Subgroups — #14539347

Order 2176.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763345	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763340	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763209	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763190	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763165	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763162	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763048	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763043	C₂	1	1	yes	maximal, characteristic
9	#11763038	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762901	C₂	1	1	yes	maximal, characteristic, complemented
11	SD₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762896	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762889	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#465	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
19	C₁₇×#121	C₂²	1	1	yes	characteristic
20	#10587633	C₂²	1	1	yes	characteristic
21	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
22	C₂×#63231	C₂²	1	1	yes	characteristic
23	#10587622	C₂²	1	1	yes	characteristic
24	#10587621	—	2	1	no	—
25	D₄×Dic₁₇	—	2	1	no	—
26	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
27	C₂×#63228	C₂²	1	1	yes	characteristic
28	#10587616	C₂²	1	1	yes	characteristic
29	#10587610	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
31	C₂²×#3622	C₂²	1	1	yes	characteristic
32	C₂×#63219	C₂²	1	1	yes	characteristic
33	#10587584	—	2	1	no	—
34	#10587579	C₂²	1	1	yes	characteristic
35	#10587578	—	2	1	no	—
36	#10587576	C₂²	1	1	yes	characteristic
37	#10587576	C₂²	1	1	yes	characteristic
38	#10587573	C₂²	1	1	yes	characteristic
39	#10587572	—	2	1	no	—
40	#10587567	—	2	1	no	—
41	#10587565	—	2	1	no	—
42	#10587563	—	2	1	no	—
43	C₁₇×#106	C₂²	1	1	yes	characteristic
44	C₁₇×#103	C₂²	1	1	yes	characteristic
45	C₁₇×#102	C₂²	1	1	yes	characteristic
46	#10587506	C₂²	1	1	yes	characteristic
47	#10587503	C₂²	1	1	yes	characteristic
48	#10587492	C₂²	1	1	yes	characteristic
49	#10587488	C₂²	1	1	yes	characteristic
50	#10587487	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
52	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
53	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
54	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
55	#10587481	C₂²	1	1	yes	characteristic
56	#10587480	C₂²	1	1	yes	characteristic
57	#10587478	C₂²	1	1	yes	characteristic
58	#10587478	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₇	—	2	1	no	—
61	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₇	—	2	1	no	—
64	C₂²×Dic₁₇	—	2	1	no	—
65	C₂²×Dic₁₇	D₄	1	1	yes	characteristic
66	C₂²×Dic₁₇	D₄	1	1	yes	characteristic
67	C₂×D₆₈	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₁₇	—	2	1	no	—
69	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
70	SD₁₆×C₁₇	—	2	2	no	—
71	C₂×C₈×C₁₇	—	2	2	no	—
72	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₇	—	2	1	no	—
75	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₁₇	—	2	1	no	—
78	C₂²⋊C₄×C₁₇	—	2	1	no	—
79	C₂²⋊Dic₁₇	—	2	1	no	—
80	C₂²⋊Dic₁₇	—	2	1	no	—
81	C₂²⋊Dic₁₇	—	2	1	no	—
82	C₂²⋊Dic₁₇	—	2	1	no	—
83	#63231	—	2	2	no	—
84	#63228	—	2	2	no	—
85	#63227	—	2	1	no	—
86	#63227	—	2	1	no	—
87	#63227	—	2	1	no	—
88	#63227	—	2	1	no	—
89	C₄⋊Dic₁₇	—	2	1	no	—
90	C₄⋊Dic₁₇	—	2	1	no	—
91	C₄⋊Dic₁₇	—	2	1	no	—
92	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
93	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
94	#63225	—	2	1	no	—
95	#63225	—	2	1	no	—
96	#63225	—	2	1	no	—
97	#63225	—	2	1	no	—
98	C₄×Dic₁₇	—	2	1	no	—
99	C₄×Dic₁₇	—	2	1	no	—
100	C₄×Dic₁₇	D₄	1	1	yes	characteristic
101	C₄×Dic₁₇	D₄	1	1	yes	characteristic
102	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
103	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
104	C₂×#3622	—	2	2	no	—
105	C₂×#3622	C₂³	1	1	yes	characteristic
106	C₂×#3622	C₂³	1	1	yes	characteristic
107	#63219	—	2	2	no	—
108	C₂³×C₁₇	—	2	1	no	—
109	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₁₇	—	2	1	no	—
111	Q₈×C₁₇	—	2	1	no	—
112	Q₈×C₁₇	—	2	1	no	—
113	D₄×C₁₇	—	2	1	no	—
114	D₄×C₁₇	—	2	1	no	—
115	D₄×C₁₇	—	4	2	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	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
123	C₂²⋊D₁₇	—	4	2	no	—
124	C₂×Dic₁₇	—	2	2	no	—
125	C₂×Dic₁₇	—	2	2	no	—
126	C₂×Dic₁₇	—	2	1	no	—
127	C₂×Dic₁₇	—	2	1	no	—
128	C₂×Dic₁₇	—	2	1	no	—
129	C₂×Dic₁₇	—	4	2	no	—
130	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
134	D₆₈	—	2	1	no	—
135	D₆₈	—	2	1	no	—
136	C₄×D₁₇	—	4	2	no	—
137	Dic₃₄	—	2	1	no	—
138	Dic₃₄	—	2	1	no	—
139	C₈×C₁₇	—	2	1	no	—
140	C₈×C₁₇	#41	1	2	yes	—
141	#3622	—	2	1	no	—
142	#3622	#41	1	2	yes	—
143	#3168	—	17	1	no	maximal, Sylow
144	C₂²×C₁₇	—	2	1	no	—
145	C₂²×C₁₇	—	2	1	no	—
146	C₂²×C₁₇	—	4	1	no	—
147	C₂²×C₁₇	—	4	1	no	—
148	C₂²×C₁₇	—	4	1	no	—
149	C₂²×C₁₇	#135	1	2	yes	—
150	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₁₇	—	4	1	no	—
152	C₂×D₁₇	—	4	1	no	—
153	C₂×D₁₇	—	4	1	no	—
154	C₄×C₁₇	—	2	1	no	—
155	C₄×C₁₇	—	4	1	no	—
156	C₄×C₁₇	—	4	1	no	—
157	C₄×C₁₇	—	4	1	no	—
158	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
159	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
160	Dic₁₇	—	2	1	no	—
161	Dic₁₇	—	2	1	no	—
162	Dic₁₇	—	2	1	no	—
163	Dic₁₇	—	4	1	no	—
164	Dic₁₇	—	4	1	no	—
165	Dic₁₇	—	4	1	no	—
166	Dic₁₇	#135	1	2	yes	—
167	#529	—	17	1	no	—
168	#529	—	17	1	no	—
169	#499	—	17	1	no	—
170	#495	—	17	1	no	—
171	#488	—	17	1	no	—
172	#488	—	17	1	no	—
173	#467	—	17	1	no	—
174	#466	—	17	1	no	—
175	#465	—	17	1	no	—
176	#465	D₁₇	1	1	yes	characteristic, complemented
177	C₄×Q₁₆	—	17	1	no	—
178	C₄×SD₁₆	—	17	1	no	—
179	C₄×SD₁₆	—	17	1	no	—
180	C₄×D₈	—	17	1	no	—
181	C₂×C₄×C₈	—	17	1	no	—
182	C₂×C₁₇	—	2	2	no	—
183	C₂×C₁₇	—	8	1	no	—
184	C₂×C₁₇	#532	1	1	yes	characteristic
185	C₂×C₁₇	C₂×#135	1	1	yes	characteristic
186	C₂×C₁₇	C₂×#135	1	1	yes	characteristic
187	D₁₇	—	8	1	no	—
188	C₂×Q₁₆	—	17	1	no	—
189	C₂×SD₁₆	—	17	1	no	—
190	C₂×SD₁₆	C₂×D₁₇	1	1	yes	characteristic
191	C₂×D₈	—	17	1	no	—
192	C₂²×C₈	—	17	1	no	—
193	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
194	#126	—	34	1	no	—
195	#126	—	34	1	no	—
196	#125	—	17	1	no	—
197	#125	—	17	1	no	—
198	#124	—	17	1	no	—
199	#124	—	17	1	no	—
200	#123	—	34	1	no	—
201	#123	—	34	1	no	—
202	C₂²⋊Q₈	—	17	1	no	—
203	C₂²⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
204	#121	—	17	1	no	—
205	#121	C₂×D₁₇	1	1	yes	characteristic
206	C₄×Q₈	—	17	1	no	—
207	C₄×Q₈	—	17	1	no	—
208	C₄×D₄	—	17	1	no	—
209	C₄×D₄	—	17	1	no	—
210	C₄×D₄	—	34	1	no	—
211	C₄×D₄	—	34	1	no	—
212	#117	—	34	1	no	—
213	#117	—	34	1	no	—
214	C₂×C₄²	—	17	1	no	—
215	#107	—	17	1	no	—
216	#107	—	17	1	no	—
217	#106	—	17	1	no	—
218	#106	C₂×D₁₇	1	1	yes	characteristic
219	#103	—	17	1	no	—
220	#103	—	17	1	no	—
221	#103	—	17	1	no	—
222	#103	C₂×D₁₇	1	1	yes	characteristic
223	#102	—	17	1	no	—
224	#102	—	17	1	no	—
225	#102	—	17	1	no	—
226	#102	C₂×D₁₇	1	1	yes	characteristic
227	C₄×C₈	—	17	1	no	—
228	C₄×C₈	—	17	1	no	—
229	C₄×C₈	—	17	1	no	—
230	C₄×C₈	—	17	1	no	—
231	C₁₇	#3168	1	1	yes	characteristic, complemented, Sylow
232	C₂×Q₈	—	17	1	no	—
233	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	17	1	no	—
236	C₂×D₄	—	34	1	no	—
237	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
238	C₂²×C₄	—	17	1	no	—
239	C₂²×C₄	—	17	1	no	—
240	C₂²×C₄	—	34	1	no	—
241	C₂²×C₄	—	34	1	no	—
242	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
243	Q₁₆	—	34	2	no	—
244	SD₁₆	—	2	2	no	—
245	SD₁₆	—	34	2	no	—
246	D₈	—	34	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	17	1	no	—
249	C₂×C₈	—	17	1	no	—
250	C₂×C₈	—	34	2	no	—
251	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	17	1	no	—
255	C₄⋊C₄	—	17	1	no	—
256	C₄⋊C₄	—	34	1	no	—
257	C₄⋊C₄	—	34	1	no	—
258	C₄⋊C₄	—	34	1	no	—
259	C₄⋊C₄	—	34	1	no	—
260	C₄⋊C₄	—	34	1	no	—
261	C₄⋊C₄	—	34	1	no	—
262	C₄⋊C₄	—	34	1	no	—
263	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	34	1	no	—
268	C₂²⋊C₄	—	34	1	no	—
269	C₂²⋊C₄	—	34	1	no	—
270	C₂²⋊C₄	—	34	1	no	—
271	C₂²⋊C₄	—	34	1	no	—
272	C₂²⋊C₄	—	34	1	no	—
273	C₂²⋊C₄	—	34	1	no	—
274	C₂²⋊C₄	—	34	1	no	—
275	C₄²	—	17	1	no	—
276	C₄²	—	17	1	no	—
277	C₄²	—	17	1	no	—
278	C₄²	—	17	1	no	—
279	C₄²	—	34	1	no	—
280	C₄²	—	34	1	no	—
281	C₂³	—	2	1	no	—
282	C₂³	—	34	1	no	—
283	C₂³	D₄×D₁₇	1	1	yes	characteristic
284	Q₈	—	2	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	34	1	no	—
287	Q₈	—	34	1	no	—
288	D₄	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	4	2	no	—
291	D₄	—	34	1	no	—
292	D₄	—	34	1	no	—
293	D₄	—	68	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	17	1	no	—
300	C₂×C₄	—	17	1	no	—
301	C₂×C₄	—	17	1	no	—
302	C₂×C₄	—	17	1	no	—
303	C₂×C₄	—	34	2	no	—
304	C₂×C₄	—	34	2	no	—
305	C₂×C₄	—	34	1	no	—
306	C₂×C₄	—	34	1	no	—
307	C₂×C₄	—	34	1	no	—
308	C₂×C₄	—	68	2	no	—
309	C₂×C₄	—	68	2	no	—
310	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	17	2	no	—
314	C₈	—	34	1	no	—
315	C₈	#63254	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	68	1	no	—
322	C₂²	—	68	1	no	—
323	C₂²	—	68	1	no	—
324	C₂²	#10587602	1	2	yes	—
325	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	17	2	no	—
331	C₄	—	34	1	no	—
332	C₄	—	34	1	no	—
333	C₄	—	34	1	no	—
334	C₄	—	68	1	no	—
335	C₄	—	68	1	no	—
336	C₄	—	68	1	no	—
337	C₄	C₂×#63254	1	1	yes	characteristic
338	C₄	#41×D₁₇	1	1	yes	characteristic
339	C₂	—	2	2	no	—
340	C₂	—	8	1	no	—
341	C₂	—	136	1	no	—
342	C₂	#11763354	1	1	yes	characteristic
343	C₂	#135×D₁₇	1	1	yes	characteristic
344	C₂	C₂×#10587602	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×#465	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763043	C₂
hypercenter	#465	D₁₇
hypocenter	C₁₇	#3168
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #465

Complements

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

N	H ≅ G/N	type
1. #11763345	C₂	semidirect
2. #11763340	C₂	semidirect
3. #11763209	C₂	semidirect
4. #11763190	C₂	semidirect
5. #11763165	C₂	semidirect
6. #11763162	C₂	semidirect
7. #11763048	C₂	semidirect
9. #11763038	C₂	semidirect
10. #11762901	C₂	semidirect
11. SD₁₆×Dic₁₇	C₂	semidirect
12. #11762896	C₂	semidirect
13. #11762889	C₂	semidirect
14. C₂×C₈×Dic₁₇	C₂	semidirect
15. C₁₇×#465	C₂	semidirect
176. #465	D₁₇	semidirect
231. C₁₇	#3168	semidirect

Maximal subgroups

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

subgroup	index
#11763345	2
#11763340	2
#11763209	2
#11763190	2
#11763165	2
#11763162	2
#11763048	2
#11763043	2
#11763038	2
#11762901	2
SD₁₆×Dic₁₇	2
#11762896	2
#11762889	2
C₂×C₈×Dic₁₇	2
C₁₇×#465	2
#3168	17