Subgroups — #14540022

Order 2176.

328 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763362	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763340	C₂	1	1	yes	maximal, characteristic
3	#11763217	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763216	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763167	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763167	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763151	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763150	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763130	C₂	1	1	yes	maximal, characteristic
10	#11763119	C₂	1	1	yes	maximal, characteristic, complemented
11	#11763115	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763111	C₂	1	1	yes	maximal, characteristic, complemented
13	#11763107	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762863	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#482	C₂	1	1	yes	maximal, characteristic
16	C₁₇×#121	C₂²	1	1	yes	characteristic
17	C₁₇×#117	C₂²	1	1	yes	characteristic
18	#10587622	C₂²	1	1	yes	characteristic
19	#10587621	—	2	1	no	—
20	D₄×Dic₁₇	—	2	1	no	—
21	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
22	#10587612	C₂²	1	1	yes	characteristic
23	#10587610	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
25	#10587576	C₂²	1	1	yes	characteristic
26	#10587576	C₂²	1	1	yes	characteristic
27	#10587576	C₂²	1	1	yes	characteristic
28	#10587574	C₂²	1	1	yes	characteristic
29	#10587574	C₂²	1	1	yes	characteristic
30	#10587573	C₂²	1	1	yes	characteristic
31	#10587565	—	2	1	no	—
32	#10587564	—	2	1	no	—
33	#10587564	—	2	1	no	—
34	#10587563	—	2	1	no	—
35	C₄×Dic₃₄	—	2	1	no	—
36	C₄×Dic₃₄	—	2	1	no	—
37	C₁₇×#107	C₂²	1	1	yes	characteristic
38	C₁₇×#106	C₂²	1	1	yes	characteristic
39	C₁₇×#102	C₂²	1	1	yes	characteristic
40	C₁₇×#102	C₂²	1	1	yes	characteristic
41	C₁₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	#10587503	C₂²	1	1	yes	characteristic
43	#10587503	C₂²	1	1	yes	characteristic
44	C₂²⋊#3622	C₂²	1	1	yes	characteristic
45	#10587489	C₂²	1	1	yes	characteristic
46	#10587488	C₂²	1	1	yes	characteristic
47	#10587487	C₂²	1	1	yes	characteristic
48	#10587487	C₂²	1	1	yes	characteristic
49	#10587486	C₂²	1	1	yes	characteristic
50	#10587486	C₂²	1	1	yes	characteristic
51	#10587485	C₂²	1	1	yes	characteristic
52	#10587485	C₂²	1	1	yes	characteristic
53	#10587481	C₂²	1	1	yes	characteristic
54	#10587481	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	C₂×D₄×C₁₇	—	2	1	no	—
60	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂²×Dic₁₇	—	2	1	no	—
63	C₂²×Dic₁₇	D₄	1	1	yes	characteristic
64	C₂²×Dic₁₇	D₄	1	1	yes	characteristic
65	C₂×Dic₃₄	—	2	1	no	—
66	C₂×Dic₃₄	—	2	1	no	—
67	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
73	C₂²⋊C₄×C₁₇	—	2	1	no	—
74	C₂²⋊C₄×C₁₇	—	2	1	no	—
75	C₄²×C₁₇	—	2	1	no	—
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	C₄⋊Dic₁₇	—	2	1	no	—
82	C₄⋊Dic₁₇	—	2	1	no	—
83	C₄⋊Dic₁₇	—	2	1	no	—
84	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
87	#63225	—	2	1	no	—
88	#63225	—	2	1	no	—
89	#63225	—	2	1	no	—
90	#63225	—	2	1	no	—
91	#63225	—	2	1	no	—
92	#63225	—	2	1	no	—
93	#63225	—	2	1	no	—
94	C₄×Dic₁₇	—	2	1	no	—
95	C₄×Dic₁₇	—	2	1	no	—
96	C₄×Dic₁₇	D₄	1	1	yes	characteristic
97	C₄×Dic₁₇	D₄	1	1	yes	characteristic
98	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
104	D₄×C₁₇	—	2	1	no	—
105	D₄×C₁₇	—	2	1	no	—
106	D₄×C₁₇	—	4	2	no	—
107	C₂×C₄×C₁₇	—	2	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	C₂⁴	1	1	yes	characteristic
114	C₂×Dic₁₇	—	2	1	no	—
115	C₂×Dic₁₇	—	2	1	no	—
116	C₂×Dic₁₇	—	2	1	no	—
117	C₂×Dic₁₇	—	2	1	no	—
118	C₂×Dic₁₇	—	4	1	no	—
119	C₂×Dic₁₇	—	4	1	no	—
120	C₂×Dic₁₇	—	4	2	no	—
121	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
125	Dic₃₄	—	2	1	no	—
126	Dic₃₄	—	2	1	no	—
127	Dic₃₄	—	4	2	no	—
128	Dic₃₄	—	4	2	no	—
129	C₈×C₁₇	—	2	1	no	—
130	C₈×C₁₇	—	2	1	no	—
131	#3622	—	2	1	no	—
132	#3622	—	2	1	no	—
133	#3240	—	17	1	no	maximal, Sylow
134	C₂²×C₁₇	—	2	1	no	—
135	C₂²×C₁₇	—	2	1	no	—
136	C₂²×C₁₇	—	2	1	no	—
137	C₂²×C₁₇	—	4	1	no	—
138	C₂²×C₁₇	—	4	1	no	—
139	C₂²×C₁₇	—	4	1	no	—
140	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
141	C₄×C₁₇	—	2	1	no	—
142	C₄×C₁₇	—	4	1	no	—
143	C₄×C₁₇	—	4	1	no	—
144	C₄×C₁₇	—	4	1	no	—
145	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
146	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
147	Dic₁₇	—	2	1	no	—
148	Dic₁₇	—	2	1	no	—
149	Dic₁₇	—	2	1	no	—
150	Dic₁₇	—	2	1	no	—
151	Dic₁₇	—	4	1	no	—
152	Dic₁₇	—	4	1	no	—
153	Dic₁₇	—	4	1	no	—
154	Dic₁₇	—	4	1	no	—
155	#533	—	17	1	no	—
156	#529	—	17	1	no	—
157	#501	—	17	1	no	—
158	#501	—	17	1	no	—
159	#489	—	17	1	no	—
160	#489	—	17	1	no	—
161	#485	—	17	1	no	—
162	#485	—	17	1	no	—
163	#482	—	17	1	no	—
164	#482	D₁₇	1	1	yes	characteristic
165	#479	—	17	1	no	—
166	#479	—	17	1	no	—
167	#478	—	17	1	no	—
168	#478	—	17	1	no	—
169	#432	—	17	1	no	—
170	C₂×C₁₇	—	4	1	no	—
171	C₂×C₁₇	—	8	1	no	—
172	C₂×C₁₇	#536	1	1	yes	characteristic
173	C₂×C₁₇	#575	1	1	yes	characteristic
174	C₂×C₁₇	#575	1	1	yes	characteristic
175	C₄⋊Q₈	—	17	1	no	—
176	C₄⋊Q₈	—	17	1	no	—
177	#126	—	34	1	no	—
178	#125	—	17	1	no	—
179	#125	—	17	1	no	—
180	#125	—	17	1	no	—
181	#124	—	17	1	no	—
182	#123	—	34	1	no	—
183	C₂²⋊Q₈	—	17	1	no	—
184	C₂²⋊Q₈	—	34	1	no	—
185	C₂²⋊Q₈	—	34	1	no	—
186	#121	C₂×D₁₇	1	1	yes	characteristic
187	C₄×Q₈	—	17	1	no	—
188	C₄×Q₈	—	34	1	no	—
189	C₄×Q₈	—	34	1	no	—
190	C₄×D₄	—	17	1	no	—
191	C₄×D₄	—	34	1	no	—
192	#117	—	17	1	no	—
193	#117	—	34	1	no	—
194	#117	C₂×D₁₇	1	1	yes	characteristic
195	C₂×C₄²	—	17	1	no	—
196	#107	—	17	1	no	—
197	#107	—	17	1	no	—
198	#107	—	17	1	no	—
199	#107	C₂×D₁₇	1	1	yes	characteristic
200	#106	—	17	1	no	—
201	#106	—	17	1	no	—
202	#106	—	17	1	no	—
203	#106	C₂×D₁₇	1	1	yes	characteristic
204	C₄⋊C₈	—	17	1	no	—
205	C₄⋊C₈	—	17	1	no	—
206	#103	—	17	1	no	—
207	#103	—	17	1	no	—
208	#103	—	17	1	no	—
209	#103	—	17	1	no	—
210	#102	—	17	1	no	—
211	#102	—	17	1	no	—
212	#102	C₂×D₁₇	1	1	yes	characteristic
213	#102	C₂×D₁₇	1	1	yes	characteristic
214	C₂²⋊C₈	—	17	1	no	—
215	C₂²⋊C₈	C₂×D₁₇	1	1	yes	characteristic
216	#97	—	17	1	no	—
217	#97	—	17	1	no	—
218	C₁₇	#3240	1	1	yes	characteristic, complemented, Sylow
219	C₂×Q₈	—	17	1	no	—
220	C₂×Q₈	—	34	1	no	—
221	C₂×Q₈	—	34	1	no	—
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
224	C₂²×C₄	—	17	1	no	—
225	C₂²×C₄	—	17	1	no	—
226	C₂²×C₄	—	34	1	no	—
227	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
228	C₂×C₈	—	17	1	no	—
229	C₂×C₈	—	17	1	no	—
230	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
232	C₄⋊C₄	—	17	1	no	—
233	C₄⋊C₄	—	17	1	no	—
234	C₄⋊C₄	—	17	1	no	—
235	C₄⋊C₄	—	34	1	no	—
236	C₄⋊C₄	—	34	1	no	—
237	C₄⋊C₄	—	34	1	no	—
238	C₄⋊C₄	—	34	1	no	—
239	C₄⋊C₄	—	34	1	no	—
240	C₄⋊C₄	—	34	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	C₂²×D₁₇	1	1	yes	characteristic
246	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
247	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
248	C₂²⋊C₄	—	2	1	no	—
249	C₂²⋊C₄	—	2	1	no	—
250	C₂²⋊C₄	—	34	1	no	—
251	C₂²⋊C₄	—	34	1	no	—
252	C₂²⋊C₄	—	34	1	no	—
253	C₂²⋊C₄	—	34	1	no	—
254	C₂²⋊C₄	—	34	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	17	1	no	—
257	C₄²	—	17	1	no	—
258	C₄²	—	17	1	no	—
259	C₄²	—	17	1	no	—
260	C₄²	—	34	1	no	—
261	C₄²	—	34	1	no	—
262	C₂³	—	2	1	no	—
263	C₂³	D₄×D₁₇	1	1	yes	characteristic
264	Q₈	—	34	1	no	—
265	Q₈	—	34	1	no	—
266	Q₈	—	68	2	no	—
267	Q₈	—	68	2	no	—
268	D₄	—	2	1	no	—
269	D₄	—	2	1	no	—
270	D₄	—	4	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	17	1	no	—
277	C₂×C₄	—	17	1	no	—
278	C₂×C₄	—	17	1	no	—
279	C₂×C₄	—	17	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	—	68	1	no	—
285	C₂×C₄	—	68	1	no	—
286	C₂×C₄	—	68	2	no	—
287	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
288	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	34	1	no	—
292	C₈	—	34	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	34	1	no	—
305	C₄	—	34	1	no	—
306	C₄	—	34	1	no	—
307	C₄	—	34	1	no	—
308	C₄	—	68	1	no	—
309	C₄	—	68	1	no	—
310	C₄	—	68	1	no	—
311	C₄	—	68	1	no	—
312	C₄	#10587697	1	1	yes	characteristic
313	C₄	#10587697	1	1	yes	characteristic
314	C₂	—	4	1	no	—
315	C₂	—	8	1	no	—
316	C₂	#11763381	1	1	yes	characteristic
317	C₂	#11763605	1	1	yes	characteristic
318	C₂	#11763606	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×#482	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₁₇×#121	C₂²
hypercenter	#482	D₁₇
hypocenter	C₁₇	#3240
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #482

Complements

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

N	H ≅ G/N	type
1. #11763362	C₂	semidirect
3. #11763217	C₂	semidirect
4. #11763216	C₂	semidirect
5. #11763167	C₂	semidirect
6. #11763167	C₂	semidirect
7. #11763151	C₂	semidirect
8. #11763150	C₂	semidirect
10. #11763119	C₂	semidirect
11. #11763115	C₂	semidirect
12. #11763111	C₂	semidirect
13. #11763107	C₂	semidirect
14. #11762863	C₂	semidirect
218. C₁₇	#3240	semidirect

Maximal subgroups

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

subgroup	index
#11763362	2
#11763340	2
#11763217	2
#11763216	2
#11763167	2
#11763151	2
#11763150	2
#11763130	2
#11763119	2
#11763115	2
#11763111	2
#11763107	2
#11762863	2
C₁₇×#482	2
#3240	17