Subgroups — #14540032

Order 2176.

332 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763439	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763413	C₂	1	1	yes	maximal, characteristic
3	#11763218	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763208	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763155	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763132	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763131	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763126	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763121	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763111	C₂	1	1	yes	maximal, characteristic, complemented
11	#11763107	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763097	C₂	1	1	yes	maximal, characteristic, complemented
13	#11763087	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762871	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#490	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₇×#127	C₂²	1	1	yes	characteristic
17	C₁₇×#125	C₂²	1	1	yes	characteristic
18	#10587625	—	2	1	no	—
19	#10587625	—	2	1	no	—
20	#10587624	C₂²	1	1	yes	characteristic
21	#10587624	C₂²	1	1	yes	characteristic
22	#10587621	—	2	1	no	—
23	#10587621	—	2	1	no	—
24	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
25	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
26	#10587583	—	2	1	no	—
27	#10587582	—	2	1	no	—
28	#10587578	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
30	#10587576	C₂²	1	1	yes	characteristic
31	#10587575	—	2	1	no	—
32	#10587574	—	2	1	no	—
33	#10587574	C₂²	1	1	yes	characteristic
34	#10587558	C₂²	1	1	yes	characteristic
35	#10587556	C₂²	1	1	yes	characteristic
36	#10587555	C₂²	1	1	yes	characteristic
37	C₁₇×#102	C₂²	1	1	yes	characteristic
38	C₁₇×#102	C₂²	1	1	yes	characteristic
39	C₁₇×#102	C₂²	1	1	yes	characteristic
40	C₁₇×#102	C₂²	1	1	yes	characteristic
41	C₁₇×#97	C₂²	1	1	yes	characteristic
42	#10587503	C₂²	1	1	yes	characteristic
43	#10587503	C₂²	1	1	yes	characteristic
44	#10587503	C₂²	1	1	yes	characteristic
45	#10587503	C₂²	1	1	yes	characteristic
46	#10587491	C₂²	1	1	yes	characteristic
47	#10587491	C₂²	1	1	yes	characteristic
48	#10587489	C₂²	1	1	yes	characteristic
49	#10587489	C₂²	1	1	yes	characteristic
50	#10587488	C₂²	1	1	yes	characteristic
51	#10587488	C₂²	1	1	yes	characteristic
52	#10587485	C₂²	1	1	yes	characteristic
53	#10587485	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×D₄×C₁₇	—	2	1	no	—
60	C₂×D₄×C₁₇	—	2	1	no	—
61	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₇	—	2	1	no	—
64	C₂×C₂²⋊D₁₇	—	2	1	no	—
65	C₂²×Dic₁₇	—	2	1	no	—
66	C₂²×Dic₁₇	—	2	1	no	—
67	C₂×C₄×D₁₇	—	2	1	no	—
68	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
69	C₂×Dic₃₄	—	2	1	no	—
70	C₂×Dic₃₄	—	2	1	no	—
71	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
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	C₄²×C₁₇	C₂³	1	1	yes	characteristic
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	#63227	—	2	1	no	—
83	#63227	D₄	1	1	yes	characteristic
84	#63227	D₄	1	1	yes	characteristic
85	C₄⋊Dic₁₇	—	2	1	no	—
86	C₄⋊Dic₁₇	—	2	1	no	—
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	D₄	1	1	yes	characteristic
97	#63225	D₄	1	1	yes	characteristic
98	C₄×Dic₁₇	—	2	1	no	—
99	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
100	C₂×#3622	C₂³	1	1	yes	characteristic
101	C₂×#3622	C₂³	1	1	yes	characteristic
102	C₂³×C₁₇	—	2	1	no	—
103	C₂³×C₁₇	—	2	1	no	—
104	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
105	D₄×C₁₇	—	2	1	no	—
106	D₄×C₁₇	—	2	1	no	—
107	D₄×C₁₇	—	2	1	no	—
108	D₄×C₁₇	—	2	1	no	—
109	D₄×C₁₇	—	8	1	no	—
110	D₄×C₁₇	—	8	1	no	—
111	C₂×C₄×C₁₇	—	2	1	no	—
112	C₂×C₄×C₁₇	—	2	1	no	—
113	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
116	C₂²⋊D₁₇	—	8	1	no	—
117	C₂²⋊D₁₇	—	8	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	—	8	1	no	—
123	C₂×Dic₁₇	—	8	1	no	—
124	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
127	C₄×D₁₇	—	2	1	no	—
128	C₄×D₁₇	—	2	1	no	—
129	C₄×D₁₇	—	8	1	no	—
130	Dic₃₄	—	8	1	no	—
131	Dic₃₄	—	8	1	no	—
132	C₈×C₁₇	—	2	1	no	—
133	C₈×C₁₇	—	2	1	no	—
134	#3622	—	2	1	no	—
135	#3622	—	2	1	no	—
136	#3241	—	17	1	no	maximal, Sylow
137	C₂²×C₁₇	—	4	1	no	—
138	C₂²×C₁₇	—	4	1	no	—
139	C₂²×C₁₇	—	4	1	no	—
140	C₂²×C₁₇	—	4	1	no	—
141	C₂²×C₁₇	—	4	1	no	—
142	C₂²×C₁₇	—	4	1	no	—
143	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₁₇	—	2	1	no	—
145	C₂×D₁₇	—	2	1	no	—
146	C₂×D₁₇	—	2	1	no	—
147	C₄×C₁₇	—	2	1	no	—
148	C₄×C₁₇	—	2	1	no	—
149	C₄×C₁₇	—	4	1	no	—
150	C₄×C₁₇	—	4	1	no	—
151	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
152	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
153	Dic₁₇	—	2	1	no	—
154	Dic₁₇	—	2	1	no	—
155	Dic₁₇	—	2	1	no	—
156	Dic₁₇	—	4	1	no	—
157	Dic₁₇	—	4	1	no	—
158	Dic₁₇	—	4	1	no	—
159	Dic₁₇	—	4	1	no	—
160	#544	—	17	1	no	—
161	#540	—	17	1	no	—
162	#501	—	17	1	no	—
163	#499	—	17	1	no	—
164	#490	D₁₇	1	1	yes	characteristic, complemented
165	#486	—	17	1	no	—
166	#482	—	17	1	no	—
167	#482	—	17	1	no	—
168	#481	—	17	1	no	—
169	#480	—	17	1	no	—
170	#478	—	17	1	no	—
171	#478	—	17	1	no	—
172	#476	—	17	1	no	—
173	#474	—	17	1	no	—
174	#433	—	17	1	no	—
175	C₂×C₁₇	—	8	1	no	—
176	C₂×C₁₇	—	8	1	no	—
177	C₂×C₁₇	#536	1	1	yes	characteristic
178	C₂×C₁₇	#576	1	1	yes	characteristic
179	C₂×C₁₇	#577	1	1	yes	characteristic
180	D₁₇	—	4	1	no	—
181	C₄⋊Q₈	—	17	1	no	—
182	C₄⋊Q₈	—	17	1	no	—
183	C₄⋊Q₈	—	34	1	no	—
184	#127	C₂×D₁₇	1	1	yes	characteristic
185	#125	—	17	1	no	—
186	#125	—	17	1	no	—
187	#125	—	34	1	no	—
188	#125	C₂×D₁₇	1	1	yes	characteristic
189	#123	—	34	1	no	—
190	#123	—	34	1	no	—
191	C₂²⋊Q₈	—	34	1	no	—
192	C₂²⋊Q₈	—	34	1	no	—
193	#121	—	17	1	no	—
194	#121	—	17	1	no	—
195	#121	—	34	1	no	—
196	#121	—	34	1	no	—
197	C₄×D₄	—	17	1	no	—
198	C₄×D₄	—	17	1	no	—
199	#117	—	17	1	no	—
200	#117	—	17	1	no	—
201	C₂×C₄⋊C₄	—	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	—	17	1	no	—
211	C₄⋊C₈	—	17	1	no	—
212	#102	—	17	1	no	—
213	#102	—	17	1	no	—
214	#102	—	17	1	no	—
215	#102	—	17	1	no	—
216	#102	C₂×D₁₇	1	1	yes	characteristic
217	#102	C₂×D₁₇	1	1	yes	characteristic
218	#102	C₂×D₁₇	1	1	yes	characteristic
219	#102	C₂×D₁₇	1	1	yes	characteristic
220	C₂²⋊C₈	—	17	1	no	—
221	C₂²⋊C₈	—	17	1	no	—
222	#97	C₂×D₁₇	1	1	yes	characteristic
223	C₄×C₈	—	17	1	no	—
224	C₁₇	#3241	1	1	yes	characteristic, complemented, Sylow
225	C₂×Q₈	—	34	1	no	—
226	C₂×Q₈	—	34	1	no	—
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	—	2	1	no	—
229	C₂×D₄	—	34	1	no	—
230	C₂×D₄	—	34	1	no	—
231	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
232	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
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	C₂×C₈	—	17	1	no	—
238	C₂×C₈	—	17	1	no	—
239	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	17	1	no	—
244	C₄⋊C₄	—	17	1	no	—
245	C₄⋊C₄	—	17	1	no	—
246	C₄⋊C₄	—	17	1	no	—
247	C₄⋊C₄	—	17	1	no	—
248	C₄⋊C₄	—	17	1	no	—
249	C₄⋊C₄	—	34	1	no	—
250	C₄⋊C₄	—	34	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	C₂²⋊C₄	—	17	1	no	—
259	C₂²⋊C₄	—	17	1	no	—
260	C₂²⋊C₄	—	34	1	no	—
261	C₂²⋊C₄	—	34	1	no	—
262	C₂²⋊C₄	—	34	1	no	—
263	C₂²⋊C₄	—	34	1	no	—
264	C₂²⋊C₄	—	34	1	no	—
265	C₄²	—	17	1	no	—
266	C₄²	—	34	1	no	—
267	C₄²	C₂²×D₁₇	1	1	yes	characteristic
268	C₂³	—	2	1	no	—
269	C₂³	—	2	1	no	—
270	C₂³	—	17	1	no	—
271	Q₈	—	136	1	no	—
272	Q₈	—	136	1	no	—
273	D₄	—	2	1	no	—
274	D₄	—	2	1	no	—
275	D₄	—	2	1	no	—
276	D₄	—	2	1	no	—
277	D₄	—	8	1	no	—
278	D₄	—	8	1	no	—
279	D₄	—	136	1	no	—
280	D₄	—	136	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	17	1	no	—
284	C₂×C₄	—	17	1	no	—
285	C₂×C₄	—	17	1	no	—
286	C₂×C₄	—	34	1	no	—
287	C₂×C₄	—	34	1	no	—
288	C₂×C₄	—	34	1	no	—
289	C₂×C₄	—	34	1	no	—
290	C₂×C₄	—	34	1	no	—
291	C₂×C₄	—	34	1	no	—
292	C₂×C₄	—	136	1	no	—
293	C₂×C₄	—	136	1	no	—
294	C₂×C₄	—	136	1	no	—
295	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
296	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	34	1	no	—
301	C₈	—	34	1	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	34	1	no	—
309	C₂²	—	34	1	no	—
310	C₂²	—	34	1	no	—
311	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	34	1	no	—
317	C₄	—	34	1	no	—
318	C₄	—	34	1	no	—
319	C₄	—	68	1	no	—
320	C₄	—	68	1	no	—
321	C₄	—	68	1	no	—
322	C₄	—	68	1	no	—
323	C₄	#10587697	1	1	yes	characteristic
324	C₄	#10587697	1	1	yes	characteristic
325	C₂	—	8	1	no	—
326	C₂	—	8	1	no	—
327	C₂	—	68	1	no	—
328	C₂	#11763379	1	1	yes	characteristic
329	C₂	#11763613	1	1	yes	characteristic
330	C₂	#11763619	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×#490	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763413	C₂
hypercenter	#490	D₁₇
hypocenter	C₁₇	#3241
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #490

Complements

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

N	H ≅ G/N	type
1. #11763439	C₂	semidirect
3. #11763218	C₂	semidirect
4. #11763208	C₂	semidirect
5. #11763155	C₂	semidirect
6. #11763132	C₂	semidirect
7. #11763131	C₂	semidirect
8. #11763126	C₂	semidirect
9. #11763121	C₂	semidirect
10. #11763111	C₂	semidirect
11. #11763107	C₂	semidirect
12. #11763097	C₂	semidirect
13. #11763087	C₂	semidirect
14. #11762871	C₂	semidirect
15. C₁₇×#490	C₂	semidirect
164. #490	D₁₇	semidirect
224. C₁₇	#3241	semidirect

Maximal subgroups

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

subgroup	index
#11763439	2
#11763413	2
#11763218	2
#11763208	2
#11763155	2
#11763132	2
#11763131	2
#11763126	2
#11763121	2
#11763111	2
#11763107	2
#11763097	2
#11763087	2
#11762871	2
C₁₇×#490	2
#3241	17