Subgroups — #16927118

Order 4352.

516 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10587555	C₂	1	1	yes	maximal, characteristic
2	#14528936	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762556	C₂	1	2	yes	maximal
4	#14526668	C₂	1	2	yes	maximal
5	C₁₇×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587555	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₃₄	—	2	1	no	—
8	C₂×C₄⋊#3622	—	2	1	no	—
9	C₂×C₄⋊#3622	—	2	1	no	—
10	C₂×C₄⋊#3622	C₂²	1	2	yes	—
11	#11762715	—	2	1	no	—
12	#11762707	—	2	1	no	—
13	#11762707	C₄	1	2	yes	—
14	#11762679	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊Dic₁₇	—	2	1	no	—
16	C₄×C₄⋊Dic₁₇	C₄	1	2	yes	—
17	#11762556	—	2	4	no	—
18	C₂×C₁₇×C₄⋊C₈	—	2	1	no	—
19	C₂×C₁₇×C₄⋊C₈	—	2	1	no	—
20	C₂×C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
21	C₄³×C₁₇	C₂²	1	1	yes	characteristic
22	C₂²×Dic₃₄	—	2	1	no	—
23	C₂²×C₈×C₁₇	—	2	2	no	—
24	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊Dic₁₇	—	2	1	no	—
32	C₂×C₄⋊Dic₁₇	—	2	1	no	—
33	C₂×C₄⋊Dic₁₇	—	2	1	no	—
34	C₂×C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
36	C₂×#63225	—	2	2	no	—
37	C₂×C₄×Dic₁₇	—	2	2	no	—
38	C₂²×#3622	—	2	2	no	—
39	#10587555	—	2	1	no	—
40	#10587555	—	2	1	no	—
41	#10587555	D₄	1	2	yes	—
42	#10587555	Q₈	1	2	yes	—
43	C₄×Dic₃₄	—	4	2	no	—
44	C₄×Dic₃₄	—	4	2	no	—
45	C₁₇×C₄⋊C₈	—	2	2	no	—
46	C₁₇×C₄⋊C₈	—	2	2	no	—
47	C₁₇×C₄⋊C₈	—	4	2	no	—
48	C₁₇×C₄⋊C₈	—	4	2	no	—
49	#10587502	—	2	2	no	—
50	#10587502	—	2	2	no	—
51	C₄⋊#3622	—	2	2	no	—
52	C₄⋊#3622	—	2	2	no	—
53	C₄⋊#3622	—	4	2	no	—
54	C₄⋊#3622	—	4	2	no	—
55	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₁₇	—	2	2	no	—
63	C₂²×Dic₁₇	—	2	2	no	—
64	C₂×Dic₃₄	—	2	2	no	—
65	C₂×Dic₃₄	—	2	2	no	—
66	C₂×Dic₃₄	—	4	1	no	—
67	C₂×Dic₃₄	—	4	1	no	—
68	C₂×Dic₃₄	—	4	1	no	—
69	C₂×Dic₃₄	—	4	1	no	—
70	C₂×C₈×C₁₇	—	4	2	no	—
71	C₂×C₈×C₁₇	—	4	2	no	—
72	C₂×C₈×C₁₇	—	4	2	no	—
73	C₄²×C₁₇	—	2	2	no	—
74	C₄²×C₁₇	—	2	1	no	—
75	C₄²×C₁₇	—	2	1	no	—
76	C₄²×C₁₇	—	2	1	no	—
77	C₄²×C₁₇	—	2	1	no	—
78	C₄²×C₁₇	—	4	1	no	—
79	C₄²×C₁₇	D₈	1	2	yes	—
80	C₄²×C₁₇	SD₁₆	1	2	yes	—
81	C₄²×C₁₇	SD₁₆	1	2	yes	—
82	C₄²×C₁₇	Q₁₆	1	2	yes	—
83	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
85	C₄²×C₁₇	#41	1	1	yes	characteristic
86	C₄²×C₁₇	#41	1	1	yes	characteristic
87	C₄⋊Dic₁₇	—	2	2	no	—
88	C₄⋊Dic₁₇	—	2	2	no	—
89	C₄⋊Dic₁₇	—	2	2	no	—
90	C₄⋊Dic₁₇	—	4	1	no	—
91	C₄⋊Dic₁₇	—	4	1	no	—
92	C₄⋊Dic₁₇	—	4	1	no	—
93	C₄⋊Dic₁₇	—	4	1	no	—
94	C₄⋊Dic₁₇	—	4	2	no	—
95	C₄⋊Dic₁₇	C₄⋊C₄	1	4	yes	—
96	#63225	—	4	4	no	—
97	C₄×Dic₁₇	—	4	4	no	—
98	C₂×#3622	—	4	2	no	—
99	C₂×#3622	—	4	2	no	—
100	C₂×#3622	—	4	2	no	—
101	#8586	—	17	1	no	maximal, Sylow
102	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₇	—	2	1	no	—
104	C₂×C₄×C₁₇	—	2	1	no	—
105	C₂×C₄×C₁₇	—	2	1	no	—
106	C₂×C₄×C₁₇	—	2	1	no	—
107	C₂×C₄×C₁₇	—	2	1	no	—
108	C₂×C₄×C₁₇	—	2	1	no	—
109	C₂×C₄×C₁₇	—	2	1	no	—
110	C₂×C₄×C₁₇	—	2	1	no	—
111	C₂×C₄×C₁₇	—	2	1	no	—
112	C₂×C₄×C₁₇	#101	1	2	yes	—
113	C₂×C₄×C₁₇	#102	1	2	yes	—
114	C₂×C₄×C₁₇	#102	1	2	yes	—
115	C₂×C₄×C₁₇	#103	1	2	yes	—
116	C₂×C₄×C₁₇	#103	1	2	yes	—
117	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
119	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
120	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
127	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
130	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
131	C₂×Dic₁₇	—	2	4	no	—
132	C₂×Dic₁₇	—	4	2	no	—
133	C₂×Dic₁₇	—	4	2	no	—
134	C₂×Dic₁₇	—	4	2	no	—
135	C₂×Dic₁₇	—	4	2	no	—
136	C₂×Dic₁₇	—	4	2	no	—
137	Dic₃₄	—	4	2	no	—
138	Dic₃₄	—	4	2	no	—
139	Dic₃₄	—	4	2	no	—
140	Dic₃₄	—	4	2	no	—
141	C₈×C₁₇	—	8	2	no	—
142	#3622	—	8	2	no	—
143	C₄×C₄⋊Q₈	—	17	1	no	—
144	#1842	—	17	1	no	—
145	#1842	D₁₇	1	1	yes	characteristic
146	C₂×#332	—	17	2	no	—
147	#1296	—	17	2	no	—
148	C₂²×C₁₇	#380	1	1	yes	characteristic
149	C₂²×C₁₇	C₂×#101	1	1	yes	characteristic
150	C₂²×C₁₇	#413	1	1	yes	characteristic
151	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
152	C₂²×C₁₇	C₂×#103	1	1	yes	characteristic
153	C₂²×C₁₇	#417	1	1	yes	characteristic
154	C₂²×C₁₇	#417	1	1	yes	characteristic
155	C₄×C₁₇	—	2	1	no	—
156	C₄×C₁₇	—	2	1	no	—
157	C₄×C₁₇	—	2	2	no	—
158	C₄×C₁₇	—	2	2	no	—
159	C₄×C₁₇	—	4	1	no	—
160	C₄×C₁₇	#332	1	4	yes	complemented
161	C₄×C₁₇	#459	1	1	yes	characteristic
162	C₄×C₁₇	#460	1	1	yes	characteristic
163	C₄×C₁₇	#461	1	1	yes	characteristic
164	C₄×C₁₇	#462	1	1	yes	characteristic
165	C₄×C₁₇	#474	1	1	yes	characteristic
166	C₄×C₁₇	#475	1	1	yes	characteristic
167	C₄×C₁₇	#476	1	1	yes	characteristic
168	C₄×C₁₇	#477	1	1	yes	characteristic
169	Dic₁₇	—	4	4	no	—
170	Dic₁₇	—	8	2	no	—
171	C₂×C₄⋊Q₈	—	17	1	no	—
172	C₂×C₄×Q₈	—	34	1	no	—
173	C₂×C₄⋊C₈	—	2	1	no	—
174	C₂×C₄⋊C₈	—	2	1	no	—
175	C₂×C₄⋊C₈	—	17	2	no	—
176	C₂×C₄⋊C₈	—	34	1	no	—
177	C₂×C₄⋊C₈	—	34	1	no	—
178	C₂×C₄⋊C₈	C₂×D₁₇	1	2	yes	—
179	#391	—	34	1	no	—
180	#389	—	17	2	no	—
181	#389	—	34	1	no	—
182	#384	—	17	1	no	—
183	C₄×C₄⋊C₄	—	17	2	no	—
184	C₄×C₄⋊C₄	—	34	1	no	—
185	C₄³	C₂×D₁₇	1	1	yes	characteristic
186	#332	—	34	4	no	—
187	C₂×C₁₇	C₂×#332	1	1	yes	characteristic
188	C₂×C₁₇	#1547	1	1	yes	characteristic
189	C₂×C₁₇	#1550	1	1	yes	characteristic
190	C₂×C₁₇	#1561	1	1	yes	characteristic
191	C₂×C₁₇	#1801	1	1	yes	characteristic
192	C₂×C₁₇	#1802	1	1	yes	characteristic
193	C₂×C₁₇	#1803	1	1	yes	characteristic
194	C₂²×Q₈	—	34	1	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	34	2	no	—
197	C₄⋊Q₈	—	17	2	no	—
198	C₄⋊Q₈	—	17	2	no	—
199	C₄⋊Q₈	—	34	1	no	—
200	C₄⋊Q₈	—	34	1	no	—
201	C₄×Q₈	—	68	2	no	—
202	C₄×Q₈	—	68	2	no	—
203	C₂×C₄⋊C₄	—	17	2	no	—
204	C₂×C₄⋊C₄	—	17	2	no	—
205	C₂×C₄⋊C₄	—	34	1	no	—
206	C₂×C₄⋊C₄	—	34	1	no	—
207	C₂×C₄⋊C₄	—	34	1	no	—
208	C₂×C₄⋊C₄	—	34	2	no	—
209	C₂×C₄²	—	34	2	no	—
210	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
211	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
212	C₂×C₄²	D₆₈	1	1	yes	characteristic
213	C₂×C₄²	D₆₈	1	1	yes	characteristic
214	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
215	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
216	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
217	C₄⋊C₈	—	2	2	no	—
218	C₄⋊C₈	—	2	2	no	—
219	C₄⋊C₈	—	4	2	no	—
220	C₄⋊C₈	—	4	2	no	—
221	C₄⋊C₈	—	34	2	no	—
222	C₄⋊C₈	—	34	2	no	—
223	C₄⋊C₈	—	68	2	no	—
224	C₄⋊C₈	—	68	2	no	—
225	#95	—	34	2	no	—
226	#95	—	34	2	no	—
227	C₁₇	#8586	1	1	yes	characteristic, complemented, Sylow
228	C₂×Q₈	—	34	2	no	—
229	C₂×Q₈	—	34	2	no	—
230	C₂×Q₈	—	68	1	no	—
231	C₂×Q₈	—	68	1	no	—
232	C₂×Q₈	—	68	1	no	—
233	C₂×Q₈	—	68	1	no	—
234	C₂²×C₄	—	34	2	no	—
235	C₂²×C₄	—	34	2	no	—
236	C₂²×C₄	#63227	1	1	yes	characteristic
237	C₂²×C₄	#63227	1	1	yes	characteristic
238	C₂²×C₄	#63227	1	1	yes	characteristic
239	C₂²×C₄	#63227	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	68	2	no	—
247	C₂×C₈	—	68	2	no	—
248	C₂×C₈	—	68	2	no	—
249	C₄⋊C₄	—	17	4	no	—
250	C₄⋊C₄	—	34	2	no	—
251	C₄⋊C₄	—	34	2	no	—
252	C₄⋊C₄	—	34	2	no	—
253	C₄⋊C₄	—	68	2	no	—
254	C₄⋊C₄	—	68	4	no	—
255	C₄⋊C₄	—	68	1	no	—
256	C₄⋊C₄	—	68	1	no	—
257	C₄⋊C₄	—	68	1	no	—
258	C₄⋊C₄	—	68	1	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	4	1	no	—
265	C₄²	—	68	4	no	—
266	C₄²	#63219	1	2	yes	—
267	C₄²	Dic₆₈	1	2	yes	—
268	C₄²	#63228	1	2	yes	—
269	C₄²	#63230	1	2	yes	—
270	C₄²	D₄×D₁₇	1	1	yes	characteristic
271	C₄²	#63254	1	1	yes	characteristic
272	C₄²	Q₈×D₁₇	1	1	yes	characteristic
273	C₄²	#63256	1	1	yes	characteristic
274	C₂³	C₂×#63227	1	1	yes	characteristic
275	Q₈	—	68	2	no	—
276	Q₈	—	68	2	no	—
277	Q₈	—	68	2	no	—
278	Q₈	—	68	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	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	—	34	4	no	—
289	C₂×C₄	—	68	2	no	—
290	C₂×C₄	—	68	2	no	—
291	C₂×C₄	—	68	2	no	—
292	C₂×C₄	—	68	2	no	—
293	C₂×C₄	—	68	2	no	—
294	C₂×C₄	#10587480	1	2	yes	—
295	C₂×C₄	#10587480	1	2	yes	—
296	C₂×C₄	#10587487	1	2	yes	—
297	C₂×C₄	#10587487	1	2	yes	—
298	C₂×C₄	#10587495	1	2	yes	—
299	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
300	C₂×C₄	#10587578	1	1	yes	characteristic
301	C₂×C₄	#10587581	1	1	yes	characteristic
302	C₂×C₄	#10587583	1	1	yes	characteristic
303	C₂×C₄	C₂×#63219	1	1	yes	characteristic
304	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
305	C₂×C₄	#10587594	1	1	yes	characteristic
306	C₂×C₄	#10587595	1	1	yes	characteristic
307	C₂×C₄	C₂×#63228	1	1	yes	characteristic
308	C₂×C₄	#10587618	1	1	yes	characteristic
309	C₂×C₄	#10587624	1	1	yes	characteristic
310	C₂×C₄	C₂×#63230	1	1	yes	characteristic
311	C₂×C₄	#10587628	1	1	yes	characteristic
312	C₂×C₄	#10587632	1	1	yes	characteristic
313	C₈	—	8	2	no	—
314	C₈	—	136	2	no	—
315	C₂²	#11762666	1	1	yes	characteristic
316	C₂²	C₂×#10587495	1	1	yes	characteristic
317	C₂²	#11762791	1	1	yes	characteristic
318	C₂²	C₂×#10587480	1	1	yes	characteristic
319	C₂²	C₂×#10587487	1	1	yes	characteristic
320	C₂²	#11762808	1	1	yes	characteristic
321	C₂²	#11762811	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	4	1	no	—
327	C₄	—	68	4	no	—
328	C₄	—	136	2	no	—
329	C₄	#11762556	1	4	yes	complemented
330	C₄	#11763001	1	1	yes	characteristic
331	C₄	#11763010	1	1	yes	characteristic
332	C₄	#11763011	1	1	yes	characteristic
333	C₄	#11763020	1	1	yes	characteristic
334	C₄	#11763086	1	1	yes	characteristic
335	C₄	#11763095	1	1	yes	characteristic
336	C₄	#11763096	1	1	yes	characteristic
337	C₄	#11763105	1	1	yes	characteristic
338	C₂	C₂×#11762556	1	1	yes	characteristic
339	C₂	#14527462	1	1	yes	characteristic
340	C₂	#14527472	1	1	yes	characteristic
341	C₂	#14527521	1	1	yes	characteristic
342	C₂	#14528777	1	1	yes	characteristic
343	C₂	#14528781	1	1	yes	characteristic
344	C₂	#14528790	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#63227
commutator	C₂²×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1842	C₂
socle, prime radical	C₂³×C₁₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1842	D₁₇
hypocenter	C₁₇	#8586
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
160. C₄×C₁₇	#332	semidirect
227. C₁₇	#8586	semidirect
329. C₄	#11762556	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10587555	2
#14528936	2
C₂×#11762556	2
#14526668	2
C₁₇×#1842	2
#8586	17