Subgroups — #14539448

Order 2176.

350 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763424	C₂	1	1	yes	maximal, characteristic
2	#11763417	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763216	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763200	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763162	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763158	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763068	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763052	C₂	1	1	yes	maximal, characteristic
9	#11763049	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763037	C₂	1	1	yes	maximal, characteristic
11	#11762914	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762909	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×Dic₆₈	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762763	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
19	#10587632	C₂²	1	1	yes	characteristic
20	#10587630	C₂²	1	1	yes	characteristic
21	C₂×#63231	C₂²	1	1	yes	characteristic
22	#10587624	C₂²	1	1	yes	characteristic
23	#10587622	C₂²	1	1	yes	characteristic
24	C₂×#63229	C₂²	1	1	yes	characteristic
25	#10587615	—	2	1	no	—
26	#10587610	—	2	1	no	—
27	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
28	C₂×#63218	C₂²	1	1	yes	characteristic
29	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
30	#10587584	—	2	1	no	—
31	#10587583	C₂²	1	1	yes	characteristic
32	#10587583	C₂²	1	1	yes	characteristic
33	#10587576	C₂²	1	1	yes	characteristic
34	#10587575	—	2	1	no	—
35	#10587574	C₂²	1	1	yes	characteristic
36	#10587571	—	2	1	no	—
37	#10587565	—	2	1	no	—
38	#10587562	—	2	1	no	—
39	#10587558	C₂²	1	1	yes	characteristic
40	#10587556	—	2	1	no	—
41	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
42	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
43	C₁₇×#106	C₂²	1	1	yes	characteristic
44	C₁₇×#103	C₂²	1	1	yes	characteristic
45	C₁₇×#102	C₂²	1	1	yes	characteristic
46	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
47	#10587506	C₂²	1	1	yes	characteristic
48	#10587503	C₂²	1	1	yes	characteristic
49	#10587489	C₂²	1	1	yes	characteristic
50	#10587487	C₂²	1	1	yes	characteristic
51	#10587487	C₂²	1	1	yes	characteristic
52	#10587486	C₂²	1	1	yes	characteristic
53	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
54	#10587481	C₂²	1	1	yes	characteristic
55	#10587481	C₂²	1	1	yes	characteristic
56	#10587479	C₂²	1	1	yes	characteristic
57	#10587478	C₂²	1	1	yes	characteristic
58	#10587474	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	—	2	1	no	—
62	C₂×C₂²⋊D₁₇	—	2	1	no	—
63	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
64	C₂×Dic₃₄	—	2	1	no	—
65	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
66	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
67	SD₁₆×C₁₇	—	2	2	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	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
73	C₂²⋊C₄×C₁₇	—	2	1	no	—
74	C₄²×C₁₇	—	2	1	no	—
75	C₄²×C₁₇	C₂³	1	1	yes	characteristic
76	C₂²⋊Dic₁₇	—	2	1	no	—
77	C₂²⋊Dic₁₇	—	2	1	no	—
78	C₂²⋊Dic₁₇	—	2	1	no	—
79	#63231	—	2	2	no	—
80	#63229	—	2	2	no	—
81	#63227	—	2	1	no	—
82	#63227	—	2	1	no	—
83	#63227	—	2	1	no	—
84	#63227	D₄	1	1	yes	characteristic
85	#63227	D₄	1	1	yes	characteristic
86	C₄⋊Dic₁₇	—	2	1	no	—
87	C₄⋊Dic₁₇	—	2	1	no	—
88	C₄⋊Dic₁₇	—	2	1	no	—
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	—	2	1	no	—
97	#63225	—	2	1	no	—
98	#63225	D₄	1	1	yes	characteristic
99	#63225	D₄	1	1	yes	characteristic
100	C₄×Dic₁₇	—	2	1	no	—
101	C₄×Dic₁₇	—	2	1	no	—
102	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
103	C₂×#3622	C₂³	1	1	yes	characteristic
104	C₂×#3622	C₂³	1	1	yes	characteristic
105	Dic₆₈	—	2	2	no	—
106	#63218	—	2	2	no	—
107	C₈×D₁₇	—	2	2	no	—
108	C₂³×C₁₇	—	2	1	no	—
109	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₁₇	—	2	1	no	—
111	Q₈×C₁₇	—	2	1	no	—
112	D₄×C₁₇	—	2	1	no	—
113	D₄×C₁₇	—	2	1	no	—
114	C₂×C₄×C₁₇	—	2	1	no	—
115	C₂×C₄×C₁₇	—	2	1	no	—
116	C₂×C₄×C₁₇	—	2	1	no	—
117	C₂×C₄×C₁₇	—	8	1	no	—
118	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₁₇	—	8	1	no	—
122	C₂×Dic₁₇	—	2	1	no	—
123	C₂×Dic₁₇	—	2	1	no	—
124	C₂×Dic₁₇	—	2	1	no	—
125	C₂×Dic₁₇	—	2	1	no	—
126	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
129	C₄×D₁₇	—	2	1	no	—
130	C₄×D₁₇	—	2	1	no	—
131	Dic₃₄	—	2	1	no	—
132	Dic₃₄	—	2	1	no	—
133	Dic₃₄	—	2	1	no	—
134	Dic₃₄	—	2	1	no	—
135	Dic₃₄	—	8	1	no	—
136	C₈×C₁₇	—	2	1	no	—
137	C₈×C₁₇	#41	1	2	yes	—
138	#3622	—	2	1	no	—
139	#3622	#41	1	2	yes	—
140	#3178	—	17	1	no	maximal, Sylow
141	C₂²×C₁₇	—	4	1	no	—
142	C₂²×C₁₇	—	4	1	no	—
143	C₂²×C₁₇	—	4	1	no	—
144	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₁₇	—	2	1	no	—
146	C₂×D₁₇	—	2	1	no	—
147	C₂×D₁₇	—	2	1	no	—
148	C₄×C₁₇	—	2	1	no	—
149	C₄×C₁₇	—	2	1	no	—
150	C₄×C₁₇	—	4	1	no	—
151	C₄×C₁₇	—	4	1	no	—
152	C₄×C₁₇	—	4	1	no	—
153	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
154	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
155	Dic₁₇	—	2	1	no	—
156	Dic₁₇	—	2	1	no	—
157	Dic₁₇	—	2	1	no	—
158	Dic₁₇	—	4	1	no	—
159	Dic₁₇	—	4	1	no	—
160	Dic₁₇	—	4	1	no	—
161	Dic₁₇	—	4	1	no	—
162	#542	—	17	1	no	—
163	#541	—	17	1	no	—
164	#501	—	17	1	no	—
165	#497	—	17	1	no	—
166	#488	—	17	1	no	—
167	#487	—	17	1	no	—
168	#470	—	17	1	no	—
169	#468	—	17	1	no	—
170	#467	—	17	1	no	—
171	#465	—	17	1	no	—
172	#441	—	17	1	no	—
173	#440	—	17	1	no	—
174	C₄×Q₁₆	—	17	1	no	—
175	C₄×SD₁₆	D₁₇	1	1	yes	characteristic, complemented
176	#405	—	17	1	no	—
177	C₂×C₁₇	—	8	1	no	—
178	C₂×C₁₇	#532	1	1	yes	characteristic
179	C₂×C₁₇	#577	1	1	yes	characteristic
180	C₂×C₁₇	#578	1	1	yes	characteristic
181	D₁₇	—	4	1	no	—
182	C₂×Q₁₆	—	17	1	no	—
183	C₂×Q₁₆	—	17	1	no	—
184	C₂×SD₁₆	—	17	1	no	—
185	C₂×SD₁₆	C₂×D₁₇	1	1	yes	characteristic
186	C₂×M₁₆	—	17	1	no	—
187	C₂²×C₈	—	17	1	no	—
188	C₄⋊Q₈	—	17	1	no	—
189	C₄⋊Q₈	—	17	1	no	—
190	#126	—	34	1	no	—
191	#126	—	34	1	no	—
192	#126	—	34	1	no	—
193	#125	—	17	1	no	—
194	#125	—	34	1	no	—
195	#125	—	34	1	no	—
196	#124	—	17	1	no	—
197	#124	—	34	1	no	—
198	#123	—	34	1	no	—
199	C₂²⋊Q₈	—	17	1	no	—
200	C₂²⋊Q₈	—	17	1	no	—
201	C₂²⋊Q₈	—	17	1	no	—
202	C₂²⋊Q₈	—	34	1	no	—
203	#121	—	17	1	no	—
204	C₄×Q₈	—	17	1	no	—
205	C₄×Q₈	—	17	1	no	—
206	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
207	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
208	#117	—	17	1	no	—
209	#107	—	17	1	no	—
210	#107	—	17	1	no	—
211	#106	—	17	1	no	—
212	#106	C₂×D₁₇	1	1	yes	characteristic
213	#103	—	17	1	no	—
214	#103	—	17	1	no	—
215	#103	—	17	1	no	—
216	#103	—	17	1	no	—
217	#103	—	17	1	no	—
218	#103	C₂×D₁₇	1	1	yes	characteristic
219	#102	—	17	1	no	—
220	#102	C₂×D₁₇	1	1	yes	characteristic
221	#97	—	17	1	no	—
222	#97	—	17	1	no	—
223	C₄×C₈	—	17	1	no	—
224	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
225	C₁₇	#3178	1	1	yes	characteristic, complemented, Sylow
226	C₂×Q₈	—	17	1	no	—
227	C₂×Q₈	—	17	1	no	—
228	C₂×Q₈	—	34	1	no	—
229	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
230	C₂×D₄	—	34	1	no	—
231	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	17	1	no	—
234	Q₁₆	—	34	2	no	—
235	Q₁₆	—	34	2	no	—
236	SD₁₆	—	2	2	no	—
237	SD₁₆	—	34	2	no	—
238	M₁₆	—	34	2	no	—
239	C₂×C₈	—	17	1	no	—
240	C₂×C₈	—	17	1	no	—
241	C₂×C₈	—	34	2	no	—
242	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
244	C₄⋊C₄	—	2	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	—	34	1	no	—
257	C₄⋊C₄	—	34	1	no	—
258	C₄⋊C₄	—	34	1	no	—
259	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	17	1	no	—
263	C₂²⋊C₄	—	17	1	no	—
264	C₂²⋊C₄	—	34	1	no	—
265	C₂²⋊C₄	—	34	1	no	—
266	C₂²⋊C₄	—	34	1	no	—
267	C₂²⋊C₄	—	34	1	no	—
268	C₂²⋊C₄	—	34	1	no	—
269	C₂²⋊C₄	—	34	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	17	1	no	—
272	C₄²	—	34	1	no	—
273	C₄²	—	34	1	no	—
274	C₄²	C₂²×D₁₇	1	1	yes	characteristic
275	C₂³	—	2	1	no	—
276	C₂³	—	17	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	34	1	no	—
280	Q₈	—	34	1	no	—
281	Q₈	—	34	1	no	—
282	Q₈	—	34	1	no	—
283	Q₈	—	136	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	136	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	17	1	no	—
292	C₂×C₄	—	17	1	no	—
293	C₂×C₄	—	17	1	no	—
294	C₂×C₄	—	34	1	no	—
295	C₂×C₄	—	34	1	no	—
296	C₂×C₄	—	34	1	no	—
297	C₂×C₄	—	34	1	no	—
298	C₂×C₄	—	34	1	no	—
299	C₂×C₄	—	34	1	no	—
300	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
301	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
302	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
303	C₈	—	2	1	no	—
304	C₈	—	17	2	no	—
305	C₈	—	34	1	no	—
306	C₈	#63252	1	2	yes	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	34	1	no	—
311	C₂²	—	34	1	no	—
312	C₂²	—	34	1	no	—
313	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	34	1	no	—
320	C₄	—	34	1	no	—
321	C₄	—	34	1	no	—
322	C₄	—	68	1	no	—
323	C₄	—	68	1	no	—
324	C₄	—	68	1	no	—
325	C₄	—	68	1	no	—
326	C₄	C₂×#63252	1	1	yes	characteristic
327	C₄	#41×D₁₇	1	1	yes	characteristic
328	C₂	—	8	1	no	—
329	C₂	—	68	1	no	—
330	C₂	#11763356	1	1	yes	characteristic
331	C₂	#11763615	1	1	yes	characteristic
332	C₂	#11763623	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₄×SD₁₆×C₁₇	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#10587624	C₂²
hypercenter	C₄×SD₁₆	D₁₇
hypocenter	C₁₇	#3178
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
2. #11763417	C₂	semidirect
3. #11763216	C₂	semidirect
4. #11763200	C₂	semidirect
5. #11763162	C₂	semidirect
6. #11763158	C₂	semidirect
7. #11763068	C₂	semidirect
9. #11763049	C₂	semidirect
11. #11762914	C₂	semidirect
12. #11762909	C₂	semidirect
13. C₄×Dic₆₈	C₂	semidirect
14. #11762763	C₂	semidirect
15. C₄×SD₁₆×C₁₇	C₂	semidirect
175. C₄×SD₁₆	D₁₇	semidirect
225. C₁₇	#3178	semidirect

Maximal subgroups

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

subgroup	index
#11763424	2
#11763417	2
#11763216	2
#11763200	2
#11763162	2
#11763158	2
#11763068	2
#11763052	2
#11763049	2
#11763037	2
#11762914	2
#11762909	2
C₄×Dic₆₈	2
#11762763	2
C₄×SD₁₆×C₁₇	2
#3178	17