Subgroups — #14541162

Order 2176.

354 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763512	C₂	1	1	yes	maximal, characteristic
2	#11763497	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763152	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763146	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763136	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763124	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763113	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763106	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763039	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763039	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#63219	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈×D₆₈	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762844	C₂	1	1	yes	maximal, characteristic, complemented
14	#106×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#499	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₇×#125	C₂²	1	1	yes	characteristic
17	C₁₇×#125	C₂²	1	1	yes	characteristic
18	C₂×#63219	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
21	#10587584	—	2	1	no	—
22	#10587584	—	2	1	no	—
23	#10587583	C₂²	1	1	yes	characteristic
24	#10587583	C₂²	1	1	yes	characteristic
25	#10587581	—	2	1	no	—
26	#10587581	C₂²	1	1	yes	characteristic
27	#10587581	C₂²	1	1	yes	characteristic
28	#10587580	—	2	1	no	—
29	#10587580	—	2	1	no	—
30	#10587579	—	2	1	no	—
31	#10587578	—	2	1	no	—
32	#10587578	C₂²	1	1	yes	characteristic
33	#10587578	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
36	#10587576	—	2	1	no	—
37	C₄×D₆₈	C₂²	1	1	yes	characteristic
38	C₄×D₆₈	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
40	C₁₇×#107	C₂²	1	1	yes	characteristic
41	C₁₇×#107	C₂²	1	1	yes	characteristic
42	C₁₇×#106	C₂²	1	1	yes	characteristic
43	C₁₇×#106	C₂²	1	1	yes	characteristic
44	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
45	#10587492	C₂²	1	1	yes	characteristic
46	#10587491	C₂²	1	1	yes	characteristic
47	#10587491	C₂²	1	1	yes	characteristic
48	#10587488	C₂²	1	1	yes	characteristic
49	#10587487	C₂²	1	1	yes	characteristic
50	#10587481	C₂²	1	1	yes	characteristic
51	#10587481	C₂²	1	1	yes	characteristic
52	#10587480	C₂²	1	1	yes	characteristic
53	#10587480	C₂²	1	1	yes	characteristic
54	#10587479	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	C₄⋊#3622	C₂²	1	1	yes	characteristic
59	C₂×D₆₈	—	2	1	no	—
60	C₂×D₆₈	—	2	1	no	—
61	C₂×D₆₈	D₄	1	1	yes	characteristic
62	C₂×D₆₈	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₁₇	—	2	1	no	—
64	C₂×C₄×D₁₇	—	2	1	no	—
65	C₂×C₄×D₁₇	—	2	1	no	—
66	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
68	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₇	—	2	1	no	—
72	C₄⋊C₄×C₁₇	—	2	1	no	—
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	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
79	C₄²×C₁₇	C₂³	1	1	yes	characteristic
80	#63227	—	2	1	no	—
81	#63227	—	2	1	no	—
82	#63227	—	2	1	no	—
83	#63227	—	2	1	no	—
84	#63227	—	2	1	no	—
85	#63227	—	2	1	no	—
86	#63227	—	2	1	no	—
87	#63227	D₄	1	1	yes	characteristic
88	#63227	D₄	1	1	yes	characteristic
89	C₄⋊Dic₁₇	—	2	1	no	—
90	C₄⋊Dic₁₇	—	2	1	no	—
91	C₄⋊Dic₁₇	D₄	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	C₄×Dic₁₇	—	2	1	no	—
98	C₄×Dic₁₇	—	2	1	no	—
99	C₄×Dic₁₇	—	2	1	no	—
100	C₂×#3622	C₂³	1	1	yes	characteristic
101	C₂×#3622	C₂³	1	1	yes	characteristic
102	#63219	—	2	2	no	—
103	C₈×D₁₇	—	2	2	no	—
104	C₈×D₁₇	—	2	2	no	—
105	C₂²×D₁₇	—	2	1	no	—
106	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₁₇	—	2	1	no	—
116	C₂×Dic₁₇	—	2	1	no	—
117	C₂×Dic₁₇	—	2	1	no	—
118	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
120	D₆₈	—	2	2	no	—
121	D₆₈	—	2	1	no	—
122	D₆₈	—	2	1	no	—
123	D₆₈	—	4	2	no	—
124	D₆₈	—	8	1	no	—
125	C₄×D₁₇	—	2	1	no	—
126	C₄×D₁₇	—	2	1	no	—
127	C₄×D₁₇	—	2	1	no	—
128	C₄×D₁₇	—	2	1	no	—
129	C₄×D₁₇	—	4	2	no	—
130	C₄×D₁₇	—	8	1	no	—
131	C₄×D₁₇	—	8	1	no	—
132	Dic₃₄	—	2	1	no	—
133	Dic₃₄	—	2	1	no	—
134	C₈×C₁₇	—	2	1	no	—
135	C₈×C₁₇	#41	1	2	yes	—
136	#3622	—	2	1	no	—
137	#3622	—	2	1	no	—
138	#3337	—	17	1	no	maximal, Sylow
139	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×D₁₇	—	2	1	no	—
141	C₂×D₁₇	—	2	1	no	—
142	C₂×D₁₇	—	2	1	no	—
143	C₂×D₁₇	—	2	1	no	—
144	C₂×D₁₇	—	2	1	no	—
145	C₂×D₁₇	—	4	1	no	—
146	C₂×D₁₇	—	4	1	no	—
147	C₂×D₁₇	—	4	1	no	—
148	C₂×D₁₇	#135	1	2	yes	—
149	C₄×C₁₇	—	2	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	C₂×#41	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	—	4	1	no	—
160	Dic₁₇	—	4	1	no	—
161	Dic₁₇	—	4	1	no	—
162	#553	—	17	1	no	—
163	#552	—	17	1	no	—
164	#499	D₁₇	1	1	yes	characteristic, complemented
165	#485	—	17	1	no	—
166	#484	—	17	1	no	—
167	#482	—	17	1	no	—
168	#480	—	17	1	no	—
169	#479	—	17	1	no	—
170	#478	—	17	1	no	—
171	#465	—	17	1	no	—
172	#465	—	17	1	no	—
173	C₄×SD₁₆	—	17	1	no	—
174	C₈×D₄	—	17	1	no	—
175	#427	—	17	1	no	—
176	C₂×#106	—	17	1	no	—
177	C₂×C₁₇	#547	1	1	yes	characteristic
178	C₂×C₁₇	C₂×#135	1	1	yes	characteristic
179	C₂×C₁₇	#577	1	1	yes	characteristic
180	D₁₇	—	2	2	no	—
181	D₁₇	—	4	1	no	—
182	D₁₇	—	8	1	no	—
183	C₂×SD₁₆	—	17	1	no	—
184	C₂²×C₈	—	17	1	no	—
185	C₂²×C₈	—	17	1	no	—
186	#126	—	34	1	no	—
187	#126	—	34	1	no	—
188	#125	—	34	1	no	—
189	#125	C₂×D₁₇	1	1	yes	characteristic
190	#125	C₂×D₁₇	1	1	yes	characteristic
191	#123	—	34	1	no	—
192	#123	—	34	1	no	—
193	C₂²⋊Q₈	—	17	1	no	—
194	C₂²⋊Q₈	—	17	1	no	—
195	#121	—	17	1	no	—
196	#121	—	17	1	no	—
197	#121	—	34	1	no	—
198	C₄×Q₈	—	17	1	no	—
199	C₄×D₄	—	17	1	no	—
200	C₄×D₄	—	17	1	no	—
201	C₄×D₄	—	34	1	no	—
202	#117	—	17	1	no	—
203	#117	—	17	1	no	—
204	#117	—	34	1	no	—
205	C₂×C₄⋊C₄	—	17	1	no	—
206	C₂×C₄⋊C₄	—	17	1	no	—
207	#107	—	17	1	no	—
208	#107	—	17	1	no	—
209	#107	C₂×D₁₇	1	1	yes	characteristic
210	#107	C₂×D₁₇	1	1	yes	characteristic
211	#106	—	17	1	no	—
212	#106	—	17	1	no	—
213	#106	—	17	1	no	—
214	#106	C₂×D₁₇	1	1	yes	characteristic
215	#106	C₂×D₁₇	1	1	yes	characteristic
216	C₄⋊C₈	—	17	1	no	—
217	#103	—	17	1	no	—
218	#103	—	17	1	no	—
219	#103	—	17	1	no	—
220	#102	—	17	1	no	—
221	#102	—	17	1	no	—
222	#102	—	17	1	no	—
223	C₂²⋊C₈	—	17	1	no	—
224	C₂²⋊C₈	—	17	1	no	—
225	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
226	C₁₇	#3337	1	1	yes	characteristic, complemented, Sylow
227	C₂×Q₈	—	17	1	no	—
228	C₂×D₄	—	17	1	no	—
229	C₂×D₄	—	17	1	no	—
230	C₂×D₄	—	34	1	no	—
231	C₂×D₄	—	34	1	no	—
232	C₂²×C₄	—	17	1	no	—
233	C₂²×C₄	—	17	1	no	—
234	C₂²×C₄	—	34	1	no	—
235	C₂²×C₄	—	34	1	no	—
236	C₂²×C₄	—	34	1	no	—
237	SD₁₆	—	34	2	no	—
238	C₂×C₈	—	17	1	no	—
239	C₂×C₈	—	17	1	no	—
240	C₂×C₈	—	34	2	no	—
241	C₂×C₈	—	34	2	no	—
242	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	17	1	no	—
249	C₄⋊C₄	—	17	1	no	—
250	C₄⋊C₄	—	17	1	no	—
251	C₄⋊C₄	—	34	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	C₂²×D₁₇	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
258	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
260	C₂²⋊C₄	—	17	1	no	—
261	C₂²⋊C₄	—	17	1	no	—
262	C₂²⋊C₄	—	34	1	no	—
263	C₂²⋊C₄	—	34	1	no	—
264	C₂²⋊C₄	—	34	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	C₄²	—	34	1	no	—
270	C₄²	—	34	1	no	—
271	C₄²	—	34	1	no	—
272	C₄²	C₂²×D₁₇	1	1	yes	characteristic
273	C₂³	—	17	1	no	—
274	C₂³	—	17	1	no	—
275	C₂³	—	34	1	no	—
276	Q₈	—	34	1	no	—
277	Q₈	—	34	1	no	—
278	D₄	—	34	2	no	—
279	D₄	—	34	1	no	—
280	D₄	—	34	1	no	—
281	D₄	—	68	2	no	—
282	D₄	—	136	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	17	1	no	—
288	C₂×C₄	—	17	1	no	—
289	C₂×C₄	—	34	1	no	—
290	C₂×C₄	—	34	1	no	—
291	C₂×C₄	—	34	1	no	—
292	C₂×C₄	—	34	1	no	—
293	C₂×C₄	—	34	1	no	—
294	C₂×C₄	—	34	1	no	—
295	C₂×C₄	—	34	1	no	—
296	C₂×C₄	—	68	2	no	—
297	C₂×C₄	—	136	1	no	—
298	C₂×C₄	—	136	1	no	—
299	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
300	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
301	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	34	1	no	—
304	C₈	—	34	1	no	—
305	C₈	#63256	1	2	yes	—
306	C₂²	—	17	2	no	—
307	C₂²	—	34	1	no	—
308	C₂²	—	34	1	no	—
309	C₂²	—	34	1	no	—
310	C₂²	—	34	1	no	—
311	C₂²	—	34	1	no	—
312	C₂²	—	68	1	no	—
313	C₂²	—	68	1	no	—
314	C₂²	—	68	1	no	—
315	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	34	1	no	—
323	C₄	—	34	1	no	—
324	C₄	—	68	1	no	—
325	C₄	—	68	1	no	—
326	C₄	—	68	1	no	—
327	C₄	C₂×#63256	1	1	yes	characteristic
328	C₄	#10587694	1	1	yes	characteristic
329	C₂	—	34	2	no	—
330	C₂	—	68	1	no	—
331	C₂	—	136	1	no	—
332	C₂	#11763462	1	1	yes	characteristic
333	C₂	#135×D₁₇	1	1	yes	characteristic
334	C₂	#11763616	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×#499	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763512	C₂
hypercenter	#499	D₁₇
hypocenter	C₁₇	#3337
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

Complements

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

N	H ≅ G/N	type
2. #11763497	C₂	semidirect
3. #11763152	C₂	semidirect
4. #11763146	C₂	semidirect
5. #11763136	C₂	semidirect
6. #11763124	C₂	semidirect
7. #11763113	C₂	semidirect
8. #11763106	C₂	semidirect
9. #11763039	C₂	semidirect
10. #11763039	C₂	semidirect
11. C₄×#63219	C₂	semidirect
12. C₈×D₆₈	C₂	semidirect
13. #11762844	C₂	semidirect
14. #106×D₁₇	C₂	semidirect
15. C₁₇×#499	C₂	semidirect
164. #499	D₁₇	semidirect
226. C₁₇	#3337	semidirect

Maximal subgroups

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

subgroup	index
#11763512	2
#11763497	2
#11763152	2
#11763146	2
#11763136	2
#11763124	2
#11763113	2
#11763106	2
#11763039	2
C₄×#63219	2
C₈×D₆₈	2
#11762844	2
#106×D₁₇	2
C₁₇×#499	2
#3337	17