Subgroups — #14541316

Order 2176.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763529	C₂	1	1	yes	maximal, characteristic
2	#11763499	C₂	1	1	yes	maximal, characteristic
3	#11763218	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763151	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763151	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763151	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763145	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763117	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763105	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763067	C₂	1	1	yes	maximal, characteristic, complemented
11	#11763050	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762917	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762851	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762847	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#435	C₂	1	1	yes	maximal, characteristic
16	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
19	#10587622	—	2	1	no	—
20	#10587621	—	2	1	no	—
21	#10587612	C₂²	1	1	yes	characteristic
22	#10587612	C₂²	1	1	yes	characteristic
23	#10587612	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
25	#10587610	C₂²	1	1	yes	characteristic
26	#10587610	C₂²	1	1	yes	characteristic
27	#10587610	C₂²	1	1	yes	characteristic
28	#10587610	C₂²	1	1	yes	characteristic
29	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
30	#10587575	—	2	1	no	—
31	#10587573	—	2	1	no	—
32	#10587565	—	2	1	no	—
33	#10587565	—	2	1	no	—
34	#10587565	—	2	1	no	—
35	#10587563	—	2	1	no	—
36	#10587556	C₂²	1	1	yes	characteristic
37	#10587555	C₂²	1	1	yes	characteristic
38	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
40	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
43	C₁₇×#97	C₂²	1	1	yes	characteristic
44	#10587489	C₂²	1	1	yes	characteristic
45	#10587489	C₂²	1	1	yes	characteristic
46	#10587489	C₂²	1	1	yes	characteristic
47	#10587489	C₂²	1	1	yes	characteristic
48	#10587489	C₂²	1	1	yes	characteristic
49	#10587488	C₂²	1	1	yes	characteristic
50	#10587488	C₂²	1	1	yes	characteristic
51	#10587488	C₂²	1	1	yes	characteristic
52	#10587488	C₂²	1	1	yes	characteristic
53	#10587487	C₂²	1	1	yes	characteristic
54	#10587487	C₂²	1	1	yes	characteristic
55	#10587487	C₂²	1	1	yes	characteristic
56	#10587487	C₂²	1	1	yes	characteristic
57	#10587487	C₂²	1	1	yes	characteristic
58	#10587487	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₇	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂²×Dic₁₇	—	2	1	no	—
63	C₂×Dic₃₄	—	2	1	no	—
64	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
65	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
66	M₁₆×C₁₇	—	2	2	no	—
67	C₂×C₈×C₁₇	—	2	2	no	—
68	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
73	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
74	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
75	C₄²×C₁₇	C₂³	1	1	yes	characteristic
76	C₂²⋊Dic₁₇	—	2	1	no	—
77	C₂²⋊Dic₁₇	—	2	1	no	—
78	C₂²⋊Dic₁₇	—	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	C₄⋊Dic₁₇	—	2	1	no	—
84	C₄⋊Dic₁₇	—	2	1	no	—
85	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
87	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
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	#63225	—	2	1	no	—
98	#63225	—	2	1	no	—
99	C₄×Dic₁₇	—	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	Dic₆₈	—	2	2	no	—
105	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
106	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
107	D₄×C₁₇	—	4	1	no	—
108	C₂×C₄×C₁₇	—	2	1	no	—
109	C₂×C₄×C₁₇	—	2	1	no	—
110	C₂×C₄×C₁₇	—	2	1	no	—
111	C₂×C₄×C₁₇	—	2	1	no	—
112	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
117	C₂×Dic₁₇	—	2	1	no	—
118	C₂×Dic₁₇	—	2	1	no	—
119	C₂×Dic₁₇	—	2	1	no	—
120	C₂×Dic₁₇	—	2	1	no	—
121	C₂×Dic₁₇	—	2	1	no	—
122	C₂×Dic₁₇	—	2	1	no	—
123	C₂×Dic₁₇	—	2	1	no	—
124	C₂×Dic₁₇	—	2	1	no	—
125	C₂×Dic₁₇	—	4	2	no	—
126	Dic₃₄	—	2	1	no	—
127	Dic₃₄	—	2	1	no	—
128	Dic₃₄	—	2	1	no	—
129	Dic₃₄	—	2	1	no	—
130	Dic₃₄	—	4	2	no	—
131	C₈×C₁₇	—	2	1	no	—
132	C₈×C₁₇	—	2	1	no	—
133	C₈×C₁₇	—	2	1	no	—
134	C₈×C₁₇	#41	1	2	yes	—
135	#3349	—	17	1	no	maximal, Sylow
136	C₂²×C₁₇	—	2	1	no	—
137	C₂²×C₁₇	—	2	1	no	—
138	C₂²×C₁₇	—	2	1	no	—
139	C₂²×C₁₇	—	2	1	no	—
140	C₂²×C₁₇	—	2	1	no	—
141	C₂²×C₁₇	—	2	1	no	—
142	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₁₇	—	2	1	no	—
144	C₄×C₁₇	—	2	1	no	—
145	C₄×C₁₇	—	2	1	no	—
146	C₄×C₁₇	—	2	1	no	—
147	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
148	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
149	Dic₁₇	—	4	1	no	—
150	Dic₁₇	—	4	1	no	—
151	Dic₁₇	—	4	1	no	—
152	Dic₁₇	—	4	1	no	—
153	Dic₁₇	—	4	1	no	—
154	Dic₁₇	—	4	1	no	—
155	Dic₁₇	—	4	1	no	—
156	Dic₁₇	—	4	1	no	—
157	#556	—	17	1	no	—
158	#552	—	17	1	no	—
159	#501	—	17	1	no	—
160	#485	—	17	1	no	—
161	#485	—	17	1	no	—
162	#485	—	17	1	no	—
163	#484	—	17	1	no	—
164	#479	—	17	1	no	—
165	#477	—	17	1	no	—
166	#470	—	17	1	no	—
167	#467	—	17	1	no	—
168	#441	—	17	1	no	—
169	#435	D₁₇	1	1	yes	characteristic
170	#428	—	17	1	no	—
171	#427	—	17	1	no	—
172	C₂×C₁₇	—	4	1	no	—
173	C₂×C₁₇	—	4	1	no	—
174	C₂×C₁₇	#547	1	1	yes	characteristic
175	C₂×C₁₇	#575	1	1	yes	characteristic
176	C₂×C₁₇	#578	1	1	yes	characteristic
177	C₂×Q₁₆	—	17	1	no	—
178	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
179	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
180	C₄⋊Q₈	—	17	1	no	—
181	#126	—	34	1	no	—
182	#126	—	34	1	no	—
183	#126	—	34	1	no	—
184	#125	—	17	1	no	—
185	#125	—	34	1	no	—
186	#124	—	34	1	no	—
187	#123	—	34	1	no	—
188	C₂²⋊Q₈	—	17	1	no	—
189	C₂²⋊Q₈	—	17	1	no	—
190	C₂²⋊Q₈	—	17	1	no	—
191	C₂²⋊Q₈	—	17	1	no	—
192	C₄×Q₈	—	17	1	no	—
193	C₄×Q₈	—	17	1	no	—
194	C₄×Q₈	—	34	1	no	—
195	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
196	#117	—	17	1	no	—
197	#117	—	17	1	no	—
198	#117	—	17	1	no	—
199	#117	—	34	1	no	—
200	C₂×C₄⋊C₄	—	17	1	no	—
201	#107	—	17	1	no	—
202	#107	—	17	1	no	—
203	#107	—	17	1	no	—
204	#107	—	17	1	no	—
205	#107	—	17	1	no	—
206	#106	—	17	1	no	—
207	#106	—	17	1	no	—
208	#106	—	17	1	no	—
209	#106	—	17	1	no	—
210	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
211	#103	—	17	1	no	—
212	#103	—	17	1	no	—
213	#103	—	17	1	no	—
214	#103	—	17	1	no	—
215	#103	—	17	1	no	—
216	#103	—	17	1	no	—
217	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
218	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
219	#97	C₂×D₁₇	1	1	yes	characteristic
220	C₁₇	#3349	1	1	yes	characteristic, complemented, Sylow
221	C₂×Q₈	—	17	1	no	—
222	C₂×Q₈	—	17	1	no	—
223	C₂×Q₈	—	34	1	no	—
224	C₂×D₄	D₆₈	1	1	yes	characteristic
225	C₂²×C₄	—	34	1	no	—
226	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
227	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
228	Q₁₆	—	34	2	no	—
229	M₁₆	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
232	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
233	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
235	C₄⋊C₄	—	17	1	no	—
236	C₄⋊C₄	—	17	1	no	—
237	C₄⋊C₄	—	17	1	no	—
238	C₄⋊C₄	—	17	1	no	—
239	C₄⋊C₄	—	17	1	no	—
240	C₄⋊C₄	—	17	1	no	—
241	C₄⋊C₄	—	34	1	no	—
242	C₄⋊C₄	—	34	1	no	—
243	C₄⋊C₄	—	34	1	no	—
244	C₄⋊C₄	—	34	1	no	—
245	C₄⋊C₄	—	34	1	no	—
246	C₄⋊C₄	—	34	1	no	—
247	C₄⋊C₄	—	34	1	no	—
248	C₄⋊C₄	—	34	1	no	—
249	C₄⋊C₄	—	34	1	no	—
250	C₄⋊C₄	—	34	1	no	—
251	C₄⋊C₄	D₆₈	1	1	yes	characteristic
252	C₂²⋊C₄	—	34	1	no	—
253	C₂²⋊C₄	—	34	1	no	—
254	C₂²⋊C₄	—	34	1	no	—
255	C₂²⋊C₄	—	34	1	no	—
256	C₂²⋊C₄	—	34	1	no	—
257	C₂²⋊C₄	—	34	1	no	—
258	C₂²⋊C₄	—	34	1	no	—
259	C₂²⋊C₄	D₆₈	1	1	yes	characteristic
260	C₂²⋊C₄	D₆₈	1	1	yes	characteristic
261	C₄²	—	34	1	no	—
262	C₄²	—	34	1	no	—
263	C₄²	—	34	1	no	—
264	C₄²	—	34	1	no	—
265	C₄²	—	34	1	no	—
266	C₄²	C₂²×D₁₇	1	1	yes	characteristic
267	C₂³	C₂×D₆₈	1	1	yes	characteristic
268	C₂³	C₂×D₆₈	1	1	yes	characteristic
269	Q₈	—	34	1	no	—
270	Q₈	—	34	1	no	—
271	Q₈	—	34	1	no	—
272	Q₈	—	34	1	no	—
273	Q₈	—	68	2	no	—
274	D₄	—	4	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	34	1	no	—
280	C₂×C₄	—	34	1	no	—
281	C₂×C₄	—	34	1	no	—
282	C₂×C₄	—	34	1	no	—
283	C₂×C₄	—	34	1	no	—
284	C₂×C₄	—	34	1	no	—
285	C₂×C₄	—	34	1	no	—
286	C₂×C₄	—	34	1	no	—
287	C₂×C₄	—	68	2	no	—
288	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
289	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
290	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
291	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
292	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	#63254	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	C₂²×D₆₈	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	68	1	no	—
309	C₄	—	68	1	no	—
310	C₄	—	68	1	no	—
311	C₄	—	68	1	no	—
312	C₄	—	68	1	no	—
313	C₄	—	68	1	no	—
314	C₄	—	68	1	no	—
315	C₄	—	68	1	no	—
316	C₄	C₂×#63254	1	1	yes	characteristic
317	C₄	#10587697	1	1	yes	characteristic
318	C₂	—	4	1	no	—
319	C₂	—	4	1	no	—
320	C₂	#11763463	1	1	yes	characteristic
321	C₂	#11763609	1	1	yes	characteristic
322	C₂	#11763626	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₆₈
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#435	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×D₄×C₁₇	D₄
hypercenter	#435	D₁₇
hypocenter	C₁₇	#3349
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #435

Complements

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

N	H ≅ G/N	type
3. #11763218	C₂	semidirect
4. #11763151	C₂	semidirect
5. #11763151	C₂	semidirect
6. #11763151	C₂	semidirect
7. #11763145	C₂	semidirect
8. #11763117	C₂	semidirect
9. #11763105	C₂	semidirect
10. #11763067	C₂	semidirect
11. #11763050	C₂	semidirect
12. #11762917	C₂	semidirect
13. #11762851	C₂	semidirect
14. #11762847	C₂	semidirect
220. C₁₇	#3349	semidirect

Maximal subgroups

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

subgroup	index
#11763529	2
#11763499	2
#11763218	2
#11763151	2
#11763145	2
#11763117	2
#11763105	2
#11763067	2
#11763050	2
#11762917	2
#11762851	2
#11762847	2
C₁₇×#435	2
#3349	17