Subgroups — #17079449

Order 4352.

656 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763079	C₂	1	1	yes	maximal, characteristic
2	#14537637	C₂	1	1	yes	maximal, characteristic, complemented
3	#14530247	C₂	1	2	yes	maximal, complemented
4	#14530057	C₂	1	2	yes	maximal, complemented
5	#14529694	C₂	1	1	yes	maximal, characteristic, complemented
6	#14529574	C₂	1	1	yes	maximal, characteristic, complemented
7	#14529274	C₂	1	2	yes	maximal, complemented
8	#14528908	C₂	1	1	yes	maximal, characteristic, complemented
9	#14528894	C₂	1	2	yes	maximal, complemented
10	#14528838	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₇×#1757	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10587635	C₂²	1	2	yes	—
13	C₂²×D₁₃₆	C₂²	1	1	yes	characteristic
14	#11763283	C₂²	1	2	yes	—
15	#11763079	—	2	2	no	—
16	#11763079	—	2	2	no	—
17	#11762922	—	2	2	no	—
18	C₄×D₁₃₆	—	2	2	no	—
19	#11762857	C₂²	1	2	yes	—
20	C₂×#10587490	C₂²	1	1	yes	characteristic
21	C₂×#10587490	C₂²	1	1	yes	characteristic
22	#11762847	C₂²	1	1	yes	characteristic
23	#11762826	C₂²	1	2	yes	—
24	#11762815	C₂²	1	2	yes	—
25	#11762813	C₂²	1	2	yes	—
26	C₂×#10587494	C₂²	1	2	yes	—
27	C₂×#10587494	C₂²	1	2	yes	—
28	#11762581	C₂²	1	2	yes	—
29	#11762580	C₂²	1	2	yes	—
30	#11762579	C₂²	1	2	yes	—
31	#11762565	C₂²	1	2	yes	—
32	C₂×C₁₇×#131	C₂²	1	1	yes	characteristic
33	C₁₇×#413	C₂²	1	1	yes	characteristic
34	C₁₇×#410	C₂²	1	1	yes	characteristic
35	C₁₇×#405	C₂²	1	1	yes	characteristic
36	C₁₇×#405	C₂²	1	1	yes	characteristic
37	C₁₇×#344	C₂²	1	1	yes	characteristic
38	C₁₇×#344	C₂²	1	1	yes	characteristic
39	C₂×#41×C₁₇	C₂³	1	1	yes	characteristic
40	C₂²×D₆₈	C₂³	1	2	yes	—
41	C₁₇×#131	—	2	1	no	—
42	C₁₇×#131	—	2	1	no	—
43	C₁₇×#131	—	2	1	no	—
44	C₁₇×#131	—	2	1	no	—
45	C₂×M₁₆×C₁₇	—	2	1	no	—
46	C₂×M₁₆×C₁₇	C₂³	1	1	yes	characteristic
47	C₂×M₁₆×C₁₇	C₂³	1	1	yes	characteristic
48	C₂×M₁₆×C₁₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₁₇	—	2	1	no	—
50	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
51	C₁₇×#117	C₂³	1	1	yes	characteristic
52	C₁₇×#117	C₂³	1	1	yes	characteristic
53	#10587635	—	2	4	no	—
54	#10587635	—	4	2	no	—
55	C₂×#63230	—	2	2	no	—
56	C₂×#63228	—	2	2	no	—
57	#10587612	C₂³	1	2	yes	—
58	C₂×#63223	C₂³	1	2	yes	—
59	C₂×#63223	C₂³	1	2	yes	—
60	C₂×D₁₃₆	—	2	1	no	—
61	C₂×D₁₃₆	—	2	1	no	—
62	C₂×D₁₃₆	—	2	1	no	—
63	C₂×D₁₃₆	—	2	1	no	—
64	C₂×D₁₃₆	D₄	1	2	yes	—
65	C₂×D₁₃₆	D₄	1	2	yes	—
66	#10587579	—	4	2	no	—
67	C₂²⋊C₄×D₁₇	—	2	4	no	—
68	C₄×D₆₈	—	2	4	no	—
69	C₁₇×#101	—	2	1	no	—
70	C₁₇×#100	—	2	1	no	—
71	C₁₇×#99	—	2	2	no	—
72	C₁₇×#97	—	2	1	no	—
73	C₁₇×#97	—	2	1	no	—
74	C₄×C₈×C₁₇	—	2	1	no	—
75	C₄×C₈×C₁₇	—	2	1	no	—
76	#10587508	—	4	2	no	—
77	#10587494	—	2	4	no	—
78	#10587494	—	4	2	no	—
79	#10587493	—	4	2	no	—
80	#10587492	—	2	4	no	—
81	#10587490	—	2	1	no	—
82	#10587490	—	2	1	no	—
83	#10587490	—	2	1	no	—
84	#10587490	—	2	1	no	—
85	#10587489	—	2	1	no	—
86	#10587488	—	2	1	no	—
87	#10587480	—	4	2	no	—
88	C₂³×D₁₇	—	2	2	no	—
89	#41×C₁₇	—	2	1	no	—
90	#41×C₁₇	—	2	1	no	—
91	#41×C₁₇	—	2	1	no	—
92	#41×C₁₇	—	2	1	no	—
93	C₂×Q₈×C₁₇	#41	1	1	yes	characteristic
94	C₂×D₄×C₁₇	—	2	1	no	—
95	C₂×D₄×C₁₇	#41	1	1	yes	characteristic
96	C₂²×C₄×C₁₇	—	2	1	no	—
97	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
98	C₂×D₆₈	—	2	2	no	—
99	C₂×D₆₈	—	2	2	no	—
100	C₂×D₆₈	—	2	2	no	—
101	C₂×D₆₈	—	2	2	no	—
102	C₂×D₆₈	—	2	2	no	—
103	C₂×D₆₈	C₂×D₄	1	4	yes	—
104	C₂×C₄×D₁₇	—	4	4	no	—
105	M₁₆×C₁₇	—	2	1	no	—
106	M₁₆×C₁₇	—	2	1	no	—
107	M₁₆×C₁₇	—	2	1	no	—
108	M₁₆×C₁₇	—	2	1	no	—
109	M₁₆×C₁₇	—	2	1	no	—
110	M₁₆×C₁₇	—	2	1	no	—
111	M₁₆×C₁₇	—	4	1	no	—
112	M₁₆×C₁₇	—	4	1	no	—
113	C₂×C₈×C₁₇	—	2	1	no	—
114	C₂×C₈×C₁₇	—	2	1	no	—
115	C₂×C₈×C₁₇	—	2	1	no	—
116	C₂×C₈×C₁₇	—	4	1	no	—
117	C₂×C₈×C₁₇	—	4	1	no	—
118	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
121	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
122	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
123	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
124	C₄⋊C₄×C₁₇	—	2	1	no	—
125	C₄⋊C₄×C₁₇	—	2	1	no	—
126	C₂²⋊C₄×C₁₇	—	2	1	no	—
127	C₂²⋊C₄×C₁₇	#41	1	2	yes	—
128	C₄²×C₁₇	—	2	1	no	—
129	C₄²×C₁₇	—	2	1	no	—
130	C₂²⋊Dic₁₇	—	2	2	no	—
131	#63230	—	4	4	no	—
132	#63228	—	4	4	no	—
133	#63227	—	4	4	no	—
134	C₄⋊Dic₁₇	—	2	2	no	—
135	C₄×Dic₁₇	—	2	2	no	—
136	#63223	—	2	2	no	—
137	#63223	—	2	2	no	—
138	#63223	—	2	2	no	—
139	#63223	—	2	2	no	—
140	C₂×#3622	—	2	2	no	—
141	C₂×#3622	—	2	2	no	—
142	D₁₃₆	—	2	2	no	—
143	D₁₃₆	—	2	2	no	—
144	D₁₃₆	—	2	2	no	—
145	D₁₃₆	—	2	2	no	—
146	#24988	—	17	1	no	maximal, Sylow
147	C₂³×C₁₇	—	2	1	no	—
148	C₂³×C₁₇	C₂×#41	1	1	yes	characteristic
149	C₂²×D₁₇	—	2	4	no	—
150	C₂²×D₁₇	—	4	2	no	—
151	C₂²×D₁₇	—	4	2	no	—
152	C₂²×D₁₇	—	16	2	no	—
153	Q₈×C₁₇	—	2	1	no	—
154	Q₈×C₁₇	—	2	1	no	—
155	D₄×C₁₇	—	2	1	no	—
156	D₄×C₁₇	—	2	1	no	—
157	D₄×C₁₇	—	4	1	no	—
158	D₄×C₁₇	—	4	1	no	—
159	C₂×C₄×C₁₇	—	2	1	no	—
160	C₂×C₄×C₁₇	—	2	2	no	—
161	C₂×C₄×C₁₇	—	4	1	no	—
162	C₂×C₄×C₁₇	—	4	1	no	—
163	C₂×C₄×C₁₇	—	4	1	no	—
164	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
166	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
167	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
168	C₂×C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂×C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
170	C₂×Dic₁₇	—	4	2	no	—
171	D₆₈	—	2	4	no	—
172	D₆₈	—	2	4	no	—
173	D₆₈	—	4	2	no	—
174	D₆₈	—	4	2	no	—
175	C₄×D₁₇	—	16	4	no	—
176	C₈×C₁₇	—	4	1	no	—
177	C₈×C₁₇	—	4	1	no	—
178	C₈×C₁₇	—	4	1	no	—
179	C₈×C₁₇	#123	1	1	yes	characteristic
180	C₈×C₁₇	#123	1	1	yes	characteristic
181	C₈×C₁₇	#123	1	1	yes	characteristic
182	C₈×C₁₇	#123	1	1	yes	characteristic
183	#3622	—	4	2	no	—
184	#3622	—	4	2	no	—
185	C₂×#472	—	17	1	no	—
186	#2948	—	17	1	no	—
187	#2081	—	17	2	no	—
188	#2048	—	17	2	no	—
189	#1978	—	17	1	no	—
190	#1953	—	17	1	no	—
191	#1904	—	17	2	no	—
192	#1833	—	17	1	no	—
193	#1829	—	17	2	no	—
194	#1814	—	17	1	no	—
195	#1757	D₁₇	1	1	yes	characteristic, complemented
196	C₂²×C₁₇	—	4	1	no	—
197	C₂²×C₁₇	—	4	1	no	—
198	C₂²×C₁₇	—	8	1	no	—
199	C₂²×C₁₇	#529	1	1	yes	characteristic
200	C₂²×C₁₇	#546	1	1	yes	characteristic
201	C₂²×C₁₇	#555	1	1	yes	characteristic
202	C₂×D₁₇	—	4	4	no	—
203	C₂×D₁₇	—	8	4	no	—
204	C₂×D₁₇	—	16	2	no	—
205	C₂×D₁₇	—	16	2	no	—
206	C₄×C₁₇	—	4	2	no	—
207	C₄×C₁₇	—	4	1	no	—
208	C₄×C₁₇	—	8	1	no	—
209	C₄×C₁₇	C₂×#123	1	1	yes	characteristic
210	C₄×C₁₇	#540	1	1	yes	characteristic
211	C₄×C₁₇	#546	1	1	yes	characteristic
212	C₄×C₁₇	#553	1	1	yes	characteristic
213	Dic₁₇	—	8	2	no	—
214	C₂×AGL(1,ℤ/8)	—	17	2	no	—
215	C₂²×D₈	—	17	1	no	—
216	C₂×#131	C₂×D₁₇	1	1	yes	characteristic
217	#518	—	17	2	no	—
218	#472	—	34	2	no	—
219	#472	—	34	2	no	—
220	#442	—	34	2	no	—
221	C₄×D₈	—	34	2	no	—
222	#430	—	17	2	no	—
223	C₂×#108	—	17	1	no	—
224	C₂×#108	—	17	1	no	—
225	#427	—	17	1	no	—
226	#421	—	17	2	no	—
227	#418	—	17	2	no	—
228	#418	—	17	2	no	—
229	#413	C₂×D₁₇	1	1	yes	characteristic
230	C₂×#100	—	17	2	no	—
231	C₂×#100	—	17	2	no	—
232	#410	C₂×D₁₇	1	1	yes	characteristic
233	#405	C₂×D₁₇	1	1	yes	characteristic
234	#405	C₂×D₁₇	1	1	yes	characteristic
235	#344	—	17	2	no	—
236	#344	—	17	2	no	—
237	#344	—	17	2	no	—
238	#344	C₂×D₁₇	1	1	yes	characteristic
239	#344	C₂×D₁₇	1	1	yes	characteristic
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	—	8	1	no	—
245	C₂×C₁₇	#2420	1	1	yes	characteristic
246	D₁₇	—	8	4	no	—
247	C₂×#41	C₂²×D₁₇	1	1	yes	characteristic
248	C₂²×D₄	—	17	2	no	—
249	AGL(1,ℤ/8)	—	34	4	no	—
250	AGL(1,ℤ/8)	—	68	2	no	—
251	C₂×SD₁₆	—	34	2	no	—
252	C₂×D₈	—	17	2	no	—
253	C₂×D₈	—	17	2	no	—
254	C₂×D₈	—	34	2	no	—
255	C₂×D₈	—	34	1	no	—
256	C₂×D₈	—	34	1	no	—
257	C₂×D₈	—	34	1	no	—
258	C₂×D₈	—	34	1	no	—
259	#131	—	2	1	no	—
260	#131	—	2	1	no	—
261	#131	—	2	1	no	—
262	#131	—	2	1	no	—
263	C₂×M₁₆	—	2	1	no	—
264	C₂×M₁₆	—	17	2	no	—
265	C₂×M₁₆	—	17	2	no	—
266	C₂×M₁₆	C₂²×D₁₇	1	1	yes	characteristic
267	C₂×M₁₆	C₂²×D₁₇	1	1	yes	characteristic
268	C₂×M₁₆	C₂²×D₁₇	1	1	yes	characteristic
269	C₂²×C₈	—	2	1	no	—
270	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
271	C₄×D₄	—	34	4	no	—
272	C₄×D₄	—	68	2	no	—
273	#117	—	17	2	no	—
274	#117	C₂²×D₁₇	1	1	yes	characteristic
275	#117	C₂²×D₁₇	1	1	yes	characteristic
276	C₂×C₂²⋊C₄	—	34	4	no	—
277	#108	—	34	1	no	—
278	#108	—	34	1	no	—
279	#108	—	34	1	no	—
280	#108	—	34	1	no	—
281	#108	—	68	2	no	—
282	#107	—	34	1	no	—
283	#106	—	34	1	no	—
284	#104	—	68	2	no	—
285	#102	—	34	4	no	—
286	#102	—	68	2	no	—
287	#101	—	2	1	no	—
288	#100	—	2	1	no	—
289	#100	—	34	4	no	—
290	#100	—	68	2	no	—
291	#99	—	2	2	no	—
292	#97	—	2	1	no	—
293	#97	—	2	1	no	—
294	C₄×C₈	—	2	1	no	—
295	C₄×C₈	—	2	1	no	—
296	C₁₇	#24988	1	1	yes	characteristic, complemented, Sylow
297	C₂⁴	—	34	2	no	—
298	#41	—	2	1	no	—
299	#41	—	2	1	no	—
300	#41	—	2	1	no	—
301	#41	—	2	1	no	—
302	C₂×Q₈	#63252	1	1	yes	characteristic
303	C₂×D₄	—	2	1	no	—
304	C₂×D₄	—	17	4	no	—
305	C₂×D₄	—	34	2	no	—
306	C₂×D₄	—	34	2	no	—
307	C₂×D₄	—	34	2	no	—
308	C₂×D₄	—	34	2	no	—
309	C₂×D₄	—	34	2	no	—
310	C₂×D₄	#63252	1	1	yes	characteristic
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	68	4	no	—
313	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
314	SD₁₆	—	68	4	no	—
315	D₈	—	34	2	no	—
316	D₈	—	34	2	no	—
317	D₈	—	34	2	no	—
318	D₈	—	34	2	no	—
319	D₈	—	68	4	no	—
320	M₁₆	—	2	1	no	—
321	M₁₆	—	2	1	no	—
322	M₁₆	—	2	1	no	—
323	M₁₆	—	2	1	no	—
324	M₁₆	—	2	1	no	—
325	M₁₆	—	2	1	no	—
326	M₁₆	—	4	1	no	—
327	M₁₆	—	4	1	no	—
328	M₁₆	—	34	2	no	—
329	M₁₆	—	34	2	no	—
330	M₁₆	—	34	2	no	—
331	M₁₆	—	34	2	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	4	1	no	—
336	C₂×C₈	—	4	1	no	—
337	C₂×C₈	—	34	2	no	—
338	C₂×C₈	—	34	2	no	—
339	C₂×C₈	#63252	1	1	yes	characteristic
340	C₂×C₈	#63252	1	1	yes	characteristic
341	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
342	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
343	C₂×C₈	#63254	1	1	yes	characteristic
344	C₂×C₈	#63254	1	1	yes	characteristic
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	34	2	no	—
348	C₂²⋊C₄	—	2	1	no	—
349	C₂²⋊C₄	—	34	2	no	—
350	C₂²⋊C₄	—	68	4	no	—
351	C₂²⋊C₄	#63252	1	2	yes	—
352	C₄²	—	2	1	no	—
353	C₄²	—	2	1	no	—
354	C₄²	—	34	2	no	—
355	C₂³	—	2	1	no	—
356	C₂³	—	272	2	no	—
357	C₂³	—	34	4	no	—
358	C₂³	—	68	2	no	—
359	C₂³	—	68	2	no	—
360	C₂³	C₂×#63252	1	1	yes	characteristic
361	Q₈	—	2	1	no	—
362	Q₈	—	2	1	no	—
363	D₄	—	2	1	no	—
364	D₄	—	2	1	no	—
365	D₄	—	4	1	no	—
366	D₄	—	4	1	no	—
367	D₄	—	34	4	no	—
368	D₄	—	34	4	no	—
369	D₄	—	68	2	no	—
370	D₄	—	68	2	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	—	272	4	no	—
377	C₂×C₄	—	68	2	no	—
378	C₂×C₄	C₂×#63252	1	1	yes	characteristic
379	C₂×C₄	C₂×#63252	1	1	yes	characteristic
380	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
381	C₂×C₄	C₂×#63254	1	1	yes	characteristic
382	C₂×C₄	#10587696	1	1	yes	characteristic
383	C₂×C₄	#10587696	1	1	yes	characteristic
384	C₈	—	4	1	no	—
385	C₈	—	4	1	no	—
386	C₈	—	4	1	no	—
387	C₈	—	68	2	no	—
388	C₈	—	68	2	no	—
389	C₈	#10587569	1	1	yes	characteristic
390	C₈	#10587569	1	1	yes	characteristic
391	C₈	#10587569	1	1	yes	characteristic
392	C₈	#10587569	1	1	yes	characteristic
393	C₂²	—	4	1	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	8	1	no	—
396	C₂²	—	272	2	no	—
397	C₂²	—	272	2	no	—
398	C₂²	—	68	4	no	—
399	C₂²	—	136	4	no	—
400	C₂²	#11763343	1	1	yes	characteristic
401	C₂²	#11763455	1	1	yes	characteristic
402	C₂²	#11763524	1	1	yes	characteristic
403	C₄	—	4	2	no	—
404	C₄	—	4	1	no	—
405	C₄	—	8	1	no	—
406	C₄	—	136	2	no	—
407	C₄	C₂×#10587569	1	1	yes	characteristic
408	C₄	#11763414	1	1	yes	characteristic
409	C₄	#11763455	1	1	yes	characteristic
410	C₄	#11763505	1	1	yes	characteristic
411	C₂	—	2	1	no	—
412	C₂	—	2	1	no	—
413	C₂	—	2	1	no	—
414	C₂	—	8	1	no	—
415	C₂	—	136	4	no	—
416	C₂	#14532276	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14532276
commutator, cosocle	C₂²×C₄×C₁₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	C₁₇×#1757	C₂
socle	C₂×C₁₇	#2420
prime radical	C₂×#11763079	C₂
hypercenter	#1757	D₁₇
hypocenter	C₁₇	#24988
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757

Complements

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

N	H ≅ G/N	type
2. #14537637	C₂	semidirect
3. #14530247	C₂	semidirect
4. #14530057	C₂	semidirect
5. #14529694	C₂	semidirect
6. #14529574	C₂	semidirect
7. #14529274	C₂	semidirect
8. #14528908	C₂	semidirect
9. #14528894	C₂	semidirect
10. #14528838	C₂	semidirect
11. C₁₇×#1757	C₂	semidirect
195. #1757	D₁₇	semidirect
296. C₁₇	#24988	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763079	2
#14537637	2
#14530247	2
#14530057	2
#14529694	2
#14529574	2
#14529274	2
#14528908	2
#14528894	2
#14528838	2
C₁₇×#1757	2
#24988	17