Subgroups — #14541656

Order 2176.

338 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763536	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763515	C₂	1	1	yes	maximal, characteristic
3	#11763209	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763206	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763115	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763113	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763112	C₂	1	1	yes	maximal, characteristic
8	#11763106	C₂	1	1	yes	maximal, characteristic
9	#11763103	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763088	C₂	1	1	yes	maximal, characteristic
11	C₈×Dic₃₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#63219	C₂	1	1	yes	maximal, characteristic
13	#11762896	C₂	1	1	yes	maximal, characteristic
14	#11762896	C₂	1	1	yes	maximal, characteristic
15	C₁₇×#499	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₇×#125	C₂²	1	1	yes	characteristic
17	C₁₇×#125	C₂²	1	1	yes	characteristic
18	C₂×#63219	C₂²	1	1	yes	characteristic
19	#10587583	—	2	1	no	—
20	#10587582	—	2	1	no	—
21	#10587582	—	2	1	no	—
22	#10587579	C₂²	1	1	yes	characteristic
23	#10587579	C₂²	1	1	yes	characteristic
24	C₄⋊C₄×D₁₇	—	2	1	no	—
25	#10587576	C₂²	1	1	yes	characteristic
26	#10587576	C₂²	1	1	yes	characteristic
27	#10587575	—	2	1	no	—
28	#10587575	—	2	1	no	—
29	#10587574	—	2	1	no	—
30	#10587574	C₂²	1	1	yes	characteristic
31	#10587574	C₂²	1	1	yes	characteristic
32	#10587573	—	2	1	no	—
33	#10587573	C₂²	1	1	yes	characteristic
34	#10587573	C₂²	1	1	yes	characteristic
35	C₄×D₆₈	C₂²	1	1	yes	characteristic
36	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
37	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
38	C₁₇×#107	C₂²	1	1	yes	characteristic
39	C₁₇×#107	C₂²	1	1	yes	characteristic
40	C₁₇×#106	C₂²	1	1	yes	characteristic
41	C₁₇×#106	C₂²	1	1	yes	characteristic
42	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
43	#10587492	C₂²	1	1	yes	characteristic
44	#10587488	C₂²	1	1	yes	characteristic
45	#10587487	C₂²	1	1	yes	characteristic
46	#10587485	C₂²	1	1	yes	characteristic
47	#10587485	C₂²	1	1	yes	characteristic
48	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
49	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
50	#10587481	C₂²	1	1	yes	characteristic
51	#10587481	C₂²	1	1	yes	characteristic
52	#10587480	C₂²	1	1	yes	characteristic
53	#10587480	C₂²	1	1	yes	characteristic
54	#10587479	C₂²	1	1	yes	characteristic
55	#10587479	C₂²	1	1	yes	characteristic
56	#10587478	C₂²	1	1	yes	characteristic
57	#10587478	C₂²	1	1	yes	characteristic
58	C₄⋊#3622	C₂²	1	1	yes	characteristic
59	C₂×D₆₈	C₂³	1	1	yes	characteristic
60	C₂×C₄×D₁₇	—	2	1	no	—
61	C₂×C₄×D₁₇	—	2	1	no	—
62	C₂×C₄×D₁₇	—	2	1	no	—
63	C₂×Dic₃₄	—	2	1	no	—
64	C₂×Dic₃₄	—	2	1	no	—
65	C₂×Dic₃₄	D₄	1	1	yes	characteristic
66	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
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	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	C₄⋊Dic₁₇	—	2	1	no	—
82	C₄⋊Dic₁₇	—	2	1	no	—
83	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
84	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
86	#63225	—	2	1	no	—
87	#63225	—	2	1	no	—
88	#63225	—	2	1	no	—
89	#63225	—	2	1	no	—
90	#63225	—	2	1	no	—
91	#63225	—	2	1	no	—
92	#63225	—	2	1	no	—
93	#63225	D₄	1	1	yes	characteristic
94	#63225	D₄	1	1	yes	characteristic
95	C₄×Dic₁₇	—	2	1	no	—
96	C₄×Dic₁₇	—	2	1	no	—
97	C₄×Dic₁₇	—	2	1	no	—
98	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
99	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
100	C₂×#3622	C₂³	1	1	yes	characteristic
101	C₂×#3622	C₂³	1	1	yes	characteristic
102	#63219	Q₈	1	4	yes	complemented
103	C₂²×D₁₇	—	2	1	no	—
104	C₂×C₄×C₁₇	—	2	1	no	—
105	C₂×C₄×C₁₇	—	2	1	no	—
106	C₂×C₄×C₁₇	—	2	1	no	—
107	C₂×C₄×C₁₇	—	2	1	no	—
108	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
111	C₂×Dic₁₇	—	2	1	no	—
112	C₂×Dic₁₇	—	2	1	no	—
113	C₂×Dic₁₇	—	2	1	no	—
114	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
115	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
118	D₆₈	—	2	1	no	—
119	D₆₈	C₂×Q₈	1	2	yes	—
120	C₄×D₁₇	—	4	2	no	—
121	C₄×D₁₇	—	8	1	no	—
122	C₄×D₁₇	—	8	1	no	—
123	Dic₃₄	—	2	1	no	—
124	Dic₃₄	—	2	2	no	—
125	Dic₃₄	—	4	2	no	—
126	Dic₃₄	—	8	1	no	—
127	Dic₃₄	C₂×Q₈	1	2	yes	—
128	C₈×C₁₇	—	2	1	no	—
129	C₈×C₁₇	C₂×Q₈	1	2	yes	—
130	#3622	—	2	1	no	—
131	#3622	—	2	1	no	—
132	#3381	—	17	1	no	maximal, Sylow
133	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
134	C₂×D₁₇	—	2	2	no	—
135	C₂×D₁₇	—	4	1	no	—
136	C₂×D₁₇	—	4	1	no	—
137	C₄×C₁₇	—	2	1	no	—
138	C₄×C₁₇	—	2	1	no	—
139	C₄×C₁₇	—	4	1	no	—
140	C₄×C₁₇	—	4	1	no	—
141	C₄×C₁₇	—	4	1	no	—
142	C₄×C₁₇	—	4	1	no	—
143	C₄×C₁₇	C₂²×Q₈	1	1	yes	characteristic
144	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
145	Dic₁₇	—	2	2	no	—
146	Dic₁₇	—	2	1	no	—
147	Dic₁₇	—	2	1	no	—
148	Dic₁₇	—	2	1	no	—
149	Dic₁₇	—	4	1	no	—
150	Dic₁₇	—	4	1	no	—
151	Dic₁₇	#135	1	2	yes	—
152	#557	—	17	1	no	—
153	#554	—	17	1	no	—
154	#499	—	17	1	no	—
155	#499	D₁₇	1	1	yes	characteristic, complemented
156	#498	—	17	1	no	—
157	#479	—	17	1	no	—
158	#479	—	17	1	no	—
159	#478	—	17	1	no	—
160	#478	—	17	1	no	—
161	#477	—	17	1	no	—
162	#474	—	17	1	no	—
163	C₈×Q₈	—	17	1	no	—
164	C₄×SD₁₆	—	17	1	no	—
165	C₄×SD₁₆	—	17	1	no	—
166	C₄×SD₁₆	—	17	1	no	—
167	C₂×C₁₇	D₄×Q₈	1	1	yes	characteristic
168	C₂×C₁₇	C₂×#135	1	1	yes	characteristic
169	C₂×C₁₇	#577	1	1	yes	characteristic
170	D₁₇	—	4	2	no	—
171	C₂×SD₁₆	—	17	1	no	—
172	C₄⋊Q₈	—	17	1	no	—
173	C₄⋊Q₈	—	17	1	no	—
174	C₄⋊Q₈	—	34	1	no	—
175	#125	—	17	1	no	—
176	#125	—	17	1	no	—
177	#125	—	34	1	no	—
178	#125	—	34	1	no	—
179	#125	C₂×D₁₇	1	1	yes	characteristic
180	#125	C₂×D₁₇	1	1	yes	characteristic
181	C₂²⋊Q₈	—	34	1	no	—
182	C₂²⋊Q₈	—	34	1	no	—
183	C₂²⋊Q₈	—	34	1	no	—
184	C₄×Q₈	—	17	1	no	—
185	C₄×Q₈	—	17	1	no	—
186	C₄×Q₈	—	17	1	no	—
187	C₄×Q₈	—	17	1	no	—
188	C₄×Q₈	—	34	1	no	—
189	C₄×D₄	—	17	1	no	—
190	C₄×D₄	—	17	1	no	—
191	C₄×D₄	—	17	1	no	—
192	C₂×C₄⋊C₄	—	34	1	no	—
193	#107	—	17	1	no	—
194	#107	—	17	1	no	—
195	#107	C₂×D₁₇	1	1	yes	characteristic
196	#107	C₂×D₁₇	1	1	yes	characteristic
197	#106	—	17	1	no	—
198	#106	—	17	1	no	—
199	#106	—	17	1	no	—
200	#106	C₂×D₁₇	1	1	yes	characteristic
201	#106	C₂×D₁₇	1	1	yes	characteristic
202	C₄⋊C₈	—	17	1	no	—
203	C₄⋊C₈	—	17	1	no	—
204	C₄⋊C₈	—	17	1	no	—
205	#103	—	17	1	no	—
206	#103	—	17	1	no	—
207	#103	—	17	1	no	—
208	#102	—	17	1	no	—
209	#102	—	17	1	no	—
210	#102	—	17	1	no	—
211	C₄×C₈	—	17	1	no	—
212	C₄×C₈	—	17	1	no	—
213	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
214	C₁₇	#3381	1	1	yes	characteristic, complemented, Sylow
215	C₂×Q₈	—	17	1	no	—
216	C₂×Q₈	—	17	1	no	—
217	C₂×Q₈	—	34	1	no	—
218	C₂×Q₈	—	34	1	no	—
219	C₂×D₄	—	17	1	no	—
220	C₂²×C₄	—	34	1	no	—
221	C₂²×C₄	—	34	1	no	—
222	C₂²×C₄	—	34	1	no	—
223	SD₁₆	—	17	4	no	—
224	C₂×C₈	—	17	1	no	—
225	C₂×C₈	—	17	1	no	—
226	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
227	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	17	1	no	—
233	C₄⋊C₄	—	17	1	no	—
234	C₄⋊C₄	—	17	1	no	—
235	C₄⋊C₄	—	17	1	no	—
236	C₄⋊C₄	—	17	1	no	—
237	C₄⋊C₄	—	34	1	no	—
238	C₄⋊C₄	—	34	1	no	—
239	C₄⋊C₄	—	34	1	no	—
240	C₄⋊C₄	—	34	1	no	—
241	C₄⋊C₄	—	34	1	no	—
242	C₄⋊C₄	—	34	1	no	—
243	C₄⋊C₄	—	34	1	no	—
244	C₄⋊C₄	—	34	1	no	—
245	C₄⋊C₄	—	34	1	no	—
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	—	34	1	no	—
251	C₂²⋊C₄	—	34	1	no	—
252	C₂²⋊C₄	—	34	1	no	—
253	C₄²	—	17	1	no	—
254	C₄²	—	17	1	no	—
255	C₄²	—	34	1	no	—
256	C₄²	—	34	1	no	—
257	C₄²	—	34	1	no	—
258	C₄²	C₂²×D₁₇	1	1	yes	characteristic
259	C₂³	—	34	1	no	—
260	Q₈	—	17	2	no	—
261	Q₈	—	34	2	no	—
262	Q₈	—	34	1	no	—
263	Q₈	—	68	2	no	—
264	Q₈	—	136	1	no	—
265	D₄	—	17	2	no	—
266	D₄	—	34	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	17	1	no	—
272	C₂×C₄	—	17	1	no	—
273	C₂×C₄	—	17	1	no	—
274	C₂×C₄	—	17	1	no	—
275	C₂×C₄	—	34	1	no	—
276	C₂×C₄	—	34	1	no	—
277	C₂×C₄	—	34	1	no	—
278	C₂×C₄	—	68	2	no	—
279	C₂×C₄	—	136	1	no	—
280	C₂×C₄	—	136	1	no	—
281	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
282	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
283	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	34	1	no	—
286	C₈	—	34	1	no	—
287	C₈	Q₈×D₁₇	1	2	yes	—
288	C₂²	—	34	2	no	—
289	C₂²	—	68	1	no	—
290	C₂²	—	68	1	no	—
291	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	—	4	1	no	—
298	C₄	—	17	2	no	—
299	C₄	—	34	2	no	—
300	C₄	—	34	1	no	—
301	C₄	—	34	1	no	—
302	C₄	—	34	1	no	—
303	C₄	—	68	1	no	—
304	C₄	—	68	1	no	—
305	C₄	C₂×Q₈×D₁₇	1	1	yes	characteristic
306	C₄	#10587694	1	1	yes	characteristic
307	C₂	—	68	2	no	—
308	C₂	#11763474	1	1	yes	characteristic
309	C₂	#135×D₁₇	1	1	yes	characteristic
310	C₂	#11763616	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×#499	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×D₆₈	C₂³
hypercenter	#499	D₁₇
hypocenter	C₁₇	#3381
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

Complements

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

N	H ≅ G/N	type
1. #11763536	C₂	semidirect
3. #11763209	C₂	semidirect
4. #11763206	C₂	semidirect
5. #11763115	C₂	semidirect
6. #11763113	C₂	semidirect
9. #11763103	C₂	semidirect
11. C₈×Dic₃₄	C₂	semidirect
15. C₁₇×#499	C₂	semidirect
102. #63219	Q₈	semidirect
155. #499	D₁₇	semidirect
214. C₁₇	#3381	semidirect

Maximal subgroups

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

subgroup	index
#11763536	2
#11763515	2
#11763209	2
#11763206	2
#11763115	2
#11763113	2
#11763112	2
#11763106	2
#11763103	2
#11763088	2
C₈×Dic₃₄	2
C₄×#63219	2
#11762896	2
C₁₇×#499	2
#3381	17