Subgroups — #17065203

Order 4352.

616 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14537688	C₂	1	1	yes	maximal, characteristic
2	#14530364	C₂	1	1	yes	maximal, characteristic, complemented
3	#14530359	C₂	1	1	yes	maximal, characteristic, complemented
4	#14530235	C₂	1	2	yes	maximal, complemented
5	#14530090	C₂	1	1	yes	maximal, characteristic
6	#14530071	C₂	1	1	yes	maximal, characteristic
7	#14529953	C₂	1	2	yes	maximal, complemented
8	#14528621	C₂	1	2	yes	maximal, complemented
9	#14528585	C₂	1	2	yes	maximal, complemented
10	#14528539	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763609	C₂²	1	1	yes	characteristic
13	C₂×#10587587	C₂²	1	2	yes	—
14	#11762932	—	2	2	no	—
15	#11762927	—	2	2	no	—
16	#11762858	C₂²	1	1	yes	characteristic
17	#11762827	C₂²	1	2	yes	—
18	C₂×#10587476	C₂²	1	2	yes	—
19	#11762806	C₂²	1	2	yes	—
20	#11762791	C₂²	1	2	yes	—
21	#11762782	C₂²	1	2	yes	—
22	#11762764	C₂²	1	2	yes	—
23	#11762581	C₂²	1	2	yes	—
24	#11762579	C₂²	1	2	yes	—
25	#11762574	C₂²	1	2	yes	—
26	#11762569	C₂²	1	2	yes	—
27	#11762567	C₂²	1	1	yes	characteristic
28	#11762566	C₂²	1	1	yes	characteristic
29	#11762565	C₂²	1	2	yes	—
30	C₁₇×#533	C₂²	1	1	yes	characteristic
31	C₁₇×#446	—	2	2	no	—
32	C₈×Q₈×C₁₇	—	2	2	no	—
33	C₁₇×#424	C₂²	1	1	yes	characteristic
34	C₁₇×#424	C₂²	1	1	yes	characteristic
35	C₁₇×#423	C₂²	1	1	yes	characteristic
36	C₁₇×#405	C₂²	1	1	yes	characteristic
37	C₁₇×#405	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
39	C₂×#63252	C₂³	1	2	yes	—
40	C₂×M₁₆×C₁₇	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₁₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
44	C₁₇×C₄⋊Q₈	—	2	1	no	—
45	C₁₇×#125	—	2	1	no	—
46	C₁₇×C₂²⋊Q₈	—	2	2	no	—
47	C₄×Q₈×C₁₇	—	2	2	no	—
48	C₁₇×#117	C₂³	1	1	yes	characteristic
49	C₁₇×#117	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
51	#10587612	C₂³	1	2	yes	—
52	C₂×#63223	C₂³	1	2	yes	—
53	C₂×#63223	C₂³	1	2	yes	—
54	#10587595	—	2	2	no	—
55	#10587594	—	2	2	no	—
56	#10587590	—	2	2	no	—
57	#10587587	—	2	2	no	—
58	#10587587	—	2	2	no	—
59	#10587587	—	2	2	no	—
60	#10587587	—	2	2	no	—
61	C₂×#63218	—	2	2	no	—
62	C₂×C₈×D₁₇	—	2	2	no	—
63	C₁₇×C₄⋊C₈	—	2	2	no	—
64	C₁₇×C₄⋊C₈	—	2	2	no	—
65	C₁₇×C₄⋊C₈	—	2	2	no	—
66	C₁₇×#97	—	2	1	no	—
67	C₁₇×#97	D₄	1	2	yes	—
68	C₄×C₈×C₁₇	—	2	1	no	—
69	C₄×C₈×C₁₇	—	2	1	no	—
70	C₄×C₈×C₁₇	—	2	1	no	—
71	C₄×C₈×C₁₇	D₄	1	2	yes	—
72	#10587495	—	2	2	no	—
73	#10587494	—	2	2	no	—
74	#10587492	—	2	2	no	—
75	#10587490	—	2	2	no	—
76	#10587487	—	2	2	no	—
77	#10587486	—	2	2	no	—
78	C₈×Dic₁₇	—	2	2	no	—
79	#10587477	—	2	4	no	—
80	#10587476	—	2	4	no	—
81	#10587476	—	2	2	no	—
82	#10587476	—	2	2	no	—
83	C₂×Q₈×C₁₇	—	2	2	no	—
84	C₂²×C₄×C₁₇	—	2	1	no	—
85	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
86	C₂×C₂²⋊D₁₇	—	2	2	no	—
87	#63252	—	2	2	no	—
88	#63252	—	2	2	no	—
89	#63252	—	2	2	no	—
90	#63252	—	2	2	no	—
91	C₂×D₆₈	#41	1	2	yes	—
92	C₂×C₄×D₁₇	—	2	2	no	—
93	C₂×Dic₃₄	#41	1	2	yes	—
94	M₁₆×C₁₇	—	2	2	no	—
95	M₁₆×C₁₇	—	2	2	no	—
96	C₂×C₈×C₁₇	—	2	1	no	—
97	C₂×C₈×C₁₇	—	2	1	no	—
98	C₂×C₈×C₁₇	—	2	2	no	—
99	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₇	#41	1	2	yes	—
103	C₂×C₈×C₁₇	#41	1	2	yes	—
104	C₄⋊C₄×C₁₇	—	2	4	no	—
105	C₄⋊C₄×C₁₇	—	2	2	no	—
106	C₄⋊C₄×C₁₇	—	2	1	no	—
107	C₄⋊C₄×C₁₇	—	2	1	no	—
108	C₂²⋊C₄×C₁₇	—	2	1	no	—
109	C₂²⋊C₄×C₁₇	#41	1	2	yes	—
110	C₄²×C₁₇	—	2	1	no	—
111	C₄²×C₁₇	—	2	1	no	—
112	C₄²×C₁₇	C₂×D₄	1	2	yes	—
113	C₄²×C₁₇	#41	1	1	yes	characteristic
114	C₄²×C₁₇	#41	1	1	yes	characteristic
115	C₂²⋊Dic₁₇	—	2	2	no	—
116	C₄⋊Dic₁₇	—	2	2	no	—
117	C₄×Dic₁₇	—	2	2	no	—
118	#63223	—	2	2	no	—
119	#63223	—	2	2	no	—
120	#63223	—	2	2	no	—
121	#63223	—	2	2	no	—
122	C₂×#3622	—	2	2	no	—
123	C₂×#3622	—	2	2	no	—
124	Dic₆₈	—	2	2	no	—
125	D₁₃₆	—	2	2	no	—
126	#63219	—	2	4	no	—
127	#63218	—	4	2	no	—
128	#63218	—	4	2	no	—
129	C₈×D₁₇	—	4	2	no	—
130	C₈×D₁₇	—	4	2	no	—
131	#23833	—	17	1	no	maximal, Sylow
132	C₂³×C₁₇	C₂×#41	1	1	yes	characteristic
133	C₂²×D₁₇	—	2	2	no	—
134	Q₈×C₁₇	—	4	4	no	—
135	C₂×C₄×C₁₇	—	2	1	no	—
136	C₂×C₄×C₁₇	—	2	1	no	—
137	C₂×C₄×C₁₇	—	2	2	no	—
138	C₂×C₄×C₁₇	—	2	2	no	—
139	C₂×C₄×C₁₇	—	4	2	no	—
140	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₁₇	C₂×#41	1	2	yes	—
143	C₂×C₄×C₁₇	C₂×#41	1	2	yes	—
144	C₂²⋊D₁₇	—	4	2	no	—
145	C₂²⋊D₁₇	—	4	2	no	—
146	C₂×Dic₁₇	—	2	2	no	—
147	C₂×Dic₁₇	—	4	2	no	—
148	D₆₈	—	2	2	no	—
149	D₆₈	—	2	2	no	—
150	C₄×D₁₇	—	4	2	no	—
151	C₄×D₁₇	—	4	2	no	—
152	Dic₃₄	—	2	2	no	—
153	Dic₃₄	—	2	2	no	—
154	C₈×C₁₇	—	2	2	no	—
155	C₈×C₁₇	—	4	1	no	—
156	C₈×C₁₇	—	4	1	no	—
157	C₈×C₁₇	#124	1	2	yes	—
158	C₈×C₁₇	#124	1	1	yes	characteristic
159	C₈×C₁₇	#124	1	1	yes	characteristic
160	#3622	—	4	2	no	—
161	#3622	—	4	2	no	—
162	#2960	D₁₇	1	1	yes	characteristic, complemented
163	#2955	—	17	1	no	—
164	#2103	—	17	1	no	—
165	#2102	—	17	1	no	—
166	#2080	—	17	2	no	—
167	#2053	—	17	1	no	—
168	#2050	—	17	1	no	—
169	#2031	—	17	2	no	—
170	#1764	—	17	2	no	—
171	#1757	—	17	2	no	—
172	#1747	—	17	1	no	—
173	C₂²×C₁₇	—	4	1	no	—
174	C₂²×C₁₇	#525	1	1	yes	characteristic
175	C₂²×C₁₇	#529	1	2	yes	—
176	C₂×D₁₇	—	4	2	no	—
177	C₂×D₁₇	—	8	2	no	—
178	C₄×C₁₇	—	2	2	no	—
179	C₄×C₁₇	—	4	1	no	—
180	C₄×C₁₇	—	4	2	no	—
181	C₄×C₁₇	—	4	2	no	—
182	C₄×C₁₇	#525	1	1	yes	characteristic
183	C₄×C₁₇	C₂×#124	1	1	yes	characteristic
184	C₄×C₁₇	#529	1	2	yes	—
185	Dic₁₇	—	4	2	no	—
186	Dic₁₇	—	8	2	no	—
187	#575	—	17	1	no	—
188	C₂×#131	—	17	2	no	—
189	#533	C₂×D₁₇	1	1	yes	characteristic
190	#446	—	2	2	no	—
191	C₈×Q₈	—	2	2	no	—
192	#444	—	34	2	no	—
193	#443	—	34	2	no	—
194	#430	—	17	1	no	—
195	#424	C₂×D₁₇	1	1	yes	characteristic
196	#424	C₂×D₁₇	1	1	yes	characteristic
197	#423	C₂×D₁₇	1	1	yes	characteristic
198	#421	—	17	2	no	—
199	C₂×#104	—	17	2	no	—
200	#416	—	17	2	no	—
201	#413	—	17	2	no	—
202	#410	—	17	2	no	—
203	#405	—	17	2	no	—
204	#405	C₂×D₁₇	1	1	yes	characteristic
205	#405	C₂×D₁₇	1	1	yes	characteristic
206	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
207	#344	—	17	2	no	—
208	#344	—	17	2	no	—
209	#341	—	17	2	no	—
210	#339	—	17	2	no	—
211	#338	—	17	1	no	—
212	H(3,ℤ/4)	—	17	1	no	—
213	H(3,ℤ/4)	—	17	2	no	—
214	C₂×C₁₇	—	2	2	no	—
215	C₂×C₁₇	—	2	1	no	—
216	C₂×C₁₇	#2398	1	1	yes	characteristic
217	D₁₇	—	8	2	no	—
218	C₂×#41	—	17	2	no	—
219	#137	—	34	2	no	—
220	AGL(1,ℤ/8)	—	34	2	no	—
221	#135	—	34	2	no	—
222	#131	—	34	2	no	—
223	#131	—	34	2	no	—
224	#131	—	34	2	no	—
225	#131	—	34	2	no	—
226	C₂×M₁₆	—	17	2	no	—
227	C₂×M₁₆	—	17	2	no	—
228	C₂×M₁₆	—	34	2	no	—
229	C₂×M₁₆	C₂²×D₁₇	1	1	yes	characteristic
230	C₂×M₁₆	C₂²×D₁₇	1	1	yes	characteristic
231	C₂²×C₈	—	34	2	no	—
232	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
234	C₄⋊Q₈	—	2	1	no	—
235	#125	—	2	1	no	—
236	C₂²⋊Q₈	—	2	2	no	—
237	C₄×Q₈	—	2	2	no	—
238	#117	—	17	2	no	—
239	#117	C₂²×D₁₇	1	1	yes	characteristic
240	#117	C₂²×D₁₇	1	1	yes	characteristic
241	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
242	#108	—	34	2	no	—
243	C₄⋊C₈	—	2	2	no	—
244	C₄⋊C₈	—	2	2	no	—
245	C₄⋊C₈	—	2	2	no	—
246	#104	—	34	2	no	—
247	#104	—	34	2	no	—
248	#104	—	34	4	no	—
249	#103	—	34	2	no	—
250	#102	—	34	2	no	—
251	#101	—	34	2	no	—
252	#100	—	34	2	no	—
253	#99	—	34	4	no	—
254	#97	—	2	1	no	—
255	#97	—	34	2	no	—
256	#97	C₂²⋊D₁₇	1	2	yes	—
257	C₄×C₈	—	2	1	no	—
258	C₄×C₈	—	2	1	no	—
259	C₄×C₈	—	2	1	no	—
260	C₄×C₈	—	34	2	no	—
261	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
262	C₁₇	#23833	1	1	yes	characteristic, complemented, Sylow
263	#41	—	34	2	no	—
264	#41	—	34	2	no	—
265	#41	—	34	2	no	—
266	#41	—	34	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	17	2	no	—
269	C₂×D₄	—	17	2	no	—
270	C₂×D₄	—	34	2	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	34	2	no	—
273	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
274	Q₁₆	—	34	2	no	—
275	SD₁₆	—	34	4	no	—
276	D₈	—	34	2	no	—
277	M₁₆	—	2	2	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	34	2	no	—
280	M₁₆	—	34	2	no	—
281	M₁₆	—	34	2	no	—
282	M₁₆	—	34	2	no	—
283	M₁₆	—	68	2	no	—
284	M₁₆	—	68	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	34	2	no	—
289	C₂×C₈	—	34	2	no	—
290	C₂×C₈	—	68	2	no	—
291	C₂×C₈	—	68	2	no	—
292	C₂×C₈	#63256	1	2	yes	—
293	C₂×C₈	#63256	1	2	yes	—
294	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
295	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
296	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
297	C₄⋊C₄	—	2	4	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	34	2	no	—
302	C₂²⋊C₄	—	2	1	no	—
303	C₂²⋊C₄	—	34	2	no	—
304	C₂²⋊C₄	#63252	1	2	yes	—
305	C₄²	—	2	1	no	—
306	C₄²	—	2	1	no	—
307	C₄²	—	34	2	no	—
308	C₄²	#63252	1	1	yes	characteristic
309	C₄²	#63252	1	1	yes	characteristic
310	C₄²	C₂×C₂²⋊D₁₇	1	2	yes	—
311	C₂³	—	34	2	no	—
312	C₂³	C₂×#63252	1	1	yes	characteristic
313	Q₈	—	4	4	no	—
314	Q₈	—	34	2	no	—
315	Q₈	—	34	2	no	—
316	D₄	—	34	2	no	—
317	D₄	—	34	2	no	—
318	D₄	—	68	2	no	—
319	D₄	—	68	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	34	2	no	—
326	C₂×C₄	—	68	2	no	—
327	C₂×C₄	—	68	2	no	—
328	C₂×C₄	—	68	2	no	—
329	C₂×C₄	C₂×#63252	1	1	yes	characteristic
330	C₂×C₄	C₂×#63256	1	2	yes	—
331	C₂×C₄	#41×D₁₇	1	2	yes	—
332	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	—	4	1	no	—
335	C₈	—	4	1	no	—
336	C₈	—	68	2	no	—
337	C₈	—	68	2	no	—
338	C₈	#10587633	1	2	yes	—
339	C₈	#10587633	1	1	yes	characteristic
340	C₈	#10587633	1	1	yes	characteristic
341	C₂²	—	4	1	no	—
342	C₂²	—	68	2	no	—
343	C₂²	—	136	2	no	—
344	C₂²	#11763320	1	1	yes	characteristic
345	C₂²	#11763346	1	2	yes	—
346	C₄	—	2	2	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	2	no	—
349	C₄	—	4	2	no	—
350	C₄	—	68	2	no	—
351	C₄	—	136	2	no	—
352	C₄	#11763316	1	1	yes	characteristic
353	C₄	C₂×#10587633	1	1	yes	characteristic
354	C₄	#11763346	1	2	yes	—
355	C₂	—	2	1	no	—
356	C₂	—	2	2	no	—
357	C₂	—	136	2	no	—
358	C₂	#14532067	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#10587633
commutator, cosocle	C₂²×C₄×C₁₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₇
Fitting	C₁₇×#2960	C₂
socle	C₂×C₁₇	#2398
prime radical	#11763609	C₂²
hypercenter	#2960	D₁₇
hypocenter	C₁₇	#23833
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
2. #14530364	C₂	semidirect
3. #14530359	C₂	semidirect
4. #14530235	C₂	semidirect
7. #14529953	C₂	semidirect
8. #14528621	C₂	semidirect
9. #14528585	C₂	semidirect
10. #14528539	C₂	semidirect
11. C₁₇×#2960	C₂	semidirect
162. #2960	D₁₇	semidirect
262. C₁₇	#23833	semidirect

Maximal subgroups

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

subgroup	index
#14537688	2
#14530364	2
#14530359	2
#14530235	2
#14530090	2
#14530071	2
#14529953	2
#14528621	2
#14528585	2
#14528539	2
C₁₇×#2960	2
#23833	17