—	17	1	no	—
291	C₂³	—	34	1	no	—
292	C₂³	D₄×D₁₇	1	1	yes	characteristic
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	17	2	no	—
296	Q₈	—	34	1	no	—
297	Q₈	—	34	1	no	—
298	Q₈	—	34	1	no	—
299	Q₈	—	68	2	no	—
300	D₄	—	34	2	no	—
301	D₄	—	34	1	no	—
302	D₄	—	34	1	no	—
303	D₄	—	68	2	no	—
304	D₄	—	68	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	17	1	no	—
312	C₂×C₄	—	17	1	no	—
313	C₂×C₄	—	17	1	no	—
314	C₂×C₄	—	34	1	no	—
315	C₂×C₄	—	34	1	no	—
316	C₂×C₄	—	34	1	no	—
317	C₂×C₄	—	34	1	no	—
318	C₂×C₄	—	34	1	no	—
319	C₂×C₄	—	68	2	no	—
320	C₂×C₄	—	68	2	no	—
321	C₂×C₄	—	68	2	no	—
322	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
323	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	34	1	no	—
327	C₈	—	34	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	17	2	no	—
332	C₂²	—	34	1	no	—
333	C₂²	—	34	1	no	—
334	C₂²	—	68	1	no	—
335	C₂²	—	68	1	no	—
336	C₂²	—	68	1	no	—
337	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	34	1	no	—
344	C₄	—	34	1	no	—
345	C₄	—	34	1	no	—
346	C₄	—	34	2	no	—
347	C₄	—	68	1	no	—
348	C₄	—	68	1	no	—
349	C₄	#41×D₁₇	1	1	yes	characteristic
350	C₄	#10587696	1	1	yes	characteristic
351	C₂	—	4	1	no	—
352	C₂	—	34	2	no	—
353	C₂	—	136	1	no	—
354	C₂	#11763451	1	1	yes	characteristic
355	C₂	#135×D₁₇	1	1	yes	characteristic
356	C₂	#11763625	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#485	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763469	C₂
hypercenter	#485	D₁₇
hypocenter	C₁₇	#3300
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
1. #11763498	C₂	semidirect
3. #11763165	C₂	semidirect
4. #11763124	C₂	semidirect
5. #11763119	C₂	semidirect
6. #11763048	C₂	semidirect
7. #11763039	C₂	semidirect
8. #11763034	C₂	semidirect
9. #11762973	C₂	semidirect
10. #11762966	C₂	semidirect
11. #11762903	C₂	semidirect
12. #11762872	C₂	semidirect
13. #11762803	C₂	semidirect
14. #103×D₁₇	C₂	semidirect
15. C₁₇×#485	C₂	semidirect
176. #485	D₁₇	semidirect
237. C₁₇	#3300	semidirect

Maximal subgroups

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

subgroup	index
#11763498	2
#11763469	2
#11763165	2
#11763124	2
#11763119	2
#11763048	2
#11763039	2
#11763034	2
#11762973	2
#11762966	2
#11762903	2
#11762872	2
#11762803	2
#103×D₁₇	2
C₁₇×#485	2
#3300	17