Subgroups — #14540037

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	#11763414	C₂	1	1	yes	maximal, characteristic
2	#11763215	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763208	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763175	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763157	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763133	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763133	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763127	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763122	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763106	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	#11763086	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	#10587616	C₂²	1	1	yes	characteristic
29	#10587580	—	2	1	no	—
30	#10587580	—	2	1	no	—
31	#10587570	—	2	1	no	—
32	#10587570	—	2	1	no	—
33	#10587560	C₂²	1	1	yes	characteristic
34	C₄×D₆₈	C₂²	1	1	yes	characteristic
35	C₄×D₆₈	C₂²	1	1	yes	characteristic
36	#10587556	C₂²	1	1	yes	characteristic
37	C₂²⋊#3622	C₂²	1	1	yes	characteristic
38	C₂²⋊#3622	C₂²	1	1	yes	characteristic
39	#10587480	C₂²	1	1	yes	characteristic
40	#10587480	C₂²	1	1	yes	characteristic
41	#10587480	C₂²	1	1	yes	characteristic
42	#10587480	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	#10587478	C₂²	1	1	yes	characteristic
52	#10587478	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×C₂²⋊D₁₇	—	2	1	no	—
65	C₂×D₆₈	—	2	1	no	—
66	C₂×D₆₈	—	2	1	no	—
67	C₂×D₆₈	C₂³	1	1	yes	characteristic
68	C₂×D₆₈	C₂³	1	1	yes	characteristic
69	C₂×C₄×D₁₇	—	2	1	no	—
70	C₂×C₄×D₁₇	—	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	—	2	1	no	—
76	C₄⋊C₄×C₁₇	—	2	1	no	—
77	C₄⋊C₄×C₁₇	—	2	1	no	—
78	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
79	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
80	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
84	C₂²⋊C₄×C₁₇	—	2	1	no	—
85	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
86	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
87	C₄²×C₁₇	—	2	1	no	—
88	C₄²×C₁₇	C₂³	1	1	yes	characteristic
89	C₄²×C₁₇	C₂³	1	1	yes	characteristic
90	#63227	—	2	1	no	—
91	#63227	—	2	1	no	—
92	#63227	—	2	1	no	—
93	#63227	—	2	1	no	—
94	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
96	#63225	—	2	1	no	—
97	#63225	—	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	C₂²×D₁₇	—	2	1	no	—
105	Q₈×C₁₇	—	8	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	—	2	1	no	—
112	C₂×C₄×C₁₇	—	2	1	no	—
113	C₂×C₄×C₁₇	—	8	1	no	—
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	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
120	C₂²⋊D₁₇	—	8	1	no	—
121	C₂²⋊D₁₇	—	8	1	no	—
122	C₂×Dic₁₇	—	2	1	no	—
123	C₂×Dic₁₇	—	2	1	no	—
124	D₆₈	—	2	1	no	—
125	D₆₈	—	2	1	no	—
126	D₆₈	—	2	1	no	—
127	D₆₈	—	2	1	no	—
128	D₆₈	—	8	1	no	—
129	D₆₈	—	8	1	no	—
130	C₄×D₁₇	—	8	1	no	—
131	C₄×D₁₇	—	8	1	no	—
132	#3622	—	2	1	no	—
133	#3622	—	2	1	no	—
134	#3622	—	2	1	no	—
135	#3622	—	2	1	no	—
136	#3241	—	17	1	no	maximal, Sylow
137	C₂²×C₁₇	—	2	1	no	—
138	C₂²×C₁₇	—	2	1	no	—
139	C₂²×C₁₇	—	2	1	no	—
140	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
141	C₂×D₁₇	—	4	1	no	—
142	C₂×D₁₇	—	4	1	no	—
143	C₂×D₁₇	—	4	1	no	—
144	C₂×D₁₇	—	4	1	no	—
145	C₂×D₁₇	—	4	1	no	—
146	C₂×D₁₇	—	4	1	no	—
147	C₄×C₁₇	—	2	1	no	—
148	C₄×C₁₇	—	2	1	no	—
149	C₄×C₁₇	—	2	1	no	—
150	C₄×C₁₇	—	2	1	no	—
151	C₄×C₁₇	—	2	1	no	—
152	C₄×C₁₇	—	4	1	no	—
153	C₄×C₁₇	—	4	1	no	—
154	C₄×C₁₇	—	4	1	no	—
155	C₄×C₁₇	—	4	1	no	—
156	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
157	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
158	Dic₁₇	—	4	1	no	—
159	Dic₁₇	—	4	1	no	—
160	#544	D₁₇	1	1	yes	characteristic, complemented
161	#540	—	17	1	no	—
162	#501	—	17	1	no	—
163	#499	—	17	1	no	—
164	#490	—	17	1	no	—
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	—	4	1	no	—
176	C₂×C₁₇	#536	1	1	yes	characteristic
177	C₂×C₁₇	#576	1	1	yes	characteristic
178	C₂×C₁₇	#577	1	1	yes	characteristic
179	D₁₇	—	8	1	no	—
180	D₁₇	—	8	1	no	—
181	C₄⋊Q₈	—	2	1	no	—
182	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
183	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
184	#127	—	17	1	no	—
185	#125	—	2	1	no	—
186	#125	—	17	1	no	—
187	#125	C₂×D₁₇	1	1	yes	characteristic
188	#125	C₂×D₁₇	1	1	yes	characteristic
189	#123	—	34	1	no	—
190	#123	—	34	1	no	—
191	C₂²⋊Q₈	—	2	1	no	—
192	C₂²⋊Q₈	—	2	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	C₂×D₁₇	1	1	yes	characteristic
200	#117	C₂×D₁₇	1	1	yes	characteristic
201	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
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	—	17	1	no	—
217	#102	—	17	1	no	—
218	#102	—	17	1	no	—
219	#102	—	17	1	no	—
220	C₂²⋊C₈	—	17	1	no	—
221	C₂²⋊C₈	—	17	1	no	—
222	#97	—	17	1	no	—
223	C₄×C₈	—	17	1	no	—
224	C₁₇	#3241	1	1	yes	characteristic, complemented, Sylow
225	C₂×Q₈	—	2	1	no	—
226	C₂×Q₈	—	2	1	no	—
227	C₂×D₄	—	17	1	no	—
228	C₂×D₄	—	17	1	no	—
229	C₂×D₄	—	34	1	no	—
230	C₂×D₄	—	34	1	no	—
231	C₂×D₄	—	34	1	no	—
232	C₂×D₄	—	34	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	34	1	no	—
235	C₂²×C₄	—	34	1	no	—
236	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
237	C₂×C₈	—	17	1	no	—
238	C₂×C₈	—	17	1	no	—
239	C₂×C₈	—	17	1	no	—
240	C₂×C₈	—	17	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
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	—	34	1	no	—
251	C₄⋊C₄	—	34	1	no	—
252	C₄⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
253	C₄⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
254	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	34	1	no	—
260	C₂²⋊C₄	—	34	1	no	—
261	C₂²⋊C₄	—	34	1	no	—
262	C₂²⋊C₄	—	34	1	no	—
263	C₂²⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
264	C₂²⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
265	C₄²	—	2	1	no	—
266	C₄²	C₂²×D₁₇	1	1	yes	characteristic
267	C₄²	C₂²×D₁₇	1	1	yes	characteristic
268	C₂³	—	34	1	no	—
269	C₂³	—	34	1	no	—
270	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
271	Q₈	—	8	1	no	—
272	Q₈	—	8	1	no	—
273	D₄	—	34	1	no	—
274	D₄	—	34	1	no	—
275	D₄	—	34	1	no	—
276	D₄	—	34	1	no	—
277	D₄	—	136	1	no	—
278	D₄	—	136	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	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	8	1	no	—
288	C₂×C₄	—	34	1	no	—
289	C₂×C₄	—	34	1	no	—
290	C₂×C₄	—	136	1	no	—
291	C₂×C₄	—	136	1	no	—
292	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
293	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
294	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
295	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
296	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
298	C₈	—	34	1	no	—
299	C₈	—	34	1	no	—
300	C₈	—	34	1	no	—
301	C₈	—	34	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	68	1	no	—
306	C₂²	—	68	1	no	—
307	C₂²	—	68	1	no	—
308	C₂²	—	68	1	no	—
309	C₂²	—	68	1	no	—
310	C₂²	—	68	1	no	—
311	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	68	1	no	—
322	C₄	—	68	1	no	—
323	C₄	#10587694	1	1	yes	characteristic
324	C₄	#10587694	1	1	yes	characteristic
325	C₂	—	4	1	no	—
326	C₂	—	136	1	no	—
327	C₂	—	136	1	no	—
328	C₂	#11763383	1	1	yes	characteristic
329	C₂	#11763612	1	1	yes	characteristic
330	C₂	#11763617	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	#11763414	C₂
hypercenter	#544	D₁₇
hypocenter	C₁₇	#3241
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. #11763215	C₂	semidirect
3. #11763208	C₂	semidirect
4. #11763175	C₂	semidirect
5. #11763157	C₂	semidirect
6. #11763133	C₂	semidirect
7. #11763133	C₂	semidirect
8. #11763127	C₂	semidirect
9. #11763122	C₂	semidirect
10. #11763106	C₂	semidirect
11. #11763106	C₂	semidirect
12. #11763096	C₂	semidirect
13. #11763086	C₂	semidirect
14. #11762870	C₂	semidirect
15. C₁₇×#544	C₂	semidirect
160. #544	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
#11763414	2
#11763215	2
#11763208	2
#11763175	2
#11763157	2
#11763133	2
#11763127	2
#11763122	2
#11763106	2
#11763096	2
#11763086	2
#11762870	2
C₁₇×#544	2
#3241	17