Subgroups — #14541238

Order 2176.

344 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763520	C₂	1	1	yes	maximal, characteristic
2	#11763497	C₂	1	1	yes	maximal, characteristic
3	#11763152	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763146	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763141	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763141	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763113	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763091	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763068	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763051	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762917	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762882	C₂	1	1	yes	maximal, characteristic
13	#11762850	C₂	1	1	yes	maximal, characteristic, complemented
14	#106×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#501	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₇×#125	C₂²	1	1	yes	characteristic
18	#10587632	—	2	1	no	—
19	#10587632	—	2	1	no	—
20	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
21	C₂×#63218	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
23	#10587584	—	2	1	no	—
24	#10587583	C₂²	1	1	yes	characteristic
25	#10587583	C₂²	1	1	yes	characteristic
26	#10587583	C₂²	1	1	yes	characteristic
27	#10587583	C₂²	1	1	yes	characteristic
28	#10587580	—	2	1	no	—
29	#10587579	—	2	1	no	—
30	#10587578	—	2	1	no	—
31	#10587578	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
35	#10587576	—	2	1	no	—
36	#10587576	—	2	1	no	—
37	C₄×D₆₈	C₂²	1	1	yes	characteristic
38	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
40	C₁₇×#107	C₂²	1	1	yes	characteristic
41	C₁₇×#107	C₂²	1	1	yes	characteristic
42	C₁₇×#106	C₂²	1	1	yes	characteristic
43	C₁₇×#106	C₂²	1	1	yes	characteristic
44	C₁₇×#97	C₂²	1	1	yes	characteristic
45	#10587491	C₂²	1	1	yes	characteristic
46	#10587491	C₂²	1	1	yes	characteristic
47	#10587488	C₂²	1	1	yes	characteristic
48	#10587487	C₂²	1	1	yes	characteristic
49	#10587487	C₂²	1	1	yes	characteristic
50	#10587481	C₂²	1	1	yes	characteristic
51	#10587481	C₂²	1	1	yes	characteristic
52	#10587481	C₂²	1	1	yes	characteristic
53	#10587481	C₂²	1	1	yes	characteristic
54	#10587479	C₂²	1	1	yes	characteristic
55	#10587479	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	—	2	1	no	—
60	C₂×D₆₈	D₄	1	1	yes	characteristic
61	C₂×C₄×D₁₇	—	2	1	no	—
62	C₂×C₄×D₁₇	—	2	1	no	—
63	C₂×C₄×D₁₇	—	2	1	no	—
64	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
66	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
67	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
73	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
77	C₄²×C₁₇	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
87	C₄⋊Dic₁₇	—	2	1	no	—
88	C₄⋊Dic₁₇	—	2	1	no	—
89	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
90	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
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	C₄×Dic₁₇	—	2	1	no	—
98	C₄×Dic₁₇	—	2	1	no	—
99	C₄×Dic₁₇	—	2	1	no	—
100	C₂×#3622	C₂³	1	1	yes	characteristic
101	C₂×#3622	C₂³	1	1	yes	characteristic
102	Dic₆₈	—	2	2	no	—
103	#63218	—	2	2	no	—
104	C₈×D₁₇	—	2	2	no	—
105	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
106	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
107	Q₈×C₁₇	—	8	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	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₁₇	—	2	1	no	—
116	C₂×Dic₁₇	—	2	1	no	—
117	C₂×Dic₁₇	—	2	1	no	—
118	C₂×Dic₁₇	—	2	1	no	—
119	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
121	D₆₈	—	2	2	no	—
122	C₄×D₁₇	—	2	1	no	—
123	C₄×D₁₇	—	2	1	no	—
124	C₄×D₁₇	—	2	1	no	—
125	C₄×D₁₇	—	2	1	no	—
126	C₄×D₁₇	—	4	2	no	—
127	C₄×D₁₇	—	8	1	no	—
128	C₄×D₁₇	—	8	1	no	—
129	Dic₃₄	—	2	1	no	—
130	Dic₃₄	—	2	1	no	—
131	Dic₃₄	—	2	1	no	—
132	Dic₃₄	—	2	1	no	—
133	C₈×C₁₇	—	2	1	no	—
134	C₈×C₁₇	#41	1	2	yes	—
135	#3622	—	2	1	no	—
136	#3622	—	2	1	no	—
137	#3344	—	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	C₂×D₁₇	—	2	1	no	—
143	C₂×D₁₇	—	2	1	no	—
144	C₂×D₁₇	#137	1	2	yes	—
145	C₄×C₁₇	—	2	1	no	—
146	C₄×C₁₇	—	2	1	no	—
147	C₄×C₁₇	—	4	1	no	—
148	C₄×C₁₇	—	4	1	no	—
149	C₄×C₁₇	—	4	1	no	—
150	C₄×C₁₇	—	4	1	no	—
151	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
152	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
153	Dic₁₇	—	2	1	no	—
154	Dic₁₇	—	2	1	no	—
155	Dic₁₇	—	4	1	no	—
156	Dic₁₇	—	4	1	no	—
157	Dic₁₇	—	4	1	no	—
158	Dic₁₇	—	4	1	no	—
159	#554	—	17	1	no	—
160	#552	—	17	1	no	—
161	#501	D₁₇	1	1	yes	characteristic, complemented
162	#485	—	17	1	no	—
163	#484	—	17	1	no	—
164	#483	—	17	1	no	—
165	#483	—	17	1	no	—
166	#479	—	17	1	no	—
167	#475	—	17	1	no	—
168	#470	—	17	1	no	—
169	#467	—	17	1	no	—
170	#441	—	17	1	no	—
171	#435	—	17	1	no	—
172	#428	—	17	1	no	—
173	C₂×#106	—	17	1	no	—
174	C₂×C₁₇	#547	1	1	yes	characteristic
175	C₂×C₁₇	C₂×#137	1	1	yes	characteristic
176	C₂×C₁₇	#577	1	1	yes	characteristic
177	D₁₇	—	2	2	no	—
178	D₁₇	—	4	1	no	—
179	C₂×Q₁₆	—	17	1	no	—
180	C₂×M₁₆	—	17	1	no	—
181	C₂²×C₈	—	17	1	no	—
182	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
183	#126	—	34	1	no	—
184	#125	—	34	1	no	—
185	#125	—	34	1	no	—
186	#125	C₂×D₁₇	1	1	yes	characteristic
187	#123	—	34	1	no	—
188	C₂²⋊Q₈	—	17	1	no	—
189	C₂²⋊Q₈	—	17	1	no	—
190	C₂²⋊Q₈	—	17	1	no	—
191	C₂²⋊Q₈	—	17	1	no	—
192	C₂²⋊Q₈	—	34	1	no	—
193	C₂²⋊Q₈	—	34	1	no	—
194	C₄×Q₈	—	17	1	no	—
195	C₄×Q₈	—	17	1	no	—
196	C₄×D₄	—	17	1	no	—
197	C₄×D₄	—	34	1	no	—
198	#117	—	17	1	no	—
199	#117	—	34	1	no	—
200	C₂×C₄⋊C₄	—	17	1	no	—
201	C₂×C₄⋊C₄	—	17	1	no	—
202	C₂×C₄⋊C₄	—	17	1	no	—
203	#107	—	17	1	no	—
204	#107	C₂×D₁₇	1	1	yes	characteristic
205	#107	C₂×D₁₇	1	1	yes	characteristic
206	#106	—	17	1	no	—
207	#106	—	17	1	no	—
208	#106	—	17	1	no	—
209	#106	—	17	1	no	—
210	#106	C₂×D₁₇	1	1	yes	characteristic
211	#106	C₂×D₁₇	1	1	yes	characteristic
212	C₄⋊C₈	—	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	—	17	1	no	—
218	#103	—	17	1	no	—
219	C₂²⋊C₈	—	17	1	no	—
220	C₂²⋊C₈	—	17	1	no	—
221	#97	C₂×D₁₇	1	1	yes	characteristic
222	C₁₇	#3344	1	1	yes	characteristic, complemented, Sylow
223	C₂×Q₈	—	2	1	no	—
224	C₂×Q₈	—	17	1	no	—
225	C₂×Q₈	—	17	1	no	—
226	C₂×D₄	—	17	1	no	—
227	C₂²×C₄	—	17	1	no	—
228	C₂²×C₄	—	17	1	no	—
229	C₂²×C₄	—	34	1	no	—
230	C₂²×C₄	—	34	1	no	—
231	C₂²×C₄	—	34	1	no	—
232	Q₁₆	—	34	2	no	—
233	M₁₆	—	34	2	no	—
234	C₂×C₈	—	17	1	no	—
235	C₂×C₈	—	17	1	no	—
236	C₂×C₈	—	34	2	no	—
237	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	17	1	no	—
243	C₄⋊C₄	—	17	1	no	—
244	C₄⋊C₄	—	17	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	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	—	17	1	no	—
259	C₂²⋊C₄	—	17	1	no	—
260	C₂²⋊C₄	—	34	1	no	—
261	C₂²⋊C₄	—	34	1	no	—
262	C₂²⋊C₄	—	34	1	no	—
263	C₂²⋊C₄	—	34	1	no	—
264	C₂²⋊C₄	—	34	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	C₂³	—	17	1	no	—
271	Q₈	—	8	1	no	—
272	Q₈	—	34	1	no	—
273	Q₈	—	34	1	no	—
274	Q₈	—	34	1	no	—
275	Q₈	—	34	1	no	—
276	D₄	—	34	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	17	1	no	—
282	C₂×C₄	—	17	1	no	—
283	C₂×C₄	—	34	1	no	—
284	C₂×C₄	—	34	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	—	68	2	no	—
292	C₂×C₄	—	136	1	no	—
293	C₂×C₄	—	136	1	no	—
294	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
295	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
296	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
297	C₈	—	2	1	no	—
298	C₈	—	34	1	no	—
299	C₈	—	34	1	no	—
300	C₈	#63256	1	2	yes	—
301	C₂²	—	17	2	no	—
302	C₂²	—	34	1	no	—
303	C₂²	—	34	1	no	—
304	C₂²	—	34	1	no	—
305	C₂²	—	34	1	no	—
306	C₂²	—	34	1	no	—
307	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	34	1	no	—
315	C₄	—	34	1	no	—
316	C₄	—	68	1	no	—
317	C₄	—	68	1	no	—
318	C₄	—	68	1	no	—
319	C₄	—	68	1	no	—
320	C₄	C₂×#63256	1	1	yes	characteristic
321	C₄	#10587694	1	1	yes	characteristic
322	C₂	—	34	2	no	—
323	C₂	—	68	1	no	—
324	C₂	#11763462	1	1	yes	characteristic
325	C₂	#137×D₁₇	1	1	yes	characteristic
326	C₂	#11763619	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×#501	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×D₆₈	D₄
hypercenter	#501	D₁₇
hypocenter	C₁₇	#3344
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #501

Complements

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

N	H ≅ G/N	type
3. #11763152	C₂	semidirect
4. #11763146	C₂	semidirect
5. #11763141	C₂	semidirect
6. #11763141	C₂	semidirect
7. #11763113	C₂	semidirect
8. #11763091	C₂	semidirect
9. #11763068	C₂	semidirect
10. #11763051	C₂	semidirect
11. #11762917	C₂	semidirect
13. #11762850	C₂	semidirect
14. #106×D₁₇	C₂	semidirect
15. C₁₇×#501	C₂	semidirect
161. #501	D₁₇	semidirect
222. C₁₇	#3344	semidirect

Maximal subgroups

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

subgroup	index
#11763520	2
#11763497	2
#11763152	2
#11763146	2
#11763141	2
#11763113	2
#11763091	2
#11763068	2
#11763051	2
#11762917	2
#11762882	2
#11762850	2
#106×D₁₇	2
C₁₇×#501	2
#3344	17