Subgroups — #14537620

Order 2176.

470 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10587601	C₂	1	1	yes	maximal, characteristic
2	#11763288	C₂	1	1	yes	maximal, characteristic
3	#11763285	C₂	1	1	yes	maximal, characteristic, complemented
4	#11762915	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762914	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762910	C₂	1	1	yes	maximal, characteristic, complemented
7	#11762909	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Dic₆₈	C₂	1	1	yes	maximal, characteristic, complemented
9	#11762901	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762900	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762848	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762802	C₂	1	1	yes	maximal, characteristic
13	#103×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762763	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₇	C₂²	1	1	yes	characteristic
17	C₂×#63254	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
21	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
22	C₂×#63231	C₂²	1	1	yes	characteristic
23	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
24	C₂×#63229	C₂²	1	1	yes	characteristic
25	C₄×C₂²⋊D₁₇	—	2	1	no	—
26	C₂×#63225	—	2	1	no	—
27	#10587601	C₄	1	8	yes	complemented
28	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
29	C₂×#63218	C₂²	1	1	yes	characteristic
30	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
31	#10587578	—	2	1	no	—
32	#10587578	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
34	#10587573	—	2	1	no	—
35	#10587573	C₂²	1	1	yes	characteristic
36	#10587573	C₂²	1	1	yes	characteristic
37	#10587567	—	2	1	no	—
38	#10587563	—	2	1	no	—
39	#10587558	—	2	1	no	—
40	#10587558	C₂²	1	1	yes	characteristic
41	C₄×Dic₃₄	—	2	1	no	—
42	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
43	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
44	C₁₇×#106	C₂²	1	1	yes	characteristic
45	C₁₇×#103	C₂²	1	1	yes	characteristic
46	C₁₇×#102	C₂²	1	1	yes	characteristic
47	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
48	#10587506	C₂²	1	1	yes	characteristic
49	#10587503	C₂²	1	1	yes	characteristic
50	#10587489	C₂²	1	1	yes	characteristic
51	#10587487	C₂²	1	1	yes	characteristic
52	#10587487	C₂²	1	1	yes	characteristic
53	#10587486	C₂²	1	1	yes	characteristic
54	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
55	#10587481	C₂²	1	1	yes	characteristic
56	#10587481	C₂²	1	1	yes	characteristic
57	#10587479	C₂²	1	1	yes	characteristic
58	#10587478	C₂²	1	1	yes	characteristic
59	#10587474	C₂²	1	1	yes	characteristic
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₇	—	2	1	no	—
63	C₂×C₂²⋊D₁₇	—	2	1	no	—
64	C₂²×Dic₁₇	—	2	1	no	—
65	C₂²×Dic₁₇	—	2	1	no	—
66	Q₈×D₁₇	—	2	2	no	—
67	Q₈×D₁₇	C₂×C₄	1	4	yes	complemented
68	#63254	—	2	2	no	—
69	#63254	C₂×C₄	1	4	yes	—
70	C₂×C₄×D₁₇	—	2	1	no	—
71	C₂×C₄×D₁₇	—	2	1	no	—
72	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
73	C₂×Dic₃₄	—	2	1	no	—
74	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
75	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
76	SD₁₆×C₁₇	C₂×C₄	1	4	yes	complemented
77	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₁₇	—	2	1	no	—
80	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
82	C₂²⋊C₄×C₁₇	—	2	1	no	—
83	C₄²×C₁₇	—	2	1	no	—
84	C₄²×C₁₇	C₂³	1	1	yes	characteristic
85	C₂²⋊Dic₁₇	—	2	1	no	—
86	#63231	C₂×C₄	1	4	yes	complemented
87	#63229	C₂×C₄	1	4	yes	complemented
88	#63227	—	2	1	no	—
89	#63227	D₄	1	1	yes	characteristic
90	#63227	D₄	1	1	yes	characteristic
91	C₄⋊Dic₁₇	—	2	1	no	—
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	D₄	1	1	yes	characteristic
100	#63225	D₄	1	1	yes	characteristic
101	C₄×Dic₁₇	—	2	1	no	—
102	C₄×Dic₁₇	—	2	1	no	—
103	C₄×Dic₁₇	—	2	1	no	—
104	C₄×Dic₁₇	—	2	1	no	—
105	C₄×Dic₁₇	—	2	1	no	—
106	C₄×Dic₁₇	—	2	1	no	—
107	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
108	C₂×#3622	C₂³	1	1	yes	characteristic
109	C₂×#3622	C₂³	1	1	yes	characteristic
110	Dic₆₈	C₂×C₄	1	4	yes	complemented
111	#63218	C₂×C₄	1	4	yes	complemented
112	C₈×D₁₇	—	2	2	no	—
113	C₂³×C₁₇	—	2	1	no	—
114	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₁₇	—	2	1	no	—
116	Q₈×C₁₇	C₂²×C₄	1	2	yes	—
117	D₄×C₁₇	—	2	1	no	—
118	D₄×C₁₇	C₂²×C₄	1	2	yes	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	—	2	1	no	—
121	C₂×C₄×C₁₇	—	2	1	no	—
122	C₂×C₄×C₁₇	—	8	1	no	—
123	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
126	C₂²⋊D₁₇	—	2	4	no	—
127	C₂×Dic₁₇	—	2	1	no	—
128	C₂×Dic₁₇	—	2	1	no	—
129	C₂×Dic₁₇	—	2	4	no	—
130	C₂×Dic₁₇	—	2	1	no	—
131	C₂×Dic₁₇	—	2	1	no	—
132	C₂×Dic₁₇	—	8	1	no	—
133	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
136	C₄×D₁₇	—	2	4	no	—
137	C₄×D₁₇	—	8	1	no	—
138	C₄×D₁₇	C₂²×C₄	1	2	yes	—
139	C₄×D₁₇	#41	1	2	yes	—
140	Dic₃₄	—	2	1	no	—
141	Dic₃₄	—	2	1	no	—
142	Dic₃₄	—	2	4	no	—
143	Dic₃₄	C₂²×C₄	1	2	yes	—
144	Dic₃₄	C₂²×C₄	1	2	yes	—
145	C₈×C₁₇	—	2	1	no	—
146	C₈×C₁₇	C₂²×C₄	1	2	yes	—
147	#3622	—	2	1	no	—
148	#3622	C₂²×C₄	1	2	yes	—
149	#2947	—	17	1	no	maximal, Sylow
150	C₂²×C₁₇	—	2	2	no	—
151	C₂²×C₁₇	—	4	1	no	—
152	C₂²×C₁₇	—	4	1	no	—
153	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₁₇	—	2	1	no	—
155	C₂×D₁₇	—	2	1	no	—
156	C₂×D₁₇	C₄×D₄	1	2	yes	—
157	C₄×C₁₇	—	2	2	no	—
158	C₄×C₁₇	—	2	1	no	—
159	C₄×C₁₇	—	2	1	no	—
160	C₄×C₁₇	—	4	1	no	—
161	C₄×C₁₇	—	4	1	no	—
162	C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
163	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
164	Dic₁₇	—	2	2	no	—
165	Dic₁₇	—	2	2	no	—
166	Dic₁₇	—	2	1	no	—
167	Dic₁₇	—	2	1	no	—
168	Dic₁₇	—	4	1	no	—
169	Dic₁₇	—	4	1	no	—
170	Dic₁₇	C₄×D₄	1	2	yes	—
171	C₂×#137	—	17	1	no	—
172	#520	—	17	1	no	—
173	#519	—	17	1	no	—
174	#441	—	17	1	no	—
175	#441	—	17	1	no	—
176	#440	—	17	1	no	—
177	#440	—	17	1	no	—
178	C₄×Q₁₆	—	17	1	no	—
179	C₄×Q₁₆	—	17	1	no	—
180	C₄×SD₁₆	—	17	1	no	—
181	C₄×SD₁₆	D₁₇	1	1	yes	characteristic, complemented
182	#428	—	17	1	no	—
183	#416	—	17	1	no	—
184	C₂×#103	—	17	1	no	—
185	#405	—	17	1	no	—
186	C₂×C₁₇	—	4	2	no	—
187	C₂×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
188	C₂×C₁₇	#577	1	1	yes	characteristic
189	C₂×C₁₇	#578	1	1	yes	characteristic
190	D₁₇	—	2	2	no	—
191	C₂×#41	—	17	1	no	—
192	C₂²×Q₈	—	17	1	no	—
193	#137	—	17	8	no	—
194	C₂×Q₁₆	—	17	1	no	—
195	C₂×Q₁₆	—	17	1	no	—
196	C₂×SD₁₆	—	17	1	no	—
197	C₂×SD₁₆	C₂×D₁₇	1	1	yes	characteristic
198	C₂×M₁₆	—	17	1	no	—
199	C₂²×C₈	—	17	1	no	—
200	C₄×Q₈	—	17	1	no	—
201	C₄×Q₈	—	17	1	no	—
202	C₄×Q₈	—	17	1	no	—
203	C₄×Q₈	—	17	1	no	—
204	C₄×Q₈	—	17	1	no	—
205	C₄×Q₈	—	34	1	no	—
206	C₄×Q₈	—	34	1	no	—
207	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
208	C₄×D₄	—	17	1	no	—
209	C₄×D₄	—	34	1	no	—
210	C₄×D₄	—	34	1	no	—
211	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
212	#117	—	17	1	no	—
213	#117	—	17	1	no	—
214	#117	—	34	1	no	—
215	#117	—	34	1	no	—
216	#117	—	34	1	no	—
217	C₂×C₄⋊C₄	—	17	1	no	—
218	C₂×C₄⋊C₄	—	34	1	no	—
219	#107	—	17	1	no	—
220	#107	—	17	1	no	—
221	#106	—	17	1	no	—
222	#106	C₂×D₁₇	1	1	yes	characteristic
223	#103	—	17	1	no	—
224	#103	—	17	1	no	—
225	#103	—	17	1	no	—
226	#103	—	17	1	no	—
227	#103	—	17	1	no	—
228	#103	C₂×D₁₇	1	1	yes	characteristic
229	#102	—	17	1	no	—
230	#102	C₂×D₁₇	1	1	yes	characteristic
231	#97	—	17	1	no	—
232	#97	—	17	1	no	—
233	C₄×C₈	—	17	1	no	—
234	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
235	C₁₇	#2947	1	1	yes	characteristic, complemented, Sylow
236	#41	—	17	4	no	—
237	#41	—	34	2	no	—
238	C₂×Q₈	—	17	4	no	—
239	C₂×Q₈	—	17	1	no	—
240	C₂×Q₈	—	17	1	no	—
241	C₂×Q₈	—	34	1	no	—
242	C₂×Q₈	—	34	2	no	—
243	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
244	C₂×D₄	—	34	1	no	—
245	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	17	1	no	—
248	C₂²×C₄	—	34	1	no	—
249	C₂²×C₄	—	34	1	no	—
250	C₂²×C₄	—	34	1	no	—
251	C₂²×C₄	—	34	1	no	—
252	Q₁₆	—	17	4	no	—
253	Q₁₆	—	17	4	no	—
254	SD₁₆	—	17	4	no	—
255	SD₁₆	C₄×D₁₇	1	4	yes	complemented
256	M₁₆	—	17	4	no	—
257	C₂×C₈	—	17	1	no	—
258	C₂×C₈	—	17	1	no	—
259	C₂×C₈	—	34	2	no	—
260	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	17	1	no	—
264	C₄⋊C₄	—	17	1	no	—
265	C₄⋊C₄	—	17	1	no	—
266	C₄⋊C₄	—	17	1	no	—
267	C₄⋊C₄	—	34	1	no	—
268	C₄⋊C₄	—	34	1	no	—
269	C₄⋊C₄	—	34	1	no	—
270	C₄⋊C₄	—	34	1	no	—
271	C₄⋊C₄	—	34	1	no	—
272	C₄⋊C₄	—	34	1	no	—
273	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	17	1	no	—
277	C₂²⋊C₄	—	17	1	no	—
278	C₂²⋊C₄	—	34	1	no	—
279	C₂²⋊C₄	—	34	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	17	1	no	—
282	C₄²	—	34	1	no	—
283	C₄²	—	34	1	no	—
284	C₄²	—	34	1	no	—
285	C₄²	—	34	1	no	—
286	C₄²	—	34	1	no	—
287	C₄²	—	34	1	no	—
288	C₄²	C₂²×D₁₇	1	1	yes	characteristic
289	C₂³	—	2	1	no	—
290	C₂³	—	17	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	17	2	no	—
293	Q₈	—	17	2	no	—
294	Q₈	—	34	4	no	—
295	Q₈	—	34	1	no	—
296	Q₈	—	34	1	no	—
297	Q₈	C₂×C₄×D₁₇	1	2	yes	—
298	D₄	—	2	1	no	—
299	D₄	—	34	4	no	—
300	D₄	C₂×C₄×D₁₇	1	2	yes	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	8	1	no	—
305	C₂×C₄	—	17	1	no	—
306	C₂×C₄	—	17	2	no	—
307	C₂×C₄	—	17	2	no	—
308	C₂×C₄	—	17	1	no	—
309	C₂×C₄	—	17	1	no	—
310	C₂×C₄	—	34	4	no	—
311	C₂×C₄	—	34	4	no	—
312	C₂×C₄	—	34	1	no	—
313	C₂×C₄	—	34	1	no	—
314	C₂×C₄	—	34	1	no	—
315	C₂×C₄	—	34	1	no	—
316	C₂×C₄	—	136	1	no	—
317	C₂×C₄	—	136	1	no	—
318	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
319	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
320	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	17	2	no	—
323	C₈	—	34	1	no	—
324	C₈	C₂×C₄×D₁₇	1	2	yes	—
325	C₂²	—	2	2	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	17	2	no	—
329	C₂²	—	34	1	no	—
330	C₂²	—	34	1	no	—
331	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	17	2	no	—
338	C₄	—	34	2	no	—
339	C₄	—	34	2	no	—
340	C₄	—	34	1	no	—
341	C₄	—	34	1	no	—
342	C₄	—	68	1	no	—
343	C₄	—	68	1	no	—
344	C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
345	C₄	#41×D₁₇	1	1	yes	characteristic
346	C₂	—	4	2	no	—
347	C₂	—	34	2	no	—
348	C₂	C₄×D₄×D₁₇	1	1	yes	characteristic
349	C₂	#11763615	1	1	yes	characteristic
350	C₂	#11763623	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₇
commutator	C₄×C₁₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₄×SD₁₆×C₁₇	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×#63254	C₂²
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	C₄×SD₁₆	D₁₇
hypocenter	C₁₇	#2947
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
3. #11763285	C₂	semidirect
4. #11762915	C₂	semidirect
5. #11762914	C₂	semidirect
6. #11762910	C₂	semidirect
7. #11762909	C₂	semidirect
8. C₄×Dic₆₈	C₂	semidirect
9. #11762901	C₂	semidirect
10. #11762900	C₂	semidirect
11. #11762848	C₂	semidirect
13. #103×D₁₇	C₂	semidirect
14. #11762763	C₂	semidirect
15. C₄×SD₁₆×C₁₇	C₂	semidirect
27. #10587601	C₄	semidirect
67. Q₈×D₁₇	C₂×C₄	semidirect
76. SD₁₆×C₁₇	C₂×C₄	semidirect
86. #63231	C₂×C₄	semidirect
87. #63229	C₂×C₄	semidirect
110. Dic₆₈	C₂×C₄	semidirect
111. #63218	C₂×C₄	semidirect
181. C₄×SD₁₆	D₁₇	semidirect
235. C₁₇	#2947	semidirect
255. SD₁₆	C₄×D₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10587601	2
#11763288	2
#11763285	2
#11762915	2
#11762914	2
#11762910	2
#11762909	2
C₄×Dic₆₈	2
#11762901	2
#11762900	2
#11762848	2
#11762802	2
#103×D₁₇	2
#11762763	2
C₄×SD₁₆×C₁₇	2
#2947	17