Subgroups — #14537640

Order 2176.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
2	#11763285	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763281	C₂	1	1	yes	maximal, characteristic
4	#11762911	C₂	1	1	yes	maximal, characteristic, complemented
5	#11762910	C₂	1	1	yes	maximal, characteristic, complemented
6	#11762908	C₂	1	1	yes	maximal, characteristic, complemented
7	#11762900	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#63230	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#63229	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762894	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762846	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762816	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762812	C₂	1	1	yes	maximal, characteristic
14	#11762766	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#440	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₇	C₂²	1	1	yes	characteristic
17	C₂×D₄×D₁₇	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic, complemented
20	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
21	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic, complemented
22	C₂×#63230	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×#63227	—	2	1	no	—
27	SD₁₆×D₁₇	C₄	1	8	yes	complemented
28	C₂×#63219	C₂²	1	1	yes	characteristic
29	C₂×#63218	C₂²	1	1	yes	characteristic
30	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
31	#10587579	C₂²	1	1	yes	characteristic
32	#10587578	—	2	1	no	—
33	#10587578	C₂²	1	1	yes	characteristic
34	#10587578	C₂²	1	1	yes	characteristic
35	#10587573	—	2	1	no	—
36	#10587573	C₂²	1	1	yes	characteristic, complemented
37	#10587567	—	2	1	no	—
38	C₂²⋊C₄×D₁₇	—	2	1	no	—
39	C₄×D₆₈	C₂²	1	1	yes	characteristic
40	#10587558	—	2	1	no	—
41	#10587558	C₂²	1	1	yes	characteristic
42	C₄×Dic₃₄	—	2	1	no	—
43	C₄×Dic₃₄	C₂²	1	1	yes	characteristic, complemented
44	C₁₇×#107	C₂²	1	1	yes	characteristic, complemented
45	C₁₇×#103	C₂²	1	1	yes	characteristic, complemented
46	C₁₇×#102	C₂²	1	1	yes	characteristic
47	C₁₇×#97	C₂²	1	1	yes	characteristic, complemented
48	#10587506	C₂²	1	1	yes	characteristic, complemented
49	#10587503	C₂²	1	1	yes	characteristic
50	#10587492	C₂²	1	1	yes	characteristic
51	#10587489	C₂²	1	1	yes	characteristic, complemented
52	#10587487	C₂²	1	1	yes	characteristic, complemented
53	#10587486	C₂²	1	1	yes	characteristic, complemented
54	C₈×Dic₁₇	C₂²	1	1	yes	characteristic, complemented
55	#10587481	C₂²	1	1	yes	characteristic, complemented
56	#10587480	C₂²	1	1	yes	characteristic
57	#10587479	C₂²	1	1	yes	characteristic, complemented
58	#10587479	C₂²	1	1	yes	characteristic, complemented
59	C₄×#3622	C₂²	1	1	yes	characteristic, complemented
60	C₂³×D₁₇	—	2	1	no	—
61	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₇	—	2	1	no	—
64	C₂×C₂²⋊D₁₇	—	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	D₄×D₁₇	—	2	2	no	—
69	D₄×D₁₇	C₂×C₄	1	4	yes	—
70	C₂×D₆₈	C₂³	1	1	yes	characteristic
71	C₂×C₄×D₁₇	—	2	1	no	—
72	C₂×C₄×D₁₇	—	2	1	no	—
73	C₂×C₄×D₁₇	—	2	1	no	—
74	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
75	C₂×Dic₃₄	—	2	1	no	—
76	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
77	SD₁₆×C₁₇	C₂×C₄	1	4	yes	complemented
78	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₁₇	—	2	1	no	—
81	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₁₇	—	2	1	no	—
84	C₄²×C₁₇	—	2	1	no	—
85	C₄²×C₁₇	C₂³	1	1	yes	characteristic
86	C₂²⋊Dic₁₇	—	2	1	no	—
87	#63230	C₂×C₄	1	4	yes	complemented
88	#63229	C₂×C₄	1	4	yes	complemented
89	#63227	—	2	1	no	—
90	#63227	—	2	1	no	—
91	#63227	—	2	1	no	—
92	#63227	—	2	1	no	—
93	#63227	D₄	1	1	yes	characteristic
94	#63227	D₄	1	1	yes	characteristic
95	C₄⋊Dic₁₇	—	2	1	no	—
96	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
97	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
98	#63225	—	2	1	no	—
99	#63225	—	2	1	no	—
100	#63225	D₄	1	1	yes	characteristic
101	#63225	D₄	1	1	yes	characteristic
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	#63219	C₂×C₄	1	4	yes	complemented
111	#63218	—	2	2	no	—
112	C₈×D₁₇	C₂×C₄	1	4	yes	complemented
113	C₂³×C₁₇	—	2	1	no	—
114	C₂²×D₁₇	—	2	1	no	—
115	C₂²×D₁₇	—	2	4	no	—
116	C₂²×D₁₇	—	8	1	no	—
117	C₂²×D₁₇	—	8	1	no	—
118	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
119	Q₈×C₁₇	—	2	1	no	—
120	Q₈×C₁₇	C₂²×C₄	1	2	yes	—
121	D₄×C₁₇	—	2	1	no	—
122	D₄×C₁₇	C₂²×C₄	1	2	yes	—
123	C₂×C₄×C₁₇	—	2	1	no	—
124	C₂×C₄×C₁₇	—	2	1	no	—
125	C₂×C₄×C₁₇	—	2	1	no	—
126	C₂×C₄×C₁₇	—	8	1	no	—
127	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₁₇	—	2	4	no	—
131	C₂×Dic₁₇	—	2	1	no	—
132	C₂×Dic₁₇	—	2	1	no	—
133	C₂×Dic₁₇	—	2	1	no	—
134	C₂×Dic₁₇	—	8	1	no	—
135	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
138	D₆₈	—	2	1	no	—
139	D₆₈	C₂²×C₄	1	2	yes	—
140	C₄×D₁₇	—	2	4	no	—
141	C₄×D₁₇	—	8	1	no	—
142	C₄×D₁₇	—	8	1	no	—
143	C₄×D₁₇	C₂²×C₄	1	2	yes	—
144	C₄×D₁₇	#41	1	2	yes	—
145	Dic₃₄	—	2	1	no	—
146	Dic₃₄	—	2	4	no	—
147	Dic₃₄	C₂²×C₄	1	2	yes	—
148	C₈×C₁₇	—	2	1	no	—
149	C₈×C₁₇	C₂²×C₄	1	2	yes	—
150	#3622	—	2	1	no	—
151	#3622	C₂²×C₄	1	2	yes	—
152	#2949	—	17	1	no	maximal, Sylow
153	C₂²×C₁₇	—	2	2	no	—
154	C₂²×C₁₇	—	4	1	no	—
155	C₂²×C₁₇	—	4	1	no	—
156	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×D₁₇	—	2	1	no	—
158	C₂×D₁₇	—	2	1	no	—
159	C₂×D₁₇	—	2	2	no	—
160	C₂×D₁₇	—	4	1	no	—
161	C₂×D₁₇	—	4	1	no	—
162	C₂×D₁₇	—	8	4	no	—
163	C₂×D₁₇	C₄×D₄	1	2	yes	—
164	C₄×C₁₇	—	2	2	no	—
165	C₄×C₁₇	—	2	1	no	—
166	C₄×C₁₇	—	2	1	no	—
167	C₄×C₁₇	—	4	1	no	—
168	C₄×C₁₇	—	4	1	no	—
169	C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
170	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
171	Dic₁₇	—	2	2	no	—
172	Dic₁₇	—	2	1	no	—
173	Dic₁₇	—	2	1	no	—
174	Dic₁₇	—	4	1	no	—
175	Dic₁₇	—	4	1	no	—
176	Dic₁₇	C₄×D₄	1	2	yes	—
177	C₂²×SD₁₆	—	17	1	no	—
178	#519	—	17	1	no	—
179	#518	—	17	1	no	—
180	#440	—	17	1	no	—
181	#440	—	17	1	no	—
182	#440	—	17	1	no	—
183	#440	D₁₇	1	1	yes	characteristic, complemented
184	C₄×SD₁₆	—	17	1	no	—
185	C₄×SD₁₆	—	17	1	no	—
186	C₄×SD₁₆	—	17	1	no	—
187	C₄×SD₁₆	—	17	1	no	—
188	#427	—	17	1	no	—
189	#419	—	17	1	no	—
190	#418	—	17	1	no	—
191	#405	—	17	1	no	—
192	C₂×C₁₇	—	4	2	no	—
193	C₂×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
194	C₂×C₁₇	#577	1	1	yes	characteristic
195	C₂×C₁₇	#577	1	1	yes	characteristic
196	D₁₇	—	2	2	no	—
197	D₁₇	—	4	2	no	—
198	C₂²×Q₈	—	17	1	no	—
199	C₂²×D₄	—	17	1	no	—
200	C₂×SD₁₆	—	17	1	no	—
201	C₂×SD₁₆	—	17	1	no	—
202	C₂×SD₁₆	—	17	8	no	—
203	C₂×SD₁₆	—	17	1	no	—
204	C₂×SD₁₆	C₂×D₁₇	1	1	yes	characteristic
205	C₂×M₁₆	—	17	1	no	—
206	C₂²×C₈	—	17	1	no	—
207	C₄×Q₈	—	17	1	no	—
208	C₄×Q₈	—	17	1	no	—
209	C₄×Q₈	—	17	1	no	—
210	C₄×Q₈	—	34	1	no	—
211	C₄×Q₈	—	34	1	no	—
212	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic, complemented
213	C₄×D₄	—	17	1	no	—
214	C₄×D₄	—	17	1	no	—
215	C₄×D₄	—	17	1	no	—
216	C₄×D₄	—	34	1	no	—
217	C₄×D₄	—	34	1	no	—
218	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
219	#117	—	17	1	no	—
220	#117	—	17	1	no	—
221	#117	—	17	1	no	—
222	#117	—	34	1	no	—
223	#117	—	34	1	no	—
224	C₂×C₂²⋊C₄	—	34	1	no	—
225	C₂×C₂²⋊C₄	—	34	1	no	—
226	#107	—	17	1	no	—
227	#107	C₂×D₁₇	1	1	yes	characteristic, complemented
228	#106	—	17	1	no	—
229	#106	—	17	1	no	—
230	#103	—	17	1	no	—
231	#103	—	17	1	no	—
232	#103	—	17	1	no	—
233	#103	C₂×D₁₇	1	1	yes	characteristic, complemented
234	#102	—	17	1	no	—
235	#102	—	17	1	no	—
236	#102	—	17	1	no	—
237	#102	C₂×D₁₇	1	1	yes	characteristic
238	#97	—	17	1	no	—
239	#97	C₂×D₁₇	1	1	yes	characteristic, complemented
240	C₄×C₈	—	17	1	no	—
241	C₄×C₈	—	17	1	no	—
242	C₁₇	#2949	1	1	yes	characteristic, complemented, Sylow
243	C₂⁴	—	34	1	no	—
244	C₂×Q₈	—	17	4	no	—
245	C₂×Q₈	—	17	1	no	—
246	C₂×Q₈	—	34	1	no	—
247	C₂×Q₈	—	34	2	no	—
248	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
249	C₂×D₄	—	17	4	no	—
250	C₂×D₄	—	17	1	no	—
251	C₂×D₄	—	34	2	no	—
252	C₂×D₄	—	34	1	no	—
253	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	17	1	no	—
256	C₂²×C₄	—	34	1	no	—
257	C₂²×C₄	—	34	1	no	—
258	C₂²×C₄	—	34	1	no	—
259	C₂²×C₄	—	34	1	no	—
260	SD₁₆	—	17	4	no	—
261	SD₁₆	—	17	4	no	—
262	SD₁₆	—	17	4	no	—
263	SD₁₆	C₄×D₁₇	1	4	yes	complemented
264	M₁₆	—	34	2	no	—
265	C₂×C₈	—	17	1	no	—
266	C₂×C₈	—	17	4	no	—
267	C₂×C₈	—	17	1	no	—
268	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
269	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	17	1	no	—
272	C₄⋊C₄	—	17	1	no	—
273	C₄⋊C₄	—	17	1	no	—
274	C₄⋊C₄	—	17	1	no	—
275	C₄⋊C₄	—	34	1	no	—
276	C₄⋊C₄	—	34	1	no	—
277	C₄⋊C₄	—	34	1	no	—
278	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
279	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	17	1	no	—
282	C₂²⋊C₄	—	17	1	no	—
283	C₂²⋊C₄	—	34	1	no	—
284	C₂²⋊C₄	—	34	1	no	—
285	C₂²⋊C₄	—	34	1	no	—
286	C₂²⋊C₄	—	34	1	no	—
287	C₂²⋊C₄	—	34	1	no	—
288	C₄²	—	2	1	no	—
289	C₄²	—	17	1	no	—
290	C₄²	—	34	1	no	—
291	C₄²	—	34	1	no	—
292	C₄²	—	34	1	no	—
293	C₄²	—	34	1	no	—
294	C₄²	—	34	1	no	—
295	C₄²	C₂²×D₁₇	1	1	yes	characteristic
296	C₂³	—	2	1	no	—
297	C₂³	—	17	1	no	—
298	C₂³	—	34	4	no	—
299	C₂³	—	34	1	no	—
300	C₂³	—	136	1	no	—
301	C₂³	—	136	1	no	—
302	Q₈	—	2	1	no	—
303	Q₈	—	17	2	no	—
304	Q₈	—	34	4	no	—
305	Q₈	—	34	1	no	—
306	Q₈	C₂×C₄×D₁₇	1	2	yes	—
307	D₄	—	2	1	no	—
308	D₄	—	17	2	no	—
309	D₄	—	34	4	no	—
310	D₄	—	34	1	no	—
311	D₄	C₂×C₄×D₁₇	1	2	yes	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	8	1	no	—
316	C₂×C₄	—	17	1	no	—
317	C₂×C₄	—	17	2	no	—
318	C₂×C₄	—	17	2	no	—
319	C₂×C₄	—	17	1	no	—
320	C₂×C₄	—	17	1	no	—
321	C₂×C₄	—	34	4	no	—
322	C₂×C₄	—	34	1	no	—
323	C₂×C₄	—	34	1	no	—
324	C₂×C₄	—	34	1	no	—
325	C₂×C₄	—	136	1	no	—
326	C₂×C₄	—	136	1	no	—
327	C₂×C₄	—	136	1	no	—
328	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
329	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
330	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	17	2	no	—
333	C₈	—	34	1	no	—
334	C₈	C₂×C₄×D₁₇	1	2	yes	—
335	C₂²	—	2	2	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	17	2	no	—
339	C₂²	—	34	1	no	—
340	C₂²	—	34	1	no	—
341	C₂²	—	34	2	no	—
342	C₂²	—	68	1	no	—
343	C₂²	—	68	1	no	—
344	C₂²	—	136	4	no	—
345	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
346	C₄	—	2	2	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	4	1	no	—
350	C₄	—	4	1	no	—
351	C₄	—	17	2	no	—
352	C₄	—	34	2	no	—
353	C₄	—	34	1	no	—
354	C₄	—	34	1	no	—
355	C₄	—	68	1	no	—
356	C₄	—	68	1	no	—
357	C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
358	C₄	#41×D₁₇	1	1	yes	characteristic
359	C₂	—	4	2	no	—
360	C₂	—	34	2	no	—
361	C₂	—	68	2	no	—
362	C₂	C₄×D₄×D₁₇	1	1	yes	characteristic
363	C₂	#11763618	1	1	yes	characteristic
364	C₂	#11763619	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₇
commutator	C₄×C₁₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#440	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×D₄×D₁₇	C₂²
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	#440	D₁₇
hypocenter	C₁₇	#2949
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
2. #11763285	C₂	semidirect
4. #11762911	C₂	semidirect
5. #11762910	C₂	semidirect
6. #11762908	C₂	semidirect
7. #11762900	C₂	semidirect
8. C₄×#63230	C₂	semidirect
9. C₄×#63229	C₂	semidirect
10. #11762894	C₂	semidirect
11. #11762846	C₂	semidirect
12. #11762816	C₂	semidirect
14. #11762766	C₂	semidirect
15. C₁₇×#440	C₂	semidirect
19. C₄×Q₈×C₁₇	C₂²	semidirect
21. Q₈×Dic₁₇	C₂²	semidirect
27. SD₁₆×D₁₇	C₄	semidirect
36. #10587573	C₂²	semidirect
43. C₄×Dic₃₄	C₂²	semidirect
44. C₁₇×#107	C₂²	semidirect
45. C₁₇×#103	C₂²	semidirect
47. C₁₇×#97	C₂²	semidirect
48. #10587506	C₂²	semidirect
51. #10587489	C₂²	semidirect
52. #10587487	C₂²	semidirect
53. #10587486	C₂²	semidirect
54. C₈×Dic₁₇	C₂²	semidirect
55. #10587481	C₂²	semidirect
57. #10587479	C₂²	semidirect
58. #10587479	C₂²	semidirect
59. C₄×#3622	C₂²	semidirect
67. Q₈×D₁₇	C₂×C₄	semidirect
77. SD₁₆×C₁₇	C₂×C₄	semidirect
87. #63230	C₂×C₄	semidirect
88. #63229	C₂×C₄	semidirect
110. #63219	C₂×C₄	semidirect
112. C₈×D₁₇	C₂×C₄	semidirect
183. #440	D₁₇	semidirect
212. C₄×Q₈	C₂×D₁₇	semidirect
227. #107	C₂×D₁₇	semidirect
233. #103	C₂×D₁₇	semidirect
239. #97	C₂×D₁₇	semidirect
242. C₁₇	#2949	semidirect
263. 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×SD₁₆×D₁₇	2
#11763285	2
#11763281	2
#11762911	2
#11762910	2
#11762908	2
#11762900	2
C₄×#63230	2
C₄×#63229	2
#11762894	2
#11762846	2
#11762816	2
#11762812	2
#11762766	2
C₁₇×#440	2
#2949	17