Subgroups — #14541405

Order 2176.

354 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763523	C₂	1	1	yes	maximal, characteristic
2	#11763517	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763213	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763148	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763147	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763121	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763120	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763102	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763097	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763052	C₂	1	1	yes	maximal, characteristic, complemented
11	SD₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762885	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762851	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762850	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#468	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
19	C₁₇×#121	C₂²	1	1	yes	characteristic
20	#10587632	C₂²	1	1	yes	characteristic
21	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
22	#10587625	—	2	1	no	—
23	#10587624	C₂²	1	1	yes	characteristic
24	#10587622	—	2	1	no	—
25	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
26	C₄×C₂²⋊D₁₇	C₂²	1	1	yes	characteristic
27	#10587612	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
29	C₂×#63218	C₂²	1	1	yes	characteristic
30	#10587583	—	2	1	no	—
31	#10587578	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
33	#10587575	—	2	1	no	—
34	#10587574	C₂²	1	1	yes	characteristic
35	#10587574	C₂²	1	1	yes	characteristic
36	#10587571	—	2	1	no	—
37	#10587567	—	2	1	no	—
38	#10587564	—	2	1	no	—
39	#10587563	—	2	1	no	—
40	C₁₇×#107	C₂²	1	1	yes	characteristic
41	C₁₇×#103	C₂²	1	1	yes	characteristic
42	C₁₇×#102	C₂²	1	1	yes	characteristic
43	#10587506	C₂²	1	1	yes	characteristic
44	#10587506	C₂²	1	1	yes	characteristic
45	#10587503	C₂²	1	1	yes	characteristic
46	#10587503	C₂²	1	1	yes	characteristic
47	C₂²⋊#3622	C₂²	1	1	yes	characteristic
48	#10587491	C₂²	1	1	yes	characteristic
49	#10587489	C₂²	1	1	yes	characteristic
50	#10587489	C₂²	1	1	yes	characteristic
51	#10587488	C₂²	1	1	yes	characteristic
52	#10587488	C₂²	1	1	yes	characteristic
53	#10587485	C₂²	1	1	yes	characteristic
54	C₈×Dic₁₇	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	#10587478	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₇	—	2	1	no	—
61	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₇	—	2	1	no	—
64	C₂×C₂²⋊D₁₇	D₄	1	1	yes	characteristic
65	C₂²×Dic₁₇	—	2	1	no	—
66	C₂²×Dic₁₇	—	2	1	no	—
67	C₂×C₄×D₁₇	—	2	1	no	—
68	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
69	C₂×Dic₃₄	—	2	1	no	—
70	C₂×Dic₃₄	—	2	1	no	—
71	SD₁₆×C₁₇	—	2	2	no	—
72	M₁₆×C₁₇	—	2	2	no	—
73	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₁₇	—	2	1	no	—
76	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₇	—	2	1	no	—
79	C₂²⋊C₄×C₁₇	—	2	1	no	—
80	C₂²⋊Dic₁₇	—	2	1	no	—
81	C₂²⋊Dic₁₇	—	2	1	no	—
82	C₂²⋊Dic₁₇	—	2	1	no	—
83	C₂²⋊Dic₁₇	D₄	1	1	yes	characteristic
84	#63227	—	2	1	no	—
85	#63227	—	2	1	no	—
86	#63227	D₄	1	1	yes	characteristic
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	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
93	#63225	—	2	1	no	—
94	#63225	—	2	1	no	—
95	#63225	—	2	1	no	—
96	#63225	—	2	1	no	—
97	#63225	D₄	1	1	yes	characteristic
98	C₄×Dic₁₇	—	2	1	no	—
99	C₄×Dic₁₇	—	2	1	no	—
100	C₄×Dic₁₇	—	2	1	no	—
101	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
102	C₂×#3622	C₂³	1	1	yes	characteristic
103	C₂×#3622	C₂³	1	1	yes	characteristic
104	#63218	—	2	2	no	—
105	C₂³×C₁₇	—	2	1	no	—
106	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
108	Q₈×C₁₇	—	2	1	no	—
109	Q₈×C₁₇	—	2	1	no	—
110	D₄×C₁₇	—	2	1	no	—
111	D₄×C₁₇	—	2	1	no	—
112	D₄×C₁₇	—	8	1	no	—
113	C₂×C₄×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	—	2	1	no	—
118	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
120	C₂²⋊D₁₇	—	2	2	no	—
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	—	4	2	no	—
127	C₂×Dic₁₇	—	8	1	no	—
128	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
131	C₄×D₁₇	—	2	1	no	—
132	C₄×D₁₇	—	2	1	no	—
133	C₄×D₁₇	—	8	1	no	—
134	Dic₃₄	—	8	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	—	2	1	no	—
140	#3357	—	17	1	no	maximal, Sylow
141	C₂²×C₁₇	—	2	1	no	—
142	C₂²×C₁₇	—	2	1	no	—
143	C₂²×C₁₇	—	2	1	no	—
144	C₂²×C₁₇	—	4	1	no	—
145	C₂²×C₁₇	—	4	1	no	—
146	C₂²×C₁₇	—	4	1	no	—
147	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₁₇	—	2	1	no	—
149	C₂×D₁₇	—	2	1	no	—
150	C₂×D₁₇	—	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	C₂×#41	1	1	yes	characteristic
156	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
157	Dic₁₇	—	2	1	no	—
158	Dic₁₇	—	2	1	no	—
159	Dic₁₇	—	4	1	no	—
160	Dic₁₇	—	4	1	no	—
161	Dic₁₇	—	4	1	no	—
162	Dic₁₇	—	4	1	no	—
163	Dic₁₇	#137	1	2	yes	—
164	#555	—	17	1	no	—
165	#554	—	17	1	no	—
166	#500	—	17	1	no	—
167	#485	—	17	1	no	—
168	#485	—	17	1	no	—
169	#480	—	17	1	no	—
170	#480	—	17	1	no	—
171	#477	—	17	1	no	—
172	#476	—	17	1	no	—
173	#468	—	17	1	no	—
174	#468	D₁₇	1	1	yes	characteristic, complemented
175	C₄×SD₁₆	—	17	1	no	—
176	#436	—	17	1	no	—
177	#428	—	17	1	no	—
178	#428	—	17	1	no	—
179	C₂×C₁₇	—	4	1	no	—
180	C₂×C₁₇	—	8	1	no	—
181	C₂×C₁₇	#547	1	1	yes	characteristic
182	C₂×C₁₇	C₂×#137	1	1	yes	characteristic
183	C₂×C₁₇	#576	1	1	yes	characteristic
184	D₁₇	—	4	1	no	—
185	C₂×SD₁₆	C₂×D₁₇	1	1	yes	characteristic
186	C₂×M₁₆	—	17	1	no	—
187	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
188	C₄⋊Q₈	—	17	1	no	—
189	C₄⋊Q₈	—	17	1	no	—
190	#125	—	34	1	no	—
191	#124	—	34	1	no	—
192	#124	—	34	1	no	—
193	C₂²⋊Q₈	—	17	1	no	—
194	C₂²⋊Q₈	—	34	1	no	—
195	C₂²⋊Q₈	—	34	1	no	—
196	C₂²⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
197	#121	—	17	1	no	—
198	#121	—	34	1	no	—
199	#121	C₂×D₁₇	1	1	yes	characteristic
200	C₄×Q₈	—	17	1	no	—
201	C₄×D₄	—	17	1	no	—
202	C₄×D₄	—	17	1	no	—
203	C₄×D₄	—	34	1	no	—
204	#117	—	17	1	no	—
205	#117	—	17	1	no	—
206	#117	—	34	1	no	—
207	C₂×C₄⋊C₄	—	17	1	no	—
208	C₂×C₄⋊C₄	—	17	1	no	—
209	#107	—	17	1	no	—
210	#107	—	17	1	no	—
211	#107	—	17	1	no	—
212	#107	—	17	1	no	—
213	#107	—	17	1	no	—
214	#107	C₂×D₁₇	1	1	yes	characteristic
215	#106	—	17	1	no	—
216	#106	—	17	1	no	—
217	#106	—	17	1	no	—
218	C₄⋊C₈	—	17	1	no	—
219	#103	—	17	1	no	—
220	#103	—	17	1	no	—
221	#103	C₂×D₁₇	1	1	yes	characteristic
222	#102	—	17	1	no	—
223	#102	—	17	1	no	—
224	#102	C₂×D₁₇	1	1	yes	characteristic
225	C₂²⋊C₈	—	17	1	no	—
226	C₂²⋊C₈	—	17	1	no	—
227	C₄×C₈	—	17	1	no	—
228	C₁₇	#3357	1	1	yes	characteristic, complemented, Sylow
229	C₂×Q₈	—	34	1	no	—
230	C₂×Q₈	—	34	1	no	—
231	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	—	17	1	no	—
234	C₂×D₄	—	34	1	no	—
235	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
236	C₂²×C₄	—	17	1	no	—
237	C₂²×C₄	—	34	1	no	—
238	C₂²×C₄	—	34	1	no	—
239	C₂²×C₄	—	34	1	no	—
240	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
241	SD₁₆	—	2	2	no	—
242	M₁₆	—	2	2	no	—
243	M₁₆	—	34	2	no	—
244	C₂×C₈	—	17	1	no	—
245	C₂×C₈	—	17	1	no	—
246	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	17	1	no	—
250	C₄⋊C₄	—	17	1	no	—
251	C₄⋊C₄	—	17	1	no	—
252	C₄⋊C₄	—	17	1	no	—
253	C₄⋊C₄	—	17	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	—	34	1	no	—
260	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
261	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	17	1	no	—
265	C₂²⋊C₄	—	17	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	C₂²⋊C₄	—	34	1	no	—
271	C₄²	—	17	1	no	—
272	C₄²	—	34	1	no	—
273	C₄²	—	34	1	no	—
274	C₄²	—	34	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	17	1	no	—
277	C₂³	D₄×D₁₇	1	1	yes	characteristic
278	Q₈	—	2	1	no	—
279	Q₈	—	2	1	no	—
280	Q₈	—	136	1	no	—
281	Q₈	—	136	1	no	—
282	D₄	—	2	1	no	—
283	D₄	—	2	1	no	—
284	D₄	—	8	1	no	—
285	D₄	—	34	2	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	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	17	1	no	—
293	C₂×C₄	—	17	1	no	—
294	C₂×C₄	—	17	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	—	34	1	no	—
301	C₂×C₄	—	68	2	no	—
302	C₂×C₄	—	136	1	no	—
303	C₂×C₄	—	136	1	no	—
304	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
305	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	34	1	no	—
308	C₈	—	34	1	no	—
309	C₈	#63254	1	2	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	34	1	no	—
317	C₂²	—	34	1	no	—
318	C₂²	—	34	1	no	—
319	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	17	2	no	—
325	C₄	—	34	1	no	—
326	C₄	—	34	1	no	—
327	C₄	—	68	1	no	—
328	C₄	—	68	1	no	—
329	C₄	—	68	1	no	—
330	C₄	—	68	1	no	—
331	C₄	C₂×#63254	1	1	yes	characteristic
332	C₄	#10587697	1	1	yes	characteristic
333	C₂	—	4	1	no	—
334	C₂	—	8	1	no	—
335	C₂	—	68	1	no	—
336	C₂	#11763459	1	1	yes	characteristic
337	C₂	#137×D₁₇	1	1	yes	characteristic
338	C₂	#11763613	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×#468	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763523	C₂
hypercenter	#468	D₁₇
hypocenter	C₁₇	#3357
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #468

Complements

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

N	H ≅ G/N	type
2. #11763517	C₂	semidirect
3. #11763213	C₂	semidirect
4. #11763148	C₂	semidirect
5. #11763147	C₂	semidirect
6. #11763121	C₂	semidirect
7. #11763120	C₂	semidirect
8. #11763102	C₂	semidirect
9. #11763097	C₂	semidirect
10. #11763052	C₂	semidirect
11. SD₁₆×Dic₁₇	C₂	semidirect
12. #11762885	C₂	semidirect
13. #11762851	C₂	semidirect
14. #11762850	C₂	semidirect
15. C₁₇×#468	C₂	semidirect
174. #468	D₁₇	semidirect
228. C₁₇	#3357	semidirect

Maximal subgroups

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

subgroup	index
#11763523	2
#11763517	2
#11763213	2
#11763148	2
#11763147	2
#11763121	2
#11763120	2
#11763102	2
#11763097	2
#11763052	2
SD₁₆×Dic₁₇	2
#11762885	2
#11762851	2
#11762850	2
C₁₇×#468	2
#3357	17