Subgroups — #14540236

Order 2176.

322 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763420	C₂	1	1	yes	maximal, characteristic
2	#11763417	C₂	1	1	yes	maximal, characteristic
3	#11763118	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763117	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763116	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763114	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763066	C₂	1	1	yes	maximal, characteristic
8	#11763064	C₂	1	1	yes	maximal, characteristic
9	#11763049	C₂	1	1	yes	maximal, characteristic
10	#11763047	C₂	1	1	yes	maximal, characteristic
11	#11763017	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763017	C₂	1	1	yes	maximal, characteristic, complemented
13	#11763016	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762834	C₂	1	1	yes	maximal, characteristic
15	C₁₇×#462	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
20	#10587632	C₂²	1	1	yes	characteristic
21	#10587632	C₂²	1	1	yes	characteristic
22	#10587632	C₂²	1	1	yes	characteristic
23	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
24	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
25	#10587630	C₂²	1	1	yes	characteristic
26	C₂×#63231	C₂²	1	1	yes	characteristic
27	C₂×#63231	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	#10587584	—	2	1	no	—
32	#10587584	—	2	1	no	—
33	#10587583	C₂²	1	1	yes	characteristic
34	#10587576	C₂²	1	1	yes	characteristic
35	#10587575	—	2	1	no	—
36	#10587575	—	2	1	no	—
37	#10587575	—	2	1	no	—
38	#10587562	—	2	1	no	—
39	#10587558	C₂²	1	1	yes	characteristic
40	#10587556	—	2	1	no	—
41	#10587556	C₂²	1	1	yes	characteristic
42	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
43	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₁₇×#103	C₂²	1	1	yes	characteristic
45	C₁₇×#103	C₂²	1	1	yes	characteristic
46	#10587506	C₂²	1	1	yes	characteristic
47	#10587506	C₂²	1	1	yes	characteristic
48	#10587506	C₂²	1	1	yes	characteristic
49	#10587506	C₂²	1	1	yes	characteristic
50	#10587489	C₂²	1	1	yes	characteristic
51	#10587488	C₂²	1	1	yes	characteristic
52	#10587487	C₂²	1	1	yes	characteristic
53	#10587487	C₂²	1	1	yes	characteristic
54	#10587485	C₂²	1	1	yes	characteristic
55	#10587485	C₂²	1	1	yes	characteristic
56	#10587479	C₂²	1	1	yes	characteristic
57	#10587478	C₂²	1	1	yes	characteristic
58	C₄⋊#3622	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
63	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
64	Q₁₆×C₁₇	—	4	1	no	—
65	Q₁₆×C₁₇	—	4	1	no	—
66	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₇	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	C₂³	1	1	yes	characteristic
72	C₄²×C₁₇	—	2	1	no	—
73	C₄²×C₁₇	C₂³	1	1	yes	characteristic
74	#63231	—	4	1	no	—
75	#63231	—	4	1	no	—
76	#63227	—	2	1	no	—
77	#63227	—	2	1	no	—
78	#63227	—	2	1	no	—
79	#63227	—	2	1	no	—
80	#63227	D₄	1	1	yes	characteristic
81	#63227	D₄	1	1	yes	characteristic
82	C₄⋊Dic₁₇	—	2	1	no	—
83	C₄⋊Dic₁₇	—	2	1	no	—
84	C₄⋊Dic₁₇	—	2	1	no	—
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	—	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	—	2	1	no	—
103	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
104	C₂×#3622	C₂³	1	1	yes	characteristic
105	C₂×#3622	C₂³	1	1	yes	characteristic
106	#63218	—	4	1	no	—
107	C₈×D₁₇	—	4	1	no	—
108	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
109	Q₈×C₁₇	—	2	1	no	—
110	Q₈×C₁₇	—	2	1	no	—
111	Q₈×C₁₇	—	2	1	no	—
112	Q₈×C₁₇	—	2	1	no	—
113	Q₈×C₁₇	—	2	1	no	—
114	Q₈×C₁₇	—	2	1	no	—
115	C₂×C₄×C₁₇	—	2	1	no	—
116	C₂×C₄×C₁₇	—	2	1	no	—
117	C₂×C₄×C₁₇	—	2	1	no	—
118	C₂×C₄×C₁₇	—	2	1	no	—
119	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
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	C₈×C₁₇	—	2	1	no	—
134	C₈×C₁₇	—	2	1	no	—
135	#3622	—	2	1	no	—
136	#3622	—	2	1	no	—
137	#3261	—	17	1	no	maximal, Sylow
138	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₁₇	—	2	1	no	—
140	C₂×D₁₇	—	2	1	no	—
141	C₂×D₁₇	—	2	1	no	—
142	C₄×C₁₇	—	2	1	no	—
143	C₄×C₁₇	—	2	1	no	—
144	C₄×C₁₇	—	4	1	no	—
145	C₄×C₁₇	—	4	1	no	—
146	C₄×C₁₇	—	4	1	no	—
147	C₄×C₁₇	—	4	1	no	—
148	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
149	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
150	Dic₁₇	—	2	1	no	—
151	Dic₁₇	—	2	1	no	—
152	Dic₁₇	—	2	1	no	—
153	Dic₁₇	—	4	1	no	—
154	Dic₁₇	—	4	1	no	—
155	Dic₁₇	—	4	1	no	—
156	Dic₁₇	—	4	1	no	—
157	#541	—	17	1	no	—
158	#541	—	17	1	no	—
159	#479	—	17	1	no	—
160	#479	—	17	1	no	—
161	#479	—	17	1	no	—
162	#479	—	17	1	no	—
163	#470	—	17	1	no	—
164	#470	—	17	1	no	—
165	#467	—	17	1	no	—
166	#467	—	17	1	no	—
167	#462	—	17	1	no	—
168	#462	—	17	1	no	—
169	#462	—	17	1	no	—
170	#462	D₁₇	1	1	yes	characteristic, complemented
171	#424	—	17	1	no	—
172	C₂×C₁₇	#537	1	1	yes	characteristic
173	C₂×C₁₇	#578	1	1	yes	characteristic
174	C₂×C₁₇	#578	1	1	yes	characteristic
175	D₁₇	—	4	1	no	—
176	C₂×Q₁₆	—	17	1	no	—
177	C₂×Q₁₆	—	17	1	no	—
178	C₂×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
179	C₂×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
180	C₂×M₁₆	—	17	1	no	—
181	C₂²×C₈	—	17	1	no	—
182	C₄⋊Q₈	—	17	1	no	—
183	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
184	#126	—	34	1	no	—
185	#126	—	34	1	no	—
186	#126	—	34	1	no	—
187	#126	—	34	1	no	—
188	#125	—	17	1	no	—
189	#125	—	17	1	no	—
190	#125	—	34	1	no	—
191	#125	—	34	1	no	—
192	#125	—	34	1	no	—
193	#125	—	34	1	no	—
194	C₂²⋊Q₈	—	17	1	no	—
195	C₂²⋊Q₈	—	17	1	no	—
196	C₂²⋊Q₈	—	17	1	no	—
197	C₂²⋊Q₈	—	17	1	no	—
198	C₄×Q₈	—	17	1	no	—
199	C₄×Q₈	—	17	1	no	—
200	C₄×Q₈	—	17	1	no	—
201	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
202	#117	—	17	1	no	—
203	#107	—	17	1	no	—
204	#107	—	17	1	no	—
205	#106	—	17	1	no	—
206	#106	—	17	1	no	—
207	C₄⋊C₈	—	17	1	no	—
208	C₄⋊C₈	—	17	1	no	—
209	C₄⋊C₈	—	17	1	no	—
210	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
211	#103	—	17	1	no	—
212	#103	—	17	1	no	—
213	#103	—	17	1	no	—
214	#103	—	17	1	no	—
215	#103	—	17	1	no	—
216	#103	—	17	1	no	—
217	#103	C₂×D₁₇	1	1	yes	characteristic
218	#103	C₂×D₁₇	1	1	yes	characteristic
219	C₁₇	#3261	1	1	yes	characteristic, complemented, Sylow
220	C₂×Q₈	—	17	1	no	—
221	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
222	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
223	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
224	C₂²×C₄	—	17	1	no	—
225	Q₁₆	—	4	1	no	—
226	Q₁₆	—	4	1	no	—
227	Q₁₆	—	68	1	no	—
228	Q₁₆	—	68	1	no	—
229	M₁₆	—	68	1	no	—
230	C₂×C₈	—	17	1	no	—
231	C₂×C₈	—	17	1	no	—
232	C₂×C₈	—	68	1	no	—
233	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	17	1	no	—
239	C₄⋊C₄	—	17	1	no	—
240	C₄⋊C₄	—	17	1	no	—
241	C₄⋊C₄	—	17	1	no	—
242	C₄⋊C₄	—	17	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	—	34	1	no	—
248	C₄⋊C₄	—	34	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	C₂²×D₁₇	1	1	yes	characteristic
257	C₂²⋊C₄	—	17	1	no	—
258	C₂²⋊C₄	—	17	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	C₄²	—	2	1	no	—
264	C₄²	—	17	1	no	—
265	C₄²	—	34	1	no	—
266	C₄²	—	34	1	no	—
267	C₄²	—	34	1	no	—
268	C₄²	C₂²×D₁₇	1	1	yes	characteristic
269	C₂³	—	17	1	no	—
270	Q₈	—	2	1	no	—
271	Q₈	—	2	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	2	1	no	—
276	Q₈	—	34	1	no	—
277	Q₈	—	34	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	17	1	no	—
283	C₂×C₄	—	17	1	no	—
284	C₂×C₄	—	17	1	no	—
285	C₂×C₄	—	34	1	no	—
286	C₂×C₄	—	34	1	no	—
287	C₂×C₄	—	34	1	no	—
288	C₂×C₄	—	34	1	no	—
289	C₂×C₄	—	34	1	no	—
290	C₂×C₄	—	34	1	no	—
291	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
292	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	34	1	no	—
297	C₈	—	34	1	no	—
298	C₂²	—	34	1	no	—
299	C₂²	—	34	1	no	—
300	C₂²	—	34	1	no	—
301	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	34	1	no	—
309	C₄	—	34	1	no	—
310	C₄	—	34	1	no	—
311	C₄	—	68	1	no	—
312	C₄	—	68	1	no	—
313	C₄	—	68	1	no	—
314	C₄	—	68	1	no	—
315	C₄	#10587691	1	1	yes	characteristic
316	C₄	#10587697	1	1	yes	characteristic
317	C₂	—	68	1	no	—
318	C₂	#11763388	1	1	yes	characteristic
319	C₂	#11763622	1	1	yes	characteristic
320	C₂	#11763625	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×#462	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂²×D₁₇	C₂×D₄
hypercenter	#462	D₁₇
hypocenter	C₁₇	#3261
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

Complements

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

N	H ≅ G/N	type
3. #11763118	C₂	semidirect
4. #11763117	C₂	semidirect
5. #11763116	C₂	semidirect
6. #11763114	C₂	semidirect
11. #11763017	C₂	semidirect
12. #11763017	C₂	semidirect
13. #11763016	C₂	semidirect
15. C₁₇×#462	C₂	semidirect
170. #462	D₁₇	semidirect
219. C₁₇	#3261	semidirect

Maximal subgroups

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

subgroup	index
#11763420	2
#11763417	2
#11763118	2
#11763117	2
#11763116	2
#11763114	2
#11763066	2
#11763064	2
#11763049	2
#11763047	2
#11763017	2
#11763016	2
#11762834	2
C₁₇×#462	2
#3261	17