Subgroups — #14540227

Order 2176.

346 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	#11763420	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763410	C₂	1	1	yes	maximal, characteristic
3	#11763118	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763114	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763110	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763109	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763058	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763052	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763041	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763035	C₂	1	1	yes	maximal, characteristic, complemented
11	#11763011	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763007	C₂	1	1	yes	maximal, characteristic, complemented
13	#11763007	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762834	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#461	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
20	#10587632	C₂²	1	1	yes	characteristic
21	#10587632	C₂²	1	1	yes	characteristic
22	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
23	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
24	C₂×#63230	C₂²	1	1	yes	characteristic
25	C₂×#63230	C₂²	1	1	yes	characteristic
26	#10587626	C₂²	1	1	yes	characteristic
27	#10587624	C₂²	1	1	yes	characteristic
28	#10587616	—	2	1	no	—
29	#10587615	—	2	1	no	—
30	C₂×#63218	C₂²	1	1	yes	characteristic
31	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
32	#10587584	—	2	1	no	—
33	#10587584	—	2	1	no	—
34	#10587581	C₂²	1	1	yes	characteristic
35	#10587576	C₂²	1	1	yes	characteristic
36	#10587575	—	2	1	no	—
37	#10587575	—	2	1	no	—
38	#10587570	—	2	1	no	—
39	#10587569	—	2	1	no	—
40	C₄×D₆₈	C₂²	1	1	yes	characteristic
41	#10587558	C₂²	1	1	yes	characteristic
42	#10587556	C₂²	1	1	yes	characteristic
43	C₁₇×C₄⋊C₈	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	#10587503	C₂²	1	1	yes	characteristic
49	#10587503	C₂²	1	1	yes	characteristic
50	#10587492	C₂²	1	1	yes	characteristic
51	#10587492	C₂²	1	1	yes	characteristic
52	#10587489	C₂²	1	1	yes	characteristic
53	#10587488	C₂²	1	1	yes	characteristic
54	#10587485	C₂²	1	1	yes	characteristic
55	#10587485	C₂²	1	1	yes	characteristic
56	#10587479	C₂²	1	1	yes	characteristic
57	#10587478	C₂²	1	1	yes	characteristic
58	C₄⋊#3622	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	—	2	1	no	—
63	C₂×C₂²⋊D₁₇	—	2	1	no	—
64	C₂×C₂²⋊D₁₇	—	2	1	no	—
65	C₂×C₂²⋊D₁₇	—	2	1	no	—
66	C₂×D₆₈	—	2	1	no	—
67	C₂×D₆₈	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₁₇	—	2	1	no	—
69	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
70	SD₁₆×C₁₇	—	4	1	no	—
71	SD₁₆×C₁₇	—	4	1	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	—	2	1	no	—
76	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₁₇	—	2	1	no	—
78	C₄²×C₁₇	C₂³	1	1	yes	characteristic
79	C₂²⋊Dic₁₇	—	2	1	no	—
80	#63230	—	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	#63227	D₄	1	1	yes	characteristic
87	#63227	D₄	1	1	yes	characteristic
88	C₄⋊Dic₁₇	—	2	1	no	—
89	C₄⋊Dic₁₇	—	2	1	no	—
90	C₄⋊Dic₁₇	—	2	1	no	—
91	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
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	#63225	—	2	1	no	—
99	#63225	D₄	1	1	yes	characteristic
100	#63225	D₄	1	1	yes	characteristic
101	C₄×Dic₁₇	—	2	1	no	—
102	C₄×Dic₁₇	—	2	1	no	—
103	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
104	C₂×#3622	C₂³	1	1	yes	characteristic
105	C₂×#3622	C₂³	1	1	yes	characteristic
106	#63218	—	4	1	no	—
107	C₈×D₁₇	—	4	1	no	—
108	C₂³×C₁₇	—	2	1	no	—
109	C₂²×D₁₇	—	2	1	no	—
110	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
111	Q₈×C₁₇	—	2	1	no	—
112	Q₈×C₁₇	—	2	1	no	—
113	Q₈×C₁₇	—	2	1	no	—
114	Q₈×C₁₇	—	2	1	no	—
115	D₄×C₁₇	—	2	1	no	—
116	D₄×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	—	8	1	no	—
121	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
124	C₂²⋊D₁₇	—	8	1	no	—
125	C₂²⋊D₁₇	—	8	1	no	—
126	C₂²⋊D₁₇	—	8	1	no	—
127	C₂×Dic₁₇	—	2	1	no	—
128	C₂×Dic₁₇	—	2	1	no	—
129	C₂×Dic₁₇	—	2	1	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	D₆₈	—	2	1	no	—
134	D₆₈	—	2	1	no	—
135	D₆₈	—	8	1	no	—
136	C₄×D₁₇	—	2	1	no	—
137	C₄×D₁₇	—	2	1	no	—
138	C₄×D₁₇	—	8	1	no	—
139	C₈×C₁₇	—	2	1	no	—
140	C₈×C₁₇	—	2	1	no	—
141	#3622	—	2	1	no	—
142	#3622	—	2	1	no	—
143	#3260	—	17	1	no	maximal, Sylow
144	C₂²×C₁₇	—	4	1	no	—
145	C₂²×C₁₇	—	4	1	no	—
146	C₂²×C₁₇	—	4	1	no	—
147	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₁₇	—	2	1	no	—
149	C₂×D₁₇	—	2	1	no	—
150	C₂×D₁₇	—	2	1	no	—
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	—	2	1	no	—
156	C₄×C₁₇	—	4	1	no	—
157	C₄×C₁₇	—	4	1	no	—
158	C₄×C₁₇	—	4	1	no	—
159	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
160	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
161	Dic₁₇	—	2	1	no	—
162	Dic₁₇	—	2	1	no	—
163	Dic₁₇	—	2	1	no	—
164	Dic₁₇	—	4	1	no	—
165	Dic₁₇	—	4	1	no	—
166	Dic₁₇	—	4	1	no	—
167	#541	—	17	1	no	—
168	#540	—	17	1	no	—
169	#479	—	17	1	no	—
170	#479	—	17	1	no	—
171	#478	—	17	1	no	—
172	#478	—	17	1	no	—
173	#468	—	17	1	no	—
174	#468	—	17	1	no	—
175	#465	—	17	1	no	—
176	#465	—	17	1	no	—
177	#461	—	17	1	no	—
178	#461	D₁₇	1	1	yes	characteristic, complemented
179	#460	—	17	1	no	—
180	#460	—	17	1	no	—
181	#424	—	17	1	no	—
182	C₂×C₁₇	—	8	1	no	—
183	C₂×C₁₇	#537	1	1	yes	characteristic
184	C₂×C₁₇	#577	1	1	yes	characteristic
185	C₂×C₁₇	#577	1	1	yes	characteristic
186	D₁₇	—	4	1	no	—
187	D₁₇	—	8	1	no	—
188	C₂×SD₁₆	—	17	1	no	—
189	C₂×SD₁₆	—	17	1	no	—
190	C₂×SD₁₆	C₂×D₁₇	1	1	yes	characteristic
191	C₂×SD₁₆	C₂×D₁₇	1	1	yes	characteristic
192	C₂×M₁₆	—	17	1	no	—
193	C₂²×C₈	—	17	1	no	—
194	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
195	#127	—	17	1	no	—
196	#126	—	34	1	no	—
197	#126	—	34	1	no	—
198	#125	—	17	1	no	—
199	#125	—	17	1	no	—
200	#125	—	34	1	no	—
201	#125	—	34	1	no	—
202	#123	—	34	1	no	—
203	#123	—	34	1	no	—
204	C₂²⋊Q₈	—	17	1	no	—
205	C₂²⋊Q₈	—	17	1	no	—
206	#121	—	17	1	no	—
207	#121	—	17	1	no	—
208	#121	—	34	1	no	—
209	#121	—	34	1	no	—
210	C₄×Q₈	—	17	1	no	—
211	C₄×Q₈	—	17	1	no	—
212	C₄×D₄	—	17	1	no	—
213	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
214	#117	—	17	1	no	—
215	#107	—	17	1	no	—
216	#107	—	17	1	no	—
217	#106	—	17	1	no	—
218	#106	—	17	1	no	—
219	C₄⋊C₈	—	17	1	no	—
220	C₄⋊C₈	—	17	1	no	—
221	C₄⋊C₈	—	17	1	no	—
222	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
223	#103	—	17	1	no	—
224	#103	—	17	1	no	—
225	#103	C₂×D₁₇	1	1	yes	characteristic
226	#103	C₂×D₁₇	1	1	yes	characteristic
227	#102	—	17	1	no	—
228	#102	—	17	1	no	—
229	#102	—	17	1	no	—
230	#102	—	17	1	no	—
231	C₁₇	#3260	1	1	yes	characteristic, complemented, Sylow
232	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
233	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
234	C₂×D₄	—	17	1	no	—
235	C₂×D₄	—	34	1	no	—
236	C₂×D₄	—	34	1	no	—
237	C₂×D₄	—	34	1	no	—
238	C₂×D₄	—	34	1	no	—
239	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	17	1	no	—
242	C₂²×C₄	—	34	1	no	—
243	SD₁₆	—	4	1	no	—
244	SD₁₆	—	4	1	no	—
245	SD₁₆	—	68	1	no	—
246	SD₁₆	—	68	1	no	—
247	M₁₆	—	68	1	no	—
248	C₂×C₈	—	17	1	no	—
249	C₂×C₈	—	17	1	no	—
250	C₂×C₈	—	68	1	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	—	2	1	no	—
255	C₄⋊C₄	—	17	1	no	—
256	C₄⋊C₄	—	17	1	no	—
257	C₄⋊C₄	—	17	1	no	—
258	C₄⋊C₄	—	17	1	no	—
259	C₄⋊C₄	—	17	1	no	—
260	C₄⋊C₄	—	34	1	no	—
261	C₄⋊C₄	—	34	1	no	—
262	C₄⋊C₄	—	34	1	no	—
263	C₄⋊C₄	—	34	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	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	17	1	no	—
271	C₂²⋊C₄	—	17	1	no	—
272	C₂²⋊C₄	—	34	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	C₄²	—	17	1	no	—
278	C₄²	—	34	1	no	—
279	C₄²	—	34	1	no	—
280	C₄²	C₂²×D₁₇	1	1	yes	characteristic
281	C₂³	—	2	1	no	—
282	C₂³	—	17	1	no	—
283	C₂³	—	34	1	no	—
284	Q₈	—	2	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	2	1	no	—
287	Q₈	—	2	1	no	—
288	D₄	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	34	1	no	—
291	D₄	—	34	1	no	—
292	D₄	—	136	1	no	—
293	D₄	—	136	1	no	—
294	D₄	—	136	1	no	—
295	D₄	—	136	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	—	17	1	no	—
301	C₂×C₄	—	17	1	no	—
302	C₂×C₄	—	17	1	no	—
303	C₂×C₄	—	34	1	no	—
304	C₂×C₄	—	34	1	no	—
305	C₂×C₄	—	34	1	no	—
306	C₂×C₄	—	34	1	no	—
307	C₂×C₄	—	34	1	no	—
308	C₂×C₄	—	136	1	no	—
309	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
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	—	2	1	no	—
314	C₈	—	34	1	no	—
315	C₈	—	34	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	34	1	no	—
320	C₂²	—	34	1	no	—
321	C₂²	—	34	1	no	—
322	C₂²	—	68	1	no	—
323	C₂²	—	68	1	no	—
324	C₂²	—	68	1	no	—
325	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	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	#10587691	1	1	yes	characteristic
338	C₄	#10587697	1	1	yes	characteristic
339	C₂	—	8	1	no	—
340	C₂	—	68	1	no	—
341	C₂	—	136	1	no	—
342	C₂	#11763388	1	1	yes	characteristic
343	C₂	#11763615	1	1	yes	characteristic
344	C₂	#11763618	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×#461	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763410	C₂
hypercenter	#461	D₁₇
hypocenter	C₁₇	#3260
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #461

Complements

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

N	H ≅ G/N	type
1. #11763420	C₂	semidirect
3. #11763118	C₂	semidirect
4. #11763114	C₂	semidirect
5. #11763110	C₂	semidirect
6. #11763109	C₂	semidirect
7. #11763058	C₂	semidirect
8. #11763052	C₂	semidirect
9. #11763041	C₂	semidirect
10. #11763035	C₂	semidirect
11. #11763011	C₂	semidirect
12. #11763007	C₂	semidirect
13. #11763007	C₂	semidirect
14. #11762834	C₂	semidirect
15. C₁₇×#461	C₂	semidirect
178. #461	D₁₇	semidirect
231. C₁₇	#3260	semidirect

Maximal subgroups

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

subgroup	index
#11763420	2
#11763410	2
#11763118	2
#11763114	2
#11763110	2
#11763109	2
#11763058	2
#11763052	2
#11763041	2
#11763035	2
#11763011	2
#11763007	2
#11762834	2
C₁₇×#461	2
#3260	17