Subgroups — #17034688

Order 4352.

616 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10587587	C₂	1	1	yes	maximal, characteristic
2	#14528930	C₂	1	2	yes	maximal, complemented
3	#14528909	C₂	1	1	yes	maximal, characteristic, complemented
4	#14528907	C₂	1	1	yes	maximal, characteristic, complemented
5	#14528887	C₂	1	2	yes	maximal
6	#14528831	C₂	1	1	yes	maximal, characteristic
7	#14528827	C₂	1	1	yes	maximal, characteristic
8	#14528663	C₂	1	1	yes	maximal, characteristic, complemented
9	#14528585	C₂	1	2	yes	maximal
10	#14528539	C₂	1	2	yes	maximal, complemented
11	C₁₇×#1772	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10587587	C₂²	1	1	yes	characteristic
13	C₂×#10587587	C₂²	1	1	yes	characteristic
14	C₄×#63252	C₂²	1	1	yes	characteristic
15	C₂×#10587490	C₂²	1	1	yes	characteristic
16	#11762847	C₂²	1	1	yes	characteristic
17	#11762791	C₂²	1	2	yes	—
18	#11762782	C₂²	1	2	yes	—
19	#11762764	C₂²	1	2	yes	—
20	#11762764	C₂²	1	1	yes	characteristic
21	#11762764	C₂²	1	1	yes	characteristic
22	#11762763	C₄	1	2	yes	complemented
23	#11762763	C₄	1	2	yes	complemented
24	C₄×#63223	C₂²	1	1	yes	characteristic
25	C₄×#63218	C₄	1	2	yes	complemented
26	C₄×C₈×D₁₇	C₄	1	2	yes	complemented
27	#11762581	C₂²	1	2	yes	—
28	#11762579	C₂²	1	2	yes	—
29	#11762569	C₂²	1	2	yes	—
30	#11762567	C₂²	1	2	yes	—
31	#11762566	C₂²	1	2	yes	—
32	#11762565	C₂²	1	2	yes	—
33	#11762565	C₂²	1	2	yes	—
34	C₂×C₁₇×#108	C₂²	1	1	yes	characteristic
35	C₁₇×#427	C₂²	1	1	yes	characteristic
36	C₁₇×#405	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
38	C₁₇×#339	C₂²	1	2	yes	—
39	C₂×#63252	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₁₇	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
43	C₁₇×#117	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
45	C₄×C₂²⋊D₁₇	—	2	2	no	—
46	#10587612	C₂³	1	2	yes	—
47	#10587612	C₂³	1	1	yes	characteristic
48	C₂×#63223	C₂³	1	2	yes	—
49	C₂×#63223	C₂³	1	1	yes	characteristic
50	C₂×#63223	C₂³	1	1	yes	characteristic
51	#10587587	—	2	1	no	—
52	#10587587	—	2	1	no	—
53	#10587587	—	2	1	no	—
54	#10587587	—	2	1	no	—
55	#10587587	—	2	1	no	—
56	#10587587	—	2	1	no	—
57	#10587587	—	2	1	no	—
58	#10587587	—	2	1	no	—
59	C₂×#63218	C₂×C₄	1	2	yes	—
60	C₂×#63218	C₂×C₄	1	2	yes	—
61	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
62	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
63	C₄×D₆₈	—	2	1	no	—
64	#10587558	C₂×C₄	1	2	yes	—
65	C₄²×D₁₇	C₂×C₄	1	2	yes	—
66	C₄×Dic₃₄	—	2	1	no	—
67	C₁₇×#108	—	2	1	no	—
68	C₁₇×#108	—	2	1	no	—
69	C₁₇×#107	—	2	1	no	—
70	C₁₇×#106	—	2	1	no	—
71	C₁₇×#97	—	2	2	no	—
72	C₄×C₈×C₁₇	—	2	2	no	—
73	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
75	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
77	#10587495	—	2	2	no	—
78	#10587494	—	2	2	no	—
79	#10587490	—	2	1	no	—
80	#10587490	—	2	1	no	—
81	#10587489	—	2	1	no	—
82	#10587488	—	2	1	no	—
83	#10587486	—	2	2	no	—
84	#10587486	C₂×C₄	1	2	yes	—
85	#10587486	C₂×C₄	1	2	yes	—
86	C₈×Dic₁₇	—	2	2	no	—
87	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
88	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
89	#10587477	—	2	4	no	—
90	#10587474	C₂×C₄	1	2	yes	—
91	C₄×#3622	C₂×C₄	1	2	yes	—
92	C₂²×C₄×C₁₇	—	2	1	no	—
93	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
94	C₂×C₂²⋊D₁₇	#41	1	2	yes	—
95	#63252	—	2	1	no	—
96	#63252	—	2	1	no	—
97	#63252	—	2	1	no	—
98	#63252	—	2	1	no	—
99	C₂×D₆₈	#41	1	1	yes	characteristic
100	C₂×C₄×D₁₇	—	2	2	no	—
101	C₂×C₄×D₁₇	C₂²×C₄	1	2	yes	—
102	C₂×Dic₃₄	#41	1	1	yes	characteristic
103	M₁₆×C₁₇	—	2	2	no	—
104	M₁₆×C₁₇	—	2	2	no	—
105	C₂×C₈×C₁₇	—	2	2	no	—
106	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
111	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
112	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
113	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
114	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
115	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
116	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
117	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
118	C₄⋊C₄×C₁₇	—	2	2	no	—
119	C₂²⋊C₄×C₁₇	—	2	2	no	—
120	C₄²×C₁₇	—	2	1	no	—
121	C₄²×C₁₇	—	2	2	no	—
122	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
124	C₂²⋊Dic₁₇	—	2	1	no	—
125	C₂²⋊Dic₁₇	—	2	2	no	—
126	#63227	—	2	2	no	—
127	C₄⋊Dic₁₇	—	2	1	no	—
128	C₄⋊Dic₁₇	—	2	2	no	—
129	#63225	—	2	2	no	—
130	C₄×Dic₁₇	—	2	2	no	—
131	C₄×Dic₁₇	—	2	2	no	—
132	C₄×Dic₁₇	C₂²×C₄	1	2	yes	—
133	#63223	—	2	1	no	—
134	#63223	—	2	1	no	—
135	#63223	—	2	1	no	—
136	#63223	—	2	1	no	—
137	#63223	—	2	2	no	—
138	#63223	—	2	2	no	—
139	C₂×#3622	—	2	2	no	—
140	C₂×#3622	C₂²×C₄	1	2	yes	—
141	C₂×#3622	C₂²×C₄	1	2	yes	—
142	#63218	—	2	2	no	—
143	#63218	—	2	2	no	—
144	#63218	—	2	2	no	—
145	#63218	—	2	2	no	—
146	C₈×D₁₇	—	2	2	no	—
147	C₈×D₁₇	—	2	2	no	—
148	C₈×D₁₇	—	2	2	no	—
149	C₈×D₁₇	—	2	2	no	—
150	#20787	—	17	1	no	maximal, Sylow
151	C₂³×C₁₇	C₂×#41	1	1	yes	characteristic
152	C₂²×D₁₇	#117	1	2	yes	—
153	C₂×C₄×C₁₇	—	2	1	no	—
154	C₂×C₄×C₁₇	—	2	1	no	—
155	C₂×C₄×C₁₇	—	4	2	no	—
156	C₂×C₄×C₁₇	—	4	2	no	—
157	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
158	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
159	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
160	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
161	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
162	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
163	C₂²⋊D₁₇	—	2	2	no	—
164	C₂²⋊D₁₇	—	2	2	no	—
165	C₂×Dic₁₇	—	2	2	no	—
166	C₂×Dic₁₇	—	4	2	no	—
167	C₂×Dic₁₇	#117	1	2	yes	—
168	D₆₈	—	2	1	no	—
169	D₆₈	—	2	1	no	—
170	C₄×D₁₇	—	2	2	no	—
171	C₄×D₁₇	—	2	2	no	—
172	C₄×D₁₇	—	4	4	no	—
173	Dic₃₄	—	2	1	no	—
174	Dic₃₄	—	2	1	no	—
175	C₈×C₁₇	—	4	2	no	—
176	C₈×C₁₇	#117	1	1	yes	characteristic
177	C₈×C₁₇	#117	1	1	yes	characteristic
178	C₈×C₁₇	#117	1	1	yes	characteristic
179	C₈×C₁₇	#117	1	1	yes	characteristic
180	C₈×C₁₇	#117	1	1	yes	characteristic
181	C₈×C₁₇	#117	1	1	yes	characteristic
182	C₈×C₁₇	#117	1	1	yes	characteristic
183	C₈×C₁₇	#117	1	1	yes	characteristic
184	#3622	—	2	2	no	—
185	#3622	—	2	2	no	—
186	#3622	—	4	2	no	—
187	C₄×#131	—	17	1	no	—
188	#1841	—	17	2	no	—
189	#1833	—	17	1	no	—
190	#1833	—	17	1	no	—
191	#1827	—	17	2	no	—
192	#1813	—	17	1	no	—
193	#1812	—	17	1	no	—
194	#1772	—	17	1	no	—
195	#1772	D₁₇	1	1	yes	characteristic, complemented
196	#1757	—	17	2	no	—
197	#1747	—	17	2	no	—
198	C₂²×C₁₇	—	4	1	no	—
199	C₂²×C₁₇	C₂×#117	1	1	yes	characteristic
200	C₂²×C₁₇	#529	1	1	yes	characteristic
201	C₂²×C₁₇	#529	1	1	yes	characteristic
202	C₂×D₁₇	—	2	2	no	—
203	C₂×D₁₇	—	4	2	no	—
204	C₄×C₁₇	—	2	2	no	—
205	C₄×C₁₇	—	4	1	no	—
206	C₄×C₁₇	—	8	2	no	—
207	C₄×C₁₇	C₂×#117	1	1	yes	characteristic
208	C₄×C₁₇	C₂×#117	1	1	yes	characteristic
209	C₄×C₁₇	#529	1	1	yes	characteristic
210	C₄×C₁₇	#529	1	1	yes	characteristic
211	Dic₁₇	—	2	2	no	—
212	Dic₁₇	—	4	2	no	—
213	Dic₁₇	—	8	2	no	—
214	C₂×#131	—	17	1	no	—
215	C₂×#131	—	17	1	no	—
216	C₄×#41	—	17	1	no	—
217	C₂×#108	—	17	1	no	—
218	C₂×#108	C₂×D₁₇	1	1	yes	characteristic
219	#427	—	17	1	no	—
220	#427	C₂×D₁₇	1	1	yes	characteristic
221	#413	—	17	2	no	—
222	#410	—	17	2	no	—
223	#405	—	17	1	no	—
224	#405	—	17	1	no	—
225	#405	—	17	2	no	—
226	#405	—	17	2	no	—
227	#405	—	17	2	no	—
228	#405	C₂×D₁₇	1	2	yes	—
229	C₄×M₁₆	—	17	1	no	—
230	C₄×M₁₆	—	17	2	no	—
231	C₂×C₄×C₈	—	17	2	no	—
232	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
233	#344	—	17	2	no	—
234	#344	—	17	2	no	—
235	#339	—	17	2	no	—
236	#339	C₂×D₁₇	1	2	yes	—
237	#338	—	17	2	no	—
238	H(3,ℤ/4)	—	17	2	no	—
239	H(3,ℤ/4)	—	17	2	no	—
240	H(3,ℤ/4)	—	17	2	no	—
241	C₂×C₁₇	—	2	1	no	—
242	C₂×C₁₇	—	2	1	no	—
243	C₂×C₁₇	—	2	1	no	—
244	C₂×C₁₇	#2283	1	1	yes	characteristic
245	D₁₇	—	4	2	no	—
246	C₂×#41	—	17	1	no	—
247	#131	—	34	1	no	—
248	#131	—	34	1	no	—
249	#131	—	34	1	no	—
250	#131	—	34	1	no	—
251	#131	—	34	1	no	—
252	#131	—	34	1	no	—
253	#131	—	34	1	no	—
254	#131	—	34	1	no	—
255	C₂×M₁₆	—	17	1	no	—
256	C₂×M₁₆	—	17	1	no	—
257	C₂×M₁₆	—	17	2	no	—
258	C₂×M₁₆	—	17	2	no	—
259	C₂×M₁₆	—	17	2	no	—
260	C₂×M₁₆	C₂²×D₁₇	1	2	yes	—
261	C₂²×C₈	—	17	2	no	—
262	C₂²×C₈	—	17	2	no	—
263	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
264	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
265	C₄×Q₈	—	34	1	no	—
266	C₄×D₄	—	34	1	no	—
267	C₄×D₄	—	34	2	no	—
268	#117	—	17	1	no	—
269	#117	—	17	2	no	—
270	#117	—	17	2	no	—
271	#117	C₂²×D₁₇	1	2	yes	—
272	C₂×C₄²	—	17	2	no	—
273	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
274	#108	—	2	1	no	—
275	#108	—	2	1	no	—
276	#108	—	34	1	no	—
277	#108	—	34	1	no	—
278	#107	—	2	1	no	—
279	#107	—	34	1	no	—
280	#106	—	2	1	no	—
281	#106	—	34	1	no	—
282	#101	—	34	2	no	—
283	#100	—	34	2	no	—
284	#99	—	34	4	no	—
285	#97	—	2	2	no	—
286	#97	—	17	2	no	—
287	#97	—	17	2	no	—
288	#97	—	17	2	no	—
289	#97	—	34	2	no	—
290	C₄×C₈	—	2	2	no	—
291	C₄×C₈	—	17	2	no	—
292	C₄×C₈	—	17	2	no	—
293	C₄×C₈	—	17	2	no	—
294	C₄×C₈	—	34	2	no	—
295	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
296	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
297	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
298	C₄×C₈	C₄×D₁₇	1	1	yes	characteristic
299	C₁₇	#20787	1	1	yes	characteristic, complemented, Sylow
300	#41	—	34	1	no	—
301	#41	—	34	1	no	—
302	#41	—	34	1	no	—
303	#41	—	34	1	no	—
304	C₂×Q₈	—	17	1	no	—
305	C₂×D₄	—	17	2	no	—
306	C₂×D₄	—	17	1	no	—
307	C₂²×C₄	—	2	1	no	—
308	C₂²×C₄	—	17	2	no	—
309	C₂²×C₄	—	34	2	no	—
310	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
311	M₁₆	—	2	2	no	—
312	M₁₆	—	2	2	no	—
313	M₁₆	—	34	1	no	—
314	M₁₆	—	34	1	no	—
315	M₁₆	—	34	1	no	—
316	M₁₆	—	34	1	no	—
317	M₁₆	—	34	2	no	—
318	M₁₆	—	34	2	no	—
319	M₁₆	—	34	2	no	—
320	M₁₆	—	34	2	no	—
321	M₁₆	—	34	2	no	—
322	M₁₆	—	34	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	17	2	no	—
325	C₂×C₈	—	17	2	no	—
326	C₂×C₈	—	34	2	no	—
327	C₂×C₈	—	34	2	no	—
328	C₂×C₈	—	34	2	no	—
329	C₂×C₈	—	34	2	no	—
330	C₂×C₈	—	34	2	no	—
331	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
332	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
333	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
334	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
335	C₂×C₈	#63252	1	1	yes	characteristic
336	C₂×C₈	#63252	1	1	yes	characteristic
337	C₂×C₈	#63252	1	1	yes	characteristic
338	C₂×C₈	#63252	1	1	yes	characteristic
339	C₂×C₈	#63254	1	1	yes	characteristic
340	C₂×C₈	#63254	1	1	yes	characteristic
341	C₂×C₈	#63256	1	1	yes	characteristic
342	C₂×C₈	#63256	1	1	yes	characteristic
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	34	2	no	—
345	C₄⋊C₄	—	34	1	no	—
346	C₄⋊C₄	—	34	2	no	—
347	C₂²⋊C₄	—	2	2	no	—
348	C₂²⋊C₄	—	34	2	no	—
349	C₂²⋊C₄	—	34	1	no	—
350	C₂²⋊C₄	—	34	2	no	—
351	C₄²	—	2	1	no	—
352	C₄²	—	2	2	no	—
353	C₄²	—	17	2	no	—
354	C₄²	—	34	2	no	—
355	C₄²	—	34	2	no	—
356	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
357	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
358	C₂³	—	17	2	no	—
359	C₂³	#41×D₁₇	1	1	yes	characteristic
360	Q₈	—	34	1	no	—
361	Q₈	—	34	1	no	—
362	D₄	—	34	2	no	—
363	D₄	—	34	2	no	—
364	D₄	—	34	1	no	—
365	D₄	—	34	1	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	17	2	no	—
371	C₂×C₄	—	34	2	no	—
372	C₂×C₄	—	34	2	no	—
373	C₂×C₄	—	34	2	no	—
374	C₂×C₄	—	68	4	no	—
375	C₂×C₄	—	68	2	no	—
376	C₂×C₄	C₂²×C₄×D₁₇	1	1	yes	characteristic
377	C₂×C₄	C₂×#63252	1	1	yes	characteristic
378	C₂×C₄	C₂×#63252	1	1	yes	characteristic
379	C₂×C₄	C₂×#63254	1	1	yes	characteristic
380	C₂×C₄	C₂×#63256	1	1	yes	characteristic
381	C₂×C₄	#41×D₁₇	1	1	yes	characteristic
382	C₈	—	4	2	no	—
383	C₈	—	34	2	no	—
384	C₈	—	34	2	no	—
385	C₈	—	68	2	no	—
386	C₈	#10587558	1	1	yes	characteristic
387	C₈	#10587558	1	1	yes	characteristic
388	C₈	#10587558	1	1	yes	characteristic
389	C₈	#10587558	1	1	yes	characteristic
390	C₈	#10587578	1	1	yes	characteristic
391	C₈	#10587578	1	1	yes	characteristic
392	C₈	#10587578	1	1	yes	characteristic
393	C₈	#10587578	1	1	yes	characteristic
394	C₂²	—	4	1	no	—
395	C₂²	—	34	2	no	—
396	C₂²	—	68	2	no	—
397	C₂²	#117×D₁₇	1	1	yes	characteristic
398	C₂²	#11763341	1	1	yes	characteristic
399	C₂²	#11763346	1	1	yes	characteristic
400	C₄	—	2	2	no	—
401	C₄	—	4	1	no	—
402	C₄	—	8	2	no	—
403	C₄	—	34	2	no	—
404	C₄	—	68	2	no	—
405	C₄	—	136	2	no	—
406	C₄	C₂×#10587558	1	1	yes	characteristic
407	C₄	C₂×#10587578	1	1	yes	characteristic
408	C₄	#11763341	1	1	yes	characteristic
409	C₄	#11763346	1	1	yes	characteristic
410	C₂	—	2	1	no	—
411	C₂	—	2	1	no	—
412	C₂	—	2	1	no	—
413	C₂	—	68	2	no	—
414	C₂	#14531180	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#10587578
commutator	C₂×C₄×C₁₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	C₁₇×#1772	C₂
socle	C₂×C₁₇	#2283
prime radical	C₂×#63252	C₂³
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	#1772	D₁₇
hypocenter	C₁₇	#20787
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
2. #14528930	C₂	semidirect
3. #14528909	C₂	semidirect
4. #14528907	C₂	semidirect
8. #14528663	C₂	semidirect
10. #14528539	C₂	semidirect
11. C₁₇×#1772	C₂	semidirect
22. #11762763	C₄	semidirect
23. #11762763	C₄	semidirect
25. C₄×#63218	C₄	semidirect
26. C₄×C₈×D₁₇	C₄	semidirect
195. #1772	D₁₇	semidirect
299. C₁₇	#20787	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10587587	2
#14528930	2
#14528909	2
#14528907	2
#14528887	2
#14528831	2
#14528827	2
#14528663	2
#14528585	2
#14528539	2
C₁₇×#1772	2
#20787	17