Subgroups — #14541250

Order 2176.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763507	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763460	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763216	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763135	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763129	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763127	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763122	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763108	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763098	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763063	C₂	1	1	yes	maximal, characteristic, complemented
11	#11763045	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762920	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762880	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762848	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₁₇×#106	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#63252	—	2	1	no	—
17	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
20	#10587616	—	2	1	no	—
21	#10587616	C₂²	1	1	yes	characteristic
22	#10587616	C₂²	1	1	yes	characteristic
23	#10587615	—	2	1	no	—
24	C₄×C₂²⋊D₁₇	—	2	1	no	—
25	C₄×C₂²⋊D₁₇	C₂²	1	1	yes	characteristic
26	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
27	C₂×#63218	C₂²	1	1	yes	characteristic
28	#10587584	—	2	1	no	—
29	#10587582	—	2	1	no	—
30	#10587581	C₂²	1	1	yes	characteristic
31	#10587581	C₂²	1	1	yes	characteristic
32	#10587580	—	2	1	no	—
33	#10587580	—	2	1	no	—
34	#10587579	C₂²	1	1	yes	characteristic
35	#10587579	C₂²	1	1	yes	characteristic
36	#10587578	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
38	#10587576	C₂²	1	1	yes	characteristic
39	#10587574	C₂²	1	1	yes	characteristic
40	C₁₇×#106	C₂²	1	1	yes	characteristic
41	C₁₇×#106	C₂²	1	1	yes	characteristic
42	C₁₇×#106	C₂²	1	1	yes	characteristic
43	C₁₇×#106	C₂²	1	1	yes	characteristic
44	C₂²⋊#3622	C₂²	1	1	yes	characteristic
45	#10587492	C₂²	1	1	yes	characteristic
46	#10587492	C₂²	1	1	yes	characteristic
47	#10587491	C₂²	1	1	yes	characteristic
48	#10587489	C₂²	1	1	yes	characteristic
49	#10587486	C₂²	1	1	yes	characteristic
50	#10587485	C₂²	1	1	yes	characteristic
51	#10587480	C₂²	1	1	yes	characteristic
52	#10587480	C₂²	1	1	yes	characteristic
53	#10587480	C₂²	1	1	yes	characteristic
54	#10587480	C₂²	1	1	yes	characteristic
55	#10587479	C₂²	1	1	yes	characteristic
56	#10587479	C₂²	1	1	yes	characteristic
57	#10587478	C₂²	1	1	yes	characteristic
58	#10587478	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₁₇	—	2	1	no	—
60	C₂²×C₄×C₁₇	—	2	1	no	—
61	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₇	—	2	1	no	—
63	C₂×C₂²⋊D₁₇	—	2	1	no	—
64	C₂×C₂²⋊D₁₇	D₄	1	1	yes	characteristic
65	#63252	—	4	4	no	—
66	C₂×D₆₈	—	2	1	no	—
67	C₂×D₆₈	—	2	1	no	—
68	C₂×D₆₈	C₂³	1	1	yes	characteristic
69	C₂×D₆₈	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₁₇	—	2	1	no	—
71	C₂×C₄×D₁₇	—	2	1	no	—
72	C₂×C₄×D₁₇	—	2	1	no	—
73	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
74	C₂×Dic₃₄	—	2	1	no	—
75	C₂×C₈×C₁₇	—	2	2	no	—
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	—	2	1	no	—
80	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
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⋊Dic₁₇	D₄	1	1	yes	characteristic
85	#63227	—	2	1	no	—
86	#63227	—	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	#63227	D₄	1	1	yes	characteristic
93	C₄⋊Dic₁₇	—	2	1	no	—
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	D₄	1	1	yes	characteristic
100	C₄×Dic₁₇	—	2	1	no	—
101	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
102	C₂×#3622	C₂³	1	1	yes	characteristic
103	C₂×#3622	C₂³	1	1	yes	characteristic
104	D₁₃₆	—	2	2	no	—
105	#63218	—	2	2	no	—
106	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₁₇	—	2	1	no	—
108	C₂²×D₁₇	—	2	1	no	—
109	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₇	—	2	1	no	—
111	C₂×C₄×C₁₇	—	2	1	no	—
112	C₂×C₄×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	—	4	2	no	—
117	C₂×C₄×C₁₇	—	8	1	no	—
118	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
120	C₂²⋊D₁₇	—	2	2	no	—
121	C₂²⋊D₁₇	—	4	2	no	—
122	C₂²⋊D₁₇	—	4	2	no	—
123	C₂×Dic₁₇	—	2	1	no	—
124	C₂×Dic₁₇	—	2	1	no	—
125	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
128	D₆₈	—	2	1	no	—
129	D₆₈	—	2	1	no	—
130	D₆₈	—	2	1	no	—
131	D₆₈	—	2	1	no	—
132	D₆₈	—	4	2	no	—
133	D₆₈	—	8	1	no	—
134	C₄×D₁₇	—	2	1	no	—
135	C₄×D₁₇	—	2	1	no	—
136	C₄×D₁₇	—	4	2	no	—
137	C₄×D₁₇	—	4	2	no	—
138	C₄×D₁₇	—	8	1	no	—
139	Dic₃₄	—	4	2	no	—
140	C₈×C₁₇	—	2	1	no	—
141	C₈×C₁₇	#41	1	2	yes	complemented
142	#3622	—	2	1	no	—
143	#3622	—	2	1	no	—
144	#3345	—	17	1	no	maximal, Sylow
145	C₂²×C₁₇	—	2	1	no	—
146	C₂²×C₁₇	—	2	1	no	—
147	C₂²×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
148	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₁₇	—	2	1	no	—
150	C₂×D₁₇	—	2	1	no	—
151	C₂×D₁₇	—	2	1	no	—
152	C₂×D₁₇	—	4	1	no	—
153	C₂×D₁₇	—	4	1	no	—
154	C₂×D₁₇	—	4	1	no	—
155	C₂×D₁₇	—	4	1	no	—
156	C₂×D₁₇	—	4	1	no	—
157	C₂×D₁₇	—	4	1	no	—
158	C₄×C₁₇	—	2	1	no	—
159	C₄×C₁₇	—	4	1	no	—
160	C₄×C₁₇	—	4	1	no	—
161	C₄×C₁₇	—	4	1	no	—
162	C₄×C₁₇	—	4	1	no	—
163	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
164	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
165	Dic₁₇	—	2	1	no	—
166	Dic₁₇	—	2	1	no	—
167	Dic₁₇	—	2	1	no	—
168	Dic₁₇	—	4	1	no	—
169	Dic₁₇	—	4	1	no	—
170	#553	—	17	1	no	—
171	#547	—	17	1	no	—
172	#501	—	17	1	no	—
173	#482	—	17	1	no	—
174	#481	—	17	1	no	—
175	#481	—	17	1	no	—
176	#480	—	17	1	no	—
177	#478	—	17	1	no	—
178	#476	—	17	1	no	—
179	#469	—	17	1	no	—
180	#466	—	17	1	no	—
181	#442	—	17	1	no	—
182	#435	—	17	1	no	—
183	#428	—	17	1	no	—
184	C₂×#106	D₁₇	1	1	yes	characteristic, complemented
185	C₂×C₁₇	—	2	2	no	—
186	C₂×C₁₇	#547	1	1	yes	characteristic
187	C₂×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
188	C₂×C₁₇	#577	1	1	yes	characteristic
189	D₁₇	—	4	1	no	—
190	D₁₇	—	8	1	no	—
191	D₁₇	—	8	1	no	—
192	C₂×#41	—	34	1	no	—
193	C₂×D₈	—	17	1	no	—
194	C₂×M₁₆	—	17	1	no	—
195	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
196	C₄⋊Q₈	—	17	1	no	—
197	#126	—	34	1	no	—
198	#125	—	17	1	no	—
199	#123	—	34	1	no	—
200	#123	—	34	1	no	—
201	#123	—	34	1	no	—
202	C₂²⋊Q₈	—	34	1	no	—
203	#121	—	17	1	no	—
204	#121	—	17	1	no	—
205	#121	—	17	1	no	—
206	#121	—	17	1	no	—
207	#121	—	34	1	no	—
208	C₄×D₄	—	17	1	no	—
209	C₄×D₄	—	17	1	no	—
210	C₄×D₄	—	17	1	no	—
211	C₄×D₄	—	34	1	no	—
212	#117	—	17	1	no	—
213	C₂×C₄⋊C₄	—	17	1	no	—
214	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
215	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
216	#107	—	17	1	no	—
217	#107	—	17	1	no	—
218	#107	—	17	1	no	—
219	#106	—	17	1	no	—
220	#106	—	17	1	no	—
221	#106	C₂×D₁₇	1	1	yes	characteristic
222	#106	C₂×D₁₇	1	1	yes	characteristic
223	#106	C₂×D₁₇	1	1	yes	characteristic
224	#106	C₂×D₁₇	1	1	yes	characteristic
225	C₄⋊C₈	—	17	1	no	—
226	#102	—	17	1	no	—
227	#102	—	17	1	no	—
228	#102	—	17	1	no	—
229	#102	—	17	1	no	—
230	#102	—	17	1	no	—
231	#102	—	17	1	no	—
232	C₂²⋊C₈	—	17	1	no	—
233	C₂²⋊C₈	—	17	1	no	—
234	#97	—	17	1	no	—
235	C₁₇	#3345	1	1	yes	characteristic, complemented, Sylow
236	#41	—	68	4	no	—
237	C₂×Q₈	—	34	1	no	—
238	C₂×D₄	—	17	1	no	—
239	C₂×D₄	—	17	1	no	—
240	C₂×D₄	—	17	1	no	—
241	C₂×D₄	—	34	1	no	—
242	C₂×D₄	—	34	1	no	—
243	C₂×D₄	—	34	1	no	—
244	C₂×D₄	—	34	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	17	1	no	—
248	C₂²×C₄	—	34	1	no	—
249	C₂²×C₄	—	34	1	no	—
250	C₂²×C₄	—	34	1	no	—
251	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
252	D₈	—	34	2	no	—
253	M₁₆	—	34	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	17	1	no	—
256	C₂×C₈	—	17	1	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	C₂²×D₁₇	1	1	yes	characteristic
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	—	17	1	no	—
273	C₂²⋊C₄	—	17	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	C₂²⋊C₄	—	34	1	no	—
281	C₄²	—	17	1	no	—
282	C₄²	—	34	1	no	—
283	C₂³	—	17	1	no	—
284	C₂³	—	34	1	no	—
285	C₂³	—	34	1	no	—
286	C₂³	D₄×D₁₇	1	1	yes	characteristic
287	Q₈	—	68	2	no	—
288	D₄	—	34	2	no	—
289	D₄	—	34	1	no	—
290	D₄	—	34	1	no	—
291	D₄	—	34	1	no	—
292	D₄	—	34	1	no	—
293	D₄	—	68	2	no	—
294	D₄	—	68	2	no	—
295	D₄	—	68	2	no	—
296	D₄	—	136	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	—	2	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	8	1	no	—
305	C₂×C₄	—	17	1	no	—
306	C₂×C₄	—	17	1	no	—
307	C₂×C₄	—	17	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	—	68	2	no	—
313	C₂×C₄	—	68	2	no	—
314	C₂×C₄	—	136	1	no	—
315	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	34	1	no	—
319	C₈	—	34	1	no	—
320	C₈	#63252	1	2	yes	complemented
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	34	1	no	—
324	C₂²	—	34	1	no	—
325	C₂²	—	34	1	no	—
326	C₂²	—	68	1	no	—
327	C₂²	—	68	1	no	—
328	C₂²	—	68	1	no	—
329	C₂²	—	68	1	no	—
330	C₂²	—	68	1	no	—
331	C₂²	—	68	1	no	—
332	C₂²	#10587600	1	2	yes	—
333	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	34	1	no	—
340	C₄	—	34	1	no	—
341	C₄	—	34	1	no	—
342	C₄	—	68	1	no	—
343	C₄	—	68	1	no	—
344	C₄	C₂×#63252	1	1	yes	characteristic
345	C₄	#10587694	1	1	yes	characteristic
346	C₂	—	2	2	no	—
347	C₂	—	68	1	no	—
348	C₂	—	136	1	no	—
349	C₂	—	136	1	no	—
350	C₂	#11763460	1	1	yes	characteristic
351	C₂	C₂×#10587600	1	1	yes	characteristic
352	C₂	#11763615	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₂×C₁₇×#106	C₂
socle	C₂²×C₁₇	C₂²×D₄
hypercenter	C₂×#106	D₁₇
hypocenter	C₁₇	#3345
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

Complements

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

N	H ≅ G/N	type
1. #11763507	C₂	semidirect
2. #11763460	C₂	semidirect
3. #11763216	C₂	semidirect
4. #11763135	C₂	semidirect
5. #11763129	C₂	semidirect
6. #11763127	C₂	semidirect
7. #11763122	C₂	semidirect
8. #11763108	C₂	semidirect
9. #11763098	C₂	semidirect
10. #11763063	C₂	semidirect
11. #11763045	C₂	semidirect
12. #11762920	C₂	semidirect
13. #11762880	C₂	semidirect
14. #11762848	C₂	semidirect
15. C₂×C₁₇×#106	C₂	semidirect
141. C₈×C₁₇	#41	semidirect
184. C₂×#106	D₁₇	semidirect
235. C₁₇	#3345	semidirect
320. C₈	#63252	semidirect

Maximal subgroups

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

subgroup	index
#11763507	2
#11763460	2
#11763216	2
#11763135	2
#11763129	2
#11763127	2
#11763122	2
#11763108	2
#11763098	2
#11763063	2
#11763045	2
#11762920	2
#11762880	2
#11762848	2
C₂×C₁₇×#106	2
#3345	17