Subgroups — #14541329

Order 2176.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763505	C₂	1	1	yes	maximal, characteristic
2	#11763215	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763147	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763137	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763133	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763133	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763116	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763096	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763057	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763040	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762907	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762881	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762852	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762845	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#556	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₇×#126	—	2	1	no	—
18	C₁₇×#126	—	2	1	no	—
19	C₁₇×#124	—	2	1	no	—
20	C₁₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
21	C₁₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₁₇	—	2	1	no	—
23	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
25	C₁₇×#117	C₂²	1	1	yes	characteristic
26	C₁₇×#117	C₂²	1	1	yes	characteristic
27	C₂×#63230	C₂²	1	1	yes	characteristic
28	#10587625	—	2	1	no	—
29	#10587616	C₂²	1	1	yes	characteristic
30	#10587616	C₂²	1	1	yes	characteristic
31	#10587612	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
33	C₂×#63223	C₂²	1	1	yes	characteristic
34	C₂²×#3622	C₂²	1	1	yes	characteristic
35	#10587584	—	2	1	no	—
36	#10587569	—	2	1	no	—
37	#10587567	—	2	1	no	—
38	C₄×D₆₈	C₂²	1	1	yes	characteristic
39	#10587556	C₂²	1	1	yes	characteristic
40	#10587506	C₂²	1	1	yes	characteristic
41	#10587506	C₂²	1	1	yes	characteristic
42	#10587506	C₂²	1	1	yes	characteristic
43	C₂²⋊#3622	C₂²	1	1	yes	characteristic
44	C₂²⋊#3622	C₂²	1	1	yes	characteristic
45	#10587480	C₂²	1	1	yes	characteristic
46	#10587480	C₂²	1	1	yes	characteristic
47	#10587480	C₂²	1	1	yes	characteristic
48	#10587479	C₂²	1	1	yes	characteristic
49	#10587479	C₂²	1	1	yes	characteristic
50	#10587479	C₂²	1	1	yes	characteristic
51	#10587479	C₂²	1	1	yes	characteristic
52	#10587479	C₂²	1	1	yes	characteristic
53	#10587478	C₂²	1	1	yes	characteristic
54	#10587478	C₂²	1	1	yes	characteristic
55	#10587478	C₂²	1	1	yes	characteristic
56	#10587478	C₂²	1	1	yes	characteristic
57	C₄⋊#3622	C₂²	1	1	yes	characteristic
58	#10587474	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	—	2	1	no	—
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₁₇	D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₇	—	2	1	no	—
65	C₂×C₂²⋊D₁₇	—	2	1	no	—
66	C₂²×Dic₁₇	—	2	1	no	—
67	C₂×D₆₈	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₁₇	—	2	1	no	—
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	D₄	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	C₂²⋊C₄×C₁₇	—	2	1	no	—
78	C₂²⋊C₄×C₁₇	—	2	1	no	—
79	C₂²⋊C₄×C₁₇	—	2	1	no	—
80	C₂²⋊C₄×C₁₇	—	2	1	no	—
81	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
82	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
83	C₄²×C₁₇	—	2	1	no	—
84	C₄²×C₁₇	—	2	1	no	—
85	C₄²×C₁₇	—	2	1	no	—
86	C₄²×C₁₇	C₂³	1	1	yes	characteristic
87	C₂²⋊Dic₁₇	—	2	1	no	—
88	#63230	—	2	2	no	—
89	#63227	—	2	1	no	—
90	#63227	—	2	1	no	—
91	#63227	—	2	1	no	—
92	C₄⋊Dic₁₇	—	2	1	no	—
93	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
94	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
96	#63225	—	2	1	no	—
97	#63225	—	2	1	no	—
98	C₄×Dic₁₇	—	2	1	no	—
99	#63223	—	2	2	no	—
100	C₂×#3622	—	2	2	no	—
101	C₂×#3622	C₂³	1	1	yes	characteristic
102	C₂×#3622	C₂³	1	1	yes	characteristic
103	C₂×#3622	C₂³	1	1	yes	characteristic
104	C₂×#3622	C₂³	1	1	yes	characteristic
105	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
106	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₁₇	—	2	1	no	—
108	Q₈×C₁₇	—	2	1	no	—
109	Q₈×C₁₇	—	2	1	no	—
110	Q₈×C₁₇	—	4	2	no	—
111	D₄×C₁₇	—	4	1	no	—
112	C₂×C₄×C₁₇	—	2	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	—	2	1	no	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
125	C₂²⋊D₁₇	—	4	2	no	—
126	C₂²⋊D₁₇	—	8	1	no	—
127	C₂×Dic₁₇	—	2	1	no	—
128	C₂×Dic₁₇	—	2	1	no	—
129	C₂×Dic₁₇	—	2	1	no	—
130	C₂×Dic₁₇	—	8	1	no	—
131	D₆₈	—	2	1	no	—
132	D₆₈	—	2	1	no	—
133	C₄×D₁₇	—	8	1	no	—
134	#3622	—	2	1	no	—
135	#3622	—	2	1	no	—
136	#3622	—	2	1	no	—
137	#3622	#41	1	2	yes	—
138	#3350	—	17	1	no	maximal, Sylow
139	C₂²×C₁₇	—	2	1	no	—
140	C₂²×C₁₇	—	2	1	no	—
141	C₂²×C₁₇	—	2	1	no	—
142	C₂²×C₁₇	—	2	1	no	—
143	C₂²×C₁₇	—	2	1	no	—
144	C₂²×C₁₇	—	2	1	no	—
145	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₁₇	—	4	1	no	—
147	C₂×D₁₇	—	4	1	no	—
148	C₂×D₁₇	—	4	1	no	—
149	C₄×C₁₇	—	2	1	no	—
150	C₄×C₁₇	—	2	1	no	—
151	C₄×C₁₇	—	2	1	no	—
152	C₄×C₁₇	—	2	1	no	—
153	C₄×C₁₇	—	4	1	no	—
154	C₄×C₁₇	—	4	1	no	—
155	C₄×C₁₇	—	4	1	no	—
156	C₄×C₁₇	—	4	1	no	—
157	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
158	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
159	Dic₁₇	—	4	1	no	—
160	Dic₁₇	—	4	1	no	—
161	Dic₁₇	—	4	1	no	—
162	#556	D₁₇	1	1	yes	characteristic, complemented
163	#553	—	17	1	no	—
164	#501	—	17	1	no	—
165	#485	—	17	1	no	—
166	#483	—	17	1	no	—
167	#482	—	17	1	no	—
168	#482	—	17	1	no	—
169	#479	—	17	1	no	—
170	#476	—	17	1	no	—
171	#468	—	17	1	no	—
172	#465	—	17	1	no	—
173	#440	—	17	1	no	—
174	#435	—	17	1	no	—
175	#428	—	17	1	no	—
176	#427	—	17	1	no	—
177	C₂×C₁₇	—	4	1	no	—
178	C₂×C₁₇	—	4	1	no	—
179	C₂×C₁₇	#547	1	1	yes	characteristic
180	C₂×C₁₇	#575	1	1	yes	characteristic
181	C₂×C₁₇	#577	1	1	yes	characteristic
182	D₁₇	—	8	1	no	—
183	C₂×SD₁₆	—	17	1	no	—
184	C₂×M₁₆	—	17	1	no	—
185	C₂²×C₈	—	17	1	no	—
186	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
187	#126	—	2	1	no	—
188	#126	—	2	1	no	—
189	#126	—	34	1	no	—
190	#125	—	17	1	no	—
191	#124	—	2	1	no	—
192	#123	—	34	1	no	—
193	C₂²⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
194	C₂²⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
195	#121	—	17	1	no	—
196	#121	—	17	1	no	—
197	#121	—	34	1	no	—
198	C₄×Q₈	—	2	1	no	—
199	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
200	C₄×D₄	—	17	1	no	—
201	C₄×D₄	—	34	1	no	—
202	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
203	#117	—	17	1	no	—
204	#117	C₂×D₁₇	1	1	yes	characteristic
205	#117	C₂×D₁₇	1	1	yes	characteristic
206	C₂×C₄⋊C₄	—	17	1	no	—
207	#107	—	17	1	no	—
208	#107	—	17	1	no	—
209	#107	—	17	1	no	—
210	#107	—	17	1	no	—
211	#106	—	17	1	no	—
212	#106	—	17	1	no	—
213	#106	—	17	1	no	—
214	#106	—	17	1	no	—
215	#106	—	17	1	no	—
216	C₄⋊C₈	—	17	1	no	—
217	#103	—	17	1	no	—
218	#103	—	17	1	no	—
219	#103	—	17	1	no	—
220	#102	—	17	1	no	—
221	#102	—	17	1	no	—
222	#102	—	17	1	no	—
223	C₂²⋊C₈	—	17	1	no	—
224	C₂²⋊C₈	—	17	1	no	—
225	#97	—	17	1	no	—
226	C₁₇	#3350	1	1	yes	characteristic, complemented, Sylow
227	C₂×Q₈	—	2	1	no	—
228	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
229	C₂×D₄	—	17	1	no	—
230	C₂×D₄	—	34	1	no	—
231	C₂×D₄	—	34	1	no	—
232	C₂×D₄	C₂²⋊D₁₇	1	1	yes	characteristic
233	C₂²×C₄	—	34	1	no	—
234	C₂²×C₄	—	34	1	no	—
235	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
236	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
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	—	17	1	no	—
242	C₂×C₈	—	17	1	no	—
243	C₂×C₈	—	34	2	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	17	1	no	—
249	C₄⋊C₄	—	17	1	no	—
250	C₄⋊C₄	—	17	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	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	34	1	no	—
263	C₂²⋊C₄	—	34	1	no	—
264	C₂²⋊C₄	—	34	1	no	—
265	C₂²⋊C₄	—	34	1	no	—
266	C₂²⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
267	C₂²⋊C₄	C₂²⋊D₁₇	1	1	yes	characteristic
268	C₄²	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	34	1	no	—
272	C₄²	C₂²×D₁₇	1	1	yes	characteristic
273	C₂³	—	34	1	no	—
274	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
275	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	4	2	no	—
279	D₄	—	4	1	no	—
280	D₄	—	34	1	no	—
281	D₄	—	34	1	no	—
282	D₄	—	68	2	no	—
283	D₄	—	136	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	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	—	34	1	no	—
293	C₂×C₄	—	34	1	no	—
294	C₂×C₄	—	34	1	no	—
295	C₂×C₄	—	136	1	no	—
296	C₂×C₄	—	136	1	no	—
297	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
298	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
299	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
300	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
301	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
302	C₈	—	17	2	no	—
303	C₈	—	34	1	no	—
304	C₈	—	34	1	no	—
305	C₈	—	34	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	68	1	no	—
313	C₂²	—	68	1	no	—
314	C₂²	—	68	1	no	—
315	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	68	1	no	—
325	C₄	—	68	1	no	—
326	C₄	—	68	1	no	—
327	C₄	#41×D₁₇	1	1	yes	characteristic
328	C₄	#10587697	1	1	yes	characteristic
329	C₂	—	4	1	no	—
330	C₂	—	4	1	no	—
331	C₂	—	136	1	no	—
332	C₂	#11763458	1	1	yes	characteristic
333	C₂	#11763607	1	1	yes	characteristic
334	C₂	#11763617	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#556	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763505	C₂
hypercenter	#556	D₁₇
hypocenter	C₁₇	#3350
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #556

Complements

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

N	H ≅ G/N	type
2. #11763215	C₂	semidirect
3. #11763147	C₂	semidirect
4. #11763137	C₂	semidirect
5. #11763133	C₂	semidirect
6. #11763133	C₂	semidirect
7. #11763116	C₂	semidirect
8. #11763096	C₂	semidirect
9. #11763057	C₂	semidirect
10. #11763040	C₂	semidirect
11. #11762907	C₂	semidirect
12. #11762881	C₂	semidirect
13. #11762852	C₂	semidirect
14. #11762845	C₂	semidirect
15. C₁₇×#556	C₂	semidirect
162. #556	D₁₇	semidirect
226. C₁₇	#3350	semidirect

Maximal subgroups

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

subgroup	index
#11763505	2
#11763215	2
#11763147	2
#11763137	2
#11763133	2
#11763116	2
#11763096	2
#11763057	2
#11763040	2
#11762907	2
#11762881	2
#11762852	2
#11762845	2
C₁₇×#556	2
#3350	17