Subgroups — #14540072

Order 2176.

320 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	#11763436	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763431	C₂	1	1	yes	maximal, characteristic
3	#11763217	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763213	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763170	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763165	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763148	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763143	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763136	C₂	1	1	yes	maximal, characteristic
10	#11763124	C₂	1	1	yes	maximal, characteristic
11	#11763116	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763110	C₂	1	1	yes	maximal, characteristic, complemented
13	#11763086	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762869	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#477	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
18	#10587633	C₂²	1	1	yes	characteristic
19	#10587632	—	2	1	no	—
20	#10587632	C₂²	1	1	yes	characteristic
21	#10587630	—	2	1	no	—
22	#10587582	—	2	1	no	—
23	#10587582	—	2	1	no	—
24	#10587581	C₂²	1	1	yes	characteristic
25	#10587580	—	2	1	no	—
26	#10587578	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
28	#10587576	C₂²	1	1	yes	characteristic
29	#10587575	—	2	1	no	—
30	#10587574	C₂²	1	1	yes	characteristic
31	#10587574	C₂²	1	1	yes	characteristic
32	#10587562	—	2	1	no	—
33	#10587561	—	2	1	no	—
34	C₄×D₆₈	C₂²	1	1	yes	characteristic
35	#10587558	C₂²	1	1	yes	characteristic
36	#10587556	C₂²	1	1	yes	characteristic
37	C₁₇×#107	C₂²	1	1	yes	characteristic
38	C₁₇×#107	C₂²	1	1	yes	characteristic
39	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₁₇×#103	C₂²	1	1	yes	characteristic
41	C₁₇×#103	C₂²	1	1	yes	characteristic
42	#10587506	C₂²	1	1	yes	characteristic
43	#10587506	C₂²	1	1	yes	characteristic
44	#10587492	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	#10587489	C₂²	1	1	yes	characteristic
49	#10587488	C₂²	1	1	yes	characteristic
50	#10587486	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₇	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	#10587478	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×Q₈×C₁₇	—	2	1	no	—
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₂×D₆₈	—	2	1	no	—
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	C₂³	1	1	yes	characteristic
66	C₂×Dic₃₄	—	2	1	no	—
67	C₂×Dic₃₄	—	2	1	no	—
68	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₇	—	2	1	no	—
71	C₄⋊C₄×C₁₇	—	2	1	no	—
72	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
75	C₄²×C₁₇	—	2	1	no	—
76	C₄²×C₁₇	C₂³	1	1	yes	characteristic
77	#63227	—	2	1	no	—
78	#63227	—	2	1	no	—
79	#63227	—	2	1	no	—
80	#63227	—	2	1	no	—
81	#63227	—	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	C₂³	1	1	yes	characteristic
87	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
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	#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	C₂²×D₁₇	—	2	1	no	—
103	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
104	Q₈×C₁₇	—	2	1	no	—
105	Q₈×C₁₇	—	2	1	no	—
106	Q₈×C₁₇	—	8	1	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	C₂×D₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
120	D₆₈	—	2	1	no	—
121	D₆₈	—	2	1	no	—
122	D₆₈	—	8	1	no	—
123	C₄×D₁₇	—	2	1	no	—
124	C₄×D₁₇	—	2	1	no	—
125	C₄×D₁₇	—	8	1	no	—
126	C₄×D₁₇	—	8	1	no	—
127	Dic₃₄	—	8	1	no	—
128	Dic₃₄	—	8	1	no	—
129	C₈×C₁₇	—	2	1	no	—
130	C₈×C₁₇	—	2	1	no	—
131	#3622	—	2	1	no	—
132	#3622	—	2	1	no	—
133	#3243	—	17	1	no	maximal, Sylow
134	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×D₁₇	—	2	1	no	—
136	C₂×D₁₇	—	2	1	no	—
137	C₂×D₁₇	—	2	1	no	—
138	C₂×D₁₇	—	4	1	no	—
139	C₂×D₁₇	—	4	1	no	—
140	C₂×D₁₇	—	4	1	no	—
141	C₄×C₁₇	—	2	1	no	—
142	C₄×C₁₇	—	2	1	no	—
143	C₄×C₁₇	—	4	1	no	—
144	C₄×C₁₇	—	4	1	no	—
145	C₄×C₁₇	—	4	1	no	—
146	C₄×C₁₇	—	4	1	no	—
147	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
148	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
149	Dic₁₇	—	2	1	no	—
150	Dic₁₇	—	2	1	no	—
151	Dic₁₇	—	2	1	no	—
152	Dic₁₇	—	4	1	no	—
153	Dic₁₇	—	4	1	no	—
154	Dic₁₇	—	4	1	no	—
155	#544	—	17	1	no	—
156	#542	—	17	1	no	—
157	#501	—	17	1	no	—
158	#500	—	17	1	no	—
159	#489	—	17	1	no	—
160	#488	—	17	1	no	—
161	#485	—	17	1	no	—
162	#484	—	17	1	no	—
163	#482	—	17	1	no	—
164	#480	—	17	1	no	—
165	#479	—	17	1	no	—
166	#478	—	17	1	no	—
167	#477	D₁₇	1	1	yes	characteristic, complemented
168	#474	—	17	1	no	—
169	#433	—	17	1	no	—
170	C₂×C₁₇	#536	1	1	yes	characteristic
171	C₂×C₁₇	#576	1	1	yes	characteristic
172	C₂×C₁₇	#578	1	1	yes	characteristic
173	D₁₇	—	4	1	no	—
174	D₁₇	—	8	1	no	—
175	C₄⋊Q₈	—	17	1	no	—
176	C₄⋊Q₈	—	17	1	no	—
177	C₄⋊Q₈	—	34	1	no	—
178	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
179	#126	—	34	1	no	—
180	#125	—	17	1	no	—
181	#125	—	17	1	no	—
182	#125	—	34	1	no	—
183	#124	—	17	1	no	—
184	#124	—	34	1	no	—
185	#123	—	34	1	no	—
186	C₂²⋊Q₈	—	17	1	no	—
187	C₂²⋊Q₈	—	34	1	no	—
188	C₂²⋊Q₈	—	34	1	no	—
189	C₂²⋊Q₈	—	34	1	no	—
190	#121	—	17	1	no	—
191	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
192	C₄×D₄	—	17	1	no	—
193	#117	—	17	1	no	—
194	#117	—	17	1	no	—
195	C₂×C₄⋊C₄	—	17	1	no	—
196	#107	—	17	1	no	—
197	#107	—	17	1	no	—
198	#107	—	17	1	no	—
199	#107	—	17	1	no	—
200	#107	C₂×D₁₇	1	1	yes	characteristic
201	#107	C₂×D₁₇	1	1	yes	characteristic
202	#106	—	17	1	no	—
203	#106	—	17	1	no	—
204	C₄⋊C₈	—	17	1	no	—
205	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
206	#103	—	17	1	no	—
207	#103	—	17	1	no	—
208	#103	C₂×D₁₇	1	1	yes	characteristic
209	#103	C₂×D₁₇	1	1	yes	characteristic
210	#102	—	17	1	no	—
211	#102	—	17	1	no	—
212	#102	—	17	1	no	—
213	#102	—	17	1	no	—
214	C₂²⋊C₈	—	17	1	no	—
215	C₂²⋊C₈	—	17	1	no	—
216	#97	—	17	1	no	—
217	C₄×C₈	—	17	1	no	—
218	C₁₇	#3243	1	1	yes	characteristic, complemented, Sylow
219	C₂×Q₈	—	2	1	no	—
220	C₂×Q₈	—	34	1	no	—
221	C₂×Q₈	—	34	1	no	—
222	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
223	C₂×D₄	—	17	1	no	—
224	C₂×D₄	—	34	1	no	—
225	C₂²×C₄	—	17	1	no	—
226	C₂²×C₄	—	34	1	no	—
227	C₂²×C₄	—	34	1	no	—
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	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	17	1	no	—
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	—	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	—	34	1	no	—
246	C₄⋊C₄	—	34	1	no	—
247	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
248	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
250	C₂²⋊C₄	—	17	1	no	—
251	C₂²⋊C₄	—	17	1	no	—
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	C₄²	—	2	1	no	—
259	C₄²	—	17	1	no	—
260	C₄²	—	34	1	no	—
261	C₄²	C₂²×D₁₇	1	1	yes	characteristic
262	C₂³	—	17	1	no	—
263	C₂³	—	34	1	no	—
264	Q₈	—	2	1	no	—
265	Q₈	—	2	1	no	—
266	Q₈	—	8	1	no	—
267	Q₈	—	136	1	no	—
268	Q₈	—	136	1	no	—
269	D₄	—	34	1	no	—
270	D₄	—	34	1	no	—
271	D₄	—	136	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	—	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	—	136	1	no	—
285	C₂×C₄	—	136	1	no	—
286	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
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	—	34	1	no	—
294	C₂²	—	34	1	no	—
295	C₂²	—	34	1	no	—
296	C₂²	—	68	1	no	—
297	C₂²	—	68	1	no	—
298	C₂²	—	68	1	no	—
299	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	34	1	no	—
307	C₄	—	34	1	no	—
308	C₄	—	34	1	no	—
309	C₄	—	68	1	no	—
310	C₄	—	68	1	no	—
311	C₄	—	68	1	no	—
312	C₄	#10587694	1	1	yes	characteristic
313	C₄	#10587697	1	1	yes	characteristic
314	C₂	—	68	1	no	—
315	C₂	—	136	1	no	—
316	C₂	#11763382	1	1	yes	characteristic
317	C₂	#11763613	1	1	yes	characteristic
318	C₂	#11763622	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×#477	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#10587581	C₂²
hypercenter	#477	D₁₇
hypocenter	C₁₇	#3243
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

Complements

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

N	H ≅ G/N	type
1. #11763436	C₂	semidirect
3. #11763217	C₂	semidirect
4. #11763213	C₂	semidirect
5. #11763170	C₂	semidirect
6. #11763165	C₂	semidirect
7. #11763148	C₂	semidirect
8. #11763143	C₂	semidirect
11. #11763116	C₂	semidirect
12. #11763110	C₂	semidirect
13. #11763086	C₂	semidirect
14. #11762869	C₂	semidirect
15. C₁₇×#477	C₂	semidirect
167. #477	D₁₇	semidirect
218. C₁₇	#3243	semidirect

Maximal subgroups

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

subgroup	index
#11763436	2
#11763431	2
#11763217	2
#11763213	2
#11763170	2
#11763165	2
#11763148	2
#11763143	2
#11763136	2
#11763124	2
#11763116	2
#11763110	2
#11763086	2
#11762869	2
C₁₇×#477	2
#3243	17