Subgroups — #14540082

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	#11763430	C₂	1	1	yes	maximal, characteristic
2	#11763219	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763205	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763171	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763166	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763150	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763139	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763133	C₂	1	1	yes	maximal, characteristic
9	#11763127	C₂	1	1	yes	maximal, characteristic
10	#11763113	C₂	1	1	yes	maximal, characteristic, complemented
11	#11763106	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763096	C₂	1	1	yes	maximal, characteristic, complemented
13	#11763091	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762870	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#544	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₇×C₄⋊Q₈	—	2	1	no	—
17	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₁₇×#125	—	2	1	no	—
20	C₁₇×#125	C₂²	1	1	yes	characteristic
21	C₁₇×#125	C₂²	1	1	yes	characteristic
22	C₁₇×C₂²⋊Q₈	—	2	1	no	—
23	C₁₇×C₂²⋊Q₈	—	2	1	no	—
24	C₁₇×#117	C₂²	1	1	yes	characteristic
25	C₁₇×#117	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
27	#10587616	C₂²	1	1	yes	characteristic
28	#10587610	C₂²	1	1	yes	characteristic
29	#10587584	—	2	1	no	—
30	#10587582	—	2	1	no	—
31	#10587571	—	2	1	no	—
32	#10587569	—	2	1	no	—
33	#10587561	C₂²	1	1	yes	characteristic
34	C₄×D₆₈	C₂²	1	1	yes	characteristic
35	#10587555	C₂²	1	1	yes	characteristic
36	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
37	C₂²⋊#3622	C₂²	1	1	yes	characteristic
38	C₂²⋊#3622	C₂²	1	1	yes	characteristic
39	#10587481	C₂²	1	1	yes	characteristic
40	#10587481	C₂²	1	1	yes	characteristic
41	#10587481	C₂²	1	1	yes	characteristic
42	#10587481	C₂²	1	1	yes	characteristic
43	#10587480	C₂²	1	1	yes	characteristic
44	#10587480	C₂²	1	1	yes	characteristic
45	#10587480	C₂²	1	1	yes	characteristic
46	#10587480	C₂²	1	1	yes	characteristic
47	#10587479	C₂²	1	1	yes	characteristic
48	#10587479	C₂²	1	1	yes	characteristic
49	#10587479	C₂²	1	1	yes	characteristic
50	#10587479	C₂²	1	1	yes	characteristic
51	#10587479	C₂²	1	1	yes	characteristic
52	#10587479	C₂²	1	1	yes	characteristic
53	#10587478	C₂²	1	1	yes	characteristic
54	#10587478	C₂²	1	1	yes	characteristic
55	C₄⋊#3622	C₂²	1	1	yes	characteristic
56	C₄⋊#3622	C₂²	1	1	yes	characteristic
57	#10587474	C₂²	1	1	yes	characteristic
58	C₄×#3622	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	—	2	1	no	—
60	C₂×Q₈×C₁₇	—	2	1	no	—
61	C₂²×C₄×C₁₇	—	2	1	no	—
62	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₇	—	2	1	no	—
64	C₂×D₆₈	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₇	—	2	1	no	—
66	C₂×Dic₃₄	—	2	1	no	—
67	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₁₇	—	2	1	no	—
69	C₄⋊C₄×C₁₇	—	2	1	no	—
70	C₄⋊C₄×C₁₇	—	2	1	no	—
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	D₄	1	1	yes	characteristic
76	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
81	C₂²⋊C₄×C₁₇	—	2	1	no	—
82	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
83	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
84	C₄²×C₁₇	—	2	1	no	—
85	C₄²×C₁₇	C₂³	1	1	yes	characteristic
86	C₄²×C₁₇	C₂³	1	1	yes	characteristic
87	C₂²⋊Dic₁₇	—	2	1	no	—
88	#63227	—	2	1	no	—
89	#63227	—	2	1	no	—
90	#63227	—	2	1	no	—
91	#63227	—	2	1	no	—
92	C₄⋊Dic₁₇	—	2	1	no	—
93	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
94	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
95	#63225	—	2	1	no	—
96	#63225	—	2	1	no	—
97	C₄×Dic₁₇	—	2	1	no	—
98	C₂×#3622	C₂³	1	1	yes	characteristic
99	C₂×#3622	C₂³	1	1	yes	characteristic
100	C₂×#3622	C₂³	1	1	yes	characteristic
101	C₂×#3622	C₂³	1	1	yes	characteristic
102	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×D₁₇	—	2	1	no	—
104	Q₈×C₁₇	—	8	1	no	—
105	Q₈×C₁₇	—	8	1	no	—
106	C₂×C₄×C₁₇	—	2	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	—	2	1	no	—
112	C₂×C₄×C₁₇	—	8	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	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₁₇	—	8	1	no	—
120	C₂×Dic₁₇	—	2	1	no	—
121	C₂×Dic₁₇	—	2	1	no	—
122	C₂×Dic₁₇	—	2	1	no	—
123	D₆₈	—	2	1	no	—
124	D₆₈	—	2	1	no	—
125	C₄×D₁₇	—	8	1	no	—
126	Dic₃₄	—	2	1	no	—
127	Dic₃₄	—	2	1	no	—
128	Dic₃₄	—	8	1	no	—
129	#3622	—	2	1	no	—
130	#3622	—	2	1	no	—
131	#3622	—	2	1	no	—
132	#3622	—	2	1	no	—
133	#3244	—	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	C₂²×D₄	1	1	yes	characteristic
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	—	2	1	no	—
144	C₄×C₁₇	—	2	1	no	—
145	C₄×C₁₇	—	2	1	no	—
146	C₄×C₁₇	—	4	1	no	—
147	C₄×C₁₇	—	4	1	no	—
148	C₄×C₁₇	—	4	1	no	—
149	C₄×C₁₇	—	4	1	no	—
150	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
151	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
152	Dic₁₇	—	4	1	no	—
153	Dic₁₇	—	4	1	no	—
154	Dic₁₇	—	4	1	no	—
155	#544	D₁₇	1	1	yes	characteristic, complemented
156	#542	—	17	1	no	—
157	#501	—	17	1	no	—
158	#498	—	17	1	no	—
159	#489	—	17	1	no	—
160	#488	—	17	1	no	—
161	#485	—	17	1	no	—
162	#483	—	17	1	no	—
163	#482	—	17	1	no	—
164	#481	—	17	1	no	—
165	#479	—	17	1	no	—
166	#478	—	17	1	no	—
167	#476	—	17	1	no	—
168	#475	—	17	1	no	—
169	#433	—	17	1	no	—
170	C₂×C₁₇	—	4	1	no	—
171	C₂×C₁₇	#536	1	1	yes	characteristic
172	C₂×C₁₇	#577	1	1	yes	characteristic
173	C₂×C₁₇	#577	1	1	yes	characteristic
174	D₁₇	—	8	1	no	—
175	C₄⋊Q₈	—	2	1	no	—
176	C₄⋊Q₈	—	17	1	no	—
177	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
178	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
179	#126	—	34	1	no	—
180	#125	—	2	1	no	—
181	#125	C₂×D₁₇	1	1	yes	characteristic
182	#125	C₂×D₁₇	1	1	yes	characteristic
183	#124	—	17	1	no	—
184	#124	—	34	1	no	—
185	#123	—	34	1	no	—
186	C₂²⋊Q₈	—	2	1	no	—
187	C₂²⋊Q₈	—	2	1	no	—
188	C₂²⋊Q₈	—	17	1	no	—
189	C₂²⋊Q₈	—	34	1	no	—
190	#121	—	17	1	no	—
191	C₄×Q₈	—	17	1	no	—
192	C₄×D₄	—	17	1	no	—
193	#117	C₂×D₁₇	1	1	yes	characteristic
194	#117	C₂×D₁₇	1	1	yes	characteristic
195	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
196	#107	—	17	1	no	—
197	#107	—	17	1	no	—
198	#106	—	17	1	no	—
199	#106	—	17	1	no	—
200	#106	—	17	1	no	—
201	#106	—	17	1	no	—
202	#106	—	17	1	no	—
203	#106	—	17	1	no	—
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	—	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	#3244	1	1	yes	characteristic, complemented, Sylow
219	C₂×Q₈	—	2	1	no	—
220	C₂×Q₈	—	2	1	no	—
221	C₂×Q₈	—	17	1	no	—
222	C₂×Q₈	—	34	1	no	—
223	C₂×D₄	—	17	1	no	—
224	C₂×D₄	—	34	1	no	—
225	C₂²×C₄	—	2	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	—	17	1	no	—
231	C₂×C₈	—	17	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	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	C₂²⋊D₁₇	1	1	yes	characteristic
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	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
250	C₂²⋊C₄	—	2	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	C₂²⋊C₄	—	34	1	no	—
256	C₂²⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
257	C₂²⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
258	C₄²	—	2	1	no	—
259	C₄²	—	34	1	no	—
260	C₄²	C₂²×D₁₇	1	1	yes	characteristic
261	C₄²	C₂²×D₁₇	1	1	yes	characteristic
262	C₂³	—	34	1	no	—
263	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
264	Q₈	—	8	1	no	—
265	Q₈	—	8	1	no	—
266	Q₈	—	34	1	no	—
267	Q₈	—	34	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	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	8	1	no	—
279	C₂×C₄	—	34	1	no	—
280	C₂×C₄	—	34	1	no	—
281	C₂×C₄	—	34	1	no	—
282	C₂×C₄	—	136	1	no	—
283	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
284	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
285	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
286	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
287	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
288	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
289	C₈	—	34	1	no	—
290	C₈	—	34	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	—	68	1	no	—
297	C₂²	—	68	1	no	—
298	C₂²	—	68	1	no	—
299	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	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	#10587694	1	1	yes	characteristic
314	C₂	—	4	1	no	—
315	C₂	—	136	1	no	—
316	C₂	#11763383	1	1	yes	characteristic
317	C₂	#11763617	1	1	yes	characteristic
318	C₂	#11763620	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#544	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#10587616	C₂²
hypercenter	#544	D₁₇
hypocenter	C₁₇	#3244
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #544

Complements

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

N	H ≅ G/N	type
2. #11763219	C₂	semidirect
3. #11763205	C₂	semidirect
4. #11763171	C₂	semidirect
5. #11763166	C₂	semidirect
6. #11763150	C₂	semidirect
7. #11763139	C₂	semidirect
10. #11763113	C₂	semidirect
11. #11763106	C₂	semidirect
12. #11763096	C₂	semidirect
13. #11763091	C₂	semidirect
14. #11762870	C₂	semidirect
15. C₁₇×#544	C₂	semidirect
155. #544	D₁₇	semidirect
218. C₁₇	#3244	semidirect

Maximal subgroups

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

subgroup	index
#11763430	2
#11763219	2
#11763205	2
#11763171	2
#11763166	2
#11763150	2
#11763139	2
#11763133	2
#11763127	2
#11763113	2
#11763106	2
#11763096	2
#11763091	2
#11762870	2
C₁₇×#544	2
#3244	17