Subgroups — #14541613

Order 2176.

346 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763532	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763519	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763178	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763169	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763164	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763117	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763110	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763095	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763025	C₂	1	1	yes	maximal, characteristic
10	#11763025	C₂	1	1	yes	maximal, characteristic
11	#11763015	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762922	C₂	1	1	yes	maximal, characteristic
13	#11762911	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#63219	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
17	#10587632	—	2	1	no	—
18	#10587632	—	2	1	no	—
19	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
20	C₂×#63219	C₂²	1	1	yes	characteristic
21	C₂×#63219	C₂²	1	1	yes	characteristic
22	#10587584	—	2	1	no	—
23	#10587584	—	2	1	no	—
24	#10587582	—	2	1	no	—
25	#10587582	—	2	1	no	—
26	#10587578	—	2	1	no	—
27	C₄⋊C₄×D₁₇	—	2	1	no	—
28	#10587561	C₂²	1	1	yes	characteristic
29	#10587561	C₂²	1	1	yes	characteristic
30	C₄×D₆₈	C₂²	1	1	yes	characteristic
31	C₄×D₆₈	C₂²	1	1	yes	characteristic
32	C₄×D₆₈	C₂²	1	1	yes	characteristic
33	C₄×D₆₈	C₂²	1	1	yes	characteristic
34	#10587556	C₂²	1	1	yes	characteristic
35	#10587556	C₂²	1	1	yes	characteristic
36	#10587555	C₂²	1	1	yes	characteristic
37	#10587555	C₂²	1	1	yes	characteristic
38	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
40	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₁₇×#97	C₂²	1	1	yes	characteristic
44	C₁₇×#97	C₂²	1	1	yes	characteristic
45	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
46	#10587492	C₂²	1	1	yes	characteristic
47	#10587492	C₂²	1	1	yes	characteristic
48	#10587492	C₂²	1	1	yes	characteristic
49	#10587492	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	#10587487	C₂²	1	1	yes	characteristic
54	#10587487	C₂²	1	1	yes	characteristic
55	#10587487	C₂²	1	1	yes	characteristic
56	#10587487	C₂²	1	1	yes	characteristic
57	#10587487	C₂²	1	1	yes	characteristic
58	#10587487	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	D₄	1	1	yes	characteristic
60	C₂×D₆₈	C₂³	1	1	yes	characteristic
61	C₂×D₆₈	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₁₇	—	2	1	no	—
63	C₂×C₄×D₁₇	—	2	1	no	—
64	C₂×C₄×D₁₇	—	2	1	no	—
65	C₂×C₄×D₁₇	—	2	1	no	—
66	C₂×Dic₃₄	—	2	1	no	—
67	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
68	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
76	C₄²×C₁₇	C₂³	1	1	yes	characteristic
77	C₄²×C₁₇	C₂³	1	1	yes	characteristic
78	C₄²×C₁₇	C₂³	1	1	yes	characteristic
79	#63227	—	2	1	no	—
80	#63227	—	2	1	no	—
81	#63227	—	2	1	no	—
82	#63227	—	2	1	no	—
83	#63227	—	2	1	no	—
84	#63227	—	2	1	no	—
85	#63227	—	2	1	no	—
86	#63227	—	2	1	no	—
87	C₄⋊Dic₁₇	—	2	1	no	—
88	C₄⋊Dic₁₇	—	2	1	no	—
89	C₄⋊Dic₁₇	—	2	1	no	—
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	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
94	#63225	—	2	1	no	—
95	#63225	—	2	1	no	—
96	#63225	—	2	1	no	—
97	#63225	—	2	1	no	—
98	#63225	—	2	1	no	—
99	#63225	—	2	1	no	—
100	C₄×Dic₁₇	—	2	1	no	—
101	C₄×Dic₁₇	—	2	1	no	—
102	D₁₃₆	—	2	2	no	—
103	#63219	—	2	2	no	—
104	#63219	—	2	2	no	—
105	C₂²×D₁₇	—	2	1	no	—
106	C₂²×D₁₇	—	2	1	no	—
107	Q₈×C₁₇	—	4	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	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
120	C₂×Dic₁₇	—	2	1	no	—
121	D₆₈	—	2	1	no	—
122	D₆₈	—	2	1	no	—
123	D₆₈	—	2	1	no	—
124	D₆₈	#41	1	2	yes	—
125	C₄×D₁₇	—	4	2	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	Dic₃₄	—	8	1	no	—
134	C₈×C₁₇	—	2	1	no	—
135	C₈×C₁₇	—	2	1	no	—
136	C₈×C₁₇	—	2	1	no	—
137	C₈×C₁₇	—	2	1	no	—
138	#3376	—	17	1	no	maximal, Sylow
139	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×D₁₇	—	2	2	no	—
141	C₂×D₁₇	—	4	1	no	—
142	C₂×D₁₇	—	4	1	no	—
143	C₂×D₁₇	—	4	1	no	—
144	C₂×D₁₇	—	4	1	no	—
145	C₂×D₁₇	—	4	1	no	—
146	C₄×C₁₇	—	2	1	no	—
147	C₄×C₁₇	—	2	1	no	—
148	C₄×C₁₇	—	2	1	no	—
149	C₄×C₁₇	—	2	1	no	—
150	C₄×C₁₇	—	2	1	no	—
151	C₄×C₁₇	—	2	1	no	—
152	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
153	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
154	Dic₁₇	—	4	1	no	—
155	Dic₁₇	—	4	1	no	—
156	Dic₁₇	—	4	1	no	—
157	Dic₁₇	—	4	1	no	—
158	Dic₁₇	—	4	1	no	—
159	Dic₁₇	—	4	1	no	—
160	#556	—	17	1	no	—
161	#554	—	17	1	no	—
162	#491	—	17	1	no	—
163	#489	—	17	1	no	—
164	#488	—	17	1	no	—
165	#479	—	17	1	no	—
166	#478	—	17	1	no	—
167	#475	—	17	1	no	—
168	#463	—	17	1	no	—
169	#463	—	17	1	no	—
170	#461	—	17	1	no	—
171	#446	D₁₇	1	1	yes	characteristic, complemented
172	#442	—	17	1	no	—
173	#440	—	17	1	no	—
174	C₄×SD₁₆	—	17	1	no	—
175	C₂×C₁₇	#548	1	1	yes	characteristic
176	C₂×C₁₇	#575	1	1	yes	characteristic
177	C₂×C₁₇	#577	1	1	yes	characteristic
178	D₁₇	—	4	2	no	—
179	D₁₇	—	8	1	no	—
180	C₂×SD₁₆	—	17	1	no	—
181	C₂×SD₁₆	—	17	1	no	—
182	C₂×D₈	—	17	1	no	—
183	C₄⋊Q₈	—	17	1	no	—
184	C₄⋊Q₈	—	17	1	no	—
185	#126	—	34	1	no	—
186	#126	—	34	1	no	—
187	#125	—	17	1	no	—
188	#125	—	17	1	no	—
189	#124	—	17	1	no	—
190	#124	—	17	1	no	—
191	C₂²⋊Q₈	—	34	1	no	—
192	C₂²⋊Q₈	—	34	1	no	—
193	C₂²⋊Q₈	—	34	1	no	—
194	C₂²⋊Q₈	—	34	1	no	—
195	C₄×Q₈	—	17	1	no	—
196	C₄×Q₈	—	17	1	no	—
197	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
198	C₄×D₄	—	17	1	no	—
199	C₄×D₄	—	17	1	no	—
200	C₄×D₄	—	17	1	no	—
201	C₄×D₄	—	17	1	no	—
202	#117	—	34	1	no	—
203	C₂×C₄⋊C₄	—	34	1	no	—
204	#107	—	17	1	no	—
205	#106	—	17	1	no	—
206	#106	—	17	1	no	—
207	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
208	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
209	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
210	#103	—	17	1	no	—
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	#102	—	17	1	no	—
217	#102	—	17	1	no	—
218	#102	—	17	1	no	—
219	#102	—	17	1	no	—
220	#97	C₂×D₁₇	1	1	yes	characteristic
221	#97	C₂×D₁₇	1	1	yes	characteristic
222	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
223	C₁₇	#3376	1	1	yes	characteristic, complemented, Sylow
224	C₂×Q₈	—	17	1	no	—
225	C₂×Q₈	—	17	1	no	—
226	C₂×Q₈	—	34	1	no	—
227	C₂×Q₈	D₆₈	1	1	yes	characteristic
228	C₂×D₄	—	17	1	no	—
229	C₂×D₄	—	17	1	no	—
230	C₂²×C₄	—	34	1	no	—
231	C₂²×C₄	—	34	1	no	—
232	C₂²×C₄	—	34	1	no	—
233	C₂²×C₄	—	34	1	no	—
234	SD₁₆	—	34	2	no	—
235	SD₁₆	—	34	2	no	—
236	D₈	—	34	2	no	—
237	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
239	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
241	C₄⋊C₄	—	17	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	D₆₈	1	1	yes	characteristic
255	C₄⋊C₄	D₆₈	1	1	yes	characteristic
256	C₄⋊C₄	D₆₈	1	1	yes	characteristic
257	C₂²⋊C₄	—	34	1	no	—
258	C₂²⋊C₄	—	34	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	C₂²⋊C₄	—	34	1	no	—
264	C₂²⋊C₄	—	34	1	no	—
265	C₄²	—	34	1	no	—
266	C₄²	—	34	1	no	—
267	C₄²	C₂²×D₁₇	1	1	yes	characteristic
268	C₄²	C₂²×D₁₇	1	1	yes	characteristic
269	C₄²	C₂²×D₁₇	1	1	yes	characteristic
270	C₂³	—	34	1	no	—
271	C₂³	—	34	1	no	—
272	Q₈	—	4	1	no	—
273	Q₈	—	34	1	no	—
274	Q₈	—	34	1	no	—
275	Q₈	—	34	1	no	—
276	Q₈	—	34	1	no	—
277	Q₈	—	136	1	no	—
278	D₄	—	17	2	no	—
279	D₄	—	34	1	no	—
280	D₄	—	34	1	no	—
281	D₄	—	34	1	no	—
282	C₂×C₄	—	34	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	—	68	2	no	—
289	C₂×C₄	—	68	2	no	—
290	C₂×C₄	—	136	1	no	—
291	C₂×C₄	—	136	1	no	—
292	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
293	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
294	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
295	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
296	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
297	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₂²	—	34	2	no	—
304	C₂²	—	68	1	no	—
305	C₂²	—	68	1	no	—
306	C₂²	—	68	1	no	—
307	C₂²	—	68	1	no	—
308	C₂²	—	68	1	no	—
309	C₂²	C₂²×D₆₈	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	68	1	no	—
317	C₄	—	68	1	no	—
318	C₄	—	68	1	no	—
319	C₄	—	68	1	no	—
320	C₄	—	68	1	no	—
321	C₄	—	68	1	no	—
322	C₄	#10587694	1	1	yes	characteristic
323	C₄	#41×D₁₇	1	1	yes	characteristic
324	C₂	—	68	2	no	—
325	C₂	—	136	1	no	—
326	C₂	#11763471	1	1	yes	characteristic
327	C₂	#11763609	1	1	yes	characteristic
328	C₂	#11763621	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₆₈
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#446	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×D₁₃₆	C₂²
hypercenter	#446	D₁₇
hypocenter	C₁₇	#3376
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

Complements

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

N	H ≅ G/N	type
1. #11763532	C₂	semidirect
2. #11763519	C₂	semidirect
3. #11763178	C₂	semidirect
4. #11763169	C₂	semidirect
5. #11763164	C₂	semidirect
6. #11763117	C₂	semidirect
7. #11763110	C₂	semidirect
8. #11763095	C₂	semidirect
11. #11763015	C₂	semidirect
13. #11762911	C₂	semidirect
14. C₄×#63219	C₂	semidirect
15. C₁₇×#446	C₂	semidirect
171. #446	D₁₇	semidirect
223. C₁₇	#3376	semidirect

Maximal subgroups

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

subgroup	index
#11763532	2
#11763519	2
#11763178	2
#11763169	2
#11763164	2
#11763117	2
#11763110	2
#11763095	2
#11763025	2
#11763015	2
#11762922	2
#11762911	2
C₄×#63219	2
C₁₇×#446	2
#3376	17