Subgroups — #14538746

Order 2176.

386 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763382	C₂	1	1	yes	maximal, characteristic
2	#11763290	C₂	1	1	yes	maximal, characteristic
3	#11763153	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763151	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763148	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763147	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763144	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763143	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763141	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762848	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762844	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762816	C₂	1	1	yes	maximal, characteristic, complemented
13	#103×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762774	C₂	1	1	yes	maximal, characteristic
15	C₁₇×#483	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	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
20	#10587632	C₂²	1	1	yes	characteristic
21	C₄×C₂²⋊D₁₇	—	2	1	no	—
22	C₄×C₂²⋊D₁₇	—	2	1	no	—
23	#10587612	C₂²	1	1	yes	characteristic
24	#10587610	C₂²	1	1	yes	characteristic
25	C₂×#63218	C₂²	1	1	yes	characteristic
26	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
27	#10587583	C₂²	1	1	yes	characteristic
28	#10587580	—	2	1	no	—
29	#10587579	—	2	1	no	—
30	#10587578	C₂²	1	1	yes	characteristic
31	#10587578	C₂²	1	1	yes	characteristic
32	#10587578	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
35	#10587574	—	2	1	no	—
36	#10587573	—	2	1	no	—
37	#10587571	—	2	1	no	—
38	#10587569	—	2	1	no	—
39	C₁₇×#106	C₂²	1	1	yes	characteristic
40	C₁₇×#106	C₂²	1	1	yes	characteristic
41	C₁₇×#103	C₂²	1	1	yes	characteristic
42	C₁₇×#103	C₂²	1	1	yes	characteristic
43	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	#10587506	C₂²	1	1	yes	characteristic
45	#10587506	C₂²	1	1	yes	characteristic
46	C₂²⋊#3622	C₂²	1	1	yes	characteristic
47	#10587491	C₂²	1	1	yes	characteristic
48	#10587491	C₂²	1	1	yes	characteristic
49	#10587489	C₂²	1	1	yes	characteristic
50	#10587488	C₂²	1	1	yes	characteristic
51	#10587487	C₂²	1	1	yes	characteristic
52	#10587487	C₂²	1	1	yes	characteristic
53	#10587481	C₂²	1	1	yes	characteristic
54	#10587481	C₂²	1	1	yes	characteristic
55	#10587479	C₂²	1	1	yes	characteristic
56	#10587478	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	C₂²×C₄×C₁₇	—	2	1	no	—
61	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₇	D₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₇	D₄	1	1	yes	characteristic
64	Q₈×D₁₇	—	2	2	no	—
65	Q₈×D₁₇	—	2	2	no	—
66	#63252	—	2	2	no	—
67	#63252	—	2	2	no	—
68	C₂×D₆₈	D₄	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×C₄×D₁₇	C₂³	1	1	yes	characteristic
74	C₂×Dic₃₄	—	2	1	no	—
75	C₂×Dic₃₄	D₄	1	1	yes	characteristic
76	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₁₇	—	2	1	no	—
80	C₄⋊C₄×C₁₇	—	2	1	no	—
81	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
84	C₂²⋊C₄×C₁₇	—	2	1	no	—
85	C₂²⋊Dic₁₇	—	2	1	no	—
86	C₂²⋊Dic₁₇	—	2	1	no	—
87	#63227	—	2	1	no	—
88	#63227	—	2	1	no	—
89	#63227	—	2	1	no	—
90	#63227	—	2	1	no	—
91	#63227	—	2	1	no	—
92	C₄⋊Dic₁₇	—	2	1	no	—
93	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
94	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
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	C₄×Dic₁₇	—	2	1	no	—
101	C₄×Dic₁₇	—	2	1	no	—
102	C₄×Dic₁₇	—	2	1	no	—
103	C₄×Dic₁₇	—	2	1	no	—
104	C₂×#3622	C₂³	1	1	yes	characteristic
105	C₂×#3622	C₂³	1	1	yes	characteristic
106	#63218	—	2	2	no	—
107	C₈×D₁₇	—	2	2	no	—
108	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
111	Q₈×C₁₇	—	2	1	no	—
112	Q₈×C₁₇	—	2	1	no	—
113	C₂×C₄×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	—	8	1	no	—
120	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₁₇	—	2	2	no	—
123	C₂²⋊D₁₇	—	2	2	no	—
124	C₂×Dic₁₇	—	2	1	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	D₆₈	—	2	2	no	—
131	C₄×D₁₇	—	2	1	no	—
132	C₄×D₁₇	—	2	1	no	—
133	C₄×D₁₇	—	4	2	no	—
134	C₄×D₁₇	—	8	1	no	—
135	C₄×D₁₇	—	8	1	no	—
136	C₄×D₁₇	#41	1	2	yes	—
137	C₄×D₁₇	#41	1	2	yes	—
138	Dic₃₄	—	2	2	no	—
139	Dic₃₄	—	2	1	no	—
140	Dic₃₄	—	2	1	no	—
141	Dic₃₄	—	4	2	no	—
142	C₈×C₁₇	—	2	1	no	—
143	C₈×C₁₇	—	2	1	no	—
144	#3622	—	2	1	no	—
145	#3622	—	2	1	no	—
146	#3097	—	17	1	no	maximal, Sylow
147	C₂²×C₁₇	—	2	1	no	—
148	C₂²×C₁₇	—	2	1	no	—
149	C₂²×C₁₇	—	2	1	no	—
150	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₁₇	—	2	1	no	—
152	C₂×D₁₇	—	2	1	no	—
153	C₂×D₁₇	—	2	1	no	—
154	C₂×D₁₇	—	2	1	no	—
155	C₂×D₁₇	—	2	1	no	—
156	C₂×D₁₇	#123	1	2	yes	—
157	C₄×C₁₇	—	2	1	no	—
158	C₄×C₁₇	—	4	1	no	—
159	C₄×C₁₇	—	4	1	no	—
160	C₄×C₁₇	—	4	1	no	—
161	C₄×C₁₇	—	4	1	no	—
162	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
163	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
164	Dic₁₇	—	2	1	no	—
165	Dic₁₇	—	4	1	no	—
166	Dic₁₇	—	4	1	no	—
167	Dic₁₇	—	4	1	no	—
168	Dic₁₇	—	4	1	no	—
169	Dic₁₇	#123	1	2	yes	—
170	#536	—	17	1	no	—
171	#520	—	17	1	no	—
172	#485	—	17	1	no	—
173	#485	—	17	1	no	—
174	#485	—	17	1	no	—
175	#485	—	17	1	no	—
176	#484	—	17	1	no	—
177	#484	—	17	1	no	—
178	#483	—	17	1	no	—
179	#483	D₁₇	1	1	yes	characteristic, complemented
180	#428	—	17	1	no	—
181	#427	—	17	1	no	—
182	#419	—	17	1	no	—
183	C₂×#103	—	17	1	no	—
184	#408	—	17	1	no	—
185	C₂×C₁₇	—	4	1	no	—
186	C₂×C₁₇	C₂×#123	1	1	yes	characteristic
187	C₂×C₁₇	#577	1	1	yes	characteristic
188	C₂×C₁₇	#578	1	1	yes	characteristic
189	D₁₇	—	2	2	no	—
190	D₁₇	—	4	1	no	—
191	C₂×#41	—	17	1	no	—
192	C₂²×Q₈	—	17	1	no	—
193	C₂×M₁₆	—	17	1	no	—
194	C₂²×C₈	—	17	1	no	—
195	C₄⋊Q₈	—	34	1	no	—
196	#124	—	34	1	no	—
197	#123	—	34	1	no	—
198	#123	—	34	1	no	—
199	C₂²⋊Q₈	—	17	1	no	—
200	C₂²⋊Q₈	—	17	1	no	—
201	C₂²⋊Q₈	—	17	1	no	—
202	C₂²⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
203	C₄×Q₈	—	34	1	no	—
204	C₄×D₄	—	34	1	no	—
205	C₄×D₄	—	34	1	no	—
206	C₄×D₄	—	34	1	no	—
207	#117	—	17	1	no	—
208	#117	—	17	1	no	—
209	#117	—	17	1	no	—
210	#117	—	17	1	no	—
211	C₂×C₄⋊C₄	—	17	1	no	—
212	C₂×C₄⋊C₄	—	17	1	no	—
213	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
214	#107	—	17	1	no	—
215	#107	—	17	1	no	—
216	#107	—	17	1	no	—
217	#107	—	17	1	no	—
218	#106	—	17	1	no	—
219	#106	—	17	1	no	—
220	#106	C₂×D₁₇	1	1	yes	characteristic
221	#106	C₂×D₁₇	1	1	yes	characteristic
222	#103	—	17	1	no	—
223	#103	—	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	C₂×D₁₇	1	1	yes	characteristic
229	#103	C₂×D₁₇	1	1	yes	characteristic
230	C₂²⋊C₈	—	17	1	no	—
231	C₂²⋊C₈	—	17	1	no	—
232	C₂²⋊C₈	—	17	1	no	—
233	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
234	C₁₇	#3097	1	1	yes	characteristic, complemented, Sylow
235	#41	—	34	2	no	—
236	#41	—	34	2	no	—
237	C₂×Q₈	—	17	1	no	—
238	C₂×Q₈	—	17	1	no	—
239	C₂×Q₈	—	34	1	no	—
240	C₂×Q₈	—	34	2	no	—
241	C₂×Q₈	—	34	2	no	—
242	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
243	C₂×D₄	—	17	1	no	—
244	C₂×D₄	—	17	1	no	—
245	C₂×D₄	—	17	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	17	1	no	—
248	C₂²×C₄	—	17	1	no	—
249	C₂²×C₄	—	34	1	no	—
250	C₂²×C₄	—	34	1	no	—
251	C₂²×C₄	—	34	1	no	—
252	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
253	M₁₆	—	34	2	no	—
254	C₂×C₈	—	17	1	no	—
255	C₂×C₈	—	17	1	no	—
256	C₂×C₈	—	34	2	no	—
257	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	17	1	no	—
262	C₄⋊C₄	—	17	1	no	—
263	C₄⋊C₄	—	17	1	no	—
264	C₄⋊C₄	—	34	1	no	—
265	C₄⋊C₄	—	34	1	no	—
266	C₄⋊C₄	—	34	1	no	—
267	C₄⋊C₄	—	34	1	no	—
268	C₄⋊C₄	—	34	1	no	—
269	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
270	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
271	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	34	1	no	—
274	C₂²⋊C₄	—	34	1	no	—
275	C₂²⋊C₄	—	34	1	no	—
276	C₂²⋊C₄	—	34	1	no	—
277	C₂²⋊C₄	—	34	1	no	—
278	C₂²⋊C₄	—	34	1	no	—
279	C₂²⋊C₄	—	34	1	no	—
280	C₄²	—	34	1	no	—
281	C₄²	—	34	1	no	—
282	C₄²	—	34	1	no	—
283	C₄²	—	34	1	no	—
284	C₂³	—	17	1	no	—
285	C₂³	—	17	1	no	—
286	C₂³	D₄×D₁₇	1	1	yes	characteristic
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	34	2	no	—
290	Q₈	—	34	1	no	—
291	Q₈	—	34	1	no	—
292	Q₈	—	68	2	no	—
293	D₄	—	34	2	no	—
294	D₄	—	34	2	no	—
295	D₄	—	34	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	—	17	1	no	—
304	C₂×C₄	—	17	2	no	—
305	C₂×C₄	—	17	2	no	—
306	C₂×C₄	—	17	1	no	—
307	C₂×C₄	—	34	1	no	—
308	C₂×C₄	—	34	1	no	—
309	C₂×C₄	—	34	1	no	—
310	C₂×C₄	—	34	1	no	—
311	C₂×C₄	—	34	1	no	—
312	C₂×C₄	—	34	1	no	—
313	C₂×C₄	—	68	2	no	—
314	C₂×C₄	—	136	1	no	—
315	C₂×C₄	—	136	1	no	—
316	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
317	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	34	1	no	—
321	C₈	—	34	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	17	2	no	—
326	C₂²	—	34	1	no	—
327	C₂²	—	34	1	no	—
328	C₂²	—	34	1	no	—
329	C₂²	—	34	1	no	—
330	C₂²	—	34	1	no	—
331	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	17	2	no	—
338	C₄	—	34	1	no	—
339	C₄	—	68	1	no	—
340	C₄	—	68	1	no	—
341	C₄	—	68	1	no	—
342	C₄	—	68	1	no	—
343	C₄	#41×D₁₇	1	1	yes	characteristic
344	C₄	#41×D₁₇	1	1	yes	characteristic
345	C₂	—	4	1	no	—
346	C₂	—	34	2	no	—
347	C₂	—	68	1	no	—
348	C₂	#123×D₁₇	1	1	yes	characteristic
349	C₂	#11763615	1	1	yes	characteristic
350	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×#483	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×#63252	C₂²
hypercenter	#483	D₁₇
hypocenter	C₁₇	#3097
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #483

Complements

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

N	H ≅ G/N	type
3. #11763153	C₂	semidirect
4. #11763151	C₂	semidirect
5. #11763148	C₂	semidirect
6. #11763147	C₂	semidirect
7. #11763144	C₂	semidirect
8. #11763143	C₂	semidirect
9. #11763141	C₂	semidirect
10. #11762848	C₂	semidirect
11. #11762844	C₂	semidirect
12. #11762816	C₂	semidirect
13. #103×D₁₇	C₂	semidirect
15. C₁₇×#483	C₂	semidirect
179. #483	D₁₇	semidirect
234. C₁₇	#3097	semidirect

Maximal subgroups

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

subgroup	index
#11763382	2
#11763290	2
#11763153	2
#11763151	2
#11763148	2
#11763147	2
#11763144	2
#11763143	2
#11763141	2
#11762848	2
#11762844	2
#11762816	2
#103×D₁₇	2
#11762774	2
C₁₇×#483	2
#3097	17