Subgroups — #14540116

Order 2176.

354 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	#11763361	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763345	C₂	1	1	yes	maximal, characteristic
3	#11763115	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763114	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763092	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763088	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763048	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763040	C₂	1	1	yes	maximal, characteristic
9	#11763038	C₂	1	1	yes	maximal, characteristic
10	#11763034	C₂	1	1	yes	maximal, characteristic, complemented
11	#11763033	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763017	C₂	1	1	yes	maximal, characteristic, complemented
13	#11763013	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#10587485	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#467	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₁₆×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×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
20	#10587633	C₂²	1	1	yes	characteristic
21	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
22	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
23	#10587630	C₂²	1	1	yes	characteristic
24	C₂×#63231	C₂²	1	1	yes	characteristic
25	C₂×#63230	C₂²	1	1	yes	characteristic
26	#10587616	C₂²	1	1	yes	characteristic
27	#10587610	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
29	C₂²×#3622	C₂²	1	1	yes	characteristic
30	C₂×#63219	C₂²	1	1	yes	characteristic
31	#10587584	—	2	1	no	—
32	#10587579	C₂²	1	1	yes	characteristic
33	#10587578	—	2	1	no	—
34	#10587576	C₂²	1	1	yes	characteristic
35	#10587575	—	2	1	no	—
36	#10587574	C₂²	1	1	yes	characteristic
37	#10587573	—	2	1	no	—
38	#10587573	C₂²	1	1	yes	characteristic
39	#10587572	—	2	1	no	—
40	#10587567	—	2	1	no	—
41	#10587564	—	2	1	no	—
42	#10587563	—	2	1	no	—
43	C₁₇×#107	C₂²	1	1	yes	characteristic
44	C₁₇×#103	C₂²	1	1	yes	characteristic
45	C₁₇×#103	C₂²	1	1	yes	characteristic
46	#10587506	C₂²	1	1	yes	characteristic
47	#10587506	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	#10587485	C₂²	1	1	yes	characteristic
52	#10587485	C₂²	1	1	yes	characteristic
53	#10587485	C₂²	1	1	yes	characteristic
54	#10587485	C₂²	1	1	yes	characteristic
55	#10587481	C₂²	1	1	yes	characteristic
56	#10587480	C₂²	1	1	yes	characteristic
57	#10587479	C₂²	1	1	yes	characteristic
58	#10587478	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₇	—	2	1	no	—
63	C₂²×Dic₁₇	D₄	1	1	yes	characteristic
64	C₂²×Dic₁₇	D₄	1	1	yes	characteristic
65	C₂×D₆₈	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₁₇	—	2	1	no	—
67	C₂×Dic₃₄	—	2	1	no	—
68	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
69	Q₁₆×C₁₇	—	4	1	no	—
70	C₂×C₈×C₁₇	—	4	1	no	—
71	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₇	—	2	1	no	—
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	#63231	—	4	1	no	—
81	#63230	—	4	1	no	—
82	#63227	—	2	1	no	—
83	#63227	—	2	1	no	—
84	#63227	—	2	1	no	—
85	#63227	—	2	1	no	—
86	C₄⋊Dic₁₇	—	2	1	no	—
87	C₄⋊Dic₁₇	—	2	1	no	—
88	C₄⋊Dic₁₇	—	2	1	no	—
89	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
91	#63225	—	2	1	no	—
92	#63225	—	2	1	no	—
93	#63225	—	2	1	no	—
94	#63225	—	2	1	no	—
95	#63225	—	2	1	no	—
96	#63225	—	2	1	no	—
97	C₄×Dic₁₇	—	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	—	4	1	no	—
105	C₂×#3622	C₂³	1	1	yes	characteristic
106	C₂×#3622	C₂³	1	1	yes	characteristic
107	#63219	—	4	1	no	—
108	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₁₇	—	2	1	no	—
110	Q₈×C₁₇	—	2	1	no	—
111	Q₈×C₁₇	—	2	1	no	—
112	Q₈×C₁₇	—	2	1	no	—
113	Q₈×C₁₇	—	2	1	no	—
114	C₂×C₄×C₁₇	—	2	1	no	—
115	C₂×C₄×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	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₁₇	—	4	2	no	—
123	C₂×Dic₁₇	—	2	2	no	—
124	C₂×Dic₁₇	—	2	2	no	—
125	C₂×Dic₁₇	—	2	1	no	—
126	C₂×Dic₁₇	—	2	1	no	—
127	C₂×Dic₁₇	—	2	1	no	—
128	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
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	C₄×D₁₇	—	4	2	no	—
135	Dic₃₄	—	2	1	no	—
136	Dic₃₄	—	2	1	no	—
137	Dic₃₄	—	4	2	no	—
138	C₈×C₁₇	—	2	1	no	—
139	C₈×C₁₇	—	2	1	no	—
140	#3622	—	2	1	no	—
141	#3622	—	2	1	no	—
142	#3248	—	17	1	no	maximal, Sylow
143	C₂²×C₁₇	—	2	1	no	—
144	C₂²×C₁₇	—	2	1	no	—
145	C₂²×C₁₇	#137	1	2	yes	—
146	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×D₁₇	—	4	1	no	—
148	C₂×D₁₇	—	4	1	no	—
149	C₂×D₁₇	—	4	1	no	—
150	C₄×C₁₇	—	2	1	no	—
151	C₄×C₁₇	—	4	1	no	—
152	C₄×C₁₇	—	4	1	no	—
153	C₄×C₁₇	—	4	1	no	—
154	C₄×C₁₇	—	4	1	no	—
155	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
156	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
157	Dic₁₇	—	2	1	no	—
158	Dic₁₇	—	2	1	no	—
159	Dic₁₇	—	2	1	no	—
160	Dic₁₇	—	4	1	no	—
161	Dic₁₇	—	4	1	no	—
162	Dic₁₇	—	4	1	no	—
163	Dic₁₇	#135	1	2	yes	—
164	#533	—	17	1	no	—
165	#529	—	17	1	no	—
166	#479	—	17	1	no	—
167	#479	—	17	1	no	—
168	#475	—	17	1	no	—
169	#474	—	17	1	no	—
170	#467	—	17	1	no	—
171	#467	D₁₇	1	1	yes	characteristic, complemented
172	#465	—	17	1	no	—
173	#465	—	17	1	no	—
174	#464	—	17	1	no	—
175	#464	—	17	1	no	—
176	#462	—	17	1	no	—
177	#461	—	17	1	no	—
178	C₂×C₄⋊C₈	—	17	1	no	—
179	C₂×C₁₇	—	2	2	no	—
180	C₂×C₁₇	#537	1	1	yes	characteristic
181	C₂×C₁₇	C₂×#135	1	1	yes	characteristic
182	C₂×C₁₇	C₂×#137	1	1	yes	characteristic
183	D₁₇	—	8	1	no	—
184	C₂×Q₁₆	—	17	1	no	—
185	C₂×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
186	C₂×SD₁₆	—	17	1	no	—
187	C₂×SD₁₆	—	17	1	no	—
188	C₂²×C₈	—	17	1	no	—
189	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
190	C₄⋊Q₈	—	17	1	no	—
191	C₄⋊Q₈	—	17	1	no	—
192	#126	—	34	1	no	—
193	#125	—	17	1	no	—
194	#125	—	34	1	no	—
195	#124	—	17	1	no	—
196	#123	—	34	1	no	—
197	C₂²⋊Q₈	—	17	1	no	—
198	C₂²⋊Q₈	—	34	1	no	—
199	C₂²⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
200	C₂²⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
201	#121	—	17	1	no	—
202	C₄×Q₈	—	17	1	no	—
203	C₄×Q₈	—	17	1	no	—
204	C₄×Q₈	—	17	1	no	—
205	C₄×Q₈	—	34	1	no	—
206	C₄×D₄	—	17	1	no	—
207	C₄×D₄	—	34	1	no	—
208	#117	—	34	1	no	—
209	#117	—	34	1	no	—
210	C₂×C₄²	—	17	1	no	—
211	#107	—	17	1	no	—
212	#107	C₂×D₁₇	1	1	yes	characteristic
213	#106	—	17	1	no	—
214	#106	—	17	1	no	—
215	C₄⋊C₈	—	17	1	no	—
216	C₄⋊C₈	—	17	1	no	—
217	C₄⋊C₈	—	17	1	no	—
218	C₄⋊C₈	—	17	1	no	—
219	#103	—	17	1	no	—
220	#103	—	17	1	no	—
221	#103	—	17	1	no	—
222	#103	—	17	1	no	—
223	#103	C₂×D₁₇	1	1	yes	characteristic
224	#103	C₂×D₁₇	1	1	yes	characteristic
225	#102	—	17	1	no	—
226	#102	—	17	1	no	—
227	C₁₇	#3248	1	1	yes	characteristic, complemented, Sylow
228	C₂×Q₈	—	17	1	no	—
229	C₂×Q₈	—	34	1	no	—
230	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
231	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
232	C₂×D₄	—	17	1	no	—
233	C₂×D₄	—	34	1	no	—
234	C₂²×C₄	—	17	1	no	—
235	C₂²×C₄	—	17	1	no	—
236	C₂²×C₄	—	34	1	no	—
237	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
238	Q₁₆	—	4	1	no	—
239	Q₁₆	—	68	1	no	—
240	SD₁₆	—	68	1	no	—
241	SD₁₆	—	68	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	17	1	no	—
244	C₂×C₈	—	17	1	no	—
245	C₂×C₈	—	68	1	no	—
246	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	17	1	no	—
251	C₄⋊C₄	—	17	1	no	—
252	C₄⋊C₄	—	34	1	no	—
253	C₄⋊C₄	—	34	1	no	—
254	C₄⋊C₄	—	34	1	no	—
255	C₄⋊C₄	—	34	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	C₂²×D₁₇	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	34	1	no	—
266	C₂²⋊C₄	—	34	1	no	—
267	C₂²⋊C₄	—	34	1	no	—
268	C₂²⋊C₄	—	34	1	no	—
269	C₂²⋊C₄	—	34	1	no	—
270	C₄²	—	17	1	no	—
271	C₄²	—	17	1	no	—
272	C₄²	—	17	1	no	—
273	C₄²	—	17	1	no	—
274	C₄²	—	34	1	no	—
275	C₄²	—	34	1	no	—
276	C₄²	—	34	1	no	—
277	C₂³	—	34	1	no	—
278	C₂³	D₄×D₁₇	1	1	yes	characteristic
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	34	1	no	—
284	Q₈	—	34	1	no	—
285	Q₈	—	68	2	no	—
286	D₄	—	34	1	no	—
287	D₄	—	34	1	no	—
288	D₄	—	68	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	17	1	no	—
296	C₂×C₄	—	17	1	no	—
297	C₂×C₄	—	17	1	no	—
298	C₂×C₄	—	17	1	no	—
299	C₂×C₄	—	34	2	no	—
300	C₂×C₄	—	34	2	no	—
301	C₂×C₄	—	34	1	no	—
302	C₂×C₄	—	34	1	no	—
303	C₂×C₄	—	34	1	no	—
304	C₂×C₄	—	68	2	no	—
305	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	34	1	no	—
310	C₈	—	34	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	68	1	no	—
314	C₂²	—	68	1	no	—
315	C₂²	—	68	1	no	—
316	C₂²	#10587604	1	2	yes	—
317	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	17	2	no	—
324	C₄	—	34	1	no	—
325	C₄	—	34	1	no	—
326	C₄	—	34	1	no	—
327	C₄	—	68	1	no	—
328	C₄	—	68	1	no	—
329	C₄	—	68	1	no	—
330	C₄	#10587691	1	1	yes	characteristic
331	C₄	#10587697	1	1	yes	characteristic
332	C₂	—	2	2	no	—
333	C₂	—	136	1	no	—
334	C₂	#11763386	1	1	yes	characteristic
335	C₂	#135×D₁₇	1	1	yes	characteristic
336	C₂	C₂×#10587604	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×#467	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#10587616	C₂²
hypercenter	#467	D₁₇
hypocenter	C₁₇	#3248
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #467

Complements

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

N	H ≅ G/N	type
1. #11763361	C₂	semidirect
3. #11763115	C₂	semidirect
4. #11763114	C₂	semidirect
5. #11763092	C₂	semidirect
6. #11763088	C₂	semidirect
7. #11763048	C₂	semidirect
10. #11763034	C₂	semidirect
11. #11763033	C₂	semidirect
12. #11763017	C₂	semidirect
13. #11763013	C₂	semidirect
14. C₂×#10587485	C₂	semidirect
15. C₁₇×#467	C₂	semidirect
171. #467	D₁₇	semidirect
227. C₁₇	#3248	semidirect

Maximal subgroups

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

subgroup	index
#11763361	2
#11763345	2
#11763115	2
#11763114	2
#11763092	2
#11763088	2
#11763048	2
#11763040	2
#11763038	2
#11763034	2
#11763033	2
#11763017	2
#11763013	2
C₂×#10587485	2
C₁₇×#467	2
#3248	17