Subgroups — #17978472

Order 4352.

598 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545152	C₂	1	2	yes	maximal, complemented
2	#14545150	C₂	1	2	yes	maximal, complemented
3	#14545143	C₂	1	2	yes	maximal, complemented
4	#14545141	C₂	1	2	yes	maximal, complemented
5	#14545120	C₂	1	2	yes	maximal, complemented
6	#14545015	C₂	1	2	yes	maximal
7	C₄×#10587530	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#10587529	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#10587593	C₂	1	1	yes	maximal, characteristic
10	C₂×#10587530	C₂²	1	2	yes	—
11	C₂×#10587529	C₂²	1	2	yes	—
12	#11763730	C₂²	1	4	yes	—
13	C₄⋊#63216	C₂²	1	4	yes	—
14	#11763704	C₂²	1	4	yes	—
15	C₂²⋊#63216	C₂²	1	4	yes	—
16	#11763698	C₂²	1	2	yes	—
17	#11763695	C₂²	1	2	yes	—
18	C₄×#63216	C₂²	1	2	yes	—
19	C₄×#63216	C₂²	1	2	yes	—
20	C₂×#10587593	C₄	1	1	yes	characteristic
21	C₂×#10587593	C₂²	1	1	yes	characteristic
22	C₄×#63252	C₄	1	1	yes	characteristic
23	#11762767	C₄	1	1	yes	characteristic
24	#11762767	C₂²	1	1	yes	characteristic
25	#11762766	C₄	1	1	yes	characteristic
26	#11762766	C₄	1	1	yes	characteristic
27	#11762766	C₂²	1	1	yes	characteristic
28	#11762766	C₂²	1	1	yes	characteristic
29	C₄×#63218	C₄	1	1	yes	characteristic
30	C₄×#63218	C₄	1	1	yes	characteristic
31	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
32	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
34	C₄×M₁₆×C₁₇	C₄	1	1	yes	characteristic
35	C₂×#63252	C₂×C₄	1	1	yes	characteristic
36	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
37	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₂²⋊D₁₇	C₈	1	2	yes	—
40	C₄×C₂²⋊D₁₇	C₈	1	2	yes	—
41	#10587612	C₂×C₄	1	1	yes	characteristic
42	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
43	C₂²×#3622	C₂³	1	1	yes	characteristic
44	#10587593	—	2	4	no	—
45	#10587593	—	2	4	no	—
46	C₂×#63218	C₂×C₄	1	1	yes	characteristic
47	C₂×#63218	C₂×C₄	1	1	yes	characteristic
48	C₂×#63218	C₂×C₄	1	1	yes	characteristic
49	C₂×#63218	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
53	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
54	C₄×D₆₈	C₈	1	2	yes	—
55	#10587558	C₂×C₄	1	1	yes	characteristic
56	#10587558	C₂×C₄	1	1	yes	characteristic
57	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
58	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₃₄	C₈	1	2	yes	—
60	C₂×#63216	C₂³	1	4	yes	—
61	C₂×#63216	C₂³	1	4	yes	—
62	#10587530	—	2	4	no	—
63	#10587529	—	2	4	no	—
64	C₁₇×#97	C₂×C₄	1	1	yes	characteristic
65	C₁₇×#97	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
68	#10587486	C₂×C₄	1	1	yes	characteristic
69	#10587486	C₂×C₄	1	1	yes	characteristic
70	#10587486	C₂×C₄	1	1	yes	characteristic
71	#10587486	C₂×C₄	1	1	yes	characteristic
72	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
73	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
74	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
75	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
76	C₄×#3622	C₂×C₄	1	1	yes	characteristic
77	C₄×#3622	C₂×C₄	1	1	yes	characteristic
78	C₄×#3622	C₂³	1	1	yes	characteristic
79	C₄×#3622	C₂³	1	1	yes	characteristic
80	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
81	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₂²⋊D₁₇	C₂×C₈	1	1	yes	characteristic
83	C₂×C₂²⋊D₁₇	C₂×C₈	1	1	yes	characteristic
84	#63252	—	2	2	no	—
85	#63252	—	2	2	no	—
86	C₂×D₆₈	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
88	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
89	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₃₄	C₂×C₈	1	1	yes	characteristic
92	M₁₆×C₁₇	—	2	2	no	—
93	M₁₆×C₁₇	—	2	2	no	—
94	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₁₇	C₂×C₈	1	2	yes	—
99	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
101	C₂²⋊Dic₁₇	C₂×C₈	1	2	yes	—
102	#63227	C₂×C₈	1	2	yes	—
103	#63227	C₂×C₈	1	2	yes	—
104	C₄⋊Dic₁₇	C₂×C₈	1	2	yes	—
105	#63225	C₂×C₈	1	2	yes	—
106	#63225	C₂×C₈	1	2	yes	—
107	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
108	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
109	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
110	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×#3622	—	2	2	no	—
112	C₂×#3622	—	2	2	no	—
113	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
114	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
115	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
116	C₂×#3622	C₂⁴	1	1	yes	characteristic
117	#63218	—	2	2	no	—
118	#63218	—	2	2	no	—
119	#63218	—	2	2	no	—
120	#63218	—	2	2	no	—
121	C₈×D₁₇	—	2	2	no	—
122	C₈×D₁₇	—	2	2	no	—
123	C₈×D₁₇	M₁₆	1	4	yes	—
124	C₈×D₁₇	#41	1	4	yes	—
125	#63216	—	2	4	no	—
126	#63216	—	2	4	no	—
127	#33420	—	17	1	no	maximal, Sylow
128	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
129	C₂²×D₁₇	C₂²×C₈	1	1	yes	characteristic
130	C₂²×D₁₇	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₁₇	—	2	1	no	—
132	C₂×C₄×C₁₇	—	2	1	no	—
133	C₂×C₄×C₁₇	—	2	4	no	—
134	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
137	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
138	C₂²⋊D₁₇	—	2	2	no	—
139	C₂²⋊D₁₇	—	2	2	no	—
140	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
141	C₂×Dic₁₇	C₂²×C₈	1	2	yes	—
142	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
143	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
144	D₆₈	—	2	2	no	—
145	C₄×D₁₇	—	2	1	no	—
146	C₄×D₁₇	—	2	1	no	—
147	C₄×D₁₇	—	2	4	no	—
148	C₄×D₁₇	—	2	4	no	—
149	C₄×D₁₇	#117	1	2	yes	—
150	C₄×D₁₇	C₂×M₁₆	1	2	yes	—
151	Dic₃₄	—	2	2	no	—
152	C₈×C₁₇	—	2	1	no	—
153	C₈×C₁₇	—	2	1	no	—
154	C₈×C₁₇	#117	1	2	yes	—
155	C₈×C₁₇	C₂×M₁₆	1	2	yes	—
156	#3622	—	2	1	no	—
157	#3622	—	2	1	no	—
158	#3622	C₂×M₁₆	1	2	yes	—
159	#3622	C₂×#41	1	2	yes	—
160	C₄×#131	—	17	1	no	—
161	#2164	—	17	2	no	—
162	#2164	—	17	2	no	—
163	#2162	—	17	2	no	—
164	#2162	—	17	2	no	—
165	#2151	—	17	2	no	—
166	#2113	—	17	2	no	—
167	C₄×M₃₂	—	17	1	no	—
168	C₂×C₄×C₁₆	—	17	1	no	—
169	C₂²×C₁₇	—	2	2	no	—
170	C₂²×C₁₇	—	2	1	no	—
171	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
172	C₂×D₁₇	—	2	2	no	—
173	C₂×D₁₇	—	2	2	no	—
174	C₂×D₁₇	—	2	1	no	—
175	C₂×D₁₇	#431	1	2	yes	—
176	C₄×C₁₇	—	2	2	no	—
177	C₄×C₁₇	—	2	1	no	—
178	C₄×C₁₇	#504	1	4	yes	—
179	C₄×C₁₇	C₂×#117	1	1	yes	characteristic
180	C₄×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
181	Dic₁₇	—	2	2	no	—
182	Dic₁₇	—	2	2	no	—
183	Dic₁₇	—	2	1	no	—
184	Dic₁₇	#431	1	2	yes	—
185	C₂×#131	—	17	1	no	—
186	C₂×#131	—	17	1	no	—
187	C₄×#41	—	17	1	no	—
188	C₂×M₃₂	—	17	2	no	—
189	C₂²×C₁₆	—	17	2	no	—
190	#405	—	17	1	no	—
191	#405	—	17	1	no	—
192	#405	—	17	1	no	—
193	#405	—	17	1	no	—
194	#405	—	17	1	no	—
195	#405	—	17	1	no	—
196	C₄×M₁₆	—	17	1	no	—
197	C₄×M₁₆	—	17	1	no	—
198	C₄×M₁₆	#595	1	1	yes	characteristic
199	C₂×C₄×C₈	—	17	1	no	—
200	C₂×C₄×C₈	—	17	1	no	—
201	C₂×C₄×C₈	—	17	1	no	—
202	C₄⋊C₁₆	—	17	4	no	—
203	C₄⋊C₁₆	—	17	4	no	—
204	C₂²⋊C₁₆	—	17	4	no	—
205	C₂²⋊C₁₆	—	17	4	no	—
206	#346	—	17	2	no	—
207	C₄×C₁₆	—	17	2	no	—
208	C₄×C₁₆	—	17	2	no	—
209	C₄×C₁₆	—	17	2	no	—
210	C₂×C₁₇	—	4	1	no	—
211	C₂×C₁₇	C₂×#431	1	1	yes	characteristic
212	C₂×C₁₇	C₂×#504	1	2	yes	—
213	D₁₇	—	2	2	no	—
214	D₁₇	—	4	1	no	—
215	C₂×#41	—	17	1	no	—
216	#131	—	34	4	no	—
217	#131	—	34	4	no	—
218	C₂×M₁₆	—	17	1	no	—
219	C₂×M₁₆	—	17	1	no	—
220	C₂×M₁₆	—	17	1	no	—
221	C₂×M₁₆	—	17	1	no	—
222	C₂×M₁₆	C₂×#595	1	1	yes	characteristic
223	C₂×M₁₆	C₂×#595	1	1	yes	characteristic
224	C₂²×C₈	—	17	1	no	—
225	C₂²×C₈	—	17	1	no	—
226	C₂²×C₈	—	17	1	no	—
227	C₂²×C₈	—	17	1	no	—
228	C₂²×C₈	—	17	1	no	—
229	C₂²×C₈	—	17	1	no	—
230	C₄×Q₈	—	17	2	no	—
231	C₄×D₄	—	17	2	no	—
232	C₄×D₄	—	17	2	no	—
233	C₄×D₄	—	17	2	no	—
234	#117	—	17	1	no	—
235	#117	—	17	1	no	—
236	#117	—	17	1	no	—
237	C₂×C₄²	—	17	1	no	—
238	C₂×C₄²	—	17	1	no	—
239	C₂×C₄²	C₂×#595	1	1	yes	characteristic
240	M₃₂	—	34	4	no	—
241	C₂×C₁₆	—	17	4	no	—
242	C₂×C₁₆	—	17	4	no	—
243	C₂×C₁₆	—	34	4	no	—
244	#97	—	17	1	no	—
245	#97	—	17	1	no	—
246	#97	—	17	1	no	—
247	#97	—	17	1	no	—
248	#97	C₂×#595	1	1	yes	characteristic
249	#97	C₂×#595	1	1	yes	characteristic
250	C₄×C₈	—	17	1	no	—
251	C₄×C₈	—	17	1	no	—
252	C₄×C₈	—	17	1	no	—
253	C₄×C₈	—	17	1	no	—
254	C₄×C₈	—	17	1	no	—
255	C₄×C₈	—	17	1	no	—
256	C₄×C₈	—	17	1	no	—
257	C₄×C₈	—	17	1	no	—
258	C₄×C₈	C₂×#595	1	1	yes	characteristic
259	C₄×C₈	C₂×#595	1	1	yes	characteristic
260	C₁₇	#33420	1	1	yes	characteristic, complemented, Sylow
261	#41	—	34	2	no	—
262	#41	—	34	2	no	—
263	C₂×Q₈	—	17	1	no	—
264	C₂×D₄	—	17	1	no	—
265	C₂×D₄	—	17	1	no	—
266	C₂×D₄	—	17	1	no	—
267	C₂²×C₄	—	17	1	no	—
268	C₂²×C₄	—	17	2	no	—
269	C₂²×C₄	—	17	2	no	—
270	C₂²×C₄	—	17	1	no	—
271	C₂²×C₄	#63242	1	2	yes	—
272	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
273	M₁₆	—	2	2	no	—
274	M₁₆	—	2	2	no	—
275	M₁₆	—	34	2	no	—
276	M₁₆	—	34	2	no	—
277	M₁₆	—	34	2	no	—
278	M₁₆	—	34	2	no	—
279	C₂×C₈	—	17	1	no	—
280	C₂×C₈	—	17	1	no	—
281	C₂×C₈	—	17	4	no	—
282	C₂×C₈	—	17	4	no	—
283	C₂×C₈	—	17	1	no	—
284	C₂×C₈	—	17	1	no	—
285	C₂×C₈	—	34	2	no	—
286	C₂×C₈	—	34	2	no	—
287	C₂×C₈	—	34	2	no	—
288	C₂×C₈	—	34	2	no	—
289	C₂×C₈	C₂²×#595	1	1	yes	characteristic
290	C₂×C₈	C₂²×#595	1	1	yes	characteristic
291	C₂×C₈	C₂²×#595	1	1	yes	characteristic
292	C₂×C₈	C₂²×#595	1	1	yes	characteristic
293	C₄⋊C₄	—	17	2	no	—
294	C₄⋊C₄	—	17	2	no	—
295	C₄⋊C₄	—	17	2	no	—
296	C₂²⋊C₄	—	17	2	no	—
297	C₂²⋊C₄	—	17	2	no	—
298	C₂²⋊C₄	—	17	2	no	—
299	C₄²	—	17	2	no	—
300	C₄²	—	17	2	no	—
301	C₄²	—	17	1	no	—
302	C₄²	—	17	1	no	—
303	C₄²	#63242	1	2	yes	—
304	C₄²	C₂²×#595	1	1	yes	characteristic
305	C₄²	C₂²×#595	1	1	yes	characteristic
306	C₁₆	—	34	4	no	—
307	C₁₆	—	34	4	no	—
308	C₂³	—	17	1	no	—
309	C₂³	—	17	1	no	—
310	C₂³	C₂×#63242	1	1	yes	characteristic
311	Q₈	—	34	2	no	—
312	D₄	—	34	2	no	—
313	D₄	—	34	2	no	—
314	D₄	—	34	2	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	17	1	no	—
319	C₂×C₄	—	17	2	no	—
320	C₂×C₄	—	17	2	no	—
321	C₂×C₄	—	17	2	no	—
322	C₂×C₄	—	17	2	no	—
323	C₂×C₄	—	17	1	no	—
324	C₂×C₄	—	34	4	no	—
325	C₂×C₄	—	34	4	no	—
326	C₂×C₄	—	34	1	no	—
327	C₂×C₄	—	34	1	no	—
328	C₂×C₄	C₂×#63242	1	1	yes	characteristic
329	C₂×C₄	C₂×#63242	1	2	yes	—
330	C₂×C₄	C₂×#63242	1	2	yes	—
331	C₂×C₄	C₂³×#595	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	17	2	no	—
335	C₈	—	17	2	no	—
336	C₈	—	34	1	no	—
337	C₈	—	34	1	no	—
338	C₈	D₁₇⋊M₁₆	1	2	yes	—
339	C₈	#10587677	1	2	yes	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	17	2	no	—
343	C₂²	—	34	2	no	—
344	C₂²	—	34	1	no	—
345	C₂²	—	34	2	no	—
346	C₂²	C₂²×#63242	1	1	yes	characteristic
347	C₄	—	2	1	no	—
348	C₄	—	2	2	no	—
349	C₄	—	17	2	no	—
350	C₄	—	34	1	no	—
351	C₄	—	34	2	no	—
352	C₄	—	34	2	no	—
353	C₄	#11763868	1	4	yes	—
354	C₄	C₂×#10587677	1	1	yes	characteristic
355	C₄	C₂×D₁₇⋊M₁₆	1	1	yes	characteristic
356	C₂	—	4	1	no	—
357	C₂	—	34	2	no	—
358	C₂	—	68	1	no	—
359	C₂	#14545448	1	1	yes	characteristic
360	C₂	C₂×#11763868	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²×#595
commutator, socle	C₂²×C₁₇	C₂³×C₈
Frattini	C₂×C₄	C₂³×#595
Fitting	C₄×M₁₆×C₁₇	C₄
prime radical	C₂×#63252	C₂×C₄
cosocle	C₂×#3622	C₂⁴
hypercenter	C₄×M₁₆	#595
hypocenter	C₁₇	#33420
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
1. #14545152	C₂	semidirect
2. #14545150	C₂	semidirect
3. #14545143	C₂	semidirect
4. #14545141	C₂	semidirect
5. #14545120	C₂	semidirect
7. C₄×#10587530	C₂	semidirect
8. C₄×#10587529	C₂	semidirect
260. C₁₇	#33420	semidirect

Maximal subgroups

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

subgroup	index
#14545152	2
#14545150	2
#14545143	2
#14545141	2
#14545120	2
#14545015	2
C₄×#10587530	2
C₄×#10587529	2
C₄×#10587593	2
#33420	17